Dynamic reset output feedback with guaranteed convergence rate ⁎

11th IFAC Symposium on Nonlinear Control Systems 11th IFAC Symposium on Nonlinear Control Systems Vienna, Austria, Sept. 4-6, 2019 11th IFAC Symposium on Nonlinear Control Systems Vienna, Austria, Sept. 4-6, 2019 Available online at www.sciencedirect.com 11th IFAC Symposium on Nonlinear Control Systems Vienna, Austria, Sept. 4-6, 2019 Vienna, Austria, Sept. 4-6, 2019

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IFAC PapersOnLine 52-16 (2019) 102–107

Dynamic Dynamic reset reset output output feedback feedback with with Dynamic reset output feedback with ⋆⋆ guaranteed rate Dynamic reset convergence output feedback with guaranteed convergence rate guaranteed convergence rate ⋆⋆ guaranteed convergence rate∗∗,∗∗∗ Francesco Ferrante ∗∗ and Luca Zaccarian ∗∗,∗∗∗

Francesco Ferrante ∗∗∗ and Luca Zaccarian ∗∗,∗∗∗ ∗∗,∗∗∗ Francesco Ferrante ∗ and Luca Zaccarian ∗∗,∗∗∗ ∗∗,∗∗∗ Francesco Ferrante and Luca Zaccarian ∗ Grenoble Alpes, CNRS, GIPSA-lab, F-38000 Grenoble, France ∗ Univ. Univ. Grenoble Alpes, CNRS, GIPSA-lab, F-38000 Grenoble, France ∗ ∗ Univ. Grenoble Alpes,[email protected]). CNRS, GIPSA-lab, F-38000 Grenoble, France (e-mail: (e-mail: [email protected]). ∗ ∗∗ Univ. Grenoble Alpes, CNRS, GIPSA-lab, F-38000 Grenoble, France Universit´ ∗∗ LAAS-CNRS, (e-mail: [email protected]). Universit´ee de de Toulouse, Toulouse, France. France. ∗∗ LAAS-CNRS, ∗∗∗ (e-mail: [email protected]). ∗∗ di Industriale, Trento, ∗∗∗ Dipartimento e deUniversity Toulouse, of France. di Ing. Ing. Universit´ Industriale, Trento, Italy Italy ∗∗ LAAS-CNRS, ∗∗∗ Dipartimento LAAS-CNRS, Universit´ e deUniversity Toulouse, of France. ∗∗∗ Dipartimento di Ing. Industriale, University of Trento, Italy (e-mail: [email protected]) (e-mail: [email protected]) ∗∗∗ Dipartimento di Ing. Industriale, University of Trento, Italy (e-mail: [email protected]) (e-mail: [email protected]) Abstract: Abstract: We We propose propose a a new new output output feedback feedback reset reset control control architecture architecture for for linear linear continuouscontinuoustime plants, where the dwell-time parameter corresponds to a quantified interval of time Abstract: We propose a new output feedback reset control architecture for linear time plants,We where the dwell-time parameter corresponds to aarchitecture quantified interval of continuoustime where where Abstract: propose a new output feedback reset control for linear continuousthe controller state and the plant input are freezed, thus providing a constant plant input. For time plants, where the dwell-time parameter corresponds to a quantified interval of time where the controller state and the plant input are freezed, thus providing a constant plant input. For time plants, where the dwell-time parameter corresponds to a quantified interval of time where this new architecture we first prove a stability and performance analysis test, based on certain the controller state and the plant input are freezed, thus providing a constant plant input. For this new architecture we first prove a stability and performance analysis test, based on certain the controller state and the plant input are freezed, thus providing a using constant plant input. For this new architecture we first prove a stability and performance analysis test, based on certain strict Lyapunov conditions, and show that these reduce to LMIs when quadratic Lyapunov strict Lyapunov conditions, and showathat theseand reduce to LMIs when usingtest, quadratic Lyapunov this new architecture we first prove stability performance analysis based on certain certificates. Then we follow typical LMI-based characterizations of plant-order linear output strict Lyapunov conditions, and show that these reduce to LMIs when using quadratic Lyapunov certificates. Then we followand typical LMI-based characterizations of using plant-order linear output strict Lyapunov conditions, show that these reduce to LMIs when quadratic Lyapunov feedback controller designs providing an optimality-based procedure for the design of certificates. Then we followthus typical LMI-based characterizations of plant-order linear output feedback controller designs thus providing an optimality-based procedure for the design of the the certificates. Then we follow typical LMI-based characterizations of plant-order linear output controller parameters involving a set of LMIs coupled with a line search of a few parameters. feedback controller designs thus providing an optimality-based procedure for the design of the controller parameters involving a set of LMIs coupled with a line search of a few parameters. feedback controller designs thus aproviding an optimality-based procedure forathe of the controller parameters involving set of LMIs coupled with a line search of fewdesign parameters. The proposed construction is on example. The proposed construction is illustrated illustrated on aa numerical numerical example. controller parameters involving a set of LMIs coupled with a line search of a few parameters. The proposed construction is illustrated on a numerical example. © 2019, IFAC (International of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. The proposed constructionFederation is illustrated on a numerical example. Keywords: Reset Reset control control systems, systems, hybrid hybrid systems, systems, stabilization, stabilization, LMIs. LMIs. Keywords: Keywords: Reset control systems, hybrid systems, stabilization, LMIs. Keywords: Reset control systems, hybrid systems, stabilization, LMIs. 1. INTRODUCTION INTRODUCTION are 1. are investigated, investigated, Tarbouriech Tarbouriech et et al. al. (2011) (2011) where where resets resets 1. INTRODUCTION addressed in a context with saturation, Zhao and are investigated, Tarbouriech et al. (2011) where resets are addressed in a context with saturation, Zhao and 1. INTRODUCTION are investigated, Tarbouriech et first-order al.saturation, (2011) reset where resets are addressed in a context with Zhao and Hua (2017) where a generalized element Reset controllers were proposed for the first time more (2017) where a generalized first-order reset element Reset controllers were proposed for the first time more Hua are addressed in aa proposed context with saturation, Zhao and Hua (2017) where generalized first-order reset element (GFORE) has been and characterized, Heemels Reset controllers were proposed for the first time more than 50 years ago by Clegg (1958), with the aim at prohas beena proposed andfirst-order characterized, Heemels than years agowere by Clegg (1958), aim pro- (GFORE) Hua (2017) where generalized reset element Reset50 controllers proposed forwith the the first timeat more et al. (2016) where a lifting approach is used for the case than 50 years ago by Clegg (1958), with the aim at providing more flexibility in linear controller designs and at (GFORE) has been proposed and characterized, Heemels al. (2016)has where a proposed lifting approach is used for theHeemels case of of viding flexibility in linear controller designs andproat et (GFORE) been and characterized, than 50more years ago byfundamental Clegg (1958), with the aim at periodic resets, Zhao et al. (2019) that especially focuses viding more flexibility in linear controller designs and at potentially removing performance limitations et al. (2016) where a lifting approach is used for the case of periodic resets, Zhao et al. (2019) that especially focuses potentially removing fundamental performance limitations et al. (2016) where a lifting is used for limitations the focuses case of viding flexibility in linear controller andand at on periodic resets, Zhao et al. approach (2019) that especially the of the potentially removing fundamental performance limitations of linearmore controllers. After Horowitz’s worksdesigns Horowitz the goal goal of characterizing characterizing the performance performance limitations of linear controllers. After Horowitz’s works Horowitz and on periodic resets, Zhao by et al. (2019) thatand especially focuses potentially removing fundamental performance limitations that can be overcome reset control, van Loon of linear controllers. After Horowitz’s works Horowitz and Rosenbaum (1975), which generalized the corresponding on the goal of characterizing the performance limitations that cangoal be overcome by reset the control, and van limitations Loon et et al. al. Rosenbaum (1975), After which Horowitz’s generalizedworks the corresponding on the offrequency-domain characterizing performance of linear controllers. Horowitz and (2017) where tools for stability analysis Rosenbaum (1975), which generalized the corresponding idea from mere reset integrators to first order reset elethat can be overcome by reset control, and van Loon et al. (2017) where frequency-domain tools for stability analysis idea from mere reset integrators to first order reset elecan be overcome by reset control, and van Loon et al. Rosenbaum (1975), theorder corresponding of reset control system. action resets idea from mere resetwhich integrators to first ele- that ments (FORE), research hasgeneralized been quiescent quiescent until reset the early early (2017) frequency-domain tools for stability of resetwhere control system. The The combined combined action of ofanalysis resets ments (FORE), research has been until the (2017) where frequency-domain tools for stability analysis idea from mere reset integrators to first order reset eleof reset control system. The combined action of resets and time delays has been extensively studied in Dav´ o ments when (FORE), research has been quiescent the early 2000, when Hollot and co-authors co-authors produced aa number of timecontrol delays system. has been extensively studied in resets Dav´ o 2000, Hollot and produceduntil number of and of reset The combined action of ments when (FORE), research hasBeker been et quiescent until the early et al. (2017) and references therein. Parallel to the above2000, Hollot and co-authors produced a number of relevant results (see, e.g., al. (2001, 2004) and and time delays has been extensively studied in Dav´ o et al. (2017) and references therein. Parallel to the aboverelevant results (see, e.g., Beker et al. (2001, 2004) and time delays has been extensively studied inaboveDav´ o 2000, when Hollot and co-authors produced a(2001) number of and mentioned works, many relevant and successful industrial relevant results (see, e.g., Beker et al. (2001, 2004) and references therein). In particular Beker et al. first et al. (2017) and references therein. Parallel to the mentioned works, many relevant andParallel successful industrial references therein). Ine.g., particular Beker et al. (2001) first et al. (2017) and references therein. to the aboverelevant results (see, Beker et al. (2001, 2004) and applications of control can found in literature references therein). In that particular et al.may (2001) first mentioned rigorously established that reset Beker controllers may achieve works, many relevant and successful applications of reset reset control can be be found in the the industrial literature rigorously established reset controllers achieve mentioned works, relevant and successful industrial references therein). In(inthat particular Beker et al.may (2001) first applications of reset control can be found in n the literature (just to mention aa many few, see Carrasco and Ba˜ os (2012); Li rigorously established reset controllers achieve design specifications terms of overshoot) that are im(just to mention few, see Carrasco and Ba˜ n os (2012); Li design specifications (in terms of overshoot) that are imapplications of reset control can be found in the literature rigorously established that reset controllers may achieve et al. (2011); Panni et al. (2014) and references therein). design specifications (in terms of overshoot) that are impossible to achieve by any linear controller. (just to mention a few, see Carrasco and Ba˜ n os (2012); Li et al. (2011); Panni et al. (2014) and references therein). possible to achieve by any linear controller. to mention a few, see Carrasco and Ba˜ nos (2012); Li design specifications terms overshoot) that are im- (just A more comprehensive overview of these methods can be possible to achieve by(in any linearofcontroller. et al. (2011); Panni et al. (2014) and references therein). A more comprehensive overview ofand these methodstherein). can be More recently, reset controller were addressed using the et al. (2011); Panni et al. (2014) references possible to achieve by any linear controller. found the Ba˜ n Barreiro (2011) More recently, reset controller were addressed using the A morein overview of these methods canand be incomprehensive the monograph monograph Ba˜ nos os and and Barreiro (2011) More recently, controller were addressed usingthus the found hybrid systems reset framework of Goebel Goebel et al. al. (2012), (2012), thus A more overview ofal.these methods canand be found incomprehensive the monograph Ba˜ noset and Barreiro (2011) and the recent survey paper Prieur (2019). hybrid systems framework of et More recently, reset controller were addressed using the the recent survey paper Prieur et al. (2019). hybrid systems framework conditions of Goebel for et L al.2 stability (2012), thus providing Lyapunov-based conditions for L and found in the monograph Ba˜ n os and Barreiro (2011) and providing Lyapunov-based stability and the recent survey paper Prieur et al. (2019). 2 (2012), thus hybrid systems framework ofsystems Goebel Neˇ et al. In this paper we address aa specific generalization providing Lyapunov-based conditions exponential stability i´ al. stability and the recent survey Prieur et al. (2019). this paper wepaper address specific generalization the the exponential stability of of reset reset systems for NeˇssL i´cc22 et et al. (2008); (2008); providing Lyapunov-based conditions for L stability and In In this mechanism, paper we address aaa specific generalization the 2 exponentially FORE where plant-order controller is exponential stability of reset systems Neˇ s i´ c et al. (2008); Zaccarian et al. (2005) possibly including an FORE mechanism, where plant-order controller is Zaccarian et al. (2005) possibly including an exponentially In this paper we address a specific generalization the exponential stability of reset systems Neˇ san i´c exponentially et al. a(2008); equipped with a reset mechanism. This generalization Zaccarian et al. (2005) possibly including unstable FORE. The suggestive idea of stabilizing plant FORE mechanism, where a plant-order controller is equipped with a reset mechanism. This generalization unstable FORE. The suggestive idea of stabilizing a plant FORE mechanism, where a plant-order controller is Zaccarian et loops al. (2005) possibly including an exponentially stemmed from some initial ideas reported in Prieur et al. using closed whose continuous-time dynamics is exequipped with a reset mechanism. This generalization unstable FORE. The suggestive idea of stabilizing a plant from some initial ideas reported ingeneralization Prieur et al. using closed loopsThe whose continuous-time dynamicsa is ex- stemmed equipped with a reset mechanism. This unstable FORE. suggestive idea of stabilizing plant stemmed from some initial ideas reported in Prieur et al. and further developed in Fichera al. ponentially (and which by using closedunstable loops whose dynamics is ex- (2013) (2013) and further developed inreported Fichera inet etPrieur al. (2016), (2016), ponentially unstable (and continuous-time which is is stabilized stabilized by the the resets) resets) stemmed from some initial ideas et al. using closed loops whose continuous-time dynamics isetex(2013) further developed in Fichera etresets al. (2016), where aaand specific choice of the region where should ponentially unstable (and which is stabilized by the resets) has been then pursued in subsequent works like Neˇ s i´ c al. where specific choice of the region where resets should has been then pursued in subsequent works like Neˇ s i´ c et al. (2013) and further developed in Fichera et al. (2016), ponentially unstable (and which is stabilized by Neˇ thesi´ resets) be enforced (the so-called jump set) allows for establishing has been then pursued in subsequent works like c et al. (2011) and shown to be experimentally successful, e.g., in where a specific choice of the region where resets should enforced (the so-called set) allows forresets establishing (2011) and shown to beinexperimentally successful, e.g., in be a specific choice ofjump the region where should has been then pursued subsequent works like Neˇ si´ c et al. designing easily (through some LMI conditions with Panni et al. (2014). these works, the scientific be enforced (the so-called jump set) allows for establishing (2011) and shown toParalleling be experimentally successful, e.g., in where designing easily (through some LMI conditions with a a Panni et al. (2014). Paralleling these works, the scientific be enforced (the so-called jump set) allows for establishing (2011) and shown to be experimentally successful, e.g., in designing easily (through some LMI conditions with a line search) all the controller matrices (comprising how community addressed in multiple ways the goal of Panni et al. has (2014). Paralleling these works, the scientific line search) all the controller matrices (comprising how community has addressed in multiple ways the goal of designing easily (through some LMI conditions with a Panni et al. has (2014). Paralleling these works, the scientific line search) all the controller matrices (comprising how the controller state should continuously evolve, how and community addressed in multiple ways the goal of generalizing the concept of reset systems to broader classes controller should continuously evolve, how how and generalizing the concept of reset systems ways to broader classes line search) allstate the controller matrices (comprising community has addressed in multiple the goal of the when it should be reset). The optimality-based design of controllers reaching beyond classical control solutions. the controller state should continuously evolve, how and generalizing the concept of reset systems to broader classes when it should be should reset). continuously The optimality-based design of controllersthe reaching beyond control solutions. the controller state evolve, how and generalizing concept of resetclassical systems to broader classes also allowed guaranteeing stability and L Some key works with relevant references can be found when it should be reset).useful The optimality-based design 2 perforof controllers reaching beyond classical control solutions. also allowed guaranteeing useful stability and L perforSome key works with relevant references can be found 2 when it should be reset). The optimality-based design of controllers reaching beyond classical control solutions. also allowed guaranteeing useful stability and L performance properties of the reset closed loop. The problem in Aangenent et (2010) where L Some key works with relevant references can be found mance properties of the reset closed loop. The 22problem 2 properties in Aangenent et al. al. (2010) where L22 and and H H 2 properties allowed guaranteeing useful stability L2 perforSome key works with relevant references can be found also with these works checking whether resets should in Aangenent et al. (2010) where L22 and H mance properties of that the reset closed loop.and The 2 properties 2 with these works is is that checking whether resetsproblem should ⋆ mance properties of the reset closed loop. The problem in Aangenent et al. (2010) where L and H properties funded in part by ANR via project HANDY, number 2 2 be enforced (namely whether the controller state ⋆ Research with these works is that checking whether resets should Research funded in part by ANR via project HANDY, number be enforced (namely whether the controller state should with these works is that checking whether resets should ⋆ ANR-18-CE40-0010. Research funded in part by ANR via project HANDY, number be enforced (namely whether the controller state should ANR-18-CE40-0010. ⋆ Research funded in part by ANR via project HANDY, number be enforced (namely whether the controller state should ANR-18-CE40-0010.

ANR-18-CE40-0010. 2405-8963 © © 2019 2019, IFAC IFAC (International Federation of Automatic Control) Copyright 134 Hosting by Elsevier Ltd. All rights reserved. Copyright 2019 IFAC 134 Control. Peer review© under responsibility of International Federation of Automatic Copyright © 2019 IFAC 134 10.1016/j.ifacol.2019.11.763 Copyright © 2019 IFAC 134

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be reset) required knowledge of the plant state, thereby destroying the “output feedback” nature of the generalized H∞ design. This problem had been partially addressed in Fichera et al. (2013) using Luenberger observers to estimate online the unavailable portion of the state. Follow-up works in Yuan and Wu (2014) and Satoh (2011) discussed two possible (and different) LMI-based designs of a full reset output feedback control, providing relevant research direction. However, due to certain subtle aspects discussed in our Remark 6, those designs are not viable and prone to producing ill-posed conditions. This paper provides a further advancement in the direction of reset output feedback reset control analysis and design by relying on the novel idea of holding the plant input constant during the dwell-time that must be inevitably enforced in output feedback reset control (see the discussions in (Prieur et al., 2019, Ch. 4)). To this end, we propose a novel reset controller architecture in Section 2, we provide LMI-based conditions for stability analysis and guaranteed convergence rate in Section 3, we provide a controller design technique in Section 4 and discuss a simulation example in Section 5. Conclusions are drawn in Section 6. Due to space constraints, the proofs of Proposition 1, Theorems 1 and 2 are here omitted. Notation: The symbols R and R≥0 denote, respectively, the set of real numbers and the set of nonnegative real numbers. The symbol N≥0 represents the set of nonnegative integers. The symbol Rn×m represents the set of n × m real matrices, while Sn (Sn+ ) stands for the set of real n × n symmetric (positive definite) matrices. For a matrix A ∈ Rn×m , AT denotes the transpose of A, and when n = m, He(A) = A + AT . For a vector x ∈ Rn , |x| denotes the Euclidean norm. Given two vectors x, y, we denote by (x, y) = [xT yT ]T . For a symmetric matrix A, positive definiteness (negative definiteness) and positive semidefiniteness (negative semidefiniteness) are denoted, respectively, by A ≻ 0 (A ≺ 0) and A � 0 (A � 0). Given a symmetric matrix A, λmax (A) and λmin (A) stand, respectively, for the largest and the smallest eigenvalue of A. Given a vector x ∈ Rn and a closed set A, the distance of x to A is defined as |x|A = inf y∈A |x − y|. Given a set S, we denote by S the closure of S. For any s ∈ R, the function dz : R → R is defined as dz(s) = 0 if |s| ≤ 1 and dz(s) = sign(s)(|s| − 1) otherwise. Given a hybrid signal u, we denote by domt u := {t ∈ R≥0 : ∃j ∈ N≥0 s.t. (t, j) ∈ dom u}. 2. PROBLEM STATEMENT

103

We consider the following dwell-time reset output feedback controller architecture   x˙ c = (1 − q)(Ac xc + Bc y)  v˙ = −(1 − q)λv � � [ xyc ]T M [ xyc ] ≤ 0∨τ ∈ [0, ρ] τ   τ˙ = 1 − dz ρ  +  xc = Kc xc + Gc y T v + = Kv xc + Gv y [ xyc ] M [ xyc ] ≥ 0 ∧ τ ∈ [ρ, 2ρ]  + τ =0 u = (Cc xc + Dc y)(1 − q) + vq 1 q = (1 − sign(τ − ρ)) 2

(2) where (xc , τ, v) ∈ Rnc × [0, 2ρ] × Rnu is the controller state, and λ, ρ ∈ R>0 , Ac , Bc , Cc , Dc , Gc , Kc , Gv , Kv , M = M T have appropriate dimensions, all of them being design parameters of the controller. Controller (2) comprises a continuous-time dynamics always enforced when timer τ is smaller than ρ, and a jump dynamics that is enabled as long as τ ≥ ρ and the accessible quantities [ xyc ] satisfy the jump condition T [ xyc ] M [ xyc ] ≥ 0. As such, one clearly sees that parameter ρ structurally enforces a minimum dwell-time between consecutive jumps of the controller state, and matrix M = M T is a design parameter, typically corresponding to a sign indefinite matrix, so that both flowing and jumping is possible, depending on the value of the accessible variables (xc , y). By taking x = (xp , xc , v) ∈ Rnp +nc +nu , the interconnection of controller (2) with plant (1) can be represented by the following hybrid system with state ζ := (x, τ )   x˙ = A(τ )x � � �x ≤ 0 ∨ τ ∈ [0, ρ] τ xT M  τ˙ = 1 − dz ρ (3a) � + x = Gx �x ≥ 0 ∧ τ ∈ [ρ, 2ρ] xT M τ+ = 0 where (1 − sign(τ − ρ))A1 (1 + sign(τ − ρ))A0 + τ �→ A(τ ) := 2 � � 2 �T � � � I 0 0 0 0 C C 00 p � := � := C �TM C G := Gc Cp Kc 0 , M M p 0 I 0 0 I 0 Gv Cp Kv 0 � �� � (3b) � C � � � � Ap + Bp Dc Cp Bp Cc 0 Ap 0 Bp Bc Cp Ac 0 , A1 := 0 0 0 A0 := 0 0 −λI 0 0 0

By denoting

2.1 Plant-controller architecture Consider the following plant x˙ p = Ap xp + Bp u (1) y = Cp xp where xp ∈ Rnp is the plant state, u ∈ Rnu is the control input, and y ∈ Rny is the measured output. Remark 1. In this paper we assume that the plant is strictly proper from the control input u to the output y. This assumption does not lead to any loss of generality since any linear plant can be put into this form via an output transformation. 135

�x ≤ 0} ×[0, 2ρ])∪ C := ({x ∈ Rnp +nc +nu : xT M �� � � F

(Rnp +nc +nu × [0, ρ])

D := (Rnp +nc +nu \ F ) × [ρ, 2ρ]   � � A(τ � )x � Gx   τ C ∋ ζ �→ f (ζ) := , D ∋ ζ �→ g(ζ) := 0 1 − dz ρ (4a) system (3) can be rewritten in a more compact form as follows

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�

ζ˙ = f (ζ), ζ ∈ C (4b) ζ + = g(ζ), ζ ∈ D Remark 2. Controller (2) is a hybrid controller in sense of the monograph Goebel et al. (2012) meaning that the solutions to (2) (and (4)) are maps defined on twodimensional (hybrid) time domains. The goal of this paper is to characterize exponential stability of the x substate of the closed loop (3), uniformly in τ . The fact that τ evolves in the compact set [0, 2ρ] simplifies this task and allows stating it as a suitable stability property for the compact set (attractor) A := {0} × [0, 2ρ] ⊂ Rnp +nc +nu × [0, 2ρ] (5) Due to the above mentioned dwell-time property, complete solutions (namely solutions defined over an unbounded domain), are guaranteed to have an unbounded domain in the ordinary time direction t; see Proposition 1 for more details about the properties of the solutions to (4). It makes therefore sense to characterize exponential convergence to A by specifically concentrating on the flowing feature of solutions. In particular, following Fichera et al. (2016), in this paper we consider the following notion of exponential stability Definition 1. (Global t-exponential stability). Given system (4) and λ ∈ R>0 . A closed set A ⊂ Rn is said to be globally t-exponentially stable for (4) with t-decay rate α if there exits a positive real number κ such that for any maximal solution φ to (4), supt dom φ = ∞ and |φ(t, j)|A ≤ κe−αt |φ(0, 0)|A ∀(t, j) ∈ dom φ (6) ✸ Remark 3. By virtue of the dwell time property enjoyed by solutions to (4) wherein jumps are forbidden unless ρ ordinary time has elapsed since the previous jump, bound (6) can be easily converted in a hybrid class KL bound involving both t and j as characterized in (Goebel et al., 2012, Thm 3.40). In particular, the dwell time property states that for any solution φ to (4), (t, j) ∈ dom φ implies t ≥ ρj − 1, which allows proving e−αt = αρ αρ α α e− 1+ρ t e− 1+ρ t ≤ e 1+ρ e− 1+ρ (t+j) , namely the following version of (6) involving the elapsed hybrid time t + j: ¯ ¯e−α(t+j) |φ(0, 0)|A |φ(t, j)|A ≤ κ α 1+ρ

∀(t, j) ∈ dom φ

αρ 1+ρ .

κ and α ¯ := The standard hybrid class where κ ¯ := e KL bound given above, together with the well posedness proven in the next section is key to ensuring that the stated stability property enjoys the desirable robustness features well characterized in (Goebel et al., 2012, Ch. 7). 2.2 Structural properties In this subsection, we provide some preliminary results characterizing the dynamics of (3). The first result pertains to the well-posednesness of the closed-loop system. Specifically, due to the definition of τ �→ A(τ ), the flow map of (3) is discontinuous. This prevents hybrid system (3) from fulfilling the hybrid basic assumptions in (Goebel et al., 2012, As. 6.5). However, we show below that, despite the discontinuity of the flow map, system (3) is well-posed. Lemma 1. Hybrid system (3) (equivalently, (4)) is wellposed. 136

Proof. To prove the above result, let us consider the Krasovskii regularization of (4), which is given as follows � ζ˙ ∈ F� (ζ), ζ ∈ C (7) ζ + = g(ζ), ζ ∈ D where C, D, g are defined in (4), and for all ζ ∈ C 

 1 1 (1 + Sign(τ − ρ))A0 + (1 − Sign(τ − ρ))A1  2 �2 � F� (ζ) :=    τ 1 − dz ρ

with Sign : R ⇒ [−1, 1] defined as � sign(h) if h �= 0 Sign(h) := [−1, 1] elsewhere

From (Goebel et al., 2012, Theorem 6.31), the Krasovskii regularization (7) is well posed. The only difference between (4) and (7) is in the flow map when τ = ρ. Since τ = ρ implies τ˙ = 1, then flowing solutions to (4) (satisfying ζ˙ = f (ζ) almost everywhere in the flowing domain) coincide with flowing solutions to (7), because they transit through the set τ = ρ. Since the rest of the dynamics coincide, then solutions to (4) and (7) coincide and also (4) (equivalently, (3)) is well posed. � The result given next provides indications about the existence and the properties of the solutions to (3). Proposition 1. The following properties hold: (i) For any ξ ∈ C ∪ D, there exists a nontrivial solution to (3); (ii) let φ be a maximal solution to (3), then φ is complete and in particular supt dom φ = ∞. � 3. LYAPUNOV-BASED STABILITY ANALYSIS We state a general result providing sufficient conditions for global t-exponential stability of the compact set A in (5) for system (3). The results that we present hinge upon the following property. Property 1. Consider system (3). There exist a continuously differentiable function V : Rnp +nc +nu → R≥0 and positive real numbers ω1 , ω2 , χ, α, such that ω1 |x|2χ ≤ V (x) ≤ ω2 |x|2χ ∀x ∈ Rnp +nc +nu (8) ∀x ∈ F (9) �∇V (x), A0 x� ≤ −2αV (x) V (eA1 ρ Gx) ≤ e−2αρ V (x)

∀x ∈ Rnp +nc +nu \ F . (10) △ Theorem 1. Let Property 1 hold. Then the set A in (5) is globally t-exponentially stable for (3) with t-decay rate α/χ. � Remark 4. Notice that hybrid system (3) is well-posed in the sense of (Goebel et al., 2012, Definition 6.2). Among other things, this implies that nominal GAS of the set A (established in Theorem 1) is structurally robust with respect to small perturbations, in the sense of (Goebel et al., 2012, Corollary 7.23). This makes the use of our results appealing in practice. The applicability of Theorem 1 requires one to search for a function V fulfilling Property 1. This is in general a nontrivial task. To overcome this problem, we select V as a quadratic function and recast the conditions in Property 1

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into some matrix inequalities. This approach is formalized in the corollary below. n +n +n Corollary 1. (Quadratic Conditions). Let P ∈ S+p c u , α, ρ ∈ R>0 , and σF , σJ ∈ R≥0 be such that �≺0 �T M C He(P A0 ) + 2αP − σF C (11)

M ∈ Snp +ny , λ ∈ R, and ρ ∈ R>0 such that the set A in (5) is globally t-exponentially stable with t-decay rate α for the closed-loop system (3) (equivalently, (4)). ◦

Proof. For all x ∈ Rnp +nc +nu , define V (x) = xT P x. Now we show that under (11)–(12), V satisfies (8), (9), and (10). Specifically, relation (8) holds with ω1 = λmin (P ), ω2 = λmax (P ), and χ = 1. Moreover, by the S-procedure, it follows that (11) and (12) imply (9) and (10). The result then follows from Theorem 1.  Remark 5. It is worthwhile to observe that (11) is a linear matrix inequality (LMI ) in the decision variables P and σF . As such, the feasibility of (11) can be checked via numerically efficient algorithms; see Boyd et al. (1997). On the contrary, (12), which is again linear in P and in σJ , is actually nonlinear in the decision variable ρ. However, such a nonlinearity can be handled via a line search for the variable ρ. Remark 6. A novelty of the conditions in Corollary 1, as compared to previous works, is that the jump condition (12) assumes here a new form due to the flow occurring during the “hold” phase of our controller (when q = 1), which allows injecting matrix eA1 ρ in the first term of the jump condition (12). Interestingly, in Fichera et al. (2016) the jump condition was not needed because it was automatically enforced by the choice of the jump and flow sets. However, that solution had the drawback of not being implementable as an output feedback architecture. The jump condition was instead used (and needed) in Yuan and Wu (2014); Satoh (2011) where it was essentially formalized as (12) with ρ = 0 (because no holding mechanism during the dwell time period is enforced in those papers). Such a condition is unfortunately impossible to satisfy due to the peculiar block structures of G (due to � (due the fact that the plant states are never reset) and C to the assumption that only output y is accessible). More specifically, denoting by Cp⊥ the orthogonal complement � ⊥ �T C of Cp one may easily compute 0 ≻ 0p (GT P G − P + 0 � ⊥� � C0p = (Cp⊥ )T (P11 − P11 )Cp⊥ = 0, which is �T M C) σJ C

(i) There exist M = M T , σJ , σF ∈ R≥0 such that (11) and (12) hold strictly; (ii) There exists M = M T , σ ∈ R≥0 such that

T � ≺ 0, �T M C GT eA1 ρ P eA1 ρ G − e−2αρ P + σJ C

(12) � where C is defined in (3b). Then, Property 1 holds with x �→ V (x) := xT P x, χ = 1, and set A in (5) is globally t-exponentially stable for (3) with t-decay rate α.

0

evidently infeasible whenever Cp is not full column rank (namely the plant state is accessible).

Before stating the main result of this section, let us consider the following preliminary result that enables one to eliminate one of the two scalar variables introduced by the S-procedure in (11)-(12). n Lemma 2. Let α, ρ ∈ R>0 , P ∈ S+p be given and A0 , A1 , and G be defined in (3b). The following items are equivalent:

�T M C �≺0 He(AT0 P ) + 2αP − σ C T �T M C � ≺ 0, GT eA1 ρ P eA1 ρ G − e−2αρ P + C

In this section, we propose a methodology for the design of hybrid controller (2). The approach we pursue hinges upon Corollary 1 and an extension of the change of variables proposed in Scherer et al. (1997). Specifically, the problem we solve can be formalized as follows: Problem 1. Consider plant (1) and hybrid controller (2) and let α ∈ R>0 be given. Design � � � � Ac Bc Gc Kc , Σd := Σc := Cc Dc Gv Kv 137

(14)

� is defined in (3b). where C

Proof. The implication (ii) =⇒ (i) is trivial. To show that (i) =⇒ (ii), first note that if (12) holds with σJ ≥ 0, then it must hold with σJ′ := σJ + σε > 0 where σε > 0 small enough exists due to the strict inequality in (12). As a consequence, (ii) holds with M ′ = σJ′ M and σ ′ = σˆσ′FJ . This concludes the proof.  We may now state our main controller design result providing a design procedure in terms of bilinear matrix inequalities arising from a line search and a remaining set of linear matrix inequalities. To simplify the notation, given plant (1), and a dwell time parameter ρ > 0, define the following sampled-data input matrix: Aρ := eAp ρ ,

Bρ :=

�

ρ

eAp s Bp ds.

(15)

0

Theorem 2. Given plant (1), a dwell time parameter ρ > 0, and the sampled-data matrices in (15). Assume that n there exist X, Y ∈ S+p , K ∈ Rnp ×np , L ∈ Rnp ×nu , R ∈ � ∈ Snp +ny , S ∈ Rnu ×np , Q ∈ Rny ×np , N ∈ Rny ×nu , M nu ×ny np ×nu R ,T ∈R , J ∈ Rnp ×ny , σ > 0 such that � F := He(Γ) + 2αΥ − σ C˘T M �C˘ ≺ 0 (16) Π   −2αρ T�˘ T T ˘ −e Υ + C MC Λ Ψ � J :=  (17) Π Λ −Υ 0  ≺ 0 Ψ 0 −I where � � Y I Υ := I X �

�

+ Bp R Ap + Bp N Cp , Γ := Ap Y K XA + LC

Λ :=

4. CONTROLLER DESIGN

(13)

�

p

p

�

Aρ Y + Bρ S Aρ + Bρ QCp , T XAρ + JCp

� � I 0 , C˘ := 0 Cp

(18)

Ψ := [S QCp ].

Then, matrix I − XY is nonsingular. Let U, V ∈ Rnp ×np be any pair of nonsingular matrices such that (19) XY + U V T = I Then, for any λ > α, Σc , Σd , and M as in (20) solve Problem 1.  5. SIMULATION EXAMPLE In this section, we showcase the applicability of the proposed design methodology to a double integrator system,

2019 IFAC NOLCOS 106 Vienna, Austria, Sept. 4-6, 2019

�

Francesco Ferrante et al. / IFAC PapersOnLine 52-16 (2019) 102–107

� �� K − XAp Y L 0 V −T , −T R N I −Cp Y V � � � −T I V � 0 M −V −1 Y CT I −Cp Y V −T p

−1 −1 Σc = U0 −U I XBp

�

0

M = V −1 whose data is: �

��

� Ap Bp CT p =

�

0 10 1 0 01 0

�

As mentioned earlier, when the scalars ρ and σ are fixed, the conditions given in Theorem 2 are genuine linear matrix inequalities. As such, Theorem 2 can be employed to design the controller parameters by performing a line search on the scalars ρ and σ. In this example, numerical R solutions to LMIs are obtained in Matlab using the YALMIP package Lofberg (2004) combined with the solver SDPT3 T¨ ut¨ unc¨ u et al. (2003). With the objective of avoiding the occurrence of controllers characterized by overly fast continuous-time dynamics, we consider the following additional constraint: −βΥ � He(Γ) � βΥ, which ensures that the eigenvalues of the matrix characterizing the closed-loop continuous-time dynamics, are contained in the set {z ∈ C : | Re(z)| ≤ β}; see Scherer et al. (1997). For design purposes, we select

xp1

2 0 -2 0

1

2

3

xp2

6

7

4

5

6

7

4

5

6

7

4

5

6

7

4

5

6

7

4

5

6

7

0 -2 0

1

2

3

t

1

xc1

5

t

2

0 -1 0

1

2

3

t

2

xc2

4

0 -2 0

1

2

3

t

0.2

τ

0.1 0 0

1

2

3

t

q

1

�

−1 −1 Σd = −U IXBρ U0

��

Q S T − XAρ Y J

� �� 0 V −T −T I −Cp Y V

(20)



   2.11 3.9 −0.57 0.11 −0.12 −1.81 Σc =  −8.84 −5.49 −2.45  , Σd =  0.37 −0.33 0.34  10.3 8.51 −0.1737 −10.31 −5.66 4.66 � � 44.38 106.3 46.19 M = 106.3 262.5 115.5 46.19 115.5 50.53

(21) It is worthwhile to observe that spec(M ) ≈ {356.3, 1.4, −0.3}, so that int D �= ∅. This implies that our design returns a genuinely hybrid controller for this example. To assess the benefit of the proposed hybrid controller against a standard LTI continuous-time output feedback controller, we compare the responses of the closed-loop system obtained from different initial conditions obtained when using: 1) the reset controller, 2) the “base” dynamic output feedback controller defined by the parameters Σc in (21), and 3) an LTI observer-based controller designed by placing the poles of the closed-loop system in −1 ± i. For this example, we simulate the closed-loop response from 100 initial conditions in which the controller initial state is set to zero and the plant state is randomly picked on the unit circle 1 . To somehow quantify the performance of each design, for each simulation, we compute the L2 -norm of the plant state response (projected onto the ordinary time); smaller values of this criterion are generally associated to an improved transient response. Table 1 gives an overview on the percentage improvement (decrease) of the L2 -norm of the plant response provided by the reset controller as compared to the above mentioned continuous-time LTI compensators. In these simulations, λ = 2, yet we recall that this parameter has no role in the plant response. Table 1 clearly shows that the proposed controller overall performs the best. To further emphasize the benefits of the proposed hybrid architecture, in Fig.1 we compare the responses of the closed-loop system obtained for the three different designs described above from the initial condition ζ(0, 0) = (0.8540, 0.5202, 0) ∈ R6 . The evolution of the variables τ and q reveals the occurrence of resets in the controller state and the activation of the “freezing” mechanism. Fig.1 clearly shows that the proposed reset controller outperforms the two LTI controllers in terms of transient response. Similar results can be also appreciated from alternative initial conditions.

0.5

Improvement max min mean

0 0

1

2

3

t Fig. 1. The response of the closed-loop system for: reset control (solid line), base linear controller (dashed line), and pole-placement observer-based controller (dotted line). α = 1 and perform a line search on the parameters σ, ρ, and β, while minimizing the value of β. Numerical experiments show that, in this example, for β ≥ 2.3, the values of σ and ρ are not crucial to ensure the feasibility of the conditions in Theorem 2. By selecting σ = 10 and ρ = 0.1, one gets the following data for the controller: 138

Base Contr. 27.4% 4% 17.9%

Pole-plac. 52.49% 11.93% 28.42%

Table 1. Improvement in terms of L2 -norm of the plant response.

6. CONCLUSION We proposed a novel reset control architecture for linear continuous-time plants when only plant output information is available for the feedback action. The new architecture stems from the intuitive idea of freezing the controller state and the plant input during the necessary dwell time 1

Code at https://github.com/f-ferrante/NOLCOS19Reset

2019 IFAC NOLCOS Vienna, Austria, Sept. 4-6, 2019

Francesco Ferrante et al. / IFAC PapersOnLine 52-16 (2019) 102–107

imposed after each controller reset. With this architecture in place, we provide Lyapunov-based proofs of stability and guaranteed exponential convergence rate, which corresponds to LMI conditions when performing quadratic stability/performance analysis and line search on LMIs when also designing the controller parameters, following a typical LMI-based plant-order dynamic output feedback design formulation. A simulation example illustrates the effectiveness of the proposed approach. REFERENCES Aangenent, W., Witvoet, G., Heemels, W., van de Molengraft, M., and Steinbuch, M. (2010). Performance analysis of reset control systems. International Journal of Robust and Nonlinear Control, 20(11), 1213–1233. Ba˜ nos, A. and Barreiro, A. (2011). Reset control systems. Springer. Beker, O., Hollot, C., and Chait, Y. (2001). Plant with integrator: an example of reset control overcoming limitations of linear feedback. IEEE Transactions on Automatic Control, 46(11), 1797–1799. Beker, O., Hollot, C., Chait, Y., and Han, H. (2004). Fundamental properties of reset control systems. Automatica, 40(6), 905–915. Boyd, S., Ghaoui, L.E., Feron, E., and Balakrishnan, V. (1997). Linear Matrix Inequalities in System and Control Theory. Society for Industrial and Applied Mathematics. Carrasco, J. and Ba˜ nos, A. (2012). Reset control of an industrial in-line pH process. IEEE Transactions on Control Systems Technology, 20(4), 1100–1106. Clegg, J. (1958). A nonlinear integrator for servomechanisms. Trans. A. I. E. E., 77 (Part II), 41–42. Dav´o, M., Ba˜ nos, A., Gouaisbaut, F., Tarbouriech, S., and Seuret, A. (2017). Stability analysis of linear impulsive delay dynamical systems via looped-functionals. Automatica, 81, 107–114. Fichera, F., Prieur, C., Tarbouriech, S., and Zaccarian, L. (2013). Using Luenberger observers and dwell-time logic for feedback hybrid loops in continuous-time control systems. International Journal of Robust and Nonlinear Control, 23(10), 1065–1086. Fichera, F., Prieur, C., Tarbouriech, S., and Zaccarian, L. (2016). LMI-based reset H∞ design for linear continuous-time plants. IEEE Transactions on Automatic Control, 61(12), 4157–4163. Goebel, R., Sanfelice, R.G., and Teel, A.R. (2012). Hybrid Dynamical Systems: Modeling, Stability, and Robustness. Princeton University Press. Heemels, W., Dullerud, G., and Teel, A. (2016). L2 -gain analysis for a class of hybrid systems with applications to reset and event-triggered control: A lifting approach. IEEE Transactions on Automatic Control, 61(10), 2766– 2781. Horowitz, I. and Rosenbaum, P. (1975). Non-linear design for cost of feedback reduction in systems with large parameter uncertainty. International Journal of Control, 21, 977–1001.

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