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### Observers/Estimators

Professor Walter W. Olson

Department of Mechanical, Industrial and Manufacturing Engineering

University of Toledo

…

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bn-1

b2

b1

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y

z2

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zn-1

…

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S

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an-1

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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

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

Disturbance

Controller

u

Plant/Process

Input

r

Output

y

S

S

kr

State Controller

Prefilter

x

-K

State Feedback

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

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

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.

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 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

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

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

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Example

Using the electric motor developed

in Lecture 5, develop the

Observability Canonical form

using the values

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 / Estimators

Output y(t)

Input u(t)

y

+

+

B

L

A

B

C

C

A

u

Noise

_

+

+

+

+

+

Observer/Estimator

State

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 / Estimators

- To compute L in
we need to compute the observable canonical form with

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

- 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

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

- 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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