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:

${\begin{aligned}{\dot {x}}&=Ax+Bw,x(0)=x_{0}\\\end{aligned}}$

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

${\begin{aligned}J_{i}(u)=\int _{0}^{\infty }{\begin{bmatrix}x^{T}&u^{T}\end{bmatrix}}{\begin{bmatrix}Q_{i}&C_{i}\\C_{i}^{T}&R_{i}\end{bmatrix}}{\begin{bmatrix}x\\u\end{bmatrix}}dt,\\i=0,...,p\end{aligned}}$

Here the symmetric matrices,

${\begin{aligned}{\begin{bmatrix}Q_{i}&C_{i}\\C_{i}^{T}&R_{i}\end{bmatrix}},i=0,...,p\end{aligned}}$,

are not necessarily positive-definite.

The Data

The matrices $A,B,C$.

The Optimization Problem

The constrained optimal control problem is:

${\begin{aligned}\max :J_{0},\\\end{aligned}}$

subject to

${\begin{aligned}J_{i}\leq \gamma _{i},i=1,...,p,x\rightarrow 0,t\rightarrow \infty \end{aligned}}$

The LMI: Nonconvex Multi-Criterion Quadratic Problems

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

${\begin{aligned}Q=Q_{0}+\sum _{i=1}^{p}\tau _{i}Q_{i},\\R=R_{0}+\sum _{i=1}^{p}\tau _{i}R_{i},\\C=C_{0}+\sum _{i=1}^{p}\tau _{i}C_{i},\\\end{aligned}}$

where $\tau _{i}\geq 0$ and for every $\tau _{i}$, we define

${\begin{aligned}S=J_{0}+\sum _{i=1}^{p}\tau _{i}J_{i}-\sum _{i=1}^{p}\tau _{i}\gamma _{i}\end{aligned}}$

then, the solution can be found by:

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

subject to

${\begin{aligned}{\begin{bmatrix}A^{T}P+PA+Q&PQ+C^{T}\\B^{T}P+C&R\end{bmatrix}}&\geq 0\\\tau _{i}&\geq 0\end{aligned}}$

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.