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# pdlstab

Assess robust stability of polytopic or parameter-dependent system

## Synopsis

[tau,Q0,Q1,...] = pdlstab(pds,options)

## Description

pdlstab uses parameter-dependent Lyapunov functions to establish the stability of uncertain state-space models over some parameter range or polytope of systems. Only sufficient conditions for the existence of such Lyapunov functions are available in general. Nevertheless, the resulting robust stability tests are always less conservative than quadratic stability tests when the parameters are either time-invariant or slowly varying.

For an affine parameter-dependent system

E(p) = A(p)x + B(p)u

y = C(p)x + D(p)u

with p = (p1, . . ., pn) ∊ Rn, pdlstab seeks a Lyapunov function of the form

V(xp, ) = xTQ(p)–1x, Q(p) = Q0 + p1Q1 + . . .pnQn

such that dV(x, p)/dt < 0 along all admissible parameter trajectories. The system description pds is specified with psys and contains information about the range of values and rate of variation of each parameter pi.

For a time-invariant polytopic system

E = Ax + Bu

y = Cx + Du

with

 (2-17)

pdlstab seeks a Lyapunov function of the form

V(x, α) = xTQ(α)–1x, Q(α) = α1Q1 + . . .+ αnQn

such that dV(x, α)/dt < 0 for all polytopic decompositions of the form Equation 2-17.

Several options and control parameters are accessible through the optional argument options:

• Setting options(1)=0 tests robust stability (default)

• When options(2)=0, pdlstab uses simplified sufficient conditions for faster running times. Set options(2)=1 to use the least conservative conditions

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### Tips

For affine parameter-dependent systems with time-invariant parameters, there is equivalence between the robust stability of

 $E\left(p\right)\stackrel{˙}{x}=A\left(p\right)x$ (2-18)

and that of the dual system

 $E{\left(p\right)}^{T}\stackrel{˙}{z}=A{\left(p\right)}^{T}z$ (2-19)

However, the second system may admit an affine parameter-dependent Lyapunov function while the first does not.

In such case, pdlstab automatically restarts and tests stability on the dual system Equation 2-19 when it fails on Equation 2-18.