This is a repository copy of Robust sampled-data implementation of PID controller. White Rose Research Online URL for this paper: http://eprints.whiterose.ac.uk/153375/ Version: Accepted Version Proceedings Paper: Selivanov, A. orcid.org/0000-0001-5075-7229 and Fridman, E. (2019) Robust sampled-data implementation of PID controller. In: 2018 IEEE Conference on Decision and Control (CDC). 2018 IEEE Conference on Decision and Control (CDC), 17-19 Dec 2018, Miami Beach, FL, USA. IEEE , pp. 932-936. ISBN 9781538613962 https://doi.org/10.1109/cdc.2018.8619030 © 2018 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other users, including reprinting/ republishing this material for advertising or promotional purposes, creating new collective works for resale or redistribution to servers or lists, or reuse of any copyrighted components of this work in other works. Reproduced in accordance with the publisher's self-archiving policy. 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Since the derivative is hard to measure directly, allows to reduce the amount of sampled control signals it is approximated using a finite difference giving rise to a used for stabilization [7], [8]. Some of the ideas presented delayed sampled-data controller. We suggest a novel method for the analysis of the resulting closed-loop system that allows to use here have been generalized in [9] to study time-delayed and only the last two measurements, while the existing results used sampled-data implementation of control depending on high- a history of measurements. This method also leads to essentially order output derivatives. larger sampling period. We show that, if the sampling period is The analysis will employ the following inequalities. small enough, then the performance of the closed-loop system Lemma 1 (Exponential Wirtinger’s inequality [10]): Let under the sampled-data PID controller is preserved close to the Rn one under the continuous-time PID controller. The maximum f : [a,b] → be an absolutely continuous function with a sampling period is obtained from LMIs derived using an square integrable first order derivative such that f(a) = 0 appropriate Lyapunov-Krasovskii functional. These LMIs allow or f(b) = 0. Then, for any α ∈ R and 0 ≤ W ∈ Rn×n, to consider systems with uncertain parameters. Finally, we b develop an event-triggering mechanism that allows to reduce 2αt T the amount of sampled control signals used for stabilization. e f (t)Wf(t) dt Za I. INTRODUCTION 4(b − a)2 b ≤ e2|α|(b−a) e2αtf˙T (t)W f˙(t) dt Proportional integral derivative (PID) controllers are ex- π2 Za tremely popular in the control engineering practice. These R controllers depend on the output derivative that can hardly be Lemma 2 (Jensen’s inequality [11]): If f : [a,b] → measured in practice. Instead, the derivative can be approxi- and ρ: [a,b] → [0, ∞) are such that the integration con- mated using the finite difference y˙ ≈ (y(t)−y(t−τ))/τ. This cerned is well-defined, then 2 gives rise to a time-delayed controller, which was studied, b b b e.g., in [1], [2] using the frequency domain approach. ρ(s)f(s) ds ≤ ρ(s) ds ρ(s)f 2(s) ds. In this paper, we study the sampled-data implementation of "Za # Za Za PID. This problem has been recently considered in [3]. One of the ideas (originated in [4], [5]) was to use the Taylor’s II. SAMPLED-DATA PID CONTROL expansion for the delayed term with the remainder in the Consider the scalar system integral form. The remainder is then compensated by an appropriate Lyapunov-Krasovskii term. The data sampling y¨(t)+ a1y˙(t)+ a2y(t)= bu(t) (1) was studied using the time-delay approach [6]. and the PID controller Here, we significantly improve the results of [3]. The key t novelty is that the sampling error is calculated for the deriva- u(t)= k¯py(t)+ k¯i y(s) ds + k¯dy˙(t). (2) tive approximation, while in [3] it was calculated for the Z0 delayed measurement. This idea allows to write the stability The controller (2) depends on the output derivative, which conditions in terms of the controller gains used in the original is hard to measure directly. Instead, the derivative can be continuous PID, while [3] used the sampling-dependent approximated by the finite-difference gains obtained after substituting the approximation into the y(t) − y(t − h) y˙(t) ≈ y (t)= , h> 0. (3) continuous PID (see Remark 3). One of the consequences is 1 h that the sampled-data implementation of PID requires to use This approximation leads to the delay-dependent control only the last two measurements, while [3] used a history of measurement whose length was increasing for a decreasing t u(t)= k¯ y(t)+ k¯ y(s) ds + k¯ y (t) sampling period. Moreover, we obtain significantly larger p i d 1 0 (4) sampling periods compared to [3] (see Example 1). The Z t stability conditions are formulated in terms of LMIs that are = kpy(t)+ ki y(s) ds + kdy(t − h), affine with respect to the system parameters. This allows to Z0 1 use them to study systems with uncertain parameters. where y(t)= y(0) for t< 0 and ¯ ¯ A. Selivanov ([email protected]) is with Royal kd kd kp = k¯p + , ki = k¯i, kd = − . (5) Institute of Technology, Sweden, and Tel Aviv University, Israel. E. Fridman h h ([email protected]) is with Tel Aviv University, Israel. Supported by Israel Science Foundation (grant No. 1128/14). 1Then y˙(t) with t ∈ [0,h) is approximated by 0 We study the sampled-data implementation of the controller Then the system (1) under the sampled-data PID control (8) (4) that is obtained using the approximations can be presented as t tk k−1 x˙ = Ax + Avv + B (κ + δ) , (9) y(s) ds ≈ y(s) ds ≈ h y(tj), y = Cx, 0 0 Z Z j=0 t ∈ [tk,tk+1), X where y(t ) − y(t ) y˙(t) ≈ y˙(t ) ≈ y (t )= k k−1 , k 1 k h 0 1 0 A = −a + bk¯ −a + bk¯ bk¯ , where h > 0 is the sampling period, t = kh, k ∈ N , 2 p 1 d i k 0 1 0 0 are the sampling instants, and y(t ) = y(t ). Substituting (10) −1 0 0 0 0 0 these approximations into (2), we obtain the sampled-data A = bk¯ 0 bk¯ , B = bk¯ , C = 1 0 0 . controller v p i d 1 0 0 0 k−1 u(t)= k¯py(tk)+ k¯ih y(tj)+ k¯dy1(tk) Theorem 1: Consider the system (1). j=0 ¯ X (i) For given sampling period h > 0, controller gains kp, k−1 (6) k¯i, k¯d, and decay rate α > 0, let there exist positive- = kpy(tk)+ kih y(tj)+ kdy(tk−1), definite matrices P,S ∈ R3×3 and nonnegative scalars j=0 2 X W , R such that Ψ ≤ 0, where Ψ = {Ψij} is the t ∈ [tk,tk+1), k ∈ N0 symmetric matrix composed from T with kp, ki, and kd defined in (5). Ψ11 = PA + A P + 2αP, Ψ12 = PAv, 2 T π We will show that the sampled-data controller (6) stabi- Ψ13 =Ψ14 = PB, Ψ15 = A G, Ψ22 = − 4 S, T T lizes the system (1) if (2) stabilizes (1) and the sampling Ψ25 = Av G, Ψ35 =Ψ45 = B G, 2 period h > 0 is small enough. Moreover, we will derive π −2αh −2αh Ψ33 = −W 4 e , Ψ44 = −Re , Ψ55 = −G, LMIs that allow to find appropriate h. αh 000 1 αh 2 2 2 010 2 G = h e S + h ( 4 R + e W ) First, we present the estimation error y˙(t) − y1(t) in a 000 h i convenient integral form. with A, Av, B, and C given in (10). Then, the sampled- Lemma 3: If y ∈ C1 and y˙ is absolutely continuous, then data PID controller (6) exponentially stabilizes the sys- y1 defined in (3) satisfies tem (1) with the decay rate α. (ii) Let there exist k¯ , k¯ , k¯ such that the PID controller (2) t t − h − s p i d y1(t)=y ˙(t)+ κ(t), κ = y¨(s) ds. (7) exponentially stabilizes the system (1) with a decay rate t−h h α′. Then, the sampled-data PID controller (6) with k , Proof: Taylor’s expansionZ with the remainder in the p k , k given by (5) exponentially stabilizes the system integral form gives i d (1) with any given decay rate α<α′ if the sampling t period h> 0 is small enough. y(t − h)= y(t) − y˙(t)h − (t − h − s)¨y(s) ds.