LMIs in Control/pages/Nonconvex Multi-Criterion Quadratic Problems

The Non-Concex Multi-Criterion Quadratic linear matrix inequality will allow one to form an optimized controller, similar to that in an LQR framework, for a non-convex state space system 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 for this LMI is a linear time invariant system that can be represented in state space as shown below:

\begin{align} \dot x &=Ax+Bw, x(0) = x_{0}\\ \end{align}$$ where the system is assumed to be controllable.

where $$x \in R^{n}$$ represents the state vector, respectively, $$w \in R^{p}$$ is the disturbance vector, and $$A, B$$ 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.

for any input, we define a set $$p+1$$ cost indices $$J_{0},...,J_{P}$$ by



\begin{align} J_{i}(u) = \int_{0}^{\infty}\begin{bmatrix} x^{T}&u^{T} \end{bmatrix}\begin{bmatrix} Q_i & C_i\\ C_i^T & R_i \end{bmatrix}\begin{bmatrix} x\\ u \end{bmatrix} dt,\\ i = 0,...,p \end{align}$$

Here the symmetric matrices,



\begin{align} \begin{bmatrix} Q_i & C_i\\ C_i^T & R_i \end{bmatrix}, i=0,...,p \end{align}$$,

are not necessarily positive-definite.

The Data
The matrices $$ A, B, C $$.

The Optimization Problem
The constrained optimal control problem is:



\begin{align} \max:     J_0,\\ \end{align}$$ subject to

\begin{align} J_i \leq \gamma_i, i=1,...,p, x \rightarrow 0, t \rightarrow \infty \end{align}$$

The LMI: Nonconvex Multi-Criterion Quadratic Problems
The solution to this problem proceeds as follows: We first define



\begin{align} Q = Q_{0} + \sum_{i=1}^{p} \tau_i Q_i ,\\ R = R_{0} + \sum_{i=1}^{p} \tau_i R_i ,\\ C = C_{0} + \sum_{i=1}^{p} \tau_i C_i ,\\ \end{align}$$

where $$\tau_i \geq 0$$ and for every $$\tau_i$$, we define



\begin{align} S = J_0 + \sum_{i=1}^{p} \tau_i J_i - \sum_{i=1}^{p} \tau_i \gamma_i \end{align}$$

then, the solution can be found by:

\begin{align} \max: x(0)^{T}Px(0) - \sum_{i=1}^{p} \tau_i \gamma_i \end{align}$$

subject to



\begin{align} \begin{bmatrix} A^TP + PA + Q & PQ+C^T \\ B^TP+C &R \end{bmatrix} &\geq 0\\ \tau_i &\geq 0 \end{align}$$

Conclusion:
If the solution exists, then $$ K $$ is the optimal controller and can be solved for via an EVP in P.

Implementation
This implementation requires Yalmip and Sedumi.

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

Related LMIs

 * Multi-Criterion LQG
 * Inverse Problem of Optimal Control
 * Nonconvex Multi-Criterion Quadratic Problems
 * Static-State Feedback Problem