Global uniform output regulation of nonlinear systems

Jetro Sihvonen GLOBAL UNIFORM OUTPUT REGULATION OF NONLINEAR SYSTEMS Master’s thesis Faculty of Engineering and Natural Sciences Supervisors: Professor Lassi Paunonen Postdoctoral Research Fellow Nicolas Vanspranghe November 2024 i ABSTRACT Jetro Sihvonen: Global uniform output regulation of nonlinear systems Master’s thesis Tampere University Master’s degree in science and engineering November 2024 The output regulation problem is a control problem that encompasses tracking, disturbance rejection, synchronization and observer design problems. The goal in output regulation is for a given system to achieve asymptotic tracking of a reference signal. In this thesis the output regulation problem is considered for nonlinear systems. We give sufficient conditions for solvability of both the local and global output regulation problem. This requires considering the solvability of the so called regulator equations and proving the existence of a stabilizing controller such that the system is made uniformly convergent. We first focus on presenting and discussing sufficient conditions for uniform convergence of systems with inputs. We discuss the Demidovich condition and extend it using contractivity to form a more general result guaranteeing exponential convergence. We consider two stabilizing controllers, one for quadratically stabilizable systems corresponding to the Demidovich condition and one for systems stabilizable using contractivity. Then we solve the regulator equations for a standard harmonic oscillator, a harmonic oscillator with nonlinear damping and some other simple systems. Also we present a known algorithm for solving the regulator equations for control affine systems and apply it to the Hopfield neural network model. Finally we apply the discussed control design methods to the harmonic oscillators and the Hopfield model to solve the uniform output regulation problem. Keywords: Output regulation, uniform convergence, uniform contractivity, output tracking, disturbance rejection, nonlinear systems, regulator equations, harmonic oscillator, Hopfield model The originality of this thesis has been checked using the Turnitin OriginalityCheck service. ii TIIVISTELMÄ Jetro Sihvonen: Epälineaaristen järjestelmien globaali tasainen lähdön regulaatio Diplomityö Tampereen yliopisto Tekniikan ja luonnontieteiden maisteri tutkinto Marraskuu 2024 Regulointi ongelma on säätöongelma, jota voidaan käyttää seuranta-, häiriönpoisto-, synkronointija tarkkailijasuunnitteluongelmien tarkasteluun. Reguloinnin tavoitteena on saada jokin annetun järjestelmän mittaus seuraamaan asymptoottisesti referenssisignaalia siten että järjestelmä samalla hylkää häiriöt. Tässä opinnäytetyössä tarkastellaan regulointi ongelmaa epälineaarisille järjestelmille. Sekä paikallisen että globaalin tasaisen lähdön regulaatio ongelman riittävät ratkaistumisedellytykset esitetään ja tarkastellaan kahdessa pääosassa. Tätä varten tarkastellaan niin sanottujen regulaattoriyhtälöiden ratkaistavuutta, sekä sellaisen stabiloivan säätimen olemassaoloa, että järjestelmästä tulee tasaisesti konvergoituva. Keskitymme ensin muodostamaan riittäviä ehtoja tasaiselle konvergenssille sisääntuloja sisältäville järjestelmille. Käsittelemme Demidovichin ehtoa ja laajennamme sitä käyttämällä kontraktiivisuuden käsitettä muodostaen yleisemmän tuloksen, joka takaa eksponentiaalisen konvergenssin. Käsittelemme kahta stabiloivaa säädintä, yhtä neliöllisesti stabiloitaville järjestelmille, joilla Demidovichin ehto täyttyy, ja toista järjestelmille, jotka voidaan stabiloida kontraktivuuden avulla. Toisekseen ratkaisemme regulaattoriyhtälöt yksinkertaiselle harmoniselle oskillaattorille, harmoniselle oskillaattorille, jossa on epälineaarinen vaimennus ja joillekin muille yksinkertaisille järjestelmille. Käsittelemme myös erästä tunnettua algoritmia regulaattoriyhtälöiden ratkaisemiseksi ohjausaffiineille järjestelmille ja sovellamme sitä Hopfieldhermoverkkomalliin. Lopuksi sovellamme käsiteltyjä ohjauksen suunnittelumenetelmiä harmonisiin oskillaattoreihin ja Hopfield-malliin tasaisen lähdön regulaatio ongelman ratkaisemiseksi. Avainsanat: Regulointi, stabiilisuus, tasainen kontraktiivisuus, mittauksen seuranta, häiriönhylkääminen, epälineaariset järjestelmät, regulaattoriyhtälöt, harmoninen oskillaattori, Hopfieldin malli Tämän julkaisun alkuperäisyys on tarkastettu Turnitin OriginalityCheck -ohjelmalla. iii PREFACE This thesis project has been exciting and full of meaningful lessons. I am grateful for the guidance, suggestions and patience of my supervisors Lassi Paunonen and Nicolas Vanspranghe. It has been a delight working with you and getting to know the field of mathematical systems theory, while sharing the frustrations of complex technical details. I wish to also thank my family and friends for their support and encouragement. This thesis could not have been possible without you. Finally thanks be to the Creator, for his marvelous design in which we always find things to ponder upon, to the Son through whom we are saved and through whom we find meaning and to the Holy Spirit who strengthens us and guides us. May all the glory be to the God in the highest. In Tampere, 18th November 2024 Jetro Sihvonen iv CONTENTS 1. Introduction ........................... 1 2. Stability of nonlinear systems .................... 5 2.1 Uniform stability........................ 5 2.2 Stability with respect to inputs .................. 7 2.3 Uniform convergence of systems with inputs ............ 9 2.4 Contractivity analysis......................11 2.5 Universal contractivity of systems with inputs ............14 3. The output regulation problem ....................20 3.1 The problem formulation ....................20 3.1.1 Harmonic oscillator....................20 3.1.2 General problem formulation ................22 3.2 Solvability of the output regulation problem using regulator equations . . 25 3.3 Methods for solving the regulator equations.............27 4. Controller design for output regulation.................34 4.1 Design strategy ........................34 4.2 Quadratically stabilizable systems .................35 4.3 Systems stabilizable using universal contractivity...........38 4.4 Locally stabilizable systems ...................42 5. Case studies of output regulation ...................43 5.1 Harmonic oscillator ......................43 5.2 Harmonic oscillator with nonlinear damping.............45 5.3 Hopfield neural networks ....................48 5.3.1 Hopfield system with control inputs on each node .......49 5.3.2 Three node Hopfield system with a single input and output . . . 50 6. Conclusion ............................55 References ..............................57 v LIST OF SYMBOLS AND ABBREVIATIONS δISS incremental input-to-state stability (A)ij The element of matrix A on row iand column j NNatural numbers, i.e., {0,1,2, . . . } C1Functions that are continuously differentiable C∞Functions that are infinitely continuously differentiable IIdentity matrix ISS Input-to-state stable J(x, t)Generalized Jacobian || · || Euclidean vector norm LMI Linear matrix inequality Ln gf(x)The n:th Lie derivative of f with respect to g MIMO Multiple input multiple output PCnClass of bounded piecewise continuous n-dimensional vector functions PC(W)Class of bounded piecewise continuous vector functions that are, for all values, in the set W PReLU Parametric Rectified Linear Unit RReal numbers R+Positive real numbers RnReal vectors of length n SISO Single input single output TAU Tampere University ATTranspose of matrix A TUNI Tampere Universities UBSS Uniformly bounded steady state URL Uniform Resource Locator 1 1. INTRODUCTION In this thesis we study system theory where the goal is to affect or control phenomena that can be modeled mathematically. We focus specifically on cases where a predefined behaviour is required and discuss the tools needed to achieve it. A system models the behaviour of some chosen variable or set of variables in specific environment. More precisely a system is a mathematical model that describes how the state of the system changes over time. The state of a system consists of chosen parameters of interest and completely describes the system at a specific instance of time. In mathematical systems theory the model consists of one or more differential equations where the solution of the equations is the state of the system and the initial value is the state of the system at zero time, that is a known starting point. Systems can have so called inputs, which model some external parameters that can be constant or varying with time and affect the system’s behavior. Likewise systems can have outputs, which describe measurable parameters or effects of the system on specific parameters of interest. Different systems can be connected using their inputs and outputs. In this way we can form larger combined systems, the states of which are the states of the connected systems considered together. To influence the behaviour of a system, inputs called control inputs, for which values can be chosen are considered. In order to control the system with these control inputs, we wish to design controllers which are systems that generate a control input as their output. This output can then be connected to the original systems control input to control the system. The controllers in most cases need information about the systems state to be able to generate a useful control input. The information we are able to measure and give to the controller can be thought as an output of the system. These outputs are called measured outputs. Our goal is to study problems where we are given a required behaviour of the system in the form of a reference signal. More precisely the reference signal can vary with time or be constant and usually describes the wanted behaviour of some output of the system. In addition to a reference signal we consider a so called disturbance signal, which models possible external disturbances that affect the system. Naturally the goal is to control the system so that it is relatively unaffected by the disturbances and the chosen output 2 Figure 1.1. The systems of the output regulation problem and their connections follows the reference signal. The way the system is required to follow the reference signal can vary. In our case we want the output of the system to approach the reference signal with time, that is we want asymptotic tracking. To measure whether asymptotic tracking is achieved we define the regulator error, a measure between the reference signal and the output of the system. We then consider this error as an output of the system writing it as eand more precisely requiring that e(t)→0as t→ ∞. Also in this thesis we consider the cases where the reference signal and the disturbances can be modeled using another system, which outputs these signals. This external system that can be said to generate the reference and disturbances will be called the exosystem and the signal it generates will be called the exosignal. Using exosystems gives us a concrete way of describing exosignals and possible assumptions regarding them and is useful in analysis. Problems that consider asymptotic tracking of a reference signal under disturbances, when more formally defined are called output regulation problems. An illustration of the different systems of the output regulation problem is shown in Figure 1.1. Depending on the form of the differential equation describing the system and possible restrictions on the initial conditions, different variants of the output regulation problem can be formulated. The system is called linear if the differential equations describing it are linear. Otherwise the system is called nonlinear. The problem is called a global problem if the goal is to solve it for all initial values, otherwise it is called a local problem. In this thesis we consider the global and the local nonlinear output regulation problem. The output regulation problem is a useful problem formulation in many practical applications. For example systems with periodic and bounded reference signals can be considered through exosystems. Synchronization problems where the goal is to synchronize two different systems as well as observer design problems for autonomous (time 3 independent) systems can be formed as output regulation problems. In this thesis we consider the output regulation of two different harmonic oscillators and a neural network model. [1] The local output regulation problem has been solved in general for all nonlinear systems and so it is more interesting to focus on the global output regulation problem [1]. Such considerations are also of interest in semiglobal cases, where the problem is solved for a predefined set of initial conditions. Linear systems have also been studied extensively with applications also in approximating nonlinear systems [1]. However, especially when considering global output regulation it is not always possible to approximate real systems using linear theory as in general linear approximations hold only locally. Therefore discussion on nonlinear systems is of interest. In this thesis we aim to discuss the nonlinear output regulation problem as defined in [1] and expand slightly on the existing theory. We will focus on the global problem, but will also give a local variant. In addition to formulating the output regulation problems, we discuss methods to show their solvability and some simple controller design methods. The conditions for solvability that we will discuss can roughly be divided into three parts. First we need the problem to be well-posed. This will be achieved through continuity and boundedness assumptions discussed shortly in Chapter 2for general systems. Second the system needs to be stable in the sense that solutions of the system’s differential equations corresponding to different initial conditions approach each other. Stability will be the focus of Chapter 2. Thirdly there must exist a state of the system and a control input where the output corresponds to the trajectory to be tracked. This requirement will be discussed in Chapter 3, where these requirements are brought together in the problem formulation. We will discuss some different methods for verifying the solvability conditions that will be formulated. Three different stability conditions are presented in Chapter 2. The first of which is a set of so called Lyapunov functions that guarantee stability, the second is the so called Demidovich condition and the third uses a stability notion called contractivity. The first two are discussed in [1], while the third we formulate using [2] and [3]. The condition using contractivity is more general than the Demidovich condition and we use it to extend the theory presented in [1]. Using contractivity is also simpler than the method using the Lyapunov functions directly and so of interest. To guarantee the existence of a state and control that correspond to the reference signal a set of differential equations called the regulator equations of the system must be solvable. We will discuss an algorithm from [4] that can be used to solve them for a specific class of systems in Section 3.3. Other methods, including direct computation will also be discussed and demonstrated. 10 convergent in Zfor every input w(·)∈ N. To use the concept of input-to-state stability together with convergence input-to-state convergence is defined. Definition 2.11 ([1]).System (2.3) is said to be input-to-state convergent if it is globally uniformly convergent for the class of inputs PCmand it is ISS with respect to each steady-state solution x¯w(t)corresponding to some input w(·)∈PCm. By ISS with respect to the steady-state solution xw(t)we mean that there exists a KL-function β and a K∞-function γsuch that any solution x¯(t)of system (2.3) corresponding to some input w(t)+∆w(t)satisfies ∥x(t)−x¯w(t)∥ ≤ β(∥x0−x¯w0∥, t) + γ(︄sup 0≤τ≤t ∥∆w(τ)∥)︄.(2.6) In general, the functions βand γin Definition 2.11 may depend on the particular input w(·). Here input-to-state stability is required with respect to the steady-states. This is similar to the more general condition, incremental input-to-state stability (δISS), where any two trajectories are required to satisfy (2.6) [8]. This extension allows us to consider systems where zero is not an equilibrium solution, that is systems where f(0,0) = 0, allowing for analysis of more general systems without a coordinate transform. Input-to-state convergence is a global property. Thus another description of stability with respect to the inputs is needed, one that is applicable to local cases in the context of convergence. We use a somewhat more general property defined in [1, Def. 2.17]. Definition 2.12. The system (2.3) that is uniformly convergent in Zfor a class of inputs N(W)⊂PC(W)is said to have the uniformly bounded steady state (UBSS) property if for any compact set C⊂Wthere exists a compact set D⊂Rnsuch that for any input w(·)∈ N(W)the following implication holds: w(t)∈C, ∀t∈R=⇒xw(t)∈D, ∀t∈R. Input-to-state convergence implies global uniform convergence with the UBSS property [1, Prop. 2.19]. We introduce two different conditions that guarantee uniform convergence with the UBSS property or input-to-state convergence. The first one is a variant of the Demidovich condition generalized for inputs in [1]. The condition’s application to stability is usually called Krasovskii’s method [5]. Theorem 2.13 ([1]).Consider system (2.3) and let the function f:Rn×R→Rnbe C1. Suppose there exist matrices P=PT>0and Q=QT>0such that P∂f ∂x(x, w) + ∂fT ∂x (x, w)P≤ −Q, ∀x∈Rn,∀w∈W⊂Rm. 11 Then system (2.3) is globally exponentially convergent with the UBSS property for the class of inputs PC(W). If W=Rm, then system (2.3) is input-to-state convergent. The second condition uses Lyapunov-like functions and is more general than the Demidovich condition, which can be seen as a particular case corresponding to quadratic Lyapunov-like functions. Theorem 2.14 ([1]).Consider system (2.3). Suppose there exist C1functions V1: Rn×Rn→R+and V2:Rn→R+,K-functions α2,α3,α5,γ, and K∞-functions α1, α4satisfying the conditions α1(∥x1−x2∥)≤V1(x1, x2)≤α2(∥x1−x2∥), ∂V1 ∂x1 (x1, x2)F(x1, w)+∂V1 ∂x2 (x1, x2)F(x2, w)≤ −α3(∥x1−x2∥), α4(∥x∥)≤V2(x)≤α5(∥z∥), ∂V2 ∂x (x)F(x, w)≤0for ∥x∥ ≥ γ(∥w∥), for all x1, x2,x∈Rnand all w∈Rm. Then system (2.3) is globally uniformly convergent and has the UBSS property for the class of inputs PCm. Finally we give a condition for local exponential convergence where there exists some neighborhood of the origin where the system in question is exponentially convergent. Theorem 2.15 ([1]).Consider system (2.3) with f(0,0) = 0 and fbeing C1in some neighborhood of the origin, (x, w) = (0,0). Let N ⊆ PCmbe some class of inputs containing the zero input, w(t)=0. The following statements are equivalent: (i) System (2.3) is locally exponentially convergent for the class of inputs N (ii) The matrix Jacobian matrix at the origin, (∂F/∂x)(0,0) is Hurwitz. 2.4 Contractivity analysis Contractivity is a stability notion that describes a set of initial conditions for which trajectories of a system converge to each other. In application it is similar to the concept of exponential convergence, but with the difference that contractivity does not require bounded steady states. However as will be seen, contractivity, when assuming bounded trajectories is in fact equivalent to exponential convergence. Thus it may be used to achieve another condition for uniform convergence. Contractivity can be defined in numerous ways as demonstrated in [9]. Following the technique in [2], we consider the solutions of (2.1), x(t), as trajectories and the analyze the relationship between two neighboring trajectories. To describe the distance between these neighboring trajectories we consider a virtual or infinitesimal displacement δx. The 12 dynamics of virtual displacements are described by the first variation equation δx˙ = ∂f ∂x(x, t)δx. (2.7) In using virtual displacements a quadratic Lyapunov function of the form V(x, δx, t) = δxTM(x, t)δx, where M:Rn×R→Rn×nis a matrix function can be considered. This Lyapunov function can be used to analyze the dynamics of the trajectories of system (2.1) with the goal of forming a condition using Mthat guarantees that neighboring trajectories approach each other. First the matrix function M:Rn×R→Rn×n is required to be uniformly positive definite. Secondly it it must satisfy M(x, t) = Θ(x, t)TΘ(x, t), for all x∈Rn,t∈Rfor some invertible Θ : Rn×R→Rn×n. By uniformly positive definite Mit is meant that there exists α > 0such that for all x∈Rnand t∈R,1 2(M(x, t)T+M(x, t)) ≤αI. Since Mis symmetric, this is equivalent to M(x, t)≤αI for all x∈Rn,t∈R. The matrix function Mis formally a time-dependent Riemannian metric. By the notation M ˙ we mean the matrix function M ˙:Rn×R→Rn×nwith entries M ˙(x, t)ij =∂Mij ∂t (x, t) + ∇xMij(x, t)·f(x, t)(2.8) for all x∈Rn,t∈Rand i, j ∈[1, . . . , n]. [9] Using the above assumptions and notations the Lyapunov function V(x, δx, t) = δxTM(x, t)δx (2.9) can be considered with the local coordinate transform z= Θ(x, t)x, which leads to a coordinate change of the virtual displacements δz = Θ(x, t)δx. (2.10) Then analyzing the time derivative of this transformed virtual displacement along the trajectories x(t)of system (2.1) leads to a sufficient condition guaranteeing that different trajectories converge to each other. Next we use this condition derived in [2] to define contractivity. Definition 2.16 ([2]).Given a system x˙ = f(x, t), a set of initial conditions Z⊂Rnis called a contraction region with respect to a uniformly positive definite metric M(x, t) = Θ(x, t)TΘ(x, t)if there exists βM>0such that J(x, t) := (︄Θ ˙(x, t) + Θ(x, t)∂f ∂x(x, t))︄Θ−1(x, t)≤ −βMI 13 or equivalently ∂fT ∂x (x, t)M(x, t) + M(x, t)∂f ∂x(x, t) + M ˙(x, t)≤ −2βMM(x, t)(2.11) for all x∈Zand all t∈R. In this case, Jis called the generalized Jacobian. The constant βMis the contraction rate. Theorem 2.17 ([2]).Consider the system (2.1) and a solution of that system x¯(t). Any solution x(t)which starts in a ball, with respect to metric Mcentered around x¯(t) and contained at all times in a contraction region with respect to M, remains in that ball and converges exponentially to x¯(t). Furthermore global exponential convergence to the given solution x¯(t)is guaranteed if the whole state space is a contraction region with respect to the metric M. Theorem 2.18 ([2]).Consider the system (2.1) and assume that it is exponentially convergent. Then the system is contracting with respect to a uniformly positive definite metric M. These theorems connect contractivity to exponential convergence and vice versa. Thus, when there exists a bounded trajectory, the system is exponentially convergent if and only if there exists a metric Msatisfying the conditions in Definition 2.16. This metric is called a contraction metric. Compared to the Lyapunov functions in Theorem 2.14 guaranteeing convergence, the contractivity approach seems appealing. If we know that trajectories are bounded, we simply need to solve for a contraction metric Mor show its existence. This can be simpler than the task of finding a Lyapunov function. Naturally this simplification leads to decreased generality as Theorem 2.14 guarantees uniform convergence, while Theorem 2.17 gives exponential convergence. However in situations where exponential convergence is required, the contractivity condition is necessary and no generality is lost. On the other hand compared to the Demidovich condition, Theorem 2.13, the contractivity condition is clearly more general and complex. This can be seen by considering (2.11) with a metric that is constant with respect to both state and time. The condition becomes ∂fT ∂x M+M∂f ∂x ≤ −2βMM, (2.12) because M ˙= 0. This can be seen to directly correspond to the Demidovich condition in Theorem 2.13. Contraction metrics can be guessed or in some simple cases computed directly. Also a numerical approach could be used, with one such presented for example in [10]. Another definition of contractivity using one sided Lipschitz constants, discussed in [5], allows for handling of other norms in addition to the Euclidean norm. However the discussion is on the cases corresponding to constant contraction metrics, and thus the 14 Euclidean norm case discussed in [5] corresponds to Theorem 2.13. Other variants of contractivity and historical connections to similar ideas are discussed extensively in both [5] and [9]. Further considering the relationship between convergence and contractivity the fact that they are not equivalent is demonstrated in [9]. However as shown, the concepts are complementary. In fact the use of contractivity to achieve convergence is a common method of analysis used especially with autonomous systems; first contractivity analysis is used to show that a system is incrementally stable and then convergent dynamics are constructed [9]. 2.5 Universal contractivity of systems with inputs In this section we consider contractivity in the context of systems with inputs. For this we consider system (2.3). The goal is to achieve input-to-state convergence, and since Theorem 2.17 guarantees uniform convergence, the focus will be on connecting contracting systems with inputs to ISS. Contractivity does not directly imply ISS, but with some simple assumptions, namely universal contractivity, there is a direct link to incremental input-to-state stability (δISS) [3], which as discussed in Section 2.3 is similar to the type of ISS needed for input-to-state convergence. The connection is formulated precisely here. First we define universal contractivity following [3]. Definition 2.19. The system (2.3) is uniformly contracting in a set of initial conditions Z⊂Rnand for a set of inputs Wr⊂Rmwhen ∂f ∂w is bounded and (2.3) is contracting in Zwith respect to a metric M= ΘTΘfor all w∈Wr, that is there exists βM>0 such that J(x, w, t) := (︄Θ ˙(x, w, t) + Θ(x, t)∂f ∂x(x, w, t))︄Θ−1(x, t)≤ −βMI, for all x∈Z,w∈Wrand t∈R. The constant βMis the contraction rate and J is the generalized Jacobian of the system. If Z=Rn, the system is globally uniformly contracting. When considering the system (2.3), we define the matrix function M ˙:Rn+m×R→ Rn×nsuch that its entries are M ˙ij(x, w, t) = ∂Mij ∂t (x, t) + ∇xMij(x, t)·f(x, w)(2.13) for all x∈Rn,w∈Rm,t∈Rand i, j ∈[1, . . . , n]. This notation corresponds to (2.8) and explains the reason why we write Θ ˙with a dependency on w. The Grönwall’s Lemma is a highly useful result in contracting applications. The variant we use can be found for example in [11, Lemma 2.1]. 15 Lemma 2.20. Suppose that a C1function v:R+→Rsatisfies v˙(t)≤ −γv(t) + c, v(0) = v0,∀t∈R+, where 0< γ ∈R,c∈Rand v0∈R. Then the inequality v(t)≤v0e−γt +c γ(1 −e−γt), holds ∀t∈R+ Another lemma that will be used is Young’s inequality with ϵ. Lemma 2.21. Let a, b ∈Rnthen for arbitrary ϵ > 0 aTb≤aTa 2ϵ+ϵbTb 2 Lemma 2.22. Consider system (2.3), a class of inputs Nand a set Wr⊂Rmsuch that for all w(·)∈ N,w(t)∈Wrfor all t∈R. Suppose there exists a metric M:Rn×R→Rn×nsuch that the system (2.3) is globally uniformly contracting for the set of inputs Wrat a rate λwith respect to M. Suppose also that Mis bounded in the sense that there exists σmin, σmax >0such that σ2 minI≤M(x, t)≤σ2 maxI, ∀x∈Rn,∀t∈R.(2.14) Finally assume that for an input w(·)∈ N, there exists a corresponding bounded trajectory xw(·)of (2.3). Then the system is input-to-state stable with respect to xw(·). Proof. The proof is based on the techniques in the proof of [12, Lemma 2.1]. We consider the steady state xw(·)corresponding to input w(·)and a solution of the system (2.3), x(·), corresponding to input w(·)+∆w(·)and with initial values xw(0) = xw0 and x(0) = x0. First we form line segments sbetween x0and xw0, and ubetween w(·) and w(·)+∆w(·). These are defined by s(α) = αx0+ (1 −α)xw0, α ∈[0,1], u(β, t) = w(t) + β∆w(t), β ∈[0,1] for all t > 0. The goal is to use the lengths of these line segments, ∥x0−xw0∥and ∥∆w(t)∥to estimate ∥x(t)−xw(t)∥,∀t > 0. To do this we consider two curves (2.3), generated using s(·),u(·,·)and solutions of (2.3) written here as x(x0, w(t), t),∀t > 0. They are defined by ϕ1(α, t) = x(s(α), u(0, t), t), α∈[0,1] and ϕ2(β, t) = x(s(1), u(β, t), t),β∈[0,1], for t > 0. These two curves 16 combined defines a curve between x(t)and xw(t)for t > 0whose length satisfies L(t)≤∫︂1 0 ∂x(s(α), u(0, t), t) ∂α  dα +∫︂1 0 ∂x(s(1), u(β, t), t) ∂β  dβ, t > 0 This length then satisfies ∥x(t)−xw(t)∥ ≤ L(t), t > 0.(2.15) We can further estimate the length L(t)for t > 0by defining v1and v2as v1(α, t) := ∂x(s(α), u(0, t), t) ∂α =∂ϕ1 ∂α (α, t), v2(β, t) := ∂x(s(1), u(β, t), t) ∂β =∂ϕ2 ∂β (β, t). To shorten notation when necessary we omit the dependency on α,βand twhen notating ϕ1,ϕ2,v1and v2. We transform v1using the local transformation Θ(·,·)corresponding to the contraction metric Mand differentiate to get d dtΘ(ϕ1, t)v1= Θ ˙(ϕ1, u(0, t), t)v1+ Θ(ϕ1, t)d dt ∂x(s(α), u(0, t), t) ∂α = Θ ˙(ϕ1, u(0, t), t)v1+ Θ(ϕ1, t)∂ ∂αf(x(s(α), u(0, t), t), u(0, t), t) = Θ ˙(ϕ1, u(0, t), t)v1+ Θ(ϕ1, t)∂f(ϕ1, u(0, t), t) ∂x ∂ϕ1 ∂α (α, t) =J(ϕ1, u(0, t), t)Θ(ϕ1, t)v1, for t > 0,α∈[0,1], where Jis the generalized Jacobian. This means that d dt(Θ(ϕ1, t)v1)TΘ(ϕ1, t)v1= 2(Θ(ϕ1, t)v1)TJ(ϕ1, u(0, t), t)Θ(ϕ1, t)v1. Since the system is uniformly contracting with rate λ,J(ϕ1(α, t), u(0, t), t)must be uniformly negative definite for all α∈[0,1],t > 0with d dt(Θ(ϕ1, t)v1)TΘ(ϕ1, t)v1≤ −2λ(Θ(ϕ1, t)v1)TΘ(ϕ1, t)v1, which using the Lemma 2.20 gives vT 1Θ(ϕ1, t)TΘ(ϕ1, t)v1≤v10(α)TΘ(0, t)TΘ(0, t)v10(α)e−2λt,(2.16) where v10(α) = v1(α, 0). Like when writing v1, to shorten notation, we notate v10 omitting the dependency on αwhen necessary. Since Θ(x, t)TΘ(x, t) = M(x, t)for all 17 x∈Rnand t∈R, estimating (2.16) using (2.14) yields σ2 minvT 1v1≤vT 1M(ϕ1, t)v1≤vT 10M(0, t)v10e−2λt ≤σ2 maxvT 10v10e−2λt. Thus v1(α, t)Tv1(α, t)≤σ2 max σ2 min v10(α)Tv10(α)e−2λt, holds, which further implies ∥v1(α, t)∥ ≤ σmax σmin ∥v1(α, 0)∥e−λt,(2.17) for t > 0,α∈[0,1]. We will similarly transform v2using the local transform Θ(·,·). To shorten and clarify notation we will notate when necessary Θ,∂f ∂w and ∂f ∂x without their dependencies on ϕ2, α,tand u. Transforming v2and differentiating yields d dtΘ(ϕ2, t)v2= Θ ˙(ϕ2, u(β, t), t)v2+ Θ(ϕ2, t)d dt ∂x(s(1), u(β, t), t) ∂β = Θ ˙(ϕ2, u(β, t), t)v2+ Θ(ϕ2, t)∂ ∂β f(x(s(1), u(β, t), t), u(β, t), t) = Θ ˙(ϕ2, u(β, t), t)v2+ Θ(ϕ2, t)(︄∂f ∂xv2+∂f ∂w ∂u ∂β (β, t))︄ =J(ϕ2, u(β, t), t)Θ(x, t)v2+ Θ(ϕ2, t)∂f ∂w∆w(t). Thus d dt(Θv2)TΘv2= 2(Θv2)TJ(ϕ2, u(β, t), t)Θv2+ 2(Θv2)TΘ∂f ∂w∆w(t),(2.18) holds. Using Young’s inequality, Lemma 2.21 with ϵ=1 λand the fact that Θand ∂f ∂w are uniformly bounded we get 2(Θv2)TΘ∂f ∂w∆w(t)≤λvT 2ΘTΘv2+1 λ∆w(t)T∂fT ∂w ΘTΘ∂f ∂w∆w(t). ≤λ(Θv2)TΘv2+csup 0≤τ≤t ∥∆w(t)∥2,(2.19) for some c > 0dependent on λand the upper bounds of Θand ∂f ∂w . Again since the system is globally contracting with rate λ > 0,J(ϕ2, u(β, t), t)must be uniformly negative definite with (Θv2)TJ(ϕ2, u(β, t), t)Θv2≤ −λ(Θv2)TΘv2for all t > 0and 18 β∈[0,1]. This with (2.18) and (2.19) implies that d dt(v2Θ)TΘv2= 2(v2Θ)J(ϕ2, u(β, t), t)Θv2+ 2(v2Θ)Θ ∂f ∂w∆w(t) ≤ −2λ(Θv2)TΘv2+λ(Θv2)TΘv2+csup 0≤τ≤t ∥∆w(t)∥2 =−λ(Θv2)TΘv2+csup 0≤τ≤t ∥∆w(t)∥2. Thus using Lemma 2.20, vT 2Θ(ϕ2, t)TΘ(ϕ2, t)v2≤v20(β)TΘ(0, t)TΘ(0, t)v20(β)e−λt +γ( sup 0≤τ≤t ∥∆w(t)∥), where v20(β) = v2(β, 0) and γ(r) = cr2 λfor r > 0implying that γis a class Kfunction. Since Θ(x, t)TΘ(x, t) = M(x, t)for all x∈Rnand t > 0, estimating this using (2.14) yields σ2 minv2(β, t)Tv2(β, t)≤σ2 maxv20(β)Tv20(β)e−λt +γ( sup 0≤τ≤t ∥∆w(t)∥). Thus v2(β, t)Tv2(β, t)≤σ2 max σ2 min v20(β)Tv20(β)e−λt +1 σ2 min γ( sup 0≤τ≤t ∥∆w(t)∥), holds, which implies ∥v2(β, t)∥ ≤ σmax σmin ∥v2(β, 0)∥e−λ 2t+γ∗( sup 0≤τ≤t ∥∆w(t)∥),(2.20) where γ∗(y) = 1 σ2 min |√︂γ(y)|for y > 0, implying that γ∗is a class Kfunction. Finally taking (2.15), estimating with (2.17) and (2.20), and integrating with respect to αand βyields ∥x(t)−xw(t)∥ ≤ L1(t) + L2(t) ≤σ2 max σ2 min L1(0)e−λt +σ2 max σ2 min L2(0)e−λ 2t+γ∗( sup t0≤τ≤t ∥∆w(t)∥) =σ2 max σ2 min ∥x0−xw0∥e−λt +σ2 max σ2 min ∥∆w(0)∥e−λ 2t+γ∗( sup t0≤τ≤t ∥∆w(t)∥) ≤σ2 max σ2 min ∥x0−xw0∥e−λt +σ2 max σ2 min sup 0≤τ≤t ∥∆w(τ)∥+γ∗( sup 0≤τ≤t ∥∆w(t)∥), for all t > 0Thus the system is input-to-state stable with respect to xw(·)as defining γ0(y) = σ2 max σ2 min y+γ∗(y)for y > 0implies that γ0a class Kfunction. Theorem 2.23. Consider system (2.3), a class of inputs Nand a set Wr⊂Rmsuch 19 that for all w(·)∈ N,w(t)∈Wrfor all t∈R. Suppose the system (2.3), is globally uniformly contracting with respect to a bounded metric for the set of inputs Wrand that for some input w(·)∈ N, there exists a corresponding bounded solution xw(t). Then the system is input-to-state convergent for the class of inputs N. Proof. This is a direct consequence of Theorem 2.17 and Lemma 2.22. While the generalization of allowing non-constant contraction metrics, compared to Demidovich’s condition in Theorem 2.13, allows for analysis of a wider class of systems the conditions of Theorem 2.23 require additional knowledge of the system. Of these while the boundedness of ∂f ∂w may be simple to determine, the existence of a bounded solution for some input may require computing solutions or the use of another Lyapunov function. Regardless this third applicable condition to achieve input-to-state convergence appears to have some potential in systems where these conditions are simple to verify. It is possible to further generalize the results by allowing the metric Mto depend on the input was well. In this case M ˙will also depend on the time derivative of w(·)and thus considering the class of inputs Is(W)with an exosystem would be of interest. However for simplicity we do not consider this generalization. 26 global uniform output regulation problem. Theorem 3.5 ([1, Thm. 4.16]).Consider the system (3.3) and the exosystem (3.4) satisfying Assumption 3.1. The global uniform output regulation problem is solvable if and only if the following conditions are satisfied: (i) There exist continuous functions π:Rm→Rnand c:Rm→Rddefined in some neighborhood of Ω(W)and satisfying the regulator equations d dtπ(w(t)) = f(π(w(t)), c(w(t)), w(t)),(3.7a) 0 = h(π(w(t)), w(t)),(3.7b) for all t∈Rand for all solutions w(t)of exosystem (3.4) satisfying w(t)∈Ω(W)for all t∈R. (ii) There exists a controller of the form (3.5) such that for any solution of the exosystem w(·)lying in the set Ω(W)the input y(w) := k(π(w), w)induces the output u(w) = c(w), and the corresponding closed-loop system is globally uniformly convergent with the UBSS property for the class of inputs Is(W). Solvability conditions for the robust and forward time variants of the uniform output regulation problem are also found in [1]. The forward time variant replaces Assumption 3.1 on the exosystem with an assumption that makes it possible to consider signals that are well defined only for positive time. The robust variant considers an additional system parameter that represents uncertainty in system behavior and measurement. The next theorem gives a sufficient condition for the solvability of the local output regulation problem. Theorem 3.6 ([1, Thm. 4.22]).Consider system (3.3) and exosystem (3.4) that satisfies the Assumption 3.2. The local uniform output regulation problem is solvable if and only if the following conditions are satisfied: (i) There exist continuous mappings π:Rm→Rnand c:Rm→Rddefined in some neighborhood of the origin W⊂Rminvariant with respect to (3.4), such that π(0) = 0, c(0) = 0, and d dtπ(w(t)) = f(π(w(t)), c(w(t)), w(t)),(3.8a) 0 = h(π(w(t)), w(t)),(3.8b) for all solutions of exosystem (3.4) satisfying w(t)∈Wfor all t∈R. (ii) There exists a controller of the form (3.5) satisfying the following conditions: 27 a) There exists a continuous mapping σ:W→Rqsatisfying σ(0) = 0 and d dtσ(w(t)) = θ(σ(w), k(π(w), w)),(3.9a) c(w(t)) = θ(σ(w(t)), k(π(w(t)), w(t))) (3.9b) b) The closed-loop system corresponding to this controller is locally uniformly convergent for the class of inputs Is(W). In the case of a static state feedback controller the mapping σcan be chosen to be zero and thus the regulator equation (3.9) can be written as c(w(t)) = θ(k(π(w(t)), w(t))).(3.10) Thus in this case we do not need to solve the equation (3.9a). For both the local and the global uniform output regulation problem the solvability conditions consist of two central parts: the regulator equations and the existence of a controller that induces the required output and guarantees stability. Stability has been studied in Chapter 2and designing the controllers will be considered in Chapter 4. Since the solutions of the regulator equations are usually used as a part in controller design, we will consider solving the regulator equations in the next section. 3.3 Methods for solving the regulator equations The regulator equations may be solved directly in some cases, but in general it may be challenging [1]. For special classes of systems there are a number methods for solving the regulator equations of the local regulation problem (3.8). For example affine systems are studied in [14] and control affine systems in [15]. A numerical method for solving the regulator equations is presented in [16]. Various techniques are also collected in [4], one of which we follow to study control affine systems specifically. The methods presented in [4,14,15,16] are local in the sense that they only guarantee the existence of π(·)and c(·)in some neighborhood of the origin, and thus can not be applied to the global output regulation problem. However, in some cases the solutions may exist in some neighborhood of Ω(W)or they may be extended to this set and then applied directly to the global problem. Additionally local results are often given at the origin, thus requiring that f(0,0) = 0. This is true for the result we discuss as well, allowing us to apply it only to specific cases when solving the global uniform output regulation problem. Nevertheless we consider them here with the note that similar methods can likely be extended to systems not satisfying f(0,0) = 0 as well through a shift of co-ordinates. 28 An affine system is is a system that is linear with respect to some or all inputs. Restricting considerations to such systems simplifies methods significantly. A control affine system has the form x˙ = f(x, w) + g(x, w)u, (3.11a) e=h(x, w),(3.11b) y=k(x, w)(3.11c) where the state is x∈Rn, reference signal is w∈Rm, control input is u∈Rd, the measure output is y∈Rland regulated output is e∈Rp. This form of system will be analyzed in this section. It is assumed that g:Rn+m→Rn×d,f:Rn+m→Rn and h:Rn+m→Rpare such that f∈C∞,h∈C∞,g∈C1and k:Rn+m→Rl is continuous. In addition we assume that f(0,0) = 0. The system (3.11) is a special case of systems of the form (3.3). The exosystem is defined as (3.4) with a set of initial conditions W. The regulator equations (3.7) for system (3.11) have the form d dtπ(w(t)) = f(π(w(t)), w(t)) + g(π(w(t)), w(t))c(w(t)),(3.12a) 0 = h(π(w(t)), w(t)).(3.12b) We consider the system (3.11) together with the exosystem (3.4) as a composite system, x˙s=fs(xs) + gs(xs)u, (3.13a) e=h(xs),(3.13b) where xs= [x, w]T,fs(xs)=[f(x, w), s(w)]Tand gs(xs) = [g(x, w),0]T. Lie derivatives are directional derivatives of a function along a vector field. Thus we can use them to simplify the notation of derivatives of the regulator error along the vector fields defined by fand g. Definition 3.7 ([4, Def. 2.35]).The Lie derivative of a function U:Rd→Rwith U∈C1along the vector field X:Rd→Rd×mis defined by LXU(x) := ∂U ∂x X(x) = d ∑︂ i=1 ∂U ∂xi Xi(x), 29 For k∈Nwhen m= 1,U∈Ckand X∈Ck−1we denote Lk XU(x):=∂Lk−1 XU ∂x X(x), L0 XU(x):=U(x). Another notion we need is the vector relative degree, which describes whether each output is connected to an input. The least number of state elements the output is dependent on to have a dependence on an input is its relative degree. The vector relative degree combines the relative degree of each output into a vector and it is required that each output is connected to a different input. We give the definition of the vector relative degree using a decoupling matrix for control affine systems. Definition 3.8. [4, Def. 2.47] A system of the form (3.13) has a vector relative degree {r1, r2, . . . , rp}at (x, w)=0when the decoupling matrix Ds(x, w) := ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ LgsLr1−1 fsh1(x, w) LgsLr2−1 fsh2(x, w) . . . LgsLrp−1 fshp(x, w) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ has full row rank at (x, w) = 0 and for ri∈N,i∈[1,2, . . . , p] LgsLk fshi(x, w)=0 for all 0≤k < ri−1and for all x, w in an open neighbourhood of the origin. The H-vector and E-vector of the system (3.11) are also necessary to simplify notation. The zero dynamics of the composite system (3.13) are defined as the dynamics when the H-vector is restricted to be zero. The E-vector on the other hand will be used to 30 solve the state feedback control for the zero dynamics. These vectors are defined as Hs(x, w) := ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ h1(x, w) Lfsh1(x, w) . . . Lr1−1 fsh1(x, w) . . . hp(x, w) Lfshp(x, w) . . . Lrp−1 fshp(x, w) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ , Es(x, w) := ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ Lr1 fsh1(x, w) Lr2 fsh2(x, w) . . . Lrp fshp(x, w) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ . The zero dynamics correspond to the solutions of the regulator equations since Hs(x, w) = 0defines the zero dynamics so that e=h(x, w) = 0. Now to further describe the zero dynamics a partition of the system’s state is considered. The partition is made with respect to the sum r:= ∑︁p i=1 ri, where riare components of a relative degree vector of the composite system (3.13). The partition is defined as x1= [xj1, . . . , xjr]Tand x2= [xjr+1 , . . . , xjn]such that rank∂Hs ∂x1(0,0) = r. Then the Implicit Function Theorem guarantees that there exists a function σ:Rn−r+m→ Rrsuch that σ(x2, w) = x1,σ(0) = 0 and 0 = Hs(x, w)|x1=σ(x2,w) for some neighborhood of the origin. Using this partition we define δ(x2, w) = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ (fjr+1 (x, w) + gjr+1 (x, w)ue(x, w))|x1=σ(x2,w) . . . (fjn(x, w) + gjn(x, w)ue(x, w))|x1=σ(x2,w) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ . Having introduced these notations we will finally give a sufficient condition guaranteeing the existence of the regulator equations. Theorem 3.9 ([4, Thm. 3.26]).Suppose the composite system (3.13) at (x, w)=0 has a relative degree {r1, r2, . . . , rp}with ∑︁p i=1 ri=r. Assume that for some continuous 31 function ue:Rn+m→Rmsatisfying Es(x, w) + Ds(x, w)ue(x, w) = 0,∀x∈Rn,∀w∈Ω(W)(3.14) there exists a C1function π2:Rm→Rn−rwith π2(0) = 0 and ∂π2 ∂w s(w(t)) = δ(π2(w(t)), w(t)),∀t∈Rm. Then π(w) = (π1(w), π2(w)) and c(w) = ue(π(w), w)where π1(w) = σ(π2(w), w)are the solutions of the regulator equations (3.12). For p<dthe function uewhich is a feedback control is not unique. Because of this a partition can be made on the control similarly to the partition of the state and the extra components can be used to modify the zero dynamics of the system. This partition is defined as u1∈Rp, u2∈Rd−psuch that there exists a function u¯e:Rn+m+d−p→Rd satisfying Es(x, w) + Ds(x, w)u¯e(x, w, u2) = 0,∀x∈Rn,∀w∈Rm,∀u2∈Rd−p.(3.15) Naturally the extent the zero dynamics can be modified is limited to the effects of the control u2, which contains the extra components. To determine the zero dynamics uniquely we must choose some feedback control u2=θ(x)where θ:Rn→Rd−pis continuous and satisfies θ(0) = 0. Using Theorem 3.9 we can form an algorithm to solve the regulator equations. It is given below following the formulation in [4]. 1) Calculate the vector relative degree, decoupling matrix, H-vector and E-vector of the composite system (3.13). 2) Solve the equation Hs(x, w)=0 for rcomponents of xin terms of wand the remaining n−rcomponents of x. Form the partition fo state defined by some x1and x2and find the mapping σ:Rn−r+m→Rr satisfying σ(x2, w) = x1. 3) Solve ue(x, w)from equation (3.14) if it is possible to do uniquely. If necessary form the partition defined by some u1and u2, solving the equation (3.15). Then choose a control θ, and set u2=θ(x)and ue(x, w) = u¯e(x, w, θ(x)). 4) Solve the differential equation x˙ji= (fji(x, w) + gji(x, w)ue(x, w))|x1=σ(x2,w), i =r+ 1, . . . , n, 32 denoting the solution by π2(w). Denote π1(w) = σ(π2(w), w). Then the solutions of the regulator equations are π(·)and c(·)such that π(w)=[π1(w), π2(w)]Tand c(w) = ue(π(w), w)for all w∈Rm. If the mappings σand π2exist globally, the solutions of the regulator equations exist globally and can be applied to the global regulation problem when f(0,0) = 0. The next example demonstrates a simple system where this is the case. Example 3.10. Consider a simple system of the form x˙ = −⎡ ⎢ ⎣ x1 x2 ⎤ ⎥ ⎦+⎡ ⎢ ⎣ ϕ2(x2) −ϕ1(x1)⎤ ⎥ ⎦+u, (3.16a) y=x1−Fw, (3.16b) w˙ = s(w),(3.16c) where x∈R2,u∈R2,w∈Rq,q∈N,F∈R1×q,s:Rq→Rqis a sufficiently smooth vector function, and ϕ1and ϕ2are sufficiently smooth scalar functions that satisfy ϕ1(0) = 0,ϕ2(0) = 0. To solve the regulator equations the algorithm presented above is followed. 1) The composite system has a relative degree of r1=r= 1. Direct computation shows that Da(x, w) = [1,0] Hs(x, w) = x1−Fw and Es(x, w) = −x1−ϕ2(x2)−Fs(w). 2) The equality Hs(x, w) = x1−Fw = 0 implies x1=Fw. Thus the partition can be chosen as x1=x1,x2=x2and σ(x2, w) = Fw. 3) Since p<m, the control may also be partitioned as u= [u1, u2]T, where only u1is significant. Then solving equation (3.14) corresponding to the system (3.16) gives ue(x, w) = ⎡ ⎢ ⎣ Fs(w) + x1−ϕ2(x2) u2 ⎤ ⎥ ⎦, where u2can be chosen freely. 4) To determine the zero dynamics the system x˙2=−x2−ϕ1(x1) + u2|x1=F w, needs to be solved. The solution depends on the choice of u2, which for example may be chosen as u2=ϕ1(w)for simplicity. Then x2= 0 solves the system. Thus solutions 33 of the regulator equations of the system (3.16) are given by π(w) = ⎡ ⎢ ⎣ Fw 0⎤ ⎥ ⎦, c(w) = ⎡ ⎢ ⎣ Fs(w) + Fw −ϕ2(0) ϕ1(Fw)⎤ ⎥ ⎦.(3.17) 34 4. CONTROLLER DESIGN FOR OUTPUT REGULATION In this chapter different approaches to controller design are discussed. The solutions of the regulator equations are assumed to be given and will be used together with stabilizing techniques to form controllers. The focus will be on two stabilization techniques for the global output regulation problem; one using quadratic stabilizability and the other universal contractivity. A simple technique related to the local output regulation problem will also be discussed. 4.1 Design strategy In controller design a large variety of methods can be used. Commonly the solutions of the regulator equations are utilized with some appropriate stability notions. While it can be possible to formulate a full controller directly, usually to achieve the desired conditions, design is decomposed into two main parts. First the goal is to design a controller such that for any solution of exosystem (3.4) lying in the ω-limit set Ω, the input yw(t) = k(π(w(t)), w(t)) induces the output uw(t) = c(w(t)), where π(·)and c(·) are solutions of the regulator equations. Second the inducing controller is combined with the system and the resultant system extended to take an input from another controller. Then a second controller will be designed such that when it is in closed-loop with the extended system, the closed-loop system is input-to-state convergent. [1] The two parts in design can be seen to correspond to the two main conditions required for the solvability of the output regulation problem given in Theorem 3.5 and Eq. (3.8). The first part essentially ensures that the control input corresponds to the solution c(·) of the regulator equations which describes the input required at the steady-state. The second part ensures that the closed-loop system is input-to-state convergent and so approaches the steady-state. Thus it is the stabilizing part and so systems for which the second controller can be formulated are called stabilizable. This two part design is in different forms present in many methods of solving output regulation problems. For example in [15] and [14] the focus is on design related to the first controller that induces the output is focused and local stabilizability is simply 35 assumed. The approach of using quadratic stabilizability in [1] concentrates on the second part of the controller design. In the simplest case where the regulator equations can be solved and the output of the exosystem is measured, the first controller can be simply chosen as u(w) = c(w(t)). However in general solving the regulator equations is not easy and will require detailed techniques for specific systems and assumptions as seen in Chapter 3. Also, the exosystem cannot always be measured and in such a case the first controller should be a type of observer of the exosystem. In addition the choice of the first controller and its properties may affect the existence of a stabilizing controller and therefore the two parts of the overall problem are not in general disconnected. [1, pg. 75] Regardless, for simplicity we will concentrate on systems with y= (x, w), that is systems where the state and the output can be measured. In addition we will assume that the regulator equations are solvable and that the solutions are known. With these assumptions we will formulate a simple approach similar to the use of quadratic stabilizability in [1] to achieve stability through contractivity and then global output regulation. However first we will introduce quadratic stability and a controller design method for quadractically stabilizable systems. While these approaches are rather restrictive as quadratic stability and contractivity both imply an exponential form of convergence, the methods are instructive and allow to achieve regulation relatively simply. For more general systems system specific Lyapunovfunctions may be used to achieve uniform convergence using Theorem 2.14. Also global output regulation is already restrictive and many systems can only be regulated locally. In such cases controller design methods differ significantly as local convergence with the UBSS property is sufficient. Then it is sufficient to consider the linearization of the system. Still the global convergence requirements can be used to formulate estimates for the regions of admissible initial conditions as demonstrated in [1] using quadratic stability. 4.2 Quadratically stabilizable systems In this section quadratic stability and quadratically stabilizable systems are introduced. Quadratic stability is practically correspondent to the Demidovich condition, so quadratically stabilizable systems present a group of systems that can be made uniformly convergent. This fact will be used for controller design. The methods discussed here are from [1] and easily incorporate inputs and numerical computation. Assuming that f:Rn×Rd×Rm→Rnis C1, to simplify notation it is defined for 42 4.4 Locally stabilizable systems The methods for designing controllers locally for the uniform output regulation problem are relatively similar to the global methods. The regulator equations are solved and then their solutions are utilized to find an applicable controller. However as the controller needs to work only in a neighborhood of the origin, the task is simpler. We will apply Theorem 3.6 in discussing a simple linear state feedback controller, following [4]. The controller needs to make the closed-loop system locally uniformly convergent. Because Theorem 2.15 guarantees that this is the case when the Jacobian of the system is Hurwitz, we consider the linearization of the system. The linearization approximates the system’s dynamics near the origin and can be used to evaluate local stability properties. The linearization of the equation (3.3a) is x˙ = ∂f ∂x(0,0,0)x+∂f ∂u(0,0,0)u=A(0)x+B(0)u. (4.7) As the linearization describes the systems dynamics in a neighbourhood of the origin through a linear system we can use linear theory. Evaluating (4.7), the pair (A(0), B(0)) is stabilizable using a linear state feedback controller u=Kx if A(0)+B(0)Kis Hurwitz for some K∈Rd×n[4]. This fact is used to formulate the next result. Theorem 4.8. Consider system (3.3) with f(0,0) = 0,y= (x, w)and exosystem (3.4) satisfying the neutral stability assumption. Suppose the regulator equations (3.8) are solvable and the corresponding continuous solutions π(·)and c(·)are defined in a some neighborhood of the origin invariant with respect to (3.4) and π(0) = 0,c(0) = 0. If there exists a matrix K∈Rd×nsuch that A(0) + B(0)Kis Hurwitz, then the local uniform output regulation problem is solved by a controller of the form u=c(w) + Kx. (4.8) Proof. The controller u=c(w) + Kx satisfies (3.9) with σ(t)=0,∀t∈Rand θ(ξ, y) = c(w) + Kx. The closed-loop system x˙ = f(x, c(w) + Kx, w) is locally exponentially convergent by Theorem 2.15 because ∂ ∂xf(x, c(w) + Kx, w) = ∂f ∂x(ξ) + ∂f ∂u(ξ)K=A(ξ) + B(ξ)K, is Hurwitz at (x, w) = (0,0). Thus by Theorem 3.6 the control u=c(w) + Kx solves the local uniform output regulation problem. 43 5. CASE STUDIES OF OUTPUT REGULATION In this chapter we apply results on controller design from Chapter 4to two variants of a harmonic oscillator and a neural network model. First we consider a standard model of the harmonic oscillator and secondly a similar model with a nonlinear damping term. Finally, the Hopfield model [5] representing a neural network is analyzed. 5.1 Harmonic oscillator In this section we consider a simple harmonic oscillator and design a controller that achieves global output regulation. A harmonic oscillator can be modelled using a linear system [13]. As will be demonstrated, the system is quadratically stabilizable and the regulator equations are easily solvable. Knowing this we can use Theorem 4.4 to design a controller. The system to be considered is x˙ = ⎡ ⎢ ⎣ 0 1 −k m0⎤ ⎥ ⎦x+⎡ ⎢ ⎣ 0 k m ⎤ ⎥ ⎦u, (5.1a) e=Cx −Fw, (5.1b) w˙ = Sw, (5.1c) where x(t), w(t), e(t)∈R2,u(t)∈R, for all t∈R,k, m > 0and S=⎡ ⎢ ⎣ 0 1 −1 0⎤ ⎥ ⎦, C =[︃1 0]︃, F =[︃a b]︃,(5.2) for some a, b ∈R. Assume that w0∈Wr={w∈R2:∥w∥ ≤ r}for some r > 0. The solution of (5.1c) is w(t) = ⎡ ⎢ ⎣ cos(t) sin(t) −sin(t) cos(t)⎤ ⎥ ⎦⎡ ⎢ ⎣ w1(0) w2(0)⎤ ⎥ ⎦.(5.3) Thus w(·)is bounded and w(t)∈Wr∀t∈Rimplying that Wris invariant with respect to the system (5.1c). Therefore w(·)∈Is(Wr)and Assumption 3.1 is satisfied, because 44 Wris compact. The choices of Cand Frestrict the system to a case where the goal is to track the position of the oscillator to a linear combination of sine and cosine signals. This restriction is useful as it is not possible in general to control both the position and velocity of a harmonic oscillator separately due to the availability of a single control input. The regulator equations of this system are d dtπ1(w(t)) = π2(w(t)), d dtπ2(w(t)) = −k mπ1(w(t)) + k mc(w(t)), 0 = π1(w(t)) −aw1(t)−bw2(t). Thus we get π1(w(t)) = aw1(t) + bw2(t), which using (5.1) implies π2(w(t)) = d dt(aw1(t) + bw2(t)) = aw2(t)−bw1(t). Therefore we have c(w(t)) = m k(d dtπ2(w(t)) + k mπ1(w(t))) =m k(−aw1(t)−bw2(t)) + aw1(t) + bw2(t). Thus the solutions of regulator equations are π1(w(t)) = aw1(t) + bw2(t), π2(w(t)) = aw2(t)−bw1(t), c(w(t)) = (1 −m k)(aw1(t) + bw2(t)). The system is quadratically stabilizable with for example K=[︃k−m k−m k]︃, P =⎡ ⎢ ⎣ 3 1 1 2⎤ ⎥ ⎦.(5.4) Thus by Theorem 4.4 the control can be chosen as u=k−m kRw +K(x−Nw), 45 (a) State of the system (b) The regulated output of the system Figure 5.1. The simulated signals of system (5.1) where R=[︃a b]︃, Nw =⎡ ⎢ ⎣ a b −b a⎤ ⎥ ⎦w=π(w). To verify the functionality of the controller SimulinkTM is used to simulate the harmonic oscillator and its controller in a closed-loop system. For this purpose choose k= 1, m= 2,a= 3,b=−1,w0= [1,−2]Tand x0= [−1,0]T. The simulation results are shown in Figure 5.1. The closed-loop system clearly achieves asymptotic output zeroing as the regulated output tends to zero and convergence is fast. Thus we have verified that the controller works. 5.2 Harmonic oscillator with nonlinear damping In this section a nonlinear damping term is added to the oscillator considered in the previous section. As we will see, due to the damping term the system is no longer quadratically stabilizable. The regulator equations are solvable even with the damping term and thus we design a controller solving the local output regulation problem. 46 We consider the damped harmonic oscillator x˙ = ⎡ ⎢ ⎣ 0 1 −k m−d(1 + hx2 2)⎤ ⎥ ⎦x+⎡ ⎢ ⎣ 0 k m ⎤ ⎥ ⎦u(5.5) e=Cx −Fw (5.6) w˙ = Sw, (5.7) where h, m, d, k > 0,x(t), u(t), w(t), e(t)∈R2, for all t∈Rand S=⎡ ⎢ ⎣ 0 1 −1 0⎤ ⎥ ⎦, C =[︃1 0]︃, F =[︃a b]︃,(5.8) for some a, b ∈R. Thus as in the previous example w(·)∈Is(Wr)and Assumption 3.1 is satisfied. The regulator equations of the system are d dtπ1(w(t)) = π2(w(t)) d dtπ2(w(t)) = −k mπ1(w(t)) −d(π2(w(t)) + hπ2(w(t))3) + k mc(w(t)) 0 = π1(w(t)) −aw1(t)−bw2(t). Thus we get π1(w(t)) = aw1(t) + bw2(t), which implies π2(w(t)) = d dt(aw1(t) + bw2(t)) = aw2(t)−bw1(t). Therefore we have c(w(t)) =m k(d dtπ2(w(t)) + k mπ1(w(t)) + d(π2(w(t)) + hπ2(w(t))3)) =m k(−aw1(t)−bw2(t)) + aw1(t) + bw2(t) +md k(aw2(t)−bw1(t) + h(aw2(t)−bw1(t))3). Thus the solutions of the regulator equations are π1(w(t)) = aw1(t) + bw2(t), π2(w(t)) = aw2(t)−bw1(t) c(w(t)) = (1 −m k)(aw1(t) + bw2(t)) + md k(aw2(t)−bw1(t) + h(aw2(t)−bw1(t))3). 47 For the system (5.5) we have A(ζ) = ⎡ ⎢ ⎣ 0 1 −k m−d(1 + 3hx2 2).⎤ ⎥ ⎦, B(ζ) = ⎡ ⎢ ⎣ 0 k m ⎤ ⎥ ⎦, where ζ= (x, w, u). For the system to be quadratically stable the inequality PA(ζ) + A(ζ)TP < 0,∀ζ∈R6 must hold for some P=PT>0. Without loss of generality we can write, P=⎡ ⎢ ⎣ 1p p p2 ⎤ ⎥ ⎦,(5.9) where p, p2∈R. Then for quadratic stability the following inequalities must hold −pk m<0 −p2d(1 + hx2 2) + p < 0 pk m(p2d(1 + hx2 2)−p)−(1 −k mp2−d(1 + hx2 2)p)2>0 for all x2∈R. Thus p > 0and since at large values of x2, only the terms with x2 2are significant we have p2pk md(1 + hx2 2)−d2p2(1 + hx2 2)2>0, implying p2 k m−dp(1 + hx2 2)>0. This however is not possible for an arbitrarily large x2with any choice of p2as p > 0. Therefore the system is not quadratically stable. Also the system is not quadratically stabilizable as it is not possible to suppress the growing term linearly. The system is still locally exponentially convergent as A(0) = ⎡ ⎢ ⎣ 0 1 −k m−d,⎤ ⎥ ⎦ has eigenvalues λ1=1 2(−d+√︂d2−4k m),λ2=1 2(−d−√︂d2−4k m). For all values d, k, m > 0, the real parts are negative and so the matrix is Hurwitz. Thus by Theorem 4.8, the control u=c(w)solves the local uniform output regulation problem. 48 5.3 Hopfield neural networks Two models of neural circuits are analyzed in [5] using the notion of strong infinitesimal contractivity, which corresponds to the Demidovich condition, Theorem 2.13. In this section we solve the regulator equations for the Hopfield model and design a controller using quadratic stability to solve the output regulation problem. Output regulation of neural networks has not been considered widely. Examples can be found mainly for Boolean networks, which are considered for example in [17]. The Hopfield model is defined by x˙ = −x+NΨ(x) + Bu := fH(x, u)(5.10) where x∈Rnis the state, u∈Rdis an input, N∈Rn×n,B∈Rn×dand Ψ : Rn→Rn is a diagonal map Ψ(x)=[ψ1(x1), . . . , ψn(xn)]Twith component functions ψi:R→R. We assume that these component functions, called activation functions, are differentiable and that 0≤∂ψi ∂y (y)<1for all y∈R. In [5, Lemma 3.21] conditions for strong infinitesimal contraction of the other model introduced in [5], the firing rate model, are analyzed. Different norms, defined based on logarithmic functions and ℓ1,ℓ2and ℓ∞-norms, are used to formulate the conditions. Here we consider quadratic stability for the Hopfield model. This approach corresponds to strong infinitesimal contraction with an ℓ2-norm and is chosen due to the similarity of the methods for different norms and models. In the context of output tracking, we consider the system x˙ = fH(x, u)(5.11a) e=Cx −Fw (5.11b) y= (x, w)(5.11c) w˙ = s(w),(5.11d) where x∈Rnis the state, e∈Rpis the regulator error, u∈Rdis the control input, w∈Rmis the reference signal, y∈Rn+mis the measured output, C∈Rp×n, F∈Rp×m,B∈Rn×dand s:Rm→Rmdescribes the exosystem. First we consider solving the regulator equations. In general Cis not invertible and in such cases direct computation is not feasible and different methods need to be utilized depending on the properties of Cand A. We use the method outlined in Section 3.3 to solve the regulator equations for two cases, one where all nodes can be controlled, that is Chas full row rank and Bis invertible and another where the system models three nodes with a single input and a single output (SISO). 49 5.3.1 Hopfield system with control inputs on each node First assume that Chas full row rank and Bis invertible, implying n=d. Also assume that f(0,0) = 0. We now follow the steps of the algorithm discussed in Section 3.3. 1) The composite system has a relative degree of {1,...,1}with r=pbecause Da(x, w) = CB has full row rank. Then Hs(x, w) = Cx −Fw and Es(x, w) = C(−x+NΨ(x)) −Fs(w). 2) Since Chas full row rank, the equation Cx −Fw = 0 can be solved for r=pvalues of x, in terms of wand the remaining n−rcomponents of xsuch that the chosen r components of xdefine the partition x1,x2satisfying rank∂Hs ∂x1(0) = r. The solution of Cx −Fw = 0 and the partition define the mapping σ:Rn−p+m→Rpglobally. 3) Solving Es(x, w)+Ds(x, w)ue(x, w) = C(−x+NΨ(x))−Fs(w)−CBue(x, w)=0 gives ue(x, w) = B−1(x−NΨ(x) + C+Fs(w)) as a solution. The matrix C+∈Rn×p is a left inverse of Cthat exists as Cis of full row rank. Since ueis not uniquely defined for p < d, it is possible to form a partition of uwhere only the elements u1are needed for output regulation. The function uecan then depend on a control u2. For simplicity we consider the case where u2is constant. Then we can choose any C+, dependent on the wished effects on the zero dynamics of the system. 4) To determine the zero dynamics the differential equation x˙ji=−xji+ n ∑︂ k=1 Njikψk(xk) + xji− n ∑︂ k=1 Njikψk(xk) + n ∑︂ k=1 (C+)jikFksk(w) = n ∑︂ k=1 (C+)jikFksk(w), where si(w)is the i:th element of the vector function s(w)and Fithe i:th row of F, needs to be solved. Thus solutions of the zero dynamics depend on C+,Fand w, and are of the form π2(w) = C ¯+Fw, where C ¯+∈R(n−p)×pis dependent on C+and the partition defined. Then π1(w) = σ(π2(w), w)and c(w) = B−1(x−NΨ(x)+C+Fs(w)). For example a system where the goal is to control a weighted sum of pairs of nodes may be modelled with C= ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ c1c20 0 . . . 0 0 0 0 c3c4. . . 0 0 . . .. . .. . .. . ..... . .. . . 0000. . . cn−1cn ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ . with even nand c2i−1>0for i∈ {1,2,...,n 2}. Then the matrix has full row rank and r=m=n 2. The equation Hs(x, w) = Cx −Fw = 0 can be solved for xi, 50 i∈ {1,3, . . . , n −1}in terms of xj,j∈ {2,4...,n}and w. This yields xi=(Fw)k ci −ci+1xi+1 ci , where k=i+1 2,i∈ {1,3, . . . , n −1}. Using this a partition x1={x1, x3, . . . , xn−1}, x2={x2, x4, . . . , xn}, can be formed. The mapping σ:Rn→Rn 2can be globally defined componentwise with σi(x2, w) = (F w)i c2i−1−c2ix2i c2i−1,i∈ {1,2, . . . n 2}. A left inverse of Cis C+= ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 c10. . . 0 0 0 . . . 0 01 c3. . . 0 . . .. . ..... . . 0 0 . . . 1 cn−1 0 0 . . . 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ , which we choose for simplicity. The zero dynamics can be then solved from xj ˙ = ∑︁n k=1 c+ jkFksk(w)=0,j∈ {2,4...,n}and so xj=xj(0) are the solutions for this choice of C+. Then we have π2(w) = 0,π1 i(w) = (F w)i c2i−1, implying that πand care the solutions of the regulator equations where π(w(t)) = C+Fw(t)and c(w(t)) = B−1(C+Fw(t)−NΨ(C+Fw(t)) + C∗Fs(w(t))) for all t∈R. 5.3.2 Three node Hopfield system with a single input and output Next we consider a SISO example of the Hopfield model with 3 nodes. We choose N=⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 0 0 1 0 0 0 1 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ , B =⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 0 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ , C =⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 0 1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ T , F =[︃F1F2]︃.(5.12) We assume that the exosystem is w˙ = Sw, where S=⎡ ⎢ ⎣ 0 1 −1 0⎤ ⎥ ⎦,(5.13) so that as in the previous examples in Section 5.1 and Section 5.2, Assumption 3.1 is satisfied. Likewise we assume that y= (x, w). 51 Thus the composite system (3.13) for these choices is ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ x˙1 x˙2 x˙3 w˙1 w˙2 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ −x1 −x2+ψ1(x1) −x3+ψ2(x2) w2 −w1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ + ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 0 0 0 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ u, (5.14a) e=x3−F1w1−F2w2,(5.14b) y= (x, w),(5.14c) w˙ = Sw. (5.14d) We further choose the activation functions ψ1as a hyperbolic tangent function and ψ2 as a smoothened parametric rectified linear unit (PReLU) function to ensure solvability. These activation functions are used within the Hopfield model for example in [18]. The functions are defined as ψ1(x1) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ x1x1>1 mx1x1<1 ψ∗(x1)otherwise. ψ2(x2) = tanh(x2), where 0< m < 1and ψ∗(·)is a smooth approximation of the PreLU function on the interval [−1,1] constructed using mollifiers, which are smooth differentiable functions. Thus these activation functions satisfy the necessary assumptions as they are differentiable and clearly 0≤∂ψi ∂xi(xi)≤1as ∂ψ2 ∂x2(x2)=1−tanh2(x2). We follow the algorithm in Section 3.3 to solve the regulator equations of (5.14). 58 [14] Cheng, D., Tarn, T. and Spurgeon, S. On the design of output regulators for nonlinear systems. eng. Systems and control letters 43.3 (2001), pp. 167–179. issn: 0167-6911. [15] Huang, J. On the solvability of the regulator equations for a class of nonlinear systems. eng. IEEE transactions on automatic control 48.5 (2003), pp. 880–885. issn: 0018-9286. [16] Numerical method for the solution of the regulator equation with application to nonlinear tracking. eng. Automatica (Oxford) 44.5 (2008), pp. 1358–1365. issn: 0005-1098. [17] Li, H., Xie, L. and Wang, Y. Output Regulation of Boolean Control Networks. eng. IEEE transactions on automatic control 62.6 (2017), pp. 2993–2998. issn: 0018-9286. [18] Wang, C., Liang, J. and Deng, Q. 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