LMIs in Control/Stability Analysis/Continuous Time/Transient Output Bound for Non-Autonomous LTI Systems

The System
Consider a continuous-time LTI system with a state-space representation of:

$$\dot{x} = Ax + Bu,$$

$$y = Cx,$$

where $$A \in \mathbb{R}^{n \times n}$$, $$B \in \mathbb{R}^{n \times m}$$, $$C \in \mathbb{R}^{p \times n}                               $$ and $$x(0) = x_0$$,

the transient bound on the output can be evaluated with the following LMI.

The Data
$$A \in \mathbb{R}^{n \times n}$$, $$B \in \mathbb{R}^{n \times m}$$, $$C \in \mathbb{R}^{p \times n}                               $$ and $$x(0) = x_0$$.

The LMI:
The Euclidean norm of the output satisfies

$$\lVert y(T) \rVert _2^{2} \leq \gamma^{2} (\lVert x_0 \rVert _2^{2} + \lVert u \rVert _2T^{2}), \forall T \in \mathbb{R}_{+},$$

if there exists $$P \in \mathbb{S}^{p} $$ and $$\gamma \in \mathbb{R}_{++}$$, where $$P > 0$$, such that


 * $$P - \gamma I \leq 0, $$
 * $$\begin{bmatrix} P & C^T \\ * & \gamma I \end{bmatrix} \geq 0, $$
 * $$\begin{bmatrix} PA + A^TP & PB \\ * & - \gamma I \end{bmatrix} \leq 0 . $$

Conclusion
By using these LMIs, the transient output bound can be found for a given non-autonomous LTI system such that if $$x_0 = 0$$ and $$u$$ is a unit-energy input (i.e., $$\lVert u \rVert _2T^{2} \leq 1, \forall T \in \mathbb{R}_+$$), then the preceding conditions ensure that $$\lVert y(T) \rVert _2 \leq \gamma, \forall T \in \mathbb{R}_+$$.

Implementation
The LMI given above can be implemented and solved using a parser such as YALMIP, along with an LMI solver such as SeDuMi, or MOSEK. (For a detailed list of solvers, please visit this page on Relevant Solvers).