LMIs in Control/Click here to continue/Controller synthesis/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: 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 The matrices . The Optimization Problem The constrained optimal control problem is: subject to The LMI: Nonconvex Multi-Criterion Quadratic Problems 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: If the solution exists, then 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 Multi-Criterion LQG Inverse Problem of Optimal Control Nonconvex Multi-Criterion Quadratic Problems 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. Return to Main Page: