full order observer design using simulink
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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?
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