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

From Wikibooks, open books for an open world
Jump to navigation Jump to search

Robust Full State Feedback Optimal Control[edit | edit source]

Additive uncertainty[edit | edit source]

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 System[edit | edit source]

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 Data[edit | edit source]

, , , , , , , , are known.

The LMI:Full State Feedback Optimal Control LMI[edit | edit source]

There exists and and scalar such that

.

Where

And .

Conclusion:[edit | edit source]

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:

Implementation[edit | edit source]

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

Related LMIs[edit | edit source]

Full State Feedback Optimal H_inf LMI

External Links[edit | edit source]


Return to Main Page:[edit | edit source]

LMIs in Control: https://en.wikibooks.org/wiki/LMIs_in_Control