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

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 ₁(t) and ₂(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 satisfying < 0.8. We extend the bound to < 1.0 by using the more general Lyapunov functions satisfying the scalar-quadratic propertyV(ax)=a ² V(x), aR, xR ₀ ² .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 , >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 , where _max is a normalization bound for ₁(t) and ₂(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 _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 ₁>0 and ₂>0.