LMIs in Control/pages/Robust H inf State Feedback Control

Robust Full State Feedback Optimal ControlEdit

Additive uncertaintyEdit

Full State Feedback is a control technique which places a given system's closed loop system poles in locations specified by desired performance specifications. One can use   methods to turn this into an optimization problem with the goal to minimize the impact of uncertain perturbations in a closed loop system while maintaining system stability. This is done by minimizing the   norm of the uncertain closed loop system, which minimizes the worst case effect of the system disturbance or perturbation. This can be done for single-input single-output (SISO) or multiple-input multiple-output (MIMO) systems. Here we consider the case of a MIMO system with additive uncertainties.

The SystemEdit

Consider linear system with uncertainty below:

 

Where   is the state,   is the output,   is the exogenous input or disturbance vector, and   is the actuator input or control vector, at any  


  and   are real-valued matrices which represent the time-varying parameter uncertainties in the form:

 


Where

  are known matrices with appropriate dimensions and   is the uncertain parameter matrix which satisfies:  


For additive perturbations:  

Where

  are the known system matrices and

  are the perturbation parameters which satisfy  


Thus,   with

 

 

 

The DataEdit

 ,  ,  ,  ,  ,  ,  ,  ,   are known.

The LMI:Full State Feedback Optimal Control LMIEdit

There exists   and   and scalar   such that

 .

Where  

And  .

Conclusion:Edit

Once K is found from the optimization LMI above, it can be substituted into the state feedback control law   to find the robustly stabilized closed loop system as shown below:

 

where   is the state,   is the output,   is the exogenous input or disturbance vector, and   is the actuator input or control vector, at any  


Finally, the transfer function of the system is denoted as follows:

 

ImplementationEdit

This implementation requires Yalmip and Sedumi. https://github.com/rubindan/LMIcontrol/blob/master/HinfFilter.m

Related LMIsEdit

Full State Feedback Optimal H_inf LMI

External LinksEdit


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