LMIs in Control/pages/LMI for H2/Hinf Polytopic Controller for Robot Arm.

LMIs in Control/pages/LMI for H2/Hinf Polytopic Controller for Robot Arm.


The System:Edit

 



The Optimization Problem:Edit

Given a state space system of

 

where   , ,  and   form the K matrix as defined in below. This, therefore, means that the Regulator system can be re-written as:

 

With the above 9-matrix representation in mind, the we can now derive the controller needed for solving the problem, which in turn will be accomplished through the use of LMI's. Firstly, we will be taking our  / state-feedback control and make some modifications to it. More specifically, since the focus is modeling for worst-case scenario of a given parameter, we will be modifying the LMI's such that the mixed  /  controller is polytopic.

The LMI:Edit

 /  Polytopic Controller for Quadrotor with Robotic Arm.

Recall that from the 9-matrix framework ,   and   represent our process and sensor noises respectively and   represents our input channel. Suppose we were interested in modeling noise across all three of these channels. Then the best way to model uncertainty across all three cases would be modifying the   matrix to  , where (  parameters,  , and   is a constant noise value). This, in turn results in our  -  matrices to be modifified to  - 

Using the LMI's given for optimal  / -optimal state-feedback controller from Peet Lecture 11 as reference, our resulting polytopic LMI becomes:

  + 

 

CD=0

 

where i=1,..,k, &  and   and:

 


After solving for both the optimal   and   gain ratios as well as  , we can then construct our worst-case scenario controller by setting our   matrix (and consequently our   matrices) to the highest  value. This results in the controller:

 

which is constructed by setting:

 

where:

 


Conclusion:Edit

The LMI is feasible and the resulting controller is found to be stable under normal noise disturbances for all states.




ImplementationEdit

ReferencesEdit

1. An LMI-Based Approach for Altitude and Attitude Mixed H2/Hinf-Polytopic Regulator Control of a Quadrotor Manipulator by Aditya Ramani and Sudhanshu Katarey.