Event-triggered PI control of time-delay ⋆ systems with parametric uncertainties Takuya Iwaki ∗ Emilia Fridman ∗∗ Karl Henrik Johansson ∗ ∗ School of Electrical Engineering and Computer Science, KTH Royal Institute of Technology, Stockholm, Sweden (e-mail: takya, kalle.kth.se) ∗∗ School of Electrical Engineering, Tel-Aviv University, Israel (e-mail: [email protected]) Abstract: This paper studies sampled-data implementation of event-triggered PI control for time-delay systems with parametric uncertainties. The systems are given by continuous- time linear systems with parameter uncertainty polytopes. We propose an event-triggered PI controller, in which the controller transmits its signal to the actuator when its relative value goes beyond a threshold. A state-space formulation of the Smith predictor is used to compensate the time-delay. An asymptotic stability condition is derived in the form of LMIs using a Lyapunov- Krasovskii functional. Numerical examples illustrate that our proposed controller reduces the communication load without performance degradation and despite plant uncertainties. Keywords: PI control, event-triggered control, robust control, sampled-data systems, networked control, time-delay systems, linear matrix inequality 1. INTRODUCTION et al. (2010)). Event-triggered control for these architec- tures has also been proposed. In Iwaki et al. (2018), event- Control of process plants using wireless sensors and actu- triggered feedforward control is discussed. Event-triggered ators is of growing interest in process automation indus- cascade control and feedforward control where the con- tries (Isaksson et al. (2017); Park et al. (2018); Ahl´en et al. trollers are switched according to the event generation is (2019)). Wireless process control offers advantages through proposed in Iwaki et al. (2019). Time-delay compensation massive sensing, flexible deployment, operation, and effi- is important for process control applications. The Smith cient maintenance. However, there remains an important predictor (Seborg et al. (2010)) is a widely-used technique problem, which is how to limit the amount of information for this purpose. The Smith predictor compensates large that needs to be exchanged over the network, since the sys- time-delay by predicting the plant output using a simple tem performance is critically affected by network-induced plant model. Since its development in Smith (1959), mod- delay, packet dropout, and sensor energy shortage. ifications were proposed to apply the predictor to integra- tor systems (Astr¨om˚ et al. (1994)) and unstable systems In this context, event-triggered control has received a lot of (Majhi and Atherton (1999); Sanz et al. (2018)). State attention from both academia and industry as a measure ˚ predictor-based controllers are also studied in more general to reduce the communication load in networks (Astr¨om settings. In Najafi et al. (2013), the authors propose a ˚ and Bernhardsson (1999); Arz´en (1999)). Various event- sequence of subpredictors to stabilize the plant with a long triggered control architectures appeared recently (see the time-delay. Uncertain time-varying delays are investigated survey in Heemels et al. (2012) and the references therein). in L´echapp´eet al. (2018). Event-triggered PID control for industrial automation sys- tems is considered in some studies. For example, stability In this paper, we investigate event-triggered PI control conditions of PI control subject to actuator saturation are for time-delay systems. We consider continuous-time lin- derived by Kiener et al. (2014); Moreira et al. (2016). Set- ear systems with parametric uncertainties. We introduce point tracking using the PIDPLUS controller (Song et al. sample-data PI control with a predictor, as a state-space (2006)) is discussed in Tiberi et al. (2012). Experimental formulation of the Smith predictor. A stability condi- validation is carried out in Kiener et al. (2014); Lehmann tion is derived using a Lyapunov-Krasovskii functional and Lunze (2011). Implementations on a real industrial via Wirtinger’s inequality (Liu and Fridman (2012)) in plant is presented in Norgren et al. (2012); Lindberg and the form of Linear Matrix Inequalities (LMIs). An event- Isaksson (2015); Blevins et al. (2015). triggered controller which updates its signal based on a relative threshold (Heemels and Donkers (2013); Selivanov In process control systems, there are several control archi- and Fridman (2018)) is considered. The event threshold tectures, such as feedforward control, cascade control and synthesis is also proposed. Numerical examples show how ˚ decoupling control (Astr¨om and H¨agglund (2006); Seborg our proposed controller reduces the communication load ⋆ This work was supported in part by the VINNOVA PiiA project without performance degradation compared to conven- “Advancing System Integration in Process Industry,” the Knut tional sampled-data PI control. and Alice Wallenberg Foundation, the Swedish Strategic Research Foundation, and the Swedish Research Council. The remainder of the paper is organized as follows. Sec- PI Event Plant tion 2 describes the sampled-data PI control system with Controller generator the predictor. An asymptotic stability conditions are de- ZOH rived. In Section 3, we propose an event-triggered PI controller and derive a stability condition. We provide nu- Predictor merical examples in Section 4. The conclusion is presented in Section 5. Controller Notation Throughout this paper, N and R are the sets of nonnegative integers and real numbers, respectively. Fig. 1. Event-triggered PI control with the Smith predic- The set of n by n positive definite (positive semi-definite) tor. The event-generator is introduced in Section 3. Rn×n Sn Sn matrices over is denoted as ++ ( +). For simplicity, Sn Sn where A(i) and B(i), i ∈ N are the vertex matrices and we write X > Y (X ≥ Y ), X, Y ∈ ++, if X − Y ∈ ++ (X − Y ∈ Sn ) and X > 0 (X ≥ 0) if X ∈ Sn (X ∈ Sn ). vector, respectively, and λi,µi ∈ [0, 1], are constants with + ++ + N N A B λi = 1 and µi = 1. Symmetric matrices of the form are written as i=1 i=1 B⊤ C Assumption 2. The system (1)–(2) with the uncertain P P A B polytopes is (A(i),B(i)) controllable and (C , A(i)) observ- with B⊤ denoting the transpose of B. p p p p ∗ C able for all i =1,...,N. , ⊤ ⊤ ⊤ ⊤ 2. TIME-TRIGGERED PI CONTROL OF By augmenting the state x(t) [xp (t + η), xˆp (t), xc (t)] , TIME-DELAY SYSTEMS we have the following closed-loop system description x˙(t)= Ax(t)+ A1x(tk) In this paper, we consider a time-invariant linear plant + A2x(tk − η)+ BRr, t ∈ [tk,tk+1) (7) with a process time-delay. The plant is controlled by a sampled-data PI controller with the Smith predictor with (Fig. 1). In this section, we introduce the plant with Ap 0 0 0 −BpKpCp BpKi uncertain parameters, the predictor, and the sampled-data A = 0 00 , A1 = 0 Aˆp − BˆpKpCp BˆpKi , PI controller. A stability condition for the closed-loop is " 0 00# 0 −Cp 0 derived. −BpKpCp BpKpCp 0 BpKp A = −Bˆ K C Bˆ K C 0 , BR = Bˆ K . 2.1 System description 2 p p p p p p p p −Cp Cp 0 1 Consider a plant with a constant process time-delay given Remark 3. Suppose that Assumption 1 holds. Then the by matrix A, A1, A2, and BR reside in the uncertain polytope (i) x˙ p(t)= Apxp(t)+ Bpu(t − η), (1) A = λiA , 0 ≤ λi ≤ 1, λi =1, y(t)= C x (t), (2) i∈N i∈N p p X X x t Rn u t R y t R (i) where p( ) ∈ , ( ) ∈ , and ( ) ∈ are the state, A1 = µiA1 , 0 ≤ µi ≤ 1, µi =1, input, and output, respectively, and the constant η > 0, i∈N i∈N a process time-delay. We assume that the sensor samples X X A µ A(i), µ , µ , and transmits its measurement every h time interval. Let 2 = i 2 0 ≤ i ≤ 1 i =1 i∈N i∈N tk, k = 0, 1, 2,..., be the time of transmission k of the X X (i) sensor, i.e., tk+1 − tk = h for all t > 0. A sampled-data BR = µiBr , 0 ≤ µi ≤ 1, µi =1, implementation of a predictor, which updates its state i i X∈N X∈N every h time interval, is given by where xˆ˙ (t)= Aˆ xˆ (t )+ Bˆ u(t) t ∈ [t ,t ), (3) (i) (i) (i) p p p k p k k+1 Ap 0 0 0 −Bp KpCp Bp Ki (i) (i) yˆ(t)= Cpxˆp(tk), (4) A = 0 00 , A1 = 0 Aˆp − BˆpKpCp BˆpKi , n wherex ˆp(t) ∈ R andy ˆ(t) ∈ R are the predictions of the 0 00 0 −Cp 0 plant state and the output. A PI controller is given by (i) (i) (i) −Bp KpCp Bp KpCp 0 Bp Kp (i) (i) x˙ c(t)= r − e(tk) − yˆ(tk), t ∈ [tk,tk+1), (5) A2 = −Bˆ K C Bˆ K C 0 , BR = Bˆ K . p p p p p p p p u(t)= Kixc(tk)+ Kp(r − e(tk) − yˆ(tk)), (6) −Cp Cp 0 1 where xc(t) ∈ R is the controller state, r ∈ R the constant reference signal, e(t) , y(t) − yˆ(t − η) the prediction error. 2.2 Stability condition of time-triggered PI control We make the following assumptions on the uncertainty of Now, we derive the stability condition of the system (7). the plant. Theorem 4. Consider the system (7). Suppose that As- R Assumption 1. The system matrix A and the vector B sumption 1 holds. Given Kp, Ki ∈ , and decay rate p p S2n+1 reside in the uncertain polytopes α> 0, assume that there exist P, R1, R2, W1, W2 ∈ ++ , N N such that (i) (i) (i) (i)⊤ (i) Ap = λiAp , Bp = µiBp , Φ =Φ = {Φℓm} < 0, ℓ,m =1,..., 5, (8) i=1 i=1 X X where (i) (i) (i) (i) (i) ⊤ 3.2 Stability condition of event-triggered PI control Φ11 = P (A + A1 ) + (A + A1 ) P αP R −2αηR , +2 + 1 − e 2 We have the following stability condition.