User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/Discrete-Time Modified Minimum Gain Lemma

The Concept
The output of the system y(t) is fed back through a sensor measurement F to a comparison with the reference value r(t). The controller C then takes the error e (difference) between the reference and the output to change the inputs u to the system under control P. This is shown in the figure. This kind of controller is a closed-loop controller or feedback controller.

This is called a single-input-single-output (SISO) control system; MIMO (i.e., Multi-Input-Multi-Output) systems, with more than one input/output, are common. In such cases variables are represented through vectors instead of simple scalar values. For some distributed parameter systems the vectors may be infinite-dimensional (typically functions).

If we assume the controller C, the plant P, and the sensor F are linear and time-invariant (i.e., elements of their transfer function C(s), P(s), and F(s) do not depend on time), the systems above can be analysed using the Laplace transform on the variables. This gives the following relations:


 * $$Y(s) = P(s) U(s)$$
 * $$U(s) = C(s) E(s)$$
 * $$E(s) = R(s) - F(s)Y(s).$$

Solving for Y(s) in terms of R(s) gives


 * $$Y(s) = \left( \frac{P(s)C(s)}{1 + P(s)C(s)F(s)} \right) R(s) = H(s)R(s).$$

The expression $$H(s) = \frac{P(s)C(s)}{1 + F(s)P(s)C(s)}$$ is referred to as the closed-loop transfer function of the system. The numerator is the forward (open-loop) gain from r to y, and the denominator is one plus the gain in going around the feedback loop, the so-called loop gain. If $$|P(s)C(s)| \gg 1$$, i.e., it has a large norm with each value of s, and if $$|F(s)| \approx 1$$, then Y(s) is approximately equal to R(s) and the output closely tracks the reference input. This page gives an LMI to reduce the gain so that the ouput closely tracks the reference input.

The System
Consider a discrete-time LTI system, $$\mathcal{G} : \mathcal{l}_{2e} \rightarrow \mathcal{l}_{2e}$$, with minimal state-space relization $$(\mathcal{A}_{d}, \mathcal{B}_{d}, \mathcal{C}_{d}, \mathcal{D}_{d})$$, where $$\mathcal{A}_{d} \in \mathcal{R}^{n\times n}, \mathcal{B}_{d} \in \mathcal{R}^{n\times m}, \mathcal{C}_{d} \in \mathcal{R}^{p\times n},$$ and $$\mathcal{D}_{d} \in \mathcal{R}^{p\times m} $$.


 * $$x(k+1)=\mathcal{A}_{d}x(k)+\mathcal{B}_{d}u(k)$$
 * $$y(k)=\mathcal{C}_{d}x(k)+\mathcal{D}_{d}u(k), k=0,1... $$

The Data
The matrices $$\mathcal{A}_{d},\mathcal{B}_{d},\mathcal{C}_{d} $$ and $$\mathcal{D}_{d}$$

LMI : Discrete-Time Modified Minimum Gain Lemma
The system $$\mathcal{G}$$ has minimium gain γ under any of the following equivalent sufficient conditions.


 * 1. There exists $$ P \in \mathcal{S}^{n}, $$ and γ $$\in \mathcal{R}_{\ge 0}$$ where $$ P \ge 0 $$ such that


 * $$\begin{bmatrix}

A^{T}_{d}PA_{d} - P & A^{T}_{d}PB_{d} - C^{T}_{d}D_{d} \\ (A^{T}_{d}PB_{d} - C^{T}_{d}D_{d})^{T} &  B^{T}_{d}PB_{d} + \gamma^{2}I - (D^{T}_{d}+ D_{d}) \end{bmatrix}$$

$$Proof$$. The term $$ -C_{d}^{T}C_{d} $$ in Discrete Time Minimum Gain Lemma makes the matrix inequality more negative definite. Therefore,
 * $$\begin{bmatrix}

A^{T}_{d}PA_{d} - P - C_{D}^{T}C_{D} & A^{T}_{d}PB_{d} - C^{T}_{d}D_{d} \\ (A^{T}_{d}PB_{d} - C^{T}_{d}D_{d})^{T} &  B^{T}_{d}PB_{d} + \gamma^{2}I - (D^{T}_{d}+ D_{d})\end{bmatrix} \le \begin{bmatrix} A^{T}_{d}PA_{d} - P & A^{T}_{d}PB_{d} - C^{T}_{d}D_{d} \\ (A^{T}_{d}PB_{d} - C^{T}_{d}D_{d})^{T} &  B^{T}_{d}PB_{d} + \gamma^{2}I - (D^{T}_{d}+ D_{d}) \end{bmatrix}$$


 * 2. There exists $$ P \in \mathcal{S}^{n}, $$ and $$ \gamma \in \mathcal{R}_{\ge 0}$$ where $$ P \ge 0 $$ such that


 * $$\begin{bmatrix}

A^{T}_{d}PA_{d} - P &  A^{T}_{d}PB_{d} - C^{T}_{d}D_{d} & 0\\ (A^{T}_{d}PB_{d} - C^{T}_{d}D_{d})^{T} &  B^{T}_{d}PB_{d}- (D^{T}_{d}+ D_{d})  & \gamma I \\ 0 & \gamma I & I \end{bmatrix}\le 0.$$

$$proof$$ : Applying the Schur complement lemma to the γ2 term in equation 1 gives equation 2.

Conclusion:
If there exist a positive definite $$P$$ for the system $$\mathcal{G}$$, then the minimum gain of the system γ can be obtaied from above defined LMIs.

Implementation
Code for implementation of this LMI using MATLAB. https://github.com/VJanand25/LMI

Related LMIs
KYP Lemma State Space Stability KYP Lemma without Feedthrough Discrete Time Minimum Gain Lemma