Event-triggered control subject to actuator saturation GEORG A. KIENER Degree project in Automatic Control Master's thesis Stockholm, Sweden 212 XR-EE-RT 212:9
Diploma Thesis Event-triggered control subject to actuator saturation Georg A. Kiener 211-212 Examiner, KTH: Prof. Dr. Karl H. Johansson Supervisor, KTH: Dr.-Ing. Daniel Lehmann Examiner, TUM: Prof. Dr.-Ing. Sandra Hirche Supervisor, TUM: Dipl.-Ing. Adam Molin
Abstract. Event-triggered control is a recent approach in control theory which aims at reducing the communication load in networked control systems by adapting the communication among the components to the current needs. In more detail, the information exchange over the feedback link only takes place if certain event conditions, that guarantee a desired control performance, are satisfied. This thesis analyzes the consequences of actuator saturation on the stability of the event-triggered control loop. Based on linear matrix inequalities, stability criteria have been derived which can be used to determine regions in the state space that guarantee a stable behavior. Furthermore, the existence of a lower bound on the minimum inter-event time is shown. Due to integrator windup, actuator saturation may severely degrade the performance of the event-triggered closed-loop system. In order to overcome this problem, the stability criteria have been extended by incorporating a static anti-windup structure. Finally, the effects of transmission delays in the feedback link are analyzed and a procedure to deal with their consequences is proposed. The results are illustrated by simulations and by experiments with a wirelessly controlled first-order tank system.
Contents 1 Introduction 1 1.1 Event-triggered control........................... 1 1.2 Deadband sampling............................. 2 1.3 Actuator saturation............................. 3 1.4 Contribution of this thesis......................... 4 1.5 Structure of this thesis........................... 5 2 Preliminaries 6 2.1 Notation and definitions.......................... 6 2.2 Running example.............................. 8 3 Continuous-time control subject to actuator saturation 9 3.1 System representation........................... 9 3.1.1 Dynamic output feedback controller................ 1 3.1.2 PI controller............................. 11 3.1.3 Universal representation of the continuous-time control loop.. 11 3.2 Sector condition............................... 11 3.3 Stability analysis.............................. 13 3.4 Example................................... 15 4 Event-triggered control subject to actuator saturation 2 4.1 System representation........................... 2 4.1.1 Dynamic output feedback controller................ 2 4.1.2 PI controller............................. 22 4.1.3 Universal representation of the event-triggered control loop... 22 4.2 Stability analysis.............................. 23 4.3 Minimum inter-event time......................... 26 4.4 Extended stability analysis......................... 27 4.4.1 Alternative event generator.................... 27 4.4.2 Nonzero disturbance signals.................... 29 4.4.3 Nonzero reference signals...................... 31 4.4.4 Nonzero disturbance and reference signals............ 32 4.5 Example................................... 33 4.5.1 Fixed event threshold and disturbance.............. 34 4.5.2 Variable event threshold and disturbances............ 37 i
5 Anti-windup compensation 39 5.1 Introduction................................. 39 5.2 Modeling of a static anti-windup structure................ 4 5.3 Example................................... 42 6 Transmission delays 45 6.1 Modeling................................... 45 6.2 Large delays................................. 48 6.3 Small delays................................. 48 6.4 Mutual dependencies............................ 49 6.5 Example................................... 5 7 Optimization problems and practical implementation 54 7.1 Algorithms for stability analysis...................... 54 7.2 Optimization functions........................... 55 7.3 Implementation in MATLAB....................... 56 7.4 Example................................... 56 8 Experiments 62 9 Conclusion and outlook 67 List of Figures 69 List of Tables 71 A Mathematical tools 72 A.1 Ellipsoid contained in a symmetric polytope............... 72 A.2 The S-procedure.............................. 73 B Implementations 74 B.1 The event-triggered control loop...................... 74 B.2 Algorithm 1................................. 75 B.3 Algorithm 2................................. 76 B.4 Plotting of ellipsoids............................ 78 Bibliography 79 ii
1 Introduction Due to an ongoing progress in communication engineering, a new opportunity of controlling complex plants has evolved: Wireless control systems. Compared to wired installations, these systems provide advantages such as simple installation as well as maintenance or a high degree of flexibility. However, it has to be considered that sufficient network resources are an essential requirement for wireless control systems. For instance, a busy communication network may evoke delays or packet losses which degrade the control performance or can even lead to an unstable behavior of the system [1]. In order to resolve this issue, a new control scheme has gained attention, namely: Event-triggered control [2 4]. 1.1 Event-triggered control Event-triggered control is a method to reduce the communication over the feedback link by adapting the communication to the current needs [5, 6]. To describe its functionality, the event-triggered control loop as considered in this thesis is depicted in Fig. 1.1. It consists of a plant with state x p, input u and output y. The plant is controlled by a controller with controller state x c and input y which is forwarded by a wireless network. The signal d represents an unknown disturbance affecting the plant and w is a reference signal used by the controller. w(t) d(t) y(t k ) Controller x c (t) u(t) Plant y(t) Event y(t k ) x p (t) generator Wireless network Figure 1.1: The event-triggered control loop 1
1 Introduction The main characteristic of an event-triggered control loop is the event generator. The event generator forwards information y(t k ) only at certain event times t k. In Fig. 1.1, this is indicated by dashed lines. The event generator can be seen as a kind of smart sensor that forwards its information only if certain event conditions are met. The substantial difference between discrete-time and event-triggered control loops is that in discrete-time control loops, the time intervals between consecutive events are fixed. By contrast, in event-triggered control loops time intervals between consecutive events may vary, based on the conditions of the event generator. To verify these conditions, an event generator requires a processing unit. Within this thesis, it is assumed that such processing units do not pose any restrictions regarding the computational effort at the event generator. By forwarding information only at event times, network traffic is reduced. 1.2 Deadband sampling Depending on the event conditions, different types of event generators can be defined [2]. In this thesis, deadband sampling will be considered [7]. To explain deadband sampling, a first order system and an event-generator with input y(t) is used. The event-generator compares the current value y(t) with the last value forwarded to the controller y(t k ). As long as the absolute value of the difference is within the deadband B k = {y R p, y(t k ) y(t) < e} (1.1) with event threshold e, where. denotes the absolute value of scalar, no information is forwarded to the controller. When the absolute value of the difference meets the event threshold at the time t, the event condition y(t k ) y(t) = e (1.2) is satisfied. Now, the current value y(t k ) is forwarded to the controller and a new deadband B k+1 is set around the current y(t). An illustration of this behavior is shown in Fig. 1.2. The stability and communication behavior of certain event-triggered control loops are analyzed in various sources, for example in [2, 3, 8 12]. However, the effects of a very practical phenomenon actuator saturation on event-triggered control loops still needs to be studied in depth. 2
1.3 Actuator saturation y 2e y(t k ) y(t k+2 ) y(t k+1 ) t k t k+1 t k+2 t 1.3 Actuator saturation Figure 1.2: Deadband sampling Actuator saturation is a constraint that has to be considered in virtually all practical control systems. The maximum impact an actuator can exert on a system is usually limited by physical or safety constraints. For instance, the output of a valve within a process plant is limited between fully open and fully closed. If a controller demands more or less flow than the valve permits by being fully open or fully closed, actuator saturation occurs. Further examples are deflection limits in aircrafts or ships or voltage limits in electrical actuators. Actuator saturation does not only restrict achievable system performance. If it is not considered appropriately, it may even lead to instabilities within the control loop [13]. For this, an example is given in Section 3.4. In summary, saturating actuators represent nonlinearities within the control loop yielding stability problems. Especially controller with an integrator part are subject to these problems as additional integrator windup can occur [14]. In continuous-time control systems, actuator saturation and its effect is well studied [13, 14]. w(t) d(t) Controller x c (t) &(t) u(t) Plant y(t) Event y(t k ) x p (t) generator Figure 1.3: The event-triggered control loop subject to actuator saturation 3
1 Introduction However, regarding actuator saturation in event-triggered control loops, further research is necessary. Up to now, research has mostly focused on event-triggered proportional controller. Event-triggered control incorporating dynamical controllers is of recent interest. Especially the analysis of PI controllers aims to improve the applicability of event-triggered control in practical applications [15]. Within this thesis, suitable methods to deal with actuator saturation within an event-triggered control loop are developed. The respective system to be analyzed is depicted in Fig. 1.3 which includes the actuator saturation nonlinearity with input v(t) and output u(t). 1.4 Contribution of this thesis The approach presented in this thesis adapts methods from continuous-time control theory to event-triggered control. According to [13], there are two methods to deal with actuator saturation. The first is an analytical approach, in which actuator saturation is explicitly considered. In the second approach, negative effects of actuator saturation are avoided by using anti-windup techniques. Anti-windup control aims at avoiding integrator windup within the integrator parts of the controller [13, 14]. Both of these methods are considered, leading to the following contributions of this thesis: 1. LMI 1 -based stability conditions are derived which can be used to calculate regions of stability for the event-triggered control loop subject to actuator saturation (Theorem 2, Corollaries 1, 2 and 4). 2. It is shown, that Zeno behavior can be excluded by deriving lower bounds on the minimum inter-event time (Theorem 3 and Corollary 3). 3. A static anti-windup structure is used to deal with actuator saturation and to overcome potential performance degradations caused by actuator saturation (Corollary 5). 4. The effect of transmission delays in the feedback link on the behavior of the event-triggered control loop subject to actuator saturation is investigated (Propositions 1 and 2). 5. Simulations and experiments illustrate the theoretical results by showing the dependencies of the stability region and the positive effects of anti-windup compensation. Parts of this thesis were submitted for publication [17 19]. 1 For an introduction to linear matrix inequalities, see [16]. 4
1.5 Structure of this thesis 1.5 Structure of this thesis The thesis is organized as follows: Chapter 2 introduces the basic notations and definitions that are needed for this thesis. Furthermore, a running example is introduced that is used to illustrate theoretical results of the subsequent chapters. Chapter 3 presents a method to deal with actuator saturation in continuous-time control loops. Chapter 4 derives stability conditions for the event-triggered control loop subject to actuator saturation by adapting the method presented in the previous chapter. Chapter 5 extends the event-triggered control loop with a static anti-windup structure to deal with actuator saturation. Chapter 6 highlights the problems of transmission delays in the feedback link. Chapter 7 explains algorithms that are suitable to calculate regions of stability based on LMI conditions. Chapter 8 presents an experiment with a water tank to validate the theoretical results. 5
2 Preliminaries In Section 2.1, the general notation and definitions which are used throughout this thesis are introduced. Furthermore, Section 2.2 introduces a running example that is used to illustrate the results of the subsequent chapters. 2.1 Notation and definitions Notation. Throughout the thesis, a scalar is denoted by italic letters (x R), a vector by bold italic letters (x R n ), a matrix by bold capital letters (A R n n ) and a signal at time t R + by x(t), where x describes the initial signal value at time t =. Identity matrices of the size n n are denoted by I n. The i-th element of a vector x is denoted by x (i), the i-th row of a matrix A by A (i) and the transpose of a matrix or vector by () T. Symmetric matrices of the form [ ] [ ] A B T A are abbreviated by. Furthermore, the absolute value of a scalar is B C B C denoted by x, the Euclidean vector norm by x and the induced matrix norm by A. x rounds x up to the next whole number, for example.2 = 1. Regarding matrices, A > (A ) means that the matrix A is positive definite (semidefinite) and A < (A ) means that the matrix A is negative definite (semidefinite). The trace, i.e., the sum of the diagonal elements, of a matrix A is denoted by trace(a). E (P, η) describes elliptical sets that are defined by x T P x η 1 with P >. S (G, u ) describes symmetric polytopes that are defined by G (i) x u (i) for i {1,..., m}. Plant. The plant considered in this thesis is described by the linear continuous-time state representation ẋ p (t) = Ãx p (t) + Bu(t) + B D d(t), x p () = x p (2.1) y(t) = Cx p (t). (2.2) 6
2.1 Notation and definitions x p R np is the state vector, u R m is the input, y R p is the measured output vector and d R q represents disturbances at the plant. Ã, B, B D and C are real matrices of appropriate dimensions. Actuator saturation. The plant input u(t) is given by u(t) = sat(v(t)), (2.3) where sat(.) represents the nonlinear saturation function defined by u max(i) if v (i) > u max(i) sat(v (i) ) = v (i) if u min(i) v (i) u max(i) (2.4) u min(i) if v (i) < u min(i) with i {1,..., m}. Throughout the thesis, only symmetrical saturation functions with u (i) = u max(i) = u min(i) for i {1,..., m} (2.5) will be considered. This is illustrated in Fig. 2.1. Stability. A square matrix A R n n is called Hurwitz if all its eigenvalues λ i (i {1,..., n}) have strictly negative real part [2]. System (2.1), (2.2) with zero inputs (u(t) = w(t) = for all t ) is asymptotically stable, i.e., lim t x(t) =, if the matrix A is Hurwitz [2]. Since information in event-triggered control loops is only exchanged at event times t k, i.e., when a certain event threshold is met, asymptotic stability cannot be ensured for these systems. Instead a more appropriate definition for the stability of such systems is considered in this thesis: Ultimate boundedness (practical stability). Ultimate sat(v (i) ) v(t) u(t) u (i) v (i) -u (i) Figure 2.1: Circuit diagram and graph of the saturation function 7
2 Preliminaries boundedness means, that the state x(t) can be driven into a certain surrounding E (P, η) of the equilibrium point and kept there for all future times despite exogenous disturbances. Definition 1. The solution x(t) of the system (2.1), (2.2) is ultimately bounded in E (P, η) if x(t) E (P, η), t holds for every x() E (P, η). In this case, system (2.1), (2.2) is called ultimately bounded or stable [2, 2]. 2.2 Running example Within this section, an exemplary system based on the system introduced in [15] is presented. It acts as a running example and is used to illustrate the different results of this thesis at the end of each chapter. The system includes a scalar plant that is given by ẋ p (t) = ax p (t) + bu(t) + b D d(t), x p () = x p (2.6) y(t) = x p (t). The plant is controlled by a PI controller which is given by (2.7) ẋ c (t) = y(t) w(t), x c () = x c (2.8) v(t) = k I x c (t) + k P (y(t) w(t)). (2.9) The numerical values used for following simulations are a =.1, b = 1, b D =.1, k I = 1 and k P = 1.6. Furthermore, the actuator output u(t) is subject to actuator saturation. This is described by the saturation function (2.4) with u(t) = sat(v(t)) and u =.4. In Chapter 4, an event generator is introduced to analyze the event-triggered control loop. In that case, the continuous-time plant output y(t) has to be replaced by the event-triggered plant output y(t k ) within the controller equations (2.8) and (2.9). This modification will be explained in detail within Chapter 4. 8
3 Continuous-time control subject to actuator saturation This chapter introduces a stability criterion for continuous-time PI control loops subject to actuator saturation displayed in Fig. 3.1. First, an appropriate system representation is developed in Section 3.1. Section 3.2 introduces a general sector condition based on a dead zone nonlinearity. This condition is needed to derive the stability criterion for the continuous-time control loop subject to actuator saturation which is described in Section 3.3. Finally, an example illustrates the results in Section 3.4. The presented stability criterion and its derivation are explained in detail in [13]. The concepts explained in this chapter are essential for the development of the stability criterion for the event-triggered control loop subject to actuator saturation proposed in Chapter 4. 3.1 System representation This section introduces an appropriate system representation for the continuous-time control loop subject to actuator saturation [13]. In this representation, the controller description is included in an extended plant description. Thereby, general methods for analyzing the closed-loop systems become applicable. w(t) d(t) Controller x c (t) (t) u(t) Plant x p (t) y(t) Figure 3.1: The continuous-time control loop subject to actuator saturation 9
3 Continuous-time control subject to actuator saturation 3.1.1 Dynamic output feedback controller The plant (2.1), (2.2) is controlled by a general dynamic linear output feedback controller, which is described by ẋ c (t) = à c x c (t) + B c y(t) + B cw w(t), x c () = x c (3.1) v(t) = C c x c (t) + D c y(t) + D cw w(t) (3.2) with the controller state x c R nc and reference signals w R s. The matrices à c, B c, B cw, C c, D c and D cw are real and have appropriate dimensions. By considering the dynamic controller equation (3.1) and the augmented state vector x = [ ] xp x c the plant (2.1), (2.7) becomes, (3.3) ẋ(t) = Ax(t) + Bu(t) + B D d(t) + B W w(t), x() = x, (3.4) y(t) = Cx(t) (3.5) with [ ] à A =, B = B c C à c [ ] B, B D = [ ] [ ] B D, B W =, C = [ C ]. (3.6) B cw Moreover, by using equation (2.2) and the state transformation (3.3), the controller output (3.2) becomes with v(t) = Kx(t) + K W w(t) (3.7) K = [ Dc C Cc ], KW = D cw. (3.8) This transformation is possible for every linear plant that is controlled by a dynamic output feedback controller and results in a general system representation where controller dynamics are included within the plant description (3.4). Thereby, the a proportional control law results as shown in equation (3.7). 1
3.2 Sector condition 3.1.2 PI controller By setting n p = n c = s and by using as well as ẋ c (t) = y(t) w(t), x c () = x c v(t) = K I x c + K P (y(t) w(t)), instead of equations (3.1) and (3.2), PI controllers are explicitly included in this general system representation. Thereby, the respective matrices (3.6) and (3.8) become [Ã ] [ ] [ ] [ ] B B A =, B =, B C D = D, B W =, C = [ ] C, I np K = [ (3.9) ] K P C K I, KW = K P. 3.1.3 Universal representation of the continuous-time control loop Finally, by introducing the saturation function u(t) = sat(v(t)), the following representation for the continuous-time system subject to actuator saturation is obtained: ẋ(t) = Ax(t) + Bsat(Kx(t) + K W w(t)) + B D d(t) + B W w(t), x() = x, y(t) = Cx(t). (3.1) This representation is displayed in Fig. 3.2. All considerations within this chapter will be based on systems of the form (3.1) with x R n, n = n p + n c, u R m, d R q, w R s, y R p and matrices A, B, B D, B W, K, K W and C of appropriate dimensions. In Section 4.1 it will be extended to cover the event-triggered case. 3.2 Sector condition Equation (3.1) includes the saturation function as defined in equation (2.4). To apply generalized methods for stability analysis, the system has to be transformed in a Lure problem. Therefore, the saturation will be replaced by another nonlinearity, i.e., a decentralized dead zone nonlinearity [13] φ(v(t)) = sat(v(t)) v(t) (3.11) 11
3 Continuous-time control subject to actuator saturation w(t) #(t) w(t) P controller v(t) u(t) Plant x(t) y(t) x(t) Figure 3.2: Adapted system representation of the continuous-time control loop with elements φ (i) (v), where u (i) v (i) for v (i) > u (i) φ (i) (v) = φ(v (i) ) = for u min(i) v (i) u max(i) (3.12) u (i) v (i) for v (i) < u (i) for i {1,..., m}. In Fig. 3.3, the dead zone nonlinearity φ(v i ) is displayed. By using the dead zone representation, system (3.1) can be rewritten as ẋ(t) = Ax(t) + Bφ(Kx(t) + K W w(t)) + B D d(t) + (B W + BK W )w(t), x() = x, y(t) = Cx(t) (3.13) with A = A + BK, B = B. sat(v (i) ) φ(v (i) ) u (i) v (i) u (i) v (i) -u (i) -u (i) Figure 3.3: Graph of the dead zone nonlinearity 12
3.3 Stability analysis Defining u (1) u =., u (m) a generalized sector condition for symmetrical saturation functions (see equation (2.5)) can be used to derive stability criteria for the continuous-time as well as for the eventtriggered control loop subject to actuator saturation. The following lemma is proven in [13]. Lemma 1. If υ and ω are elements of the set S ( υ ω, u ) = {υ R m, ω R m, υ (i) ω (i) u (i) }, i {1,..., m} then the nonlinearity φ(υ) satisfies φ(υ) T T (φ(υ) + ω) for any diagonal positive definite matrix T R m m. 3.3 Stability analysis In this section, a criterion regarding internal asymptotic stability for system (3.1) is presented [13]. Therefore, d(t) = and w(t) = hold and the state equation from representation (3.1) turns into ẋ(t) = Ax(t) + Bsat(Kx(t)), x() = x (3.14) or equivalently ẋ(t) = Ax(t) + Bφ(Kx(t)), x() = x. (3.15) By using a quadratic Lyapunov function V (x) = x T P x, P = P T > (3.16) and applying Lemma 1, the following stability criterion can be obtained. 13
3 Continuous-time control subject to actuator saturation Theorem 1. If there exist a symmetric positive definite matrix W R n n, a positive definite diagonal matrix S R m m and a matrix Z R m n satisfying W AT + AW BS W K T Z T SB T < (3.17) KW Z 2S [ ] W Z T (i) 2, i {1,..., m} (3.18) Z (i) ηu (i) then the ellipsoid E (P, η), with P = W 1, is a region of asymptotic stability for the saturated system (3.14) or equivalently (3.15). Proof. By setting υ = Kx and ω = Kx + Gx, Lemma 1 guarantees that any x belonging to the set S (G, u ) = {x R n, G (i) x u (i) }, i 1,..., m (3.19) satisfies the inequality φ(kx) T T (φ(kx) + Kx + Gx). (3.2) Considering the quadratic Lyapunov function (3.16), the region of stability to be calculated is an ellipsoid. This ellipsoid E (P, η) is included in the set (3.19) if the inequalities [ P G T (i) G (i) ηu 2 (i) ] for i {1,..., m} P > hold (see Appendix A.1) [13]. The second inequality is fulfilled automatically by choosing a proper Lyapunov function (3.16). By setting W = P 1 and Z = GW, the first inequality can be rewritten to get equation (3.18). In summary, this equation means that the sector condition (3.2) holds for every x E (P, η). Furthermore, the time derivative of the Lyapunov function has to fulfill the following condition in order to prove asymptotic stability of system (3.14): V (x) = ẋ T P x + x T P ẋ <. By using sector condition (3.2) a quadratic term in φ can be added according to V (x) V (x) 2φ T T (φ + Kx + Gx) <. This estimate is valid for all x E (P, η). For clarity, the argument of φ is not displayed from now on. 14
3.4 Example By using state representation (3.15), the previous inequality reads [ ] T x A T P + P A P B K T T T G T T T [ ] φ B T x <. P T K T G 2T φ This inequality contains nonlinearities in decision variables, e.g., T G. By extending the states [ ] T [ ] x xp P 1 T [ ] [ x P = φ φt T 1, = 1 ] P x φ T 1 T φ and by setting W = P 1, S = T 1 and Z = GW, this problem is solved and equation (3.17) is achieved. Altogether, this means that asymptotic stability of the continuous time system subject to actuator saturation (3.14) is ensured for any x E (P, η) because V (x) < holds for all t. Moreover, global stability conditions can be derived for linear plants with a matrix A that is Hurwitz. As shown in equation (3.9), PI controllers yield a matrix A that is not Hurwitz due to the integrator state of the controller. Therefore, these conditions are not explained here and the reader is referred to [13]. Furthermore, in Chapter 4, exogenous signals as well as matrices B D, B W and K W will be considered. Results of that chapter can also be used to ensure stability within the continuous-time control loop subject to actuator saturation with d(t) and w(t). With Theorem 1, regions of stability for the continuous-time system subject to actuator saturation can be calculated which is illustrated in the next section. 3.4 Example The running example presented in Section 2.2 is used to illustrate the methods described in this chapter. First, equations (2.6)-(2.9) are translated into the form of (3.4) and (3.7). By introducing the augmented state vector x = [ ] T, x p x c these equations read [ ] [ ] [ ] [ ] a b bd ẋ(t) = x(t) + u(t) + d(t) + w(t), x() = x 1 1, y(t) = [ 1 ] x(t), v(t) = [ k P k I ] x(t) kp w(t). 15
3 Continuous-time control subject to actuator saturation 6 5 4 3 Region of asymptotic stability x = [ 1 2 ] T x = [ 2 2.5 ] T x = [ 2.5 2.5 ] T xc 2 1-1 -2-1.5-1 -.5.5 1 1.5 2 2.5 Figure 3.4: Region of asymptotic stability and trajectories for different initial states x of the continuous-time control loop x p By introducing the saturation function u(t) = sat(v(t)), the state equation [ ] [ ] a b ẋ(t) = x(t) + sat( [ [ ] [ ] ] bd k 1 P k I x(t) kp w(t)) + d(t) + w(t), 1 x() = x according to (3.1) is obtained with matrices [ ] [ ] a b A =, B =, K = [ ] k 1 P k I, KW = k P, B D = [ ] [ ] bd, B W =. 1 In this chapter, disturbances and reference signals were not considered, i.e. d(t) = w(t) =. Therefore, the representation above is simplified as [ ] [ ] a b ẋ(t) = x(t) + sat( [ ] k 1 P k I x(t)), x() = x. (3.21) 16
3.4 Example By solving the optimization problem min{ trace(w )} subject to conditions from Theorem 1 with the YALMIP toolbox [21] for τ 1 = τ 2 =.1, Theorem 1 was used to calculate an estimate for the region of asymptotic stability of this system. Furthermore, a MAT- LAB/Simulink implementation of the control loop was used to calculate trajectories of the system. These methods are described in detail in Chapter 7. Necessary numerical values of the system parameters are included in Section 2.2. Region of asymptotic stability. The region of asymptotic stability is presented in Fig. 3.4. For every initial state x() = x within the ellipsoid, the system will converge asymptotically towards the equilibrium point which is the origin. For instance, the green trajectory starts at x = [ 1 2 ] T and converges towards the origin. However, the presented approach yields some conservatism as the presented stability conditions are sufficient conditions. For example the black trajectory starts at x = [ 2 2.5 ] T, i.e., outside the region of asymptotic stability but it still converges asymptotically towards the origin. This is due to the fact that with this method only elliptical estimates can be calculated for the region of asymptotic stability. In general, these estimates do not fit exactly the actual region of stability. They are furthermore dependent on various optimization parameters which is described in Chapter 7. In general, it is not possible to analytically calculate the overall region of asymptotic stability [13]. Nevertheless, for most of the initial states outside the region of asymptotic stability, the system yields an instable behavior. This is illustrated by the red trajectory. After starting in x = [ 2.5 2.5 ] T it quickly rises to infinity. Behavior of the continuous-time control loop. In Fig. 3.5, the trajectories of the states are presented. The states converge slower towards the origin for x = [ ] T 2 2.5 than for x = [ 1 2 ] T because the initial states are farer away from the origin and actuator saturation keeps the control output within certain bounds. The red trajectories of x = [ 2.5 2.5 ] T describe a rising oscillation till infinity which is caused by the corresponding control input depicted in Fig. 3.6. In that figure, controller and actuator outputs for the different initial states are portrayed. As actuator saturation occurs, the actuator output u lies in each case between -.4 and.4. However, the controller output v may extend these borders. This is the windup effect. The integrator of the PI controller increases the controller output even if the actuator is 17
3 Continuous-time control subject to actuator saturation xp xc 2-2 -4-6 -8-1 1 2 3 4 5 6 7 8 9 15 1 5 x = [ 1 2.5 ] T x = [ 2 2.5 ] T x = [ 2.5 2.5 ] T -5 1 2 3 4 5 6 7 8 9 t Figure 3.5: States of the continuous-time control loop for different initial states x of the continuous-time control loop already saturated which affects the control performance negatively which can be seen in Fig. 3.5. For instance, controller windup depicted in the bottom graph of Fig. 3.6 leads to an increasing integrator state x c in Fig. 3.5. Thereby, the integrator state keeps the actuator output within the saturation bounds for almost all times. Finally, this leads to an unstable oscillation of the integrator state x c and the plant behavior. In summary, actuator saturation slows down the transient behavior of the controller or even leads to instability if integrator windup occurs. However, there are anti-windup methods to deal with integrator windup [14]. A static anti-windup structure is applied and analyzed in Chapter 5. 18
3.4 Example.5 v and u v and u v and u -.5 v for x = [ 1 2 ] T u for x = [ 1 2 ] T -1 1 2 3 4 5 6 7 8 9 3 2-1 1 v for x = [ 2 2.5 ] T -2-3 u for x = [ 2 2.5 ] T -4-6 -5 1 2 3 4 5 6 7 8 9 2 1-1 v for x = [ 2.5 2.5 ] T -2 u for x = [ 2.5 2.5 ] T -3-4 1 2 3 4 5 6 7 8 9 t Figure 3.6: Controller and actuator outputs for different initial states x continuous-time control loop of the 19
4 Event-triggered control subject to actuator saturation Based on the derivations from the previous chapter, stability criteria for event-triggered control loops subject to actuator saturation are developed within this chapter. The considered system is depicted in Fig. 1.3. Similar to Chapter 3, an appropriate system representation is introduced first. In Section 4.2, the basic stability criterion is presented and its derivation is explained in detail. Section 4.3 presents an approximation for the minimum inter-event time. Extensions to the stability conditions like different definitions of the event generator or disturbances as well as reference signals are discussed in Section 4.4. Finally, an example is used to illustrate the results in Section 4.5. 4.1 System representation To deal with event-triggered sampling, an appropriate system representation has to be derived first. Again, the controller dynamics will be included by using an extended plant description in order to make methods for stability analysis introduced in Section 3.1 applicable. 4.1.1 Dynamic output feedback controller The plant (2.1), (2.2) is controlled by a general dynamic linear output feedback controller. However, in difference to the continuous-time control loop, this controller receives new information about the plant output y(t) only at event times t k with k {, 1, 2,...}. Hence, the controller can be described during the time interval [t k, t k+1 ) by ẋ c (t) = Ã c x c (t) + B c y(t k ) + B cw w(t), x c (t k ) = x ck (4.1) v(t) = C c x c (t) + D c y(t k ) + D cw w(t), (4.2) where x c R nc is the integrator state and w R s represents reference signals. Ã c, B c, BcW, Cc, Dc and D cw are real matrices of appropriate dimensions. 2
4.1 System representation As this representation of the controller is only valid during the time interval [t k, t k+1 ), it is necessary to develop an appropriate model that holds for all times t in order to apply the presented methods for stability analysis. Therefore, the output error is introduced: e(t) = y(t k ) y(t). (4.3) By using this error, the controller can be presented in a way that holds for all times t according to: ẋ c (t) = Ã c x c (t) + B c y(t) + B c e(t) + B cw w(t), x c () = x c (4.4) v(t) = C c x c (t) + D c y(t) + D c e(t) + D cw w(t). (4.5) By considering equation (4.4) and by using the augmented state vector ] x = [ xp x c, (4.6) the plant (2.1), (2.2) can be rewritten as ẋ(t) = Ax(t) + Bu(t) + B D d(t) + B W w(t) + B E e(t), x() = x, (4.7) y(t) = Cx(t) (4.8) with [ ] Ã A =, B = B c C C = [ C ]. Ã c [ ] B, B D = Furthermore, the controller output (4.5) can be rewritten to with [ ] [ ] [ B D, B W =, B B E =, cw Bc] (4.9) v = Kx(t) + K W w(t) + K E e(t) (4.1) K = [ D c C C c ], KW = D cw, K E = D c. (4.11) Similar to Section 3.1, a general system representation with an extended plant description (4.7) that includes the controller dynamics is achieved. The resulting control law (4.1) is again proportional. This transformation is possible for every linear plant that is controlled by an event-triggered dynamic output feedback controller of the form (4.1), (4.2). 21
4 Event-triggered control subject to actuator saturation 4.1.2 PI controller PI controllers are explicitly included within the general system representation by setting n p = n c = s and by using ẋ c (t) = y(t k ) w(t), x c (t k ) = x ck v(t) = K I x c + K P (y(t k ) w(t)) instead of equations (4.1) and (4.2). Moreover, by introducing the output error (4.3) and by using the augmented state (4.6), the general system representation (4.7), (4.1) is obtained again. Thereby, the respective matrices (4.9), (4.11) become [Ã ] [ ] [ ] [ ] [ ] B B A =, B =, B C D = D, B W =, B I E =, np I np C = [ (4.12) [ ] C ], K = KP C K I, KW = K P, K E = K P. 4.1.3 Universal representation of the event-triggered control loop By introducing the saturation function u(t) = sat(v(t)), a general closed-loop representation for plants that are controlled by an event-triggered controller subject to actuator saturation is developed ẋ(t) = Ax(t) + Bsat(Kx(t) + K W w(t) + K E e(t))+ + B D d(t) + B W w(t) + B E e(t), x() = x, y(t) = Cx(t). (4.13) This representation depicted in Fig. 4.1. All further analysis of this thesis is based on systems that have the form of representation (4.13) with x R n, n = n p +n c, u R m, d R q, w R s, e R p, y R p and matrices A, B, B D, B W, B E, K, K W, K E and C of appropriate dimensions. Besides additional terms for the output error e(t), this representation is identical to the representation that describes continuous-time systems subject to actuator saturation (3.1). Finally, by adding the dead zone model (3.11), the following representation is developed ẋ(t) = Ax(t) + Bφ(Kx(t) + K W w(t) + K E e(t))+ + B D d(t) + (B W + BK W )w(t) + (B E + BK E )e(t), x() = x, y(t) = Cx(t) (4.14) 22
4.2 Stability analysis w(t) e(t) d(t) w(t) e(t) P controller v(t) u(t) Plant x(t) y(t) x(t) Figure 4.1: Adapted system representation of the event-triggered control loop with A = A + BK, B = B. 4.2 Stability analysis In this section, a stability criterion for the event-triggered control loop subject to actuator saturation is presented. To develop such a criterion, again the quadratic Lyapunov function (3.16) is used. Regarding exogenous signals, only the output error e(t) will be considered in this section, i.e., d(t) = w(t) =. Thereby, the state equations from (4.13) and (4.14) become and ẋ(t) = Ax(t) + Bsat(Kx(t) + K E e(t)) + B E e(t), x() = x (4.15) ẋ(t) = Ax(t) + Bφ(Kx(t) + K E e(t)) + (B E + BK E )e(t), x() = x. (4.16) The results of this section can be extended to cover additional disturbances or reference signals as well. This will be explained in Section 4.4. In [13] stability criteria for systems that are similar to equation (4.15) are described. However, only disturbances e(t) that appear either only outside or only inside the saturation term (i.e. either K E or B E are zero) are considered there. For the eventtriggered control-loop subject to actuator saturation both terms K E and B E have to be considered. Therefore, the stability criterion from [13] has to be extended. An important requirement is the boundedness of e(t). In this chapter, the output error e(t) is amplitude bounded through a quadratic norm, i.e., e(t) belongs to the following set W (R, δ) = {e R p, e T Re δ 1 }, R = R T >, δ >. (4.17) 23
4 Event-triggered control subject to actuator saturation The boundedness of the output error can be achieved by a proper definition of the event generator. An event generator which ensures that e(t) belongs to the set (4.17) for all t is defined by the following event condition e T Re = δ 1. (4.18) Event condition (1.2) is included as a special case with R = 1 and δ 1 = e 2. Methods to handle other forms of boundedness are discussed in Section 4.4. With the set (4.17), the following stability criterion can be derived. Theorem 2. If there exist a symmetric positive definite matrix W R n n, a positive definite diagonal matrix S R m m, a matrix Z R m n and three positive scalars τ 1, τ 2 and η satisfying W A T + AW + τ 1 W BS W K T Z T B E + BK E SB T KW Z 2S K E < (4.19) (B E + BK E ) T KE T τ 2 R [ ] W Z T (i) Z (i) ηu 2, i {1,..., m} (4.2) (i) τ 1 δ + τ 2 η < (4.21) then, 1. for e =, the ellipsoid E (P, η), with P = W 1, is a region of asymptotic stability of the saturated system (4.15) or equivalently (4.16). 2. for any e W (R, δ) and x E (P, η), the trajectories of the saturated system (4.15) or equivalently (4.16) are ultimately bounded and do not leave the ellipsoid E (P, η). Proof. By setting υ = Kx + K E e and ω = Kx + K E e + Gx, Lemma 1 ensures that the ellipsoid E (P, η) is included in S ( G, u ) and that the sector condition φ(kx + K E e) T T (φ(kx + K E e) + Kx + K E e + Gx) (4.22) holds for all x E (P, η). In the following, conditions are derived to show that E (P, η) is a positively invariant set regarding e(t). By applying the S-procedure (see Appendix A.2), the condition V (x) < can be extended according to V (x) + τ 1 (x T P x η 1 ) + τ 2 (δ 1 e T Re) < (4.23) with τ 1, τ 2 > [13, 16]. The relation above ensures that V (x) < holds for all x satisfying x T P x η 1 and for any e W (R, δ). Relation (4.23) has to be verified especially for x(t 1 ) E (P, η), i.e., at the boundary of E (P, η). If e(t) W (R, δ) holds, it follows V (x(t 1 )) <. Thus x(t 1 + t) will lie in 24
4.2 Stability analysis the interior of the ellipsoid for any arbitrary time step t. As a conclusion, inequality (4.23) ensures that the ellipsoid E (P, η) is a positive invariant set regarding e(t). For further analysis, inequality (4.23) is split into the two inequalities τ 1 δ + τ 2 η < (4.24) V (x) + τ 1 x T P x τ 2 e T Re <. (4.25) Inequality (4.24) transfers directly into condition (4.21) of Theorem 2. Similar to Section 3.3, inequality (4.25) can be extended by using the sector condition (4.22) for x E (P, η): V (x) + τ 1 x T P x τ 2 e T Re V (x) + τ 1 x T P x τ 2 e T Re 2φ T T (φ + Kx + K E e + Gx) <. By using the state equation from (4.16) and denoting W = P 1, S = T 1 and Z = GW, finally condition (4.19) is obtained. In summary, this procedure leads to the second statement of Theorem 2. Condition (4.2) ensures that the ellipsoid E (P, η) is included in S (G, u ) and, thereby, sector condition (4.22) holds for all x E (P, η). This can be used to derive the conditions (4.19) and (4.21). If these conditions are satisfied, inequality (4.23) holds yielding V (x) < for all x E (P, η). Thereby, it is shown that E (P, η) is an invariant set because the state trajectories of system (4.15) or equivalently (4.16) will not leave the set for any e W (R, δ). Moreover, the special case e = needs to be considered. Thereby, inequality (4.25) turns into V (x) < τ 1 x T P x < ensuring, that V (x) < holds for all x E (P, η). Hence, it can be proven that E (P, η) is a region of asymptotic stability of system (4.15) or equivalently (4.16) [13]. The result is the first statement of Theorem 2. As this case is similar to the analysis in Section 3.3 and since event-triggered systems usually have e, this is not explained in detail within this thesis. Theorem 2 can be used to calculate invariant regions for event-triggered control loops subject to actuator saturation. Suitable algorithms are portrayed in Chapter 7. In Section 4.4, Theorem 2 will be adapted to handle further extensions. 25
4 Event-triggered control subject to actuator saturation 4.3 Minimum inter-event time In event-triggered control, it is important that a lower bound for the time between two consecutive events can be found in order to exclude Zeno behavior. Zeno behavior means that an infinite amount of events could be triggered in an infinitesimal time frame. Such a control behavior could negatively affect the performance in practical applications. The lower bound for the time between two consecutive events is called minimum interevent time and is given by T min = min k {t k+1 t k }, k {, 1, 2,...}. The following theorem shows that there exists a lower bound on the minimum interevent time and, therefore, Zeno behavior can be excluded. Theorem 3. Assume that the event-triggered control loop (4.15) with output (2.2) satisfies the inequalities (4.19)-(4.21), then for x E (P, η) the minimum interevent time T min is lower bounded by with and T min T = arg min t { ẽ(t) = ẽ(t) = max C(eÃt I n ) x max + t x max = 1 } δ R (4.26) t CeÃ(t τ) B dτu max (4.27) max x p, u max = max u (i). (4.28) x p E (P,η) i {1,2,...,m} Proof. The plant is described by equations (2.1), (2.2). For d(t) = the plant output trajectory is given by y(t) = CeÃt x + C t By using the over approximation e T Re e 2 R, the norm of the output error e(t) = y(t k ) y(t) eã(t τ) Bu(τ)dτ. 26
4.4 Extended stability analysis can be upper bounded by e(t) = C ( ) eãt I n x + C t for t k = and with ẽ(t) given by (4.27). eã(t τ) Bu(τ)dτ ẽ(t) u u max holds independently of the controller output v(t) because of the actuator limitations. As events are generated whenever the event condition e T Re = δ 1 is satisfied, relation (4.26) defines a lower bound on the minimum inter-event time by means of ẽ(t). This theorem shows, that the communication exchange over the feedback link is dependent from the event generator defined by R and δ. It can be arbitrarily adapted by varying δ. However, as these parameters are also used in the stability conditions of Theorem 2, the stability of the control loop may be affected as well. 4.4 Extended stability analysis In this section, the stability conditions stated in Theorem 2 are extended. The extensions include a different definition of the event generator and the presence of disturbances d(t) and reference signals w(t). 4.4.1 Alternative event generator In Section 4.2, it is stated that the output error e(t) needs to be bounded in order to develop a stability criterion. Therefore, the event generator has to be defined in an appropriate way. In Section 4.2, the event generator triggers whenever event condition (4.18) is satisfied. However, it might be useful to define the event generator in a different way. Especially, a way to bound the amplitude of each output y (i) (i {1,..., p}) individually is interesting for practical applications. Such definitions e 2 (1) δ 1 1, e2 (2) δ 1 2,..., e2 (p) δ 1 p define a p-dimensional cuboid for the set W which is illustrated in Fig. 4.2 in comparison to the elliptical region introduced by event condition (4.18). The event generator triggers whenever the output error e(t) reaches the border of the cuboid (p = 2). 27
4 Event-triggered control subject to actuator saturation e 2 e 1 Figure 4.2: Elliptic and cubic event conditions The inequalities above have to be considered individually within the derivation. Thereby, inequality (4.23) turns into V (x) + τ 1 (x T P x η 1 ) + τ 2 (δ 1 1 e 2 1 ) + τ 3(δ 1 2 e 2 2 ) +... + τ m+1(δ 1 m e2 m ) <. Furthermore, decompositions B E = [B E1,..., B Ep ] and K E = [K E1,..., K Ep ] are introduced and inequalities (4.19), (4.21) are adapted accordingly. This leads to the following corollary that enables amplitude boundedness for each individual output. Corollary 1. If there exist a symmetric positive definite matrix W R n n, a positive definite diagonal matrix S R m m, a matrix Z R m n and positive scalars τ 1,..., τ p+1 and η satisfying W A T + AW + τ 1 W SB T KW Z 2S (B E1 + BK E1 ) T K T E1 τ 2........ < (4.29).......... (B Ep + BK Ep ) T KEp T τ p+1 [ ] W Z T (i) Z (i) ηu 2, i {1,..., m} (4.3) (i) τ 1 δ 1... δ p + τ 2 ηδ 2... δ p + + τ m+1 δ 1... δ p 1 < (4.31) then, 1. for e =, the ellipsoid E (P, η), with P = W 1, is a region of asymptotic stability of the saturated system (4.15) or equivalently (4.16). 2. for any e W = {e R p, e 2 (1) δ 1 1,..., e 2 (p) δ 1 p } with δ 1,..., δ p > and x E (P, η), the trajectories of the saturated system (4.15) or equivalently (4.16) are ultimately bounded and do not leave the ellipsoid E (P, η). 28
4.4 Extended stability analysis Zeno behavior can also be excluded if cubic event conditions are used. To calculate the minimum inter-event time T min, Theorem 3 can be used in combination with the adapted parameters δ 1....... R =....., δ = 1... δ p These parameters describe an approximation of the cubic event generator that can be used to calculate the minimum inter-event time. 4.4.2 Nonzero disturbance signals In practical applications, disturbances affecting the plant cannot be neglected. Therefore, d(t) needs to be considered within the stability analysis. The state equations are ẋ(t) = Ax(t) + Bsat(Kx(t) + K E e(t)) + B D d(t) + B E e(t) (4.32) or equivalently ẋ(t) = Ax(t) + Bφ(Kx(t) + K E e(t)) + B D d(t) + (B E + BK E )e(t) (4.33) with x() = x in both cases. For clarity, the event generator is furthermore defined as in Section 4.2. Therefore, e T Re δ 1 holds and e belongs to the ellipsoid (4.17). Similar to the output error signal e(t), the disturbance d(t) has to be bounded in order to derive stability conditions. In the following, the disturbance d(t) is assumed to be bounded by a quadratic norm. In that case, d(t) belongs to the set V D (Q D, ǫ D ) = {d R q, d T Q D d ǫ 1 D }, Q D = Q T D >, ǫ D >. (4.34) This leads to the inequality V (x) + τ 1 (x T P x η 1 ) + τ 2 (δ 1 e T Re) + τ 3 (ǫ 1 D dt Q D d) < with τ 1, τ 2, τ 3 > which is satisfied whenever the following corollary holds. 29
4 Event-triggered control subject to actuator saturation Corollary 2. If there exist a symmetric positive definite matrix W R n n, a positive definite diagonal matrix S R m m, a matrix Z R m n and four positive scalars τ 1, τ 2, τ 3 and η satisfying W A T + AW + τ 1 W SB T KW Z 2S (B E + BK E ) T KE T τ 2R < (4.35) BD T τ 3 Q D [ ] W Z T (i) Z (i) ηu 2, i {1,..., m} (4.36) (i) τ 1 δǫ D + τ 2 ηǫ D + τ 3 ηδ < (4.37) then, 1. for e = d =, the ellipsoid E (P, η), with P = W 1, is a region of asymptotic stability of the saturated system (4.32) or equivalently (4.33). 2. for any e W (R, δ), d V D (Q D, ǫ D ) and x E (P, η), the trajectories of the saturated system (4.32) or equivalently (4.33) are ultimately bounded and do not leave the ellipsoid E (P, η). Furthermore, Zeno behavior can still be excluded if disturbances d(t) occur. However, the minimum inter-event time needs to be adapted, because the output y(t) is affected by the disturbance signal d(t). Corollary 3. Assume that the event-triggered control loop (4.32) with output (2.2) satisfies the inequalities (4.35)-(4.37), then for x E (P, η) the minimum interevent time T min is lower bounded by with T min T d = arg min t { ẽ d (t) = 1 } δ R (4.38) ẽ d (t) = max t C(eÃt I n ) xmax + + t CeÃ(t τ) dτ( B u max + B D d max ), (4.39) x max and u max as defined in (4.28) and d max = max d. (4.4) d V D (Q D,ǫ D ) 3
4.4 Extended stability analysis Proof. The plant is described by equations (2.1), (2.2) yielding output trajectory y(t) = CeÃt x + C t eã(t τ) ( Bu(τ) + B D d(τ))dτ. By following the same procedure as in the proof of Theorem 3, the norm of the output error can be upper bounded by e(t) = C ( ) t eãt I n x + C eã(t τ) ( Bu(τ) + B D d(τ))dτ ẽ d(t) for t k = and with ẽ d (t) given by (4.39). Finally, relation (4.38) is used to derive a lower bound on the minimum inter-event time. In Corollary 2 and 3, the output error e and the disturbance d at the plant belong to the sets (4.17) and (4.34). These sets were chosen, because they allow a comfortable notation of the stability condition for event-triggered control loops subject to actuator saturation and disturbances. Nevertheless, output error signals or disturbances with individually bounded amplitudes can be considered straightforward as previously discussed in Section 4.4.1. In that case, the stability conditions and the minimum inter-event time need to be adapted accordingly. 4.4.3 Nonzero reference signals Finally, possible reference signals w(t) have to be considered. For simplicity reasons, disturbances will be neglected in this section, i.e., d(t) =. The state equation reads ẋ(t) = Ax(t) + Bsat(Kx(t) + K W w(t) + K E e(t)) + B W w(t) + B E e(t) (4.41) or equivalently ẋ(t) = Ax(t) + Bφ(Kx(t) + K W w(t) + K E e(t))+ + (B W + BK W )w(t) + (B E + BK E )e(t) (4.42) with x() = x in both cases. For clarity, the event generator is again defined as in Section 4.2. Hence, e T Re δ 1 holds and e belongs to the ellipsoid (4.17). In order to derive stability conditions, the reference signal w(t) has to be bounded. In the following, the reference signal w(t) is assumed to be amplitude bounded by a quadratic norm, i.e., w(t) belongs to the following set V W (Q W, ǫ W ) = {w R nc, w T Q W w ǫ 1 W }, Q W = Q T W >, ǫ W >. (4.43) 31
4 Event-triggered control subject to actuator saturation The inequality V (x) + τ 1 (x T P x η 1 ) + τ 2 (δ 1 e T Re) + τ 3 (ǫ 1 W wt Q W w) < with τ 1, τ 2, τ 3 > which is similar to inequality (4.23) is derived. It holds, whenever the following corollary is satisfied. Corollary 4. If there exist a symmetric positive definite matrix W R n n, a positive definite diagonal matrix S R m m, a matrix Z R m n and four positive scalars τ 1, τ 2, τ 3 and η satisfying W A T + AW + τ 1 W SB T KW Z 2S (B E + BK E ) T KE T τ 2 R < (4.44) (B W + BK W ) T KW T τ 3 Q W [ ] W Z T (i) Z (i) ηu 2, i {1,..., m} (4.45) (i) τ 1 δǫ W + τ 2 ηǫ W + τ 3 ηδ < (4.46) then, 1. for e = w =, the ellipsoid E (P, η), with P = W 1, is a region of asymptotic stability of the saturated system (4.41) or equivalently (4.42). 2. for any e W (R, δ), w V W (Q W, ǫ W ) and x E (P, η), the trajectories of the saturated system (4.32) or equivalently (4.33) are ultimately bounded and do not leave the ellipsoid E (P, η). The minimum inter-event time T min is not affected by reference signals w(t) because these signals only affect the controller output (4.1) which is bounded by the saturation limits and the controller state (4.2) which is not used for event generation. Therefore, the minimum inter-event time from Theorem 3 also applies for the problem formulation of this section and Zeno behavior can be excluded. Again, output error signals or reference signals with individually bounded amplitudes can be considered straightforward as in Section 4.4.1. In that case, the stability conditions and the minimum inter-event time need to be adapted accordingly. 4.4.4 Nonzero disturbance and reference signals To analyze event-triggered systems with disturbances d(t) as well as reference signals w(t), Corollaries 2 and 4 have to be combined to extend the first and the third LMI-conditions. For instance, if the event generator, disturbances and reference 32
4.5 Example signals are defined as in the previous two subsections, the first and the third LMIcondition are given by W A T + AW + τ 1 W SB T KW Z 2S (B E + BK E ) T KE T τ 2 R < BD T τ 3 Q D (B W + BK W ) T KW T τ 4 Q W τ 1 δǫ D ǫ W + τ 2 ηǫ D ǫ W + τ 3 ηδǫ W + τ 4 ηδǫ D < and the minimum inter-event time from Corollary 3 applies, excluding Zeno behavior. 4.5 Example The running example from Section 2.2 is used to illustrate the various results presented in this chapter. The main difference to Section 3.4 is the fact, that the event generator with event threshold e is introduced to the system. Information is only forwarded from the event generator to the controller, when the event condition e(t) = y(t k ) y(t) = e is met. Such an event generator is modeled by R = 1 and δ = e 2. Therefore, the controller equations (2.8) and (2.9) read ẋ c (t) = y(t k ) w(t), x c (k) = x ck v(t) = k I x c (t) + k P (y(t k ) w(t)) for t [t k, t k+1 ). By introducing the output error e(t) = y(t k ) y(t) = x p (t k ) x p (t), the augmented state vector x = [ ] T x p x c and the saturation function u(t) = sat(v(t)), the state equation of the universal system representation (4.13) reads [ ] [ ] a b ẋ(t) = x(t) + sat( [ ] k 1 P k I x(t) kp w(t) + k P e(t))+ [ ] [ ] [ ] bd + d(t) + w(t) + e(t), x() = x 1 1 which holds for all times t. A, B, K, K W, B D and B W are the same matrices as in Section 3.4. The matrices that apply to the output error e(t) are [ ] K E = k P, B E =. 1 In the examples, Theorem 2 and the corollaries proposed in Section 4.4 are used to calculate regions of stability and trajectories. Numerical values for the system parameters are given in Section 2.2. 33
4 Event-triggered control subject to actuator saturation xc 3 2.5 2 1.5 1.5 -.5-1 -1.5 x a CT ET: e =.1, d(t) = ET: e =.1, d =.1-2 -2.5-1.5-1 -.5.5 1 1.5 2 Figure 4.3: Regions of stability of the even-triggered control loop (ET) in comparison to the region of asymptotic stability of the continuous-time control loop (CT) 4.5.1 Fixed event threshold and disturbance In this section, sets of initial states are calculated that yield stability for a given event generator as well as given disturbances. Since reference signals w(t) as well as output error signals e(t) appear both inside and outside of the saturation function sat(.) within the general system representation (4.13), they have similar effects on the region of stability and, therefore, reference signals are neglected, i.e., w(t) =. The respective optimization problems min{ trace(w )} subject to conditions from Theorem 2/Corollary 2 were solved with the YALMIP toolbox for τ 1 = τ 2 = τ 3 =.1 to calculate estimates for the regions of stability. To calculate trajectories of the system, a MATLAB/Simulink implementation of the control loop is used. For further details, the reader is referred to Chapter 7 and Appendix B.1. Regions of stability. The general effects of event-triggered sampling and additional disturbances are depicted in Fig. 4.3. The blue ellipsoid describes the region of asymptotic stability for the continuous-time control loop (3.21). It is identical to the 34
4.5 Example xc 8 6 4 2 CT ET: e =.1, d(t) = [ ] T ET: x =.75 [ ] T ET: x = 1 2 [ ET: x = 2 ] T 2.5 ET: x = [ 2.5 2.5 ] T -2-2 -1 1 2 Figure 4.4: Trajectories of the event-triggered control loop (dashed, ET) and regions of stability (solid, CT and ET) x p region shown in the upper graph of Fig. 3.4. The other two ellipsoids describe the event-triggered cases. In both cases, an event generator with event threshold e =.1 is considered. If no disturbances occur, i.e., d(t) =, the resulting region of stability of the eventtriggered control loop is smaller then the region of asymptotic stability of the continuous-time control loop. This is illustrated by the pink ellipsoid in Fig. 4.3. The result is reasonable, because communication within the control loop is limited due to event generator which degrades the control performance. An additional disturbance d(t) d =.1, which is modeled by Q D = 1 and ǫ D = d 2, further decreases the region of stability which is illustrated by the brown ellipsoid in Fig. 4.3. It has to be remarked, that the ellipsoids in the event-triggered describe only regions of stability. This means, that trajectories which begin within the respective ellipsoid, i.e. x E (P, η), will not leave it for all times t. In general asymptotic stability cannot be achieved in event-triggered control loops. Fig. 4.4 illustrates trajectories of the event-triggered control loop. The event threshold is set to e =.1 and no disturbances occur, i.e. d(t) =. As calculated before, trajectories that begin within the region of stability, i.e. the pink ellipsoid, do not leave it for any time t. This is illustrated by the green trajectory, starting at x = [.75 ] T. 35
4 Event-triggered control subject to actuator saturation xp 1.5 x p (t) x p (t k ) -.5 5 1 15 2 25 3.1 e.5 e(t) Events 5 1 15 2 25 3 xc 1.5 x c (t) 5 1 15 2 25 3 t Figure 4.5: States and output error of the event-triggered control loop for x = [.75 ] T However, the conservatism introduced by the approach allows trajectories to start in or even outside the region of asymptotic stability of the continuous-time control loop represented by the blue ellipsoid to still yield a stable behavior of the event-triggered control loop. For instance, the black and purple trajectories starting in x = [ 1 2 ] T and x = [ 2 2.5 ] T, still yield a stable behavior of the event-triggered control loop as illustrated in Fig. 4.4. Nevertheless, for most initial states that are outside the region of stability, instable behavior can be observed. For instance the red trajectory starting at x = [ 2.5 2.5 ] T quickly rises to infinity. Behavior of the event-triggered control loop. The states x p (t) and x c (t), the absolute value of the output error e(t) and communication within the event-triggered control loop for the initial state x = [.75 ] T are illustrated in Fig. 4.5. 36
4.5 Example v and u -.5-1 v(t) u(t) -1.5 5 1 15 2 25 3 t Figure 4.6: Controller and actuator outputs of the event-triggered control loop for x = [.75 ] T In the top graph, x p (t) and x p (t k ) are drawn. According to their difference, the output error e(t) is defined. Its absolute value e(t) is depicted in the middle graph. Whenever e(t) = e holds, the event condition is satisfied and new information is forwarded to the controller. This is indicated by the pink stars. According to Theorem 3, the minimum inter-event time can be calculated. x max =.77 obtained from Fig. 4.4, T min T = 1 ) (1 a log ae + =.21 < T min,sim =.29 ax max u max With holds for this example which is smaller than the smallest time T min,sim observed between two consecutive events in Fig. 4.5. Furthermore, the bottom graph shows x c (t). As mentioned before, asymptotic stability can usually not be achieved with event-triggered control. Therefore, x p (t) and x c (t) are both forced into stable oscillations around zero. Finally, the controller output v(t) and the actuator output u(t) are depicted in Fig. 4.6. Both graphs are unsteady due to the event-triggered communication between the event generator and the controller. Actuator saturation only occurs during the first three seconds. 4.5.2 Variable event threshold and disturbances In the following, the effects of different event thresholds is analyzed. To calculate the maximal event threshold leading to a feasible problem, R = 1 has been set and the 37
4 Event-triggered control subject to actuator saturation xc 2.5 2 1.5 1.5 -.5-1 -1.5-2 x p e =.5 e =.1 e = e max =.148-2.5-1.5-1 -.5.5 1 1.5 Figure 4.7: Regions of stability for different event thresholds optimization problem min{δ} subject to conditions from Theorem 2 was solved with the YALMIP toolbox and τ 1 = τ 2 =.1. The exact procedure is explained in Chapter 7. Fig. 4.7 shows various regions of stability depending on the event threshold e of the event-triggered control loop without disturbance or reference signals d(t) = w(t) =. In general, it can be said that: The bigger the event threshold, the smaller the region of stability. This is logical because a bigger event threshold yields fewer communication between the controller and the event generator affecting the control performance. In Chapter 7 it is shown, how the maximum event threshold e max can be calculated. For this example this value is e max =.148. The graphs for different reference signals would look similar to Fig. 4.7. With the same methods it would also be possible to calculate the maximal upper bounds for the disturbance d max and the reference signal w max. Since the procedure is the same, this is omitted here. 38
5 Anti-windup compensation The examples of the previous chapters showed, that integrator windup cannot be neglected within event-triggered control. Therefore, methods to deal with integrator windup are introduced in this chapter. First, the general motivation of anti-windup control is described in Section 5.1. In Section 5.2, a static anti-windup structure is introduced. Its effects are illustrated in Section 5.3 by using the running example. 5.1 Introduction Integrator windup is an effect, that can generally occur whenever actuators and either the plant or the controller include integrator dynamics. An example for this effect is displayed in Fig. 3.6. The saturation nonlinearity creates a difference between the controller output v(t) and the actuator output u(t). This slows down the response of the feedback loop and causes the integrator state x c to wind up. In Fig. 3.5, it is shown, that a saturation nonlinearity can cause a severe windup of the controller state which even leads to an unstable system behavior. Furthermore, negative effects of windup may even increase in event-triggered control because only limited communication takes place between the controller and the plant. Thereby, old information is used by the controller. A possible way to deal with these issues is to enhance the control loop with an antiwindup structure which aims at avoiding integrator windup within the integrator part of the controller. Many anti-windup structures are realized by feeding back the difference u(t) v(t) of the actuator and the controller output. When no actuator saturation occurs, this difference is zero and the structure has no effect on the closedloop system. However, if actuator saturation occurs, this difference becomes nonzero and corrective actions to reduce integrator windup are carried out. In the following section, a static anti-windup structure is used to enhance the system performance. For further references regarding anti-windup and a wide variety of different anti-windup structures, the reader is referred to [14]. 39
5 Anti-windup compensation w(t) d(t) Controller x c (t) (t) - u(t) Plant y(t) Event y(t k ) x p (t) generator K AW Figure 5.1: Event-triggered control loop with a static anti-windup extension 5.2 Modeling of a static anti-windup structure In this section, the static anti-windup structure depicted in Fig. 5.1 is introduced. The considered system is the same as in Section 4.1. The difference between the actuator and the controller output corresponds to the dead zone nonlinearity (3.11) according to u(t) v(t) = sat(v(t)) v(t) = φ(v(t)). Using static gain K AW, this difference is fed back to the controller state. Thereby, the general controller state (4.4) is extended in the following way ẋ c (t) = Ã c x c (t) + B c y(t) + B c e(t) + B cw w(t) + K AW φ(v(t)), x c () = x c. By using the adapted controller state, the following system representation is developed ẋ(t) = Ax(t) + Bφ(Kx(t) + K W w(t) + K E e(t))+ + B D d(t) + (B W + BK W )w(t) + (B E + BK E )e(t), x() = x, y(t) = Cx(t) (5.1) with A = A + BK, B = [ B K AW ]. 4
5.3 Example The only difference between the representation of the event-triggered control loop with anti-windup compensation (5.1) and the description without anti-windup compensation (4.14) is the different definition of the matrix B including new degrees of freedom for the design of the closed-loop system through the feedback gain K AW. These additional degrees of freedom can be used to improve the control performance. For example, the size of the region of stability can be increased by choosing an appropriate K AW which is shown in Section 5.3. After adapting matrix B, all the tools presented in Chapter 4 can be used straightforward to calculate regions of stability for control loops that are extended by a static anti-windup structure. For instance, the following corollary adds the presented anti-windup compensation to Theorem 2. Corollaries 1, 2 and 4 can be adapted accordingly. Corollary 5. If there exist a symmetric positive definite matrix W R n n, a positive definite diagonal matrix S R m m, a matrix Z R m n and three positive scalars τ 1, τ 2 and η satisfying W A T + AW + τ 1 W BS W K T Z T B E + BK E SB T KW Z 2S K E < (5.2) (B E + BK E ) T KE T τ 2 R [ ] W Z T (i) Z (i) ηu 2, i {1,..., m} (5.3) (i) τ 1 δ + τ 2 η < (5.4) then, 1. for e =, the ellipsoid E (P, η), with P = W 1, is a region of asymptotic stability of the saturated system (5.1). 2. for any e W (R, δ) and x E (P, η), the trajectories of the saturated system (5.1) are ultimately bounded and do not leave the ellipsoid E(P, η). The presented anti-windup structure does not affect the minimum inter-event time T min. It only affects the controller whose state x c (t) is not used for event generation and whose output v(t) is bounded by the saturation limits anyway. Therefore, the minimum inter-event time from Theorem 3 applies, excluding Zeno behavior for the system considered in this section. 41
5 Anti-windup compensation xc 8 6 4 2 CT ET: no AW ET: k AW =.5 ET: k AW = 1 ET: k AW = 5-2 -4-6 -3-2 -1 1 2 3 4 Figure 5.2: Regions of stability of the event-triggered control loop (ET, e =.1 and d(t) = ) with and without anti-windup extension (AW) in comparison to the region of asymptotic stability of the continuous-time control loop (CT) 5.3 Example The example from the previous chapter is now extended by a static anti-windup structure. The difference k AW (u(t) v(t)) is fed back to the controller state. After introducing the dead zone φ(v(t)) = sat(v(t)) v(t), the state equation is given by ẋ(t) = x p [ ] [ ] a + bkp bk I b x(t) + φ( [ ] k 1 k P k I x(t) kp w(t) + k P e(t))+ AW [ ] [ ] [ ] bd bkp bkp + d(t) + w(t) + e(t), x() = x 1 1 with the same matrices A, B, K, K W, B D, B W, K E and B E as in Section 4.5 and A = [ ] [ ] a + bkp bk I b, B =, B 1 k W +BK W = AW [ ] bkp, B 1 E +BK E = To calculate estimates for the regions of stability, the optimization problem min{ trace(w )} subject to conditions from Theorem 2/Corollary 5 [ ] bkp. 1 42
5.3 Example is used again and solved with the YALMIP toolbox for τ 1 = τ 2 =.1. Furthermore, trajectories of the system are calculated again with a MATLAB/Simulink implementation of the control loop (see Chapter 7 and Appendix B.1). The numerical values for the system parameters are given in Section 2.2. Regions of stability. In Fig. 5.2, regions of stability for different values k AW are depicted. The regions are significantly bigger than the region obtained for the eventtriggered control loop without anti-windup augmentation and even include the region of asymptotic stability for the continuous-time control loop. Therefore, it can be concluded, that anti-windup compensation is able to significantly increase the region of stability also in event-triggered control. Behavior of the adapted event-triggered control loop. The trajectories of the states for x = [ 2.5 2.5 ] T are depicted in Fig. 5.3. The red graphs describes the states of the event-triggered control loop without anti-windup compensation which refers to the red trajectory in Fig. 4.4. Without anti-windup compensation the initial value x = [ 2.5 2.5 ] T yields an unstable system behavior. If anti-windup compensation is applied, the initial state x = [ 2.5 2.5 ] T lies within the region of stability (see Fig. 5.2). Therefore, the system behavior is stable which is illustrated by the black graphs in Fig. 5.3. This behavior can also be observed at the controller output which is depicted in Fig. 5.4. The upper graph shows that windup rises quickly within the event-triggered control loop without anti-windup compensation. This yields an unstable system behavior because integrator windup occurs within the controller that slows its transient behavior. In the lower graph, the anti-windup extension is applied and windup is compensated quickly. The actuator output is driven into a stable oscillation within the saturation bounds. As a conclusion, anti-windup compensation did not only increase the size of the region of stability, but also improve the performance of the event-triggered control loop significantly. 43
5 Anti-windup compensation xp xc 6 4 2-2 ET: no AW ET: k AW = 1 1 2 3 4 5 6 7 8 9 t 15 1 5 1 2 3 4 5 6 7 8 9 t Figure 5.3: Behavior of the event-triggered control loop with and without anti-windup extension for x = [ 2.5 2.5 ] T v and u -5-1 ET: no AW, v ET: no AW, u v and u -15 1 2 3 4 5 6 7 8 9 t.5 -.5-1 ET: k AW = 1, v -1.5 ET: k AW = 1, u -2-2.5 1 2 3 4 5 6 7 8 9 t Figure 5.4: Controller and actuator outputs with and without anti-windup compensation for x = [ 2.5 2.5 ] T 44
6 Transmission delays This chapter will focus on communication imperfections that can occur if a wireless network is used to realize the feedback link. The focus of this chapter are transmission delays, which are modeled by a constant dead time T. This is portrayed in Fig. 6.1. Thereby, messages y(t k ) get delayed causing the controller to rely on old information longer than intended by the event-triggered control scheme which may affect the control performance. In order to investigate the problems of dead times, an appropriate modeling is necessary which is presented in Section 6.1. In the subsequent Sections 6.2 and 6.3, two cases will be analyzed: Dead times T bigger or smaller than the minimum inter-event time T min. The problem of mutual dependencies between different variables is highlighted in Section 6.4. Finally the effects of delays in the feedback link will be illustrated in Section 6.5. 6.1 Modeling Unlike in the rest of the thesis, this chapter will rely on a scalar problem formulation. The plant is given by ẋ p (t) = ax p (t) + bu(t) + b D d(t), x p () = x p (6.1) y(t) = cx p (t). (6.2) w(t) $(t) Controller x! (t) v(t) "(t) Plant y(t) Event y(t k ) x p (t) generator Dead time % Figure 6.1: The event-triggered control loop subject to transmission delays 45
6 Transmission delays An event is generated whenever the event condition y(t k ) y(t) = e (6.3) is satisfied. In that case, y(t k ) is forwarded to the controller. The controller, without considering the dead time T, is given by ẋ c (t) = a c x c (t) + b c y(t k ) + b cw w(t), x c (t k ) = x ck (6.4) v(t) = c c x c (t) + d c y(t k ) + d cw w(t) (6.5) for t [t k, t k+1 ). This representation of the controller with y(t k ) is rather inadequate to model dead time delays because it is not possible to include delays T properly. Nevertheless, since the controller uses values y(t k ) until a new one y(t k+1 ) is transmitted, it is possible to introduce a piecewise defined signal y tk (t) = y(t k ), t [t k, t k+1 ). (6.6) By using this signal, it is possible to rewrite the controller equations in a way that includes the constant dead time T and holds for all t T: ẋ c (t) = a c x c (t) + b c y tk (t T) + b cw w(t), x c (T) = x ct (6.7) v(t) = c c x c (t) + d c y tk (t T) + d cw w(t). (6.8) To continue the modeling process, it makes sense to study the impact of the dead time T on the signal that the controller receives. Therefore, exemplary trajectories are drawn in Fig. 6.2. The signal y(t), which is drawn in the upper graph, is the regular plant output (6.2) which is subject to event-triggered sampling. The event generator forwards messages y(t k ) whenever the event condition (6.3) is met, i.e., when y(t) reaches the horizontal dashed borders around its trajectory. In the middle graph, y tk (t) is displayed. This signal describes the values the controller would use if no delays would occur. The difference between y tk (t) and y(t) is the output error e(t) = y(t k ) y(t) which is known from the previous chapters (see relation (4.3)). However, delays occur. Therefore, the signal which the controller receives is y tk (t T). This signal may vary from y tk (t). To model the overall difference between y tk (t T) and y(t), the extended output error e e (t) = y tk (t T) y(t) = e(t) + e T (t) (6.9) is introduced. This signal consists of the regular output error signal e(t) caused by the event generation and an additional error signal e T (t) caused by the time delay. 46
6.1 Modeling y(t) t k t k+1 t k+2 2e t y tk (t) y tk (t-t) t T t Figure 6.2: Comparison of y(t), y tk (t) and y tk (t T) Similar to the previous chapters, plant and controller equations can be combined by using the extended output error (6.9) as well as an augmented state vector x = [ xp x c ] T. The universal system representation reads ẋ(t) = Ax(t) + Bsat(Kx(t) + K W w(t) + K E e e (t))+ + B D d(t) + B W w(t) + B E e e (t), x() = x, y(t) = Cx(t). (6.1) This system representation is distinguished from (4.13) by the error signal e e (t) defined in equation (6.9). If the extended error signal e e (t) can be bounded, the stability analyses as presented in the previous chapters can be also used in the case of transmission delays. 47
6 Transmission delays 6.2 Large delays Dead times T are considered as large, if they are bigger as the minimum inter-event time T min. In that case, several events can be generated during the dead time T. Due to the event condition (6.3), the output error e(t) is upper bounded by e(t) e and the additional error caused by the delay e T (t) can be upper bounded by T e T (t) T min e. Altogether, these results can be used to bound the extended error signal according to the following proposition. Proposition 1. For transmission delays T > T min, the norm of the extended output error e e (t) is upper bounded by ( T e e (t) e e,max = T min ) + 1 e. (6.11) The problem with this bound is, that it includes the minimum inter-event time T min which is defined in Corollary 3. This definition includes x max from (4.28) which depends on the region of stability E (P, η). Thereby, e e,max depends on E (P, η) which depends on e e,max itself. This problem is discussed in Section 6.4. 6.3 Small delays For dead times T that are smaller than the minimum inter-event time T min, relation (6.11) can be used to immediately receive an upper bound for the extended error signal e e (t) 2e. However, it is possible to receive an even more precise bound, since the output error can only increase to a certain level e e (t) = e(t) + e T (t) { ẽdmax + e for t [t k, t k + T) e + for t [t k + T, t k+1 ) (6.12) depending on the dead time T. During the first interval between two consecutive events, the controller still uses the old value y(t k 1 ). Therefore, e T (t) = e holds. 48
6.4 Mutual dependencies However, the output error e(t) can only rise to a certain level ẽ dmax during that time. Equation (4.39) can be used to calculate this value ẽ dmax = c(e at 1) x max + T ce a(t τ) dτ( b umax + b D d max ) (6.13) with u max and d max as defined in equations (4.28) and (4.4). During the second time interval, the controller already received the updated value y(t k+1 ). Therefore, e T (t) becomes zero. The output error is bounded by e(t) e during that time interval since a new event is about to occur. The results above are summarized in the following proposition. Proposition 2. For transmission delays T < T min, the norm of the extended output error e e (t) is upper bounded by e e (t) e e,max = ẽ dmax + e (6.14) with ẽ dmax as defined in equation (6.13). However, relation (6.13) uses x max again which depends on the region of stability E (P, η). Thereby, e e,max depends on E (P, η) which relies on e e,max itself. This problem is discusses in the following section. 6.4 Mutual dependencies The last two sections revealed a problem for calculating bounds on the extended output error e e (t): The bounds rely on x max from relation (4.28). However, x max can only be obtained if the region of stability E (P, η) is known and for that, the extended output error e e (t) has to be bounded. Furthermore, the definition of the minimum inter-event time in Corollary 3 also relies on x max which makes a clear a priori distinction between large and small delays impossible. The mutual dependencies of the different variables, considered that u max and d max are known, are portrayed in Fig. 6.3. To overcome these issues further investigations are necessary. However, the following procedure can be used to show stability for practical systems subject to time delays. 1. Fix e. 2. Calculate the region of stability by using either Theorem 2 or Corollary 2, 4 or 5 without considering the dead time T. 3. Calculate x max and T min. 4. Calculate e e,max according to Proposition 1 or 2. 49
6 Transmission delays Event generator e x max T T mi e~ max Bounds for e(t) Bounds for e T (t) Bounds for e e (t) Figure 6.3: Mutual dependencies of different variables 5. Calculate the region of stability for e e (t) by using either Theorem 2 or Corollary 2, 4 or 5. 6. Validate if Proposition 1 or 2 still holds for the new x max and T min. If not, iterate adapting E (P, η) by fixing x max. 6.5 Example In this section, transmission delays within the feedback link of the system from Section 4.5 with e =.5 are considered. Behavior of the event-triggered control loop. The effects of different transmission delays are highlighted in Fig. 6.4. The trajectories of x = [.75 ] T for T =.1 and T =.8 yield a stable system behavior. However, compared to the trajectory without transmission delays T =, the control performance degrades. If the delays are too large, the system behavior becomes unstable which can be seen by the trajectory for T = 1. Different trajectories of the event-triggered control loop subject to time delays for x = [.75 ] T are depicted in Fig. 6.5. The upper plot depicts xp (t) with events indicated by stars and circles. Furthermore, the middle plot depicts x c (t) and the lower plot depicts u(t) (solid) and v(t) (dashed). It is clearly demonstrated, that transmission delays deteriorate the control performance in terms of the amount of events, the stationary behavior and occurring actuator saturation. 5
.6 6.5 Example xc 2.5 2 1.5 1.5 -.5-1 -1.5-2 -2.5 ROS for T = T = T =.1 T =.8 T = 1-1.5-1 -.5.5 1 1.5 Figure 6.4: Trajectories of the event-triggered control loop subject to transmission delays and region of stability (ROS) for e =.5 x p Regions of stability. The minimum inter-event time of the control loop without transmission delays T = can be calculated according to Theorem 3. With x max =.81 obtained from Fig. 6.4, the minimum inter-event time T min T = 1 ) (1 a log ae + =.134 < T min,sim =.14 ax max u max is obtained for this example, which is smaller than the smallest time T min,sim observed between two consecutive events in Fig. 6.5. Furthermore, for small delays T < T min, ẽ dmax is defined by ( ẽ dmax = x max + u ) max (e at 1 ), a where x max is defined by the region of stability that belongs to the control loop with the respective transmission delay T. For x max = x max =.81, ẽ dmax =.483 is obtained. In Fig. 6.6, the region of stability for e e (t) ẽ dmax +e =.983 is depicted. Despite the fact that x max =.81 does not belong to the new region anymore, it is still a region of stability for the event-control loop with e =.5 subject to transmission delays T =.1, because the x max obtained from that region leads to a T min =.143 that still fulfills T < T min. Furthermore, bigger x max lead to a more conservative estimate for ẽ dmax and, hence, the region of stability. Exemplary trajectories in Fig. 6.6 for e =.5, T =.1 as well as x = [.75 ] T, x = [.7 1.81 ] T or x = [.4 1.7 ] T illustrate that the trajectories do not leave the respective region of stability for e =.983. 51
for e = 6 Transmission delays u and v xc xp 1-1 1 2 3 4 5 6 7 8 9 1 2 T = T =.8-2 1 2 3 4 5 6 7 8 9 1 2-2 1 2 3 4 5 6 7 8 9 1 t Figure 6.5: Behavior of the event-triggered control loop subject to transmission delays for x = [.75 ] T xc 2.5 2 1.5 1.5 -.5-1 -1.5-2 ROS for e =.5 ROS for e =.983-2.5-1.5-1 -.5.5 1 1.5 Figure 6.6: Region of stability for e =.5 and e =.983. The second region also refers to a control loop with e =.5 and T =.1. x p 52
6.5 Example Large delays T > T min yield bounds according to equation (6.11) that are at least 3e =.15. Since the maximum event threshold for that a region of stability can be calculated is e max =.148 (see Section 4.5.2), stability cannot be shown for large delays T > T min with the presented methods. 53
7 Optimization problems and practical implementation Theorems 1 and 2 as well as Corollaries 1, 2, 4 and 5 can be used to test if given sets of initial conditions E (P, η) yield a stable behavior of the respective systems. To do so, P and η are put to verify the stability conditions of the respective theorem or corollary. However, testing random sets of initial conditions is quite inefficient and yields only few practical value. Therefore, the following two optimization problems are considered in this thesis: Maximize the size of the set of initial conditions E (P, η) that yield a stable system behavior for a given event generator and/or exogenous signals. Maximize the size of the set of the output error W (R, δ) and/or the set(s) of exogenous signals V D (Q D, ǫ D )/V W (Q W, ǫ W ) that yield a stable system behavior. Suitable algorithms for both problems are presented in Section 7.1. In order to execute these algorithms, it is necessary to determine a suitable size criterion of the sets that should be maximized. There are several possibilities that translate into different objective functions f(e (P, η)). Some examples are given in Section 7.2. The algorithms can be implemented with various tools. In Section 7.3, the approach of this thesis is described, i.e., the usage of specific toolboxes for solving linear matrix inequalities in MATLAB. Finally, examples are given in Section 7.4 to illustrate the results of different objective functions and parameters. 7.1 Algorithms for stability analysis The stability conditions presented in this thesis include linear decision variables W, S, Z and η if the parameters τ i are fixed. The influence of the τ i is discussed in Section 7.4. Furthermore, parameters that describe the sets of the output error (R, δ), the disturbance signals (Q D, ǫ D ) and the reference signals (Q W, ǫ W ) are either a priori given or additional decision variables. 54
7.2 Optimization functions For clarity and as this thesis focuses on event-triggered control, the sets of exogenous signals V D (Q D, ǫ D ), V W (Q W, ǫ W ) are fixed within the following algorithms. However, the sets of admissible disturbance signals V D (Q D, ǫ D ) and/or reference signals V W (Q W, ǫ W ) could also be considered or maximized accordingly after adapting the algorithms with the respective decision variables (Q D, ǫ D and/or Q W, ǫ W ). The first algorithm can be used to calculate the maximal region of stability E (P, η) for a given event generator (fixed R, δ). Algorithm 1. Maximization of the set of initial conditions E (P, η) [13]: 1. Given R and δ, choose a suitable objective function f(e (P, η)). 2. Fix τ 1, τ 2, τ 3 and τ 4. 3. Solve for W, S, Z and η the optimization problem min{f(e (P, η))} subject to inequalities of either Theorem 1, 2 or Corollary 1, 2, 4 or 5. The second algorithm can be used to calculate the maximum set of the output error W (R, δ). The parameters R and δ of this set can be used to determine event conditions that yield a stable system behavior. Algorithm 2. Maximization of the set of the output error W (R, δ) [13]: 1. Choose a suitable objective function f(w (R, δ)). 2. Fix τ 1, τ 2, τ 3 and τ 4. 3. Solve for W, S, Z, R, η and δ the optimization problem min{f(w (R, δ))} subject to inequalities of either Theorem 1, 2 or Corollary 1, 2, 4 or 5. 7.2 Optimization functions An appropriate size criterion has to be determined in order to maximize the respective set by using the algorithms introduced in the previous section. Several possibilities that translate into different objective functions f(e (P, η)) are given in [13]. Two examples are: f(e (P, η)) = n log(η) + log(det(p )) referring to volume maximization. f(e (P, η)) = β η +β 1 trace(p ) with weighting parameters β and β 1 which leads to ellipsoids that are homogeneous in all directions. 55
7 Optimization problems and practical implementation However, these objective functions become nonlinear due to the required transformation P = W 1. To avoid this, the objective functions have to be adapted accordingly [13]: f(e (W 1, η)) = n log(η) log(det(w )) f(e (W 1, η)) = β η β 1 trace(w ) The results by using different objective functions are discussed in Section 7.4. 7.3 Implementation in MATLAB MATLAB/Simulink offers convenient tools to simulate the behavior of control loops. In fact, all the trajectories in the examples of the previous chapters were calculated by using MATLAB/Simulink models of the respective control loops. An exemplary model can be found in Appendix B.1. However, the tools MATLAB offers for solving optimization problems based on LMI conditions are rather inconvenient. Additional toolboxes like YALMIP [21] or CVX [22] overcome these issues and allow for a simple implementation and execution of the algorithms described in this chapter. For this thesis, the YALMIP toolbox was used to calculate regions of stability. Exemplary implementations of the algorithms can be found in Appendix B.2 and B.3. 7.4 Example This section discusses the results obtained by using different objective functions and different τ i for the system considered in Section 4.5. Algorithm 1. Fig. 7.1 shows the results of Algorithm 1 for different objective functions, e(t) W (R, δ), d(t) = w(t) = and τ 1 = τ 2 =.1. It can be seen that the objective functions that include either the term trace(w ) or log(det(w )) yield similar results. The term η yields rather conservative results. The objective function β η β 1 trace(w ) yields the same result as trace(w ) for β = β 1 = 1. However, the results become more conservative if β increases. Unfortunately, objective functions with the term n log(η) could not be solved with the used tools. As the simple objective function trace(w ) yields good results, it was used for all the simulations within the examples of the previous chapters. 56
7.4 Example 2 1.5 1.5 trace(w ) η 6η trace(w ) η trace(w ) log(det(w )) xc -.5-1 -1.5-2 -1 -.8 -.6 -.4 -.2.2.4.6.8 1 x p Figure 7.1: Results of Algorithm 1 for different objective functions xc 2 1.5 1.5 -.5 τ 1 =.1 τ 1 =.3 τ 1 =.5 τ 1 =.7 τ 1 =.9 τ 1 =.8 τ 1 =.6 τ 1 =.4-1 -1.5-2 -1 -.8 -.6 -.4 -.2.2.4.6.8 1 x p Figure 7.2: Results of Algorithm 1 for different τ 1 57
7 Optimization problems and practical implementation 2 1.5 1.5 τ 2 =.1 τ 2 = 1 τ 2 =.1 xc -.5-1 -1.5-2 -1.5-1 -.5.5 1 1.5 x p Figure 7.3: Results of Algorithm 1 for different τ 2 The results of Fig. 7.2, 7.3 and 7.4 were calculated by using the objective function trace(w ). However, the general statements of the following paragraphs also hold for other objective functions. The influence of τ 1 is portrayed in Fig. 7.2. The algorithm is feasible for τ 1 [.4, 1.4]. The size of the region of stability varies significantly with different τ 1. For this example, there is a maximum around τ 1.8. The obtained region of stability remains the same for a wide range of τ 2 which is shown in Fig. 7.3. Furthermore, despite of changing τ 2, the domain of τ 1 [.4, 1.4] for which the algorithm is feasible remains unchanged with similar results as described in the previous paragraph. The reason for this is, that a change in τ 2 only leads to a different scaling between P and η within the inequality x T P x η 1 that describes the ellipsoid E (P, η). However, if for certain reasons η is fixed, the influence of τ 2 on the region of stability would have to be considered, too. Furthermore, the influence of τ 3 for d(t), w(t) = and τ 1 = τ 2 =.1 is portrayed in Fig. 7.4. The algorithm is feasible for τ 3 [.1,.3]. The size of the region of stability varies for different τ 3. A maximum is obtained for τ 3.1. As a summary, τ 1 and τ 3 have to be chosen carefully, while different values for τ 2 do not affect the results if P and η are both decision variables. To retrieve the largest region of stability, it is recommended to iterate Algorithm 1 on a grid of different τ i values [13]. 58
7.4 Example xp 2 1.5 1.5 -.5-1 τ 3 =.3 τ 3 =.1 τ 3 =.5 τ 3 =.1 τ 3 =.5 τ 3 =.3 τ 3 =.1-1.5-2 -1 -.8 -.6 -.4 -.2.2.4.6.8 1 x c Figure 7.4: Results of Algorithm 1 for different τ 3 Algorithm 2. Algorithm 2 is used to maximize the set W (R, δ) and thereby to retrieve parameters R and δ that can be used to define the event generator. The system of Section 4.5 uses a scalar event generator. Therefore, R = 1 is set and the target function δ is used. For the event generator, the relation e(t) e holds with e = δ 1. The influence of τ 1 on the region of stability is shown in Fig. 7.5. The algorithm is feasible for τ 1 [1 5, 1.4]. The resulting e are displayed in Table 7.1. There seems to be a maximum for e around τ 1.5. However, in some practical applications, a compromise between a large region of stability and a big event threshold might be needed. Therefore, it is recommended to try different τ 1 until satisfying sets E (P, η) and W (R, δ) are retrieved. Table 7.1: Calculated event thresholds for different τ 1 τ 1.5.1.5.1.5.9 1.4 e.377.529.1121.1485.2146.27.1195 Fig. 7.6 shows, that the results remain unchanged for a wide range of τ 2 (τ 1 =.1). Again, a change in τ 2 only leads to a different scaling between P and η. The eventthreshold e is neither affected by changing τ 2 which is documented in Table 7.2. 59
7 Optimization problems and practical implementation Table 7.2: Calculated event thresholds for different τ 2 τ 1.5.5.5.1.1.1.9.9.9 τ 2.1.1 1.1.1 1.1.1 1 e.377.377.377.1485.1485.1485.1121.1121.1121 xc 1.5 1.5 -.5 τ 1 =.1 τ 1 =.5 τ 1 =.9 τ 1 = 1.4 τ 1 =.5 τ 1 =.1 τ 1 =.5-1 -1.5-1 -.5.5 1 x p Figure 7.5: Results of Algorithm 2 for different τ 1 Controller design. Eventually, the obtained region of stability is also greatly dependent of the controller design. This is illustrated in Fig. 7.7 where regions of stability are depicted for different k P and k I values. It can be seen, that the size of the region of stability is bigger for bigger ratios k P ki (provided that the absolute values of k P and k I are sufficiently small) due to the lower influence of k I and potential integrator windup. These results show, that appropriate controller design can also be used to increase the region of stability. 6
7.4 Example 1.5 1.5 τ 2 =.1 τ 2 = 1 τ 2 =.1 xc -.5-1 -1.5 -.8 -.6 -.4 -.2.2.4.6.8 x p Figure 7.6: Results of Algorithm 2 for different τ 2 xc 6 4 2 k P = 1.6, k I = 1 k P = 1.6, k I =.5 k P = 1, k I = 1 k P = 1, k I =.5-2 -4-6 -1.5-1 -.5.5 1 1.5 x p Figure 7.7: Regions of stability for different controller parameters 61
8 Experiments The theoretical results of the previous chapters are used in a practical experiment consisting of a first-order tank system controlled by an event-triggered PI controller. The overall setup of the experiment is depicted in Fig. 8.1. The main component is a cylindric tank from Quanser Inc. [23] which can be filled with water. It is depicted in Fig. 8.2. Its inflow can be controlled by a pump with input u [, 12] V. The outflow results from a hole at the bottom of the tank and depends on the water level. The plant state x p (t) describes the water level within the tank with unit cm. It is continuously measured by a pressure sensor at the bottom of the tank. The controller, the actuator and the sensor are connected via a wireless communication network consisting of four Tmote Sky sensor nodes [24]. The event generator is implemented within the sensor node and new information is only forwarded to the controller if the event condition x p (t k ) x p (t) = e is met, whereas the controller sends its information v(t) to the actuator with a high frequency to meet the structural requirements used throughout this thesis as presented in Fig. 1.3. All the software is realized with TinyOS [25]. Tank systems usually have nonlinear dynamics [26]. Therefore, the system is linearized (t) v(t) v(t) PI controller (t k ) Wireless network (t k ) Event generator L Figure 8.1: Setup of the tank experiment 62
Figure 8.2: Water tank from Quanser Inc. around the operating point x OP = 1 cm, u OP = 5.43 V, leading to the following stable linear state-space model ẋ p (t) = 1 15 x p(t) 1 s + 4 15 u(t)cm Vs, x p() = x p with x p (t) = x p (t) x OP and u(t) = u(t) u OP. The plant is furthermore controlled by an event-triggered PI controller which is given by ẋ c (t) = x p (t k ) 1 s, x c(t k ) = x ck v(t) =.1x c (t) V cm x p(t k ) V cm for t [t k, t k+1 ). New information is forwarded to the controller by the sensor whenever the event condition x p (t k ) x p (t) = e is satisfied, where the event threshold e has been varied in the experiments. The overall non-symmetric saturation bounds u [, 12] V are artificially adapted around the operating point u OP = 5.43V according to u max = u OP + u u min = u OP u in order to obtain symmetric actuator saturation u(t) = sat(v(t)) which meets the requirements of this thesis. Similar to the event threshold e, u has also been varied in the experiments. Regions of stability. Regions of stability for different event thresholds e and different saturation limits u are depicted in Fig. 8.3. As expected, increasing the event threshold e and decreasing the saturation bound u both lead to smaller regions of stability. 63