z Transforms IIT Bombay Consider discrete time uniformly - - PowerPoint PPT Presentation

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z Transforms

4/10/2012 System Identification 43

{ }

riable. complex va a is z where ) ( ) ( ) ( as defined is )} ( {

• f

Transform

• z

} 1 : ) ( { signal discrete sampled uniformly time discrete Consider

∑

∞ = −

= ≡ Ζ =

k k

z k f z f k f k f ,.... , k k f

• perator

Laplace s interval, sampling T and : Note ≡ ≡ =

Ts

e z

). (

• f

ies singularit all encloses integral contour where ) ( i 2 1 ) ( as given is transform Inverse

1

z f dz z z f k f

k

∫

−

= π

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z Transforms : Properties

4/10/2012 System Identification 44

[ ] [ ] [ ]

g(k) f(k) ) ( f(k) Linearity Ζ + Ζ = + Ζ β α β α k g

{ } { }

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − = Ζ = Ζ

∑

− = − − − 1

) ( ) ( (k) ) ( (k) Shift Time

n j j n n n n

z j f z f z f q z f z f q

) ( z lim (0) Theorem Value Initial z f f ∞ → =

) ( )

• (1

1 z lim ) ( k lim then circle, unit the

• utside
• r
• n

poles any have not does ) ( )

• (1

If Theorem Value Final

1 1

z f z k f z f z

• →

= ∞ →

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z Transforms : Examples

4/10/2012 System Identification 45

{ }

2 2 1 2 1

) 1 ( ....) ( ...... 2 y(kT) (ramp) k with kT y(kT) 2 Example − = + + = + + + = Ζ ≥ =

− − − −

z Tz z z T Tz Tz

{ }

1 2 1

1 ...... y(kT) function) (step k with y(kT) 1 Example

− − −

− = + + + = Ζ ≥ = z z z α α α α α

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Pulse Transfer Function Matrix

4/10/2012 System Identification 46

4 43 4 42 1 4 43 4 42 1 (z) z k z z k z k z z k k k k

k k k k k k k k

u u Γ x x Φ x x x Γu Φx x ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − = + + = +

∑ ∑ ∑ ∑

∞ = − ∞ = − ∞ = − ∞ = −

) ( ) ( ) ( (0) ) ( ) 1 ( equation difference

• f

sides bothe

• n

transform

• z

Taking ) ( ) ( ) 1 (

[ ] [ ]

) (

• z

(z) ) ( (z)

• z

g Rearrangin ) ( (z) (z) have we (0) When

1

z z z z Γu Φ I x Γu x Φ I Γu Φx x x

−

= ⇒ = + = =

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Pulse Transfer Function Matrix

4/10/2012 System Identification 47

) ( ) ( ) ( ) ( ) ( sides both the

• n

transform z Taking ) ( ) ( z z (z) z k z z k k k

k k k k

Cx y x x C y y Cx y = ⇒ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ =

∑ ∑

∞ = − ∞ = −

4 43 4 42 1 4 43 4 42 1

[ ] [ ]

{ }

) ( ) ( ) (

• z

) ( have we ) ( ) ( with ) (

• z

) ( Combining

1 1

z z z z z z z z u G u Γ Φ I C y Cx y Γu Φ I x = = = =

− −

[ ] Γ

Φ I C G

1

) ( Function Transfer Pulse

−

− = z z

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4/10/2012 System Identification 48

Example: Quadruple Tank System

Sampling Time (T) = 5 sec Pulse transfer function model

Developed using discretization of linearized mechanistic model

) ( ) ( ) ( 0.9462

• z

0.1528 0.8009 + z 1.793

• z

0.005614 + z 0.006045 0.749 + z 1.734

• z

0.01034 + z 0.01138 0.9233

• z

0.2 ) ( ) ( ) (

2 1 2 2 2 1

3 2 1 4 4 4 4 4 4 4 4 4 3 4 4 4 4 4 4 4 4 4 2 1 3 2 1 (z) z v z v z z z h z h u G y ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡

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Time domain difference Equation

4/10/2012 System Identification 49

) ( z b + z a 1 bz + z b ) ( a + z a z b + z b ) (

2

• 2

1

• 1

2

• d

1

• d

1 2 1 2 2 1

z u z u z z y

d

+ = + =

−

[ ] [ ]

) ( ) ( + 1

2 1 1 2 2 1 1

z u + z z b z y z a z a

• d-
• d-
• =

+ ) 2 ( b + ) 1 ( ) 2 ( a + ) 1 ( ) (

2 1 2 1

− − − − = − − + d k u d k u b k y k y a k y ) 1 ( ) ( : Conditions Initial ,...... 4 , 3 , 2 where ) 2 ( + ) 1 ( b ) 2 ( a

• )

1 ( a

• )

( Model Equation Difference Linear

2 1 2 1

= = = − − − − + − − = y y k d k u b d k u k y k y k y

d = dead time Taking Inverse z transform on both sides, we get

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Practical Difficulty and Remedy

4/10/2012 System Identification 50

Practical Difficulty A reliable mechanistic model may not be available for the system under consideration. Instead of starting from well developed nonlinear mechanistic model, can we estimate parameters of linear perturbation directly from operating data? Remedy 1.Inject known input perturbations in the system 2.Record measured outputs generated in response to the perturbations

• 3. Estimate dynamic model parameters using optimization

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4 Tank Experimental Setup

4/10/2012 System Identification 51

Quadruple Tanks Setup

Tank 3 Tank 1 Tank 2 Tank 4 Control Valve 1 Control Valve 2

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4/10/2012 System Identification 52

Identification Experiments

• n 4 Tank Setup

Input 1 Input 2 Output 1 Output 2

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4/10/2012 System Identification 53

4 Tank Setup: Input Excitations

200 400 600 800 1000 1200

• 1

1 Input 1 (mA) Manipulated Input Sequence 200 400 600 800 1000 1200

• 1

1 Input 2 (mA) Time (sec)

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4/10/2012 System Identification 54

4 Tank Setup: Measured Output

200 400 600 800 1000 1200

• 5

5 Level 1 (mA) Meaured Output 200 400 600 800 1000 1200

• 6
• 4
• 2

2 4 Level 2 (mA) Time (sec)

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Experimental Data for Modeling

4/10/2012 System Identification 55

{ } { }

) ( ),....., 1 ( ), ( U Inputs d manipulate and/or Known ) ( ),....., 1 ( ), ( Y Outputs Measured collected data al Experiment

N N

N N U U U Y Y Y ≡ ≡

{ } { }

) ( ),....., 1 ( ), ( U Inputs d manipulate and/or Known ) ( ),....., 1 ( ), ( Y Outputs Measured Variables

• n

Perturbati Generate

N N

N N u u u y y y ≡ ≡

) , ( State Steady Operating U Y ≡

U U u Y Y y − = − = ) ( ) ( ) ( ) ( k k k k

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4/10/2012 System Identification 56

Splitting Data for Identification and Validation

500 1000

• 5

5 y1 Input and output signals 500 1000

• 0.5

0.5 1 Samples u1

Identification Data Validation data

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Quadruple Tank: Grey Box Linear Model

4/10/2012 System Identification 57

2 1 6 5 4 3 2 1 6 5 4 3 2 1

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = Γ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = Φ γ γ β β β β β β α α α α α α C

Let us treat elements of matrices in the discrete Linear state space model as unknowns Note: We are assuming that structure of matrices is known i.e. we know where zero and non-zero elements are

[ ]

T 2 1 6 1 6 1

... ... vector parameter Define γ γ β β α α = Θ

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Model Parameter Estimation

4/10/2012 System Identification 58

{ } ( ) ( ) (1)

ˆ ) C (1) ˆ (0) ) ( (0) ˆ ) ( (1) ˆ (0) ˆ ) C (0) ˆ equation difference linear using y recursivel by sequence state generate can we (0) ˆ state initial and for guess , ) 1 ( ),.... 1 ( ), ( set Given

N

x y u x x x y x u u u U Θ = Θ Γ + Θ Φ = Θ = Θ − = N

( ) (2)

ˆ ) C (2) ˆ (1) ) ( (1) ˆ ) ( (2) ˆ x y u x x Θ = Θ Γ + Θ Φ =

( ) (3)

ˆ ) C (3) ˆ (2) ) ( (2) ˆ ) ( (3) ˆ x y u x x Θ = Θ Γ + Θ Φ =

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Model Parameter Estimation

4/10/2012 System Identification 59

) ( ˆ ) ( (k) error prediction Define k k y y ε − =

1

• ....N

0,1,2,.... k where (k) ˆ ) ( C (k) ˆ (k) ) ( (k) ˆ ) ( 1) (k ˆ ubject to = Θ = Θ Γ + Θ Φ = + x y u x x S

( )

[ ]

∑

=

Θ = Θ Θ

N k T

k k Min ) ( ) ( ) ( ˆ , ) ( ˆ , ˆ problem

• n
• ptimizati

following solving by (0) ˆ state initial and Estimate Wε ε x x x

elements ve h matrix wit weighting Diagonal Let + ≡ W

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Alternate Approach

4/10/2012 System Identification 60

4 3 4 2 1 4 4 4 4 4 4 3 4 4 4 4 4 4 2 1 4 3 4 2 1 ) ( ) ( ) ( ) ( a q b a + q a q b + q b a + q a q b + q b q b ) ( ) ( ) (

2 1 6 6 5 4 2 5 4 3 2 2 3 2 1 1 2 1

k k u k u q a k k y k y u G y ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ + + + + = ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡

[ ] [ ]

T T

b b b a a a b b b a a a

6 5 4 6 5 4 2 3 2 1 3 2 1 1

vectors parameter Define = Θ = Θ

Let us treat elements of q transfer function Matrix model as unknowns

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MISO Model Identification

4/10/2012 System Identification 61

) a + q a )(1 q (1 ) ( ) b + q b )( 1 ( ) ( ) q a + q a 1 ( ) ( a + q a q b + q b ) ( q b ) ( ˆ

2 3 1

• 2

1

• 1

2 2 3 1

• 2

1 1 1 2

• 3

1

• 2

1 1 2 3 2 2 3 2 1 1 1 1 − − − −

+ + + + + = + + + = q a k u q q a k u q b k u k u a k y

( )

) 1 ( ) 3 ( ) 2 ( ) 1 ( ) 3 ( ) 2 ( ) 1 ( ) 3 ( ˆ ) 2 ( ˆ ) ( ) 1 ( ˆ ) ( ) ( ˆ

2 3 1 2 3 2 1 2 2 1 3 1 1 2 1 1 1 1 3 1 1 3 2 1 1 2 1 1

− − − − + − + + − + − + − + − + − − − + − − + − = k u b a k u b b a k u b k u a b k u a b k u b k y a a k y a a a k y a a k y Consider MISO model

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Model Parameter Estimation

4/10/2012 System Identification 62

{ }

y recursivel (1) equation difference using ) ( ˆ ),..... 3 ( ˆ ), 2 ( ˆ sequence state generate can we ) 2 ( ˆ ), 1 ( y ˆ , (0) ˆ state initial and for guesses , ) 1 ( ),.... 1 ( ), ( set Given

1 1 1 1 1 1 1 N

N y y y y y N Θ − = u u u U

) ( ˆ ) ( (k) error prediction Define

1 1 1

k y k y − = ε

( ) ( )

[ ]

∑

=

Θ = Θ Θ

N k

k y y y Min y y y y y y

2 2 1 3 2 1 1 3 2 1 1 3 2 1 1

) ( ) ( ˆ ), ( ˆ ), ( ˆ , ) ( ˆ ), ( ˆ ), ( ˆ , ˆ problem

• n
• ptimizati

following solving by ) ( ˆ ), ( ˆ ), ( ˆ , ˆ Estimate ε

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Pros and Cons

In each case, we get highly nonlinear optimization

problem

Important to give good initial guess to obtain

reliable model parameters: not an easy task

It is assumed that ‘structure’ of the linearized

state space model / q-TF matrix is know a-priori, which may not be the case

Effect of unmeasured inputs (disturbances) on

system dynamics is not captured by the model

Possible Remedy: Black box modeling 4/10/2012 System Identification 63