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

Automatic Control Theory Exercises (33) 6 — 10, 11, 12 6 — 13 ( Optional )

Review §6.4.1 Linear Time-Invariant Difference Equations (1) Definition of difference ① Forward ② Backward §6.4 Mathematical Models of Discrete-Time Systems (2) The difference equation and its solving method ① Iteration ② Z-transformation §6.4.2 Impulse-Transfer Function (1) Definition(2) Properties(3) Limitation §6.4.3 Impulse Transfer Function of Open-Loop Systems (1) Switch between factors (2) No switch between factors (3) With ZOH §6.4.4 Impulse Transfer Function of Closed-Loop Systems (1) General Method (2) Mason’s formula

Automatic Control Theory （ Lecture 33 ） 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 33 ） §6 Analysis and Design of Linear Discrete-Time Systems §6.5 Stability and Steady-state Errors of Discrete-Time Systems

§6.5 Stability and Steady-state Errors of Discrete-Time Systems §6.5.1 s-Domain to z-Domain Mappings-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. Prove ： — Sufficiency — Necessity

§6.5.3 The Stability Criterion of Discrete-Time Systems (1) §6.5.3 The Stability Criterion of Discrete-Time Systems (1) w-transformation and Routh criterion in w-domain w-transformation Suppose [w] imaginary axis [z] unit circle, in z-plane Points inside outside of the unit circle

§6.5.3 The Stability Criterion of Discrete-Time Systems (2) Example 1 Determine the stability from the characteristic equation of a discrete system. (1) Routh criterion in w-domain Unstable!

§6.5.3 The Stability Criterion of Discrete-Time Systems (3) (2) Jurry criterion in z domain Unstable! Example 2 Determine the stability from the characteristic equation of a discrete system.

§6.5.3 The Stability Criterion of Discrete-Time Systems (4) System is stable Example 3 Determine the stability from the characteristic equation of a discrete system.

§6.5.3 The Stability Criterion of Discrete-Time Systems (5) Example 4 Consider the system shown in the figure (T=1). Determine the stable range of K. Solution I — Routh criterion in w domain

§6.5.3 The Stability Criterion of Discrete-Time Systems (6)

§6.5.3 The Stability Criterion of Discrete-Time Systems (7) Solution II — Jurry criterion in z domain

§6.5.3 The Stability Criterion of Discrete-Time Systems (8) Solution III —Root-locus method in z domain Breakaway points Thus For point A We have

§6.5.3 The Stability Criterion of Discrete-Time Systems (9) Example 4 Consider the system shown in the figure (T=0.25). Determine the stable range of K.

§6.5.3 The Stability Criterion of Discrete-Time Systems (10) ① ② ③ ④

Summary §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 Exercises (33) 6 — 10, 11, 12 6 — 13 ( Optional )

§6.5 Stability and Steady-state Errors of Discrete-Time Systems §6.1 Discrete-Time Control Systems —— characteristic research method §6.2 Signal Sampling and Holding —— signal sampling ZOH §6.3 z-transform—— definition method theory inverse-transform limitation §6.4 Mathematical Models of Discrete-Time Systems —— Linear Time-Invariant Difference Equations —— Impulse-Transfer Functions —— Impulse Transfer Function of Open-Loop Systems —— Impulse Transfer Function of Closed-Loop Systems §6.5 Stability and Steady-state Errors of Discrete-Time Systems —— s-Domain to z-Domain Mapping —— Necessary and Sufficient Condition for Stability of Linear Discrete-Time Systems —— The Stability Criterion of Discrete-Time Systems

§6.5 Stability and Steady-state Errors of Discrete-Time Systems §6.5.4 The Steady-State Errors of Discrete-Time Systems 1.The general method for calculating steady-state errors （ 1 ） Determine the stability of system （ 2 ） Determine the impulse- transfer function of errors （ 3 ） Determine the steady-state errors by final value theory 2. The steady-state errors coefficient method for calculating steady-state errors §6.6 Dynamic Performance Analysis of Discrete-Time Systems § Transient Responses of Discrete-Time Systems § Correlation between Closed-Loop Poles and Dynamic Responses

§6.5 Stability and Steady-state Errors of Discrete-Time Systems