Scalar-quadratic stabilizability of the Petersen counterexample via a linear static controller

Abstract

A new property called scalar-quadratic is presented for establishing the stabilizability of linear time-varyring uncertain systems. It is applied to a well-known linear time-varying system ∑_OL which contains two uncertainties γ₁(t) and γ₂(t). Using the Lyapunov functionsV(x)=x ^T Px, whereP is a constant postitive-definite symmetric matrix, previous authors have shown that ∑_OL is stabilizable by linear static controllers when the time-varying uncertainties are bounded by a normalized bound$\bar \delta$ satisfying$\bar \delta$ < 0.8. We extend the bound to$\bar \delta$ < 1.0 by using the more general Lyapunov functions satisfying the scalar-quadratic propertyV(ax)=a ² V(x), ∇a∈R, ∇x∈R ²₀ .

Our proof uses a hexagon as a closed, convex hypersuface to construct a scalar-quadratic Lyapunov function, so that the Lyapunov time derivative satisfies the quadratic convergence condition$\dot V(x) \leqslant - \in ||x||^2$, ∈>0, for the closed-loop system ∑_CL formed from ∑_OL and a stabilizing linear static controller. The critical condition in the proof of the quaratic convergence ondition is the satisfaction of the inequality$\Delta _{\max }< {\text{ }}e_2^2 /[1 + e_1 e_2 + \sqrt {1 + } e_2^2 ]$, where Δ_max is a normalization bound for γ₁(t) and γ₂(t) and wheree ₁ ande ₂ are parameters for the controller. For the controller parametrized bye ₁=8 ande ₂=20, this inequality reduces to Δ_max < 2.2096. This result, in particular, establishes that the Petersen counterexample is stabilitzable by the linear static controller withe ₁=8 ande ₂=20. Furthermore, it establishes the amazing result that ∑_OL is stabilizable by a linear static controlle on any compact subset of the constant uncertainaty controllability space defined by γ₁>0 and γ₂>0.