On Poly-Quadratic Stabilizability and Detectability of Polytopic LPV Systems

In this technical communique, we generalize the well-known Lyapunov-based stabilizability and detectability tests for discrete-time linear time-invariant systems to polytopic linear parameter-varying systems using the class of so-called poly-quadratic Lyapunov functions.

💡 Research Summary

This paper extends the classical Lyapunov‑based stabilizability and detectability tests that are well known for discrete‑time linear time‑invariant (LTI) systems to the broader class of discrete‑time linear parameter‑varying (LPV) systems whose uncertainty set is a polytope. The authors introduce the notion of a poly‑quadratic Lyapunov function (poly‑QLF), a convex combination of several quadratic forms, each associated with a vertex of the polytope. For a polytopic LPV system
(x_{k+1}=A(p_k)x_k+Bu_k,; y_k=Cx_k)
with the parameter (p_k) belonging to a known compact set (\mathcal P), the system matrix can be expressed as (A(p)=\sum_{i=1}^N\xi_i(p)A_i) where the scalar weighting functions (\xi_i(p)) are continuous, non‑negative and sum to one. The poly‑QLF is defined as (V(p,x)=x^\top P(p)x) with (P(p)=\sum_i\xi_i(p)\bar P_i) and (\bar P_i\succ0) are design variables.

Two new concepts are defined:

1. Poly‑Q detectability – the existence of an observer gain (L(k,p)) such that the error dynamics (e_{k+1}=(A(p_k)+L(k,p)C)e_k) admits a poly‑QLF that strictly decreases.
2. Poly‑Q stabilizability – the existence of a state‑feedback gain (K(k,p)) that makes the closed‑loop dynamics (x_{k+1}=(A(p_k)+BK(k,p))x_k) poly‑QS (i.e., admits a decreasing poly‑QLF).

The main results are two theorems that give necessary and sufficient linear matrix inequality (LMI) conditions for these properties when the LPV system is strictly polytopic (i.e., each vertex of the polytope can be reached by a parameter value).

Theorem 1 (Detectability) states that the system is poly‑Q detectable iff there exist matrices (\bar P_i\succ0) satisfying
\