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

and Applied Analysis 3 see 4–6 . In general, there exists a unique solution ψ t eψ0 of the continuous base system with initial condition ψ 0 ψ0 on 0,∞ , for any ψ0 ∈ R. Informally speaking, the solution x of the IDS 2.1 from the initial condition x 0 x0 is given by x t ex0 for 0 < t ≤ t1, where t1 is the first resetting time satisfying x t1 ∈ M. Then, the state is instantaneously transferred to ARx t1 according to the resetting law. The solution x t , t1 < t ≤ t2 being t2 the second resetting time given by e t2−t1 x t1 ∈ M is given by x t e t−t1 x t1 e t−t1 ARe1x0, 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 tk, and x tk lim → 0 x tk − , x ( t k ) lim → 0 x tk ARx tk . 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 > 0 such that x t δ / ∈ M, for all δ ∈ 0, . A2 x ∈ M ⇒ ARx / ∈ M. Note that for a particular solution x · , the first resetting time t1 is well defined since min{t ∈ R : ψ t, 0, x0 ex0 ∈ M} exists and thus, it is unique by uniqueness of solutions of the base system . Analogously, for k 2, 3, . . ., the resetting time tk is well defined since again min{t ∈ R : ψ t, tk−1, ARx tk−1 ∈ M} exists, and in addition 0 t0 < t1 < t2 < · · · , for any x0 ∈ R. 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 t−t0 ψ0. Therefore, if for any initial condition x0 ∈ R the resetting times are well defined, functions τk : R → 0,∞ are defined for k 1, 2, . . ., where tk τk 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 Ix0 , and that functions τk 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 xZ · exists on the interval Ix0 0, T for some initial condition xz 0 x0 ∈ R, if there exists an infinite sequence of resetting times τk x0 ∞ k 0, and a positive number T , such as τk 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 Ix0 0,∞ . Note that conditions A1 and A2 can be interpreted as: i states that belong to the closure ofM, and does not belong toM, evolve with the continuous base dynamics for some finite time interval A1 ; ii after-reset states are not elements of the reset setM A2 . In other words, for resetting times to be well-posed a IDS system solution can only reachM through a point belonging to bothM and its boundary ∂M; and if a solution reaches a point inM that is on its boundary, then it is instantaneously removed fromM. Roughly speaking, condition A1 4 Abstract and Applied Analysis 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 matrixA ∈ R2×2, 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 Ma is the rectangle Ma {( x1 x2 ) ∈ R2 : −1 ≤ x1 ≤ 1, 0.7 < x2 ≤ 1 } , 2.3 and the after reset set is MR {( x1 x2 ) ∈ 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 Ma at some point belonging to both Ma and its boundary ∂Ma. 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 ∂Ma but not to Ma. Thus, condition A1 is not satisfied since the trajectory enters into M for any arbitrarily small time after reaching the pointC the second resetting time τ2 A is undefined . b Figure 1 b ,Mb is the grey region it contains its boundary , and the after reset set isMR MR. Note that theM∩MR {C,D}. In this case, a trajectory starting from the point A reaches Mb at the point B which belongs both to Mb 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 toMR andMb and then make an infinite number of resets condition A2 is not satisfied . 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: P : { ẋp t Apxp t Bpu t , xp 0 xp0, y t Cpxp t , 2.5 Abstract and Applied Analysis 5and Applied Analysis 5


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.

Abstract and Applied Analysis
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 x y . In addition, e i ∈ R n stands for the unit vector 0 · · · 0 1 0 · · · 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 1.1

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 forṁ 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 R x 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 x 0 → R n is a solution of the IDS system 2.1 on the interval I x 0 ⊆ R, with initial condition x 0 x 0 , if x · is left-continuous, and x t satisfies 2.1 for all t ∈ I x 0 . For further discussion on solutions to impulsive differential equations and IDS solutions, Abstract and Applied Analysis 3 see 4-6 . In general, there exists a unique solution ψ t e At ψ 0 of the continuous base system with initial condition ψ 0 ψ 0 on 0, ∞ , for any ψ 0 ∈ R n . Informally speaking, the solution x of the IDS 2.1 from the initial condition x 0 x 0 is given by x t e At x 0 for 0 < t ≤ t 1 , where t 1 is the first resetting time satisfying x t 1 ∈ M. Then, the state is instantaneously transferred to A R x t 1 according to the resetting law. The solution x t , t 1 < t ≤ t 2 being t 2 the second resetting time given by e A t 2 −t 1 x t 1 ∈ M is given by x t e A t−t 1 x t 1 e A t−t 1 A R e At 1 x 0 , 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 2.2

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 Note that for a particular solution x · , the first resetting time t 1 is well defined since min{t ∈ R : ψ t, 0, x 0 e At x 0 ∈ 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 R x t k−1 ∈ M} exists, and in addition 0 t 0 < t 1 < t 2 < · · · , for any x 0 ∈ R n . Here, ψ t, t 0 , ψ 0 is a solution of the base system with initial condition ψ t 0 ψ 0 , that is, ψ t, t 0 , ψ 0 e A t−t 0 ψ 0 . Therefore, if for any initial condition x 0 ∈ 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 x 0 is the kth resetting time, and by definition τ 0 x 0 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 x 0 , and that functions τ k x 0 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 x 0 0, T for some initial condition x z 0 x 0 ∈ R n , if there exists an infinite sequence of resetting times τ k x 0 ∞ k 0 , and a positive number T , such as τ k x 0 → 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 x 0 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 a Figure 1 a , here the reset set M a is the rectangle and the after reset set is that is, the interval −1, 1 in the x 1 -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 τ 1 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 τ 2 A is undefined .
b Figure 1 b , M b is the grey region it contains its boundary , and the after reset set 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 τ 1 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 .

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: Abstract and Applied Analysis Figure 1: Examples of ill-posed IDSs. and the single input-single output reset compensator is given by the impulsive differential equation: v t C r x 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 IDṠ Abstract and Applied Analysis where In control practice, it is required that reset control system solutions x t will be wellposed 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 x 0 ∈ R n , a solution x exists and is unique on I x 0 0, ∞ . As a state-dependent IDS system, the reset control system is wellposed 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.

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 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 x 0 ∈ M R ⇒ x t / ∈ M for t ∈ 0, and some > 0. Here depends on x 0 , but the dependence will not be explicitly shown by simplicity. In general, given x 0 ∈ M R , the first crossing with M is at the instant t 1 τ 1 x 0 , and finally A1 is equivalent to the existence of a lower bound > 0 for t 1 , for any given x 0 ∈ M R . From 3.1 , t 1 is simply given by t 1 min{t > 0 | Ce At x 0 0 ∧ e At x 0 / ∈ R A R }. If 1 > 0 is by definition a lower bound of the set {t > 0 | Ce At x 0 0}, and 2 > 0 is by definition a lower bound of the set {t > 0 | e At x 0 / ∈ R A R } both depending on x 0 , then max{ 1 , 2 } ≤ t 1 . By simplicity, consider in first instance that A has distinct eigenvalues; in this case see the Appendix Since f 1 · is a sum of exponentials in fact it is a Bohl function 13 , and thus, it is an analytical function, it is true that f 1 t is either zero for all t ≥ 0 or has isolated zeros. As a result, two options are possible as follows 1 0 if x 0 ∈ N O base f 1 t 0, for all t ∈ 0, ∞ , or there exist an interval 0, 1 in which f 1 t / 0 for some 1 > 0. Now, if condition 3.2 is satisfied then for any x 0 ∈ M R only the second option is possible, and thus, t 1 ≥ max{ 1 , 2 } ≥ 1 , 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 f 1 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 afterreset 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 αx 0 τ k x 0 for any real number α, since Ce At αx 0 αCe At x 0 0 at a resetting time t.
Example 3.5 III-posed reset system . Consider a reset system 2.7 , with the following system matrices 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 8 Abstract and Applied Analysis 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 As a result, from any initial condition x 0 0, a, 0 T ∈ M R , the condition A1 is not satisfied. Note that the origin is a stable focus in the plane x 2 − x 3 , and that the first resetting time τ 1 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 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 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 .

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.  Proof. It will be assumed that x 0 ∈ M R , and that there exist an infinite number of crossings for x 0 otherwise no Zeno solution may exist , then reset intervals are given by Δ k x 0 : In the following, the notation t k τ k x 0 is used, and the argument x 0 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 < · · · < Δ 1 , for some > 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 t 0 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, then C 0, 0, . . . , 1, 0, . . . , 0 , and O base is 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 for some values x 1 , . . . , x n p −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 x 0 x 1 , 0, 0, . . . 0 T , it is clear that τ 1 x 0 Δ for some constant Δ > 0, since τ 1 · is homogenous see Remark 3.4 . In addition, 2Δ since A R e At 1 x 0 αx 0 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 If the solution x t, 0, x 0 crosses the reset surface M R at time t 1 1 for some 1 > 0 arbitrarily small, and thus, Δ 1 1 , then 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 x 0 ∈ M R . Now, using the special structure given in 4.2 , it is obtained that for arbitrarily small 1 > 0. In addition, the following after-reset state is Repeating the above argument, the solution x t, t 1 , x 1 will cross again M at the instant t 2 t 1 Δ 2 . If Δ 2 2 ≤ 1 for some 2 > 0, then where the properties O 2 2 O 2 1 for 2 ≤ 1 and O k O , for a real constant k, have been used. Now, using 4.5 and 4.6 , the result is that given some 1 > 0 arbitrarily small, then 2 − 1 O 2 1 < 0, which is absurd. Thus, by contradiction, it is true that 2 > 1 , and thus any initial condition in the set M R that produces a first reset interval 1 > 0 arbitrarily small, gives a larger second reset interval 2 > 0. Thus, Zeno solutions does not exist in this case either.  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 n p stands for the states of the plant, x ρ ∈ R n ρ for the nonresetting compensator states and x ρ ∈ R n ρ for the resetting compensator states. Define the linear transformation T from R n to R n such that 0 n ρ ×n p I n ρ ×n ρ 0 n ρ ×n ρ I n p ×n p 0 n p ×n ρ 0 n p ×n ρ 0 n ρ ×n p 0 n ρ ×n ρ I n ρ ×n ρ ⎞ ⎟ ⎠.

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 : where A TAT T , A R TA R T T , and C CT T , and in addition, the reset surface is transformed into M {z ∈ R n : T T z ∈ M}. Note that C CT T e n p n ρ so that the output is not changed by the transformation, that is, y t z n p n ρ 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 z 1 x ρ , x p . In general, there always exists a state transformation of z z 1 , x ρ to w w 1 , x ρ , such that the state submatrix corresponding to z 1 is in the observer form, and thus, the observability matrix has the structure given in 4.2 once unobservable states are eliminated.

Example: Well-Posed Reset Control System with Partial Reset
Consider a reset control system, where the plant, with state x p x 1 x 2 T , is given by and the reset compensator, with state x r x 3 x 4 T , by 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 x 4 is set to zero at the resetting times. The closed-loop system is given by the matrices  conditions that produces a crossing in a arbitrarily small time > 0 are of the form z 1 1 − /2 0 0 T or equivalently

4.15
Now, the second after-reset state is given by Ce At α 0 1 0 T for t, given α ∈ R. The solution is shown in Figure 3, where t τ 1 α 0 1 0 T is given.
Note that for t to be arbitrarily small, the initial condition x 1 in the after-reset surface must be given by 4.15 , that is, α − /2. Then, as a result, the state after the first reset x 2 is given by 4.16 . And then the value of the second resetting time can be obtained from Figure 4 with α /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 .

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 wellposed, 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 afterreset 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 afterreset 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.