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

**Robust Full State Feedback Optimal Control**Edit

#### 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 System**Edit

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

, , , , , , , , 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:

**Implementation**Edit

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

**Related LMIs**Edit

**External Links**Edit

- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
- LMIs in Control Systems: Analysis, Design and Applications - by Guang-Ren Duan and Hai-Hua Yu, CRC Press, Taylor & amp; Francis Group, 2013.

**Return to Main Page:**Edit

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