User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/Discrete-time Static Output Feedback Stabilizability

The static output feedback (SOF) problem has been investigated and analyzed by many people and the literature concerning this topic is vast. In practicality, it is not always possible to have full access to the state vector and only a partial information through a measured output is available. This explains why this problem has challenged many researchers in control theory. Here is a systematic approach for the SOF control design for discrete time linear systems.

The System
Consider a discrete-time LTI system, with state-space realization $$(Ad, Bd,Cd, 0)$$,


 * $$\begin{align}

x_{k+1} = A_dx_k+B_du_k\\ y_k=C_dx_k \end{align}$$

$$x_k\in\mathbb{R}^n$$ is the state, $$y_k\in\mathbb{R}^p$$ is the measured output, $$u_k\in\mathbb{R}^m$$ is the control input.

The Data
$$A_d\in\mathbb{R}^{n\times n}, B_d\in\mathbb{R}^{n\times m}, C_d\in\mathbb{R}^{p\times n}$$ are the constant matrices of appropriate dimensions.

The Optimization Problem
The full state is not measurable and only partial information is available through $$y_k$$ which can used for control purposes.

We have to find a static output feedback gain with respect to
 * $$u_k = -K_dy_k,$$

where $$K_d\in\mathbb{R}^{m\times p}$$ is the output feedback gain such that the final closed loop system is asymptotically stable.

The LMI: LMI for Discrete-Time Static Output Feedback Stabilizability
The discrete time system considered is static output feedback stabilizable under any of the following equivalent necessary or sufficient conditions.


 * There exists a $$K_d\in\mathbb{R}^{m\times p}$$ and $$P \in\mathbb{S}^n$$ where $$P>0$$ such that

\begin{bmatrix} -P & (A_d+B_dK_dC_d)P \\ ((A_d+B_dK_dC_d)P)' & -P \end{bmatrix}$$$$\begin{align}< 0\end{align}.$$


 * There exists a $$K_d\in\mathbb{R}^{m\times p}$$ and $$P \in\mathbb{S}^n$$ where $$P>0$$ such that

\begin{bmatrix} -A_dPPA_d^T & A_dP+B_dK_dC_d & A_dP \\ (A_dP+B_dK_dC_d)' & -1 & 0 \\ PA_d^T & 0 & -P \end{bmatrix}$$$$\begin{align}< 0\end{align}.$$

Conclusion
If it is feasible we obtain a output feedback gain matrix $$K_d$$ such that the closed loop system is asymptotically stable. While implementing the optimization problem the following conditions are assumed to be satisfied
 * The Transfer matrix and its inverse are both analytical a s=0
 * The matrix $$C_dA_d^{-1}B_d$$ is non-singular
 * The triple $$(A_d,B_d,C_d)$$ are reachable and observable.

Implementation
A link to the Matlab code for a simple implementation of this problem in the Github repository:

https://github.com/yashgvd/LMI_wikibooks

Related LMIs
Continuous-time Static Output Feedback Stabilizability