1i_statespacemodeloverview.pdf

7/29/2019 1i_StateSpaceModelOverview.pdf

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Module 1Matrix Analysis


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State-space models

James, 2011

pp. 39-42, 82-86


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1. Set up a state-space model

2. Diagonalize the state matrix

3. Discuss the controllability andobservability of a SISO systemusing a state-space model

OUTCOMES


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Pros and cons

Popular method in digital systems analysis

Based on matrix algebra Software perform matrix calculations fast

Calculations limited to second and third ordersystems

Calculations by hand using a hand-held calculator

Intermediate steps information on state ofthe system


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Motivation

First-order linear ODE

Use an integration factor

Second-order ODE

Laplace transforms/undetermined coefficients

Fourth-order ODE Calculations complicated

Know input & output

Internal state at time t?


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Motivation

A simple LCR circuit A more complex circuit

2

2( )

d q dq qL R v t

dt dt C

State-space model!

Single differential equation?


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Motivation

Look for a method that

handles linear differential equations of anyorder with relative ease,

involves calculations that are time-efficientwhen using a CAD, and

provides information about the state of the

system at any time.


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Determine the response of a

SISO systemSingle input Single output

State of system


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Steps1. Define the state vector and its derivative.

2. Determine the state matrixA.

3. Write down the state and space equations.

4. Calculate the eigenvalues and eigenvectors ofA.

5. Write down the modal matrix M and spectral matrix.

6. Calculate M-1, M-1B and CM.

7. Use the transformationX= MZZ(0) = M-1X(0)

8. Write down et.

9. Evaluate

10. Write downZ= et + I

11. Determine y= CM-1Z

( ) 1

0

( )t

tI e M Bu d


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Good news!!!!!!!


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Steps1. Define the state vector and its derivative.

2. Determine the state matrixA.

3. Write down the state and space equations.

4. Calculate the eigenvalues and eigenvectors ofA.

5. Write down the modal matrix M and spectral matrix.

6. Calculate M-1, M-1B and CM.

7. Use the transformationX= MZZ(0) = M-1X(0)

8. Write down et.

9. Evaluate

10. Write downZ= et + I

11. Determine y= CM-1Z

( ) 1

0

( )t

tI e M Bu d


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In this module

You will use matrices to:

1. Set up a state-space model for the given

initial-value problem; and

2. Write the problem in canonical form using

diagonalization.


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An illustrative example

A SISO system is characterized by the IVP

1. Set up a state-space model for this system.

2. Diagonalize the state matrix.

2

23 2 2 , (0) 0, '(0) 1t

d y dyy e y y

dt dt


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Solution

1. Set up a state-space model for this system.

2

23 2 2 , (0) 0, '(0) 1t

d y dyy e y y

dt dt

1 1

2 2

1

2

1

2

0 1 0

22 3 1

1 0

(0) 0

(0) 1

tx x

ex x

xy

x

x

x

State vectorState matrix


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Solution

2. Diagonalize the state matrix.

2

23 2 2 , (0) 0, '(0) 1t

d y dyy e y y

dt dt

1 1 2 0,2 1 0 1

M

Eigenvectors Eigenvalues

Solution to this problem available as Example 2.3 onmyTUTor.


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What do you need?

1. Matrix calculations By hand

Calculator

2. Eigenvalues and eigenvectors


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Two lessons

1. Set up a state-space modelExample M1.1

Example M1.2

2. Diagonalize the state matrixExample M2.1

Example M2.2

YouMUST work through the theory in James(2011)!!!!!


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Let's get YOUworking!
http://localhost/var/www/apps/conversion/tmp/scratch_9/1h_MatrixCalculations.pptxhttp://localhost/var/www/apps/conversion/tmp/scratch_9/1h_MatrixCalculations.pptx

1i_statespacemodeloverview.pdf

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