A Novel Integral-Based Event Triggering Control for Linear Time-Invariant Systems

Transcription

1 53rd IEEE Conference on Decision and Control December 15-17, Los Angeles, California, USA A Novel Integral-Based Event Triggering Control for Linear Time-Invariant Systems Seyed Hossein Mousavi 1, Mohsen Ghodrat 2 and Horacio J. Marquez 3, IEEE Senior Member Abstract In this paper, a new integral-based triggering scheme is introduced for LTI systems. The main feature of the novel triggering condition is that does not require the derivative of Lyapunov function to be negative at the all time instants and so, is less conservative than the corresponding available conditions in the literature. Morever, a well-proved comparison with traditional triggering condition and simulation results for a satellite system are given to show the effectiveness of the proposed system. I. INTRODUCTION In this paper we study stability and design of eventtriggered control systems. Event-triggered systems have been growing in popularity over the last two decades, owing to their ability to deliver performance comparable to the classical periodic sampled-data systems (see for example, [1]) but with reduced transmission of information between plant and controller. In an event-triggered system information is exchanged between the different subsystems in a feedback loop only when a pre-established triggering condition is violated. Early work on the event-triggered control systems began in late 1990 s [2, 3]. In [2], Arzen defined a simple triggering condition (TC) based on the tracking error and converted the standard PID control algorithm to an event based form. Comparison with the corresponding time-triggered scheme confirmed that using an event based mechanism, is possible to obtain large reductions in the control execution with only minor control performance degradation. Event-triggered systems have received much attention in the recent years. One of the most significant and central ideas in eventtriggered control systems design was presented in [4]. In this paper, Tabuada proposed an input-to-state (ISS) Lyapunov function framework for TC design for a general class of nonlinear systems. Assuming that a continuous-time feedback control has been already designed, he introduced a TC to maintain the Lyapunov function decreasing along the system trajectories. Using a similar idea and following Tabuada s work, various event based schemes have been designed for various systems. In [5], Mazo and Tabuada investigated the event-based control problem over wireless sensor/actuator networks and presented a decentralized event-triggered implementation over a network of nonlinear systems. In [6], 1 S. H. Mousavi is with the Dept. of Electrical and Computer mousavi3@ualberta.ca 2 M. Ghodras with the Dept. of Electrical and Computer ghodrat@ualberta.ca 3 H. J. Marquez is with the Dept. of Electrical and Computer hmarquez@ualberta.ca Wang and Lemmon, examined event-triggered data transmission in NCS in the presence of packet loss and transmission delay. In [7], Donkers and Heemels developed the event-driven control scheme for output feedback based LTI systems and used several approaches to analyse stability and performance of the system. In [8], the authors considered a class of affine nonlinear systems and using a similar idea to [4] proposed an event based control algorithm to make the system states track a reference dynamical system. As mentioned earlier, most references in the literature (including those above listed) follow the same idea in [4] to design the TC, which as we will discuss, is somewhat conservative. In this paper, we propose a novel integralbased triggering condition (IBTC) for LTI systems that guarantees asymptotic stability of the event-based system while lessening the conservatism of [4]. This concept was recently explored by the authors in reference [9], in the more general case of nonlinear plants. In section II, a quick review on the traditional TC and the motivation for the IBTC are given. section III contains the main results. In this section the structure of the novel triggering condition is presented and a theoretical comparison with the traditional TC is provided. Simulation results for a satellite system are represented in Section IV to validate the privilege of our proposed scheme. Finally, concluding remarks are summarized in section V. A. Notation and Definitions Throughout the paper, R, R + and N denote the set of real numbers, set of positive real numbers and the set of natural numbers, respectively. Moreover, R n represents n dimensional Euclidean space and for x R n the Euclidean norm is defined as x = x x2 n. For mtarix A = [a ij ] m n the induced norm A is defined as: A = sup{ Ax x : x R n with x 0} Beside the above notation, the following definition is also needed: Definition 1: A continuous function α : [0, a) [0, ) is said to belong to class K if is strictly increasing and α(0) = 0. Is said to be of class K if a = and α(r) as r. II. PROBLEM STATEMENT Consider the following LTI system ẋ = Ax + Bu (1) /14/$ IEEE 1239

2 where x(t) R n is the state vector, u(t) R m is the control input, and A and B are system matrices of appropriate dimensions. Consider the event based control law where: u = kˆx(t) (2) ˆx(t) = x( ) t [, +1 ) (3) and T = {t 1, t 2,... } are the triggering instants generated by a TC, to be designed. The closed-loop event based system can be written in the following form: ẋ = Ax + Bk(x + e) (4) where, the measurement error is defined as the last state information sent to the controller and the current state value: e = x( ) x(t) t +1 (5) Now suppose that the state feedback gain k has been already designed such that the closed-loop system (4) is asymptotically stable and so ISS with respect to measurement error e. Therefore, there exists a Lyapunov function V = x T P x, satisfying the following properties [4]: c s x V (x) c s x (6) V x (Ax + Bk(x + e)) c s x 2 + e x. (7) where c s, c s, c s and are some positive constants. The key method for TC design, introduced in [4], is to maintain the Lyapunov function V decreasing along the system trajectories. To this aim, the following TC was proposed: e σc s x 0 < σ < 1 (8) Substituting (8) in (7), is readily obtained: V x ((A + Bk)x + ke) c s(σ 1) x 2 (9) since σ < 1, the time derivative of V is negative and so the system is stable. In this work, we propose a novel TC, improving the existing TC in the sense that demands less data communication between the plant and controller to guarantee the asymptotic stability of the system. The main idea to this goal lies in the result of the following lemma. Lemma 1: Consider the linear system (4) and let T = {t 1, t 2,..., } be an infinite sequence of triggering instants. If there is a Lyapunov function satisfying (6) and the following conditions: V (+1 ) V ( ) = i+1 V (τ)dτ < 0, (10) V ( ) > V (t) t (, +1 ] (11) then the origin is the asymptotic stable equilibrium point of the system. Proof: See Appendix A. Remark 1: The foregoing Lemma is based on the assumption that the number of triggering instants tends to infinity. To guarantee this, the following algorithm is applied along with the designed triggering mechanism. Define T s as an arbitrary time index and suppose that t j is the latest triggering instant generated by an existing triggering mechanism. A fresh data is automatically transmitted from plant to the controller at the instant t j +T s if no trigger happens in the interval [t j, t j +T s ]. See [9] for further details. Based on the lemma 1, to guarantee closed-loop stability, it is sufficient to design a TC which satisfies the conditions (10) and (11). These conditions ensure that the Lyapunov function decreases from one triggering instant to the next, regardless of the sign of V in between. Therefore, the proposed IBTC is less conservative than the existing TCs in the sense that it lets the derivative of Lyapunov function become positive in some time intervals and consequently, the time interval between consecutive event times increases in this scheme. III. INTEGRAL-BASED TRIGGERING CONDITION Consider the following proposed IBTC: e(t) x(t) dt σc s x(t) 2 dt for t (12) in which, 0 < σ < 1 is some arbitrary adjustable parameter. The next update for the control law is generated at +1, the time when the above condition is violated. i.e.: i+1 e(t) x(t) dt = σc s i+1 x(t) 2 dt (13) Note that to initiate the event based mechanism, t 1 is set to 0 and then, the next execution times are generated based on the above triggering condition. In the next theorem is shown that how the proposed IBTC ensures the asymptotic stability and existence of a minimum inter-event time for the closed-loop system. Theorem 1: Consider the linear system (1), where the state feedback gain k has been already designed such that the system is ISS with respect to e and there exists a Lyapunov function V = x T P x, satisfying equations (6) and (7). Now, suppose that the system is controlled by the event based control law (2) and the IBTC (12). Then, the following properties are guaranteed for the closed-loop event based system: (I) The origin is the asymptotic stable equilibrium point of the system. (II) The length of time intervals between consecutive triggering instants are bounded below by a positive value. Proof. (I) Integrating the inequality (7) from to t and using (12), we obtain: V (t) V ( ) (1 σ)c s x(t) 2 dt (14) Then, for 0 < σ < 1 asymptotic stability is guaranteed using Lemma

3 (II) To prove the existence of minimum inter-event time, we introduce the following auxiliary system: ż = Az + Bk(z + e ) (15) implemented by the traditional TC (8): e σc s z (16) Suppose that both systems (4) and (15) trigger at and also have the same triggering state values: z( ) = x( ) (17) Denote the next triggering time of the system (15) by t i+1. Then, from the TC (16), we have: e (t) z(t) < σc s z(t) 2 t [, t i+1) (18) e (t i+1) z(t i+1) = σc s z(t i+1) 2 (19) On the other hand, based on the IBTC (12) is clear that: i+1 e(t) x(t) dt = σc s i+1 Using equation (17) is easily found that; e (t) = e(t) z(t) = x(t) x(t) 2 dt (20) t [, t i+1) t [, t i+1) Now, substituting above equations in (18) and integrating from to t i+1 we obtain: i+1 e(t) x(t) dt < σc s i+1 x(t) 2 dt (21) Comparing the lasnequality with (20), we conclude that +1 > t i+1. Since the auxliary system has a minimum enter-event time [4], so does our proposed system with the IBTC (12). Contrary to the traditional TC, the proposed IBTC stabilizes the closed-loops system without restricting the derivative of Lyapunov function to be negative at all time instants. In the next subsection is proved that how IBTC preforms better from data transmition reduction view point, rather than the corresponding TC, proposed by [4]. A. Comparison With Traditional TC To have a fair comparison between the performance of our proposed scheme and the traditional one, we consider two distinct systems with the same dynamics, one of which is implemented by the IBTC and the other one uses the traditional TC. In this regard, consider the dynamic (15) driven by the TC (16) with the triggering instants T = {t 1, t 2,... }. On the other hand, the dynamic (4) is driven by the IBTC (12) with the triggering instants T = {t 1, t 2,... }. Here, suppose that at some triggering instants t p and t q (for some p, q N) the two systems have the same state values; i.e. z(t p) = x(t q ) (22) Since the dynamic (15) is time-invariant over the interval [t p, t p+1) in the sense that e only depends on z(t p) and z(t), without loss of generality assume t p = t q and so z(t q ) = x(t q ). Because the systems have the same dynamics and same initial conditions at t = t q, one can easily say that : e (t) = e(t) t [t q, t p+1) z(t) = x(t) t [t q, t p+1) Next, taking the same approach given in proof of Theorem 1 (II), is easily porved that t q+1 > t p+1 and so our proposed IBTC takes more time to reach its next execution time. Remark 2: In the comparison given above, is assumed that the initial values of the both systems at a triggering instant are equal. The privilege of our proposed event-based scheme over the traditional version can also be proved even in the case: z(t p) x(t q ) ( i.e. there is some difference between initial triggering instant values) For detailed proof of this, one can refer to [9]. IV. SIMULATION RESULTS In this section the efficiency of the proposed mechanism is shown through numerical simulation. Consider a satellite stabilizing problem, discussed in [11]. The satellite system is comprised of two rigid bodies, connected through a flexible link. Modelling the link as a spring with torque constant k s and viscous damping f, the motion equations are given as follows: J 1 θ1 + f( θ 1 θ 2 ) + k s (θ 1 θ 2 ) = u(t) J 2 θ2 + f( θ 2 θ 1 ) + k s (θ 2 θ 1 ) = 0 (23) where, J 1 and J 2 represent the moment of inertia of two bodies, θ 1 and θ 2 denote the yaw angle of two bodies and u(t) is the control torque. Now, define the state variables as x = [ ] θ 1 θ 2 θ1 θ2 and select J1 = J 2 = 1, f = 0.09 and k s = Then, the state space equation is expressed in the following form: ẋ = x u (24) 0 Suppose that this system has been stabilized by the following control law: u = [ ] x (25) Using the Lyapunov function V = x T P x with P =

4 Fig. 1: x 1 and x 2 trajectories of the plant, controlled by the classic state feedback controller (dashed), traditional event based controller (dotted) and the proposed event based controller (solid). Fig. 2: x 3 and x 4 trajectories of the plant, controlled by the classic state feedback controller (dashed), traditional event based controller (dotted) and the proposed event based controller (solid). the inequalities (6) and (7) are satisfied with the following parameters: c s = λ min (P ) = , c s = λ max (P ) = , c s = λ min (Q) = 1, = P Bk + k T B T P = (26) where Q = (A + Bk) T P + P (A + Bk). Now, the initial condition is chosen as x 0 = [ ] and the simulations are implemented for three different controllers: Classic continuous time state-feedback controller with the feedback law u = kx. Our proposed event based scheme with the following IBTC: e(τ) x(τ) dτ σc s x(τ) 2 dτ (27) Traditional event triggering mechanism proposed in [4] with the following TC: e(t) σc s x(t) (28) Note that σ is selected as 0.75 for both triggering conditions (27) and (28). Simulation results are shown in Figs As seen in Figs. 1 and 2, all four states for the three systems asymptotically approach to the origin with similar performances. In addition, the control signals are depicted in Figs 3 and 4, where all three signals have similar trends. However, the traditional and our proposed mechanism have transmitted 100 and 50 data samples over 15 seconds respectively. It means that the IBTC triggers 50% less than the traditional TC to generate a similar performance. This shows that the integral-based event triggering mechanism acts much more effective than the traditional scheme in data transmission reduction between the plant and controller. V. CONCLUSION A new integral-based triggering scheme was proposed for linear systems. The proposed IBTC is less conservative than Fig. 3: Control signal generated by the classic state feedback controller (dashed trajectory), traditional event based controller (dotted spikes) and the proposed event based controller (solid spikes). Fig. 4: Control signals generated by the three controllers over the time interval [2, 9]. As seen, for each two samples update, made by the traditional event based controller, averagely just one sample update is made by our proposed controller. 1242

5 the existing triggering conditions in the literature and so more effective from data transmission viewpoint. Moreover, both theoretical and numerical comparison with traditional TC were provided to show the efficiency of the proposed scheme. APPENDIX Proof of Lemma 1. Suppose that the system initial condition is x(t 1 ) at the initial time t 1. To prove the stability of the system, assume that ε V is given. Based on the equations (10) and (11), and using (6) one can easily obtain: c s x(t) c s x(t 1 ) t > t 1. (A-1) Hence, for any initial condition, satisfying x(t 1 ) δ V = c s ε V / c s, we would have: x(t) ε V t > t 1 (A-2) and so, the system is stable. In the next step the convergence of the state trajectories to the origin is proved: Condition (11) implies that for any T, there exists ξ i > 0 such that I i = {x V (x(t)) + ξ i V (x( ))} (A-3) is an invariant set containing the origin. Moreover, (10) implies that I i+1 I i i N. It follows that the intervals I i s, so defined, constitute a shrinking sequence of invariant sets, denoted by I = {I i : i N}. To prove asymptotic convergence of state trajectories, it suffices to show I i C as i (A-4) where C = {0} is the singleton containing the origin. We prove it by contradiction. Let us assume that C contains at least a point υ 0 and that V (x( )) V (υ) > 0 as i. Consider now the compact set Ω 0 = {x V (x(t)) V (x(t 1 ))} and define r > 0 large enough such that On the other hand, consider and define d > 0 such that Ω 0 B r = {x x r}. Ω f = {x V (x(t)) V (υ)} B d = {x x d} Ω f (A-5) (A-6) (A-7) (A-8) (A-9) From (A-5), we have that x( ) remains inside B r and outside the ball B d as i N. i.e. d x( ) r for i = 1, 2,... (A-10) Now define κ = max {V (x(+1)) V (x( ))}. Based d x() r on (10), we have: V (x(+1 )) V (x(t 1 )) + iκ (A-11) and since κ < 0, there is some î such that for i î the left-hand side of the above inequality is negative, which contradicts the positive definiteness of the Lyapunov function V. Hence, V (x( )) 0 as i and so, x converges asymptotically to the origin. REFERENCES [1] T. Chen and B. A. Francis, Optimal Sampled-Data Control Systems. London: Springer, [2] K. E. Arzen, A simple event-based pid controller, in Proc. of 14th IFAC World Congr., Beijing, China, Jan [3] K. J. Astrom and B.. Bernhardsson, Comparison of periodic and event based sampling for first-order stochastic systems, in Proc. IFAC World Congr, 1999, pp [4] P. Tabuada, Event-triggered real-time scheduling of stabilizing control task, IEEE Trans. Autom. Control, vol. 52, no. 9, pp , [5] M. Mazo and P. Tabuada, Decentralized eventtriggered control over wireless sensor/actuator nrtworks, IEEE Trans. Autom. Control, vol. 56, no. 10, pp , [6] X. Wang and M. D. Lemmon, Event-triggering in distributed networked control systems, IEEE Trans. Autom. Control, vol. 56, no. 3, pp , [7] W. Heemels, M. C. F. Donkers, and A. R. Teel, Periodic event-triggered control for linear systems, IEEE Trans. Autom. Control, vol. 58, no. 4, pp , [8] P. Tallapragada and N. Chopra, On event triggered trajectory tracking for control affine nonlinear systems, in CDC/ECC (Conf. Dec. Control and Eur. Control Conf), December 2011, pp [9] S. H. Mousavi, M. Ghodrat, and H. J. Marquez, A novel integral based event triggered control scheme for a general class of nonlinear systems, Submitted for Publication as Journal Paper, [10] H. Khalil, Nonlinear Systems, 3rd Edition. Englewood Cliffs, NJ: Prentice-Hall, [11] R. M. Biernacki, H. Hwang, and S. P. Battacharyya, Robust stability with structured real parameter perturbations, IEEE Trans. Autom. Control, vol. AC-32, no. 6, pp ,