LMIs in Control/Stability Analysis/Continuous Time/Hurwitz Stabilizability

This section studies the stabilizability properties of the control systems.

The System

Given a state-space representation of a linear system

{\begin{aligned}\ \rho x=Ax+Bu\\\ y=Cx+Du\\\end{aligned}} Where $\rho$ represents the differential operator ( when the system is continuous-time) or one-step forward shift operator ( Discrete-Time system). $x\in \mathbb {R} ^{n},y\in \mathbb {R} ^{m},u\in \mathbb {R} ^{r}$ are the state, output and input vectors respectively.

The Data

$A,B,C,D$ are system matrices.

Definition

The system , or the matrix pair $(A,B)$ is Hurwitz Stabilizable if there exists a real matrix $K$ such that $(A+BK)$ is Hurwitz Stable. The condition for Hurwitz Stabilizability of a given matrix pair (A,B) is given by the PBH criterion:

{\begin{aligned}\ rank{\begin{bmatrix}sI-A&B\end{bmatrix}}&=n,\forall s\in \mathbb {C} ,Re(s)\geq 0\\\end{aligned}} (1)

The PBH criterion shows that the system is Hurwitz stabilizable if all uncontrollable modes are Hurwitz stable.

LMI Condition

The system, or matrix pair $(A,B)$ is Hurwitz stabilizable if and only if there exists symmetric positive definite matrix $P$ and $W$ such that:

{\begin{aligned}AP+PA^{T}+BW+W^{T}B^{T}<0\\\end{aligned}} (2)

Following definition of Hurwitz Stabilizability and Lyapunov Stability theory, the PBH criterion is true if and only if , a matrix $K$ and a matrix $P>0$ satisfying:

{\begin{aligned}(A+BK)P+P(A+BK)^{T}<0\\\end{aligned}} (3)

Letting

{\begin{aligned}W=KP\\\end{aligned}} (4)

Putting (4) in (3) gives us (2).

Implementation

This implementation requires Yalmip and Mosek.

Conclusion

Compared with the second rank condition, LMI has a computational advantage while also maintaining numerical reliability.