EE611 Deterministic Systems State Observers Kevin D. Donohue Electrical and Computer Engineering University of Kentucky

Open-Loop Observers Block diagram of state estimator tracking only system input: ++ Actual System Simulated System (Needs model parameters and initial state)

Closed-Loop Observers Block diagram of state estimator tracking both input and output + + Actual System + - Observer

Simplified Observer Block diagram of state estimator tracking both input and output + Based on Simplified Observer Equation +

Observability and Feedback Theorem 8.O3 Given pair (A, c) and n x 1 vector l, all eigenvalues of (A-lc) can be assigned arbitrarily iff (A, c) is observable. Proved with duality: (A, c) observable iff (A', c') controllable. Controllability implies eigenvalue of (A', c'k) can be arbitrarily assigned. (A', c'k)' => (A, k'c). Note k is equivalent to l for observers, so same design procedures for k can be used for l. Note these eigenvalues relate to the convergence of the estimator, not the eigenvalues of the given system! The observer is an independent system (typically an analog or digital computer) that observers the given system's input and output, and computes the state variables.

Feedback for Observers The l vector for the observer can be determined to arbitrarily place the eigenvalues of the state estimation error iff the the original systems is observable. Show that the eigenvalues directly impact the convergence of the state estimation error: where

Observable Canonical Form Given an observable system with characteristic polynomial system can be transformed into observable canonical form by

Observable Canonical Form

Example Given system: Design a state observer with eigenvalues at -2 and Use observable canonical form to find l 2. Use duality, find l by the same process as finding k. 3. Use Lyapunov equation.

Lyapunov Method Find equivalence transformation such that desired l vector satisfies: where F is a matrix with desired eigenvalues distinct from A (i.e. the eigenvalues of the observer). Terms can be rearranged to show: Therefore, can be chosen almost arbitrarily, and similarity transform matrix T solved for to obtain

Lyapunov Method If A and F have no eigenvalues in common, then a solution for T exists in and is nonsingular iff (A, c ) is observable and (F, ) controllable. Procedure: 1. Select n x n matrix F with desired eigenvalues. 2. Select arbitrary n x 1 vector such that (F, ) controllable. 3. Solve for T in the Lyapunov equation 4. Compute observer gain vector

Reduced-Dimensional State Estimator An equivalence transformation can be found to transform an qxq partition of the C matrix to an identity, assuming C has full row rank (q). In this case q state variables are equivalent to the the output values and only n-q state variable need to be estimated. An observer using this approach is referred to as a reduced- dimensional state estimator. Consider now an SISO system where an n-1 x n-1 reduced state estimator must be derived.

Lyapunov Method for Reduce State Estimators If A and F have no eigenvalues in common, then square matrix: where T is a unique solution for, and is nonsingular iff (A, c ) is observable and (F, ) controllable. Procedure: 1. Select n-1 x n-1 matrix F with desired eigenvalues. 2. Select n-1 x 1 vector such that (F, ) controllable. 3. Solve for n-1 x n matrix T in 4. Estimator is realized in transformed basis and transformed to original basis:

Reduced-State Observers Block diagram of reduced state estimator + + Actual System Observer P -1

Example Design a reduced state estimator for Place eigenvalues of estimator error at -9, -7, -8.7  j4

Combined Observer Feedback System If the estimated states are used in the feedback the total system is expressed as: It can be expressed in terms of estimation error with equivalence transformation:

Reduced-Observer Feedback System If the states from a reduced-dimension estimator are used in the feedback the total system is expressed as: It can be expressed in terms of estimation error with equivalence transformation: