Strategie di Controllo Robusto Multivariabile

Transcription

1 UNIVERSITÀ DEGLI STUDI DI PISA Dipartimento di Ingegneria Chimica, Chimica Industriale e Scienza dei Materiali Dottorato di Ricerca in Ingegneria Chimica (XIV Ciclo) Strategie di Controllo Robusto Multivariabile Relatore: Prof. Alessandro Brambilla Candidato: Ing. Gabriele Pannocchia

2 UNIVERSITY OF PISA Department of Chemical Engineering, Industrial Chemistry and Material Science Strategies for Robust Multivariable Model-Based Control Gabriele Pannocchia Degree of Doctor of Philosophy in Chemical Engineering 2001

3 Copyright 2001 by Gabriele Pannocchia

4 To my wife Rita...

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6 Acknowledgments I would like to start by acknowledging my advisor, Prof. Alessandro Brambilla, who constantly provided me his guidance. His advice and his practical sense kept me focus on the target and, at the same time, he let wander around the mathematical and philosophical details as I wanted. Thank you for all the support you gave me. Then, I would like tho thank my former co-advisor, Daniele Semino. One of the main reasons I decided to get a PhD degree was him and his classes on advanced control. I still remember when we had pizza near the Tower and you told me about the pros of a PhD (and you let Prof. Brambilla tell me about the cons!). I also would like to thank Prof. Claudio Scali who was always able to fill up my days in Pisa with practical things to do (mostly things I tried to avoid, like the MURST final report!). Still remaining in Pisa, I would like to acknowledge my friend and colleague Gabriele Marchetti. We studied together for so many years during the MS and PhD degrees that some professors in the department called us The Twins (perhaps because we are both bald!). Your ability to solve computer problems is unique even though you are still using Microsoft products! I also would like to acknowledge Monia Cavallaro, the youngest PhD student in the group, because of her friendship and her sincerity. And now, Madison. The year and half I spent in Madison has certainly been the highest and most productive period during the graduate school and one the best experience in my life. I would to start by acknowledging Prof. James B. Rawlings for accepting me as visitor in his group. As soon as I joined the Rawlings group, he made me feel part of it, he constantly guided me and provided me an exceptional enviroment for my research activity. Jim, your unique teaching and research skills are an inspiration for me, and I often miss our morning discussions at the white board in Working in the your group has been a privilege and a honor for me, and the University of Wisconsin will remain my second

7 viii Acknowledgments alma mater. I also had the privilege of meeting and working with Prof. Stephen J. Wright, who is now at the Computer Science department of the University of Wisconsin. The mathematical rigor of his proofs and his practical sense in numerical computation are rare to be found in the same mathematician. When I arrived to Madison, Chris Rao was at the airport to pick me up (with Scott s car!). Then he brought me around Madison to find an apartment and to Wal Mart to buy the basic stuff (in his junky car!). Finally, when he graduated he left me his TV, which ran great all the time! What can I say? Thank you. The day after my arrival to Madison, I went to Engineering Hall and the first person who appeared to welcome me was Daniel Patience. He is now one of my best friends. We shared so many things together from Saturday night movies to the U2 concert in Milwaukee. Unfortunately for you, I always (i.e. sometimes) beated you at darts. You are ONE of the persons who define my stay in Madison. And now Matt Tenny, my closest office mate, certainly my English teacher and a unique friend. He was always making fun of my Italian accent, but without him I would have never written my dissertation in English. You taught me what optimization is about, how UNIX works and how to debug a code. In return, I cooked real Italian pasta for you ( illegally imported to the USA by my parents) and we spent many nights playing weird music with the guitar and talking together. I ll never forget it. Eric Haseltine has been a great office and class mate. And a sincere friend too, who also made one of my dreams come true: watching a Lakers game (live). When I think about Madison I often remember our morning walks from Engineering Hall to the Math department, our Easter lunch and my last night in Madison at your place watching game 6 of Bucks 76ers. Thanks to you and Lori for all of it. When I was about to go to Madison, I never thought that I would have played classic guitar duets with someone. Thanks to Scott Middlebrooks if this has become true. I also would like to acknowledge Jenny Wang for making me try sushi but most of all for teaching me the basics of multi-model MPC. I would like to thank John Eaton for always answering my weird questions about Octave and Linux. Then I would like to thank the other group members and visitors, Brian Odelson, Sankash Venkatesh and John Bagterp Jørgensen, for all the dart and tennis games we played together. Finally, I would like to thank my parents, my sisters, all my relatives and my friends in Italy who supported me during these years, especially when I

8 Acknowledgments ix was in Madison. You all gave me good reasons for coming back to Italy. I conclude by thanking my wife Rita for sharing these years with me. Without you I would have never done it. Gabriele Pannocchia University of Pisa Italy December 2001

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10 Contents 1 Introduction Conventional and advanced controllers Issues in model-based control Model predictive control: optimality and robustness issues Dissertation overview I Input-output models 7 2 Method of robust modified models: theory Introduction Decouplers: theory and properties Steady-state optimization technique Choice of the optimal decoupler Choice of the optimal modified model Dynamic optimization Extension to non-square systems Preliminary definitions and theorems Optimization technique Method of robust modified models: applications Case study # Decouplers IMC controllers DMC controllers Case study # Nominal and modified models DMC controllers Conclusions

11 xii Contents 4 Implications of distillation column design on ill conditioning Introduction Distillation column design: the engineering point of view Distillation column control Case study Effect of the reflux Effect of column pressure Closed-loop simulations for columns at the same pressure Closed-loop simulations for columns at different pressure Discussion of the results Conclusions Linear models in predictive control of distillation columns Introduction Control algorithms Dynamic Matrix Control Linear Weighted Dynamic Matrix Control Distillation column models The complete nonlinear model Linear models Modified models Simulations Set-point change # Set-point change # Comments Conclusions II State-space models 71 6 Disturbance models for offset-free MPC control Introduction A motivating example Plant model Estimator and regulator Disturbance rejection MPC control algorithm Preliminary definitions and notations Disturbance model and estimator Target calculation and regulator

12 Contents xiii 6.4 Main results Restrictions on the augmented system Offset-free disturbance models Remarks Case studies Revisiting the motivating example Offset-free control: the other side of the coin Conclusions Robust invariant unconstrained offset-free MPC design Introduction Basic definitions Robust unconstrained offset-free controller Model, estimator and regulator Closed-loop system description Nominal model and disturbance model search Robustly invariant set Comments Case studies Example # Example # Conclusions Robust constrained offset-free model predictive control Introduction Min-max MPC controller Properties Remarks Case studies Example # Example # Comments Conclusions Model predictive control with active steady-state constraints Introduction Formulation of the problem Infinite horizon model predictive control Finite parameterization of the control problem Optimal solution of the infinite horizon problem

13 xiv Contents Upper bound on the optimal solution Lower bound on the optimal solution Convergence of the optimal sequences Implementation issues Case studies Example # Example # Conclusions Conclusions 155 A The velocity algorithm LQR 159 A.1 Algorithm formulation A.1.1 The model A.1.2 The regulator A.1.3 The estimator A.2 Key-issues of the velocity algorithm A.2.1 Initialization of the model A.2.2 Choice of the noise model A.2.3 The standard algorithm derived from the velocity algorithm A.3 Differences between the traditional and the velocity algorithm. 168 A.3.1 Offset-free control A.3.2 Inferential control A.3.3 Control of singular systems and underdefined systems (with more outputs than inputs) A.3.4 Unreachable targets A.4 Case studies A.4.1 Example # A.4.2 Example # A.5 Observations and conclusions B Complements of Chapter B.1 Closed-loop system matrices B.2 Closed-loop objective function evaluation C Complements of Chapter C.1 Proof of Theorem C.2 Quadratic programs and sequences of sets C.2.1 Convex quadratic programs in l

14 Contents xv C.2.2 Increasing and decreasing sequences of sets C.3 Proofs of the convergence theorems Bibliography 201

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16 List of Tables 3.1 Case # 1. Characteristics of the four decouplers Case # 1. ISE for decouplers in the nominal case and for the worst uncertainty Case # 1. Parameters of the filters for the IMC controllers Case # 1. ISE for IMC controllers in the nominal case and for the worst uncertainty Case # 1. Tuning parameters of DMC controllers Case # 1. ISE for DMC controllers in the nominal case and for the worst uncertainty Case # 2. Modified transfer function matrices Case # 2. Tuning parameters of DMC controllers Case # 2. ISE for DMC controllers in the nominal case and for the worst uncertainty ISE for columns at the same pressure ISE for columns with the same number of stages ISE for columns with the same design reflux ratio Steady states and linear models Modified models Tuning parameters Integral square errors in the set-point change from c to b Integral square errors in the set-point change from d to e Parameters of the CSTR Detectability of disturbance models Example # 1: plant matrices (case 1) Example # 1: plant matrices (Case 1) Example # 1: plant matrices (Case 2) Example # 2: time-varying set points

17 xviii List of Tables A.1 Unknown variables and their dimension A.2 Independent equations

18 List of Figures 2.1 Feedback loop with a decoupler Typical behavior of IRI vs α Control scheme with input uncertainty Case # 1. Decouplers in the nominal case Case # 1. Decouplers in the uncertain case Case # 1. Open-loop behavior of nominal and modified models Case # 1. IMC controllers in the nominal case Case # 1. IMC controllers in the uncertain case Case # 1. DMC controllers in the uncertain case Case # 2. Condition number as a function of frequency Case # 2. DMC controllers in the uncertain case Distillation column scheme Reflux ratio vs number of stages (at constant pressure) Minimum condition number vs number of stages (at constant pressure) Minimum condition number vs reflux ratio (at constant pressure) Reflux ratio vs operating pressure Minimum condition number vs operating pressure Set-point change in the uncertain case for columns at the same pressure Set-point change from c to b in the uncertain case Set-point change from d to e in the uncertain case Scheme of the CSTR Linearized CSTR: disturbance on the inlet flow rate that causes offset Linearized CSTR. Rejection of the disturbance on the inlet flow rate: outputs

19 xx List of Figures 6.4 Linearized CSTR. Rejection of the disturbance on the inlet flow rate: inputs Nonlinear CSTR. Rejection of the disturbance on the inlet flow rate: outputs Nonlinear CSTR. Rejection of the disturbance on the inlet flow rate: inputs Ill-conditioned column outputs. Set-point change in the presence of gain errors Ill-conditioned column inputs. Set-point change in the presence of gain errors Projection of the invariant region onto R Example # 1: closed-loop input and output (case 1) Example # 1: closed-loop input and output (case 2) Example # 2: closed-loop outputs Example # 2: closed-loop inputs Example # 1: closed-loop input and output (Case 1) Example # 1: closed-loop input and output (Case 2) Example # 1: closed-loop input and output (Case 3) Example # 2: closed-loop input and output Horizon length vs relative difference of objective functions Closed-loop input for Example # Closed-loop output for Example # Closed-loop inputs for Example # Closed-loop outputs for Example # Closed-loop required horizon for Example # A.1 Output and input responses for each controller (Example # 1). 178 A.2 Outputs for velocity MPC controllers with different horizon and MPC with target calculation (Example # 2)

20 Chapter 1 Introduction 1.1 Conventional and advanced controllers Process control is an important and active area both in academic research and in practical applications. An efficient control system should be able to maintain the process in a safe operating regime and, possibly, close to an economically optimal steady state. Typically, several variable measurements are available to monitor the process regime, and this information is used to adjust the manipulated variables (feedback control). Conventional decentralized controllers are designed by coupling each manipulated variable with a controlled one. Integral action is often added in order to obtain offset-free control, that is to maintain each controlled variable at a desired value (set point). This approach can provide an acceptable performance when the interactions among control loops are limited and when the process disturbances have little effect on the controlled variables. Also, the application of conventional controllers is limited to systems with an equal number of controlled and manipulated variables (square systems), while in the process industries, and in particular in the chemical industries, non-square systems often arise. In order to address such issues advanced controllers can be used. This class includes a wide variety of controllers, ranging from decouplers to predictive controllers. These controllers are based on an explicit process model, which may be of different type (e.g. transfer function, impulse or step response, autoregressive or state-space models). Feedback from the process measurements is still used since modeling errors and/or unknown disturbances are always present. Clearly, the performance of a model based controller is strictly related to the accuracy of the model. Robustness of a model-based controller, which is its ability of providing a good control performance in the presence of modeling errors, is therefore a basic issue in process control, and it is often a

21 2 Chapter 1. Introduction source of skepticism about the use of advanced controllers in industry. 1.2 Issues in model-based control There are a number of issues that often limit the performance of model-based controllers, such as the process ill conditioning and nonlinearity. In many practical applications multivariable processes are ill conditioned, especially in the chemical industry. Ill conditioning occurs when some manipulated variables have similar effect on some controlled variables. Thus, these processes are characterized by large interactions between control loops. Moreover, they show a phenomenon called directionality, which is a large variation of the process gain depending on the input direction. Thus, when a disturbance affects the plant in the favorable direction (i.e. direction with high gain), a small variation of the manipulated variables is required to reject such a disturbance. On the other hand, if the disturbance is aligned with the unfavorable direction (i.e. direction with low gain), a large change in the process inputs is required. For such reasons, conventional controllers cannot provide an adequate performance with fast disturbance rejection (or set-point tracking), and a model-based strategy may be desired. Model-based controllers (e.g. feedback controllers with decoupler, IMC controllers, predictive controllers) are able to solve the problems related to interactions and directionality. However, in the case of ill-conditioned processes, they all suffer from sensitivity to uncertainties, especially to input uncertainties. Therefore, they can guarantee a good performance in the nominal case (i.e. when the uncertainties are negligible) but this performance quickly deteriorates as plant-model mismatch occurs. Model-based controllers typically use a linear model of the process but, in practice, many processes show a nonlinear behavior (e.g. reacting systems, distillation processes). Thus, the performance of a linear model-based controller can be satisfactory if the process operates in limited region, i.e. a region in which the linearized model is a good approximation of the nonlinear process. A number of nonlinear control strategies exist, which are based on first principle models (e.g. differential-algebraic equations) and on empirical models (e.g. artificial neural networks). Some processes, e.g. distillation processes, show a nonlinear behavior coupled with ill conditioning, thus making the control task even more challenging. Consider the composition control of the top and bottom product of a binary distillation column using, as manipulated variables, the top reflux and the boil-up rate (usually known as LV control structure). The nonlinear behavior

22 Chapter 1. Introduction 3 occurs when the product purity changes. In fact, when the column operates at high purity, the gains of the manipulated variables on the controlled variables are high, and when the product purity decreases the gains decrease as well. Another source of nonlinearity is represented by the fact that mole fractions have physical bounds (e.g. they must be non-negative and no greater than 1). Clearly, a linear model can represent the behavior of a distillation column only in small operating area. Moreover, the composition control in distillation columns is a typical ill-conditioned process. In fact, if one increases the top reflux rate with the boil-up rate kept constant, the top product purity increases and the bottom product purity decreases. On the other hand, if one increases the boil-up rate with the top reflux rate kept constant, the bottom product purity increases and the top product purity decreases. Thus, increasing the purity of one product at expense of the other one can be accomplished with small input changes, while increasing (or decreasing) the purity of both products can be obtained only with large changes of the manipulated variables. 1.3 Model predictive control: optimality and robustness issues In the last decades Model Predictive Controllers (MPC), which arose from the pioneering industrial applications called Identification and Command, IDCOM (Richalet et al., 1978), and Dynamic Matrix Control, DMC (Cutler and Ramaker, 1979), have become the prominent form of advanced controllers applied in the process industries. Originally, these control algorithms used finite impulse or step response models to predict the future process behavior. In order to obtain offset-free control, the model is updated with feedback information. Comparing the current measured process output and the current predicted output, a constant bias term is added to the future model forecasts. The ability of dealing with process constraints is one of the features that rendered MPC popular, both in industry and in academia. IDCOM used an iterative algorithm to handle constraints while DMC, first, used different control and prediction horizon and solved the optimization problem in a least squares sense. Next, quadratic programs were used (Garcia and Morshedi, 1986) to obtain an exact solution of the constrained optimization problem, allowing the designer to specify constraints both on manipulated inputs and on controlled outputs. However, these convolution models cannot be used with open-loop integrating (e.g. tank level control) or unstable plants (e.g. CSTR reactors may be open-loop unstable). Generalized Predictive Control, GPC (Clarke

23 4 Chapter 1. Introduction et al., 1987a; Clarke et al., 1987b), instead, used auto-regressive models. Despite their success in practical applications these first implementations of predictive controllers suffered from a lack of theoretical basis, in terms of nominal stability, offset-free properties, robustness, and on-line implementation issues. Later, state-space formulations of MPC controllers have been proposed, which handle open-loop stable and unstable systems, and many theoretical results appeared. Several recent reviews are available (Qin and Badgwell, 1997; Morari and Lee, 1999; Mayne et al., 2000), which summarize the theoretical formulations and industrial implementations of MPC. However, several issues about MPC still remain open and are interest of many researches both in academia and in industry. Several nonlinear model predictive control (NMPC) strategies are currently under investigation and issues about the computational cost are particularly relevant (Tenny et al., 2001). One of the most active fields concerns robust formulations of linear MPC. Several algorithms, typically based on a multi-model description of the plant, have been proposed. A basic task is to design a controller that stabilizes all the plants in a given region. Next, it is desirable to design a controller that guarantees offset-free performance for all plants. 1.4 Dissertation overview In this dissertation the problem of designing robust model-based controllers is considered, with particular attention to predictive control strategies, because of their practical applicability. The dissertation is organized in two parts, each of them composed of four chapters. In Part I, control strategies based on input-output models are considered. In Chapter 2 a method for designing robust model-based controllers for illconditioned systems is presented. This method is based on the search of a modified model that, used in the place of the nominal model, increases the controller robustness. Application of this method to linearized models of illconditioned distillation columns are presented in Chapter 3. In Chapter 4 a study of how the process ill conditioning varies with the main design parameters (number of stages, reflux rate, operating pressures) of a distillation column is presented. In Chapter 5, instead, the problem of controlling a nonlinear illconditioned distillation process is considered. Predictive controllers based on different kinds of linear models are compared. The nonlinear behavior is taken into account by using a number of linear models identified in different operating conditions, while the ill conditioning is faced by using modified models.

24 Chapter 1. Introduction 5 In Part II, state-space formulations of model predictive control are considered. In Chapter 6 the problem of steady-state offset is addressed, and general results on how to build an offset-free MPC controller are derived. These results are particularly useful in the common case in which only some of the measured variables need to be controlled with zero steady-state offset. In Chapters 7 and 8 a robust MPC algorithm with proved stability and offset-free performance is proposed. The plant is assumed to be an unknown time-varying convex combination of a number of plants. First, in Chapter 7 an unconstrained offset-free robust controller is designed by optimally choosing a nominal model (inside the uncertainty region) and a disturbance model. Process constraints are considered by constructing an invariant set in which the controller is guaranteed to satisfy these constraints. Next, in Chapter 8 a robust MPC controller based on a dual-mode paradigm is proposed. If the plant state is inside the invariant region, the unconstrained controller previously developed is applied. Otherwise, a min-max problem is posed and solved in such a way that the plant state eventually reaches this invariant region. In Chapter 9 the problem of finding the optimal solution of the MPC problem for the case in which constraints are active at steady state is addressed. This cases is relevant since it is frequently the case that, in order to maximize performance objectives, the MPC controller operates at the boundary of the feasible region with respect to both input and output constraints. Moreover, when a nonzero disturbance enters the process, it is often the case that one or more manipulated inputs ride at their corresponding saturation values during a period of steady-state operation. For such important problems, only suboptimal solutions were available. Finally, in Chapter 10 the main accomplishments of this work are summarized, while some complementary results can be found in the Appendices.

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26 Part I Input-output models

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28 Chapter 2 Method of robust modified models: theory In this chapter the theory of the robust modified model method is presented. This methodology is based on the search of modified models that, used place of the nominal ones, are able to increase the robustness of model-based controllers. These models are obtained as the solution of an optimization problem whose objective function contains terms related both to nominal performance and robustness to uncertainty. First, a steady-state optimization problem that finds a modified gain matrix is developed. Next, this problem is extended to find a complete dynamic modified model and to the case of non-square systems. Applications of this method are reported in Chapter Introduction Robustness is a basic issue in process control both from a theoretical and from a practical point of view. From a theoretical point of view, it is nice to manage to show that, given a nominal process model, an approximate description of uncertainty and a performance criterion, a controller that satisfies the performance criterion in the worst case included in the region of uncertainty (Morari and Zafiriou, 1989) can be built. From a practical point of view, it would be nice to observe that the performance of the designed controller are indeed satisfactory for the different operating conditions in which the plant is required to operate. A clear implication of the robustness issue is the necessity of a compromise between nominal performance and performance in unfavorable uncertain cases. Portions of this chapter were published in (Semino and Pannocchia, 1999; Pannocchia and Semino, 1999).

29 10 Chapter 2. Method of robust modified models: theory Even for single-input single-output systems, it can be shown that, as the region of uncertainty enlarges, nominal performance of robust controllers deteriorates, so that the advantages offered by advanced controllers may vanish (see e.g. (Lewin and Scali, 1988; Scali and Semino, 1991)). The problem is even more dramatic for multivariable systems. The inability of advanced controllers to give satisfactory performance in the case of uncertainties for multivariable systems is mainly related to their ill conditioning (Skogestad and Morari, 1987). From a physical point of view, a multivariable system is difficult to control when the effects of the manipulated variables on the controlled ones are almost dependent from each other. A typical example is the control of top and bottom compositions of a distillation column through reflux flow rate L and boil-up flow rate V (known as LV control structure). A variation of each of the manipulated variables with the other one kept at a constant value leads to the improvement of one product purity at the expense of a decrease in the other one; this is the reason why an improvement in the purity of both products is very hard to achieve and can be obtained only if the internal fluxes in the column are changed relevantly. From a practical point of view, the control problem is similar to the one of attempting to control strictly two different variables while acting only on one manipulated variable. Mathematically, this problem is described by the Singular Value Decomposition (SVD, see e.g. (Klema and Laub, 1980)) of the system. In particular, the process condition number, which is defined as the ratio between the maximum and the minimum singular value, measures how strongly the process gain may change with the input direction. If the process condition number is large, therefore, an advanced inverse based controller would be required in order for the control system to be effective in the presence of all kinds of disturbances and set-point variations. However, if the process is structurally ill conditioned (i.e. it has a large value of the minimum condition number so that the ill conditioning is independent of the units of measurement used for the process variables), inverse-based controllers show large sensitivity both to input uncertainties and to uncertainties in the individual elements of the transfer function matrix (Morari and Zafiriou, 1989). This is the reason why their use is not recommended unless uncertainties of all kinds are almost negligible or the controller is strongly detuned. For such processes, therefore, the use of decentralized controllers or at most of one way decouplers is usually suggested (Morari and Zafiriou, 1989); these controllers are surely robust but their performance is very sensitive to the input direction so that the overall control system effectiveness is guaranteed only if all the disturbances and set-point variations that take place during

30 Chapter 2. Method of robust modified models: theory 11 operation are aligned close to the favorable directions (i.e. the direction corresponding to the maximum gain). A number of ad hoc techniques can be found in the literature in order to design particular kinds of multivariable inverse-based controllers that are characterized by robustness in case of structural ill conditioning. Brambilla and D Elia (1992) present a short cut method for the design of a robust decoupler. In particular they propose a multivariable controller based on the singular value decomposition which is composed by a diagonal controller plus a compensator that, according to the value of a tuning parameter, moves from an inverse-based controller to a controller that does not remove the effect of directionality. Moreover, they show a criterion to guide the choice of the free parameter. Morari and Zafiriou (1989) present a robust design of multivariable IMC controllers. Their technique is based on a modification of the common IMC structure through the introduction of a second filter able to face the problems connected with ill conditioning. Moreover, they perform the design of the filters in order to satisfy a robust performance objective in terms of Structured Singular Value (Doyle, 1982). Semino et al. (1993) simplify the design of the double filter structure by introducing in the second filter a dependence on the process condition number thus reducing the number of tuning parameters. None of these techniques, however, proposes a general approach to the problem of robust design of a generic multivariable inverse-based controller in the case of an ill-conditioned process. In this chapter a method to address this issue is presented. The approach is based on the search of a surrogate model of the nominal one that is able to increase the robustness when used in any kind of inverse-based controller. The auxiliary model is found in order to minimize an objective function that is explicitely related to measures of nominal performance and robustness. The illustration of the method is presented through the design of an optimal decoupler. Thereafter, it is shown that a completely equivalent optimization can be posed in order to find the modified model to be used in an inverse-based controller of a generic kind. It is important to point out that the optimization technique does not have any pretense of optimality in the worst case sense (as e.g. the SSV theory in (Doyle, 1982)) given a description of uncertainty. Even if it is not as theoretically sound as these approaches, it does not share the limits of strong conservatism and computational complexity that characterize these mathematical theories.

31 12 Chapter 2. Method of robust modified models: theory d r - C p D u G y Figure 2.1: Feedback loop with a decoupler 2.2 Decouplers: theory and properties The control structure shown in Figure 2.1 is the common feedback loop for a multivariable process where a compensator D has been added between the diagonal controller C and the process G. The system as seen by the controller C is therefore G = GD. If D is chosen so that G is diagonal, the compensator D is called a decoupler given that interactions among the variables controlled by C are eliminated. The decoupler is therefore D = G 1 G, (2.1) in which, according to the choice of G, different decouplers can be designed (see e.g. (Ogunnaike and Ray, 1994)). The most common choice is G = diag(g) so that the design of the controller C becomes straightforward (each element can be designed for the corresponding element on the diagonal of G); D I = G 1 diag(g) is called the ideal decoupler. Alternatively, if G = [diag(g 1 )] 1, one obtains the so called simplified decoupler (the diagonal elements of D S are equal to 1). Since the inversion of the process matrix can lead to stability and causality problems, a common choice used when the static interactions are the main concern is a steady-state decoupler design as D SS = G 1 (0) diag(g(0)). A further alternative is to use a partial decoupler, designed so that only the interactions on the more important variables are eliminated (G is no longer diagonal). First, some conditions that a steady-state decoupler (and in a more general form also an ideal decoupler) must satisfy are derived. Theorem 2.1. A constant square matrix D of order n is the solution of a problem D = K 1 diag(k), in which K is a generic constant matrix of order

32 Chapter 2. Method of robust modified models: theory 13 n, if and only if the following n independent conditions are satisfied: adj(d i,i ) = det(d), i = 1,..., n. (2.2) Proof. Necessity. From the definition of inverse matrix and from the definition of decoupler, it follows: } { } K 1 adj(kj,i ) = {ˆki,j =, (2.3) det(k) { } K = (K 1 ) 1 adj(ˆk j,i ) = {k i,j } =, (2.4) det(k 1 ) { } kj,j adj(k j,i ) D = {d i,j } =. (2.5) det(k) Still from the definition of inverse matrix and using Laplace s properties of the determinants, one can write: { } { } { } D 1 adj(dj,i ) adj(ˆk j,i ) det(k 1 ) = = k k,k = k i,j k k,k, det(d) det(d) det(d) which implies: In particular when i = j: k,k i adj(d j,i ) = det(k 1 )k i,j adj(d i,i ) = 1 det(k) k i,i k,k i k,k i k k,k = k,k i (2.6) k k,k. (2.7) k k k,k det(k). (2.8) Therefore, all the principal minors of D have the same value which can be shown to be equal to det(d) by using the Binet-Cauchy theorem as follows: det(d) = det(k 1 k ) det(diag(k)) = k k,k det(k). (2.9) Sufficiency. Assume that the matrix D satisfies condition (2.2). A matrix K whose elements on the diagonal are arbitrarily chosen and whose remaining elements are chosen as det(k) k i,j = adj(d j,i ) k,k i k k,k = adj(d j,i) det(d) k i,i (2.10)

33 14 Chapter 2. Method of robust modified models: theory is such that D = K 1 diag(k). The condition is therefore sufficient; even more, there are n matrices K that satisfy the problem. These matrices can be obtained by choosing arbitrarily the elements on the diagonal and computing the remaining elements consequently. Moreover, if K is such that D = K 1 diag(k) it belongs to the set of solutions that can be computed with the above mentioned criterion. Finally, condition (2.2) implies that the space of matrices D that are steadystate decoupler of a generic process G(s) has dimension n 2 n so that, if one wants to choose an alternative steady-state decoupler to the nominal one, one has to limit the search to such a defined space. 2.3 Steady-state optimization technique Choice of the optimal decoupler The first problem addressed through the proposed optimization technique is the design of an optimal robust decoupler. The purpose is to find a steadystate decoupler that belongs to the family previously defined (and therefore satisfies condition (2.2)) and solves an optimization problem posed to give a good compromise between nominal performance and robustness, leaving some free parameters to adjust the tuning. First the criterion of choice of the optimal decoupler is defined, and then the rationale of all the choices is explained. Let K be the process gain matrix; the following optimization problem is solved: subject to: min J(D) = D αγ (D) + (1 α)γ (KD) + β K K, (2.11a) K adj(d i,i ) = det(d), i = 1,..., n, (2.11b) D = K 1 diag( K), diag( K) = diag(k), (2.11c) (2.11d) in which γ is the minimum condition number, is the Frobenius norm, α [0, 1] and β 0. It is important to notice that the optimization problem (2.11) is feasible because, for instance, the ideal steady-state decoupler satisfies all the equality constraints.

34 Chapter 2. Method of robust modified models: theory 15 The minimum condition number of D is related to the closed-loop robustness (Skogestad and Morari, 1987); a diagonal controller or a triangular controller makes this parameter minimum (equal to 1), while an inverse-based controller has a minimum condition number equal to the one of the process. The minimum condition number of KD is a measure of nominal performance; the ideal decoupler (and also all decouplers that make this product diagonal) has the minimum possible value (equal to 1). The third term is meant to guide the choice of the decoupler toward those that manage to keep low the sum of the two above mentioned minimum condition numbers while being as close as possible to the ideal one; without this term there would be infinitely many solutions to the optimization problem. In particular the parameter β is meant to be taken as c(1 α), in which c is chosen so that the weight of the different terms in the objective function is similar. It is clear, therefore, that when α 0 the optimal decoupler tends to the ideal one, while when α 1 the decoupler tends to the triangular decoupler that cancels the non-diagonal terms of the gain matrix that have the lowest values. In general, the computation of a minimum condition number requires one to solve an optimization problem. However, Grosdidier et al. (1985) show that the minimum condition number of a matrix is strictly related to the Relative Gain Array, RGA (Bristol, 1966) and, more precisely, to its norm: { γ = Λ 1 + Λ systems γ 2 max( Λ 1, Λ ) n n systems. (2.12) There are a number of reasons why the choice of the optimal modified decoupler is limited to the space defined by condition (2.2): firstly, it is nice that a modified process exists (possibly in the uncertainty region) for which the decoupler works perfectly; more relevantly, if an alternative inverse-based scheme is the final purpose of the design (IMC, predictive controller), the final result is not the decoupler but the alternative model to use in place of the nominal one in the control scheme. As stated above, the parameter c is chosen so that the third term in the summation contributes with a weight similar to the other two; in all the tests that have been accomplished, this has been obtained by choosing c in the interval [2 10]γ (K) (without a relevant influence of the precise value on the obtained results). Indeed α remains the only tuning parameter, which has to be chosen according to the expected uncertainty, the process ill conditioning and the desired performance. It may be useful to define a more direct measure of the results of the

35 16 Chapter 2. Method of robust modified models: theory IRI α Figure 2.2: Typical behavior of IRI vs α optimization as follows: IRI = γ (D) γ (K). (2.13) This parameter, referred to as Ill-conditioning Reduction Index (IRI), explicity shows how much of ill conditioning has been eliminated in the controller. It is clear that this index decreases from 1 to 1/γ (K) when α varies from 0 to 1. A typical behavior is reported in Figure Choice of the optimal modified model If the purpose of the optimization is to find a model alternative to the nominal one to be used in a model-based controller, one can easily obtain the gain matrix K of the process that has as steady-state decoupler the result of the optimization. This procedure has been outlined above when the matrix K has been defined. However, in this case the optimization problem can be restated

36 Chapter 2. Method of robust modified models: theory 17 directly as: subject to: min J( K) = αγ ( K 1 diag( K)) + (1 α)γ (K K 1 diag( K)) K +β K K K, (2.14a) diag( K) = diag(k). (2.14b) It is important to notice that the optimization is accomplished only on the off-diagonal elements since the diagonal ones are equal to those of the original gain matrix. 2.4 Dynamic optimization When the process minimum condition number varies relevantly with frequency, it is appropriate to undertake a dynamic search of the modified model, which is based on the extension of the developed technique over a range of frequencies. The difference between the nominal model and the modified model is maintained only in the off-diagonal elements. In the dynamic search it is assumed to keep the same form of parametric models for the off-diagonal elements, and the optimization is accomplished by operating on their parameters. Let G and Ḡ be the transfer function matrices of the nominal model and of the modified one, respectively; let be the set of parameters of the off-diagonal elements of Ḡ, and let J k( ) be the following: J k ( ) = αγ (D(iω k )) + (1 α)γ (G(iω k )D(iω k )) + c(1 α) G(iω k) Ḡ(iω k) G(iω k ), (2.15) in which ω k is a frequency belonging to the set [ω 1, ω 2,..., ω N ] as defined below and the other symbols follow the definitions for the steady-state case. Thus, the set of parameter is the solution of the problem: min J( ) = N [J k ( )] 2, (2.16a) k=1

37 18 Chapter 2. Method of robust modified models: theory subject to: D(iω k ) = Ḡ 1 (iω k ) diag(ḡ(iω k)), diag(ḡ(iω k) = diag(g(iω k ). (2.16b) (2.16c) The constant c is still chosen in the interval [2 10]γ (K), while the tuning parameter α is taken in the interval [0, 1] according to the characteristics of the process, the uncertainties and the required performance. It has been observed in many cases that the delays in the modified process do not differ significantly from their nominal value (due to their action on the condition number only at the high frequencies) so that the optimization problem can be simplified by eliminating them from the optimization vector. The frequency vector is chosen according to the following guidelines. ˆ The minimum frequency ω 1 conditions. must be representative of the stationary ˆ The maximum frequency ω N must be smaller than the sampling frequency ω s = 2π/T s and must not belong to the high frequency region where the minimum condition number strongly oscillates due to the presence of different time delays. ˆ A small number (5-6) of logarithmically spaced frequencies is sufficient to lead to an accurate modified model. In order to guide the choice of the parameter α, an Ill-conditioning Reduction Index (IRI) is defined for the dynamic case as follows: k ( γ Ḡ(iω k ) ) IRI = k γ (G(iω k )). (2.17) This parameter decreases from 1 to N/ k γ (G(iω k )) as α goes from 0 to 1; in particular: ˆ for α = 0 the modified model is the same as the nominal model, ˆ for α = 1 the modified model is triangular, ˆ for 0 < α < 1 the modified model has intermediate characteristics. Large values of α (corresponding to IRI= ) are convenient for badly ill-conditioned processes in the presence of strong uncertainties; small values

38 Chapter 2. Method of robust modified models: theory 19 (corresponding to IRI= ) for moderately ill-conditioned processes with small uncertainties. 2.5 Extension to non-square systems In this section the technique for finding modified models is extended to nonsquare systems with more manipulated variables than controlled variables (over-defined systems). This extension requires preliminarily the development of decouplers for non-square systems, the understanding of their properties and the extensions of well-known tools (e.g. RGA and minimum condition number) to such systems Preliminary definitions and theorems Decouplers for non-square systems with more inputs than outputs. Figure 2.1 on page 12 can also represent a feedback control scheme with a decoupler for non-square systems with more manipulated than controlled variables. In such cases, reference r and output y are vectors of dimension n, while input u is a vector of dimension m, with m > n. The controller C is a conventional diagonal controller, with integral action, i.e. a diagonal matrix of dimension n n, while the decoupler D is a constant m n matrix. It is clear that the full multivariable non-square controller is C = DC. In order to eliminate steady-state interactions the decoupler D must satisfy the following condition: G(0)D = I. (2.18) Such a condition is satisfied by (m n)n constant matrices, which can be computed using the result that follows. Given an n m constant full rank matrix, K, it can be be partitioned in the following way: K = [ K 1 K 2 ], (2.19) in which K 1 is a non-singular n n matrix and K 2 is the complementary n (m n) matrix (a rearrangement of columns can be preliminary needed). In the same way, the decoupler D of dimension m n can be partitioned as: [ ] D1 D =, (2.20) in which D 1 is a n n matrix and D 2 is its complementary one of dimension D 2