Full Order Observer Design Using Simulink

David Pyne

EE 692

Goal of the Project

To design a Simulink library block that automatically generates a Full Order Observer for a given linear dynamical system

Desired Characteristics: Scaleable Practical Easy to Use

Agenda

What is a Full Order Observer? Overview of Linear Dynamical Systems Library Block Concept Design of the Observer Simulink Implementation A Simple Example A More Complex Example Questions

What is a Full Order Observer?

A mathematical model of the entire dynamical system

An estimator of the unmeasurable states Flexible Scaleable Highly accurate

Overview of Linear Dynamical Systems

Systems characterized by the following model:

MIMOxn

x1...

ym

y1...

Overview of Linear Dynamical Systems

The concept of “State”

Key system attributesThe minimum number of measurements neededNot always consistent

Library Block Concept

Systems in Simulink A series of function blocks Connected like a circuit Used to model dynamical systems

Library Block Concept

Example system

Library Block Concept

Inputs System A and B matrices Roots of observer

Outputs Preconfigured Simulink block

Design of the Observer

The full order observer equation

K0 is to be designed by the user Eigenvalues of the characteristic error

dynamics polynomial

Design of the Observer

Define the error dynamics equation

Simulink Implementation

Design Tasks Implement the observer equation using

existing Simulink blocks Create a subsystem by grouping smaller units

Create the user interface Validate user input Load user matrices into observer equation

Verify correct output

Simulink Implementation

Simulink Implementation

Observer user interface

Requires limited knowledge of observer design by the user

Simulink Implementation

Validation of user input A must be square B must have same number of rows as A C is assumed to be of the form [1 0 0 … 0] K must be the same length as C The system must be completely observable

Simulink Implementation

Complete observability check

Complete controllability check

Simulink Implementation

Check for repeated roots in K vector If root multiplicity is greater than the rank of C

use ACKER() Not terribly reliable Breaks down for higher order systems

Otherwise use PLACE() Much more robust Based on algorithm designed by Kautsky and

Nichols

Simulink Implementation

Block inputs y Scalar system output u System control command

Block outputs Any errors are warnings for the user Command line output of the calculated K0

matrix Vector estimate of system state

A Simple Example

A Simple Example

Output from new block tracks system exactly after locking on

A Simple Example

Error between actual and estimate converges to zero

A More Complex Example

A More Complex Example

A More Complex Example

A More Complex Example

A More Complex Example

A More Complex Example

Output from new block tracks system exactly after locking on

A More Complex Example

Error between actual and estimate converges to zero

Summary

A Simulink full order observer library block was created

Accurate Easy to use Scaleable

Saves the modern control designer (or student) time

Reduces the pain and suffering inherent in the design for higher order systems

References

Dorf, R.C. and Bishop, R.H.,Modern Control Systems, Tenth Edition, New Jersey:Pearson, Prentice Hall Publishing, 2004

Johnson, C.D., “Stabilization of Linear Dynamical Systems with Respect to Arbitrary Linear Subspaces,” Journal of Mathematical Analysis and Applications, Vol. 44 (1973),

No. 1, pp. 175-185

Johnson, C.D., “A Unified Canonical Form For Controllable and Uncontrollable Linear Dynamical Systems,” International Journal of Control, Vol. 13 (1971), No. 3, pp. 497-517

Kalman, R.E., “Mathematical Description of Linear Dynamical Systems,” SIAM Journal of Controls, Vol. 1 (1963), pp. 152-192

Kautsky, J. and N.K. Nichols, "Robust Pole Assignment in Linear State Feedback," International Journal of Control, Vol. 41 (1985), pp. 1129-1155

Kolman, B and Hill, D.R., Linear Algebra With Applications, Seventh Edition, New Jersey:Pearson-Prentice Hall Publishing, 2001

Questions?