3. Systems and Transfer functionDiscrete-time system revisionDiscrete-time systemA/D and D/A convertersSampling frequency and sampling theoremNyquist frequencyAliasingsZ-transform & inverse Z-transformThe output of a D/A converter

3.1 Zero-order-hold (ZOH)A Zero-order hold in a system

3.1 Zero-order-hold (ZOH)How does a signal change its form in a discrete-time system?The input signal x(t) is sampled at discrete instants and the sampled signal is passed through the zero-order-hold (ZOH). The ZOH circuit smoothes the sampled signal to produce the signal h(t), which is a constant from the last sampled value until the next sample is available. That is

3.1 Zero-order-hold (ZOH)Transfer function of Zero-order-holdThe figure below shows a combination of a sampler and a zero-order hold.

3.1 Zero-order-hold (ZOH)Assume that the signal x(t) is zero for t

3.1 Zero-order-hold (ZOH)As

The Laplace transform of the above equation becomes

3.1 Zero-order-hold (ZOH)AsTherefore

Finally, we obtain the transfer function of a ZOH as

3.1 Zero-order-hold (ZOH)There are also first-order-hold and high-order-hold although they are not used in control system.

3.1 Zero-order-hold (ZOH)A zero-order-hold creates one sampling interval delay in input signal.

3.1 Zero-order-hold (ZOH)First-order-hold

3.1 Zero-order-hold (ZOH)First-order-hold and high-order-hold does not bring us much advantages except in some special cases. Therefore, in a control system, usually a ZOH is employed. The device to implement a ZOH is a D/A converter.

If not told, always suppose there is a ZOH in a digital control system.

3.2 Plants with ZOHGiven a discrete-time system, the transfer function of a combination of a ZOH and the plant can be written as GHP(z) in Z-domain. HP, here, means the ZOH and the Plant.

3.2 Plants with ZOHThe continuous time transfer function GHP(s)=G0(s)GP(s)The discrete time transfer function

3.2 Plants with ZOHExample 1: Given a ZOH and a plantDetermine their Z-domain transfer function.

3.2 Plants with ZOHExample 2: Given a ZOH and a plantDetermine their z-domain transfer function.

3.2 Plants with ZOHExercise 1: Given a ZOH and a plantDetermine their z-domain transfer function.Answer:

Assignment 1You are required to implement a digital PID controller which will enable a control object with a transfer function of

where K=0.2, n=10 rad/s, and =0.3.to track a) a unit step signal, and b) a unit ramp signal.1) Simulate this control object and find the responses using Matlab or other packages/computer languages.

Assignment 12) Choose a suitable sample period for a control loop for G(s) and explain your choice.3)* Derive the discrete-time system transfer function GHP(z) from G(s).4) Design a digital PID controller for the discrete-time system, and optimize its parameters with respect to the performance criterion below using steepest descent minimization process .5) Simulate the resulting closed-loop system and find the responses. Swapping the input signals a) and b), discuss the resulting responses.

3.3 Represent a system in difference equationFor we haveLet A=1-e-T and B=e-T, then the transfer function can be rewritten as

3.3 Represent a system in difference equationSimulate the above system1) Parameters and input: A=1-e-T, B=e-T , x(k)=12) initial condition: x(k-1)=0, y(k)=y(k-1)=0, k=03) SimulationWhile k

3.3 Represent a system in difference equationLet T=1, we have A=0.6321 and B=0.3679For a unit step input, the response isy(k)=0.6321x(k-1)+0.3679y(k-1)k=012345x(k)111111y(k)00.63211111

3.3 Represent a system in difference equation

Assignment 11)* Simulate this control object and find the responses using Matlab or other packages/computer languages.

Hints: Method 1

Assignment 1Hints: Method 2

3.4 System stabilityWe can rewrite the difference equation as

If A=1 and =0.9, for an impulse input we have

k01234...x(k)10000...y(k)010.90.810.729It decreases exponentially, a stable system.

3.4 System stabilityIf K=1 and =1.2, we have

k01234...x(k)10000...y(k)01 1.2 1.44 1.728 2.074It increases exponentially, an unstable system.

3.4 System stabilityIf K=1 and = -0.8, we have

k01234...x(k)10000...y(k)01-0.80.64-0.512It decays exponentially, and alternates in sign, a gradual stable system.

3.4 System stabilityIt is clear that the value of determines the system stability. Why is so important?First, let A=1, we have

From the transfer function, we can see that z= is a pole of the system. The pole of the system will determine the nature of the response.

3.4 System stabilityFor continuous system, we have stable, critical stable and unstable areas in s domain.

3.4 System stabilityWhat is the stable area, critical stable area and unstable area for a discrete system in Z domain ?Stable area: unit circleCritical stable: on the unit circleUnstable area:outside of the unit circle

3.4 System stabilityAsFor the critical stable area in s domain s=j,

As is from 0 to , then the angle will be greater than 2. That is the critical area forms a unit circle in Z domain.

3.4 System stabilityIf we choose a point from the stable area at S domain, eg s=- a + j, we have

Let eg s=- + j

The stable area in Z domain is within a unit circle around the origin.

3.4 System stabilityExercise 2: Prove that the unstable area in Z domain is the area outside the unit circle.Hint: Follow the above procedures.

3.4 System stabilityZ domain responses01

3.5 Closed-loop transfer functionComputer controlled system

3.5 Closed-loop transfer functionLets find out the closed-loop transfer function

3.5 Closed-loop transfer functionC(z): output;E(z): errorR(z): input;M(z): controller outputGC(z): controllerGP(z)/G(z): plant transfer functionGHP(z): transfer function of plant + ZOHT(z): closed-loop transfer functionGC(z)GHP(z): open-loop transfer function1+ GC(z)GHP(z)=0: characteristic equation

3.6 System block diagram

3.6 System block diagramThe difference between G(z)H(z) and GH(z)G(z)H(z)=Z[G(s)]Z[H(s)]GH(z)=Z[G(s)H(s)]Usually, G(z)H(z) GH(z)G(z)H(z) means they are connected through a sampler. Whereas GH(z) they are connected directly.

3.6 System block diagramExample: Find the closed-loop transfer function for the system below.

Solution: The open-loop is G1(z)G2H(z). The forward path is G1(z)G2(z).

3.6 System block diagram

3.6 System block diagram*Exercise 3: Find the output for the closed-loop system below.

3.6 System block diagram*Exercise 4: Find the output for the closed-loop system below.

ReadingStudy bookModule 3: Systems and transfer functions (Please try the problems on page 3.46-47)Textbook Chapter 3 : Z-plane analysis of discrete-time control system (pages 74-83 & 104-114).

TutorialExercise 1: Given a ZOH and a plantDetermine their z-domain transfer function.

TutorialYou are required to implement a digital PID controller which will enable a control object with a transfer function of

where K=0.2, n=10 rad/s, and =0.3.to track a) a unit step signal, and b) a unit ramp signal.1) Simulate this control object and find the responses using Matlab or other packages/computer languages.

Tutorial2) Choose a suitable sample period for a control loop for G(s) and explain your choice.3) Derive the discrete-time system transfer function GHP(z) from G(s).4) Design a digital PID controller for the discrete-time system, and optimize its parameters with respect to the performance criterion below using steepest descent minimization process .5) Simulate the resulting closed-loop system and find the responses. Swapping the input signals a) and b), discuss the resulting responses.

Tutorial2) Choose a suitable sample period for a control loop for G(s) and explain your choice.

Sampling theoremInput signalBandwidth of a systemBold plotsApplying sampling theoremSampling frequency