LMIs in Control/Click here to continue/Observer synthesis/Full-order state Observer
LMIs in Control/Click here to continue/Observer synthesis/Full-order state Observer
Full-Order State Observer
[edit | edit source]The problem of constructing a simple full-order state observer directly follows from the result of Hurwitz detectability LMI's, Which essentially is the dual of Hurwitz stabilizability. If a feasible solution to the first LMI for Hurwitz detectability exist then using the results we can back out a full state observer such that is Hurwitz stable.
The System
[edit | edit source]where , , , at any .
The Data
[edit | edit source]- The matrices are system matrices of appropriate dimensions and are known.
The Optimization Problem
[edit | edit source]The full-order state observer problem essential is finding a positive definite such that the following LMI conclusions hold.
The LMI:
[edit | edit source]1) The full-order state observer problem has a solution if and only if there exist a symmetric positive definite Matrix and a matrix that satisfy
Then the observer can be obtained as
2) The full-state state observer can be found if and only if there is a symmetric positive definite Matrix that satisfies the below Matrix inequality
In this case the observer can be reconstructed as . It can be seen that the second relation can be directly obtained by substituting in the first condition.
Conclusion:
[edit | edit source]Hence, both the above LMI's result in a full-order observer such that is Hurwitz stable.
External Links
[edit | edit source]A list of references documenting and validating the LMI.
- LMIs in Control Systems Analysis, Design and Applications - Duan and Yu
- 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.