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

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:

$\begin{array}{cc}\dot{x}& =Ax+Bw,x(0)=x_0\\ \end{array}$

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

$\begin{array}{c}{J}_i(u)=\underset{0}{\overset{\mathrm{\infty }}{\int }}\left[\begin{array}{cc}{x}^T& u^T\end{array}\right]\left[\begin{array}{cc}{Q}_i& C_i\\ C_i^T& R_i\end{array}\right]\left[\begin{array}{c}x\\ u\end{array}\right]dt,\\ i=0,...,p\end{array}$

Here the symmetric matrices,

$\begin{array}{c}\left[\begin{array}{cc}{Q}_i& C_i\\ C_i^T& R_i\end{array}\right],i=0,...,p\end{array}$,

are not necessarily positive-definite.

The Data

The matrices $A,B,C$.

The Optimization Problem

The constrained optimal control problem is:

$\begin{array}{c}\max :J_0,\\ \end{array}$

subject to

$\begin{array}{c}{J}_i\le {\gamma }_i,i=1,...,p,x\to 0,t\to \mathrm{\infty }\end{array}$

The LMI: Nonconvex Multi-Criterion Quadratic Problems

The solution to this problem proceeds as follows: We first define

$\begin{array}{c}Q=Q_0+\sum _{i=1}^p{\tau }_i{Q}_i,\\ R=R_0+\sum _{i=1}^p{\tau }_i{R}_i,\\ C=C_0+\sum _{i=1}^p{\tau }_i{C}_i,\\ \end{array}$

where ${\tau }_i\ge 0$ and for every ${\tau }_i$, we define

$\begin{array}{c}S=J_0+\sum _{i=1}^p{\tau }_i{J}_i-\sum _{i=1}^p{\tau }_i{\gamma }_i\end{array}$

then, the solution can be found by:

$\begin{array}{c}\max :x(0{)}^{T}Px(0)-\sum _{i=1}^p{\tau }_i{\gamma }_i\end{array}$

subject to

$\begin{array}{cc}\left[\begin{array}{cc}{A}^{T}P+PA+Q& PQ+C^T\\ B^{T}P+C& R\end{array}\right]& \ge 0\\ {\tau }_i& \ge 0\end{array}$

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

1. Multi-Criterion LQG
2. Inverse Problem of Optimal Control
3. Nonconvex Multi-Criterion Quadratic Problems
4. Static-State Feedback Problem

External Links

A list of references documenting and validating the LMI.

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.