1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal...

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1. Internal Model Principle

Transcript of 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal...

Page 1: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

1. Internal Model Principle

Page 2: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

1. Internal Model Principle

r

−G(z) =

B

AGc(z) =

Sc

Rc

ue y

v

Page 3: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

1. Internal Model Principle

r

−G(z) =

B

AGc(z) =

Sc

Rc

ue y

v

• α(z) = least common multiple of the unstable poles ofRc(z) and of V (z),

Page 4: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

1. Internal Model Principle

r

−G(z) =

B

AGc(z) =

Sc

Rc

ue y

v

• α(z) = least common multiple of the unstable poles ofRc(z) and of V (z), all polynomials in z−1

Page 5: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

1. Internal Model Principle

r

−G(z) =

B

AGc(z) =

Sc

Rc

ue y

v

• α(z) = least common multiple of the unstable poles ofRc(z) and of V (z), all polynomials in z−1

• Let there be no common factors between α(z) and B(z)

Page 6: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

1. Internal Model Principle

r

−G(z) =

B

AGc(z) =

Sc

Rc

ue y

v

• α(z) = least common multiple of the unstable poles ofRc(z) and of V (z), all polynomials in z−1

• Let there be no common factors between α(z) and B(z)

• Can find a controller Gc(z) for servo/tracking (followingRc)

Page 7: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

1. Internal Model Principle

r

−G(z) =

B

AGc(z) =

Sc

Rc

ue y

v

• α(z) = least common multiple of the unstable poles ofRc(z) and of V (z), all polynomials in z−1

• Let there be no common factors between α(z) and B(z)

• Can find a controller Gc(z) for servo/tracking (followingRc) and regulation (rejection of disturbance V )

Page 8: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

1. Internal Model Principle

r

−G(z) =

B

AGc(z) =

Sc

Rc

ue y

v

• α(z) = least common multiple of the unstable poles ofRc(z) and of V (z), all polynomials in z−1

• Let there be no common factors between α(z) and B(z)

• Can find a controller Gc(z) for servo/tracking (followingRc) and regulation (rejection of disturbance V ) if Rc con-tains α,

Page 9: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

1. Internal Model Principle

r

−G(z) =

B

AGc(z) =

Sc

Rc

ue y

v

• α(z) = least common multiple of the unstable poles ofRc(z) and of V (z), all polynomials in z−1

• Let there be no common factors between α(z) and B(z)

• Can find a controller Gc(z) for servo/tracking (followingRc) and regulation (rejection of disturbance V ) if Rc con-tains α, say, Rc = αR1:

Page 10: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

1. Internal Model Principle

r

−G(z) =

B

AGc(z) =

Sc

Rc

ue y

v

• α(z) = least common multiple of the unstable poles ofRc(z) and of V (z), all polynomials in z−1

• Let there be no common factors between α(z) and B(z)

• Can find a controller Gc(z) for servo/tracking (followingRc) and regulation (rejection of disturbance V ) if Rc con-tains α, say, Rc = αR1:

e(n)

y(n)G =

B

A

r(n)Gc =

Sc

αR1

u(n)

v(n)

Digital Control 1 Kannan M. Moudgalya, Autumn 2007

Page 11: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

2. Internal Model Principle - Regulation

Page 12: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

2. Internal Model Principle - Regulation

e(n)

y(n)G =

B

A

r(n)Gc =

Sc

αR1

u(n)

v(n)

Page 13: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

2. Internal Model Principle - Regulation

e(n)

y(n)G =

B

A

r(n)Gc =

Sc

αR1

u(n)

v(n)

Y (z) =

Sc

R1

1

α

B

A

1 +Sc

R1

1

α

B

A

R

Page 14: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

2. Internal Model Principle - Regulation

e(n)

y(n)G =

B

A

r(n)Gc =

Sc

αR1

u(n)

v(n)

Y (z) =

Sc

R1

1

α

B

A

1 +Sc

R1

1

α

B

A

R+1

1 +Sc

R1

1

α

B

A

V

Page 15: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

2. Internal Model Principle - Regulation

e(n)

y(n)G =

B

A

r(n)Gc =

Sc

αR1

u(n)

v(n)

Y (z) =

Sc

R1

1

α

B

A

1 +Sc

R1

1

α

B

A

R+1

1 +Sc

R1

1

α

B

A

V

=ScB

R1Aα+ ScBR

Page 16: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

2. Internal Model Principle - Regulation

e(n)

y(n)G =

B

A

r(n)Gc =

Sc

αR1

u(n)

v(n)

Y (z) =

Sc

R1

1

α

B

A

1 +Sc

R1

1

α

B

A

R+1

1 +Sc

R1

1

α

B

A

V

=ScB

R1Aα+ ScBR+

R1Aα

R1Aα+ ScB

bV

αaV

Page 17: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

2. Internal Model Principle - Regulation

e(n)

y(n)G =

B

A

r(n)Gc =

Sc

αR1

u(n)

v(n)

Y (z) =

Sc

R1

1

α

B

A

1 +Sc

R1

1

α

B

A

R+1

1 +Sc

R1

1

α

B

A

V

=ScB

R1Aα+ ScBR+

R1Aα

R1Aα+ ScB

bV

αaV• Unstable pole present in α gets cancelled

Page 18: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

2. Internal Model Principle - Regulation

e(n)

y(n)G =

B

A

r(n)Gc =

Sc

αR1

u(n)

v(n)

Y (z) =

Sc

R1

1

α

B

A

1 +Sc

R1

1

α

B

A

R+1

1 +Sc

R1

1

α

B

A

V

=ScB

R1Aα+ ScBR+

R1Aα

R1Aα+ ScB

bV

αaV• Unstable pole present in α gets cancelled

• Regulation problem verifiedDigital Control 2 Kannan M. Moudgalya, Autumn 2007

Page 19: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

3. Internal Model Principle - Servo

Page 20: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

3. Internal Model Principle - Servo

e(n)

y(n)G =

B

A

r(n)Gc =

Sc

αR1

u(n)

v(n)

Page 21: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

3. Internal Model Principle - Servo

e(n)

y(n)G =

B

A

r(n)Gc =

Sc

αR1

u(n)

v(n)

Y (z) =ScB

R1Aα+ ScBR+

R1Aα

R1Aα+ ScBV

Page 22: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

3. Internal Model Principle - Servo

e(n)

y(n)G =

B

A

r(n)Gc =

Sc

αR1

u(n)

v(n)

Y (z) =ScB

R1Aα+ ScBR+

R1Aα

R1Aα+ ScBV

Servo problem: assume V = 0:

Page 23: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

3. Internal Model Principle - Servo

e(n)

y(n)G =

B

A

r(n)Gc =

Sc

αR1

u(n)

v(n)

Y (z) =ScB

R1Aα+ ScBR+

R1Aα

R1Aα+ ScBV

Servo problem: assume V = 0:

E(z) = R(z)− Y (z)

Page 24: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

3. Internal Model Principle - Servo

e(n)

y(n)G =

B

A

r(n)Gc =

Sc

αR1

u(n)

v(n)

Y (z) =ScB

R1Aα+ ScBR+

R1Aα

R1Aα+ ScBV

Servo problem: assume V = 0:

E(z) = R(z)− Y (z)

=

(1− ScB

R1Aα+ ScB

)R

Page 25: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

3. Internal Model Principle - Servo

e(n)

y(n)G =

B

A

r(n)Gc =

Sc

αR1

u(n)

v(n)

Y (z) =ScB

R1Aα+ ScBR+

R1Aα

R1Aα+ ScBV

Servo problem: assume V = 0:

E(z) = R(z)− Y (z)

=

(1− ScB

R1Aα+ ScB

)R =

R1Aα

R1Aα+ ScB

bR

αaR

Page 26: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

3. Internal Model Principle - Servo

e(n)

y(n)G =

B

A

r(n)Gc =

Sc

αR1

u(n)

v(n)

Y (z) =ScB

R1Aα+ ScBR+

R1Aα

R1Aα+ ScBV

Servo problem: assume V = 0:

E(z) = R(z)− Y (z)

=

(1− ScB

R1Aα+ ScB

)R =

R1Aα

R1Aα+ ScB

bR

αaR• Unstable poles of Rc are cancelled by zeros of α.

Page 27: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

3. Internal Model Principle - Servo

e(n)

y(n)G =

B

A

r(n)Gc =

Sc

αR1

u(n)

v(n)

Y (z) =ScB

R1Aα+ ScBR+

R1Aα

R1Aα+ ScBV

Servo problem: assume V = 0:

E(z) = R(z)− Y (z)

=

(1− ScB

R1Aα+ ScB

)R =

R1Aα

R1Aα+ ScB

bR

αaR• Unstable poles of Rc are cancelled by zeros of α.

• Can choose Rc and Sc such that R1Aα+ ScB has rootswithin the unit circle

Page 28: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

3. Internal Model Principle - Servo

e(n)

y(n)G =

B

A

r(n)Gc =

Sc

αR1

u(n)

v(n)

Y (z) =ScB

R1Aα+ ScBR+

R1Aα

R1Aα+ ScBV

Servo problem: assume V = 0:

E(z) = R(z)− Y (z)

=

(1− ScB

R1Aα+ ScB

)R =

R1Aα

R1Aα+ ScB

bR

αaR• Unstable poles of Rc are cancelled by zeros of α.

• Can choose Rc and Sc such that R1Aα+ ScB has rootswithin the unit circle (pole placement)

Page 29: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

3. Internal Model Principle - Servo

e(n)

y(n)G =

B

A

r(n)Gc =

Sc

αR1

u(n)

v(n)

Y (z) =ScB

R1Aα+ ScBR+

R1Aα

R1Aα+ ScBV

Servo problem: assume V = 0:

E(z) = R(z)− Y (z)

=

(1− ScB

R1Aα+ ScB

)R =

R1Aα

R1Aα+ ScB

bR

αaR• Unstable poles of Rc are cancelled by zeros of α.

• Can choose Rc and Sc such that R1Aα+ ScB has rootswithin the unit circle (pole placement)

• IM Principle:

Page 30: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

3. Internal Model Principle - Servo

e(n)

y(n)G =

B

A

r(n)Gc =

Sc

αR1

u(n)

v(n)

Y (z) =ScB

R1Aα+ ScBR+

R1Aα

R1Aα+ ScBV

Servo problem: assume V = 0:

E(z) = R(z)− Y (z)

=

(1− ScB

R1Aα+ ScB

)R =

R1Aα

R1Aα+ ScB

bR

αaR• Unstable poles of Rc are cancelled by zeros of α.

• Can choose Rc and Sc such that R1Aα+ ScB has rootswithin the unit circle (pole placement)

• IM Principle: unstable poles of V , Rc appear in loop

Page 31: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

3. Internal Model Principle - Servo

e(n)

y(n)G =

B

A

r(n)Gc =

Sc

αR1

u(n)

v(n)

Y (z) =ScB

R1Aα+ ScBR+

R1Aα

R1Aα+ ScBV

Servo problem: assume V = 0:

E(z) = R(z)− Y (z)

=

(1− ScB

R1Aα+ ScB

)R =

R1Aα

R1Aα+ ScB

bR

αaR• Unstable poles of Rc are cancelled by zeros of α.

• Can choose Rc and Sc such that R1Aα+ ScB has rootswithin the unit circle (pole placement)

• IM Principle: unstable poles of V , Rc appear in loop thro’α

Digital Control 3 Kannan M. Moudgalya, Autumn 2007

Page 32: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

4. Internal Stability

Page 33: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

4. Internal Stability

u(n)e(n)

y(n)+

r(n)gc(n) g(n)

Page 34: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

4. Internal Stability

u(n)e(n)

y(n)+

r(n)gc(n) g(n)

• Notion in UG classes: output has to be stable

Page 35: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

4. Internal Stability

u(n)e(n)

y(n)+

r(n)gc(n) g(n)

• Notion in UG classes: output has to be stable

• Output being stable is not sufficient

Page 36: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

4. Internal Stability

u(n)e(n)

y(n)+

r(n)gc(n) g(n)

• Notion in UG classes: output has to be stable

• Output being stable is not sufficient

• Every signal in the loop should be bounded

Page 37: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

4. Internal Stability

u(n)e(n)

y(n)+

r(n)gc(n) g(n)

• Notion in UG classes: output has to be stable

• Output being stable is not sufficient

• Every signal in the loop should be bounded

• If any signal is unbounded, will result in saturation / overflow/ explosion

Page 38: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

4. Internal Stability

u(n)e(n)

y(n)+

r(n)gc(n) g(n)

• Notion in UG classes: output has to be stable

• Output being stable is not sufficient

• Every signal in the loop should be bounded

• If any signal is unbounded, will result in saturation / overflow/ explosion

• When every signal is bounded, called internal stability

Page 39: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

4. Internal Stability

u(n)e(n)

y(n)+

r(n)gc(n) g(n)

• Notion in UG classes: output has to be stable

• Output being stable is not sufficient

• Every signal in the loop should be bounded

• If any signal is unbounded, will result in saturation / overflow/ explosion

• When every signal is bounded, called internal stability

• If output is stable and if there is no unstable pole-zero can-cellation, internal stability

Digital Control 4 Kannan M. Moudgalya, Autumn 2007

Page 40: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

5. Unstb. Pole-Zero Cancel. = No Int. Stability

Page 41: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

5. Unstb. Pole-Zero Cancel. = No Int. Stability

+−

+r1

Gc =n2

d2

r2

G =n1

d1

e1 e2 y+

Page 42: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

5. Unstb. Pole-Zero Cancel. = No Int. Stability

+−

+r1

Gc =n2

d2

r2

G =n1

d1

e1 e2 y+

[E1

E2

]=

1

1 +GGc

− G

1 +GGcGc

1 +GcG

1

1 +GcG

[R1

R2

]

Page 43: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

5. Unstb. Pole-Zero Cancel. = No Int. Stability

+−

+r1

Gc =n2

d2

r2

G =n1

d1

e1 e2 y+

[E1

E2

]=

1

1 +GGc

− G

1 +GGcGc

1 +GcG

1

1 +GcG

[R1

R2

]

=

d1d2

n1n2 + d1d2

− n1d2

n1n2 + d1d2n2d1

n1n2 + d1d2

d1d2

n1n2 + d1d2

[R1

R2

]

Page 44: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

5. Unstb. Pole-Zero Cancel. = No Int. Stability

+−

+r1

Gc =n2

d2

r2

G =n1

d1

e1 e2 y+

[E1

E2

]=

1

1 +GGc

− G

1 +GGcGc

1 +GcG

1

1 +GcG

[R1

R2

]

=

d1d2

n1n2 + d1d2

− n1d2

n1n2 + d1d2n2d1

n1n2 + d1d2

d1d2

n1n2 + d1d2

[R1

R2

]Suppose d1, n2 have a common factor:

Page 45: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

5. Unstb. Pole-Zero Cancel. = No Int. Stability

+−

+r1

Gc =n2

d2

r2

G =n1

d1

e1 e2 y+

[E1

E2

]=

1

1 +GGc

− G

1 +GGcGc

1 +GcG

1

1 +GcG

[R1

R2

]

=

d1d2

n1n2 + d1d2

− n1d2

n1n2 + d1d2n2d1

n1n2 + d1d2

d1d2

n1n2 + d1d2

[R1

R2

]Suppose d1, n2 have a common factor:

d1 = (z + a)d′1n2 = (z + a)n′2

Digital Control 5 Kannan M. Moudgalya, Autumn 2007

Page 46: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

6. Unstable Pole-Zero Cancellation = No InternalStability

Page 47: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

6. Unstable Pole-Zero Cancellation = No InternalStability

Assume the cancellation of z + a,

G(z) =n1(z)

(z + a)d′1(z), Gc(z) =

(z + a)n′2(z)

d2(z)

Page 48: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

6. Unstable Pole-Zero Cancellation = No InternalStability

Assume the cancellation of z + a,

G(z) =n1(z)

(z + a)d′1(z), Gc(z) =

(z + a)n′2(z)

d2(z)

with |a| > 1.

Page 49: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

6. Unstable Pole-Zero Cancellation = No InternalStability

Assume the cancellation of z + a,

G(z) =n1(z)

(z + a)d′1(z), Gc(z) =

(z + a)n′2(z)

d2(z)

with |a| > 1. Assume stability of

TE

Page 50: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

6. Unstable Pole-Zero Cancellation = No InternalStability

Assume the cancellation of z + a,

G(z) =n1(z)

(z + a)d′1(z), Gc(z) =

(z + a)n′2(z)

d2(z)

with |a| > 1. Assume stability of

TE =1

1 +GGc

Page 51: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

6. Unstable Pole-Zero Cancellation = No InternalStability

Assume the cancellation of z + a,

G(z) =n1(z)

(z + a)d′1(z), Gc(z) =

(z + a)n′2(z)

d2(z)

with |a| > 1. Assume stability of

TE =1

1 +GGc

=d′1d2

d′1d2 + n1n′2

Page 52: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

6. Unstable Pole-Zero Cancellation = No InternalStability

Assume the cancellation of z + a,

G(z) =n1(z)

(z + a)d′1(z), Gc(z) =

(z + a)n′2(z)

d2(z)

with |a| > 1. Assume stability of

TE =1

1 +GGc

=d′1d2

d′1d2 + n1n′2

T.F. between R2 and Y can be shown to be unstable.

Page 53: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

6. Unstable Pole-Zero Cancellation = No InternalStability

Assume the cancellation of z + a,

G(z) =n1(z)

(z + a)d′1(z), Gc(z) =

(z + a)n′2(z)

d2(z)

with |a| > 1. Assume stability of

TE =1

1 +GGc

=d′1d2

d′1d2 + n1n′2

T.F. between R2 and Y can be shown to be unstable. LetR1 = 0.

Page 54: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

6. Unstable Pole-Zero Cancellation = No InternalStability

Assume the cancellation of z + a,

G(z) =n1(z)

(z + a)d′1(z), Gc(z) =

(z + a)n′2(z)

d2(z)

with |a| > 1. Assume stability of

TE =1

1 +GGc

=d′1d2

d′1d2 + n1n′2

T.F. between R2 and Y can be shown to be unstable. LetR1 = 0.

Y

R2

=G

1 +GGc

=n1d2

(d′1d2 + n1n′2)(z + a)

Page 55: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

6. Unstable Pole-Zero Cancellation = No InternalStability

Assume the cancellation of z + a,

G(z) =n1(z)

(z + a)d′1(z), Gc(z) =

(z + a)n′2(z)

d2(z)

with |a| > 1. Assume stability of

TE =1

1 +GGc

=d′1d2

d′1d2 + n1n′2

T.F. between R2 and Y can be shown to be unstable. LetR1 = 0.

Y

R2

=G

1 +GGc

=n1d2

(d′1d2 + n1n′2)(z + a)

It is unstable

Page 56: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

6. Unstable Pole-Zero Cancellation = No InternalStability

Assume the cancellation of z + a,

G(z) =n1(z)

(z + a)d′1(z), Gc(z) =

(z + a)n′2(z)

d2(z)

with |a| > 1. Assume stability of

TE =1

1 +GGc

=d′1d2

d′1d2 + n1n′2

T.F. between R2 and Y can be shown to be unstable. LetR1 = 0.

Y

R2

=G

1 +GGc

=n1d2

(d′1d2 + n1n′2)(z + a)

It is unstable and a bounded signal injected at R2 will producean unbounded signal at YDigital Control 6 Kannan M. Moudgalya, Autumn 2007

Page 57: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

7. Forbid Unstable Pole-Zero Cancellation: LoopVariable

Page 58: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

7. Forbid Unstable Pole-Zero Cancellation: LoopVariable

• Internal stability = all variables in loop are bounded for boundedexternal inputs at all locations

Page 59: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

7. Forbid Unstable Pole-Zero Cancellation: LoopVariable

• Internal stability = all variables in loop are bounded for boundedexternal inputs at all locations

• Can be checked by the following closed loop diagram

Page 60: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

7. Forbid Unstable Pole-Zero Cancellation: LoopVariable

• Internal stability = all variables in loop are bounded for boundedexternal inputs at all locations

• Can be checked by the following closed loop diagram

+−

+r1

Gc=

n2

d2

r2

G =n1

d1

e1 e2 y+

Page 61: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

7. Forbid Unstable Pole-Zero Cancellation: LoopVariable

• Internal stability = all variables in loop are bounded for boundedexternal inputs at all locations

• Can be checked by the following closed loop diagram

+−

+r1

Gc=

n2

d2

r2

G =n1

d1

e1 e2 y+

• Can show that Output is stable + no pole-zero cancellation= internal stability

Digital Control 7 Kannan M. Moudgalya, Autumn 2007

Page 62: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

8. Forbid Unstable Pole-Zero Cancellation ⇒ GetCausality

Page 63: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

8. Forbid Unstable Pole-Zero Cancellation ⇒ GetCausality

Not possible to realize this controller:

Gc =1 + z−1

z−1

Page 64: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

8. Forbid Unstable Pole-Zero Cancellation ⇒ GetCausality

Not possible to realize this controller:

Gc =1 + z−1

z−1

All sampled systems have at least one delay:

Page 65: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

8. Forbid Unstable Pole-Zero Cancellation ⇒ GetCausality

Not possible to realize this controller:

Gc =1 + z−1

z−1

All sampled systems have at least one delay:

G(z−1) = z−kB(z−1)

A(z−1)= z−k

b0 + b1z−1 + b2z

−2 + · · ·1 + a1z−1 + a2z−2 + · · ·

Page 66: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

8. Forbid Unstable Pole-Zero Cancellation ⇒ GetCausality

Not possible to realize this controller:

Gc =1 + z−1

z−1

All sampled systems have at least one delay:

G(z−1) = z−kB(z−1)

A(z−1)= z−k

b0 + b1z−1 + b2z

−2 + · · ·1 + a1z−1 + a2z−2 + · · ·

• Controller not realizable ⇒

Page 67: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

8. Forbid Unstable Pole-Zero Cancellation ⇒ GetCausality

Not possible to realize this controller:

Gc =1 + z−1

z−1

All sampled systems have at least one delay:

G(z−1) = z−kB(z−1)

A(z−1)= z−k

b0 + b1z−1 + b2z

−2 + · · ·1 + a1z−1 + a2z−2 + · · ·

• Controller not realizable ⇒ there is a common factor z−1

between plant and controller

Page 68: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

8. Forbid Unstable Pole-Zero Cancellation ⇒ GetCausality

Not possible to realize this controller:

Gc =1 + z−1

z−1

All sampled systems have at least one delay:

G(z−1) = z−kB(z−1)

A(z−1)= z−k

b0 + b1z−1 + b2z

−2 + · · ·1 + a1z−1 + a2z−2 + · · ·

• Controller not realizable ⇒ there is a common factor z−1

between plant and controller

• z−1 = 0⇒

Page 69: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

8. Forbid Unstable Pole-Zero Cancellation ⇒ GetCausality

Not possible to realize this controller:

Gc =1 + z−1

z−1

All sampled systems have at least one delay:

G(z−1) = z−kB(z−1)

A(z−1)= z−k

b0 + b1z−1 + b2z

−2 + · · ·1 + a1z−1 + a2z−2 + · · ·

• Controller not realizable ⇒ there is a common factor z−1

between plant and controller

• z−1 = 0⇒ z =∞, an unstable pole

Page 70: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

8. Forbid Unstable Pole-Zero Cancellation ⇒ GetCausality

Not possible to realize this controller:

Gc =1 + z−1

z−1

All sampled systems have at least one delay:

G(z−1) = z−kB(z−1)

A(z−1)= z−k

b0 + b1z−1 + b2z

−2 + · · ·1 + a1z−1 + a2z−2 + · · ·

• Controller not realizable ⇒ there is a common factor z−1

between plant and controller

• z−1 = 0⇒ z =∞, an unstable pole

• If unstable pole-zero cancellation is forbidden while designingcontrollers,

Page 71: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

8. Forbid Unstable Pole-Zero Cancellation ⇒ GetCausality

Not possible to realize this controller:

Gc =1 + z−1

z−1

All sampled systems have at least one delay:

G(z−1) = z−kB(z−1)

A(z−1)= z−k

b0 + b1z−1 + b2z

−2 + · · ·1 + a1z−1 + a2z−2 + · · ·

• Controller not realizable ⇒ there is a common factor z−1

between plant and controller

• z−1 = 0⇒ z =∞, an unstable pole

• If unstable pole-zero cancellation is forbidden while designingcontrollers, z−1 cannot appear in the denominator of thecontroller

Page 72: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

8. Forbid Unstable Pole-Zero Cancellation ⇒ GetCausality

Not possible to realize this controller:

Gc =1 + z−1

z−1

All sampled systems have at least one delay:

G(z−1) = z−kB(z−1)

A(z−1)= z−k

b0 + b1z−1 + b2z

−2 + · · ·1 + a1z−1 + a2z−2 + · · ·

• Controller not realizable ⇒ there is a common factor z−1

between plant and controller

• z−1 = 0⇒ z =∞, an unstable pole

• If unstable pole-zero cancellation is forbidden while designingcontrollers, z−1 cannot appear in the denominator of thecontroller - i.e., controller is realizable

Digital Control 8 Kannan M. Moudgalya, Autumn 2007

Page 73: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

9. Delay Specification for Realizability

Page 74: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

9. Delay Specification for Realizability

Closed loop delay has to be ≥ open loop delay:

Page 75: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

9. Delay Specification for Realizability

Closed loop delay has to be ≥ open loop delay:

G(z) = z−kB(z)

A(z)

Page 76: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

9. Delay Specification for Realizability

Closed loop delay has to be ≥ open loop delay:

G(z) = z−kB(z)

A(z)= z−k

b0 + b1z−1 + · · ·

1 + a1z−1 + · · ·

Page 77: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

9. Delay Specification for Realizability

Closed loop delay has to be ≥ open loop delay:

G(z) = z−kB(z)

A(z)= z−k

b0 + b1z−1 + · · ·

1 + a1z−1 + · · ·b0 6= 0.

Page 78: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

9. Delay Specification for Realizability

Closed loop delay has to be ≥ open loop delay:

G(z) = z−kB(z)

A(z)= z−k

b0 + b1z−1 + · · ·

1 + a1z−1 + · · ·b0 6= 0. Suppose that we use a feedback controller of the form

Gc(z) = z−dSc(z)

Rc(z)

Page 79: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

9. Delay Specification for Realizability

Closed loop delay has to be ≥ open loop delay:

G(z) = z−kB(z)

A(z)= z−k

b0 + b1z−1 + · · ·

1 + a1z−1 + · · ·b0 6= 0. Suppose that we use a feedback controller of the form

Gc(z) = z−dSc(z)

Rc(z)= z−d

s0 + s1z−1 + · · ·

1 + r1z−1 + · · ·

Page 80: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

9. Delay Specification for Realizability

Closed loop delay has to be ≥ open loop delay:

G(z) = z−kB(z)

A(z)= z−k

b0 + b1z−1 + · · ·

1 + a1z−1 + · · ·b0 6= 0. Suppose that we use a feedback controller of the form

Gc(z) = z−dSc(z)

Rc(z)= z−d

s0 + s1z−1 + · · ·

1 + r1z−1 + · · ·with s0 6= 0

Page 81: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

9. Delay Specification for Realizability

Closed loop delay has to be ≥ open loop delay:

G(z) = z−kB(z)

A(z)= z−k

b0 + b1z−1 + · · ·

1 + a1z−1 + · · ·b0 6= 0. Suppose that we use a feedback controller of the form

Gc(z) = z−dSc(z)

Rc(z)= z−d

s0 + s1z−1 + · · ·

1 + r1z−1 + · · ·with s0 6= 0 and d ≥ 0.

Page 82: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

9. Delay Specification for Realizability

Closed loop delay has to be ≥ open loop delay:

G(z) = z−kB(z)

A(z)= z−k

b0 + b1z−1 + · · ·

1 + a1z−1 + · · ·b0 6= 0. Suppose that we use a feedback controller of the form

Gc(z) = z−dSc(z)

Rc(z)= z−d

s0 + s1z−1 + · · ·

1 + r1z−1 + · · ·with s0 6= 0 and d ≥ 0. Closed loop transfer function:

Page 83: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

9. Delay Specification for Realizability

Closed loop delay has to be ≥ open loop delay:

G(z) = z−kB(z)

A(z)= z−k

b0 + b1z−1 + · · ·

1 + a1z−1 + · · ·b0 6= 0. Suppose that we use a feedback controller of the form

Gc(z) = z−dSc(z)

Rc(z)= z−d

s0 + s1z−1 + · · ·

1 + r1z−1 + · · ·with s0 6= 0 and d ≥ 0. Closed loop transfer function:

T =GGc

1 +GGc

Page 84: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

9. Delay Specification for Realizability

Closed loop delay has to be ≥ open loop delay:

G(z) = z−kB(z)

A(z)= z−k

b0 + b1z−1 + · · ·

1 + a1z−1 + · · ·b0 6= 0. Suppose that we use a feedback controller of the form

Gc(z) = z−dSc(z)

Rc(z)= z−d

s0 + s1z−1 + · · ·

1 + r1z−1 + · · ·with s0 6= 0 and d ≥ 0. Closed loop transfer function:

T =GGc

1 +GGc

= z−k−db0s0 + (b0s1 + b1s0)z

−1 + · · ·1 + (s1 + r1)z−1 + · · ·+ z−k−d(b0s0 + · · · )

Digital Control 9 Kannan M. Moudgalya, Autumn 2007

Page 85: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

10. Delay Specification for Realizability

Page 86: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

10. Delay Specification for Realizability

T = z−k−db0s0 + (b0s1 + b1s0)z

−1 + · · ·1 + (s1 + r1)z−1 + · · ·+ z−k−d(b0s0 + · · · )

• Closed loop delay = k + d

Page 87: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

10. Delay Specification for Realizability

T = z−k−db0s0 + (b0s1 + b1s0)z

−1 + · · ·1 + (s1 + r1)z−1 + · · ·+ z−k−d(b0s0 + · · · )

• Closed loop delay = k + d ≥ k

Page 88: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

10. Delay Specification for Realizability

T = z−k−db0s0 + (b0s1 + b1s0)z

−1 + · · ·1 + (s1 + r1)z−1 + · · ·+ z−k−d(b0s0 + · · · )

• Closed loop delay = k + d ≥ k = open loop delay.

Page 89: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

10. Delay Specification for Realizability

T = z−k−db0s0 + (b0s1 + b1s0)z

−1 + · · ·1 + (s1 + r1)z−1 + · · ·+ z−k−d(b0s0 + · · · )

• Closed loop delay = k + d ≥ k = open loop delay.

• Can make it less only by d < 0,

Page 90: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

10. Delay Specification for Realizability

T = z−k−db0s0 + (b0s1 + b1s0)z

−1 + · · ·1 + (s1 + r1)z−1 + · · ·+ z−k−d(b0s0 + · · · )

• Closed loop delay = k + d ≥ k = open loop delay.

• Can make it less only by d < 0, but controller is unrealizable.

Digital Control 10 Kannan M. Moudgalya, Autumn 2007

Page 91: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

11. Example: What if Wrong Delay is Specified?

Page 92: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

11. Example: What if Wrong Delay is Specified?

Design a controller for the plant

G = z−2 1

1− 0.5z−1

Page 93: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

11. Example: What if Wrong Delay is Specified?

Design a controller for the plant

G = z−2 1

1− 0.5z−1

so that the overall system has smaller delay:

Page 94: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

11. Example: What if Wrong Delay is Specified?

Design a controller for the plant

G = z−2 1

1− 0.5z−1

so that the overall system has smaller delay:

T = z−1 1

1− az−1

Page 95: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

11. Example: What if Wrong Delay is Specified?

Design a controller for the plant

G = z−2 1

1− 0.5z−1

so that the overall system has smaller delay:

T = z−1 1

1− az−1

Recall the standard closed loop transfer function:

Page 96: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

11. Example: What if Wrong Delay is Specified?

Design a controller for the plant

G = z−2 1

1− 0.5z−1

so that the overall system has smaller delay:

T = z−1 1

1− az−1

Recall the standard closed loop transfer function: T = GGc/(1+GGc).

Page 97: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

11. Example: What if Wrong Delay is Specified?

Design a controller for the plant

G = z−2 1

1− 0.5z−1

so that the overall system has smaller delay:

T = z−1 1

1− az−1

Recall the standard closed loop transfer function: T = GGc/(1+GGc). Solving for Gc,

Page 98: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

11. Example: What if Wrong Delay is Specified?

Design a controller for the plant

G = z−2 1

1− 0.5z−1

so that the overall system has smaller delay:

T = z−1 1

1− az−1

Recall the standard closed loop transfer function: T = GGc/(1+GGc). Solving for Gc, and substituting for T , G

Page 99: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

11. Example: What if Wrong Delay is Specified?

Design a controller for the plant

G = z−2 1

1− 0.5z−1

so that the overall system has smaller delay:

T = z−1 1

1− az−1

Recall the standard closed loop transfer function: T = GGc/(1+GGc). Solving for Gc, and substituting for T , G

Gc =1

G

T

1− T

Page 100: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

11. Example: What if Wrong Delay is Specified?

Design a controller for the plant

G = z−2 1

1− 0.5z−1

so that the overall system has smaller delay:

T = z−1 1

1− az−1

Recall the standard closed loop transfer function: T = GGc/(1+GGc). Solving for Gc, and substituting for T , G

Gc =1

G

T

1− T =1

z−1

1− 0.5z−1

1− (a+ 1)z−1

Page 101: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

11. Example: What if Wrong Delay is Specified?

Design a controller for the plant

G = z−2 1

1− 0.5z−1

so that the overall system has smaller delay:

T = z−1 1

1− az−1

Recall the standard closed loop transfer function: T = GGc/(1+GGc). Solving for Gc, and substituting for T , G

Gc =1

G

T

1− T =1

z−1

1− 0.5z−1

1− (a+ 1)z−1

This controller is unrealizable,

Page 102: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

11. Example: What if Wrong Delay is Specified?

Design a controller for the plant

G = z−2 1

1− 0.5z−1

so that the overall system has smaller delay:

T = z−1 1

1− az−1

Recall the standard closed loop transfer function: T = GGc/(1+GGc). Solving for Gc, and substituting for T , G

Gc =1

G

T

1− T =1

z−1

1− 0.5z−1

1− (a+ 1)z−1

This controller is unrealizable, no matter what a is.Digital Control 11 Kannan M. Moudgalya, Autumn 2007

Page 103: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

12. Specifications - Step Response

Page 104: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

12. Specifications - Step Response

Give unit step input in r. Output yshould have

Page 105: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

12. Specifications - Step Response

Give unit step input in r. Output yshould have

1. small rise time

Page 106: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

12. Specifications - Step Response

Give unit step input in r. Output yshould have

1. small rise time

2. small overshoot

Page 107: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

12. Specifications - Step Response

Give unit step input in r. Output yshould have

1. small rise time

2. small overshoot

3. small settling time

Page 108: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

12. Specifications - Step Response

Give unit step input in r. Output yshould have

1. small rise time

2. small overshoot

3. small settling time

4. small steady state error

Page 109: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

12. Specifications - Step Response

Give unit step input in r. Output yshould have

1. small rise time

2. small overshoot

3. small settling time

4. small steady state error

n

e(n)

n

y(n)

Page 110: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

12. Specifications - Step Response

Give unit step input in r. Output yshould have

1. small rise time

2. small overshoot

3. small settling time

4. small steady state error

n

e(n)

n

y(n)

Error e(n) of the following form sat-isfies the requirements:

Page 111: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

12. Specifications - Step Response

Give unit step input in r. Output yshould have

1. small rise time

2. small overshoot

3. small settling time

4. small steady state error

n

e(n)

n

y(n)

Error e(n) of the following form sat-isfies the requirements:

e(n) = ρn cosωn, 0 < ρ < 1

Page 112: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

12. Specifications - Step Response

Give unit step input in r. Output yshould have

1. small rise time

2. small overshoot

3. small settling time

4. small steady state error

n

e(n)

n

y(n)

Error e(n) of the following form sat-isfies the requirements:

e(n) = ρn cosωn, 0 < ρ < 1

×

×

ωρ

Im(z)

Re(z)

Digital Control 12 Kannan M. Moudgalya, Autumn 2007

Page 113: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

13. Specifications - Step Response

Page 114: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

13. Specifications - Step Response

n

e(n)

n

y(n)

×

×

ωρ

Im(z)

Re(z)

e(n) = ρn cosωn

0 < ρ < 1

Page 115: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

13. Specifications - Step Response

n

e(n)

n

y(n)

×

×

ωρ

Im(z)

Re(z)

e(n) = ρn cosωn

0 < ρ < 1

1. initial error is one

Page 116: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

13. Specifications - Step Response

n

e(n)

n

y(n)

×

×

ωρ

Im(z)

Re(z)

e(n) = ρn cosωn

0 < ρ < 1

1. initial error is one

2. decaying oscillations about zero

Page 117: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

13. Specifications - Step Response

n

e(n)

n

y(n)

×

×

ωρ

Im(z)

Re(z)

e(n) = ρn cosωn

0 < ρ < 1

1. initial error is one

2. decaying oscillations about zero

3. steady state error is zero

Page 118: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

13. Specifications - Step Response

n

e(n)

n

y(n)

×

×

ωρ

Im(z)

Re(z)

e(n) = ρn cosωn

0 < ρ < 1

1. initial error is one

2. decaying oscillations about zero

3. steady state error is zero

Procedure:

Page 119: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

13. Specifications - Step Response

n

e(n)

n

y(n)

×

×

ωρ

Im(z)

Re(z)

e(n) = ρn cosωn

0 < ρ < 1

1. initial error is one

2. decaying oscillations about zero

3. steady state error is zero

Procedure: User will specify the following:

Page 120: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

13. Specifications - Step Response

n

e(n)

n

y(n)

×

×

ωρ

Im(z)

Re(z)

e(n) = ρn cosωn

0 < ρ < 1

1. initial error is one

2. decaying oscillations about zero

3. steady state error is zero

Procedure: User will specify the following:

1. a maximum allowable fall time < Nr

Page 121: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

13. Specifications - Step Response

n

e(n)

n

y(n)

×

×

ωρ

Im(z)

Re(z)

e(n) = ρn cosωn

0 < ρ < 1

1. initial error is one

2. decaying oscillations about zero

3. steady state error is zero

Procedure: User will specify the following:

1. a maximum allowable fall time < Nr

2. a maximum allowable undershoot < ε

Page 122: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

13. Specifications - Step Response

n

e(n)

n

y(n)

×

×

ωρ

Im(z)

Re(z)

e(n) = ρn cosωn

0 < ρ < 1

1. initial error is one

2. decaying oscillations about zero

3. steady state error is zero

Procedure: User will specify the following:

1. a maximum allowable fall time < Nr

2. a maximum allowable undershoot < ε

3. a minimum required decay ratio < δ

Page 123: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

13. Specifications - Step Response

n

e(n)

n

y(n)

×

×

ωρ

Im(z)

Re(z)

e(n) = ρn cosωn

0 < ρ < 1

1. initial error is one

2. decaying oscillations about zero

3. steady state error is zero

Procedure: User will specify the following:

1. a maximum allowable fall time < Nr

2. a maximum allowable undershoot < ε

3. a minimum required decay ratio < δ

• We will develop a method to determine ρ andω satisfying the above requirements

Page 124: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

13. Specifications - Step Response

n

e(n)

n

y(n)

×

×

ωρ

Im(z)

Re(z)

e(n) = ρn cosωn

0 < ρ < 1

1. initial error is one

2. decaying oscillations about zero

3. steady state error is zero

Procedure: User will specify the following:

1. a maximum allowable fall time < Nr

2. a maximum allowable undershoot < ε

3. a minimum required decay ratio < δ

• We will develop a method to determine ρ andω satisfying the above requirements

• Calculate trans. fn. between e(n) - r(n)

Page 125: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

13. Specifications - Step Response

n

e(n)

n

y(n)

×

×

ωρ

Im(z)

Re(z)

e(n) = ρn cosωn

0 < ρ < 1

1. initial error is one

2. decaying oscillations about zero

3. steady state error is zero

Procedure: User will specify the following:

1. a maximum allowable fall time < Nr

2. a maximum allowable undershoot < ε

3. a minimum required decay ratio < δ

• We will develop a method to determine ρ andω satisfying the above requirements

• Calculate trans. fn. between e(n) - r(n)

• Back calculate the controller Gc(z)Digital Control 13 Kannan M. Moudgalya, Autumn 2007

Page 126: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

14. Small Fall Time in Error

Page 127: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

14. Small Fall Time in Error

e(n) = ρn cosωn, 0 < ρ < 1

Page 128: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

14. Small Fall Time in Error

e(n) = ρn cosωn, 0 < ρ < 1

Error becomes zero, i.e.,

Page 129: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

14. Small Fall Time in Error

e(n) = ρn cosωn, 0 < ρ < 1

Error becomes zero, i.e.,

e(n) = 0.

Page 130: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

14. Small Fall Time in Error

e(n) = ρn cosωn, 0 < ρ < 1

Error becomes zero, i.e.,

e(n) = 0.

for the first time when

Page 131: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

14. Small Fall Time in Error

e(n) = ρn cosωn, 0 < ρ < 1

Error becomes zero, i.e.,

e(n) = 0.

for the first time when

ωn =π

2

Page 132: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

14. Small Fall Time in Error

e(n) = ρn cosωn, 0 < ρ < 1

Error becomes zero, i.e.,

e(n) = 0.

for the first time when

ωn =π

2

⇒ n =π

Page 133: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

14. Small Fall Time in Error

e(n) = ρn cosωn, 0 < ρ < 1

Error becomes zero, i.e.,

e(n) = 0.

for the first time when

ωn =π

2

⇒ n =π

As want n < Nr, some given value, we get

Page 134: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

14. Small Fall Time in Error

e(n) = ρn cosωn, 0 < ρ < 1

Error becomes zero, i.e.,

e(n) = 0.

for the first time when

ωn =π

2

⇒ n =π

As want n < Nr, some given value, we get

π

2ω< Nr

Page 135: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

14. Small Fall Time in Error

e(n) = ρn cosωn, 0 < ρ < 1

Error becomes zero, i.e.,

e(n) = 0.

for the first time when

ωn =π

2

⇒ n =π

As want n < Nr, some given value, we get

π

2ω< Nr ⇒ ω >

π

2Nr

Page 136: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

14. Small Fall Time in Error

e(n) = ρn cosωn, 0 < ρ < 1

Error becomes zero, i.e.,

e(n) = 0.

for the first time when

ωn =π

2

⇒ n =π

As want n < Nr, some given value, we get

π

2ω< Nr ⇒ ω >

π

2Nr

n

e(n)

n

y(n)

Desired region:Im(z)

Re(z)

Digital Control 14 Kannan M. Moudgalya, Autumn 2007

Page 137: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

15. Small Undershoot

Page 138: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

15. Small Undershoot

e(n) = ρn cosωn

Page 139: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

15. Small Undershoot

e(n) = ρn cosωn

When does it reach first min.?

Page 140: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

15. Small Undershoot

e(n) = ρn cosωn

When does it reach first min.? de/dn = 0

Page 141: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

15. Small Undershoot

e(n) = ρn cosωn

When does it reach first min.? de/dn = 0

• Not applicable, because n is an integer

• Look for a simpler expression

Page 142: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

15. Small Undershoot

e(n) = ρn cosωn

When does it reach first min.? de/dn = 0

• Not applicable, because n is an integer

• Look for a simpler expression

• Reaches min. approx.

Page 143: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

15. Small Undershoot

e(n) = ρn cosωn

When does it reach first min.? de/dn = 0

• Not applicable, because n is an integer

• Look for a simpler expression

• Reaches min. approx. when ωn = π

Page 144: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

15. Small Undershoot

e(n) = ρn cosωn

When does it reach first min.? de/dn = 0

• Not applicable, because n is an integer

• Look for a simpler expression

• Reaches min. approx. when ωn = π

e(n)|ωn=π = ρn cosωn|ωn=π

Page 145: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

15. Small Undershoot

e(n) = ρn cosωn

When does it reach first min.? de/dn = 0

• Not applicable, because n is an integer

• Look for a simpler expression

• Reaches min. approx. when ωn = π

e(n)|ωn=π = ρn cosωn|ωn=π

= −ρn|ωn=π

Page 146: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

15. Small Undershoot

e(n) = ρn cosωn

When does it reach first min.? de/dn = 0

• Not applicable, because n is an integer

• Look for a simpler expression

• Reaches min. approx. when ωn = π

e(n)|ωn=π = ρn cosωn|ωn=π

= −ρn|ωn=π = −ρπ/ω

Page 147: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

15. Small Undershoot

e(n) = ρn cosωn

When does it reach first min.? de/dn = 0

• Not applicable, because n is an integer

• Look for a simpler expression

• Reaches min. approx. when ωn = π

e(n)|ωn=π = ρn cosωn|ωn=π

= −ρn|ωn=π = −ρπ/ω

User specified maximum deviation = ε:

Page 148: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

15. Small Undershoot

e(n) = ρn cosωn

When does it reach first min.? de/dn = 0

• Not applicable, because n is an integer

• Look for a simpler expression

• Reaches min. approx. when ωn = π

e(n)|ωn=π = ρn cosωn|ωn=π

= −ρn|ωn=π = −ρπ/ω

User specified maximum deviation = ε:

ρπ/ω < ε

Page 149: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

15. Small Undershoot

e(n) = ρn cosωn

When does it reach first min.? de/dn = 0

• Not applicable, because n is an integer

• Look for a simpler expression

• Reaches min. approx. when ωn = π

e(n)|ωn=π = ρn cosωn|ωn=π

= −ρn|ωn=π = −ρπ/ω

User specified maximum deviation = ε:

ρπ/ω < ε, ρ < εω/π

Page 150: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

15. Small Undershoot

e(n) = ρn cosωn

When does it reach first min.? de/dn = 0

• Not applicable, because n is an integer

• Look for a simpler expression

• Reaches min. approx. when ωn = π

e(n)|ωn=π = ρn cosωn|ωn=π

= −ρn|ωn=π = −ρπ/ω

User specified maximum deviation = ε:

ρπ/ω < ε, ρ < εω/π

n

e(n)

n

y(n)

Desired region:Im(z)

Re(z)

Digital Control 15 Kannan M. Moudgalya, Autumn 2007

Page 151: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

16. Small Decay Ratio

Page 152: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

16. Small Decay Ratio

e(n) = ρn cosωn

Page 153: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

16. Small Decay Ratio

e(n) = ρn cosωn

Ratio of two successive peak/trough to be small

Page 154: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

16. Small Decay Ratio

e(n) = ρn cosωn

Ratio of two successive peak/trough to be small

• First undershoot in e(n) occurs at ωn ' π

Page 155: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

16. Small Decay Ratio

e(n) = ρn cosωn

Ratio of two successive peak/trough to be small

• First undershoot in e(n) occurs at ωn ' π• First overshoot occurs at ωn ' 2π

Page 156: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

16. Small Decay Ratio

e(n) = ρn cosωn

Ratio of two successive peak/trough to be small

• First undershoot in e(n) occurs at ωn ' π• First overshoot occurs at ωn ' 2π

Want this ratio to be less than user specified δ:

Page 157: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

16. Small Decay Ratio

e(n) = ρn cosωn

Ratio of two successive peak/trough to be small

• First undershoot in e(n) occurs at ωn ' π• First overshoot occurs at ωn ' 2π

Want this ratio to be less than user specified δ:∣∣∣∣e(n)|ωn=2π

e(n)|ωn=π

∣∣∣∣ < δ

Page 158: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

16. Small Decay Ratio

e(n) = ρn cosωn

Ratio of two successive peak/trough to be small

• First undershoot in e(n) occurs at ωn ' π• First overshoot occurs at ωn ' 2π

Want this ratio to be less than user specified δ:∣∣∣∣e(n)|ωn=2π

e(n)|ωn=π

∣∣∣∣ < δ ⇒ ρn|ωn=2π

ρn|ωn=π

< δ

Page 159: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

16. Small Decay Ratio

e(n) = ρn cosωn

Ratio of two successive peak/trough to be small

• First undershoot in e(n) occurs at ωn ' π• First overshoot occurs at ωn ' 2π

Want this ratio to be less than user specified δ:∣∣∣∣e(n)|ωn=2π

e(n)|ωn=π

∣∣∣∣ < δ ⇒ ρn|ωn=2π

ρn|ωn=π

< δ

δ = 0.5 ' 1/4 decay.

Page 160: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

16. Small Decay Ratio

e(n) = ρn cosωn

Ratio of two successive peak/trough to be small

• First undershoot in e(n) occurs at ωn ' π• First overshoot occurs at ωn ' 2π

Want this ratio to be less than user specified δ:∣∣∣∣e(n)|ωn=2π

e(n)|ωn=π

∣∣∣∣ < δ ⇒ ρn|ωn=2π

ρn|ωn=π

< δ

δ = 0.5 ' 1/4 decay. δ = 0.25 ' 1/8 decay.

Page 161: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

16. Small Decay Ratio

e(n) = ρn cosωn

Ratio of two successive peak/trough to be small

• First undershoot in e(n) occurs at ωn ' π• First overshoot occurs at ωn ' 2π

Want this ratio to be less than user specified δ:∣∣∣∣e(n)|ωn=2π

e(n)|ωn=π

∣∣∣∣ < δ ⇒ ρn|ωn=2π

ρn|ωn=π

< δ

δ = 0.5 ' 1/4 decay. δ = 0.25 ' 1/8 decay.

ρ2π/ω

ρπ/ω< δ

Page 162: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

16. Small Decay Ratio

e(n) = ρn cosωn

Ratio of two successive peak/trough to be small

• First undershoot in e(n) occurs at ωn ' π• First overshoot occurs at ωn ' 2π

Want this ratio to be less than user specified δ:∣∣∣∣e(n)|ωn=2π

e(n)|ωn=π

∣∣∣∣ < δ ⇒ ρn|ωn=2π

ρn|ωn=π

< δ

δ = 0.5 ' 1/4 decay. δ = 0.25 ' 1/8 decay.

ρ2π/ω

ρπ/ω< δ ⇒ ρπ/ω < δ

Page 163: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

16. Small Decay Ratio

e(n) = ρn cosωn

Ratio of two successive peak/trough to be small

• First undershoot in e(n) occurs at ωn ' π• First overshoot occurs at ωn ' 2π

Want this ratio to be less than user specified δ:∣∣∣∣e(n)|ωn=2π

e(n)|ωn=π

∣∣∣∣ < δ ⇒ ρn|ωn=2π

ρn|ωn=π

< δ

δ = 0.5 ' 1/4 decay. δ = 0.25 ' 1/8 decay.

ρ2π/ω

ρπ/ω< δ ⇒ ρπ/ω < δ ⇒ ρ < δω/π

Page 164: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

16. Small Decay Ratio

e(n) = ρn cosωn

Ratio of two successive peak/trough to be small

• First undershoot in e(n) occurs at ωn ' π• First overshoot occurs at ωn ' 2π

Want this ratio to be less than user specified δ:∣∣∣∣e(n)|ωn=2π

e(n)|ωn=π

∣∣∣∣ < δ ⇒ ρn|ωn=2π

ρn|ωn=π

< δ

δ = 0.5 ' 1/4 decay. δ = 0.25 ' 1/8 decay.

ρ2π/ω

ρπ/ω< δ ⇒ ρπ/ω < δ ⇒ ρ < δω/π

• Small undershoot: ρ < εω/π.

Page 165: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

16. Small Decay Ratio

e(n) = ρn cosωn

Ratio of two successive peak/trough to be small

• First undershoot in e(n) occurs at ωn ' π• First overshoot occurs at ωn ' 2π

Want this ratio to be less than user specified δ:∣∣∣∣e(n)|ωn=2π

e(n)|ωn=π

∣∣∣∣ < δ ⇒ ρn|ωn=2π

ρn|ωn=π

< δ

δ = 0.5 ' 1/4 decay. δ = 0.25 ' 1/8 decay.

ρ2π/ω

ρπ/ω< δ ⇒ ρπ/ω < δ ⇒ ρ < δω/π

• Small undershoot: ρ < εω/π. Usually ε < δ

Page 166: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

16. Small Decay Ratio

e(n) = ρn cosωn

Ratio of two successive peak/trough to be small

• First undershoot in e(n) occurs at ωn ' π• First overshoot occurs at ωn ' 2π

Want this ratio to be less than user specified δ:∣∣∣∣e(n)|ωn=2π

e(n)|ωn=π

∣∣∣∣ < δ ⇒ ρn|ωn=2π

ρn|ωn=π

< δ

δ = 0.5 ' 1/4 decay. δ = 0.25 ' 1/8 decay.

ρ2π/ω

ρπ/ω< δ ⇒ ρπ/ω < δ ⇒ ρ < δω/π

• Small undershoot: ρ < εω/π. Usually ε < δ

• Small undershoot satisfies fast decay

n

e(n)

n

y(n)

Desired region:Im(z)

Re(z)

Digital Control 16 Kannan M. Moudgalya, Autumn 2007

Page 167: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

17. Overall Requirements

Page 168: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

17. Overall Requirements

Im(z)

Re(z)

Page 169: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

17. Overall Requirements

Im(z)

Re(z)

Im(z)

Re(z)

Page 170: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

17. Overall Requirements

Im(z)

Re(z)

Im(z)

Re(z) Re(z)

Im(z)

Desired region by the current approach

Page 171: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

17. Overall Requirements

Im(z)

Re(z)

Im(z)

Re(z) Re(z)

Im(z)

Desired region by the current approach

Im(z)

Re(z)

Obtained by discretization of continuous domain result(Astrom and Wittenmark)

Digital Control 17 Kannan M. Moudgalya, Autumn 2007

Page 172: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

18. Desired Transfer Function

Page 173: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

18. Desired Transfer Function

Desired error to a step input is e:

e(n) = rn cosωn

Page 174: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

18. Desired Transfer Function

Desired error to a step input is e:

e(n) = rn cosωn

Taking Z-transform,

Page 175: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

18. Desired Transfer Function

Desired error to a step input is e:

e(n) = rn cosωn

Taking Z-transform,

E(z) =z(z − r cosω)

z2 − 2zr cosω + r2

Page 176: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

18. Desired Transfer Function

Desired error to a step input is e:

e(n) = rn cosωn

Taking Z-transform,

E(z) =z(z − r cosω)

z2 − 2zr cosω + r2

For step input R(z).

Page 177: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

18. Desired Transfer Function

Desired error to a step input is e:

e(n) = rn cosωn

Taking Z-transform,

E(z) =z(z − r cosω)

z2 − 2zr cosω + r2

For step input R(z). Transfer function between R(z)-E(z):

Page 178: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

18. Desired Transfer Function

Desired error to a step input is e:

e(n) = rn cosωn

Taking Z-transform,

E(z) =z(z − r cosω)

z2 − 2zr cosω + r2

For step input R(z). Transfer function between R(z)-E(z):

TE(z) =E(z)

R(z)

Page 179: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

18. Desired Transfer Function

Desired error to a step input is e:

e(n) = rn cosωn

Taking Z-transform,

E(z) =z(z − r cosω)

z2 − 2zr cosω + r2

For step input R(z). Transfer function between R(z)-E(z):

TE(z) =E(z)

R(z)=

z(z − r cosω)

z2 − 2zr cosω + r2

z − 1

z

Page 180: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

18. Desired Transfer Function

Desired error to a step input is e:

e(n) = rn cosωn

Taking Z-transform,

E(z) =z(z − r cosω)

z2 − 2zr cosω + r2

For step input R(z). Transfer function between R(z)-E(z):

TE(z) =E(z)

R(z)=

z(z − r cosω)

z2 − 2zr cosω + r2

z − 1

z

=(z − 1)(z − r cosω)

z2 − 2zr cosω + r2

Page 181: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

18. Desired Transfer Function

Desired error to a step input is e:

e(n) = rn cosωn

Taking Z-transform,

E(z) =z(z − r cosω)

z2 − 2zr cosω + r2

For step input R(z). Transfer function between R(z)-E(z):

TE(z) =E(z)

R(z)=

z(z − r cosω)

z2 − 2zr cosω + r2

z − 1

z

=(z − 1)(z − r cosω)

z2 − 2zr cosω + r2

One approach: equate this to 1/(1 +GGc)

Page 182: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

18. Desired Transfer Function

Desired error to a step input is e:

e(n) = rn cosωn

Taking Z-transform,

E(z) =z(z − r cosω)

z2 − 2zr cosω + r2

For step input R(z). Transfer function between R(z)-E(z):

TE(z) =E(z)

R(z)=

z(z − r cosω)

z2 − 2zr cosω + r2

z − 1

z

=(z − 1)(z − r cosω)

z2 − 2zr cosω + r2

One approach: equate this to 1/(1 +GGc) and calculate GcDigital Control 18 Kannan M. Moudgalya, Autumn 2007

Page 183: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

19. Desired Transfer Function

Page 184: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

19. Desired Transfer Function

Transfer function between setpointR(z) and actual output Y (z):

Page 185: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

19. Desired Transfer Function

Transfer function between setpointR(z) and actual output Y (z):

TY (z) = 1− TE(z)

Page 186: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

19. Desired Transfer Function

Transfer function between setpointR(z) and actual output Y (z):

TY (z) = 1− TE(z)

= 1− z2 − zr cosω − z + r cosω

z2 − 2zr cosω + r2

Page 187: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

19. Desired Transfer Function

Transfer function between setpointR(z) and actual output Y (z):

TY (z) = 1− TE(z)

= 1− z2 − zr cosω − z + r cosω

z2 − 2zr cosω + r2

=z(1− r cosω) + (r2 − r cosω)

z2 − 2zr cosω + r2

Page 188: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

19. Desired Transfer Function

Transfer function between setpointR(z) and actual output Y (z):

TY (z) = 1− TE(z)

= 1− z2 − zr cosω − z + r cosω

z2 − 2zr cosω + r2

=z(1− r cosω) + (r2 − r cosω)

z2 − 2zr cosω + r2

4= z−1 ψ(z−1)

φcl(z−1)

Page 189: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

19. Desired Transfer Function

Transfer function between setpointR(z) and actual output Y (z):

TY (z) = 1− TE(z)

= 1− z2 − zr cosω − z + r cosω

z2 − 2zr cosω + r2

=z(1− r cosω) + (r2 − r cosω)

z2 − 2zr cosω + r2

4= z−1 ψ(z−1)

φcl(z−1)

• One approach: equate this to GGc/(1 +GGc),

Page 190: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

19. Desired Transfer Function

Transfer function between setpointR(z) and actual output Y (z):

TY (z) = 1− TE(z)

= 1− z2 − zr cosω − z + r cosω

z2 − 2zr cosω + r2

=z(1− r cosω) + (r2 − r cosω)

z2 − 2zr cosω + r2

4= z−1 ψ(z−1)

φcl(z−1)

• One approach: equate this to GGc/(1 +GGc), find Gc

Page 191: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

19. Desired Transfer Function

Transfer function between setpointR(z) and actual output Y (z):

TY (z) = 1− TE(z)

= 1− z2 − zr cosω − z + r cosω

z2 − 2zr cosω + r2

=z(1− r cosω) + (r2 − r cosω)

z2 − 2zr cosω + r2

4= z−1 ψ(z−1)

φcl(z−1)

• One approach: equate this to GGc/(1 +GGc), find Gc

• Can’t do it exactly - will use part of this approach

Page 192: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

19. Desired Transfer Function

Transfer function between setpointR(z) and actual output Y (z):

TY (z) = 1− TE(z)

= 1− z2 − zr cosω − z + r cosω

z2 − 2zr cosω + r2

=z(1− r cosω) + (r2 − r cosω)

z2 − 2zr cosω + r2

4= z−1 ψ(z−1)

φcl(z−1)

• One approach: equate this to GGc/(1 +GGc), find Gc

• Can’t do it exactly - will use part of this approach

• We will mainly make use of φcl and not ψ

Page 193: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

19. Desired Transfer Function

Transfer function between setpointR(z) and actual output Y (z):

TY (z) = 1− TE(z)

= 1− z2 − zr cosω − z + r cosω

z2 − 2zr cosω + r2

=z(1− r cosω) + (r2 − r cosω)

z2 − 2zr cosω + r2

4= z−1 ψ(z−1)

φcl(z−1)

• One approach: equate this to GGc/(1 +GGc), find Gc

• Can’t do it exactly - will use part of this approach

• We will mainly make use of φcl and not ψ

• TY (1) = 1

Page 194: 1. Internal Model Principle - Moudgalyamoudgalya.org/digital-slides/structure-1.pdf · 1. Internal Model Principle r G(z) = B A Gc(z) = Sc Rc e u y v (z) = least common multiple of

19. Desired Transfer Function

Transfer function between setpointR(z) and actual output Y (z):

TY (z) = 1− TE(z)

= 1− z2 − zr cosω − z + r cosω

z2 − 2zr cosω + r2

=z(1− r cosω) + (r2 − r cosω)

z2 − 2zr cosω + r2

4= z−1 ψ(z−1)

φcl(z−1)

• One approach: equate this to GGc/(1 +GGc), find Gc

• Can’t do it exactly - will use part of this approach

• We will mainly make use of φcl and not ψ

• TY (1) = 1⇒ No steady state offset between Y and RDigital Control 19 Kannan M. Moudgalya, Autumn 2007