Switched Affine Systems Using Sampled-Data Controllers: Robust and Guaranteed Stabilization

Abstract

The problem of robust and guaranteed stabilization is addressed for switched affine systems using sampled state feedback controllers. Based on the existence of a control Lyapunov function for a relaxed system, we propose three sampled-data controls. Global attracting sets, computed by solving a sequence of optimization problems, guarantee practical and global asymptotic stabilization for the whole system trajectories. In addition, robust margins with respect to parameters uncertainties and non uniform sampling are provided using input-to-state stability. Finally, a buck-boost converter is considered to illustrate the effectiveness of the proposed approaches.Index Terms-Input-to-state stability, robust control, stabilization of hybrid systems, switched systems.

State feedback ℋ ∞ control design of continuous‐time switched affine systems

“…Proof: Consider that conditions (14)- (16) are satisfied and adopt the switching function (17). The time derivative of (11) along an arbitrary trajectory of (4)-…”

“…Indeed, notice that the first diagonal block of inequalities (14) together with (16) assure that A λ is Hurwitz. We denote the set of all Hurwitz matrices A λ , λ ∈ by H. This implies that all equilibrium points satisfying (15) belong to the set…”

“…The affine part is used in (15) to calculate λ(x e ) ∈ corresponding to the equilibrium point x e ∈ X e of interest. Afterwards, the linear part represented by the linear matrix inequalities (LMIs) (14) together with (16) are equivalent to make i∈K λ i (x e )N i (ρ, P) < 0. This inequality is a convex combination of N quadratic matrix functions for which N λ (ρ, P) ≤ i∈K λ i N i (ρ, P) < 0, ∀λ ∈ and, consequently, a less conservative condition would be N λ(x e ) (ρ, P) < 0.…”

“…Hence, the control goals consist in determining a set of attainable equilibrium points and a switching rule that guides any trajectory of the system, starting from an arbitrary initial condition, to the desired equilibrium, which can not be shared by any of the subsystems. The literature presents some results about stabilisation based on optimal control, see [2,16,17], and on Lyapunov method approach, see [1,3,[18][19][20][21][22][23][24][25], where [19,25] deal with sampled-data implementation. Most of them, as for instance, [3,18] are based on a quadratic Lyapunov function and show that the existence of a stable convex combination of the subsystems state-space matrices is a sufficient condition for global asymptotic stability.…”

Chattering free control of continuous‐time switched linear systems

“…Regarding switched affine systems, the literature presents fewer but important results; see [21][22][23][24], where the three latter use the concept of sampled-data control in order to avoid chattering, making the design more attractive for practical applications. However, because of sampling, these techniques only ensure practical stability since they cannot guide the state trajectories towards the equilibrium point but only to a limit cycle or to some compact set containing it; see [24].…”