# 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**[edit]

The system for this LMI is a linear time invariant system that can be represented in state space as shown below:

where the system is assumed to be controllable.

where represents the state vector, respectively, is the disturbance vector, and are the system matrices of appropriate dimension. To further define: is and is the state vector, is and is the state matrix, is and is the input matrix, is and is the exogenous input.

for any input, we define a set cost indices by

Here the symmetric matrices,

- ,

are not necessarily positive-definite.

**The Data**[edit]

The matrices .

**The Optimization Problem**[edit]

The constrained optimal control problem is:

subject to

**The LMI:** Nonconvex Multi-Criterion Quadratic Problems[edit]

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

where and for every , we define

then, the solution can be found by:

subject to

**Conclusion:**[edit]

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

**Implementation**[edit]

This implementation requires Yalmip and Sedumi.

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

**Related LMIs**[edit]

## External Links[edit]

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.