# LMIs in Control/pages/Inverse Problem of Optimal Control

LMIs in Control/pages/Inverse Problem of Optimal Control

In some cases, it is needed to solve the inverse problem of optimal control within an LQR framework. In this inverse problem, a given controller matrix needs to be verified for the system by assuring that it is the optimal solution to some LQR optimization problem that is controllable and detectable. In other words: in this inverse problem, the controller is known and the LQR gain matrices are to be calculated such that the controller is the optimal solution.

**The System**[edit | edit source]

The system is a linear time-invariant system, that can be represented in state space as shown below:

where represent the state vector, the measured output vector, and the output vector of interest, 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, is and are the output matrices, and and are and are the output and the output of interest, respectively.

**The Data**[edit | edit source]

The matrices that define the system, and a given controller for which the inverse problem is to be solved.

**The Optimization Problem**[edit | edit source]

In this LMI, the following cost function is to be minimized for a given controller K by finding an optimal input:

the solution of the problem can be formulated as a state feedback controller given as:

**The LMI:** Inverse Problem of Optimal Control[edit | edit source]

the inverse problem of optimal control is the following: Given a matrix , determine if there exist and , such that is detectable and is the optimal control for the corresponding LQR problem. Equivalently, we seek and such that there exist nonnegative and positive-definite satisfying

**Conclusion:**[edit | edit source]

If the solution exists, then is the optimal controller for the LQR optimization on the matrices and

**Implementation**[edit | edit source]

This implementation requires Yalmip and Sedumi.

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

**Related LMIs**[edit | edit source]

## External Links[edit | edit source]

- 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.