Automatic Control Theory School of Automation NWPU Teaching Group of Automatic Control Theory

Principles of Automatic Control Exercise (34) 6 — 14, 15, 16 6 — 17 （ Optional ）

Review §6.5.1 s-Domain to z-Domain Mapping §6.5 Stability and Steady-state Errors of Discrete-Time Systems §6.5.2 Necessary and Sufficient Condition for Stability of Linear Discrete-Time Systems — All poles of  (z) lie in the unit circle of z plane §6.5.3 The Stability Criterion of Discrete-Time Systems (1) Routh criterion in w domain (2) Jurry criterion in z domain (3) Root-locus method in z domain

Automatic Control Theory （ Lecture 34 ） Chapter 6 Analysis and Design of Linear Discrete-Time Systems § 6.1 Discrete-Time Control Systems § 6.2 Signal Sampling and Holding § 6.3 z-Transform § 6.4 Mathematical Models of Discrete-Time Systems § 6.5 Stability and Steady-state Errors of Discrete-Time Systems § 6.6 Dynamic Performance Analysis of Discrete-Time Systems § 6.7 Digital Control Design for Discrete-Time Systems

Automatic Control Theory （ Lecture 34 ） §6 Analysis and Design of Linear Discrete-Time Systems §6.5 Stability and Steady-state Errors of Discrete-Time Systems §6.6 Dynamic Performance Analysis of Discrete-Time Systems

§6.5.4 General method to obtain steady-state error （ 1 ） 1. General Method to obtain steady-state error Let Algorithm: (1) Determine the stability (2) Obtain the impulse transfer function from E(z) to C(z). (3) Obtain by the final value theorem v: System type

§6.5.4 General method to obtain steady-state error （ 2 ） Example 1 Consider the discrete system shown in the figure, K=2, T=1; Obtain e(∞) for r(t)=1(t), t, t 2 /2.

§6.5.4 General method to obtain steady-state error （ 3 ） Example 1 Consider the discrete system shown in the figure, K=2, T=1; Obtain e(∞) for r(t)=1(t), t, t 2 /2.

§6.5.4 Static Error Constant Method （ 1 ） 2. Static Error Constant Method shows how e(∞) changes with r(t) (For stable linear discrete systems subject to r(t) and sampled at the error signal) Let v: System type

§6.5.4 Static Error Constant Method （ 2 ） Static position error constant Static velocity error constant Static acceleration error constant

§6.5.4 Static Error Constant Method （ 3 ）

§6.5.4 Static Error Constant Method （ 4 ） Solution. Example 2 Consider the stable discrete system shown in the figure. When r(t)=2t, obtain e(∞) with/without ZOH. no ZOH with ZOH — dependent of T — independent of T

§6.5.4 Static Error Constant Method （ 5 ） Example 3 Consider the system shown in the figure, T=0.25. When r(t)=2·1(t)+t, obtain the range of K for e(∞)<0.5. Solution. The stable range of K is

§6.6 Dynamic performance analysis of discrete systems (1) §6.6. Dynamic performance analysis of discrete systems Let 1General algorithm to obtain the dynamic performance (1)Obtain the impulse transfer function (2) Obtain (3) (4) Determine the specifications.

§6.6 Dynamic performance analysis of discrete systems (2) Example 4 Consider the system shown in the figure, T=K=1. Obtain the dynamic specifications. (σ %, t s ). Solution. Obtain the unit step response series h(k) by long division method.

§6.6 Dynamic performance analysis of discrete systems (3)

§6.6 Dynamic performance analysis of discrete systems (4) Solution. Example 4 Consider the system shown in the figure, T=K=1. Obtain the dynamic specifications. (σ%, t s ).

§6.6 Dynamic performance analysis of discrete systems (3)

§6.6 Dynamic performance analysis of discrete systems (5) 2 Relationship between dynamic response and closed-loop poles (1)Single closed-loop poles on the real axis

§6.6 Dynamic performance analysis of discrete systems (6)

§6.6 Dynamic performance analysis of discrete systems (7) (2) Closed-loop Complex conjugate poles

§6.6 Dynamic performance analysis of discrete systems (8)

§6.6 Dynamic performance analysis of discrete systems (9)

§6.6 Dynamic performance analysis of discrete systems (10) Solution (1) Example 5 Consider the system shown in the figure (T=1). (1) Sketch the root locus when (2) Determine the stable range of K (3) Determine how the dynamic performance changes when Breakaway point:

§6.6 Dynamic performance analysis of discrete systems (11) Solution (2) (3) Stable Unstable Example 5 Consider the system shown in the figure (T=1). (1) Sketch the root locus when (2) Determine the stable range of K (3) Determine how the dynamic performance changes when

Summary §6.5.4 Steady-state error of discrete systems §6.6 Analysis of discrete-time dynamic performance (1)General Method Stability (2) Static error constant Obtain → Obtain s , t s by definition (1) General method (2) Closed-loop poles Response

Principles of Automatic Control Exercise (34) 6 — 14, 15, 16 6 — 17 （ Optional ）