LMIs in Control/pages/Multi-Criterion LQG

LMIs in Control/pages/Multi-Criterion LQG

The Multi-Criterion Linear Quadratic Gaussian (LQG) linear matrix inequality will allow one to form an optimized controller, similar to that in an LQR framework, for a state space system with gaussian noise 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 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.


  and  , and the system is controllable and observable.

The DataEdit

The matrices   and the noise signals  .

The Optimization ProblemEdit

In the Linear Quadratic Gaussian (LQG) control problem, the goal is to minimize a quadratic cost function while the plant has random initial conditions and suffers white noise disturbance on the input and measurement.

There are multiple outputs of interest for this problem. They are defined by

 

For each of these outputs of interest, we associate a cost function:

 

Additionally, the matrices   and   must be found as the solutions to the following Riccati equations:

 

The optimization problem is to minimize   over u subject to the measurability condition and the constraints  . This optimization problem can be formulated as:

 

over  , with:

 

The LMI: Multi-Criterion LQGEdit

 

over  , subject to the following constraints:

 

Conclusion:Edit

The result of this LMI is the solution to the aforementioned Ricatti equations:

 

ImplementationEdit

This implementation requires Yalmip and Sedumi.

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

Related LMIsEdit

  1. Inverse Problem of Optimal Control

External LinksEdit

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