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