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3. Systems and Transfer function Discrete-time system revision Discrete-time system A/D and D/A converters Sampling frequency and sampling theorem Nyquist frequency Aliasings Z-transform & inverse Z-transform The output of a D/A converter

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3.1 Zero-order-hold (ZOH) A Zero-order hold in a system x(t)x(kT) h(t) Zero-order Hold Sampler

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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

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3.1 Zero-order-hold (ZOH) Transfer function of Zero-order-hold The figure below shows a combination of a sampler and a zero-order hold. x(t)x(kT) h(t) Zero-order Hold Sampler

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3.1 Zero-order-hold (ZOH) Assume that the signal x(t) is zero for t<0, then the output h(t)is related to x(t) as follows: h(t) t

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3.1 Zero-order-hold (ZOH) As The Laplace transform of the above equation becomes

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3.1 Zero-order-hold (ZOH) As Therefore Finally, we obtain the transfer function of a ZOH as

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

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

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

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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.

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

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3.2 Plants with ZOH The continuous time transfer function G HP (s)=G 0 (s)G P (s) The discrete time transfer function

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3.2 Plants with ZOH Example 1: Given a ZOH and a plant Determine their Z-domain transfer function.

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3.2 Plants with ZOH Example 2: Given a ZOH and a plant Determine their z-domain transfer function.

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3.2 Plants with ZOH Exercise 1: Given a ZOH and a plant Determine their z-domain transfer function. Answer:

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Assignment 1 You 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.

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Assignment 1 2) Choose a suitable sample period for a control loop for G(s) and explain your choice. 3)* Derive the discrete-time system transfer function G HP (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.

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

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3.3 Represent a system in difference equation Simulate the above system 1) Parameters and input: A=1-e -T, B=e -T, x(k)=1 2) initial condition: x(k-1)=0, y(k)=y(k-1)=0, k=0 3) Simulation While k<100 do y(k)=Ax(k-1)+By(k-1); Calculate output x(k-1)=x(k); y(k-1)=y(k); x(k)=1; k=k+1; Update data print k, x(k), y(k);Display step, input & output End

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

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3.3 Represent a system in difference equation

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Assignment 1 1)* Simulate this control object and find the responses using Matlab or other packages/computer languages. Hints: Method 1

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Assignment 1 Hints: Method 2

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3.4 System stability We can rewrite the difference equation as If A=1 and =0.9, for an impulse input we have k01234... x(k)10000... y(k)010.90.810.729 … It decreases exponentially, a stable system.

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3.4 System stability If K=1 and =1.2, we have k01234... x(k)10000... y(k)01 1.2 1.44 1.728 2.074… It increases exponentially, an unstable system.

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3.4 System stability If K=1 and = -0.8, we have k01234... x(k)10000... y(k)01-0.80.64-0.512… It decays exponentially, and alternates in sign, a gradual stable system.

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3.4 System stability It 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.

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3.4 System stability For continuous system, we have stable, critical stable and unstable areas in s domain. Stable area Unstable area Critical stable area

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3.4 System stability What is the stable area, critical stable area and unstable area for a discrete system in Z domain ? Stable area: unit circle Critical stable: on the unit circle Unstable area:outside of the unit circle

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3.4 System stability As For 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.

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3.4 System stability If 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.

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

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3.4 System stability Z domain responses 0 1

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3.5 Closed-loop transfer function Computer controlled system G c (z) ZOH G P (s) R(z) E(z) M(z) G HP (z) Computer system C(z) Plant

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3.5 Closed-loop transfer function Let’s find out the closed-loop transfer function

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3.5 Closed-loop transfer function C(z): output;E(z): error R(z): input;M(z): controller output G C (z): controller G P (z)/G(z): plant transfer function G HP (z): transfer function of plant + ZOH T(z): closed-loop transfer function G C (z)G HP (z): open-loop transfer function 1+ G C (z)G HP (z)=0: characteristic equation

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3.6 System block diagram G(s) H(s) - + R(s) C(s) C(z) G(s) H(s) - + R(s) C(z)

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3.6 System block diagram The 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.

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3.6 System block diagram Example: Find the closed-loop transfer function for the system below. Solution: The open-loop is G 1 (z)G 2 H(z). The forward path is G 1 (z)G 2 (z). G 1 (s) H(s) - + R(s) C(z) G 2 (s)

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3.6 System block diagram G 1 (s) H(s) - + R(s) C(z) G 2 (s)

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3.6 System block diagram *Exercise 3: Find the output for the closed- loop system below. G(s) H(s) - + R(s) C(s) C(z)

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3.6 System block diagram *Exercise 4: Find the output for the closed- loop system below. G 1 (s) H(s) - + R(s) C(z) G 2 (s)

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Reading Study book Module 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).

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Tutorial Exercise 1: Given a ZOH and a plant Determine their z-domain transfer function.

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Tutorial You 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.

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Tutorial 2) Choose a suitable sample period for a control loop for G(s) and explain your choice. 3) Derive the discrete-time system transfer function G HP (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.

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Tutorial 2) Choose a suitable sample period for a control loop for G(s) and explain your choice. a.Sampling theorem b.Input signal c.Bandwidth of a system d.Bold plots e.Applying sampling theorem f.Sampling frequency

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