LMIs in Control/pages/Multi-Criterion LQG

The Multi-Criterion Linear Quadratic Gaussian (LQG) linear matrix inequality will allow one to form an optimized controller, similar to that in an LQR framework, for a state space system with gaussian noise based on several different criterions defined in the Q and R matrices, that are optimized as a part of the arbitrary cost function. Just like traditional LQR, the cost matrices must be tuned in much a similar fashion as traditional gains in classical control. In the LQR and LQG framework however, the gains are more intuitive as each correlates directly to a state or an input.

The System
The system is a linear time-invariant system, that can be represented in state space as shown below:



\begin{align} \dot x &= Ax + Bu + w,\\ y &= Cx + v,\\ z &=  \begin{bmatrix} Q^{1/2} & 0 \\ 0 & R^{1/2}\end{bmatrix}\begin{bmatrix} x \\ u \end{bmatrix} \end{align}$$

where $$x \in R^{n}, y \in R^{l}, z \in R^{m}$$ represent the state vector, the measured output vector, and the output vector of interest, respectively, $$w \in R^{p}$$ is the disturbance vector, and $$A, B, C, Q, R$$ are the system matrices of appropriate dimension. To further define: $$x$$ is $$\in R^{n}$$ and is the state vector, $$A $$ is $$\in R^{n*n}$$ and is the state matrix, $$B$$ is $$\in R^{n*r}$$ and is the input matrix, $$w$$ is $$\in R^{r}$$ and is the exogenous input, $$C, Q, R $$ is $$\in R^{m*n}$$ and are the output matrices, and $$y$$ and $$z$$ are $$\in R^{m}$$ and are the output and the output of interest, respectively.

$$ Q\geq 0$$ and $$ R> 0 $$, and the system is controllable and observable.

The Data
The matrices $$ A, B, C, Q, R, W, V $$ and the noise signals $$ w, v$$.

The Optimization Problem
In the Linear Quadratic Gaussian (LQG) control problem, the goal is to minimize a quadratic cost function while the plant has random initial conditions and suffers white noise disturbance on the input and measurement.

There are multiple outputs of interest for this problem. They are defined by

\begin{align} z &=  \begin{bmatrix} Q^{1/2} & 0 \\ 0 & R^{1/2}\end{bmatrix}\begin{bmatrix} x \\ u \end{bmatrix}, Q_{i} \geq 0, R_{i}>0, i = 0,...,p. \end{align}$$

For each of these outputs of interest, we associate a cost function:

\begin{align} J^{i}_{LQG} = \lim_{t\to\infty} Ez_{i}(t)^{T}z_{i}(t), i = 0,...,p. \end{align}$$

Additionally, the matrices $$X_{LQG}$$ and $$Y_{LQG}$$ must be found as the solutions to the following Riccati equations:

\begin{align} A^{T}X_{LQG} + X_{LQG}A = X_{LQG}BR^{-1}B^{T}X_{LQG} + Q &= 0\\ AY_{LQG} + Y_{LQG}A^{T} - Y_{LQG}C^{T}V^{-1}CY_{LQG}+W &= 0\\ \end{align}$$

The optimization problem is to minimize $$J^{0}_{LQG}$$ over u subject to the measurability condition and the constraints $$J^{i}_{LQG}<\gamma_{i}, i = 0,...,p.$$. This optimization problem can be formulated as:

\begin{align} \max trace(X_{LQG}U + QY_{LQG}) - \sum_{i=1}^{p} \gamma_{i}\tau_{i}, \end{align}$$ over $$\tau_{1},...,\tau_{p}$$, with:

\begin{align} Q &= Q_{0} + \sum_{i=1}^{p} \tau_{i}Q_{i},\\ R &= R_{0} + \sum_{i=1}^{p} \tau_{i}R_{i}. \end{align}$$

The LMI: Multi-Criterion LQG


\begin{align} \max:     trace(XU + (Q_{0} + \sum_{i=1}^{p} \tau_{i} Q_{i} )Y_{LQG}) - \sum_{i=1}^{p} \gamma_{i}\tau_{i}, \end{align}$$ over $$X, \tau_1, ... \tau_{p}$$, subject to the following constraints:

\begin{align} X&>0,\\ \tau_{1} \geq 0, ..., \tau_{p} &\geq 0,\\ A^{T}X + XA - XB(R_{0} + \sum_{i=1}^{p} \tau_{i} R_{i})^{-1}B^{T}X + Q_{0} + \sum_{i=1}^{p} \tau_{i} Q_{i} &\geq 0.\\ \end{align}$$

Conclusion:
The result of this LMI is the solution to the aforementioned Ricatti equations:

\begin{align} A^{T}X_{LQG} + X_{LQG}A = X_{LQG}BR^{-1}B^{T}X_{LQG} + Q &= 0\\ AY_{LQG} + Y_{LQG}A^{T} - Y_{LQG}C^{T}V^{-1}CY_{LQG}+W &= 0\\ \end{align}$$

Implementation
This implementation requires Yalmip and Sedumi.

https://github.com/rezajamesahmed/LMImatlabcode/blob/master/multicriterionquadraticproblems.m

Related LMIs

 * Inverse Problem of Optimal Control