Quadratic stability of polytopic or affine parameter-dependent systems
[tau,P] = quadstab(ps,options)
For affine parameter-dependent systems
E(p)x˙ = A(p)x, p(t) = (p1(t), . . ., pn(t))
or polytopic systems
E(t)x˙ = A(t)x, (A, E) ∊ Co{(A1, E1), . . ., (An, En)},
quadstab seeks a fixed Lyapunov function V(x) = xTPx with P > 0 that establishes quadratic stability. The affine or polytopic model is described by ps (see psys).
The task performed by quadstab is selected by options(1):
if options(1)=0 (default), quadstab assesses quadratic stability by solving the LMI problem
Minimize τ over Q = QT such that
ATQE + EQAT < τI for all admissible values of (A, E)
Q > I
The global minimum of this problem is returned in tau and the system is quadratically stable if tau < 0.
if options(1)=1, quadstab computes the largest portion of the specified parameter range where quadratic stability holds (only available for affine models). Specifically, if each parameter pi varies in the interval
quadstab computes the largest Θ > 0 such that quadratic stability holds over the parameter box
This “quadratic stability margin” is returned in tau and ps is quadratically stable if tau ≥ 1.
Given the solution Qopt of the LMI optimization, the Lyapunov matrix P is given by P = . This matrix is returned in P.
Other control parameters can be accessed through options(2) and options(3):
if options(2)=0 (default), quadstab runs in fast mode, using the least expensive sufficient conditions. Set options(2)=1 to use the least conservative conditions
options(3) is a bound on the condition number of the Lyapunov matrix P. The default is 109.