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 SystemEdit

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 DataEdit

The matrices  .

The Optimization ProblemEdit

The constrained optimal control problem is:


subject to


The LMI: Nonconvex Multi-Criterion Quadratic ProblemsEdit

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



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


This implementation requires Yalmip and Sedumi.


Related LMIsEdit

  1. Multi-Criterion LQG
  2. Inverse Problem of Optimal Control
  3. Nonconvex Multi-Criterion Quadratic Problems
  4. Static-State Feedback Problem

External LinksEdit

A list of references documenting and validating the LMI.

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