Chapter 8 Discrete (Sampling) System 8.1 Introduction 8.2 Z-transform 8.3 Mathematical describing of the sampling systems 8.4 Time-domain analysis of the sampling systems 8.5 The root locus of the sampling control systems 8.6 The frequency response of the sampling control systems 8.7 The design of the “least-clap” sampling systems

Chapter 8 Discrete (Sampling) System 8.1 Introduction x*(t) t x(t) 8.1.1 Sampling x(t) Make a analog signal to be a discrete signal shown as in Fig.8.1 . t1 t2 t3 t4 t5 t6 x(t) —analog signal . x*(t) —discrete signal . 8.1.2 Ideal sampling switch —sampler Fig.8.1 signal sampling Sampler —the device which fulfill the sampling. Another name —the sampling switch — which works like a switch shown as in Fig.8.2 . T t x*(t) x(t) 8.1.3 Some terms 1. Sampling period T— the time interval of the signal sampling: T = ti+1 － ti . Fig.8.2 sampling switch

8.1.3 Some terms 2. Sampling frequency ωs — ωs = 2π fs = 2π / T . 3. Periodic Sampling — the sampling period Ts = constant. 4. Variable period sampling — the sampling period Ts≠constant. 5. Synchronous sampling —not only one sampling switch in a system, but all work Synchronously. 6. Multi-rate sampling. 7. Opportunity(Random) sampling. We mainly discuss the periodic and synchronous sampling in chapter 8. 8.1.4 Sampling (or discrete) control system There are one or more discrete signals in a control system — the sampling (or discrete) control system. For example the digital computer control system:

8.1 Introduction 8.1.5 Sampling analysis A/D D/A computer process measure r(t) c(t) e(t) － e*(t) u*(t) u (t) Fig.8.3 computer control system 8.1.5 Sampling analysis Expression of the sampling signal: It can be regarded as Fig.8.4:

8.1.5 Sampling analysis ＝ We have: × modulating pulse(carrier) t x*(t) x(t) T δT(t) × ＝ modulating pulse(carrier) modulated wave Modulation signal Fig.8.4 sampling process We have:

8.1.5 Sampling analysis Only: could be reproduced This means: for the frequency spectrum of x(t) shown in Fig.8.5, the frequency spectrum of x*(t) is like as Fig.8.6. Fig.8.6 Filter Fig.8.5 Only: could be reproduced

8.1 Introduction So we have: 8.1.6 Sampling theorem ( Shannon’s theorem) If the analog signal could be whole restituted from the sampling signal, the sampling frequency must be satisfied : 8.1.7 zero-order hold Usually the controlled process require the analog signals, so we need a discrete-to-analog converter shown in Fig.8.5. discrete-to-analog converter x*(t) xh(t) Fig.8.7 D/A convert

8.1.7 zero-order hold The ideal frequency response of the D/A converter is shown in Fig.8.8. ω A(ω) Fig.8.8 To put the ideal frequency response in practice is difficult, the zero -order hold is usually adopted. The action of the zero-order hold is shown in Fig.8.9. x*(t) x(t) xh(t) The mathematic expression of xh(t) : The unity pulse response of the zero-order hold is shown in Fig.8.10. Fig.8.9 T Fig.8.10 t g(t) The transfer function of the zero-order hold can be obtained from the unity pulse response:

8.1 Introduction 8.2 Z-transform 8.1.7 zero-order hold Fig.8.11 The frequency response of the zero-order hold, which is shown in Fig.8.11, is: 8.2 Z-transform 8.2.1 Definition Expression of the sampled signal: Using the Laplace transform: Define:

8.2 Z-transform We have the Z-transform: 8.2.2 Z-transforms of some common signals Table 8.1 The Z-transforms of some common signals is shown in table 8.1. 8.2.3 characteristics of Z- transform The characteristics of Z- transform is given in table 8.2.

Table 8.2

8.2.3 characteristics of Z-transform Using the characteristics of Z-transform we can conveniently deduce the Z-transforms of some signals. Such as the examples shown in table 8.3: Table 8.3

8.2 Z-transform 8.2.4 Z-transform methods 1. Partial-fraction expansion approaches Example 8.1 2. Residues approaches

8.2.4 Z-transform methods Example 8.2 8.2.5 Inverse Z-transform 1. Partial-fraction expansion approaches

8.2.5 Inverse Z-transform Example 8.3 2. Power-series approaches

8.2.5 Inverse Z-transform 3. Residues approaches Example 8.5

Chapter 8 Discrete (Sampling) System 8.3 Mathematical modeling of the sampling systems 8.3.1 Difference equation For a nth-order differential equation: Make:

8.3.1 Difference equation … Or : …

8.3.1 Difference equation A nth-order differential equation can be transformed into a nth-order difference equation by the backward or forward difference: To get the solution of the difference equation is very simple by the recursive algorithm. Fig.8.12 T Example 8.6 K = 10, T = 0.5s, r(t) = 1(t) Determine the output c*(mT). For the sampling system shown in Fig.8.12, Assume: Solution

8.3.1 Difference equation

8.3.1 Difference equation For K = 10, T = 0.5s, we have: Consider e*(k) = r(k)－c(k) = 1－c(k): If c(0) = 0, applying the recursive algorithm we have: ……

8.3 Mathematical modeling of the sampling systems 8.3.2 Z-transfer (pulse) function Definition: Z-transfer (pulse) function — the ratio of the Z-transformation of the output signal versus input signal for the linear sampling systems in the zero-initial conditions, that is: 1. The Z-transfer function of the open-loop system G1(s) G2(s) T r(t) c(t) c*(t) G1(z)G2(z) R(z) C(z) G1(z) =Z [ G1(s)] G2(z) =Z [ G2(s)] T G1(s) r(t) G2(s) c(t) c*(t) G1G2(z) R(z) C(z) G1G2(z) =Z [ G1(s)G2(s) ]

8.3.2 Z-transfer (pulse) function 2. The z-transfer function of the closed-loop system G(s) r c － H(s) r c － G(s) H(s) r － c G(s) H(s) r G2(s) c － G1(s) H(s) r － G2(s) c G1(s) H(s)

8.3.2 Z-transfer (pulse) function － G3(s) c G2(s) H(s) G1(s) r － G2(s) c G1(s) H2(s) H1(s) r － G1(s) H2(s) G2(s) c H1(s) G3(s)

Chapter 8 Discrete (Sampling) System 8.4 Time-domain analysis of the sampling systems 8.4.1 The stability analysis 1. The stability condition The characteristic equation of the sampling control systems: Suppose: In s-plane, α need to be negative for a stable system, it means: So we have: The sufficient and necessary condition of the stability for the sampling control systems is: The roots zi of the characteristic equation 1+GH(z)=0 must all be inside the unity circle of the z-plane, that is:

8.4.1 The stability analysis Im critical stability z-plane The graphic expression of the stability condition for the sampling control systems is shown in Fig.8.4.1. 1 Re Stable zone unstable zone 2. The stability criterion Fig.8.4.1 In the characteristic equation 1+GH(z)=0, substitute z with —— W (bilinear) transformation. We can analyze the stability of the sampling control systems the same as we did in chapter 3 (Routh criterion in the w-plane) .

8.4.1 The stability analysis Determine K for the stable system. Example 8.7 Solution : make We have: 0 < K < 4.33. 8.4.2 The steady state error analysis The same as the calculation of the steady state error in Chapter 3, we can use the final value theorem of the z-transform:

8.4.2 The steady state error analysis For the stable system shown in Fig.8.4.2 G(s) r c － e Fig.8.4.2

8.4.2 The steady state error analysis － G (s) c Z.o.h T e Example 8.8 Z.o.h —Zero-order hold. 2) If r(t) = 1+t, determine ess=？ 1) Determine K for the stable system. Solution 1)

8.4.2 The steady state error analysis 2)

8.4 Time-domain analysis of the sampling systems 8.4.3 The unit-step response analysis Im 1 Re Analyzing c(kT) we have the graphic expression of C(kT) is shown in Fig.8.4.3. Fig.8.4.3

Chapter 8 Discrete (Sampling) System 8.5 The root locus of the sampling control systems The plotting procedure of the root loci of the sampling systems are the same as that we introduced in chapter 4. But the analysis of the root loci of the sampling systems is different from that we discussed in chapter 4 (imaginary axis of the s-plane ←→ the unit circle of the z-plane). 8.6 The frequency response of the sampling control systems The analysis and design methods of the frequency response of the sampling systems are the same as that we discussed respect-ively in chapter 5 , chapter 6, only making: ←→ Here: v — the counterfeit frequency

Chapter 8 Discrete (Sampling) System 8.7 The design of the “least-clap” sampling systems The transition process of the sampling control systems can be finished in the minimum sampling periods—the “least-clap” systems. 8.7.1 design of D(z) r － G (s) c D(z) Fig.8.6.1 e For the system shown in Fig.8.6.1. We have:

8.7.1 design of D(z)

8.7.1 design of D(z) Proof: The responses of the “least-clap” Fig.8.6.2 1 c*(t) t T 2T 3T The responses of the “least-clap” System are shown in Fig.8.6.2.

For the system shown in Fig.8.6.3 8.7.1 design of D(z) Example 8.9 For the system shown in Fig.8.6.3 r － G (s) c Gh(s) D(z) e T Fig.8.6.3 u Determine D(z), make the system to be the “least-clap” system for r(t) = t. Solution

8.7 The design of the “least-clap” sampling systems 8.7.2 The realization of D(z) To illustrate by example 8.10 Example 8.10 To realize D(z) for Example 8.9 Solution We can program the computer in terms of above formula to realize D(z).

Chapter 8 Discrete (Sampling) System Exercises: p820～ E13.11; E13.14; P13.12; p13.18; AP13.2 In example 8.9, if and respectively for r(t)=1(t); t; t2 .