LMIs in Control/Observer Synthesis/Continuous Time/Full-Order State Observer

LMIs in Control/Observer Synthesis/Continuous Time/Full-Order State Observer

Full-Order State Observer

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 $L$ such that $A+LC$ is Hurwitz stable.

The System

${\begin{aligned}{\dot {x}}(t)&=Ax(t)+Bu(t),\\y(t)&=Cx(t)+Du(t)\\x(0)&=x_{0}\\\end{aligned}}$

where $x(t)\in \mathbb {R} ^{n}$, $y(t)\in \mathbb {R} ^{m}$, $u(t)\in \mathbb {R} ^{q}$, at any $t\in \mathbb {R}$.

The Data

• The matrices $A,B,C,D$ are system matrices of appropriate dimensions and are known.

The Optimization Problem

The full-order state observer problem essential is finding a positive definite $P$ such that the following LMI conclusions hold.

The LMI:

1) The full-order state observer problem has a solution if and only if there exist a symmetric positive definite Matrix $P$ and a matrix $W$ that satisfy

• $A^{T}P+PA+W^{T}C+C^{T}W<0.$

Then the observer can be obtained as $L=P^{-1}W$
2) The full-state state observer can be found if and only if there is a symmetric positive definite Matrix $P$ that satisfies the below Matrix inequality

• $A^{T}P+PA-C^{T}C<0$
  

In this case the observer can be reconstructed as $L=-{\frac {1}{2}}P^{-1}C^{T}$. It can be seen that the second relation can be directly obtained by substituting $W=-{\frac {1}{2}}C^{T}$ in the first condition.

Conclusion:

Hence, both the above LMI's result in a full-order observer $L$ such that $A+LC$ is Hurwitz stable.

External Links

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.