UNIVERSITÀ DEGLI STUDI DI ROMA, TOR VERGATA

Transcription

1 UNIVERSITÀ DEGLI STUDI DI ROMA, TOR VERGATA Tesi di dottorato di Ricerca in Informatica ed Ingegneria dell Automazione Uniting local and global controllers for anti-windup synthesis Luca Zaccarian Roma, Aprile 2000

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3 Uniting local and global controllers for anti-windup synthesis

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5 Uniting local and global controllers for anti-windup synthesis Luca Zaccarian Advisor Salvatore Nicosia Co-advisor Andrew R. Teel Coordinator Daniel P. Bovet April 2000

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7 Contents Acknowledgements Introduction Windup and anti-windup Early anti-windup schemes Modern anti-windup schemes Reference and measurement governors The L 2 anti-windup problem Other approaches Thesis outline Publications Uniting local and global controllers Uniting controllers for nonlinear systems Problem definition The local and global components The united controller Beyond set-point objectives Robustness analysis The L 2 anti-windup definition and solution for linear systems The L 2 anti-windup definition The L 2 anti-windup solution Case studies A turbofan engine The Caltech ducted fan The F-16 s daisy chain control allocator Anti-windup for active vibration isolation devices Introduction A simple active vibration isolation system

8 2 Contents The benchmark example The saturated closed-loop system Performances of the saturated and unsaturated systems Standard and extended L 2 anti-windup Why L 2 anti-windup? The standard L 2 anti-windup The extended L 2 anti-windup Extended L 2 anti-windup for the benchmark example The benchmark indexes The Elite 3 TM system System description The system without anti-windup protection The system with L 2 anti-windup protection Anti-windup for exponentially unstable linear systems Introduction The anti-windup scheme Problem statement The anti-windup compensator Main result Manual flight control for an unstable aircraft Problem statement Anti-windup design Nominal controller design Simulations L 2 anti-windup for bumpless transfer and reliable designs Introduction L 2 anti-windup synthesis A simple physical example The L 2 anti-windup solution Application to bumpless transfer Application to reliable control designs Concluding remarks Summary of contributions Open problems and future research

9 Contents 3 A Mathematical tools 133 A.1 Special classes of functions A.2 L Stability A.2.1 Input-to-state stability A.3 L 2 gain and the L 2 small gain theorem A.4 Stability of cascaded systems A.4.1 Background A.4.2 Main results B Proof of Theorem Bibliography 151

10 4 Acknowledgements Acknowledgements I believe that every PhD thesis represents the result of many years of work. Therefore, I would like to thank all of the people that, in these years, have supported and guided me through my doctoral studies. First, I would like to thank prof. Salvatore Nicosia, whose fatherly presence and support encouraged and guided me toward the continuation of my research in a PhD program. Deep gratitude goes to prof. Andrew Teel, who was the inspiration for all of my research on anti-windup. He has been an essential reference point for me during this work, not only for his brilliant skills in control theory, but also for his strong and constant encouragement. I would also like to thank prof. Petar Kokotovic for making my work at the University of California in Santa Barbara possible and for his suggestions and support during that time period. My sincere gratitude goes to the gang, for their insight and perseverance in the research we did together. I also thank Jack Marcinkowski, of Newport Corporation, that gave me the chance, together with prof. Teel, to be challenged by a concrete industrial control problem. Many thanks also go to Laura Menini for her invaluable support in so many situations and to the students and post-docs in Santa Barbara and in Rome, who have accompanied me throughout various phases of my PhD studies: Chris, Corneliu, Dan, Doca, Dragan, Fabrizio, Federico, Francesco, Kenan, Lena, Mariateresa, Mike, Murat, Nazir, Tonio, Romeu and Sergio. Special thanks go to prof. Antonio Tornambè. Although he made no specific contribution to this thesis, his introduction to research activity proved invaluable. This work wouldn t have been possible without the lessons that I learned from him. My final thanks go to my family, who have always supported me in whatever choices I made, and to my friends, who continually push me forward in the path of my life. Luca

11 Chapter 1 Introduction 1.1 Windup and anti-windup The dynamics of many physical processes can be represented with good approximation by linear differential equations. However, this is not the case for the forcing term corresponding to the actuator, namely the control input of the linear system. As a matter of fact, any actuator (such as a rotational or a linear motor, a piezoelectric device, a valve, et cetera) is subject to saturation, i.e., there is a maximum amount of control effort that one can obtain from this physical device. Beyond this value, either the actuator operates in an unsafe region (for instance producing high temperatures due to parasitic effects such as heating in the rotor resistance of a DC motor) or the required input is physically infeasible (e.g., due to displacement constraints, such as the fullopen or the full-close position of a valve). If the constraints on the input of linear plants with actuator saturation are not accounted for in the control design, the results can be disastrous. From a nonlinear control theorist point of view, this design approach corresponds to the sloppy solution of connecting a controller designed for a certain plant to a significantly different plant, thus causing unpredictable closed-loop behavior. Historically, the symptoms exhibited by a linear control design on a saturated plant have been called controller windup (or integral windup ). The reason for the term windup is that early industrial designs usually employed PID control schemes on stable linear plants. For these systems, the main effect of windup results in an over-charging effect of the integral term that raises to very large values when saturation is hit, thus winding up the related controller state and deteriorating the subsequent closed-loop response. A notable example of the windup phenomenon on linear plans controlled by 5

12 6 Chapter 1. Introduction PID 1 Input 1 Reference PD Controller 1 K s Ki Integrator 3 IntValue Saturation 1 s Plant 2 Output Figure 1.1: PID control scheme for a linear saturated plant. PID schemes has been given in [5]. The related control scheme is reproduced in Figure 1.1, where the plant is a simple integrator with unity gain and the PID controller is divided in the PD action (upper branch) and the integral action (lower branch) to isolate the effect of the integrator from the rest of the dynamics. The gains associated with the proportional, derivative and integral actions are K p = 1, K d = 1, K i = 1, respectively. If the plant s input saturation is neglected, the closed-loop system is linear and the related response to a unitary step reference corresponds to the dotted curves in Figure 1.2. Note that during the initial transient the plant input exhibits a large peak, whose maximum value is approximately u max = 1 (this value actually exceeds the plot s margins). Integral windup is experienced when the plant input is saturated between ±0.1 and the saturated closedloop system is driven by the same unitary step reference input. The resulting trajectories correspond to the dashed curves in Figure 1.2. The undesired effect of input saturation is to induce very large oscillations at the plant s output, that eventually converge to the desired steady-state value but that severely deteriorate the system s response for a long time (in the illustrated case, linearity is recovered approximately after 60 seconds). Note that the pseudo-periodic behavior of the plant s output mainly consists in raising and falling ramps corresponding to large time intervals in which the control input remains locked on the constant saturated value. As seen from the third plot in Figure 1.2, the main cause of performance degradation for this example relies in the large values stored in the integral part of the controller (dashed curve), as compared to the trajectory of the unsaturated system (dotted curve). In particular, the dashed curve raises to the peak value x i,max = 5 during the

13 1.1. Windup and anti-windup 7 first bump and then slowly decreases in an oscillatory way, leading the control input to bounce between the two saturation values. 2 Plant Output Plant Input Time [s] Time [s] 2 Integral Value Time [s] Figure 1.2: Responses of the control system to a unitary step reference: unsaturated response (dotted curve), response with integral windup (dashed curve), response of the system with conditional integration (dash-dotted curve) and response of the system with back-calculation (solid curve). On the basis of the major performance loss caused by the insertion of the saturation nonlinearity at the plant s input, one might conclude that such a degradation is caused by the saturated plant. However, the only performance degradation directly related to input saturation is the 10 seconds raising time at the beginning of the response. The subsequent oscillation is caused by the inappropriate structure of the controller. To validate this conclusion, we augment the control scheme in Figure 1.1 with what we will henceforth call anti-windup compensation. To simplify the discussion, we describe and implement early anti-windup compensation schemes. In the 1950 s and 1960 s, when analog control systems where predominant and digital technology was beginning to influence the industrial environment, the following three techniques where well known and employed for anti-windup compensation (see, e.g., [86] where an extensive discussion on the various techniques is reported).

14 8 Chapter 1. Introduction 1. Conditional integration. The integrators are stopped (meaning that the input of the integrator is set to zero) whenever the plant s input exceeds the saturation limits. 2. Intelligent integration. Similar to the previous one, each integrator is stopped whenever its output exceeds a maximum value. Such a value has to be tuned depending on the control system s gains and on the saturation value. 3. Back-calculation. The input of the integrator is added to a correction signal proportional to the difference between the actual and the commanded input (namely, the output and the input of the saturation block, respectively). Since the intelligent integrators method is very similar in nature to the conditional integration method, we only employ the first and the third techniques in the above list for the example in Figure 1.1. The resulting trajectories, represented respectively by the dash-dotted and the solid curves in Figure 1.2, confirm that performance can be largely recovered by anti-windup compensation. We conclude that the large oscillations of the saturated response when no anti-windup action is performed, are related to an inappropriate control strategy rather than to a structural limitation of the plant. Although conditional integration and back-calculation give a more than satisfactory performance on the example in Figure 1.1, such is not the case for general linear control systems. As a matter of fact, in some cases the application of the methods above leads to poor performance. Moreover, in many situations, it is not even clear how the methods should be applied (e.g., when the controller does not contain poles in the origin). Consequently, since the 1950 s, a great deal of attention has been devoted to the anti-windup problem, mostly arising from concrete industrial needs. Indeed, for a long time, the controller windup problem has been known qualitatively as the unpredictable effects caused by actuator s saturation on linear control designs. However, controller windup could result in very different phenomena, such as performance loss (as in the simulation in Figure 1.2), generic unreliability of the control system or even instability, thus making it hard to formally state the goals of an anti-windup design. Nevertheless, anti-windup actions were qualitatively considered as the modifications of a linear control design such that: 1. for initial conditions and reference signals that do not cause actuator saturation, the nominal linear response is retained;

15 1.2. Early anti-windup schemes 9 2. if the actuators saturate, the actual response is close (as close as possible) to the linear response. The need for more rigorous and general solutions to the anti-windup problem was pointed out only in recent years in [22], where simple examples were used to show the inadequacy of some of the early schemes. Only in the last decade has the problem been addressed in a more formal and general way. In the next sections, we overview the existing anti-windup techniques, describing their salient features. 1.2 Early anti-windup schemes The anti-windup problem has been addressed since the 1950 s in relation to the design of analog controllers. In 1956, Lozier addressed the problem of stability of saturable servo systems [49] pointing out that systems with input saturation present a dual mode of response, thus already distinguishing between linear trajectories and trajectories that activate saturation. Although the analysis method proposed in [49] was based on describing function techniques (hence, it just provided a qualitative result), Lozier already focused on the main feature required from anti-windup compensation, namely that the linear response should not be modified when saturation is inactive (the following sentence is also reported in [73]). It should be recognized that to eliminate anomalous behavior by restricting designs [...] is not realistic [..] may not be desirable or even possible. Other techniques for minimizing this anomalous behavior can then be developed where necessary to fit particular cases. For example, [...] it should be possible to greatly minimize the anomalous effects by the use of a limiter to restrict the amplitude of the voltage that can be stored on the capacitor in the integrating network. As observed in [73], the simplicity of anti-windup design for many practical situations (such as the one that Lozier refers to in the sentence reported above or the one described in the previous section) set the tone for the early antiwindup schemes. Such schemes are mostly application-dependent and rely on qualitative analyses. With the advent of digital control systems, the windup problem become more and more crucial, at least for the following two reasons. Due to physical limitations, the state of an analog controller could not raise to extremely high values (e.g., the voltage stored in a capacitor cannot exceed the supply voltage of the circuit), while the digital numbers

16 10 Chapter 1. Introduction represented in a numeric controller could raise to significantly higher peaks. The increasing computational power of digital systems allowed more and more sophisticated control laws to be implemented, thus making the effects of windup more and more unpredictable. One of the first papers on anti-windup for digital systems appeared in 1967 [24]. In this paper, Fertik and Ross introduced the back-calculation antiwindup strategy (that we briefly described in the previous section). As already mentioned, the main drawback of this method is that it can only be applied to PID controlled plants, thus being unsuitable for more complicated control designs. In subsequent years, several approaches have been proposed to solve the windup problem, all of them in line with the qualitative character of early anti-windup designs. The main ones are briefly listed in the following. In 1979, Debelle proposed a model-based scheme [20] where a direct and an inverse model of the plant were employed to achieve a dynamical reference scaling. In 1980, Hanus proposed the conditioning technique [36], which was refined and extended in the following years [38, 85]. The key idea of this anti-windup compensation was that whenever saturation is hit, the reference input is scaled to the value that would just saturate the plant s input. If y c denotes the controller output, then the conditioning technique is realized by adding the quantity r = B r Dr 1 (sat(y c ) y c ) to the reference input, where B r is the input matrix of the reference in the controller dynamics and D r is the direct link between the reference input and the controller output. Two important drawbacks of this method are that it is applicable only if D r 0 and that it requires asymptotic stability of all the eigenvalues of the matrix A c B r Dr 1 C c, where A c is the state transition matrix and C c is the output matrix of the controller. 1 By means of coprime factorizations of the controller, the generalized conditioning technique is proposed in [85], to extend the conditioning approach to MIMO systems and to remove the assumption that D r 0. In 1984, Åström introduced the so-called observer approach [6], that is still largely used in the industrial environment. In this approach, it is assumed that the controller is the cascade of an observer and of a 1 For an example where this matrix is not asymptotically stable, see Section

17 1.3. Modern anti-windup schemes 11 static state feedback (see, e.g., [18, 3.4] for a parametrization of stabilizing controllers that emphasizes the intrinsic observer dynamics) and the anti-windup action consists in giving correct information to the observer about the plant s input (if y c is the controller output, this corresponds to replacing y c with sat(y c ) in the observer dynamics). The resulting scheme has been shown to induce significant performance improvement in many cases. In 1989, Morari et al. proposed the application of internal model control [58] (see also [92]) for anti-windup design, resulting in the same modelbased scheme as the one reported by Hanus in the survey paper [37], and therein attributed to Irving. In this approach, that we will henceforth call IMC/model-based, a model of the plant is fed the amount of input chopped by the saturation nonlinearity (namely, if y c is the controller output, the model is fed y c sat(y c )). By linearity, the output of the model can be added to the plant s output to reproduce the unsaturated response. The common aspect of all these early approaches is that they are generalizations of ad hoc methods. Hence, they all present the main drawbacks of being applicable only to limited classes of systems and of not being associated with a formal stability proof for the compensated control system. Between the end of the 1980 s and the beginning of the 1990 s, attempts have been made to relate all the proposed anti-windup approaches in a unified framework. The resulting survey papers [37, 5, 86, 57, 16, 92] provide a good overview of all these initial solutions to the windup problem. 1.3 Modern anti-windup schemes In the last decade, the technology improvements and the advent of increasingly sophisticated control systems pointed to the need for more accurate and rigorous solutions to the windup problem. The inadequacy of some of the early control schemes listed in the previous section was first pointed out in [22] by means of simple examples in which the anti-windup schemes would lead to poor results. Meanwhile, the need for more formal stability results on antiwindup designs led to analysis attempts that gave some initial results with a certain degree of conservativeness (see, e.g., [43, 15]), based on separation theorems 2 applied to certain transfer functions of the saturated closed-loop 2 Under the name of separation theorems we refer to the small gain theorem, absolute stability results such as the circle criterion, passivity concepts and similar results. See,

18 12 Chapter 1. Introduction system on the basis of the sector properties of the saturation nonlinearity. In the next sections, we describe the main research directions that have arisen in the 1990 s, based on more rigorous interpretations of the qualitative requirements mentioned at the end of Section Reference and measurement governors Output admissible sets have been defined in 1991 by Gilbert and Tan [30] as subsets of the state space of the closed-loop system that keep the control signal unsaturated. In the papers [34, 29], output admissible sets are employed to adjust the reference signal of the controller and guarantee forward invariance of these sets. The resulting anti-windup schemes guarantee stability and performance recovery for any trajectory starting within these sets. These approaches go under the name of reference governors. The main disadvantages of reference governor techniques are the following. Disturbances are not taken into account in the analysis, hence making it hard to apply the schemes in situations where, for instance, no explicit bound can be given on the disturbance size. The main obstacle to this is that if occasional disturbances drive the trajectory out of the output admissible set, then convergence to this set is no more guaranteed. It is not clear how the results should be applied to trivial reference trajectories, suche as those found in the vibration attenuation problem (see Chapter 3 of this thesis), where the reference is constantly zero. Since the output admissible sets depend on the controller dynamics, they may tend to shrink more and more the operating region as the control action becomes more and more aggressive, thus possibly resulting in conservative anti-windup compensations. Despite the disadvantages listed above, reference governor schemes give satisfactory results in a large number of applications. Modifications of the reference input of the controller have also been studied in the context of the command governor, where, based on results on model predictive control, nonlinear compensation schemes have been proposed both for linear (see, e.g., [10]) and for nonlinear (see, e.g., [4]) plants. An interesting application of reference governor techniques to open-loop unstable plants has been proposed by McNamee and Pachter [50, 51]. The e.g.,[76] for an overview of these topics.

19 1.3. Modern anti-windup schemes 13 theory of output admissible sets well fits in this framework, where the nullcontrollability region is bounded and forward invariance of appropriate subsets of the state space is necessary to guarantee stability. Finally, the scheme proposed by Shamma [66] goes under the name of measurement governor. In [66], on the basis of controlled invariant sets, the controller input is modified to guarantee forward invariance of certain subsets of the state space. This approach is also applicable in the presence of disturbance inputs, but it requires a priori information about their worst case size The L 2 anti-windup problem The concept of L 2 anti-windup was introduced in 1997 by Teel and Kapoor in the companion papers [79, 78], where the anti-windup problem has been addressed with the aim of blending a local controller that guarantees a certain desired performance, but only local stability, with a global controller that guarantees stability disregarding the local performance. The combination of these two ingredients (according to the approach first proposed in [79]) is attained by augmenting the local design with extra dynamics in a scheme that retains the local controller when trajectories are small enough and activates the global controller when trajectories become too large, thus requiring its stabilizing action. The general approach in [79] has been specialized for antiwindup designs for linear systems in [78] and it has been subsequently shown to be successful in a number of case studies [75, 77, 40]. In a recent paper [74], the L 2 anti-windup construction has been extended in a non-local way to the case of exponentially unstable linear plants. The main advantage in adopting the local/global scheme for anti-windup synthesis is that, by identifying the local design with a (typically linear) controller designed disregarding the input limitation, the corresponding unsaturated closed-loop behavior (as long as it is attainable within the input constraints) can be recovered on the saturated system by means of an extra (typically nonlinear) stabilizing controller (the global controller) whose design is independent of any performance requirement. This decoupled design greatly simplifies, in some cases, the synthesis of the nonlinear controller for the saturated plant. A detailed description of this approach is given in Chapter 2 of this thesis Other approaches In 1996, Miyamoto and Vinnicombe formalized the anti-windup problem as an L 2 disturbance attenuation problem. In [56], they proposed a solution,

20 14 Chapter 1. Introduction at least for asymptotically stable plants, using H synthesis techniques. Similarly, in 1998, Edwards and Postlethwaite [23] cast the anti-windup problem as an H optimization problem and showed the design effectiveness on some examples taken from the literature. An interesting overview of the existing anti-windup techniques is also reported in [23]. Finally, anti-windup techniques have been also studied within the matrix inequalities framework. In 1996, Saeki and Wada [63] proposed a design method based on a coprime factorization of the controller dynamics, resulting in two (bilinear) matrix inequalities related to the stability and the performance of the compensated close-loop system. A constructive solution was also proposed in [63], based on the solution of linear matrix inequalities that approximate the bilinear ones. 1.4 Thesis outline This thesis proposes the L 2 anti-windup technique in [78] as a starting point for synthesizing anti-windup compensators for linear plants. Chapter 2 provides the L 2 anti-windup problem definition and the solution given in [78], and emphasizes its relation to the technique proposed in [79] for the union of local and global control designs. Three applications of this initial L 2 anti-windup technique, corresponding to publications [40, 77, 75] are then illustrated on the basis of the theoretical results. In Chapter 3, the anti-windup problem for active vibration isolation systems is described and solved. First, the windup problem is illustrated through an academic example. Then a novel anti-windup technique is developed on the basis of the L 2 anti-windup scheme, and it is tested on the example. Performance indexes are also introduced for benchmark purposes [iv]. Finally, the novel anti-windup technique is shown to be effective on the industrial active vibration isolation system Elite 3 TM of Newport Corporation. This anti-windup scheme, resulting from over one year of cooperation with Newport Corporation, is extremely appealing from an industrial viewpoint [iii,vi]. Chapter 4 studies anti-windup compensations for exponentially unstable linear systems with input limited in magnitude and rate. The combination of these two problems is a challenging framework for anti-windup designs of major interest, especially in automatic flight control. As a matter of fact, rate limitation and exponential instability are non-negligible phenomena in many modern high performance aircrafts. The resulting anti-windup scheme [ii] is applied to the linearized short-period longitudinal dynamics of a tailless aircraft, showing very satisfactory results [i].

21 1.4. Thesis outline 15 In Chapter 5, the L 2 anti-windup technique is employed to achieve bumpless transfer among controllers in multi-controller schemes for asymptotically stable linear plants. The resulting performance is tested on an example that appeared recently in the literature, showing a significant performance improvement as compared to previous techniques. Finally, the bumpless scheme is employed to guarantee fault detection and handling in control systems with high reliability via hardware redundancy. The resulting construction is successfully tested on a simple physical example [v] Publications The thesis is based on the following publications: [i] C. Barbu, R. Reginatto, A.R. Teel, and L. Zaccarian. Anti-windup design for manual flight control. In American Control Conference, pages , San Diego (CA), USA, June [ii] C. Barbu, R. Reginatto, A.R. Teel, and L. Zaccarian. Anti-windup for exponentially unstable linear systems with inputs limited in magnitude and rate. In American Control Conference, Chicago (IL), USA, June [iii] A.R. Teel, L. Zaccarian, and J. Marcinkowski. Active vibration isolation systems with nonlinear compensation to account for actuator saturation. Provisional Patent Application, August 3, [iv] L. Zaccarian and A. Teel. A benchmark example for anti-windup synthesis in active vibration isolation tasks and an L 2 anti-windup solution. European Journal of Control, Submitted. [v] L. Zaccarian and A.R. Teel. Bumpless transfer and reliable designs: the L 2 anti-windup viewpoint. In CDC, Sidney, Australia, December Submitted. [vi] L. Zaccarian, A.R. Teel, and J. Marcinkowski. Anti-windup for an active vibration isolation device: theory and experiments. In American Control Conference, Chicago (IL), USA, June 2000.

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23 Chapter 2 Uniting local and global controllers In this chapter, according to [79], an algorithm for combining predesigned local and global controllers is described. In Section the problem is defined, and a solution is given in Section on the basis of the elements introduced in Section Based on the results in [78], the algorithm is then applied to linear systems with input saturation in Section 2.2, to perform an anti-windup action on linear control systems. Finally, in Section 2.3, the resulting scheme is shown to be effective by means of three examples taken from the literature. Introduction Typically, the control system design for a nonlinear plant results from a trade-off between the requirements on the final design and the difficulty of the design itself. Indeed, several design tools are available that guarantee highly desirable performance properties for plants that satisfy restrictive assumptions (such as the linearity assumption); however, such assumptions are usually significant only in restricted operating regions, and designs that guarantee stability in the large often sacrifice local performance to achieve the stability goal. According to this, we argue that a local design can achieve better performance than a global one because certain nonlinearities (such as the actuator saturation effect) can be neglected locally. In this chapter, we describe a technique proposed in [79] for combining a local high performance design and a globally stabilizing one, with the aim of preserving the local performance achieved by the first design while guaranteeing global stability by means of the second one. On the basis of this result, 17

24 18 Chapter 2. Uniting local and global controllers the control design for a plant that requires global stabilization satisfying (at least locally) certain performance requirements is broken into two much simpler problems: the first one only addresses the local performance problem, neglecting the behavior of the control system when signals become too large; the second one only solves the global stabilization problem, disregarding the required local performance. Once the two problems are separately solved, a combined control scheme is used, that employs the local design to guarantee full performance recovery in the small signal operation, while preserving stability by employing the global design if trajectories become too large. Although this uniting technique applies to a broad number of problems, the second part of the chapter focuses on the employment of the method for anti-windup synthesis and describes a general approach based on the results in [78], for the construction of anti-windup compensators for linear systems with input saturation. At the end of the chapter, the resulting scheme is shown to be effective by means of examples taken from the literature [77, 75, 40]. 2.1 Uniting controllers for nonlinear systems In this section, the results in [79] are presented and illustrated by means of examples. Given a very general nonlinear system, on the basis of assumptions on the existence of certain stabilizing controllers, a constructive technique is proposed and proven to be effective to combine a global and a local control action in a unique controller. This methodology constitutes the starting point for the development of the anti-windup control schemes proposed in this thesis Problem definition Consider a generic nonlinear system ẋ = f(x, u) + d (2.1a) y = h(x), (2.1b) where x R n is the state, u R m is the control input, y R p is the output and d is an unmeasured disturbance. Assume that f and d are such that there exists a unique solution to the above equation (for instance, locally Lipschitz in x and u). Given a reference signal r R p, the control goal for system (2.1) is to achieve global asymptotic stability of a point x satisfying y := h(x ) = r. In the following assumptions, it will be stated that two controllers have been designed for the system (2.1). In particular, given a reference signal r R p, the first controller is a (possibly dynamical) system that guarantees

25 2.1. Uniting controllers for nonlinear systems 19 asymptotic stability of a point x satisfying y := h(x ) = r only locally (namely, when the initial conditions belong to a neighborhood of x ); whereas, the second controller is a static state feedback law that guarantees a sort of global asymptotic stability of the system in a sense that will be clarified later. Assume that, in a neighborhood of the desired equilibrium, the right hand side of (2.1a) coincides with a function F (x, u), possibly satisfying favorable properties (e.g., the linearity property) that simplify the control design. Then, if a technique exists for combining two separate designs into a single one preserving the stabilization properties, the regulation problem for the original system (2.1) can be broken down into two parts: 1. the design of a high performance (possibly dynamic) controller based on the simplified model F (x, u), according to a certain prescribed behavior for trajectories close enough to the desired equilibrium; 2. the design of a static stabilizing control law on the basis of the original model f(x, u) with the aim of stabilizing the system when the simplified model F (x, u) loses significance or, more generally, when trajectories get too far from the desired equilibrium. Once the two above ingredients are available (in a sense that will be clear later), the strategy proposed in the next sections allows to combine them in such a way that, whenever trajectories are close enough to the desired equilibrium, the dynamics of the controller in item 1 are retained, thus providing the desired local behavior; if trajectories get too far from the equilibrium, the local control action is modified (and, possibly, overridden) by the global one (introduced in item 2) and stability is recovered. As the trajectory gets closer and closer to the equilibrium (and this eventually happens due to the stability assumption in item 2), the global action decreases asymptotically to zero, thus allowing a smooth transfer of the control authority to the local controller and reestablishing the local control performance. A typical situation in which this design approach could be useful is the case when a simple linear controller has been designed for a system on the basis of a linear model that does not take into account nonlinearities that activate only when signals become large. In this case, it might be desirable to keep the linear design for almost all the operating time, by adding an extra control action (namely, the global controller) that activates only infrequently to bring trajectories back to the good region when they become too large.

26 20 Chapter 2. Uniting local and global controllers The local and global components With reference to system (2.1), the following assumptions formalize the qualitative statements in the previous section. First, the properties of the local controller are stated and some basic definitions are introduced. Assumption 2.1 There exist a function F : R n R m R n, and a controller of the form x c = g(x c, x, r) u = k(x c, x, r), (2.2) such that, for each admissible values for d and r, there exists a point (x, x c) satisfying 1. F (x, u) = f(x, u) on some (not necessarily compact) neighborhood F of (x, u ), where u := k(x c, x, r); 2. the point (x, x c) is a globally asymptotically stable equilibrium of the closed loop system ẋ = F (x, u) + d (2.3a) x c = g(x c, x, r) (2.3b) u = k(x c, x, r), (2.3c) and solves the regulation problem, namely, h(x ) = r. Remark 2.2 Assumption 2.1 is related to the desired performance for system (2.1). Via the modified system F (x, u), a simpler control problem is solved through the design of the controller (2.2) and the arising closed-loop trajectories constitute the desired trajectories for the original system (2.1). Item 1 ensures that, at least locally, the original system is well behaved under the action of the controller (2.2), while item 2 guarantees that the closed-loop (2.3) globally stabilizes the desired equilibrium x that induces the desired output h(x ) = r. Note that this stabilization problem is not the original problem associated to system (2.1) because the controller (2.2) is designed on the basis of the modified plant (2.3a). Definition 2.3 The local controller corresponds to the controller (2.2) defined on the basis of the modified plant (2.3a). The local closed-loop system corresponds to the closed-loop system in equations (2.3).

27 2.1. Uniting controllers for nonlinear systems 21 Example 2.4 Consider a linear system with input saturation; then the following choices for f and F satisfy item 1 of Assumption 2.1: f(x, u) = A x + B sat(u) F (x, u) = A x + B u, for all u satisfying sat(u ) = u. Hence, the local controller (2.2) can be chosen to be any asymptotically stabilizing linear controller that induces a desired performance in the local closed-loop system (2.3). (Such a performance can be characterized through the closed-loop poles. As a matter of fact, although the original system is nonlinear, the local closed-loop system is linear.) Once the local performance and the local controller have been characterized, we introduce the global controller to complete the necessary tools for the two step design outlined in the previous section. Assumption 2.5 Let x, u and F be defined as in Assumption 2.1. There exists a function α : R n R n R m R m such that 1. α(x, x, u) = u, x F, u R m ; 2. the point x is globally asymptotically stable for the system ( ) ẋ = f x, α(x, x, u ) + d. (2.4) Remark 2.6 Assumption 2.5 constitutes a strong assumption. As a matter of fact, the design of the function α(,, ) is not easy, in general. However, while the original control problem for system (2.1) is both a performance and stabilization problem via output feedback, the design of α(,, ) is a stabilization problem via state feedback; thus, it constitutes a significant simplification of the original problem. Assumption 2.7 Let x, u and F be defined as in Assumption 2.1 and α(,, ) be defined as in Assumption 2.5. For any choice of absolutely continuous and asymptotically vanishing signals ε 1 (t) and ε 2 (t), the trajectories of the system ( ) ẋ = f x, α(x, x + ε 1 (t), u + ε 2 (t)) + d, are bounded.

28 22 Chapter 2. Uniting local and global controllers Remark 2.8 As it will be clear later, in order to prove the statement of the following Theorem 2.12, Assumption 2.7 can be relaxed to hold only for a restricted class of asymptotically vanishing signals ε 1 (t) and ε 2 (t). In particular, it needs to hold only for signals generated by the local closed-loop system. Indeed, if the local closed-loop system is linear (for instance, as in the case described in Example 2.4), then the assumption needs to hold only for exponentially vanishing signals ε 1 (t) and ε 2 (t) with a time constant that is upper bounded by the absolute value of the inverse of the largest real part of the eigenvalues of the local closed-loop system The united controller In this section, on the basis of two controllers satisfying the properties introduced in the previous section, a composite controller is constructed, that is proven to preserve the local control action in a neighborhood of the equilibrium (x, x c), to induce local asymptotic stability of the point x with a nontrivial region of attraction if Assumptions 2.1 and 2.5 hold, and to induce global asymptotic stability of x if Assumption 2.7 holds as well. Definition 2.9 On the basis of the functions f, F, g, k and α, define the composite controller as follows: ( ) ξ = f(x, u) F x ξ, k(x c, x ξ, r) ẋ c = g(x ( c, x ξ, r) ) u = α x, x ξ, k(x c, x ξ, r), (2.5) where the additional states ξ R n have the same dimension as the state x of system (2.1). Define the composite closed-loop system as the system (2.1) under the action of the composite controller (2.5). Remark 2.10 The composite controller does not need measurement of the disturbance input d, nor it needs explicit measurement of the steady state values x, x c or u. This property is crucial to make the design approach suitable for practical implementation. Indeed, the disturbance input is typically not available for measurement; moreover, due to the presence of such a disturbance input, the steady state values of the plant state, and of the local controller state and output do not correspond to the nominal values x, x c and u, hence also these quantities are typically not available for measurement. 1 Recall that, by item 1 of Assumption 2.1, these eigenvalues have all negative real part.

29 # # " A 1 " " 2.1. Uniting controllers for nonlinear systems 23! "ED FD ( & )6 $/ 0 & $+ " <BA -. $// 0 " <CA ( & )* $%% & &', , , : ; +=<>@? Figure 2.1: The union of local and global control schemes. In Figure 2.1, a block diagram representing the controller union is reported. At the top of the figure, the local and the global controllers are separately represented and the blocks are combined at the bottom of the figure. With reference to the lower part of the figure, the three blocks on the left correspond to the composite controller equations (2.5). In the following, on the basis of Assumptions 2.1 and 2.5, the local properties of the composite controller (2.5) are stated. If the stronger Assumption 2.7 is also satisfied, the results are proven to hold globally. To simplify the statement of the following theorem, given a certain reference signal r, we define the set of initial conditions for the nominal closed-loop system that generate trajectories whose projection on the (x, u) plane does not exit F. Definition 2.11 Given r R p, define the set R R n R n c as the set of initial conditions (x 0, x c0 ) for the nominal closed-loop system (2.3) such that the trajectories ϕ x (x 0, x c0, t), ϕ xc (x 0, x c0, t), projected on the (x, u) plane

30 24 Chapter 2. Uniting local and global controllers stay in F for all times: { R := (x 0, x c0 ) R n R n c : ( ( )) ϕ x (x 0, x c0, t), k ϕ xc (x 0, x c0, t), ϕ x (x 0, x c0, t), r } F t 0. Since by Assumption 2.1 (x, x c) is an interior point of F, in view of the asymptotic stability of (x, x c) for system (2.3) we conclude that R is a neighborhood of (x, x c) and that, necessarily, (x, x c ) R implies (x, k(x c, x, r)) F. The following Theorems 2.12 and 2.16, stated and proven in [79], relate to the performance and stability properties of the uniting technique therein proposed. They are reported here, together with extensive proofs, to clarify the construction that underlies the anti-windup techniques synthesized in this thesis. Theorem 2.12 Consider the composite closed-loop system (2.1), (2.5). For each value of the reference signal r R p, 1. under Assumptions 2.1 and 2.5, a. the point (x, x c, ξ) = (x, x c, 0) is locally asymptotically stable and b. its basin of attraction includes the sets: A 1 := {(x, x c, ξ) : ξ = 0 and (x, x c ) R}, A 2 := {(x, x c, ξ) : (x ξ, x c ) = (x, x c)}; 2. under Assumptions 2.1, 2.5 and 2.7, the point (x, x c, ξ) = (x, x c, 0) is globally asymptotically stable. Proof. Define the new coordinate e := x ξ and consider the composite closed-loop system dynamics in the coordinates (e, x c, x). From equations (2.1), (2.5) we get ( ) ė = F e, k(x c, e, r) + d x c = g(x ( c, x, r) ) ẋ = f x, α(x, x + ε 1, u + ε 2 ) + d, (2.6) where ε 1 := e x and ε 2 := k(x c, e, r) u. Notice that the system is the cascade connection of the (e, x c ) subsystem, coincident with the nominal

31 2.1. Uniting controllers for nonlinear systems 25 closed-loop system (2.3), and the x subsystem. Due to the asymptotic stability of the first subsystem (stated in Assumption 2.1) and to the zero input asymptotic stability of the second subsystem (stated in Assumption 2.5), the cascade system is locally asymptotically stable by Theorem A.24, thus proving item 1.a. In addition, due to the boundedness of trajectories (stated in Assumption 2.7), Theorem A.26 can be applied and global asymptotic stability of the cascade system is guaranteed, thus proving item 2 of the theorem. To prove item 1.b, consider first the set A 1. If ξ(0) = 0 and (x(0), u(0)) F, then, by Assumption 2.1, f(x(0), u(0)) = F (x(0), u(0)) for all times and, from equation (2.5) it follows that ξ(0) = 0. By uniqueness of solutions and by definition of R, ξ(t) = 0 for all times. Hence, by definition, x(t) = e(t) and global asymptotic stability of the (e, x c ) subsystem implies that x(t) converges to x and x c (t) converges to x c. Consider now the set A 2. If e(0) = x and x c (0) = x c, then from the cascade structure e(t) = x and x c (t) = x c for all times and, consequently, ε 1 (t) = 0 and ε 2 (t) = 0. Finally, by Assumption 2.5, the x subsystem with zero input, globally asymptotically stabilizes the point x = x. Since ξ = x e, this also implies that ξ(t) converges to zero. Remark 2.13 Notice that the set A 1 corresponds to the set of all the initial conditions that generate trajectories only employing the action of the local controller. As a matter of fact, if the initial conditions belong to this set, ξ(t) is zero for all times and u(t) = k(x c (t), x(t), r) for all times. Similarly, the set A 2 corresponds to the set of all initial conditions that generate trajectories only employing the action of the global controller. As a matter of fact, in this set e(t) = x and k(x c (t), e(t), r) = u for all times, hence ε 1 (t) = 0 and ε 2 (t) = 0 for all times and only the global control action is active on the x subsystem. Example 2.14 Consider the linear system in Example 2.4. From linearity, the first equation in (2.5) reduces to a reproduction of the linear plant: ξ = A ξ + B (sat(u) u). Note that, in the particular case when the function α can be written as α(x 1, x 2, u) = α(x 1 x 2, u), the last equation of the composite system becomes ( ) u = α ξ, k(x c, x ξ, r) ; hence, although the function α is defined as a state feedback that stabilizes the origin, no state measurement is needed because α acts on the extra states

32 26 Chapter 2. Uniting local and global controllers ξ. Indeed, in this special case, the only measurements needed by the control scheme are the ones required by the local control action (which, through the functions g and k might constitute a subset of the whole state space). Example 2.15 If the nonlinear system (2.1) is subject to input rate saturation, the united scheme can be employed to recover the performance of a controller synthesized for the system without input saturation. The construction in this example was outlined in [79] and explicitly formalized in [77] and [75]. If the unsaturated plant states are denoted by x p R n p, input rate saturation can be modeled by m additional states that follow the discontinuous dynamics: u si = R i sgn(u i u si ), i = 1, 2,..., m, (2.7) where u si is the i-th component of the rate saturated input u s R m (which becomes part of the state of the plant) and R i is the corresponding maximal rate. Hence, x = [x T p u T s ] denotes the state of the whole nonlinear system (2.1), and, accordingly, the function f can be partitioned in two component functions f p and f s, where f s has components given by equations (2.7). Note that, provided u i < R i, the unique solution of the differential equation (2.7) satisfies u si u i ; hence, to reproduce the unsaturated system it is sufficient to choose F (x, u) as the limit system obtained by imposing R i +, i = 1, 2,..., m in f s. Indeed, as R i goes to infinity, equation (2.7) reduces to the identity u i = u si (see [25] for details). On the basis of the above reasoning, if the anti-windup state is partitioned in two components of dimensions n p and m, system (2.5) becomes the limit case of: 2 ξ 1 = f p (x p, u s ) F p (x p ξ 1, u s ξ 2 ) ξ 2i = R i sgn(u i u si ) β sgn(k i (x c, x p ξ 1, r) u si + ξ 2i ), i = 1,..., m, ẋ c = g(x c, x ξ, r) u = α(x p, x p ξ 1, k(x c, x ξ, r)) as β approaches +. Taking the limit for β, from sliding-mode theory, the following equalities hold [25]: k i (x c, x p ξ 1, r) u si + ξ 2i = 0, i = 1,..., m, 2 Define k i (,, ) as the i-th component of the nominal controller output function k(,, ).

33 2.1. Uniting controllers for nonlinear systems 27 which lead to a manifold reduction (the states ξ 2 disappear) and to the following simplified system: ξ 1 = f p (x p, u s ) F p (x p ξ 1, k(x c, x p ξ 1, r)) ẋ c = g(x c, x ξ, r) u = α(x p, x p ξ 1, k(x c, x ξ, r)). (2.8) Note that system (2.8) corresponds to the composite controller (2.5) if only the component f p of the plant is considered for the construction of the ξ dynamics. This example shows that, if the function α is suitably designed to stabilize the rate saturated system (in the sense of Assumptions 2.5 and/or 2.7), the uniting technique can be applied to the unsaturated system despite the dynamical nature of the rate saturation). In Theorem 2.12, under Assumptions 2.1 and 2.5, local asymptotic stability of the composite closed-loop system is proven. Moreover, the theorem assures that all trajectories starting in the set A 1 converge to the equilibrium (x, x c, 0) following the local closed-loop system dynamics in the x and x c state responses. However, the local stability result does not allow to draw any conclusion about trajectories starting from initial conditions (x 0, x c0, ξ 0 ), such that (x, x c, 0) A 1 and ξ 0 is small. A stability result in the sense of the ξ coordinate is stated in the next theorem. This result is useful for practical applications of the uniting technique where only the local results are assured (namely, Assumption 2.7 is not guaranteed). As a matter of fact from a robustness point of view, it is not reasonable in practical applications to assume that ξ 0 = 0, whereas it is reasonable to assume that ξ 0 is small. Theorem 2.16 If Assumptions 2.1 and 2.5 hold, then for any compact subset Z of R and for any ɛ > 0, there exists δ such that for the composite closed-loop system (2.1), (2.5), if (x(0), x c (0)) Z and ξ(0) < δ, then ξ(t) < ɛ for all t 0. Proof. By the local asymptotic stability result proven in Theorem 2.12, there exists δ ɛ > 0 such that (x 0, x c0, ξ 0 ) < δ ɛ ϕ((x 0, x c0, ξ 0 ), t) < ɛ, t > 0, (2.9) where ϕ((x 0, x c0, ξ 0 ), t) denotes the trajectory of the composite closed-loop system (2.1), (2.5) starting from the initial conditions (x 0, x c0, ξ 0 ) evaluated at time t.

34 28 Chapter 2. Uniting local and global controllers Moreover, since R is contained in the basin of attraction, from compactness of Z R, for each δ ɛ, there exists T δɛ /2 such that 3 (x 0, x c0 ) Z ϕ((x 0, x c0, 0), t) (x, x c, 0) < δ ɛ 2, t > T δ ɛ /2.(2.10) From continuity of solutions with respect to the initial conditions on compact time intervals (see, e.g., [42, Theorem 2.6], or in a more general framework, [25, 2.8]) applied to the compact time interval [0, T δɛ /2], there exists δ ξ such that for the composite closed-loop system, if ξ 0 < δ ξ, then ϕ((x 0, x c0, ξ 0 ), t) ϕ((x 0, x c0, 0), t) δ ɛ 2, t [0, T δ ɛ /2]. (2.11) This, together with equation (2.10) implies that ϕ((x 0, x c0, ξ 0 ), T δɛ /2) < δ ɛ ; hence, by equation (2.9), for any ξ 0 < δ ξ, the arising trajectory is smaller than ɛ for all t > T δɛ /2. For the remaining compact time interval [0, T δɛ /2], by equation (2.11) and since, necessarily δ ɛ ɛ, the result follows Beyond set-point objectives Under suitable generalizations of Assumptions 2.1, 2.5 and 2.7, the results of the previous section are applicable to problems that go beyond set point control objectives. Indeed, the global asymptotic stability required in Assumption 2.1 is highly penalizing for the class of allowable disturbances d affecting the plant dynamics. To account for such a limitation and to allow a broader class of systems to be suitable for this control strategy, item 1 in Assumption 2.1 can be relaxed to just requiring that the local controller (2.2) only drives the local closed-loop trajectories into a set X R n R n c. This relaxation allows the disturbance input d and the reference input r to be timevarying and it simply constrains their values to be small enough to keep the arising local closed-loop trajectories within X for large enough times. 4 Within this extended framework, Assumptions 2.5 and 2.7 have to be restated in accordance with the new stabilization problem. In particular, if for a set point stabilization problem Assumptions 2.5 and 2.7 require that the control law u = α(x, x, u ) robustly stabilizes 5 the point x of system 3 Note that by Theorem 2.12, the ξ component of the trajectories in equation (2.10), is identically zero. 4 This does not constrain the behavior of d and r during transients, but only at the steady-state. 5 In the sense of Assumption 2.7.

35 2.2. The L 2 anti-windup definition and solution for linear systems 29 (2.1), in this more general case, they can be restated by requiring that, with reference to the composite system (2.1), (2.5), the control law α drives ξ to zero as (x, x c ) approaches the set X. Under such an assumption, Theorem 2.12 can be restated for this more general case and, since the cascade structure (2.6) is preserved, the results can be proven to hold also for this broader class of control systems Robustness analysis In real implementations of the model-based composite controller (2.5), it is natural to assume a mismatch between the actual plant dynamics and the modeled dynamics described by f. This mismatch can be well represented by replacing equation (2.1) with ẋ = f(x, u) + d + Ψ(σ 0, x, u, d) y = h(x), where Ψ(σ 0, x, u, d) denotes a dynamic element depending (in the most general framework) on an initial condition σ 0 and on all the variables involved in the plant dynamics: x, u and d. Regarding the effects of the unmodeled dynamics Ψ on the results of Theorem 2.12, consider first the case when the initial condition of the composite closed-loop system is in A 1. In this case, if the local controller in Assumption 2.1 is robust to the uncertainty Ψ, the result of the theorem is retained. In the other cases, the presence of Ψ destroys the cascade structure of system (2.6). As a matter of fact, with reference to the composite closed-loop dynamics in the coordinates (e, x c, x), not only (e, x c ) affects the dynamics of x through ε 1 and ε 2, but also x affects the (e, x c ) dynamics through Ψ. The stability properties of this feedback system can be addressed via small-gain reasonings by imposing a sufficiently small bound to the gain of the unmodeled dynamics Ψ. See, e.g., the following Theorem 2.28 where unmodeled dynamics are accounted for in the case of linear plants with input saturation. 2.2 The L 2 anti-windup definition and solution for linear systems In this section, on the basis of the tools introduced in Section 2.1, we illustrate the contribution in [78] where the anti-windup problem described in Chapter 1 is formalized in a rigorous mathematical way and is solved in a non-local way for linear plants with no exponentially unstable modes. Several

36 30 Chapter 2. Uniting local and global controllers examples that have been studied in the literature as applications of this antiwindup strategy [77, 75, 40] are then reported in the next section. As already pointed out in Chapter 1, the windup problem arises when a controller designed on the basis of a linear model is connected to the real, saturated, plant. This situation is well-suited for the basic assumptions introduced in Section As a matter of fact, the pre-designed linear controller can constitute the local controller that, together with the unsaturated plant, forms the local closed-loop system. On the other hand, a global controller should be available for the local/global design to be complete. To this aim, a number of results on global stabilization of linear systems with bounded inputs (which can be found in the literature) can be brought to bear in the united control scheme (see, e.g., [71, 53, 67, 46, 48]) The L 2 anti-windup definition Consider a linear plant whose state-space representation is ẋ = A x + B d d + B u u + Ψ(σ 0, x, u, d) z = C z x + D dz d + D uz u y = C y x + D dy d + D uy u, (2.12) where x R n is the state, y R p is the measured output, z R r is the performance output, d is a disturbance input and u R m is the control input. Assume that due to modeling errors, the dynamic perturbation Ψ affects the system state equation, where σ 0 is the initial condition of the related state. We are interested in the design of an anti-windup scheme for this particular system, in the attempt to recover a nominal linear design when the control input u is subject to magnitude saturation. Assume that a linear controller has been designed for system (2.12) following any linear design technique and that its state-space representation is ẋ c = A c x c + B cu u c + B cr r y c = C c x c + D cu u c + D cr r, (2.13) where x c R n c is the controller state, u c R p and y c R m are the controller input and output, respectively, and r R n r is the reference input. Definition 2.17 Define the dynamical system (2.13) as the nominal controller, and define the system corresponding to the feedback connection of the linear plant (2.12) with the nominal controller (2.13) via the interconnection equations u = y c, u c = y, (2.14)

37 2.2. The L 2 anti-windup definition and solution for linear systems 31 as the nominal closed-loop system. Also define the nominal trajectory starting from (x 0, x c0 ) as the (unique) solution (x n (t), x cn (t)) of the nominal closedloop system starting from the initial conditions (x n (0), x cn (0)) = (x 0, x c0 ). Henceforth, the subscript n will be used to indicate the state and output trajectories of the nominal closed-loop system. As already anticipated in Examples 2.4 and 2.14, if saturation is added at the input u of the linear plant (2.12), the nominal controller and the related nominal closed-loop system can be thought of as a special case of the more general concepts of local controller and local closed-loop system, where the global controller is a static feedback still to be designed on the basis of the saturated plant. Consider now the anti-windup problem arising from the linear control design (2.13), when the linear plant (2.12) is subject to input saturation. The intuitive goal of the anti-windup compensation is to design a (possibly dynamical) additional system that modifies the nominal controller when connected to the saturated plant in such a way that 1. if the nominal trajectory from certain initial conditions does not exceed the saturation limits, then the system with anti-windup compensation starting from the same initial conditions reproduces the same trajectory; 2. if the nominal trajectory spends a finite amount of time outside the saturation limits, then the anti-windup compensation forces the saturated trajectory to ultimately converge to the nominal one. The above requirements have been formalized mathematically in [78] in terms of the L 2 norms (see Definition A.15) of the differences between the trajectories of the nominal closed-loop system and the trajectories of the saturated system with anti-windup compensation. Definition 2.18 Given a subset of the input space U R m, the nominal, global L 2 anti-windup problem for U is to find a (possibly dynamical) system [ ] v1 v = = Σ aw (ξ(0), y, y c ), (2.15) v 2 such that the linear plant (2.12) (with Ψ 0), (2.13), (2.15) with the interconnection conditions: satisfies u = sat(y c + v 1 ), u c = y + v 2, (2.16)

38 32 Chapter 2. Uniting local and global controllers 1. if ξ(0) = 0 and u n ( ) sat(u n ( )), then z( ) z n ( ); 2. if dist U (u n ( )) L 2 then (z z n )( ) L 2. The nominal local anti-windup problem is to satisfy the two above conditions for initial conditions, disturbances and references such that ξ(0) and dist U (u n ( )) 2 are sufficiently small. The robust anti-windup problem is to satisfy the two above conditions for all Ψ with sufficiently small incremental gain. 6 Remark 2.19 On the basis of Definition 2.18, the natural question arises whether certain conditions are necessary on the set U for the problem to be solvable. Indeed, at least when the function sat( ) is the standard decentralized function, U needs to be contained in the unsaturated region; as a matter of fact, if U contains a point ϕ for which sat(ϕ) ϕ, even for a Hurwitz plant, the equilibrium associated with the input u(t) ϕ would be unattainable for the saturated system, thus making it impossible to satisfy the second item of the definition. In the next section the set U is better characterized on the basis of the function sat( ). Once this set is properly defined, a solution to the L 2 antiwindup problem introduced in Definition 2.18 will be given on the basis of the local/global construction described in Section Remark 2.20 On the basis of the general scheme described in Section 2.1.3, the construction carried out in the following applies to any nominal controller, regardless of its linearity. A linear nominal controller will be considered in the following for the sake of clarity The L 2 anti-windup solution To solve the L 2 anti-windup problem in Definition 2.18 we need some very natural assumptions to hold, both on the nominal closed-loop system and, relative to the nature of the saturation function sat( ), on the set U. 6 The dynamical system Ψ is said to have finite incremental gain γ > 0 if Ψ(σ 0, x 1, u 1, d) Ψ(σ 0, x 2, u 2, d) γ [ x1 x 2 u 1 u 2 ], σ 0, d.

39 2.2. The L 2 anti-windup definition and solution for linear systems 33 Assumption 2.21 The set U is compact and there exist strictly positive real numbers L and b such that, for all u, w R m and all ϕ U, ( ) sat(u + ϕ) u ϕ L u T sat(u + ϕ) ϕ (2.17) sat(u + w) sat(u) min{l w, b}. (2.18) Remark 2.22 Note that choosing u = 0 in equation (2.17), the right hand side is zero, hence, necessarily, sat(ϕ) = ϕ for all ϕ U. Example 2.23 Consider the standard decentralized saturation function: sat u 1 u 2. u m = sat 1 (u 1 ) sat 2 (u 2 ). sat m (u m ), (2.19) where sat i (u i ) = u im if u i > u im u i if u im < u i < u im u im if u i < u im, and u im < u im for all i = 1, 2,..., m. When this saturation function is considered (such an assumption is satisfied in a large number of practical applications), a typical choice for U is the compact set U := [u 1m, u 1M ] [u 2m, u 2M ] [u mm, u mm ], where u im < u im < u im < u im, for all i = 1, 2,..., m. This set and the decentralized saturation function satisfy both equations (2.17) and (2.18). Indeed, equation (2.18) easily follows from the boundedness and the global Lipschitz property of the standard saturation function. Equation (2.17) can be proven as follows. Define ɛ as the distance of the set U from the unsaturated region boundaries: ɛ := min i {1,..., m} min{u im u im, u im u im }. Now, consider first the scalar case. If sat(u i + ϕ i ) = u i + ϕ i, then equation (2.17) becomes 0 L u 2 i, which is true for any L 0; on the other hand, if

40 34 Chapter 2. Uniting local and global controllers sat(u i + ϕ i ) u i + ϕ i then, since ϕ i U and by definition of ɛ, sat(u i + ϕ i ) u i ϕ i u i ɛ 1 ɛ u i ɛ 1 ɛ u i sat(u i + ϕ i ) ϕ i = 1 ɛ u i (sat(u i + ϕ i ) ϕ i ), hence the choice L = 1/ɛ satisfies equation (2.17). In the vector case, if any p-norm (with p {1,..., }) is chosen for the left hand side of equation (2.17), the decentralized saturation function can be written term by term and the following holds: u 1 + ϕ 1 u 2 + ϕ 2 sat. u m + ϕ m u ϕ = sat 1 (u 1 + ϕ 1 ) u 1 ϕ 1 sat 2 (u 2 + ϕ 2 ) u 2 ϕ 2. sat m (u m + ϕ m ) u m ϕ m u 1 (sat 1 (u 1 + ϕ 1 ) ϕ 1 ) u 2 (sat 2 (u 2 + ϕ 2 ) ϕ 2 ) L. u m (sat m (u m + ϕ m ) ϕ m ) = L u T (sat(u + ϕ) ϕ) L u T (sat(u + ϕ) ϕ), thus showing that Assumption 2.21 is fulfilled. Remark 2.24 To be able to satisfy equation (2.17), the set U introduced in Example 2.23 needs to be a strict subset of the region where sat(x) = x. 7 This is consistent with the existence of a solution for the anti-windup problem introduced in Definition As a matter of fact, take as an example the one-dimensional system ẋ = sat(u) + d, (2.20) with z = y = x and sat( ) being the unitary saturation function. Pick U = [ 1, 1] and consider the case where d = 1 is a constant disturbance and 7 It can be easily verified choosing, e.g., for the scalar case, u = ϕ = u max in equation (2.17).

41 2.2. The L 2 anti-windup definition and solution for linear systems 35 the initial condition x 0 = ɛ is arbitrarily small and positive. If the nominal controller contains an integral action to guarantee zero steady state error, due to linearity, dist U (u) is an exponentially decaying function, whose L 2 norm is finite. Conversely, it is easily seen that if x 0 = ɛ, for any choice of u, the solution x(t) of (2.20) satisfies x(t) > ɛ for all times, thus showing that (z z n )( ) / L 2 and that the local anti-windup problem is not solvable. Such is not the case if we pick U as any compact subset of [ 1, 1]. The following technical lemma constitutes an important element of the L 2 anti-windup solution introduced in the main theorem. As seen from the local/global point of view, the result of the lemma provides us with the static state feedback that serves as a stabilizer for the saturated plant (as stated in Assumption 2.5). The proof of the lemma is based on the results in [53] and [72]. Lemma 2.25 Consider a set U R m and the control system ( ) ξ = A ξ + B sat(v + ϕ(t)) ϕ(t), where (A, B) is stabilizable and the function sat( ) satisfies Assumption 2.21 for the set U. Then, 1. there exists a globally Lipschitz feedback v = k(ξ) such that if dist U (ϕ) 2 and ξ(0) are sufficiently small, then ξ( ) L 2 and the L 2 gain from dist U (ϕ( )) to ξ( ) is finite; 2. if all the eigenvalues of A have non-positive real part, then there exists a globally Lipschitz feedback v = k(ξ) such that if dist U (ϕ( )) L 2, then ξ( ) L 2 ; moreover, if A is critically stable, i.e., if there exists P = P T > 0 such that A T P + P A 0, then the function k(ξ) can be chosen as the linear function k(ξ) = B T P ξ. Finally, if A is Hurwitz, the L 2 gain from dist U (ϕ( )) to ξ( ) is finite. Assumption 2.26 The nominal closed-loop system (2.12), (2.13), (2.14) is well-posed and internally stable and the pair (C z, A) is detectable. Remark 2.27 Note that Assumption 2.26 implies Assumption 2.1 for this particular case. Indeed, picking the function F as the unsaturated dynamics (2.12) and f as the dynamics (2.12) with saturation at the input, we can choose F = R n U, with U satisfying Assumption As pointed out in Remark 2.22, for every (x, ϕ) F, sat(ϕ) = ϕ, hence f = F in F thus satisfying

42 36 Chapter 2. Uniting local and global controllers item 1 of Assumption 2.1. Item 2 is directly implied by the internal stability of the linear closed-loop system required in Assumption 2.26 that implies (by linearity) the existence of a globally asymptotically stable equilibrium. The detectability assumption is needed to guarantee detectability of x from the performance output z, thus removing unnatural solutions to the L 2 windup problem in which (x x n )( ) 2 / L 2 and (z z n )( ) 2 L 2. This assumption allows to state the necessity of some of the result in the following main theorem. The following Theorem 2.28, stated and proven in [78], provides a solution to the L 2 anti-windup problem stated in Definition It is reported here, together with an extensive proof, to clarify the construction that underlies the anti-windup techniques synthesized in the following chapters. Theorem 2.28 If Assumptions 2.21 and 2.26 hold, then 1. the nominal local and robust local L 2 anti-windup problems for U are solvable; 2. the nominal global L 2 anti-windup problem for U is solvable if and only if A has no eigenvalues with positive real part; 3. the robust global L 2 anti-windup problem for U is solvable if and only if A is Hurwitz; 4. whenever one of the problems is solvable, the following anti-windup compensation solves the problem: ξ = A ξ + B u ( sat(y c + v 1 ) y c ) v 1 = k(ξ) v 2 = C y ξ D uy ( sat(y c + v 1 ) y c ), (2.21) where k(ξ) is generated by Lemma 2.25 from the data (A, B u, sat( ), U). Figure 2.2 represents the combination of the nominal controller with the stabilizing action performed by k( ). As compared to the uniting strategy introduced in Section and represented in Figure 2.1, the nominal controller plays the role of the local one and the global controller is constituted by the simple stabilizing function k( ). In particular, the choice α(x, x, y c ) = k(x x ) + y c leads to equation (2.21).

43 !, The L 2 anti-windup definition and solution for linear systems 37! $&%('*) " " # $9%:';)=< 3 *4 - /./021 Figure 2.2: Uniting a nominal controller and a stabilizing state feedback for anti-windup design. Proof of Theorem Necessity of items 2 and 3. Item 2. Assume that the solution x n (t) of the nominal closed-loop system converges asymptotically to the set U. If A has one eigenvalue with positive real part, since from equation (2.18), sat(u) is bounded, there exists a large enough x 0 such that the unstable component of the state cannot be stabilized and goes unbounded. Hence, since x(t) diverges, (x x n )( ) / L 2. From the detectability of (C z, A) stated in Assumption 2.26, also (z z n )( ) / L 2. Item 3. Pick Ψ(σ 0, x, u, d) = g x, with g > 0 sufficiently small 8 and note that, if A has at least one eigenvalue on the imaginary axis, for any positive value of g the perturbed matrix A + g I has at least one unstable eigenvalue, and for a small enough g, (C z, A + g I) is detectable; hence, the construction in the proof of the previous item can be employed to show unsolvability of the robust anti-windup problem. 8 Note that such a choice for Ψ has arbitrarily small finite incremental gain γ = g.

44 38 Chapter 2. Uniting local and global controllers Sufficiency Item 1 of Definition Since ξ(0) = 0 and y cn (t) = sat(y cn (t)) at all times, we get y c (0) = y cn (0) = sat(y cn (0)) = sat(y c (0)) which, together with v 1 (0) = k(ξ(0)) = 0 and substituted in (2.21), gives ξ(0) = 0. Hence, repeating the argument at all times it is proven that (x(t), x c (t), ξ(t)) = (x n (t), x cn (t), 0) is a solution for the anti-windup closed-loop system. By uniqueness of solutions, this is the only solution, thus proving item 1. Item 2 of Definition To prove the L 2 stability of the anti-windup closedloop system from dist U (u n ( )) to (z z n )( ), consider the change of coordinates [ ] [ ] ɛ1 x xn ξ ɛ := :=, (2.22) ɛ 2 x c x cn and notice that, if A cl denotes the state transition matrix of the nominal closed-loop system (evaluated with Ψ 0), the dynamics in the ɛ coordinates are: 9 [ ] Ψ(x, u, d) Ψ(xn, u ɛ = A cl ɛ + n, d), 0 and by the finite incremental gain of Ψ and the asymptotic stability in Assumption 2.26, Theorem A.17 can be applied to get [ ] ɛ( ) 2 γ ɛ ( ɛ 0 ) + g (x xn )( ), (2.23) (u u n )( ) where γ ɛ is a class K function. However, by equation (2.22), since k( ) and sat( ) are globally Lipschitz (see Lemma 2.25 and equation (2.18)), x x n = ɛ 1 + ξ u u n = sat(y c + v 1 ) y cn = sat(y c + v 1 ) sat(y cn ) y cn + sat(y cn ) L y c + k(ξ) y cn + y cn sat(y cn ) L y c y cn + L k ξ) + dist U (y cn ), where L is the Lipschitz constant in equation (2.18) and L k > 0 is the Lipschitz constant of k( ). Substituting the above inequalities in equation (2.23), we get ( ɛ( ) 2 γ ɛ ( ɛ 0 ) + g ɛ 1 ( ) 2 + (1 + L k ) ξ( ) L (y c y cn )( ) 2 + dist U (y cn ( )) 2 ). dy- 9 To verify this, similarly to equation (2.6), note that, when Ψ 0, the [ ] xn namics coincide with the dynamics. x cn [ x ξ x c ]

45 2.3. Case studies 39 Moreover, since y c y cn = (I D c D 22 ) 1 (C c ɛ 2 + D c C 2 ɛ 1 ), (2.24) for a sufficiently small g there exist positive constants L 1 and L 2 such that ɛ( ) 2 γ ɛ ( ɛ 0 ) + g (L 1 ξ( ) 2 + L 2 dist U (y cn ( )) 2 ). (2.25) To be able to apply the small gain theorem, we derive a similar bound on ξ( ) 2. First notice that, by equation (2.24), there exists L 3 > 0 such that dist U (y c ) dist U (y cn ) + y c y cn dist U (y cn ) + L 3 ɛ ; then, by Lemma 2.25 applied to system (2.21) and the properties of k( ), there exist L 4 > 0 and class K functions γ ξ ( ) and γ ξ ( ) such that ξ( ) 2 γ ξ ( ξ 0 ) + γ ξ ( dist U (y c ( )) 2 ) γ ξ ( ξ 0 ) + γ ξ (2 L 3 ɛ( ) 2 ) + γ ξ (2 dist U (y cn ( )) 2 ). (2.26) Finally, there exists a small enough g such that γ ξ (2 L 3 g L 1 s) < s for all s > 0 and the small gain Theorem A.18 can be applied to inequalities (2.25) and (2.26) to conclude that ɛ( ) 2 γ ɛ ( ɛ 0 ) + γ ɛ ( dist U (y cn ( )) 2 ) ξ( ) 2 γ ξ ( ξ 0 ) + γ ξ (2 dist U (y cn ( )) 2 ) (u u n )( ) 2 L ɛ ɛ( ) 2 + L k ξ( ) 2, where L ɛ is a positive constant and γ ɛ ( ), γ ɛ ( ) are class K functions. These last inequalities substituted in the second equation of system (2.12) imply that there exist class K functions γ z ( ) and γ z ( ) such that ( ) ɛ ((z z n )( ) 2 γ 0 z ξ 0 + γ z ( dist U (y c ( )) 2 ), as to be proven. 2.3 Case studies In this section, the anti-windup solution described in Section 2.2 is employed to solve the windup problems associated with the control of three case studies. For additional details about these examples, the interested reader is referred to the corresponding publications [40, 77, 75].

46 40 Chapter 2. Uniting local and global controllers A turbofan engine As a first simple example, we illustrate here part of the results in [40] where the anti-windup scheme has been applied to a turbofan engine with input magnitude limitation. The control of this engine in the absence of saturation has been studied in [26], where an H controller is proposed. Such a control scheme is therein equipped with an anti-windup modification determined by solving an optimization problem within an observer-based structure [87]. In [40] the nominal controller is chosen as in [26] but the anti-windup action is replaced by the L 2 anti-windup scheme. The novel anti-windup structure not only exhibits improvement in the response of the anti-windup closed-loop system but also allows to formally address and solve the problem of infeasible set points (this last problem was not solved with the previous observer-based anti-windup solution). Problem statement The turbofan engine model is a linear system with three states, three inputs and three performance outputs (all of them available for measurement): ẋ = A p x + B p sat(u) y = C p x + D p sat(u), (2.27) where A p = B p = C p = D p = , ,,. The eigenvalues of A p are , and ; hence, the plant is open-loop stable. The three outputs y R 3 to be controlled are the fan

47 2.3. Case studies 41 speed (PCN2R), the core engine pressure ratio (CEPR) and the liner engine pressure ratio (LEPR), respectively. The three inputs u R 3 are the fuel flow rate (WF), the nozzle area (A8) and the bypass duct area (A16), respectively. With the primary goal of guaranteeing decoupled tracking of the set points for PCN2R and CEPR with good regulation of LEPR, in [26], the following nominal controller has been derived for the unsaturated system 10 : ẋ c = A c x c + B c (y sp u c ) y c = C c x c + D c (y sp u c ), (2.28) where y sp R 3 is the set-point value for the outputs and the matrices entries are: A c = , B c = C c = D c = , Three of the eigenvalues of A c are 0.002, while the fourth is This suggests that the nominal closed-loop system may be subject to windup (it was first noticed in [22] that slow modes in the nominal controller usually lead to poor performance in the presence of saturation). The observer based anti-windup and the conditioning technique proposed by Hanus [38] have been both applied to the control scheme resulting in the following modified controller:. ẋ c = A c x c + B c (y sp u c ) + L (sat(y c ) y c ) y c = C c x c + D c (y sp u c ), 10 Recall that, according to the definition of the nominal closed-loop system, the plant (2.27) and the controller (2.28) are to be connected via the interconnection (2.14).,

48 42 Chapter 2. Uniting local and global controllers where the two following values for L correspond to the two anti-windup schemes, respectively: L OB = , L CT = The L 2 anti-windup solution Applying the L 2 anti-windup to the turbofan engine corresponds to reproducing the plant dynamics and designing the extra stabilizing action performed by k(ξ) in equations (2.21) following the construction in Lemma 2.25 or a different construction resulting in improved L 2 performance. In the case when A is Hurwitz, we can pick any positive definite symmetric matrix P that solves a Lyapunov equation A T P + P A = Q, with Q 0, and pick K = ρ B T P, where ρ > 0. The effectiveness of the stabilizing action performed by the output v 1 of system (2.21) highly depends on the choice of Q. For the turbofan engine, the choice has been Q = C T C, thus giving the following structure for the anti-windup compensator (2.21): ξ = A p ξ + B p (sat(y c + v 1 ) y c ) v 1 = K ξ ξ v 2 = C p ξ D p (sat(y c + v 1 ) y c ), with the anti-windup interconnection (2.16) and the choice: K ξ = Figure 2.3 reports and compares the responses of the system without saturation and of the system with saturation (using the three anti-windup modifications described above). The reference input is a pulse of amplitude 1.5 and duration 3.5 seconds at the PCN2R reference input. All the inputs are assumed to be constrained between 1.5 and 1.5. These operating conditions were first adopted in [26]..

49 2.3. Case studies Step responses Unsaturated inputs Time [s] Time [s] 1.5 Step responses Saturated inputs Time [s] Time [s] Figure 2.3: Responses of the unsaturated system and of the saturated system with and without anti-windup compensation, to a pulse at the PCN2R reference input. The first plot represents the PCN2R output response of the nominal (unsaturated) system (dotted) and of the constrained system with no anti-windup modification (dashed). The second plot represents the unsaturated input profiles associated with the unsaturated response: WF, A8 and A16 (solid, dashed and dash-dotted lines, respectively). The third plot compares the nominal response (dotted) to the anti-windup responses. The L 2 anti-windup action (solid) exhibits a behavior very close to the unsaturated one, while the observer-based anti-windup is closer to the constrained response (dashdotted). The conditioning technique (dashed) leads to instability, since the matrix A c B c Dc 1 C c has unstable eigenvalues. The fourth plot represents the inputs WF, A8 and A16 (solid, dashed and dash-dotted lines, respectively) corresponding to the L 2 anti-windup response (to be compared with those of

50 44 Chapter 2. Uniting local and global controllers the second plot). In [40], an interesting discussion is presented about the projection of infeasible references onto feasible ones without compromising certain performance requirements (namely, satisfying as much as possible the requirement for the PCN2R output). This discussion is not reported here because it is not essential in illustrating the features of this anti-windup scheme The Caltech ducted fan In [77], the L 2 anti-windup scheme is applied to a linear approximation of the dynamics of the Caltech ducted fan: an experimental system whose dynamics reproduce the longitudinal motion of an aircraft. The resulting scheme is reported and commented here because it constitutes an interesting and simple application of the union of a local high-performance control design with a local low-performance one, in the presence of input rate saturation. Problem definition The Caltech ducted fan consists of a short wing and a ducted fan engine with a high-efficiency electric motor and a 6-inch diameter blade, capable of generating up to 15 Newton of thrust. Paddles on the fan allow the thrust to be vectored from side to side. Additional inputs are the motor speed and the angle of a flap attached to the wing, which acts as an elevator. The engine and wing are mounted on a stand with three degrees of freedom, which allows horizontal and vertical translations as well as an unrestricted pitch angle. Taking the linearization of the system s dynamics around a trim condition associated with a forward velocity of 8 meters per second, the following model is obtained: {}} { x x trim = A f (x x trim ) + B f (u u trim ), where the state x R 6 and the input u R 3 represent the following physical quantities, x := horizontal position [m] vertical position [m] pitch angle [rad] horizontal velocity [m/s] vertical velocity [m/s] pitch angle rate [rad/s], u := voltage to motor [V] paddle deflection [rad] elevator deflection [rad].

51 2.3. Case studies 45 The nominal trim condition associated with a forward velocity of 8 meters per second is given by x trim 8 t , u trim , (2.29) and the associated system matrices are given by A f = B f = A challenging problem associated with the Caltech ducted fan is that the second and the third control inputs are subject to rate saturation, imposed by the maximal velocity of the paddle and of the elevator. Indeed, these devices are also subject to magnitude saturation but the problems induced by the rate saturation are typically more harmful in flight control systems (see, e.g., [54]). To address the problem, following the construction described in Example 2.15, a composite controller based on two LQR designs is synthesized by uniting the two control strategies in an anti-windup framework. Controller design For the trim condition (2.29), two linear quadratic regulators have been designed to minimize two performance indexes depending both on the input u u trim and on the state x x trim. The first one leads to an aggressive control action and is chosen to be the local controller (note that no dynamics are associated with this design, hence the state x c is empty). The second

52 46 Chapter 2. Uniting local and global controllers design corresponds to an index where the control input is significantly more penalized than the state, thus leading to a less aggressive controller, suitable to be used as the global component (this last controller does not actually stabilize globally the rate saturated plant, however, since the plant is marginally stable, the basin of attraction of the resulting closed-loop system is sufficiently large to guarantee asymptotic stability under reasonable operating conditions). The resulting controller is given by u u trim = K (x x trim r), where K = K hi for the local controller and K = K lo for the global one. The entries of the two gain matrices are 11 : K hi = , K lo = According to the construction in Example 2.15, choosing α(x, x, y c ) = K lo (x x ) + y c and k(x x trim, r) = K hi (x x trim r), the anti-windup controller (2.8) becomes 12 : ξ = A f ξ + B f (u s y c ) y c = K hi (x x trim r) u = K lo ξ + y c, (2.30) where u s R 3 represents the rate saturated input and u R 3 represents the input to the saturation block. Simulation results In [77] nonlinear simulations are carried out on the system to study the behavior of the controller (2.30), and the experimental results associated with the real-time implementation of the control strategy on the Caltech ducted fan are given. We show here the results of a linear simulation in which the responses of the different controllers to a step reference of amplitude 0.5 meters in the vertical displacement (namely, r = [ ] T ). 11 See [77] for details about the design of K hi and K lo. 12 Note that equation (2.8) simplifies to equation (2.30) because of the linearity of the plant and the absence of nominal controller dynamics.

53 2.3. Case studies Responses to a step command Vertical Displacement [m] Time [s] Figure 2.4: Responses of the unsaturated system (dotted) and of the saturated system using the less aggressive LQR design (dashed), the more aggressive LQR design (dash-dotted) and the combined controller (solid) to a step reference in the vertical displacement. According to the assumptions in [77], it is assumed here that the first input is not subject to saturation and that both the second and the third input are rate saturated with a maximal rate of 0.6 rad/s (approximately corresponding to 34.4 deg/s). In Figure 2.4, the results of the simulations are compared: the dotted curve shows the desired performance, achieved by using the aggressive LQR design in the absence of saturation; the dash-dotted curve shows the instability induced by the rate saturation when the aggressive LQR design is employed with no anti-windup strategy; the dashed curve shows the performance of the less aggressive LQR design in the presence of rate saturation and the solid curve represents the performance of the anti-windup controller (2.30). Notice how the anti-windup response almost recovers the nominal performance while preserving stability.

54 48 Chapter 2. Uniting local and global controllers The F-16 s daisy chain control allocator In [75], the L 2 anti-windup algorithm has been applied to compensate for the destabilizing effect of rate saturation on daisy chained actuators. Daisy chained actuators were first studied in [35], where a linearized model about a trim condition of the short period longitudinal dynamics of the VISTA/MATV F-16 were considered. The arising linear system is driven by two inputs: elevator deflection and pitch thrust vectoring. In [35] the daisy chained allocation was compared to the pseudo-inverse method and it was shown to exhibit severe performance degradation in the presence of rate saturation, even for operating conditions where the pseudo-inverse method still performs in an acceptable way. Subsequently, in [11], an analysis based on the frequency domain properties of the rate saturated system led the authors to an interpretation of the destabilizing effects of daisy chained rate saturated actuators, in terms of additional phase lag. Indeed, in flight control literature, phase lag is often associated to rate saturation and is considered amongst the main causes of pilot induced oscillations (see e.g., [55, 62]). In [75], the daisy chained rate saturated system is equipped with L 2 anti-windup. The arising closed-loop system performs well, even in the presence of extremely large set points. Problem definition The short-period longitudinal dynamics of the VISTA/MATV F-16 at Mach 0.2 and altitude feet (corresponding to a dynamic pressure value of 40.8 psf) at a trim angle of attack of 28 degrees is described locally by the linear model [ ] α ẋ p := = A q p x p + B p δ, (2.31) where x p := [α q] T is the state representing deviations of the angle of attack and pitch rate from the trim condition, δ = [δ e δ ptv ] T is the input representing respectively the deviations of the elevator deflection and of the pitch thrust vectoring from the trim condition, and the entries of the matrices A p and B p are: A p = [ ] [ b11 b, B p = 12 b 21 b 22 ] = [ ]. (2.32) Both inputs are assumed to be subject to rate and magnitude saturation. By using the model (2.7) for the rate saturation, if u = [u e u ptv ] T is the input to

55 2.3. Case studies 49 the saturation block, the following dynamic equations complete the saturated model of the F-16: ( ( )) δ e = R e sgn δ e M e sat ue ( M e ( )) uptv (2.33) δ ptv = R ptv sgn δ ptv M ptv sat M ptv, where the values M e = 21 deg, M ptv = 17 deg, R e = 50 deg/s, R ptv = 45 deg/s have been adopted. According to the results in [35], a daisy chained allocation of the inputs can be adopted; in particular, if r R is the desired angle of attack, the desired value for [b 21 b 22 ] δ is given by where K inner = [ ] and K outer (s) = κ(x, r) = K inner x + K outer (s) (r α), (s )(s s ) (s )(s )(s )(s ) ; according to the matrices (2.32), the daisy chain allocation is given by: [ ] [ yce b 1 ] y c = = 21 κ(x, r) y cptv b 1 22 b 21 (b 1 21 κ(x, r) δ. (2.34) e) Note that, with the above allocation, in the direct connection between the controller and the plant, u = y c, if δ ptv y cptv, then [b 21 b 22 ] δ κ(x, r); moreover, if δ e y ce, then y cptv 0, namely, only the elevator deflection input is used to enforce the desired angle of attack. Anti-windup controller design Since the daisy chained scheme compensates for the saturation at the elevator deflection input, we choose the local closed-loop system as the closed-loop obtained by neglecting the saturation only at the pitch thrust vectoring input. With reference to the notation used in Section 2.1, if f(x, u) is given by equations (2.31), (2.33) (choosing x = [x T p δ T ] T ), the local approximation F of f can be chosen to be the limit system as M ptv + and R ptv +, which, due to the daisy chained structure of the local controller, guarantees a well-behaved local closed-loop system for any reference r. The global controller has been chosen in [75] as α(x, x, y c ) = K aw (x x ) + y c,

56 50 Chapter 2. Uniting local and global controllers with K aw = [ ], following an LQR design for the minimization of a cost function of the state and the input. This design technique can be employed in this case because the open-loop system is Hurwitz. Similarly to the previous example, the united controller can be determined following the approach in Example 2.15 and noticing that the manifold reduction is only associated to the second input, thus obtaining the anti-windup controller: ξ p = A p ξ p + B p (δ w) ẇ 1 = R e sgn w 2 = y ptv u = y c + K aw ξ p, ( w 1 M e sat ( ue M e )) where y c is given by equation (2.34) and w := [w 1 w 2 ] T. Note that the reduced order state of the anti-windup compensator is ξ := [ξ T p w 1 ] T. Simulation results In Figure 2.5, the performance of the anti-windup controller synthesized in the previous section is compared to the response of the saturated system with no anti-windup and to the response of the system with no saturation on the pitch thrust vectoring input. The reference signal is chosen to be r(t) = 0 deg, 0 t < 1 10 deg, 1 t < 5 10 deg, t 5. The dashed curve in the first plot represents the angle of attack of the local closed-loop system (namely, when the local controller is connected to the system with no saturation on the pitch thrust vectoring); the dotted curve represents the angle of attack for the saturated system with no anti-windup and the solid curve represents the angle of attack for the saturated system with L 2 anti-windup. The remaining three plots represent the input time histories in the three cases above, respectively. As previously remarked, these simulations confirm that rate limitation induces instability and oscillations. Magnitude limitation is actually reached

57 2.3. Case studies Angle of attack 80 Nominal inputs Time [s] Time [s] 80 Saturated inputs 80 Anti windup inputs Time [s] Time [s] Figure 2.5: Performance of the anti-windup control system on the VISTA/MATV F-16 short period longitudinal dynamics. by the saturated inputs at time t = 9 after the angle of attack has reached a value of 80 deg, which is definitely already an intolerable flight condition (and is significantly out of scale on the first plot).

58 52 Chapter 2. Uniting local and global controllers

59 Chapter 3 Anti-windup for active vibration isolation devices In this chapter, the anti-windup solution described in Chapter 2 is generalized to perform a successful anti-windup action on very aggressive linear control designs for vibration attenuation systems. In Section 3.2 we briefly describe the main features of vibration isolation systems and we introduce the benchmark example. In Section 3.3 we first resume the standard L 2 antiwindup solution proposed in [78] and then we describe its extension (proposed in [90]) to achieve successful anti-windup on vibration isolation devices. Finally, in Section 3.4, we apply the extended L 2 anti-windup compensation to the benchmark example and we show the resulting performance by means of suitable performance indexes. 3.1 Introduction In most high precision applications, vibrations arising from the external environment are comparable with the required task precision. It is then necessary to guarantee a certain level of isolation of the operating environment from the external one by means of vibration isolation devices. Such is typically the case for high precision positioning devices in aerospace applications (e.g., telescope optics alignment) as well as for common terrestrial applications (e.g., semiconductor manufacturing processes), where low amplitude vibrations can compromise the effectiveness of the tasks. Most of the isolation systems in use today are compliant passive systems. Indeed, passive isolators reduce significantly the high frequency vibrations; nevertheless, they act very poorly at low frequencies, where active systems 53

60 54 Chapter 3. Anti-windup for active vibration isolation devices achieve their best performance. Consequently, the combination of active and passive isolation control actions is typically the most desirable solution to guarantee a large isolation bandwidth. The active stage of a vibration isolation system typically consists of a sensor measuring speed or acceleration (or a combination of them), located on the vibrating structure and of an actuator, typically consisting of a voice coil or a piezoelectric device. A common strategy for addressing the control design for these system is to measure the transfer function between the actuator and the sensor, and design aggressive linear control laws to guarantee adequate isolation levels, via loop shaping techniques in which the controller gain is increased as much as possible within the stability limits. Because these controllers are linear and aggressive, large environmental disturbances, which may occur once in a while, cause the controller to command values for the system s actuators that are not realizable due to magnitude limits. This saturation problem is a major concern, especially in applications where piezoelectric actuators are employed; as a matter of fact, piezoelectric actuators are characterized by severe displacement limitations, thus presenting a significant saturation problem in the control structure. In this chapter, inspired by the experience gained in improving the antiwindup protection for the control system on Newport Corporation s Elite 3 TM active vibration isolation system [90], we introduce a simple academic example comprising the salient features of an active vibration isolation device; the controller windup exhibited by this example is representative for aggressive linear designs employed in active vibration isolation systems that extend the broadband performance to very low frequencies (such as fractions of Hertz). Subsequently, the extended L 2 anti-windup scheme (patent pending [81]) introduced in [90] is described and applied to this example, and performance indexes are introduced for benchmark purposes. The values of the indexes confirm the effectiveness of the extended L 2 anti-windup scheme but leave room for alternative techniques that provide improved performance. Finally, the extended L 2 anti-windup strategy is successfully applied to the Newport Corporation s Elite 3 TM active vibration isolation system (patent pending [81]) and experimental results confirm the effectiveness of the proposed solution. 3.2 A simple active vibration isolation system Most active vibration isolation systems are designed to minimize the longitudinal vibrations of a colocated sensor/actuator device in which either acceleration or displacement is measured. Typically, high performance isolation

61 3.2. A simple active vibration isolation system 55 platforms consist of multiple decentralized modules in which passive and active isolation technologies are combined. While the passive isolation action guarantees good isolation at high frequencies, the active isolation stage significantly improves the low frequency performance of the isolating system, where the passive stage exhibits a resonance peak; this peak is extremely harmful for a number of high precision experimental devices (see, e.g., [28, 70] for a broader discussion about these features). The typical behavior of passive-only and active/passive isolation stages is shown in Figure 3.1, where the force transmissibility corresponding to the various isolation stages is reported. (See [1] for an example of similar transfer functions arising from experimental measurements.) Figure 3.1: Typical transfer functions of passive and active/passive vibration isolation modules The benchmark example Assume that a hypothetical passive isolation stage guarantees the following fourth order linear relation between the vibration source d and the sensor

62 56 Chapter 3. Anti-windup for active vibration isolation devices measurement y: with ẋ = A x + B (u + d) y = C x + D (u + d), A = , B = 0 253, C = [ ], D = [ 0 ] (3.1a) (3.1b) (the eigenvalues of A are ( 2 ± 50i, 5, 6000)), and where the control input u of the active isolation stage corresponds to the displacement of the actuator in the isolating device and d corresponds to the effect of the unmeasured environmental vibrations on such a displacement. The magnitude of the force transmissibility associated with the SISO linear plant (3.1) corresponds to the dashed line in Figure 3.2; this transfer function has been chosen to reproduce the qualitative behavior of the typical passive isolation performance shown in Figure 3.1. Note that equations (3.1) represent the transfer function between the actuator displacement and the output measurement, while Figure 3.2 represents the force transmissibility. It is thus related to the previous one via two integrators. Assume now that a linear controller has been designed for the system following a loop shaping technique, with the aim of improving the isolation performance of the passive system in the low frequency range: with A c = ẋ c = A c x c + B c (u c + d m ) y c = C c x c + D c (u c + d m ), , B c = C c = [ ], D c = [ 0 ],, (3.2a) (3.2b) (the eigenvalues of A c are ( 0.1, 0.5, 0.5, 1000)), and where d m R denotes the measurement errors and the disturbances at the plant s output. By interconnecting the two systems (3.1), (3.2) via the interconnections u c = y, u = y c, (3.3)

63 3.2. A simple active vibration isolation system 57 active vibration isolation is obtained. The resulting performance (represented by the force transmissibility of the closed-loop system) corresponds to the solid curve in Figure 3.2. The effect of the active control action is to significantly improve isolation in the low frequency range, by shifting the resonance peak to very low frequencies, where the environmental vibrations exhibit negligible amplitudes. Compare the curves in Figure 3.2 with the qualitative behavior in Figure 3.1 (as well as with the experimental data in [1]) and observe how this academic example comprises the important features of a typical active/passive vibration isolation module Force Transmissibility [db] Frequency [Hz] Figure 3.2: Performance of the linear closed-loop vibration isolation system (3.1), (3.2), (3.3) (solid) compared to the open-loop force transmissibility associated with (3.1) The saturated closed-loop system As confirmed by the transfer function in Figure 3.2, when the linear controller (3.2) is connected to the plant (3.1) via equations (3.3), an adequate vibration isolation level is achieved by the closed-loop system. Unfortunately, this is not the case when actuator s saturation is taken into account at the plant s input; namely, when equations (3.3) are replaced by the following in-

64 58 Chapter 3. Anti-windup for active vibration isolation devices terconnections: u c = y, u = sat(y c ), (3.4) where sat( ) denotes the standard symmetric saturation function. 1 For this system, if d and/or d m assume occasional large values (e.g., due to external mechanical shocks on the isolating system), the controller commands large values for y c that possibly exceed the actuator s saturation limits. If no anti-windup protection is employed in the control scheme, when saturation is reached, the linearity of the closed-loop system is lost and the resulting performance exhibits a severe degradation Performances of the saturated and unsaturated systems To understand the effect of the saturation nonlinearity on the closedloop performance, we assume that the following disturbance inputs affect the closed-loop system described in the previous section: { ni (t) + 1, t [5, 5.2] d(t) = n i (t), otherwise { no (t) + 0.2, t [5, 5.2] d m (t) = n o (t), otherwise, (3.5) where n i (t) and n o (t) are band limited Gaussian noises with cutoff frequency 100 rad/s and variances 10 3 and 10 6, respectively. The disturbance inputs in equation (3.5) aim to represent a generic large amplitude shock on the isolating system that affects both the plant input and the plant output, superimposed to the environmental vibrations that persistently affect the system. Such disturbance profiles realistically represent, for example, a mass hitting a vibration isolation table (such as the Newport Corporation s Elite 3 TM system [90]) corresponding to disturbances coming both from the floor (the plant input) and from the table top (the plant output). Figure 3.3 shows the response of the saturated system (3.1), (3.2), (3.4) to the disturbances (3.5), when the plant input is limited in the range ±0.05 (dashed lines in the upper plot). To understand the nature of the response in Figure 3.3, compare it to the response of the linear unsaturated system (3.1), (3.2), (3.3) to the same disturbances, which is represented in Figure 3.4. With reference to the upper plot in the latter figure, the large transient exhibited by the nominal closed-loop system after the disturbance has occurred, is of no harm for the unsaturated system. As a matter of fact, when the control signal 1 For ease of exposition, the saturation function is assumed to be symmetric; however, the approach, also applies to saturation-like functions satisfying weaker conditions (see Assumption 2.21.

65 3.3. Standard and extended L 2 anti-windup Plant Input Time [s] Plant Output Time [s] Figure 3.3: Response of the saturated system to pulse disturbances. oscillates between large positive and negative values, vibration isolation is still preserved (see the lower plot in Figure 3.4). Unfortunately, when saturation is present, these large oscillations of the controller output cause the plant input to exceed the saturation limits, thus compromising vibration isolation for a long time after the disturbance has occurred (see the lower plot in Figure 3.3, where vibration isolation is lost for approximately 25 seconds). 3.3 Standard and extended L 2 anti-windup Why L 2 anti-windup? The windup problem described in the previous section is a challenging example of an aggressive linear control design that is not robust to large external disturbances when connected to a saturated plant without any special attention given to the saturation nonlinearity. To recover the nominal control system s performance, it is necessary to modify the controller with an anti-windup action, with the aim of pursuing the following goals: 1. reduce the isolation recovery time after large disturbances have occurred, guaranteeing a consistent response and ruling out possibilities of limit

66 60 Chapter 3. Anti-windup for active vibration isolation devices 0.5 Plant Input Time [s] Plant Output Time [s] Figure 3.4: Response of the linear unsaturated system to pulse disturbances. cycles that compromise the control objective; 2. maintain the aggressive design and the desired vibration isolation level during normal operation; 3. satisfy a reasonable computational time constraint to keep the isolation design effective at high frequencies by allowing high sampling rates in real-time implementations. A naive approach to try to solve this problem is to reset to zero the state of the controller every time the input saturates. Unfortunately, this strategy gives catastrophic results. As a matter of fact, as detailed in the following Remark 3.4, each time the system is switched on, it shows a transient response similar to the one in Figure 3.3. Hence, resetting the controller states primes the controller for another saturation event (caused by the new power-on transient) and the control system cycles between these two events (saturation and reset) thus compromising the desired isolation. Figure 3.5 shows a typical response of the system when this resetting strategy is adopted. In particular, the technique employed in that simulation is to reset to zero the states of the controller and wait for 0.5 seconds, each time the plant input saturates. The

67 3.3. Standard and extended L 2 anti-windup Plant Input Time [s] Plant Output Time [s] Figure 3.5: Typical behavior of the isolation system when the controller states are reset upon input saturation. resulting behavior confirms the inadequacy of this intuitive solution, indicating the need for a systematic approach to address this windup problem. Among the several possible anti-windup choices, we focus on the L 2 antiwindup approach because, appropriately generalized, it gives a very timeefficient solution to the problem, guaranteeing stability and performance recovery. Other approaches might also be suitable, however, a number of them exhibit severe limitations towards the achievement of all the three above mentioned goals. For instance, the results in [29] do not consider the presence of disturbances in the loop and are based on reference shaping; since no reference is present in this control problem, it is not clear how to apply such results. The generalization given in [66] is based on measurement shaping, which would allow application of the algorithm to this problem. However, although a simple observer is used instead of a set-valued observer in that paper to reduce the computational burden, the control algorithm still requires that an optimization problem be solved on-line. Even in the case when this optimization is solved off-line, the storage of the numerical information related to the solution requires dedicated hardware components (e.g., non-volatile storage devices) or a large delay at power-on to precompute and store all the optimal solutions. We have disregarded most of the other approaches available in the literature,

68 62 Chapter 3. Anti-windup for active vibration isolation devices due to the lack of a formal proof of stability and performance recovery. (A notable exception is the result in [56].) In addition to this, the three requirements discussed at the beginning of this section fit well into the L 2 anti-windup framework The standard L 2 anti-windup With reference to the linear time-invariant plant (3.1) and the linear controller (3.2), since the nominal closed-loop system obtained by the interconnection conditions (3.3) is well-posed (i.e., solutions exist and are unique) and internally stable, following the standard L 2 anti-windup construction introduced in Section 2.2.2, we augment the nominal closed-loop with the following anti-windup compensator that reproduces the plant dynamics: ξ = A ξ + B (u y c ) (3.6a) v 1 = k(ξ) (3.6b) v 2 = C ξ D (u y c ), (3.6c) where the function k( ) is suitably designed (see, e.g., Lemma 2.25). By interconnecting the anti-windup compensator (3.6) to the controller via the interconnection conditions u = sat(y c + v 1 ), u c = y + v 2, (3.7) the anti-windup closed loop system (3.1), (3.2), (3.6), (3.7) satisfies the requirements in Definition 2.18 that can be resumed in simple words as follows: given a reference r and certain initial conditions of the nominal closed-loop system (thus characterizing a unique nominal trajectory), call u n ( ) the corresponding plant input, as a function of time, then the anti-windup closed-loop system (starting from the same initial conditions) follows a trajectory such that: 1. if u n ( ) never exceeds the saturation limits and ξ(0) = 0, then the two trajectories are coincident, and ξ(t) = 0 for all times t 0; 2. if u n ( ) exceeds the saturation limits at some time or the initial condition ξ(0) of the anti-windup compensator is not zero, then, (u n ( ) sat(u n ( ))) L 2 (y n ( ) y( )) L 2, (3.8) where y n ( ) and y( ) represent the plant output related to the nominal and to the anti-windup trajectory, respectively.

69 3.3. Standard and extended L 2 anti-windup 63 Typical solutions of the anti-windup closed-loop system guarantee that if u n (t) sat(u n (t)) exponentially converges to zero, then also the output error y n (t) y(t) exponentially converges to zero, which is a special case of (3.8). This condition is well-suited for a framework in which rare events drive a nominal linear plant input outside what would be the saturated region only for a finite amount of time, so that such an input converges in finite time to the unsaturated region. As a matter of fact, after the input enters the unsaturated region, saturation can be disregarded and the anti-windup closed-loop system is linear, thus generating an exponentially decaying response. Such is the case for vibration isolation systems subject to large occasional disturbances. For example, in the upper plot in Figure 3.4, the nominal input stays within the saturation limits after time T = 55 s The extended L 2 anti-windup Although item 2 of the properties of the anti-windup closed-loop system guarantees that the output error (y n y)( ) belongs to the space L 2, the standard L 2 anti-windup solution given in Section does not solve the problem of making this L 2 norm small (as small as possible). This minimization problem corresponds to the goal of minimizing the nonlinear L 2 gain involved in equation (3.8). This can be accomplished via the design of the function k( ) in equation (3.6). As a matter of fact, writing the dynamics of the anti-windup closed-loop system with the change of coordinates X := x ξ, and defining Y := y + v 2, the system appears in the following cascade structure: 2 (X, x c ) subsystem ξ subsystem Ẋ = A X + B (y c + d) Y = C X + D (y c + d) ẋ c = A c x c + B c (Y + d m ) y c = C c x c + D c (Y + d m ) ξ = A ξ + B (sat(y c + v 1 ) y c ) v 1 = k(ξ) v 2 = C ξ D (sat(y c + v 1 ) y c ). (3.9) (3.10) The (X, x c ) subsystem reproduces the nominal closed-loop system dynamics (3.1), (3.2), (3.3). Hence, starting from the initial conditions ξ(0) = 0, the state response X(t) coincides with the nominal (unsaturated) state response x n (t) (see [78] for details). The second system is controlled by the action of v 1 through the time-varying saturation function sat(y c (t) + v 1 ) y c (t). Since by definition v 2 = Y y, and Y reproduces the nominal output response y n, then 2 See also the proof of the more general Theorem 2.12 for a comparison.

70 64 Chapter 3. Anti-windup for active vibration isolation devices v 2 captures the information about how far the anti-windup response y is from the nominal response y n. The goal in the optimized design of v 1 = k(ξ) is then to keep the anti-windup output v 2 = y y n small and to quickly return it to zero, so that, among other things, lim t y(t) y n (t) = 0 as desired. (Note that, when the nominal controller is linear, the choice v 1 = 0 corresponds to the internal model control (IMC) anti-windup structure (see, e.g., [15]), which is also the model based approach outlined in [37] and therein attributed to Irving.) The best that can be done for v 2 depends crucially on the size, especially in an L 2 sense, of sat(y c ) y c (equivalently, sat(u n ) u n ). When the control signal of the nominal response spends very little time in what would be the saturated region, so that sat(u n ) u n is small in an L 2 sense, (correct) intuition suggests that it is possible to induce a response that is fairly close to the nominal response even in the saturated case. This is usually not difficult to achieve by an appropriate choice of k( ) (e.g., following the guidelines given in [78]). However, in some situations, such as the one described in Section 3.2.3, where the nominal response is not feasible for the saturated system for a long time (in the case shown in Figure 3.4, it exhibits large oscillations that travel outside of the unsaturated region for approximately 50 seconds), it becomes difficult to achieve high performance anti-windup through an appropriate choice for k( ) only. In [90], a methodology is proposed (patent pending [81]) to generalize the standard L 2 anti-windup design described in the previous section, using more general designs for v 1 in equation (3.10), to solve the windup problem for an industrial vibration isolation system. The resulting extended L 2 anti-windup scheme consists of the anti-windup compensator (3.6) with k( ) 0 and the following modified controller dynamics: ẋ c = A c x c + B c u c + w y c = C c x c + D c u c, (3.11) w = e(t) M x c where the function e( ) : [0, ) [0, 1] is chosen so that: 1. the closed-loop system formed by the unsaturated plant (3.1) and the linear time-varying controller (3.11) with the interconnection (3.3) is internally stable; 2. if the controller output y c stays within (moves back into) the saturation limits, then e(t) is identically zero (converges to zero in finite time);

71 3.3. Standard and extended L 2 anti-windup upon detecting controller outputs that would exceed saturation limits, the action of e(t) is to discharge as fast as possible the nominal controller states to be able to drive the controller output into the unsaturated region. Remark 3.1 The structure of the proposed intermediate solution may seem similar to the reference governor scheme in [34, 29] or to the measurement governor scheme in [66]; however, the nature of this solution is significantly different because, by virtue of the dynamics (3.6), to be added subsequently, no set invariance properties are required on the closed-loop system (3.9) and the design of w can be carried out as a linear time-varying control design, completely disregarding the saturation nonlinearity. This decoupling property of the L 2 anti-windup solution is the aspect that makes it an appealing approach that allows for simple generalizations, such as the one adopted in this chapter. Item 3 is to be achieved via a suitable design of M. In particular, this matrix is chosen with the goal of minimizing the maximal real part of the poles of the nominal closed-loop system (3.1), (3.11), (3.3) evaluated with e(t) = 1; as a result, whenever e(t) is equal to 1, the closed-loop system (3.1), (3.11), (3.3) is highly damped. If M has been suitably chosen, then when saturation occurs, e(t) can be raised to 1 and kept to 1 for a sufficiently long time. Consequently, the system s trajectory converges rapidly to zero and the nominal controller output goes back to the unsaturated region. Once the system s states are small enough, e(t) is driven back to zero, 3 thus restoring the nominal closed-loop dynamics. Remark 3.2 The design approach described above actually fits into the standard L 2 anti-windup framework described in Section if more general choices for v 1 are allowed in equation (3.6). As a matter of fact, since the controller (3.2) is linear, the modification of the closed-loop system achieved by the insertion of the time varying term e(t) M x c (t) can be realized by first eliminating the role of k( ) in equation (3.6) and then reproducing the linear controller dynamics as follows: ẋ c = (A c + e(t) M) x c + e(t) M x c v 1 = C c x c, (3.12) 3 To avoid limit cycling, e(t) is driven to zero with a limited rate, thus performing a slow authority transfer between the highly damped controller and the aggressive controller.

72 66 Chapter 3. Anti-windup for active vibration isolation devices and adding the output v 1 of this extra dynamic element to the output of the nominal controller as in equations (3.7) (this constitutes a non-minimal realization of the linear controller dynamics). The resulting anti-windup closed-loop system (3.1), (3.2), (3.6a), (3.6c), (3.12), (3.7) coincides with the closed-loop system (3.1), (3.11), (3.6), (3.7) with k( ) 0; however, this last representation is numerically more robust and more efficient for real-time implementation, because it does not require duplication of the linear controller dynamics. Remark 3.3 To prove uniform asymptotic stability of the time-varying closedloop system (3.1), (3.11), (3.3), various techniques can be employed, based on results on linear time-varying system, uncertain systems or absolute stability criteria. However, since the conservativeness of these methods might lead to difficulties in the design of M, an effective way to address the above mentioned uniform asymptotic stability problem is to limit the rate of variation of e(t) (e.g., by filtering it via a rate saturation function) and proving the desired stability result combining results on slowly varying systems (see, e.g., [42, 5.7]) with stability tools from µ-analysis [7] that apply to linear uncertain systems with constant perturbations. In this case, if the rate of variation of e(t) is small enough, the stability of the closed-loop system for each constant value of e(t) = e [0, 1] is sufficient to imply the stability of the slowly varying system. To better understand the damping action performed by the modified controller (3.11), a simulation of the response of the unsaturated closed-loop system (3.1), (3.11), (3.3) to the same disturbance used in the simulation in Figure 3.4 is shown in Figure 3.6. Notice that the large oscillations of the original response in Figures 3.3 and 3.4 are eliminated by the discharging action performed by the time-varying solution. 3.4 Extended L 2 anti-windup for the benchmark example The extended L 2 anti-windup solution described in Section has been employed to solve the windup problem introduced in Section The arising closed-loop system is derived from the interconnection of systems (3.1), (3.6), (3.11), via equations (3.7), choosing k( ) 0, that result in

73 3.4. Extended L 2 anti-windup for the benchmark example Plant Input Time [s] Plant Output Time [s] Figure 3.6: Response of the isolating system to the disturbances in equation (3.5) when saturation is neglected and the time-varying controller (3.11) is used. the following anti-windup closed-loop dynamics: ẋ = A x + B (sat(y c ) + d) ẋ c = (A c + M e(t)) x c + B c (y C ξ + d m ) ξ = A ξ + B (sat(y c ) y c ) y = C x y c = C c x c, (3.13) corresponding to the block diagram in Figure 3.7, where four main components can be identified: 1. The plant, together with disturbance sources at its input (corresponding to the input d in equation (3.13)) and at its output (corresponding to the input d m in equation (3.13)). 2. The nominal controller (shaded blocks), which is broken into the input, state and output matrices to allow state measurement (the controller transfer function is proper for this particular application).

74 68 Chapter 3. Anti-windup for active vibration isolation devices Out uc K Bc Nominal Controller x = Ax+Bu y = Cx+Du Ac xc K Cc yc Input Noise Saturation u k*z(s) p(s) Plant y K e(t) Anti windup STAGE II matrix M Out Stage II Enable In k*z(s) p(s) Anti windup STAGE I v2 Out Measurement Noise Figure 3.7: Block diagram of the control scheme with extended L 2 anti-windup protection. 3. The Anti-Windup stage I, which is a replication of the plant transfer function (corresponding to equation (3.6) with the choice v 1 = 0), designed following the approach described in Section According to the robustness analysis carried out in [78], small uncertainties in the plant model are tolerated by the anti-windup system. 4. The Anti-Windup stage II, which is the embodiment for the extra term e(t) M x c, described in Section It is constituted by simple modifications of the nominal controller dynamics mentioned in item 2 of this list. The role of the Stage II Enable block in Figure 3.7 is to assign the following behavior to e(t). The function e(t) evaluates at zero when y c = sat(y c ) and goes continuously to one as soon as y c saturates, thus switching to the highly damped closed-loop system. After the controller output enters the unsaturated region again, e(t) is kept to one for approximately one second (to ensure adequate discharging of the controller states) and is subsequently driven back to zero with a slow variation rate. Solving a simple numerical optimization algorithm, the matrix M in equa-

75 3.4. Extended L 2 anti-windup for the benchmark example 69 tions (3.13) has been chosen, according to the dynamics (3.1), (3.2), as M = , Plant Input Time [s] 1 Plant Output Time [s] 1 Controller Input Time [s] Figure 3.8: Response to the disturbances (3.5) of the vibration isolation system with extended L 2 anti-windup protection. The results shown in Figure 3.8 confirm the effectiveness of the design. This figure represents the response of the anti-windup controller to the same disturbance that caused the undesired behavior in Figure 3.3. The first and the second plots of the figure correspond to the plant input and the plant output, respectively. The dashed lines in the first plot correspond to the saturation limits. Vibration isolation is recovered in approximately 4 seconds after the disturbance has occurred, as compared to the 25 seconds required in the experiment shown in Figure 3.3. With reference to the control scheme in Figure 3.7, and its response represented in Figure 3.8, the disturbance occurs at time t = 5 driving the controller output outside the unsaturated region. When this happens, the anti-windup stage I action keeps the nominal controller well-behaved, while the function

76 70 Chapter 3. Anti-windup for active vibration isolation devices e(t) raises continuously to 1, thus enabling the anti-windup stage II action. This last action discharges the controller states driving the controller output back into the unsaturated region. Once the controller output is within the saturation limits, the function e(t) goes back to zero 4 and the nominal control action is recovered, as well as the desired vibration isolation. The lower plot in Figure 3.8 represents the input to the nominal controller u c (see the block diagram in Figure 3.7), which corresponds to the output Y in equation (3.9). The response in the Y variable confirms the effectiveness of the anti-windup stage I. As a matter of fact, since the (X, x c ) subsystem (3.9) reproduces the nominal closed-loop dynamics, the input Y to the nominal controller corresponds to the response y l of the unsaturated plant (compare it with the middle plot, representing the actual response of the plant), thus keeping the nominal controller well-behaved. Input (SAT) Input (AW) Output (SAT) Time [s] Time [s] Output (AW) Time [s] Time [s] Figure 3.9: Power-on operation for the vibration isolation system with (lower two plots) and without (upper two plots) extended L 2 anti-windup protection. Remark 3.4 An interesting feature of the extended L 2 anti-windup protection is the ability to induce a smooth and fast power-on of the active isolation 4 It is actually kept to 1 for an additional second to ensure a sufficient discharge of the controller states.

77 3.4. Extended L 2 anti-windup for the benchmark example 71 stage. According to what the author experienced on the industrial application in [90], this feature is extremely important in isolating environments. Indeed, as confirmed by the simulation in the upper two diagrams in Figure 3.9, the power-on operation in aggressive controllers is similar in nature to the response of the system to pulse disturbances (such as the one in Figure 3.3). As a matter of fact, at power-on, the initial state of the controller (which is typically set to zero) is in general incompatible with the initial state of the plant (which is different from zero because of the disturbance input). When the extended L 2 anti-windup protection is added to the controller, the power-on transient is significantly reduced, as shown by the lower two plots in Figure 3.9. This extra feature of the anti-windup protection, in addition to reducing the settling time at power-on, is very useful in batch applications, where the active isolating system needs to be restarted at regular intervals for manufacturing needs. In such situations, the large decrease of the lag time after power-on corresponds to significant reduction of the production time and consequent increase of the system s throughput and of the related performance The benchmark indexes From the discussion in Section 3.3.3, it is clear that the standard antiwindup solution proposed in [78] and resumed in Section is not sufficient to guarantee a satisfactory performance on the class of systems addressed in this chapter. For comparison purposes, let s consider the choice k( ) 0 in equation (3.6), which (as already stated in Section 3.3.3) corresponds to the IMC/model-based strategy (see, e.g., [37, 15]). Since the plant (3.1) is asymptotically stable, this choice for k( ) is sufficient to solve the standard L 2 anti-windup problem. However, the arising solution (that corresponds to the closed-loop dynamics (3.1), (3.2), (3.6), (3.7) with k( ) 0) does not improve the undesired response in Figure 3.3 but actually induces a longer isolation recovery time. This undesired effect is reported in Figure 3.10, where the input and the output of the plant are represented in the upper and the lower plots, respectively. The dotted curve in the upper plot, that represents the nominal (linear) control input, shows that the choice k( ) 0 induces the anti-windup to keeping small (in an L 2 sense) the difference (u l u)( ) between the nominal and the saturated plant inputs. However, this corresponds to an unacceptable growth of the performance output (y l y)( ) that we want to keep small. In particular, it prevents isolation recovery up to approximately 50 seconds after the disturbance has occurred, which is an even worse performance than the one of the system without any anti-windup compensation (whose response is

78 72 Chapter 3. Anti-windup for active vibration isolation devices 0.05 Plant Input Time [s] Plant Output Time [s] Figure 3.10: Response to the disturbances (3.5) of the vibration isolation system with IMC/model-based anti-windup. shown in Figure 3.3). To compare the performances of the three control schemes (corresponding to the trajectories in Figures 3.3, 3.8 and 3.10) we introduce the following family of performance indexes: I n (f( )) = k=0 1 k! ( t 0 ) 1 (t t 0 ) (k+n) f(t) 2 dt 2, n = 0, 1,... (3.14) where f( ) : [t 0, ) R is an exponentially decaying function to be specified. The idea behind equation (3.14) is that the standard L 2 norm of f( ) is weighted by increasing powers of time as the index k of the sum increases. These increasing time powers in the infinite terms of the sum highly penalize functions f( ) that do not converge rapidly to zero. For instance, the bursts in the lower plots in Figures 3.3 and 3.10 occurring at large times cause the terms that follow the first ones in sum to assume large values, thus increasing the resulting value of the index I n. The longer the bursts will take to die away, the bigger the number of non-negligible terms in the sum will be. Via the choice of the index order number n, a trade-off must be achieved between the penalty associated with large amplitudes at small times and the penalty

79 3.4. Extended L 2 anti-windup for the benchmark example 73 associated with non-negligible amplitudes at larger times. However, we are only interested in indexes corresponding to small values of n. As a matter of fact, there exists a large enough n such that the increasing powers of time penalize too much even extremely small tails of the exponentially decaying function, consequently the values of all the indexes I n (f), n n are not significant for our benchmark purposes. From a formal point of view, the sum in equation (3.14) converges whenever the f( ) is exponentially decaying, as proven by the following lemma. Lemma 3.5 If f( ) : [t 0, ) R is an exponentially decaying function, then, for any integer n 0, the sum I n given by equation (3.14) is convergent. Proof. Since f is exponentially decaying, there exist strictly positive constants c and λ such that 1 I n k! 0 t (k+n) c 2 e λ t dt = =: k=0 k=0 c k! s(k). k=0 (k + n)! λ k+n+1 (3.15) To prove convergence of the last term in equation (3.15) we invoke the ratio test (see, e.g., [19, p. 505]), and the following equalities, s(k + 1) k + n + 1 lim = lim k s(k) k (k + 1) 2 λ = 0, imply that the sum I n in equation (3.14) is convergent. As already pointed out in Section 3.3.2, we can assume for our vibration isolation system that the output error (y l y)( ) is exponentially decaying; we choose then the function f( ) in equation (3.14) as a filtered version of (y l y)( ) via a linear filter H f (s) having transfer function H f (s) := (s )(s s ), (3.16) (s + 30)(s + 40)(s + 50) namely having zeros coincident with the slow modes of the nominal closedloop system. This last filtering action reduces the effects of the exponentially

80 74 Chapter 3. Anti-windup for active vibration isolation devices AW Figure I 0 I 1 I 2 I 3 I 4 I 5 I 6 None L IMC Table 3.1: Performance indexes corresponding to the trajectories in Figures 3.3, 3.8 and decaying tails of f(t) on the values of I n (f) (for large values of n). We also choose t 0 = 5.2, which corresponds to the time when the pulse disturbances end (see equation (3.5)). The numerical values of the indexes for the three trajectories represented in Figures 3.3, 3.8 and 3.10 have been computed numerically for n {0,..., 6}, by computing the indefinite integrals up to time t 1 = 300 and by computing the first N = 50 terms of the sum. It was observed that increasing t 1 or N led to unnoticeable changes in the index values. The results are listed in Table 3.1. From Table 3.1, the values of the performance indexes in the various cases confirm the fact that small values of n penalize large values of f( ) at small times, while as n increases, large values at large times are penalized more and more. Accordingly, the extended L 2 anti-windup protection, which presents large output oscillations in the very first seconds after t 0 (see the second plot in Figure 3.8), is not associated with a significant improvement of I 0 and I 1, but shows a larger and larger performance increase as seen by the indexes from I 2 to I 6. As already pointed out, for n 4, the tails of f(t) start affecting the value of the indefinite integral in equation (3.14) and induce a general increase of all the index values. On the basis of this observation, the indexes I n with n 7 are not interesting for this benchmark problem and are disregarded. (Note that changing the transfer function of the filter (3.16) significantly affects the tails of f(t) and, consequently, also affects the values of these indexes.) 3.5 The Elite 3 TM system System description Newport Corporation has developed an active vibration control system (ELectronic Isolation TEchnology) for isolation tables (see Figure 3.11). The active control system is designed to minimize the vertical vibrations of a work table that are due to vertical vibrations of the floor. The isola-

81 3.5. The Elite 3 TM system 75 Figure 3.11: Newport Corporation s Elite 3 TM active vibration isolation system. tion device consists, typically, of three independent sensor/actuator modules connected to a DSP (see Figure 3.12). The sensor/actuator modules support the table and react to attenuate the vibrations of the floor as viewed from the table. The DSP monitors the vertical velocities of each module by means of geophone sensors and produces voltages that control the vertical displacement of each module. These commanded voltages are translated into vertical displacements through the action of piezoelectric stack actuators. An extra feature of the Elite 3 TM system, which is necessary for an effective industrial application, is the possibility to dock and undock the isolated table, recovering the isolation as soon as possible after undocking. As a matter of fact, in a typical production application, when samples are serviced on the isolated table in a batch process, every time a new sample needs to be processed, the table is docked to allow loading and unloading and is undocked to recover the isolation. A fast recovery time is then desirable after undocking because it directly affects the throughput of the overall system. In addition to this, repeatability is desirable so that multiple vibration isolation devices can

82 76 Chapter 3. Anti-windup for active vibration isolation devices Figure 3.12: The three modules of the Elite 3 TM isolating system connected to the DSP chassis. be pipelined without a drastic decrease of the throughput The system without anti-windup protection With the aim of guaranteeing the desired attenuation performance in the frequency range 0.5 Hz to 200 Hz, an aggressive linear dynamic compensator has been designed on the basis of a linear model of each isolation module using SISO loop shaping techniques. This predesigned linear controller (similar in nature to the controller (3.2)) is effective in attenuating standard environmental vibrations. However, very large magnitude disturbances (e.g., due to shocks on the floor or on the table) cause the controller to command large vertical displacements of the modules that go beyond the piezoelectric actuators saturation limits. If no anti-windup protection is employed in the control scheme, when saturation is reached, the linearity of the closed-loop system is lost and the performance of the closed-loop system exhibits a serious degradation. To understand the effects of piezoelectric saturation on the nominal linear design, we eliminate the existing anti-windup protection from the Elite 3 TM system and we analyze the response of the saturated closed-loop system to external impulsive disturbances. Figure 3.13 shows the experimental response of one isolator to the disturbance arising from a mass falling on the floor beside the isolating table. 6 5 In a pipelined configuration, if the recovery times are not uniform, the idle times of the serving stations increase significantly as the number of cascaded stations increases. 6 For safety reasons, the experiment reported in Figure 3.13 is carried out reducing, by a factor of four, the actual operating gain of the controller. The full gain response without