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Observers/Estimators. Professor Walter W. Olson Department of Mechanical, Industrial and Manufacturing Engineering University of Toledo. …. u. b n. b n-1. b 2. b 1. D. y. z 2. z n. z n-1. …. z 1. S. S. S. S. S. a n. a n-1. a 2. a 1. -1. …. Outline of Today’s Lecture.

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Observers estimators

Observers/Estimators

Professor Walter W. Olson

Department of Mechanical, Industrial and Manufacturing Engineering

University of Toledo

u

bn

bn-1

b2

b1

D

y

z2

zn

zn-1

z1

S

S

S

S

S

an

an-1

a2

a1

-1


Outline of today s lecture
Outline of Today’s Lecture

  • Review

    • Control System Objective

    • Design Structure for State Feedback

    • State Feedback

    • 2nd Order Response

    • State Feedback using the Reachable Canonical Form

  • Observability

  • Observability Matrix

  • Observable Canonical Form

  • Use of Observers/Estimators


Control system objective
Control System Objective

Given a system with the dynamics and the output

Design a linear controller with a single input which is

stable at an equilibrium point that we define as


Our design structure
Our Design Structure

Disturbance

Controller

u

Plant/Process

Input

r

Output

y

S

S

kr

State Controller

Prefilter

x

-K

State Feedback


2 nd order response
2nd Order Response

  • As the example showed, the characteristic equation for which the roots are the eigenvalues allow us to design the reachable system dynamics

  • When we determined the natural frequency and the damping ration by the equationwe actually changed the system modes by changing the eigenvalues of the system through state feedback

wn=1

z=0.6

Im(l)

Im(l)

x

wn=4

1

1

z=0.1

x

x

x

wn=2

z=0.4

x

z=0

x

x

wn=1

z=0.6

Re(l)

Re(l)

z=1

z=1

x

x

wn

-1

-1

z

z=0.6

wn=1

x

x

z=0

x

wn=2

x

z=0.4

x

x

-1

-1

z=0.1

x

wn=4


State feedback design with the reachable canonical equation
State Feedback Design with the Reachable Canonical Equation

  • Since the reachable canonical form has the coefficients of the characteristic polynomial explicitly stated, it may be used for design purposes:


Observability
Observability

  • Can we determine what are the states that produced a certain output?

    • Perhaps

  • Consider the linear system

    We say the system is observable if for any time T>0 it is possible to determine the state vector, x(T), through the measurements of the output, y(t), as the result of input, u(t), over the period between t=0 and t=T.


Observers estimators1
Observers / Estimators

Input u(t)

Output y(t)

Noise

State

Observer/Estimator


Testing for observability
Testing for Observability

  • Since observability is a function of the dynamics, consider the following system without input:

  • The output is

  • Using the truncated series


Testing for observability1
Testing for Observability

  • For x(0) to be uniquely determined, the material in the parens must exist requiring

    to have full rank, thus also being invertible, the common test

  • Wo is called the Observability Matrix


Example inverted pendulum
Example: Inverted Pendulum

  • Determine the observabilitypf the Segway system with v as the output


Observable canonical form
Observable Canonical Form

  • A system is in Observable Canonical Form if it can be put into the form

Where ai are the

coefficients of the

characteristic equation

u

bn

bn-1

b2

b1

D

y

z2

zn

zn-1

z1

S

S

S

S

S

an

an-1

a2

a1

-1




Example
Example

Using the electric motor developed

in Lecture 5, develop the

Observability Canonical form

using the values


Observers estimators2
Observers / Estimators

Input u(t)

Output y(t)

Noise

State

Observer/Estimator

Knowing that the system is observable, how do we observe the states?


Observers estimators3
Observers / Estimators

Output y(t)

Input u(t)

y

+

+

B

L

A

B

C

C

A

u

Noise

_

+

+

+

+

+

Observer/Estimator

State


Observers estimators4
Observers / Estimators

  • The form of our observer/estimator is

    If (A-LC) has negative real parts, it is both stable and the error,

    , will go to zero.

    How fast? Depends on the eigenvaluesof (A-LC)


Observers estimators5
Observers / Estimators

  • To compute L in

    we need to compute the observable canonical form with


Example1
Example

q

  • A hot air balloon has the following equilibrium equations

  • Construct a state observer assuming that the eigenvalute to achieve are l=10:

u

w

h



Control with observers
Control with Observers

  • Previously we designed a state feedback controller where we generated the input to the system to be controlled as

  • When we did that we assumed that wse had direct access to the states. But what if we do not?

  • A possible solution is to use the observer/estimator states and generate


Control with observers1
Control with Observers

r

u

y

kr

A

C

C

A

B

L

B

+

+

+

+

-K

_

+

+

+

+

+




Example3
Example

q

  • A hot air balloon has the following equilibrium equations

  • Construct a state feedback controller with an observer to achieve and maintain a given height

u

w

h




Summary
Summary

  • Observability

    • We say the system is observable if for any time T>0 it is possible to determine the state vector, x(T), through the measurements of the output, y(t), as the result of input, u(t), over the period between t=0 and t=T.

  • ObservabilityMatrix

  • Observable Canonical Form

  • Use of Observers/Estimators

Next: Kalman Filters


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