Chapter 7 Stability and Steady-State Error Analysis

1
Chapter 7 Stability and Steady-State Error
Analysis

• 7.1 Stability of Linear Feedback Systems
• 7.2 Routh-Hurwitz Stability Test
• 7.3 System Types and Steady-State Error
• 7.4 Time-Domain Performance Indices

2

7.1 Stability of Linear Feedback Systems (1)

• Basic Concepts

(1) Equilibrium States,
(2) Stable System System response can
restore to initial equilibrium state under small
disturbance. (3) Meaning of Stable System
Energy sense Stable system with minimum
potential energy. Signal sense Output
amplitude decays or grows with different meaning.
Lyapunov sense Extension of signal and
energy sense for state evolution
in state space.
3
7.1 Stability of Linear Feedback Systems (2)

• Plant Dynamics

Regular pendulum (Linear)
Inverted pendulum (Linear)
Natural response
Natural response
4
7.1 Stability of Linear Feedback Systems (3)

• Closed-loop System

• Natural behavior of a control system, r(t)d(t)0
  

Equilibrium state Initial relaxation system,
I.C.0
No general algebraic solution for 5th-order and
above polynomial equation (Abel , Hamilton)
5
7.1 Stability of Linear Feedback Systems (4)

• Stability Problems

Stabilization of unstable system
Destabilization Effect on stable system
G(s) Unstable plant Closed-loop Stable
G(s) stable plant Closed-loop Unstable
6
7.1 Stability of Linear Feedback Systems (5)

• Stability Definition

(1) Asymptotic stability Stable system if
the transient response decays to zero
(2) BIBO stability Stable system if the
response is bounded for bounded input signal
The impulse response of a system is absolutely
integrable.
7
7.1 Stability of Linear Feedback Systems (6)
(3) S-domain stability System Transfer
Function T(s) Stable system if the poles
of T(s) all lies in the left-half s-plane.
The definitions of (1), (2), and (3) are
equivalent for LTI system.
8
7.2 Routh-Hurwitz Stability Test (1865-1905)
(1)

• Characteristic Polynomial of Closed-loop System

Hurwitz polynomial All roots of D(s) have
negative real parts. stable
system Hurwitzs necessary conditions All
coefficients (ai) are to be positive. Define
9
7.2 Routh-Hurwitz Stability Test (1865-1905)
(2)

• Routh Tabulation (array)

• Routh-Hurwitz Stability Criterion

(1) The polynomial D(s) is a stable polynomial if
are all positive, i.e.
are all positive. (2) The number of
sign changes in is
equal to the number of roots in the RH
s-plane. (3) If the first element in a row is
zero, it is replaced by a small
and the sign changes when are
counted after completing the array. (4)
If all elements in a row are zero, the system has
poles in the RH plane or on the imaginary
axis.
10
7.2 Routh-Hurwitz Stability Test (1865-1905)
(3)
For entire row is zero Identify the auxiliary
polynomial The row immediately above the
zero row. The original polynomial is with factor
of auxiliary polynomial. The roots of auxiliary
polynomial are symmetric w.r.t. the origin
11
7.2 Routh-Hurwitz Stability Test (1865-1905)
(4)
Ex For a closed-loop system with transfer
function T(s)
Ex Find stability condition for a
closed-loop system with
characteristic polynomial as Sol
12
7.2 Routh-Hurwitz Stability Test (1865-1905)
(5)
Ex For a colsed-loop system with
characteristic polynomial
Determine if the system is stable Sol
Ex For ,
determine if the system is stable Sol
13
7.2 Routh-Hurwitz Stability Test (1865-1905)
(6)

• Absolute and Relative Stability

Absolute Stability
Relative Stability
Characteristic equation R-H Test on D(s)
Characteristic equation R-H Test on D (p)
14
7.3 System Types and Steady-State Error (1)

• Steady-state error for unity feedback systems

For nonunity feedback systems
15
7.3 System Types and Steady-State Error (2)

• Fundamental Regulation and Tracking Error

Regulation s.s. error
Tracking s.s. error
16
7.3 System Types and Steady-State Error (3)

• Open-loop System Types

17
7.3 System Types and Steady-State Error (4)

• Position Control of Mechanical Systems

(1) Command signal
Region 1 and 3 Constant acceleration and
deceleration Region 2 Constant speed Region 4
Constant position
(2) Error constants
18
7.3 System Types and Steady-State Error (5)
(3) Systems control with finite steady-state
position error
Constant position for Type 0 system
Constant velocity for Type 1 system
Constant acceleration for Type 2 system
19
7.3 System Types and Steady-State Error (6)

• Steady-state position errors for different types
  of system and input signal

Output positioning in feedback control is driven
by the dynamic positional error. System
nonlinearities such as friction, dead zone,
quantization will introduce steady-state error in
closed-loop position control.
20
7.3 System Types and Steady-State Error (7)
Ex Find the value of K such that there is
10 error in the steady state
Sol System G(s) is Type 1
s.s. error in ramp input
For velocity error constant
21
7.4 Time-Domain Performance Indices (1)

• Performance of Control System

Stability Transient Response Steady-state Error

• Performance Indices (PI)

A scalar function for quantitative measure of the
performance specifications of a control system.
error
command
state
output
Use P.I. To trade off transient response and
steady-state error with sufficient stability
margin.
22
7.4 Time-Domain Performance Indices (2)

• Systems Control

(1) Classical control
Plant Input-Output Model
Controller PID Control
P-Proportional control
I-Integral control

D-Differential control
(2) Modern control Plant State-space
Model P. I. Usually Quadratic functional
Controller States feedback control
23
7.4 Time-Domain Performance Indices (3)

• Optimal Control

Given Plant model Control
configuration (Usually feedback)
Controller structure (Usually linear)
Design constraints Objective Minimize P. I. (P.
I. to be selected) Find Optimal parameters in
controller
Ex Design optimal proportional control system
Find optimal K to minimize P. I.