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Quadratic stability of polytopic or affine parameter-dependent systems

Syntax

```[tau,P] = quadstab(ps,options)
```

Description

For affine parameter-dependent systems

E(p) = A(p)x, p(t) = (p1(t), . . ., pn(t))

or polytopic systems

E(t) = 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).

• 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

${p}_{i}\in \left[{p}_{i}-{\delta }_{i},{p}_{i0}+{\delta }_{i}\right],$

quadstab computes the largest Θ > 0 such that quadratic stability holds over the parameter box

${p}_{i}\in \left[{p}_{i0}-\Theta {\delta }_{i},{p}_{i0}+\Theta {\delta }_{i}\right]$

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 = ${Q}_{\text{opt}}^{-1}$. 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.