Lyapunov Functions, Stability and Input-to-State Stability Subtleties for Discrete-Time Discontinuous Systems

Approximation of explicit model predictive control using regular piecewise affine functions: an input-to-state stability approach

“…[47], which yields ISS with respect to additive disturbances. Using the method proposed in [32] and/or [33] these results can be converted in ISS with respect to e under certain conditions.…”

“…In other words, if the one-step reachability set for the perturbed system does not intersect with discontinuities of V , this technique might still apply. See [47] for more details regarding this idea.…”

On constrained stabilization of discrete-time linear systems

“…A map f (·, ·) : R n × R l → R m is said to be positively homogeneous of the first degree with respect to both arguments if f (αx, u) = αf (x, 1 α u) and f (x, αu) = αf ( 1 α x, u), for all α ∈ R + , for all x ∈ R n and all u ∈ R l . For the standard definitions of regional asymptotic stability in the Lyapunov sense in X ⊆ R n , exponential stability, the corresponding global variants of these properties and the definition of a standard Lyapunov function, we refer the interested reader to, for example, [20].…”

“…Next, a periodic conewise linear law (19), (20) was applied to the constrained stabilization problem. To enhance computability, the low complexity (k, λ)-contractive set S 0 := {z ∈ R 2 : 0 ≤ z 1 ≤ 20, 0 ≤ z 2 ≤ 3}, shown in Figure 4, which captures most of the relevant states, was selected as the set of initial conditions for which the stabilizing control law is computed.…”

“…The simplicial decompositions {D s i } s∈N [1,4] T can be seen in Figure 4. Moreover, the control effort d t := π(z t − z s ) + d s for the zero initial state, where π(·) corresponds to the control law (19), (20), is shown in Figure 5. The worst case computational time needed to solve the point location problem for the selection of the control law (20) was found to be equal to 60µsec, which is significantly lower than the time needed to compute the periodic vertex-interpolation control law.…”