Event-Triggered PI Control of Time-Delay Systems with Parametric Uncertainties

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 architectures 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); Ahlén 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 integrator systems (Aström˚ 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 (Aström settings. In Najafi et al. (2013), the authors propose a ˚ and Bernhardsson (1999); Arzén (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 Léchappéet al. (2018). Event-triggered PID control for industrial automation systems 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 (Aström and Hägglund (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 BpKpCp 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. (i) (i) −2αη (i) (i) (i) (i) Φ12 = P A2 +e R2, Φ13 = P A1 , Φ14 = P A2 Theorem 6. Consider the system (10). Suppose that As- i i i R Φ( ) = (A(i) + A( ))⊤Q, Φ( ) = −e−2αη(R + R ), sumption 1 holds. Given Kp, Ki ∈ , and decay rate 15 1 22 1 2 S2n+1 (i) (i) (i) (i)⊤ α> 0, assume that there exist P, R1, R2, W1, W2 ∈ ++ , Φ23 =0, Φ24 =0, Φ25 = A2 Q, w> 0, and σ> 0, such that 2 (i) π (i) (i) (i)⊤ PB(i) wσK⊤ Φ33 = − W1, Φ34 =0, Φ35 = A1 Q, 1 4 0 wσK⊤ 2 2 (i) π (i) (i)⊤ (i)  (i) ⊤  Φ = − e−2αηW , Φ = A Q, Φ = −Q. Φ 0 wσK1 44 2 45 2 55 (i) ⊤ 4 Ψ =  0 wσK2  < 0, (11) , 2 2 2αh −2αη  (i)  and Q η R2+h e (W1+e W2) for all i =1,...,N.  QB 0    Then the system (7) with r = 0 is exponentially stable  ∗∗ ∗ ∗∗ −w 0    with decay rate α.  ∗∗ ∗ ∗∗ 0 −wσ  for all i =1,...,N , where  Proof. See Appendix A. K1 = [0 −KpCp Ki] ,K2 = [−KpCp KpCp 0] . 3. EVENT-TRIGGERED CONTROL OF Then the system (10) with r = 0 is exponentially stable TIME-DELAY SYSTEMS with decay rate α. Proof. See Appendix B. In this section, we discuss the event-triggered control introduced in Heemels and Donkers (2013); Selivanov and Remark 7. For r 6= 0, we need to apply a coordinate Fridman (2018) for a time-delay system with parametric transformationx ˜(t) = x(t) − xe where xe = −(A + A1 + −1 uncertainties. We derive a stability condition and propose A2) BRr is the equilibrium point. Note that A+A1 +A2 how to design the event-triggering condition with given is invertible when the continuous controller (i.e., h = 0) control parameters. stabilizes the system and therefore A+A1 +A2 is Hurwitz. The event condition (9) is replaced by 2 2 3.1 System model of event-triggered PI control (u(tk) − u˜(tk−1)) > σ(u(tk) − ue) where ue = (K1 + K2)xe is the steady-state input signal. Consider the system Thus, Theorem 6 can be applied if we know the exact ˆ ˆ x˙ p(t)= Apxp(t)+ Bpu˜(t − η) model Ap = Ap and Bp = Bp. Otherwise, we use ˆ ˆ ˆ ˆ whereu ˜(t) is the event-triggered control signal. We assume the prediction of A, A1, A2,BR denoted as A, A1, A2, BR. thatu ˜(t) is updated by checking the event condition The prediction matrices are given by replacing Ap,Bp ˆ ˆ (u(t ) − u˜(t ))2 > σu2(t ) (9) of A, A1, A2,BR by Ap, Bp. The event condition (9) is k k−1 k replaced by at every sampling time t , k = 0, 1,..., where σ ∈ [0, 1) k u t u t 2 > σ u t u 2 is a relative threshold. Thus, the event-triggered control ( ( k) − ˜( k−1)) ( ( k) − ê) −1 signal is given by whereu ê = −(K1 + K2)(Aˆ + Aˆ1 + Aˆ2) BˆRr is the prediction of steady-state input. In this case, the prediction u(tk), t ∈ [tk,tk+1), if (9) is true, u˜(t)= e , u u y t r u˜(tk−1), t ∈ [tk,tk+1) if (9) is false, error u ê − e leads to oscillation of ( ) around . withu ˜0 = u(t0). Define the control signal error as Using (11), we can tune the event threshold σ to give a minimum communication load satisfying a given stability v3(t) , u˜(t) − u(t) margin α. =u ˜(tk) − u(tk), t ∈ [tk,tk+1). Corollary 8. Suppose that Assumption 1 holds. Given Then the closed-loop system is given by Kp,Ki ∈ R, w > 0, and α > 0, if the semi-definite x˙(t)= Ax(t)+A1x(tk)+A2x(tk −η)+Bv3(t)+BRr (10) programming problem (SDP): where σ∗ , max σ (12a) Bp s.t. Ψ(i) < 0, i =1,...,N, (12b) B = Bˆp .  0  is feasible, then the closed-loop system (10) under the σ∗ Remark 5. Suppose that Assumption  1 holds. Then B event condition (9) with is exponentially stable with α resides in the uncertain polytope decay rate . N N (i) 4. NUMERICAL EXAMPLE B = µiB , 0 ≤ µi ≤ 1, µi =1, i=1 i=1 X X In this section, we provide numerical examples to illustrate where (i) our theoretical results. Consider a first-order linear system Bp (i) x˙ (t)= ax (t)+ bu˜(t − η), (13) B = Bˆ . p p  p  0 y(t)= xp(t), (14)   1 EBS Comm. Comm. SDS IAE SPI until t = 50 Reduction 0.5 Open-loop EBS (η = 1) 285 43.0% 3.39 0 SDS (η = 1) 500 0% 3.52 -0.5

SPI (η = 1) 500 0% 4.52 -1 EBS (η = 3) 316 36.8% 5.37 0 5 10 15 20 25 30 35 40 45 50 0.5 SDS (η = 3) 500 0% 5.43 SPI (η = 3) 500 0% +∞ 0 Open-loop – – 18.94 -0.5 Table 1. Number of communications, their reductions, and the IAE for each strategy with -1 0 5 10 15 20 25 30 35 40 45 50 η = 1 and η = 3. 1

1 EBS 0 SDS 0 5 10 15 20 25 30 35 40 45 50 SPI 0.5 Open-loop

0 x t , , -0.5 Fig. 3. Responses to the initial state ( ) = [1 0 0], t ∈ [−3, 0] (top: y(t), middle: u(t)) of the four cases -1 0 5 10 15 20 25 30 35 40 45 50 with time-delay η = 3. The below plot shows the event 0.5 generation at the controller.

0 1.5 -0.5 1 -1 EBS 0 5 10 15 20 25 30 35 40 45 50 0.5 SDS 1 SPI 0 0 5 10 15 20 25 30 35 40 45 50 0.5 0 0 5 10 15 20 25 30 35 40 45 50 0

Fig. 2. Responses to the initial state x(t) = [1, 0, 0], -0.5

t ∈ [−1, 0] (top: y(t), middle: u(t)) of the four cases -1

with time-delay η = 1: Sampled-data event-triggered 0 5 10 15 20 25 30 35 40 45 50 PI control with Smith predictor (EBS: red solid line), 1 Sampled-data PI control with Smith predictor (SDS: 0 blue dashed line), Sampled-data PI control without 0 5 10 15 20 25 30 35 40 45 50 Smith predictor (SPI: green dash-dot line), and open loop system (black dot line). The below plot shows Fig. 4. Responses to the setpoint tracking r(t) = 1, ∀t ≥ 0 the event generation at the controller. of the three cases with time-delay η = 1 of two where the system parameters take their values in a ∈ The below plot shows the event generation at the [−0.055, −0.045] and b ∈ [0.45, 0.55]. We use a plant model controller.

xˆ˙ p(t)= −0.05ˆxp(t)+0.5˜u(t − η). are shown in Fig. 2 and Fig. 3, respectively, where we assume that the unknown system parameters are given by The LMI 8 guarantees the asymptotic stability of the a = −0.048 and b =0.52. It can be found that the EBS and system (13)–(14) with the time-triggered PI control with the SDS well compensate the time-delay and the outputs Kp = 0.816, Ki = 0.293, the sampling interval h = 0.2, converge to the origin. However, the SPI is more oscillative the decay rate α =0.04 for the time-delay η ≤ 3.4. in Fig. 2 (η = 1) and does not stabilize the system in Fig. 3 (η = 3). In fact, the IAE for the EBS and the SDS Initial response. We first see the responses of two differ- are close as in Table 1, while those for the SPI is larger ent time delays η = 1 and η = 3 with reference r = 0 or diverges. The third plot in Fig. 2 and Fig. 3 show the and the initial state x(t) = [1, 0, 0], t ∈ [−3, 0]. By solving time instances of the control signal updates. We can see, SDP (12), we obtain the event thresholds σ∗ =0.245 and ∗ as well as Table 1, that the communications between the σ =0.014 for η = 1 and η = 3, respectively. The SDP can controller and the actuator are performed only 35 times be solved effectively by YALMIP toolbox (Löfberg (2004)). and 66 times until t = 50. Including the communications To evaluate the system performance, we use the Integral between the sensor and the controller, the EBS reduces of the Absolute Error (IAE) which is calculated as the communications by 43.0% and 36.8% compared to the +∞ SDS. IAE = |r − y(t)|dt. Z0 Setpoint tracking. Next, we show the responses with r = The results for three strategies: the sampled-data event- 1. The results are shown in Fig. 4. In setpoint tracking, we triggered PI control with Smith predictor (EBS), the need to apply a coordinate transformationx ˜(t)= x(t)−xe sampled-data PI control with Smith predictor (SDS), as in Remark 7. In Fig. 4, the EBS has similar response as and the conventional sampled-data PI control (SPI) are the SDS with only 30 samplings until t = 50, even though summarized in Table 1. The responses for η = 1 and η =3 there remains very small oscillation due to inexactu ê. 2 5. CONCLUSION 2 2αh ⊤ π ⊤ V˙ 1 +2αV 1 = h e x˙ (t)W x˙(t) − v (t)W v (t), W W 1 4 1 1 1 ˙ In this paper, we investigated the sampled-data implemen- VW2 +2αVW2 tation of PI control with the Smith predictor. The Smith 2 2 2αh ⊤ π −2αη ⊤ predictor was formulated as a form of a state predictor. = h e x˙ (t)W2x˙(t) − e v2 (t)W2v2(t), We derived the stability condition under the assumption 4 that the system parameters resided in uncertain polytopes. where v1(t) , x(tk) − x(t) and v2(t) , x(tk − η) − x(t − η). Furthermore, event-triggered control was introduced and For VR, by Jensen’s inequality (Fridman (2014)), we have the stability condition was derived. Numerical examples V˙R +2αVR showed that our proposed controller reduces the commu- ⊤ −2αη ⊤ nication load with slight performance degradation. Future = x (t)R1x(t) − e x (t − η)R1x(t − η) t work will consider uncertain time-delays and other types 2 ⊤ −2α(t−s) ⊤ of predictors. + η x˙ (t)R2x˙(t) − η e x˙ (s)R2x˙(s)ds Zt−η ⊤ −2αη ⊤ ≤ x (t)R1x(t) − e x (t − η)R1x(t − η) t t Appendix A. PROOF OF THEOREM 4 2 ⊤ −2αη ⊤ + η x˙ (t)R2x˙(t) − e x˙ (s)dsR2 x˙(s)ds Zt−η Zt−η ⊤ −2αη −2αη ⊤ Before presenting the proof, we introduce the following = x (t)(R1 − e R2)x(t)+e x (t)R2x(t − η) lemma. −2αη ⊤ 2 ⊤ +e x (t − η)R2x(t)+ η x˙ (t)R2x˙(t) Lemma 9. (Selivanov and Fridman (2016)) Let z :[a,b] → −2αη ⊤ Rn be an absolutely continuous function with a square − e x (t − η)(R1 + R2)x(t − η). integrable first order derivative such that z(a) = 0or Thus, Sn z(b) = 0. Then for any α> 0 and W ∈ ++, the following Φ¯ 11 Φ¯ 12 Φ¯ 13 Φ¯ 14 inequality holds: ∗ Φ¯ Φ¯ Φ¯ V˙ +2αV ≤ φ⊤ 22 23 24 b  ∗ ∗ Φ¯ 33 Φ¯ 34 e2αξz⊤(ξ)Wz(ξ)dξ ¯  ∗ ∗ ∗ Φ44 Za  ¯  2 b Φ15 2|α|(b−a) 4(b − a) 2αξ ⊤ ≤ e e z˙ (ξ)W z˙(ξ)dξ. Φ¯ 25 −1 2 + Φ¯ Φ¯ Φ¯ Φ¯ Φ¯ φ< 0 π a ¯  55 51 52 53 54  Z Φ35 ¯ Φ45 Now, we derive the stability condition of the system (7).    , ⊤ ⊤  ⊤ ⊤ ⊤  Consider the functional where φ [x (t), x (t − η), v1 (t), v2 (t)] and ⊤ Φ¯ 11 = P (A + A1) + (A + A1) P V = V0 + VR + VW1 + VW2 (A.1) −2αη where +2αP + R1 − e R2, −2αη ⊤ Φ¯ 12 = P A2 +e R2, Φ¯ 13 = P A1, Φ¯ 14 = P A2, V0 , x (t)P x(t) ⊤ −2αη t Φ¯ 15 = (A + A1) Q, Φ¯ 22 = −e (R1 + R2), 2α(s−t) ⊤ VR , e x (s)R1x(s)ds ¯ ¯ ¯ ⊤ Φ23 =0, Φ24 =0, Φ25 = A2 Q, t−η Z 2 0 t ¯ π ¯ ¯ ⊤ 2α(s−t) ⊤ Φ33 = − W1, Φ34 =0, Φ35 = A1 Q, + η e x˙ (s)R2x˙(s)dsdθ, 4 −η t+θ π2 t Z Z −2αη ⊤ Φ¯ 44 = − e W2, Φ¯ 45 = A Q, Φ¯ 55 = −Q. , 2 2αh ⊤ 2 VW1 h e x˙ (s)W1x˙(s)ds 4 tk The proof completes by Schur complements and since t Z ¯ ¯ π2 Φ= {Φℓm}, ℓ,m =1,..., 5, is affine in A, A1, and A2. − e2α(s−t)[x(s) − x(t )]⊤W [x(s) − x(t )]ds, 4 k 1 k Ztk Appendix B. PROOF OF THEOREM 6 t 2 t−η , 2 2αh ⊤ π 2α(s−t) VW2 h e x˙ (s)W2x˙(s)ds − e tk−η 4 tk−η First, note that by the event condition (9), for some Z Z 2 2 × [x(s) − x(t − η)]⊤W [x(s) − x(t − η)]ds. w ≥ 0, we have wσu (tk) − wv3 (t) ≥ 0. Introducing the k 2 k functional (A.1) gives Using Lemma 9 and t − tk ≤ h, we have VW1 ≥ 0 and ˙ ⊤ ¯ ⊤ ⊤ ⊤ V +2αV ≤ φ Φ1:4φ + x (t)P Bv3(t)+ v3 (t)B P x(t) VW2 ≥ 0. We take the derivatives of each term: ⊤ 2 2 +x ˙ (t)Qx˙(t)+ wσu (tk) − wv (t) V˙ +2αV = x⊤(t)P x˙(t) +x ˙ ⊤(t)P x(t)+2αx⊤(t)P x(t), 3 0 0 PB ⊤ ⊤ = x (t) P (A + A1)+ P (A + A1) +2αP x(t) 0 Φ¯ ⊤ ⊤ ⊤  1:4  ⊤ 2 + x (t)P A2x(t − η)+ x (t)P A1v1(t) = ψ 0 ψ +x ˙ (t)Qx˙(t)+ wσu (tk), 0 + x⊤(t)P A v (t)+ x⊤(t − η)A⊤P x(t)   2 2 2  B⊤P 000 −w  ⊤ ⊤ ⊤ ⊤   + v1 (t)A1 P x(t)+ v2 (t)A2 P x(t),  ⊤ ⊤  ⊤ ⊤ ⊤ ⊤ ¯ where ψ = [x (t), x (t−η), v1 (t), v2 (t), v3 (t)] and Φ1:4 and is the submatrix obtained by omitting the 5-th row and column vectors from Φ.¯ Since u(tk)= K1x(tk)+ K2x(tk − Léchappé, V., Moulay, E., and Plestan, F. (2018). η) and by Schur complements, we have that V˙ +2αV < 0 Prediction-based control for lti systems with uncertain if time-varying delays and partial state knowledge. Int. J. ⊤ PB wσK1 Control, 91(6), 1403–1414. ⊤ 0 wσK2 Lehmann, D. and Lunze, J. (2011). Extension and ex-  ¯ ⊤  perimental evaluation of an event-based state-feedback Φ 0 wσK1 ¯ , ⊤ < . approach. Control Engineering Practice, 19(2), 101–112. Ψ  0 wσK2  0   Lindberg, C.F. and Isaksson, A.J. (2015). Comparison of  QB 0    different sampling schemes for wireless control subject to  ∗∗∗∗∗−w 0   ∗∗∗∗∗ 0 −wσ  packet losses. In Proc. Int. Conf. Event-based Control,    ¯  Commun., Signal Process., 1–8. The proof completes since Ψ is affine in A, A1, A2, and B. Liu, K. and Fridman, E. (2012). Wirtingers inequality and Lyapunov-based sampled-data stabilization. Auto- REFERENCES matica, 48(1), 102–108. Ahlén, A., Akerberg,˚ J., Eriksson, M., Isakssonm, A.J., Löfberg, J. (2004). YALMIP: A toolbox for modeling Iwaki, T., Johansson, K.H., Knorn, S., Lindh, T., and and optimization in matlab. In Proc. IEEE Int. Symp. Sandberg, H. (2019). Toward wireless control in indus- Comput. Aided Control Syst. Design, 284–289. trial process automation: A case study at a paper mill. Majhi, S. and Atherton, D.P. (1999). Modified smith IEEE Control Syst. Mag., 39(5), 36–57. predictor and controller for processes with time delay. Arzén,˚ K.E. (1999). A simple event-based PID controller. IEE Proc. Control Theory Appl., 146(5), 359–366. In Proc. IFAC World Congr., volume 18, 423–428. Moreira, L.G., Groff, L.B., Gomes da Silva, J.M., and Aström,˚ K.J. and Bernhardsson, B. (1999). Comparison Tarbouriech, S. (2016). Event-triggered PI control for of periodic and event based sampling for first-order continuous plants with input saturation. In Proc. Amer. stochastic systems. In Proc. IFAC World Congr., vol- Control Conf., 4251–4256. ume 11, 301–306. Najafi, M., Hosseinnia, S., Sheikholeslam, F., and Kari- Aström,˚ K.J. and Hägglund, T. (2006). Advanced PID madini, M. (2013). Closed-loop control of dead time Control. International Society of Automation. systems via sequential sub-predictors. Int. J. Control, Aström,˚ K.J., Hang, C.C., and Lim, B.C. (1994). A 86(4), 599–609. new smith predictor for controlling a process with an Norgren, T., Styrud, J., Isaksson, A.J., Akerberg,˚ J., integrator and long dead-time. IEEE Trans. Autom. and Lindh, T. (2012). Industrial evaluation of process Control, 39(2), 343–345. control using non-periodic sampling. In Proc. IEEE Blevins, T., Chen, D., Nixon, M., and Wojsznis, W. (2015). Conf. Emerging Technol. Factory Autom., 1–8. Wireless Control Foundation: Continuous and Discrete Park, P., Ergen, S.C., Fischione, C., Lu, C., and Johans- Control for the Process Industry. International Society son, K.H. (2018). Wireless network design for control of Automation. systems: A survey. IEEE Commun. Surveys Tuts., Fridman, E. (2014). Introduction to time-delay systems: 20(2), 978–1013. Analysis and control. Springer. Sanz, R., Garc´ıa, P., and Albertos, P. (2018). A generalized Heemels, W.P.M.H. and Donkers, M.C.F. (2013). Model- smith predictor for unstable time-delay siso systems. based periodic event-triggered control for linear systems. ISA Trans., 72, 197–204. Automatica, 49(3), 698–711. Seborg, D.E., Mellichamp, D.A., Edgar, T.F., and Heemels, W.P.M.H., Johansson, K.H., and Tabuada, P. Doyle III, F.J. (2010). Process Dynamics and Control. (2012). An introduction to event-triggered and self- John Wiley & Sons. triggered control. In Proc. IEEE Conf. Decision Con- Selivanov, A. and Fridman, E. (2016). Observer-based trol, 3270–3285. input-to-state stabilization of networked control systems Isaksson, A.J., Harjunkoski, I., and Sand, G. (2017). The with large uncertain delays. Automatica, 74, 63–70. impact of digitalization on the future of control and Selivanov, A. and Fridman, E. (2018). Sampled-data operations. Comput. Chem. Eng., 122–129. implementation of derivative-dependent control using Iwaki, T., Fridman, E., and Johansson, K.H. (2019). artificial delays. IEEE Trans. Autom. Control, 63(10), Event-based switching for sampled-data output feed- 3594–3600. back control: Applications to cascade and feedforward Smith, O.J.M. (1959). A controller to overcome dead time. control. In IEEE Conf. Decision Control, 2592–2597. ISA J., 6, 28–33. Iwaki, T., Wu, J., and Johansson, K.H. (2018). Event- Song, J., Mok, A.K., Chen, D., Nixon, M., Blevins, T., and triggered feedforward control subject to actuator satu- Wojsznis, W. (2006). Improving PID control with un- ration for disturbance compensation. In Proc. Europ. reliable communications. In Proc. ISA EXPO Technical Control Conf., 501–506. Conf., 17–19. Kiener, G.A., Lehmann, D., and Johansson, K.H. (2014). Tiberi, U., Araújo, J., and Johansson, K.H. (2012). On Actuator saturation and anti-windup compensation in event-based PI control of first-order processes. In Proc. event-triggered control. Discr. Event Dyn. Syst., 24(2), IFAC Conf. Adv. PID Control, 448–453. 173–197.