Spring 2019

數位控制系統Digital Control Systems

DCS-22Stability

Feng-Li LianNTU-EE

Feb19 – Jun19

DCS22-Stability-2Feng-Li Lian © 2019

The Research Procedure in Control Science

PhysicalProcess

MathModel

SystemAnalysis

ControlDesign

Differential eqn Laplace transform Transfer function State space form

Root locus Bode diagram Nyquist plot

Stability Robustness Sensitivity Controllability Observability

Estimator Identification Regulation Tracking PID Pole placement Optimal Control

LQR/LQG Adaptive control Robust control Decentralized

(or Multi-person) Control

Difference eqn z transform Transfer function State space form

Plant Sensor Actuator Computer Communication Noise Disturbance

Differential eqn Laplace transform

Introduction: The Design Philosophy of Control Science

DCS22-Stability-3Feng-Li Lian © 2019Introduction: From CT Plant to DT Plant

G(s)u(t) y(t)

C(s)e(t)r(t)

G(z)u[k] y[k]

C(z)e[k]r[k]

Plant (CT): • Input-Output Model:

• State-Space Model:

Plant (DT): • Input-Output Model:

• State-Space Model:

2

3

1

DCS22-Stability-4Feng-Li Lian © 2019Introduction: Model and Analysis

G(s)u(t) y(t)

C(s)e(t)r(t)

G(z)u[k] y[k]

C(z)e[k]r[k]

Plant (CT): • Input-Output Model:

• State-Space Model:

Plant (DT): • Input-Output Model:

• State-Space Model:

System Properties:

Stability

Controllability and Reachability

Observability and Detectability

DCS22-Stability-5Feng-Li Lian © 2019Outline

Solution of a System Stability and Asymptotic Stability Input-Output Stability Stability Tests:

• Jury’s Stability Criterion

• Nyquist and Bode Diagrams

• Nyquist Criterion

• Relative Stability

Lyapunov’s Second Stability

DCS22-Stability-6Feng-Li Lian © 2019Solution of a System

CT:

DT:

Initial Condition:

Solution:

DCS22-Stability-7Feng-Li Lian © 2019Stability

Definition 3.1: Stability

DCS22-Stability-8Feng-Li Lian © 2019Stability

Definition 3.2: Asymptotic Stability

DCS22-Stability-9Feng-Li Lian © 2019Stability of Linear DT Systems

Stability of Linear Discrete-Time Systems

DCS22-Stability-10Feng-Li Lian © 2019Stability of Linear DT Systems

Solution of LTI DT Systems

DCS22-Stability-11Feng-Li Lian © 2019Stability of Linear DT Systems

Theorem 3.1: Asymptotic Stability of Linear Systems

DCS22-Stability-12Feng-Li Lian © 2019Stability of Linear DT Systems

Stability of Linear Continuous-Time Systems

DCS22-Stability-13Feng-Li Lian © 2019Input-Output Stability

Definition 3.3: Bounded-Input-Bounded-Output Stability

Theorem 3.2: Relation between Stability Concept

DCS22-Stability-14Feng-Li Lian © 2019Input-Output Stability

Example 3.1: Harmonic Oscillator

DCS22-Stability-15Feng-Li Lian © 2019Stability Test

DCS22-Stability-16Feng-Li Lian © 2019Stability Test: Jury’s Stability Criterion

(1918) (1922) (1961)

DCS22-Stability-17Feng-Li Lian © 2019Stability Test: Routh’s Stability Criterion (for CT)

Routh in 1874 Hurwitz in 1895

Franklin, Powell, Emami-Naeini 2002

DCS22-Stability-18Feng-Li Lian © 2019Stability Test: Jury’s Stability Criterion

Theorem 3.3: Jury's Stability Test

DCS22-Stability-19Feng-Li Lian © 2019Stability Test: Jury’s Stability Criterion

Example: Jury's Stability Test

DCS22-Stability-20Feng-Li Lian © 2019Stability Test: Nyquist and Bode Diagrams

DCS22-Stability-21Feng-Li Lian © 2019Stability Test: Nyquist and Bode Diagrams

Example 3.3: Frequency Responses

DCS22-Stability-22Feng-Li Lian © 2019Stability Test: Nyquist and Bode Diagrams

CT: - - - DT:

Example 3.3: Frequency Responses

DCS22-Stability-23Feng-Li Lian © 2019Stability Test: Nyquist Criterion

Nyquist Criterion

DCS22-Stability-24Feng-Li Lian © 2019Stability Test: Nyquist Criterion

Franklin, Powell, Emami-Naeini 2002

DCS22-Stability-25Feng-Li Lian © 2019Stability Test: Nyquist Criterion

Example 3.4: A Second-order system

DCS22-Stability-26Feng-Li Lian © 2019Stability Test: Nyquist Criterion

Example 3.4: A Second-order system

DCS22-Stability-27Feng-Li Lian © 2019Stability Test: Relative Stability

Definitions 3.4 & 3.5: Gain & Phase Margins

DCS22-Stability-28Feng-Li Lian © 2019Lyapunov’s Second Stability

Definition 3.6: Lyapunov Function

DCS22-Stability-29Feng-Li Lian © 2019Lyapunov’s Second Stability

Geometric Illustration

DCS22-Stability-30Feng-Li Lian © 2019Lyapunov’s Second Stability

Example 3.6: Lyapunov function

DCS22-Stability-31Feng-Li Lian © 2019Lyapunov’s Second Stability

Example 3.6: Lyapunov function for CT case

Khalil 2002

DCS22-Stability-32Feng-Li Lian © 2019Lyapunov’s Second Stability

Example 3.6: Lyapunov function