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

 

Conclusion:Edit

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

ImplementationEdit

This implementation requires Yalmip and Sedumi.

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

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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