LMIs in Control/Stability Analysis/Strong Stabilizability

Continuous-Time Strong Stabilizability

Definition

Consider a continuous-time LTI system, $𝒢:{ℒ}_{2e}⟶{ℒ}_{2e}$, with state-space realization ($A$, $B$, $C$, $0$), where $A\in {ℝ}^{n\times n}$, $B\in {ℝ}^{n\times m}$, and $C\in {ℝ}^{p\times n}$, and it is assumed that ($A$, $B$) is stabilizable, ($A$, $C$) is detectable, and the transfer matrix $G(s)=C(s1-A{)}^{-1}B$ has no poles on the imaginary axis.

The system $𝒢$ is strongly stabilizable if there exist $P\in {𝕊}^n$, $Z\in {ℝ}^{n\times p}$, and $\gamma \in {ℝ}_{>0}$, where $P>0$, such that

LMIs

$PA+A^{T}P+ZC+C^T{Z}^T<0$

$\left[\begin{array}{ccc}P(A+BF)+(A+BF{)}^{T}P+ZC+C^T{Z}^T& -Z& -XB\\ *& -\gamma 1& 0\\ *& *& -\gamma 1\end{array}\right]<0$

where $F=-B^{T}X$ and $X\in {𝕊}_n$, $X\ge 0$ is the solution to the Lyapunov equation given by

$XA+A^{T}X-XB{B}^{T}X=0$

Moreover, a controller that strongly stabilizes $𝒢$ is given by the state-space realization

$\dot{x_c}=(A+BF+P^{-1}ZC)x-P^{-1}Zu$

$y_c=-B^{T}Xx$

Discrete-Time Strong Stabilizability

Definition

Consider a discrete-time LTI system, $𝒢:{\ell }_{2e}⟶{\ell }_{2e}$, with state-space realization ($A_d$, $B_d$, $C_d$, $0$), where $A_d\in {ℝ}^{n\times n}$, $B_d\in {ℝ}^{n\times m}$, and $C_d\in {ℝ}^{p\times n}$, and it is assumed that ($A_d$, $B_d$) is stabilizable, ($A_d$, $C_d$) is detectable, and the transfer matrix $G(z)=C_d(s1-A_d{)}^{-1}{B}_d$ has no poles on the unit circle.

The system $𝒢$ is strongly stabilizable is there exist $P\in {𝕊}^n$, $Z\in {ℝ}^{n\times p}$, and $\gamma \in {ℝ}_{>0}$, where $P>0$, such that

LMIs

$\left[\begin{array}{cc}{A}_d^{T}P{A}_d-P-A_d^{T}Z{C}_d-C_d^T{Z}^T{A}_d& C_d^T{Z}^T\\ *& -P\end{array}\right]<0$,

$\left[\begin{array}{cccc}{N}_{11}& (A_d+B_{d}F{)}^{T}Z& X{B}_d& C_d^T{Z}^T\\ *& -\gamma 1& 0& Z^T\\ *& *& -\gamma 1& 0\\ *& *& *& -P\end{array}\right]<0$,

where

$N_{11}=(A_d+B_{d}F{)}^{T}P(A_d+B_{d}F)-P+(A_d+B_{d}F{)}^{T}Z{C}_d+C_d^T{Z}^T(A_d+B_{d}F)$,

$F=B_d^{T}X$,

$X=Y$,

$Y\in {𝕊}_n$,

$Y\ge 0$

is the solution to the discrete-time Lyapunov equation given by

$A_{d}Y{A}_d^T-Y-B_d{B}_d^T=0$.

Moreover, a discrete-time controller that strongly stabilizes $𝒢$ is given by the state-space realization

$x_{c,k+1}=(A_d+B_{d}F+P^{-1}Z{C}_d)x_k+P^{-1}Z{u}_k$,

$y_{c,k}=-B_d^{T}X{x}_k$.

Proof

$\left[\begin{array}{cc}{A}_d^{T}P{A}_d-P-A_d^{T}Z{C}_d-C_d^T{Z}^T{A}_d& C_d^T{Z}^T\\ *& -P\end{array}\right]<0$

this ensures that the feedback controller defined by $x_{c,k+1}=(A_d+B_{d}F+P^{-1}Z{C}_d)x_k+P^{-1}Z{u}_k$ and $y_{c,k}=-B_d^{T}X{x}_k$,

renders the closed-loop system asymptotically stable and

$\left[\begin{array}{cccc}{N}_{11}& (A_d+B_{d}F{)}^{T}Z& X{B}_d& C_d^T{Z}^T\\ *& -\gamma 1& 0& Z^T\\ *& *& -\gamma 1& 0\\ *& *& *& -P\end{array}\right]<0$

ensures that the feedback controller defined by $x_{c,k+1}=(A_d+B_{d}F+P^{-1}Z{C}_d)x_k+P^{-1}Z{u}_k$ and $y_{c,k}=-B_d^{T}X{x}_k$ has a finite ${\mathscr{H}}_{\mathrm{\infty }}$ norm, and thus is asymptotically stable.