LMIs in Control/pages/Nonconvex Multi-Criterion Quadratic Problems

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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


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 LMIs[edit]

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

External Links[edit]

A list of references documenting and validating the LMI.

Return to Main Page:[edit]