Well-Posedness of Reset Control Systems as State-Dependent Impulsive Dynamical Systems

Abstract

Reset control systems are a special type of state-dependent impulsive dynamic systems, in which the time evolution depends both on continuous dynamics between resets and the discrete dynamics corresponding to the resetting times. This work is devoted to investigate well-posedness of reset control systems, taking as starting point the classical definition of Clegg and Horowitz. Well-posedness is related to the existence and uniqueness of solutions, and in particular to the resetting times to be well defined and distinct. A sufficient condition is developed for a reset system to have well-posed resetting times, which is also a sufficient condition for avoiding Zeno solutions and, thus, for a reset control system to be well-posed.

1. Introduction

Reset control systems [1–3] are a type of impulsive hybrid systems, in which the system state (or part of it) is reset at the instants it crosses some reset set. Impulsive hybrid systems are an active area of systems theory that has been developed in the last years. Two classical monographs are [4, 5], where reset control systems without external inputs are a particular case of autonomous system with impulse effects.

In this work, reset control systems will be formulated as a particular type of impulsive dynamical systems (IDSs), more specifically as state-dependent IDS, following the impulsive/hybrid dynamic system framework developed in [6]. In this framework, existence and uniqueness of solutions over a forward time interval is based on the well-posedness of resetting times.

The main goal of this work is to investigate well-posedness of reset control systems taking as starting point the classical definition of Clegg and Horowitz. This formulation has been also followed in several recent works, for example [7, 8] and references therein, and also [9–12]. As it is well known, reset control systems and IDS in general can exhibit behaviors that can be pathological from a control point of view. As it has been shown in [6], definition of IDS solutions has to deal with the problem of beating, deadlock, and Zenoness. In general, a reset control system will be considered well-posed if the resetting times are well-posed (they are well defined and are distinct), meaning that a number of beating or pulse phenomena are avoided, and in addition Zeno solutions do not exist.

As it will be shown, simple geometric conditions will be derived for avoiding the presence of these pathological behaviours. In Section 2, preliminary material and basic results are given. Section 3 develops a result for reset control systems to have well-posed resetting times. Finally, Section 4 tackles with the problem of existence of Zeno solutions.

Notation. In this work, ℝ⁺ is the set of nonnegative real numbers, and (𝐱,𝐲) denotes the column vector 𝐱𝐲. In addition, 𝐞_𝑖∈ℝ^𝑛 stands for the unit vector (0⋯010⋯0)^𝑇 in which the 𝑖th-component is 1. For a set ℳ⊂ℝ^𝑛, ℳ is the closure of ℳ. On the other hand, for a linear and time-invariant system with state space matrices (𝐴,𝐶), the subspace of unobservable states is given by the null space of the observability matrix 𝑂, where 𝐶⋯𝑂=𝐶𝐴𝐶𝐴^𝑛−1. (1.1)

2. Preliminaries and Problem Setup

This work deals with a special class of hybrid systems called impulsive dynamical systems (IDSs, [6]). In particular, with state-dependent IDS having the form 𝐱̇(𝑡)(𝑡)(𝑡)𝐱𝑡=𝐴𝐱,𝐱∉ℳ,₊=𝐴_𝑅𝐱(𝑡)(𝑡)𝐱(0),𝐱∈ℳ,=𝐱₀, (2.1) where 𝐱(𝑡)∈ℝ^𝑛, 𝑡≥0, is the system state at the instant 𝑡, ℳ⊂ℝ^𝑛 is the reset set, and 𝐴 and 𝐴_𝑅 are matrices of dimension 𝑛×𝑛. The following material, including definition of IDS solutions and well-posedness of resetting times, is strongly based on [6]. The first equation in (2.1) will be referred to as the continuous-time dynamics or simply base system dynamics, while the second equation in (2.1) will be referred to as the resetting law. When at some resetting time 𝑡≥0, 𝐱(𝑡)∈ℳ is true (the reset condition is active, and a crossing is performed), the state 𝐱(𝑡) jumps to 𝐱(𝑡⁺)=𝐴_𝑅𝐱(𝑡)∈ℳ_ℛ; it will be assumed that resetting times are well-posed, that is, they are well defined and distinct for any initial condition. Otherwise, the state 𝐱(𝑡) evolves with the base system dynamics. The set ℳ_ℛ will be referred to as the after-reset set.

A function 𝐱∶𝐼_𝐱0→ℝ^𝑛 is a solution of the IDS system (2.1) on the interval 𝐼_𝐱0⊆ℝ, with initial condition 𝐱(0)=𝐱₀, if 𝐱(⋅) is left-continuous, and 𝐱(𝑡) satisfies (2.1) for all 𝑡∈𝐼_𝐱0. For further discussion on solutions to impulsive differential equations and IDS solutions, see [4–6]. In general, there exists a unique solution 𝜓(𝑡)=𝑒^𝐴𝑡𝜓₀ of the (continuous) base system with initial condition 𝜓(0)=𝜓₀ on [0,∞), for any 𝜓₀∈ℝ^𝑛. Informally speaking, the solution 𝐱 of the IDS (2.1) from the initial condition 𝐱(0)=𝐱₀ is given by 𝐱(𝑡)=𝑒^𝐴𝑡𝐱₀ for 0<𝑡≤𝑡₁, where 𝑡₁ is the first resetting time satisfying 𝐱(𝑡₁)∈ℳ. Then, the state is instantaneously transferred to 𝐴_𝑅𝐱(𝑡₁) according to the resetting law. The solution 𝐱(𝑡), 𝑡₁<𝑡≤𝑡₂ (being 𝑡₂ the second resetting time given by 𝑒^{𝐴(𝑡2−𝑡1)}𝐱(𝑡₁)∈ℳ) is given by 𝐱(𝑡)=𝑒^{𝐴(𝑡−𝑡1)}𝐱(𝑡₁)=𝑒^{𝐴(𝑡−𝑡1)}𝐴_𝑅𝑒^𝐴𝑡1𝐱₀, and so forth. Note that the solution 𝐱 of (2.1) is left-continuous, that is, it is continuous everywhere except at the resetting times 𝑡_𝑘, and 𝐱𝑡_𝑘=lim_𝜖→0+𝐱𝑡_𝑘,𝐱𝑡−𝜖_+𝑘=lim_𝜖→0+𝐱𝑡_𝑘+𝜖=𝐴_𝑅𝐱𝑡_𝑘. (2.2)

2.1. Well-Posed Resetting Times and Zeno Solutions

Two standard assumptions for well-posedness of resetting times of state-dependent IDS [6], that will be used in this work, are (A1)𝐱(𝑡)∈ℳ⧵ℳ⇒ there exists 𝜖>0 such that 𝐱(𝑡+𝛿)∉ℳ, for all 𝛿∈(0,𝜖).(A2)𝐱∈ℳ⇒𝐴_𝑅𝐱∉ℳ.

Note that for a particular solution 𝐱(⋅), the first resetting time 𝑡₁ is well defined since min{𝑡∈ℝ⁺∶𝜓(𝑡,0,𝐱₀)=𝑒^𝐴𝑡𝐱₀∈ℳ} exists (and thus, it is unique by uniqueness of solutions of the base system). Analogously, for 𝑘=2,3,…, the resetting time 𝑡_𝑘 is well defined since again min{𝑡∈ℝ⁺∶𝜓(𝑡,𝑡_𝑘−1,𝐴_𝑅𝐱(𝑡_𝑘−1))∈ℳ} exists, and in addition 0=𝑡₀<𝑡₁<𝑡₂<⋯, for any 𝐱₀∈ℝ^𝑛. Here, 𝜓(𝑡,𝑡₀,𝜓₀) is a solution of the base system with initial condition 𝜓(𝑡₀)=𝜓₀, that is, 𝜓(𝑡,𝑡₀,𝜓₀)=𝑒^{𝐴(𝑡−𝑡0)}𝜓₀. Therefore, if for any initial condition 𝐱₀∈ℝ^𝑛 the resetting times are well defined, functions 𝜏_𝑘∶ℝ^𝑛→[0,∞) are defined for 𝑘=1,2,…, where 𝑡_𝑘=𝜏_𝑘(𝐱₀) is the 𝑘th resetting time, and by definition 𝜏₀(𝐱₀)=0. Note that for a particular solution, there may exist no crossings, a finite or a infinite number of crossings, and in a finite or infinite time interval 𝐼_𝐱0, and that functions 𝜏_𝑘(𝐱₀) are single valued by uniqueness of the base system solutions.

Since by assumptions A1 and A2, the resetting times are well defined and distinct, and since for a given initial condition, the solution to the base system differential equation exists and is unique, it follows that the solution of the IDS (2.1) also exists and is unique over a forward time interval [6]. For the IDS (2.1) with well-posed resetting times, a Zeno solution 𝐱_𝒵(⋅) exists on the interval 𝐼_𝐱0=[0,𝑇] for some initial condition 𝐱_𝑧(0)=𝐱₀∈ℝ^𝑛, if there exists an infinite sequence of resetting times (𝜏_𝑘(𝐱₀))^∞_𝑘=0, and a positive number 𝑇, such as 𝜏_𝑘(𝐱₀)→𝑇 as 𝑘→∞. Note that the solution is not defined beyond the time 𝑇. If there does not exist Zeno solutions for any initial condition, then the solutions of the IDS (2.1) exists and are unique for any initial condition on 𝐼_𝐱0=[0,∞).

Note that conditions A1 and A2 can be interpreted as: (i) states that belong to the closure of ℳ, and does not belong to ℳ, evolve with the continuous base dynamics for some finite time interval (A1); (ii) after-reset states are not elements of the reset set ℳ (A2). In other words, for resetting times to be well-posed a IDS system solution can only reach ℳ through a point belonging to both ℳ and its boundary 𝜕ℳ; and if a solution reaches a point in ℳ that is on its boundary, then it is instantaneously removed from ℳ. Roughly speaking, condition A1 avoids the presence of deadlock, while condition A2 avoids beating or livelock (using these terms in the sense given in [6]).

In the following, two examples of ill-posed (not well-posed) second-order IDS are shown to illustrate conditions A1 and A2. In both cases, the base system corresponds to some matrix 𝐴∈ℝ^2×2, making the equilibrium point 𝐱=𝟎 a center, and the resetting law is 𝑥₁(𝑡⁺)=𝑥₁(𝑡),𝑥₂(𝑡⁺)=0. (a)(Figure 1(a)), here the reset set ℳ^𝑎 is the rectangle ℳ^𝑎=𝑥₁𝑥₂∈ℝ²∶−1≤𝑥₁≤1,0.7<𝑥₂≤1, (2.3) and the after reset set is ℳ^𝑎_ℛ=𝑥₁𝑥₂∈ℝ²∶−1≤𝑥₁≤1,𝑥₂=0, (2.4) that is, the interval [−1,1] in the 𝑥₁-axis. From an initial condition in the point 𝐀, the trajectory reaches the reset set ℳ^𝑎 at some point belonging to both ℳ^𝑎 and its boundary 𝜕ℳ^𝑎. Thus, the first resetting time 𝜏₁(𝐀) is well defined, and then the trajectory jumps to the point 𝐁. From the point 𝐁, the system trajectory evolves as the base system until it reaches a point 𝐂 that belongs to 𝜕ℳ^𝑎 but not to ℳ^𝑎. Thus, condition A1 is not satisfied since the trajectory enters into ℳ for any arbitrarily small time after reaching the point 𝐂 (the second resetting time 𝜏₂(𝐀) is undefined). (b)(Figure 1(b)), ℳ^𝑏 is the grey region (it contains its boundary), and the after reset set is ℳ^𝑏_ℛ=ℳ^𝑎_ℛ. Note that the ℳ^𝑏∩ℳ^𝑏_ℛ={𝐂,𝐃}. In this case, a trajectory starting from the point 𝐀 reaches ℳ^𝑏 at the point 𝐁 which belongs both to ℳ^𝑏 and its boundary (thus, A1 is satisfied, and the first resetting time 𝜏₁(𝐀) is well defined). After that, the trajectory jumps to the point 𝐂 that belongs both to ℳ^𝑏_ℛ and ℳ^𝑏 and then make an infinite number of resets (condition A2 is not satisfied).

(a)

(b)

Figure 1 

Examples of ill-posed IDSs.

2.2. Reset Control Systems

In this work, reset control systems will be represented by the state-dependent IDS (2.1). Consider the feedback system given by Figure 2, where the (single input-single ouput) plant is described by the following: 𝐱̇𝑃∶^𝑝(𝑡)=𝐴_𝑝𝐱_𝑝(𝑡)+𝐵_𝑝𝑢(𝑡),𝐱_𝑝(0)=𝐱_𝑝0,𝑦(𝑡)=𝐶_𝑝𝐱_𝑝(𝑡), (2.5) and the (single input-single output) reset compensator is given by the impulsive differential equation: 𝐱̇𝐶∶^𝑟(𝑡)=𝐴_𝑟𝐱_𝑟(𝑡)+𝐵_𝑟𝑒(𝑡),𝐱_𝑟(0)=𝐱_𝑟0(𝑡)𝐱,𝑒≠0,_𝑟(𝑡+)=𝐴_𝜌𝐱_𝑟(𝑡)(𝑡)𝑣(𝑡),𝑒=0,=𝐶_𝑟𝐱_𝑟(𝑡). (2.6) Here 𝑛_𝑝 is the dimension of the state 𝐱_𝑝, and 𝑛_𝑟 is the dimension of the state 𝐱_𝑟. 𝐴_𝜌 is a diagonal matrix with (𝐴_𝜌)_𝑖𝑖=0 if the state (𝐱_𝑟)_𝑖 of the compensator is to be reset, and (𝐴_𝜌)_𝑖𝑖=1 otherwise. In general, it is assumed that the last 𝑛_𝜌 compensator states are set to zero at the resetting times. In the case of a full-reset compensator, all the elements of 𝐴_𝜌 are 0. Consider the closed loop autonomous unforced system given by 𝑒(𝑡)=−𝑦(𝑡), 𝑢(𝑡)=𝑣(𝑡), and define the (closed loop) state 𝐱=(𝐱_𝑝,𝐱_𝑟) of dimension 𝑛=𝑛_𝑝+𝑛_𝑟, being 𝑛_𝑟=𝑛_𝜌+𝑛_𝜌. The result is that the reset control system is given by the state-dependent IDS 𝐱̇(𝑡)(𝑡)(𝑡)𝐱𝑡=𝐴𝐱,𝐱∉ℳ,₊=𝐴_𝑅𝐱(𝑡)(𝑡)𝐱(0),𝐱∈ℳ,=𝐱₀,𝑦(𝑡)=𝐶𝐱(𝑡), (2.7) where 𝐴𝐴=^𝑝𝐵_𝑝𝐶_𝑟−𝐵_𝑟𝐶_𝑝𝐴_𝑟𝐴_𝑅,=diag𝐼_𝑛𝑝,𝐴_𝜌=diag𝐼_𝑛𝑝,𝐼_𝑛𝜌,𝑂_𝑛𝜌𝐶,𝐶=_𝑝,𝑂,ℳ=𝐱∈ℝ^𝑛∶𝐶𝐱=0. (2.8)(2.9)(2.10)(2.11)

Figure 2 

Reset control system.

In control practice, it is required that reset control system solutions 𝐱(𝑡) will be well-posed in the sense that they exist and are unique for 𝑡≥0. By definition, the reset control system (2.7)–(2.11) is well-posed if for any initial condition 𝐱₀∈ℝ^𝑛, a solution 𝐱 exists and is unique on 𝐼_𝐱0=[0,∞). As a state-dependent IDS system, the reset control system is well-posed if resetting times are well-posed, and in addition, there do not exist Zeno solutions for any initial condition. In Section 3, the well-posedness of resetting times for reset control systems will be investigated. The existence of Zeno solutions is explored in Section 4.

3. Reset Control Systems with Well-Posed Resetting Times

The reset control system (2.7)–(2.11) is a particular case of the state-dependent IDS (2.1), with a reset set (it will be referred to as reset surface) ℳ=𝒩(𝐶), the null space of 𝐶, and with an after-reset set, or after-reset surface, ℳ_ℛ=𝐴_𝑅(ℳ). Since 𝐴_𝑅 is a projector, it results that in general ℳ_ℛ∩ℳ≠∅, and thus, without any modification, the reset control system (2.7)–(2.11) does not have well-posed resetting times. This problem was detected in [7, 8], and a partial solution was given by redefining both sets as ℳ={𝐱∈ℝ^𝑛⧵ℳ_𝑅∶𝐶𝐱=0}, and ℳ_ℛ={𝐱∈ℝ^𝑛∶𝐶𝐱=0,(𝐼−𝐴_𝑅)𝐱=0}, where after-reset states are simply removed from ℳ as given by (2.11). Since 𝐴_𝑅 is a projector, then the set {𝐱∈ℝ^𝑛∶(𝐼−𝐴_𝑅)𝐱=0} is the column space of 𝐴_𝑅, that is ℛ(𝐴_𝑅). Thus, the above definitions are equivalent to ℳ_ℛ=ℛ𝐴_𝑅∩𝒩(𝐶),ℳ=𝒩(𝐶)⧵ℳ_ℛ. (3.1)

Proposition 3.1. The reset system (2.7), with ℳ and ℳ_ℛ given by (3.1), has well-posed resetting times if ℳ_ℛ∩𝒩𝒪_base={𝟎}, (3.2) where 𝒪_base is the base system observability matrix.

Proof. If they do exist, by (3.1), resetting times are distinct since ℳ∩ℳ_ℛ=∅ is equivalent to A2, thus, the proof is centered in their existence. Note that by (3.1), ℳ⧵ℳ=ℳ_ℛ in A1. By time invariance, A1 is equivalent to 𝐱₀∈ℳ_ℛ⇒𝐱(𝑡)∉ℳ for 𝑡∈(0,𝜖) and some 𝜖>0. Here 𝜖 depends on 𝐱₀, but the dependence will not be explicitly shown by simplicity. In general, given 𝐱₀∈ℳ_ℛ, the first crossing with ℳ is at the instant 𝑡₁=𝜏₁(𝐱₀), and finally A1 is equivalent to the existence of a lower bound 𝜖>0 for 𝑡₁, for any given 𝐱₀∈ℳ_ℛ. From (3.1), 𝑡₁ is simply given by 𝑡₁=min{𝑡>0∣𝐶𝑒^𝐴𝑡𝐱₀=0∧𝑒^𝐴𝑡𝐱₀∉ℛ(𝐴_𝑅)}. If 𝜖₁>0 is by definition a lower bound of the set {𝑡>0∣𝐶𝑒^𝐴𝑡𝐱₀=0}, and 𝜖₂>0 is by definition a lower bound of the set {𝑡>0∣𝑒^𝐴𝑡𝐱₀∉ℛ(𝐴_𝑅)} (both depending on 𝐱₀), then 𝜖=max{𝜖₁,𝜖₂}≤𝑡₁. By simplicity, consider in first instance that 𝐴 has distinct eigenvalues; in this case (see the Appendix) 𝑡>0∣𝐶𝑒^𝐴𝑡𝐱₀=0=𝑡>0∣𝑓₁(𝑡)∶=𝑒^𝑇𝝀𝑡𝑈(𝝀)𝒪_base𝐱₀=0. (3.3) Since 𝑓₁(⋅) is a sum of exponentials (in fact it is a Bohl function [13]), and thus, it is an analytical function, it is true that 𝑓₁(𝑡) is either zero for all 𝑡≥0 or has isolated zeros. As a result, two options are possible as follows 𝜖₁=0 if 𝑥₀∈𝒩(𝒪_base) (𝑓₁(𝑡)=0, for all 𝑡∈[0,∞)), or there exist an interval (0,𝜖₁) in which 𝑓₁(𝑡)≠0 for some 𝜖₁>0. Now, if condition (3.2) is satisfied then for any 𝐱₀∈ℳ_ℛ only the second option is possible, and thus, 𝑡₁≥max{𝜖₁,𝜖₂}≥𝜖₁, that is, A1 is satisfied (condition (3.2) is sufficient for A1). Finally, in the case in which the eigenvalues of 𝐴 may be repeated, similar expressions may be found for 𝑓₁ (see the Appendix), and the above arguments are again applied.

Remark 3.2. Note that, in particular, well-posedness of resetting times is obtained if the base system is observable, since in this case 𝒩(𝒪_base)={𝟎}. But some unobservable base linear systems can also produce reset systems with well-posed resetting times, as far as the after-reset surface does not contain unobservable states (different to 𝟎).

Remark 3.3. Note that, in general, Proposition 3.1 may be applied to reset systems given by (2.7) with arbitrary values 𝐴, 𝐴_𝑅, and 𝐶 as far as the developed conditions apply (not necessarily reset control systems).

Remark 3.4. For the reset and after-reset sets given by (3.1), functions 𝜏_𝑘(⋅), 𝑘=0,1,2,… are homogenous (of degree 0), that is 𝜏_𝑘(𝛼𝐱₀)=𝜏_𝑘(𝐱₀) for any real number 𝛼, since 𝐶𝑒^𝐴𝑡(𝛼𝐱₀)=𝛼𝐶𝑒^𝐴𝑡𝐱₀=0 at a resetting time 𝑡.

Example 3.5 (III-posed reset system). Consider a reset system (2.7), with the following system matrices 𝐴=−1000−1−101−1,𝐴_𝑅=100010000,𝐶=100, (3.4) where the sets ℳ_ℛ and ℳ are defined according to (3.1) as ℳ_ℛ=ℛ(𝐴_𝑅)∩𝒩(𝐶)=span{(0,1,0)^𝑇}, and ℳ=𝒩(𝐶)⧵ℳ_ℛ=span{(0,1,0)^𝑇,(0,0,1)^𝑇}⧵span{(0,1,0)^𝑇}. This is due to the fact that the after-reset surface ℳ_ℛ is a subset of the unobservable subspace of the linear base system, which is given in this case by 𝒩𝐶𝐶𝐴𝐶𝐴²=𝒩(𝐶)100−100100=𝒩⊃ℳ_ℛ. (3.5) As a result, from any initial condition 𝐱₀=(0,𝑎,0)^𝑇∈ℳ_ℛ, the condition A1 is not satisfied. Note that the origin is a stable focus in the plane 𝑥₂−𝑥₃, and that the first resetting time 𝜏₁((0,𝑎,0)^𝑇) is not well defined; in fact, the reset system is not well-posed.

Example 3.6 (Unobservable base system and well-posed resetting times). This example, adapted from [8], shows how an unobservable base system may define a reset system with well-posed resetting times, as far as the unobservable subspace does not contain after-reset states. Consider a reset control system (2.7) with 𝐴=0011−0.210−1−1,𝐴_𝑅=100010000,𝐶=010, (3.6) that has a unobservable mode corresponding to a stable pole-zero cancellation in the linear base system, where the plant has a transfer function 𝑃(𝑠)=(𝑠+1)/(𝑠(𝑠+0.2)), and the base compensator is 𝐶(𝑠)=1/(𝑠+1) (the reset compensator is a first-order reset element -FORE). In addition, the after-reset and reset surfaces are given by ℳ_ℛ=ℛ(𝐴_𝑅)∩𝒩(𝐶)=span{(1,0,0)^𝑇} and ℳ=𝒩(𝐶)⧵ℳ_ℛ=span{(1,0,0)^𝑇,(0,0,1)^𝑇}⧵span{(1,0,0)^𝑇}, respectively. In this case, the set ℳ_ℛ is not a subset of the linear base system unobservable subspace, given by 𝒩𝐶𝐶𝐴𝐶𝐴²=𝒩)0101−0.21−0.2−0.96−0.2=span(1,0,−1^𝑇. (3.7) As a result, Proposition 3.1 may be used to ensure that the system has well-posed resetting times (see Figure 3 for system solutions corresponding to two initial conditions).

Figure 3 

System solutions for the well-posed reset control system example.

4. Zeno Solutions of Reset Control Systems

In this Section, the existence of Zeno solutions is investigated for reset control systems described by (2.7)–(2.10), and with reset and after-reset surfaces given by (3.1). In principle, as discussed in Section 2.2, the reset control system may exhibit Zeno solutions even in the case in which resetting times are well-posed, that is, they are well defined and are distinct. However, as it will be shown in the following, well-posedness of resetting times is sufficient to avoid the existence of Zeno solutions in reset control systems, and thus, for reset control systems to be well-posed.

Proposition 4.1. The reset control system (2.7)–(2.10), with reset and after-reset surfaces given by (3.1), will not have Zeno solutions if it has well-posed resetting times.

Proof. It will be assumed that 𝐱₀∈ℳ_ℛ, and that there exist an infinite number of crossings for 𝐱₀ (otherwise no Zeno solution may exist), then reset intervals are given by Δ_𝑘(𝐱₀)∶=𝜏_𝑘(𝐱₀)−𝜏_𝑘−1(𝐱₀)≠0, 𝑘=1,2,…. In the following, the notation 𝑡_𝑘=𝜏_𝑘(𝐱₀) is used, and the argument 𝐱₀ is dropped by simplicity. Note that reset intervals Δ_𝑘 are well defined and Δ_𝑘≠0, 𝑘=1,2,…, by well-posedness of the resetting times. The proof will be based on the fact that in general there can only exist a finite sequence of reset intervals (Δ_𝑘)^𝑚−1_𝑘=1 such as Δ_𝑚−1<Δ_𝑚−2<⋯<Δ₁=𝜖, for some 𝜖>0 arbitrarily small but fixed, and some finite positive integer 𝑚, being 𝑚 the dimension of the after reset surface ℳ_ℛ. Thus, the sequence of resetting times (𝑡_𝑘=𝑡_𝑘−1+Δ_𝑘)^∞_𝑘=1, with 𝑡₀=0, will not be a Cauchy sequence, and thus, 𝑡_𝑘→∞, as 𝑘→∞. As a result, Zeno solutions does not exist. Assume that the plant state equations (2.5) are given in observer form (note that the plant has not to be necessarily observable), that is, 𝐴_𝑝=00⋯0−𝑎₀10⋯0−𝑎₁⋮⋮⋯⋮⋮00⋯1−𝑎_𝑛𝑝−1,𝐵_𝑝=𝑏₀𝑏₁⋮𝑏_𝑛𝑝−1,𝐶_𝑝=00⋯1, (4.1) then 𝐶=(0,0,…,1,0,…,0), and 𝒪_base is 𝒪_base=00⋯0010⋯000⋯01−𝑎_𝑛𝑝−1⋆⋯⋆00⋯1−𝑎_𝑛𝑝−1⋆⋆⋯⋆⋮⋮⋮⋮⋮⋮⋮⋮⋮1−𝑎_𝑛𝑝−1⋯⋆⋆⋆⋆⋯⋆⋮⋮⋮⋮⋮⋮⋮⋮⋮, (4.2) where ⋆ stands for a non(necessarily) zero term. By simplicity, the case of full reset is approached in first instance. An after-reset state 𝐱∈ℳ_ℛ is given by 𝐱=𝑥₁,𝑥₂,…,𝑥_𝑛𝑝−2,𝑥_𝑛𝑝−1,0,0,…,0^𝑇, (4.3) for some values 𝑥₁,…,𝑥_𝑛𝑝−1∈ℝ, being 𝑛_𝑝 the number of plant states. Thus, 𝑚=𝑛_𝑝−1 in the case of full reset.
Let us start with the case 𝑚=1. In this case, for any 𝐱₀=(𝑥₁,0,0,…0)^𝑇, it is clear that 𝜏₁(𝐱₀)=Δ for some constant Δ>0, since 𝜏₁(⋅) is homogenous (see Remark 3.4). In addition, 𝑡₂=Δ+𝜏₁(𝐴_𝑅𝑒^𝐴𝑡1𝐱₀)=2Δ since 𝐴_𝑅𝑒^𝐴𝑡1𝐱₀=𝛼𝐱₀ for some real number 𝛼. As a result Δ_𝑘=Δ, 𝑘=0,1,…, that is, resetting times are periodic with period Δ, and no Zeno solution may exist.
The case 𝑚=2 is analyzed in the following. Consider an initial condition 𝐱(0)=𝐱₀∈ℳ_ℛ, that is, 𝐱₀=(𝑥₁,𝑥₂,0,0,…0)^𝑇. If the solution 𝐱(𝑡,0,𝐱₀) crosses the reset surface ℳ_ℛ at time 𝑡₁=𝜖₁ for some 𝜖₁>0 arbitrarily small, and thus, Δ₁=𝜖₁, then 0=𝐶𝑒^𝐴𝜖1𝐱¹=𝐶𝐱¹+𝜖₁𝐶𝐴𝐱¹+𝜖²₁2𝐶𝐴²𝐱¹+⋯. (4.4) Now, since the control system (2.7)–(2.10) has well-posed resetting times, then the right hand of (4.4) is not identically zero for any 𝐱₀∈ℳ_𝑅. Now, using the special structure given in (4.2), it is obtained that 0=𝑥₂+𝜖₁2𝑥₁+𝑂𝜖²₁, (4.5) for arbitrarily small 𝜖₁>0. In addition, the following after-reset state is 𝐱₁=𝐴_𝑅𝐱(𝑡₁,0,𝐱₀)=(𝑥₁+𝑂(𝜖²₁),𝑥₂+𝜖₁𝑥₁+𝑂(𝜖²₁),0,0,…,0)^𝑇. Repeating the above argument, the solution 𝐱(𝑡,𝑡₁,𝐱₁) will cross again ℳ at the instant 𝑡₂=𝑡₁+Δ₂. If Δ₂=𝜖₂≤𝜖₁ for some 𝜖₂>0, then 0=𝑥₂+𝜖₁𝑥₁+𝜖₂2𝑥₁+𝑂𝜖²₁, (4.6) where the properties 𝑂(𝜖²₂)=𝑂(𝜖²₁) for 𝜖₂≤𝜖₁ and 𝑂(𝑘𝜖)=𝑂(𝜖), for a real constant 𝑘, have been used. Now, using (4.5) and (4.6), the result is that given some 𝜖₁>0 arbitrarily small, then 𝜖₂=−𝜖₁+𝑂(𝜖²₁)<0, which is absurd. Thus, by contradiction, it is true that 𝜖₂>𝜖₁, and thus any initial condition in the set ℳ_ℛ that produces a first reset interval 𝜖₁>0 arbitrarily small, gives a larger second reset interval 𝜖₂>0. Thus, Zeno solutions does not exist in this case either.
In the rest of the proof, the terms 𝑂(𝜖^𝑚₁) are directly neglected by simplicity. For the general case in which the dimension of ℳ_ℛ is 𝑚, with initial state 𝐱₀=(𝑥₁,𝑥₂,…,𝑥_𝑚,0,0,…0)^𝑇, a similar reasoning results in the set of equations ^𝑚_𝑘=1𝜖₁^𝑚−1()!𝑥𝑚+1−𝑘^𝑘=0,_{𝑚𝑖𝑖=1𝑘=1}𝜖₂^𝑚−𝑖𝜖₁^𝑖−𝑘()!()!𝑥𝑚+1−𝑖𝑖−𝑘^𝑘⋯=0,_{𝑚𝑖𝑖=1𝑘=1}𝜖_𝑚^𝑚−𝑖𝜖₁+⋯+𝜖_𝑚−1^𝑖−𝑘()!()!𝑥𝑚+1−𝑖𝑖−𝑘^𝑘=0, (4.7) which leads to an algebraic equation of order 𝑚 in 𝜖_𝑚, being its solutions 𝜖_𝑚=−𝜖_𝑚−1, 𝜖_𝑚=−(𝜖_𝑚−1+𝜖_𝑚−2), …, 𝜖_𝑚=−(𝜖_𝑚−1+𝜖_𝑚−2+⋯+𝜖₁). And again, it can not exist a sequence of reset intervals (𝜖₁,𝜖₂,…,𝜖_𝑚), with 𝜖₁≥𝜖₂≥⋯≥𝜖_𝑚>0 and 𝜖₁ arbitrarily small, showing that a Zeno solution is not possible in the full-reset case.
The case of partial reset can be conveniently transformed into the full-reset form by a change of coordinates, by a simple resorting of coordinates so that the bijectivity is guaranteed. We will consider the system structure decomposition by writing the states as 𝐱=(𝐱_𝑝,𝐱_𝜌,𝐱_𝜌), where 𝐱_𝑝∈ℝ^𝑛𝑝 stands for the states of the plant, 𝐱_𝜌∈ℝ^𝑛𝜌 for the nonresetting compensator states and 𝐱_𝜌∈ℝ^𝑛𝜌 for the resetting compensator states. Define the linear transformation 𝑇 from ℝ^𝑛 toℝ^𝑛 such that 𝑇𝐱=𝑇𝐱_𝑝,𝐱_𝜌,𝐱_𝜌=𝐱_𝜌,𝐱_𝑝,𝐱_𝜌=𝐳, (4.8) that is, 0𝑇=^{𝑛𝜌×𝑛𝑝}𝐼_{𝑛𝜌×𝑛𝜌}0_{𝑛𝜌×𝑛𝜌}𝐼_{𝑛𝑝×𝑛𝑝}0_{𝑛𝑝×𝑛𝜌}0_{𝑛𝑝×𝑛𝜌}0_{𝑛𝜌×𝑛𝑝}0_{𝑛𝜌×𝑛𝜌}𝐼_{𝑛𝜌×𝑛𝜌}. (4.9) Note that 𝑇 is a square matrix, all of whose entries are 0 or 1, and in each row and column of 𝑇 there is precisely one 1. This means that 𝑇 is a permutation matrix. Clearly, such a matrix is unitary, hence orthogonal, so 𝑇^𝑇=𝑇⁻¹. The nonsingular matrix 𝑇 allows to rewrite the dynamical system via a similarity transformation (congruence transformation): 𝐳̇(𝑡)=(𝑡)(𝑡)∉𝐳𝑡𝐴𝐳,𝐳ℳ,₊=𝐴_𝑅𝐳(𝑡)(𝑡)∈𝑦(𝑡)=,𝐳ℳ,𝐶𝐳(𝑡), (4.10) where 𝐴=𝑇𝐴𝑇^𝑇, 𝐴_𝑅=𝑇𝐴_𝑅𝑇^𝑇, and 𝐶=𝐶𝑇^𝑇, and in addition, the reset surface is transformed into ℳ={𝐳∈ℝ^𝑛∶𝑇^𝑇𝑧∈ℳ}. Note that 𝐶=𝐶𝑇^𝑇=𝐞_{𝑛𝑝+𝑛𝜌} so that the output is not changed by the transformation, that is, 𝑦(𝑡)=𝑧_{𝑛𝑝+𝑛𝜌}(𝑡) as expected. Henceforth, (4.10) is in full-reset form. Since observability is invariant under similarity transformations, it is clear that (2.8) is well-posed if and only if (4.10) is well-posed. Finally, to complete the proof it is necessary to show that the observability matrix has the structure given in (4.2) (using state transformations if needed). This is simply done by considering the substate 𝐳₁=(𝐱_𝜌,𝐱_𝑝). In general, there always exists a state transformation of 𝐳=(𝐳₁,𝐱_𝜌) to 𝐰=(𝐰_𝟏,𝐱_𝜌), such that the state submatrix corresponding to 𝐳₁ is in the observer form, and thus, the observability matrix has the structure given in (4.2) once unobservable states are eliminated.

4.1. Example: Well-Posed Reset Control System with Partial Reset

Consider a reset control system, where the plant, with state 𝐱_𝑝=(𝑥₁𝑥₂)^𝑇, is given by 𝐴_𝑝=011−1,𝐵_𝑝=10,𝐶_𝑝=01, (4.11) and the reset compensator, with state 𝐱_𝑟=(𝑥₃𝑥₄)^𝑇, by 𝐴_𝑟=0000,𝐵_𝑟=11,𝐶_𝑟=11,𝐴_𝜌=1000, (4.12) that is the reset control system has a partial reset compensator: it is a parallel connection of an integrator and a Clegg integrator, where only the state 𝑥₄ is set to zero at the resetting times. The closed-loop system is given by the matrices 𝐴=0−1111−1000−1000−100,𝐴_𝑅=1000010000100000,𝐶=0100, (4.13) and the closed-loop state 𝐱=(𝑥₁𝑥₂𝑥₃𝑥₄)^𝑇. This reset control system is well-posed, since ℳ_𝑅1000,0010=span,𝒪_base001=span−1, (4.14) and then, ℳ_ℛ∩𝒩(𝒪_base)={𝟎}. Following the reasoning given in the proof of Proposition 4.1, the closed-loop state 𝐱 can be transformed into a state 𝐳 in which the observability matrix has the form (4.2). In this case, this is obtained with 𝐳=(𝑥₃𝑥₁𝑥₂𝑥₄)^𝑇. Thus, the initial conditions that produces a crossing in a arbitrarily small time 𝜖>0 are of the form 𝐳¹=(1−𝜖/200)^𝑇 or equivalently 𝐱¹=−𝜖2010^𝑇. (4.15) Now, the second after-reset state is given by 𝐱²=𝐴_𝑅𝑒^𝐴𝜖−𝜖2010^𝑇=𝜖2010^𝑇, (4.16) and according to Proposition 4.1, 𝐱² cannot produce a new crossing in a time less than or equal to 𝜖. This fact can be verified by computing solutions to the implicit equation 0=𝐶𝑒^𝐴𝑡(𝛼010)^𝑇 for 𝑡, given 𝛼∈ℝ. The solution is shown in Figure 3, where 𝑡=𝜏₁((𝛼010)^𝑇) is given.

Note that for 𝑡 to be arbitrarily small, the initial condition 𝐱¹ in the after-reset surface must be given by (4.15), that is, 𝛼=−𝜖/2. Then, as a result, the state after the first reset 𝐱² is given by (4.16). And then the value of the second resetting time can be obtained from Figure 4 with 𝛼=𝜖/2. The result is that if the first resetting time is arbitrarily small, then the second resetting time is arbitrarily close to a number 𝑡^∗=3.1698⋯ (it can be approximately computed by numerically solving the implicit equation).

Figure 4 

First resetting time as a function of 𝛼.

5. Conclusions

Well-posedness of reset control system has been investigated using a state-dependent impulsive dynamic system (IDS) representation. Reset systems have been shown to be well-posed, in the sense that resetting times of the IDS are well defined and are distinct for any initial condition, and in addition, no Zeno solutions do exist. A sufficient condition for the well-posedness of resetting times has been elaborated, based on the observability of after-reset states. In addition, it has been shown that reset control systems do no exhibit Zeno solutions if resetting times are well-posed. On the other hand, it has also been revealed several properties related with the structure of the resetting times: (i) an initial condition in the after-reset surface (having dimension 𝑚) will have sequences of decreasing reset intervals with length at most 𝑚−1; (ii) resetting times as a function of the initial condition, is in general a discontinuous mapping, which explains to certain extent the complexity in the analysis of reset control systems.

Appendix

In the following, it is shown that 𝐶𝑒^𝐴𝑡𝐱₀=𝑒^{𝝀𝑡1𝑇}𝑈(𝝀)𝒪_base𝐱₀.

By simplicity, consider in first instance that the closed-loop state matrix 𝐴 has distinct eigenvalues, then the matrix exponential may be computed by use the Caley-Hamilton method, that is, 𝑒^𝐴𝑡1=𝛼₀𝐼+𝛼₁𝐴+⋯+𝛼_𝑛−1𝐴^𝑛−1, (A.1) where 𝛼_𝑖, 𝑖=0,…,𝑛−1, are given by 𝑒^𝜆1𝑡1=𝛼₀+𝛼₁𝜆₁+⋯+𝛼_𝑛−1𝜆₁^𝑛−1,𝑒^𝜆2𝑡1=𝛼₀+𝛼₁𝜆₂+⋯+𝛼_𝑛−1𝜆₂^𝑛−1,⋯𝑒^{𝜆𝑛𝑡1}=𝛼₀+𝛼₁𝜆_𝑛+⋯+𝛼_𝑛−1𝜆_𝑛^𝑛−1. (A.2) Using the notation 𝝀^𝑇=(𝜆₁𝜆₂⋯𝜆_𝑛), 𝜶^𝑇=(𝛼₀𝛼₁⋯𝛼_𝑛−1) and 𝑒^𝝀𝑡1=∑^𝑛_𝑖=1𝑒^{𝜆𝑖𝑡𝑖}𝐞_𝑖, where 𝐞_𝑖 stands for the unit (0⋯010⋯0)^𝑇 in which the 𝑖-component is 1, (A.2) can be compactly written as 𝑒^𝝀𝑡1=𝑉(𝝀)^𝑇𝜶, (A.3) where 𝑉(𝝀) is a (nonsingular) Vandermonde matrix. Now, by eliminating 𝜶 from (A.1) and (A.3), the equation 𝐶𝑒^𝐴𝑡1𝐱₀ is transformed into 𝜶^𝑇𝐶⋯𝐶𝐴𝐶𝐴^𝑛−1𝐱₀=𝑒^{𝝀𝑡1𝑇}𝑈(𝝀)𝒪_base𝐱₀=𝑒^{𝝀𝑡1𝑇}𝑈(𝝀)𝒪_base𝐱₀, (A.4) where 𝑈(𝝀)=𝑉(𝝀)⁻¹.

In the case in which the eigenvalues of 𝐴 may be repeated, a similar argument may be applied. Note that 𝑒^𝐴𝑡 may be written as the infinite series 𝐷(𝐴)=∑^∞_𝑖=0𝐴^𝑖𝑡^𝑖/𝑖!. Thus, the polynomial 𝐷(𝜆)=∑^∞_𝑖=0𝜆^𝑖𝑡^𝑖/𝑖! can be factorized by 𝐷(𝜆)=𝑄(𝜆)𝑃(𝜆)+𝑅(𝜆), with 𝑅(𝜆)=0 or degree(𝑅)<degree(𝑃)=𝑛. In addition, 𝑅 has degree no greater than 𝑛−1, and thus, 𝑅(𝜆)=∑^𝑛−1_𝑗=0𝛼_𝑗𝜆^𝑗. Since the characteristic polynomial is zero for the eigenvalues of 𝐴, then 𝐷(𝜆_𝑘)=𝑅(𝜆_𝑘) for 𝑘=0,1,…,𝑛−1. And then 𝐷(𝜆_𝑘)=∑^∞_𝑖=0𝜆^𝑖_𝑘𝑡^𝑖₁/𝑖!=𝑒^{𝜆𝑘𝑡1}=𝑅(𝜆_𝑘)=∑^𝑛−1_𝑗=0𝛼_𝑗𝜆^𝑗_𝑘para𝑘=0,1,…,𝑛−1. This can be compactly expressed as 𝑉^𝑇(𝝀)𝜶=𝑒^𝝀𝑡, and the expression (A.3) is obtained. Now, if 𝐴 has 𝑟 different eigenvalues with multiplicity order 𝑛_𝑟, and as a consequence, the characteristic polynomial is 𝑝(𝜆)=∏^𝑟_𝑖=1(𝜆−𝜆_𝑖)^𝑛𝑖. Again, there exists unique polynomials 𝑄 and 𝑅 such as 𝐷(𝜆)=𝑄(𝜆)𝑃(𝜆)+𝑅(𝜆), where 𝐷(𝜆)=𝑒^𝜆𝑡1 y 𝑅=0 or deg(𝑅)<deg(𝑃). Here, 𝑅 can be expressed as 𝑅(𝜆)=∑^𝑛−1_𝑖=0𝛼_𝑖𝜆^𝑖, where the coefficients are unique. Since 𝑝 and its derivatives up to order 𝑛_𝑟 are zero at 𝜆_𝑖, then 𝑑^𝑗𝐷𝜆_𝑖𝑑𝜆^𝑗=𝑑^𝑗𝑅𝜆_𝑖𝑑𝜆^𝑗∀𝑖=1,2,…,𝑟,∀𝑗=0,1,…,𝑛_𝑖−1. (A.5) This can be expressed by 𝝁=𝑊𝜶, where 𝝁=^𝑟𝑛_𝑖=1^𝑖−1_𝑗=0𝐞_𝑖𝑒^{𝜆𝑖𝑡1}⊗𝐞_𝑗𝜆^𝑗_𝑖,𝑊=^𝑟𝑛_𝑖=1^𝑖−1_𝑗=0𝐞_𝑖⊗𝐞_𝑗𝐞^𝑇_𝑖𝜕^𝑗𝑉(𝝀)𝜕𝜆^𝑗_𝑖. (A.6) By using arguments based on the Lagrange-Hermite interpolation problem, it can be shown that in fact the matrix 𝑊 is invertible. And then, an expression similar to (A.4) may be obtained.

Acknowledgment

This work has been supported in part by FEDER funds and the Spanish Government (Ministerio de Ciencia y Innovacion) under project DPI2010-20466-C02-02.