Digital Control of Dynamic Systems - SUPSI

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Digital Control of Dynamic Systems Silvano Balemi University of Applied Sciences of Southern Switzerland Manno, 2004

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Discrete-time signals

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Step response of a sampled system

Step Response From: U(1)

0.35

0.3

To: Y(1)

0.25

Amplitude

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0.2

0.15

0.1

0.05

0

0

0.1

0.2

0.3

0.4

0.5

Time (sec.)

0.6

0.7

0.8

0.9

1

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Sample and hold

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Sampling

Multiplication with a train of unit impulses (operation is linear but time-variant)

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Train of impulses and its Fourier expansion

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Sampled signal

with

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Spectrum of Sampled signal

SPECTRUM OF A SIGNAL 2 |R|

1.5

1

0.5

0 -10

1.5

-8

-6

-4

-2

0

2

4

6

8

10

SPECTRUM OF THE SAMPLED SIGNAL |R*| AND ITS COMPONENTS

1

w1

0.5

0 -15

-10

-5

0

5

10

15

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Hold

Linear operation 1(t)

1(t − T)

Impulse response of a ZOH

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Z transform

Laplace transformation with

where The z transform corresponds to the sequence with the function

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Relation between different transforms

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Z transform: Examples and properties

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Examples of z transforms

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Some transformations

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Linearity Delay Anticipation Damping Product Initial value End value

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Properties of the z transform

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z transform

1. From the Laplace transformation Factorization

Using „primitives“ k G(s) = s+a

g(t) = k á e àaát

G(z) = k á zàezàaáT

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Inverse z transform

1. Inverse trasform via factorization

2. Inverse transform via recursion

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Sampled Systems

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Discrete-time Transfer function from time domain

No transfer function between u and y but between u* and y*

and with variable substitution l=k-m

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Discrete-time Transfer function from frequency domain

with variable substitution m=k+n

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Transfer function with ZOH

Gzoh(z)

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Example Transfer function with ZOH

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State space representation

u constant from 0 to T from

Transfer function

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Description of Linear Time-invariant Discrete-time Systems

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Stability of sampled systems

x

x x x

x x

x

x

x

x x

x

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Step responses

x

x

x

x

x

x

x

x

x

x

x x x

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Digital controller

Digital part

Closed-loop sampled systems

Cont.-time process

Analog part

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model of program

Closed-loop Discrete-time system (2)

model of D/A conv

model of process

Gzoh(z)

model of A/D conv

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Example: system stability

≈ 0.09

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Example of a program for a controller

U(z)

G c(z) = E(z) = z 2z+3 +z+1 (z 2 + z + 1) á U(z) = (z + 3) á E(z) (1 + z à1 + z à2) á U(z) = (z à1 + 3 á z à2) á E(z) U(z) + z à1 á U(z) + z à2 á U(z) = z à1 á E(z) + 3 á z à2 á E(z) {u k} + {u kà1} + {u kà2} = {e kà1} + 3 á {e kà2} u k + u kà1 + u kà2 = e kà1 + 3 á e kà2 u k = à u kà1 à u kà2 + e kà1 + 3 á e kà2

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Example of a program for a controller: C-code

C-code ek_1=0; ek_2=0; uk_1=0; uk_2=0; while TRUE { yk=read_yk(); ek=yrefk-yk; uk=-uk_1-uk_2+ek_1-3*ek_2; write(uk); uk_2=uk_1; uk_1=uk; ek_2=ek_1; ek_1=ek; }

Minimize control delay!

ek_1=0; ek_2=0; uk_1=0; uk_2=0; while TRUE { uk=-uk_1-uk_2+ek_1-3*ek_2; yk=read_yk(); write(uk); ek=yrefk-yk; uk_2=uk_1; uk_1=uk; ek_2=ek_1; ek_1=ek; }