1.1 Introduction Comparison between ACS and CCS

ACS CCS Process Actuator Measure Controller (correcting network) Structure: Process Actuator Measure Controller (digital computer) Adapter (A/D, D/A) Parts: Analog Analog + Digital Signals: Analog Continuous analog Discrete analog Discrete digital

Discrete (Sampling) System 1 Introduction 2 Z-transform 3 Mathematical describing of the sampling systems 4 Time-domain analysis of the sampling systems

Chapter Discrete (Sampling) System Make a analog signal to be a discrete signal shown as in Fig.1. t x(t)x(t) 0 t1t1 t2t2 t3t3 t4t4 t5t5 t6t6 Fig.1 signal sampling x(t)x(t) x*(t)x*(t) x(t) —analog signal. x * (t) —discrete signal. 1.2 Ideal sampling switch —sampler Sampler —the device which fulfill the sampling. Another name —the sampling switch — which works like a switch shown as in Fig.2. T t x*(t)x*(t) 0 t x(t)x(t) 0 Fig.2 sampling switch 1.3 Some terms 1. Sampling period T— the time interval of the signal sampling: T = t i+1 － t i. 1 Introduction 1.1 Sampling

1.3 Some terms 2. Sampling frequency ω s — ω s = 2π f s = 2π / T. 3. Periodic Sampling — the sampling period T s = constant. 4. Variable period sampling — the sampling period T s ≠ constant. 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:

A/D D/A computer process measure r(t) c(t)c(t) e(t)e(t) － e*(t) u*(t) u (t) Fig.3 computer control system 1.5 Sampling analysis Expression of the sampling signal: 1 Introduction It can be regarded as Fig.4:

0 t x*(t)x*(t) t x(t)x(t) 0 T t δT(t)δT(t) 0 × ＝ modulating pulse(carrier) modulated wave Modulation signal Fig.4 sampling process x(t)x(t)x*(t) 1.5 Sampling analysis

If the analog signal could be whole restituted from the sampling signal, the sampling frequency must be satisfied : 1.7 zero-order hold Usually the controlled process require the analog signals, so we need a discrete-to-analog converter shown in Fig.7. discrete-to-analog converter x*(t)xh(t)xh(t) Fig.7 D/A convert So we have: 1.6 Sampling theorem ( Shannon’s theorem) 1 Introduction

x*(t) x(t)x(t) xh(t)xh(t) Fig.9 The action of the zero-order hold is shown in Fig.9. The unity pulse response of the zero- order hold is shown in Fig.10. The mathematic expression of x h (t) : The transfer function of the zero-order hold can be obtained from the unity pulse response: T Fig.10 t g(t)g(t) ω A(ω)A(ω) Fig.8 To put the ideal frequency response in practice is difficult, the zero -order hold is usually adopted. 1.7 zero-order hold The ideal frequency response of the D/A converter is shown in Fig.8.

2 Z-transform 2.1 Definition Expression of the sampled signal: Using the Laplace transform: Define:

We have the Z-transform:.2.2 Z-transforms of some common signals The Z-transforms of some common signals is shown in table Z-transform Table characteristics of Z- transform The characteristics of Z- transform is given in table.2.

Table.2

Table.3 Using the characteristics of Z-transform we can conveniently deduce the Z-transforms of some signals. Such as the examples shown in table.3: 2.3 characteristics of Z-transform

Example.1 2. Residues approaches.2 Z-transform 2.4 Z-transform methods 1. Partial-fraction expansion approaches

2.5 Inverse Z-transform 1. Partial-fraction expansion approaches Example Z-transform methods

2. Power-series approaches Example.4 Example Inverse Z-transform

3. Residues approaches 2.5 Inverse Z-transform Example 5

.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 T G 1 (s) r(t) G 2 (s) c(t) c*(t) G 1 (z)G 2 (z) R(z) C(z) G 1 (s) G 2 (s) TT r(t) c(t) c*(t) G1G2(z)G1G2(z) R(z) C(z) G 1 G 2 (z) =Z [ G 1 (s)G 2 (s) ].3 Mathematical modeling of the sampling systems G 1 (z) =Z [ G 1 (s)] G 2 (z) =Z [ G 2 (s)]

G(s) rc － H(s) r G 2 (s) c － G 1 (s) H(s) r － G 2 (s) c G 1 (s) H(s) rc － G(s) H(s) r － c G(s) H(s).3.2 Z-transfer (pulse) function 2. The z-transfer function of the closed-loop system

r － G 3 (s) c G 2 (s) H(s) G 1 (s).3.2 Z-transfer (pulse) function

Chapter Discrete (Sampling) System 4 Time-domain analysis of the sampling systems 4.1 The stability analysis 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 z i of the characteristic equation 1+GH(z)=0 must all be inside the unity circle of the z-plane, that is: 1. The stability condition

1 Re Im z-plane Stable zone Fig..4.1 The graphic expression of the stability condition for the sampling control systems is shown in Fig The stability criterion 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 (Routh criterion in the w-plane). 4.1 The stability analysis unstable zone critical stability

Determine K for the stable system. Solution : We have: 0 < K < make 4.2 The steady state error analysis The same as the calculation of the steady state, we can use the final value theorem of the z-transform: 4.1 The stability analysis Example.7

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

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

4.2 The steady state error analysis 2)

4 Time-domain analysis of the sampling systems 4.3 The unit-step response analysis Fig.4.3 Analyzing c(kT) we have the graphic expression of C(kT) is shown in Fig.4.3. Im Re 1

Chapter Discrete (Sampling) System 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 continuous system. But the analysis of the root loci of the sampling systems is different from that we discussed in continuous system. (imaginary axis of the s-plane ←→ the unit circle of the z-plane)..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 continuous system, only making: Here: v :frequency

Stability of Digital Control System

Stability is a basic requirement for digital and analog control systems Asymptotic Stability Bounded Input Bounded Output

Observe

Asymptotic Stability for Digital Control System BIBO Stability

Internal Stability

Routh-Hurwitz Criterion Bilinear transformation transforms the inside of the unit circle to the LHP (Left Half Plane). Bilinear Transformation Example : Substituting the bilinear transformation stability conditions

Jury Criterion

Example : We compute the entries of the Jury table using the coefficients of the polynomial

Transient and Steady State Analysis Transient Analysis

Steady State Error Analysis Final Value Theorem

1) For unity feedback in figure below, 2)