# 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:

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

where the system is assumed to be controllable.

where ${\displaystyle x\in R^{n}}$ represents the state vector, respectively, ${\displaystyle w\in R^{p}}$ is the disturbance vector, and ${\displaystyle A,B}$ are the system matrices of appropriate dimension. To further define: ${\displaystyle x}$ is ${\displaystyle \in R^{n}}$ and is the state vector, ${\displaystyle A}$ is ${\displaystyle \in R^{n*n}}$ and is the state matrix, ${\displaystyle B}$ is ${\displaystyle \in R^{n*r}}$ and is the input matrix, ${\displaystyle w}$ is ${\displaystyle \in R^{r}}$ and is the exogenous input.

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

{\displaystyle {\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,

{\displaystyle {\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 ${\displaystyle A,B,C}$.

## The Optimization Problem

The constrained optimal control problem is:

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

subject to

{\displaystyle {\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

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

{\displaystyle {\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:

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

subject to

{\displaystyle {\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 ${\displaystyle K}$ is the optimal controller and can be solved for via an EVP in P.

## Implementation

This implementation requires Yalmip and Sedumi.