automatic control theory school of automation nwpu teaching group of automatic control theory
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Automatic Control Theory Automatic Control Theory
School of Automation NWPU
Teaching Group of Automatic Control Theory
Automatic Control Theory
Exercises (36)
7 — 1,4
Automatic Control TheoryAutomatic Control Theory
（ Lecture 36 ）
§7 Nonlinear Systems §7.1 Introduction
§7.2 Phase Plane Method
§7.3 Describing Function Method
§7.4 Methods to Improve the Performance of
Nonlinear Control System
Automatic Control TheoryAutomatic Control Theory
（ Lecture 36 ）
§7 Nonlinear Systems
§7.1 Introduction to Nonlinear Control System
§7.2 Phase Plane Method
§7 Nonlinear Systems （ 1 ） §7.1 Introduction to Nonlinear Control System §7.1.1 Nonlinearities in Physics Systems
Nonlinearity is the universal law in the universe
There are a lot of kinds of nonlinear systems and responses.
The linear model is the approximate description of practical systems under the specific conditions.
§7.1.2 Typical Nonlinear Factors in Control Systems
Saturation Dead Zone Clearance Relay characteristic
§7 Nonlinear Systems （ 2 ）
§7.1.3 Characteristics of Nonlinear Control System (1) Does not satisfy Superposition principle—The linear theory does not apply.
(2) Stability — Not only depends on the structure and parameters, but also the
input and initial condition. The equilibriums may not be unique.
(3) Self-Excited Oscillation — The unique motion of nonlinear systems
(4) Complex in frequency response — Frequency hopping ， frequency
division/double Frequency, chaos.
§7.1.4 Methods to Analyze Nonlinear Control System (1) Linearization by Taylor’s Expansion
(2) The research method for nonlinear system
(3) Simulation method: Digital simulation, Hardware-in-loop simulation
Phase Plane Describing functionPopov methodFeedback linearization Differential geometry method
§7 Nonlinear Systems （ 3 ）
Analysis of nonlinear characteristics
Saturation Dead zone relay characteristic
Nonlinear
Characteristics
Equivalent K*
Affection on the system
Example
Oscillation↓,s↓Bounded tracking velocity Transistor
Steady state error↑
Remove small gain disturbance
Electromotor
Restrain divergence Self-excited oscillation
Switches
§7 Nonlinear Systems （ 4 ）
Analysis of nonlinear characteristics
Relay and its equivalent gain
§7.2 Phase Plane Method （ 1 ） §7.2.1 Phase Plane
Phase Plane:
Phase locus ：
)1(
5)(
sssG
The track of the system variable and its derivative varing with time in the phase plane.
Example 1 Unity feedback system
2236.0
236.2
n
)(1)( ttr
(1) Phase plane and phase locus
The phase plane, which can describe the state of system, is constructed by the system variable and its derivative ( ) ,c c
§7.2 Phase Plane Method （ 2 ）
(2) Features of phase locus
0
0
x
x( , ) 0
0
dx dx dt f x x
dx dx dt x
For linear time-invariant system, the origin is the unique equilibrium point.
The directionof movement
When the phase locus intersects with x axis, it always passes through with an angle of 90°
Singular point (Equilibrium point):
0),( xxfx Suppose the system equation is ：
Points on the phase locus with uncertain slop
upper half plane — moving to the right0x
Clockwise movement
under half plane — moving to the left0x
§7.2 Phase Plane Method （ 3 ）
Example 2 Consider the system Sketch the phase locus for the system
02 xx n
td
xd
xd
xd
td
xdx
Solution:
xxd
xdx n
2
xdxxdx n 2
Cxx n
22
2
22
1
222
22 2
ACx
xnn
122
2
2
2
nA
x
A
x
— Elliptic Equation
§7.2 Phase Plane Method （ 4 ）
Location of poles
(3) Phase locus of second order linear systems
Singular point
Phase locus
center point
stable focus
stable node
saddle point
unstable focus
unstable node
Location of poles
Singular point
Phase locus
§7.2 Phase Plane Method （ 5 ）Example 3 Consider the system Obtain the equilibriums xe and determine the characteristic of phase locus around the equilibriums
0)5.03( 2 xxxxx
0xx Solution. Let
0)1(2 xxxx
1
0
2
1
e
e
x
x
Unstable focus
By linearization
0)1()1(5.0
05.02xxxx
xxx
12
1
xxxx
xxxx
e
e
05.0
05.0
xxx
xxx
015.0
015.02
2
ss
ssCharacteristic equation
0.5 0.97
0.78
1.28
s j
s
Saddle point
§7.2 Phase Plane Method （ 6 ）
0xx Solution. Let
Whenxxxkx sin)2sin(sin
0
0
xx
xx
01
012
2
s
sCharacteristic equation
1
1
s
js
0sin x kxe
exk2
xxx sinsin)12( k
Linearization
Center Point
Saddle Point
0sin xxExample 4 Consider the system . Obtain the equilibriums xe and determine the characteristic of phase locus around the equilibriums
§7.2 Phase Plane Method （ 7 ）
Example 5 Consider the system . Analyze its free response.0 xxx
Solution.
2
2
I 1 0
II 1 0
s s
s s
62.1
62.0
866.05.0
2,1
2,1
s
js
Characteristic equation
Stable focus
Saddle point
Analyze a class of nonlinear systems by the phase locus of 2nd order systems.
0 0 I
0 0 II
x x x x
x x x x
1
2
I 0
II 0e
e
x
x
Singular point
PolesSwitch Line
§7.2 Phase Plane Method （ 8 ）
Solution.
2
2
I 1 0
II 1 0
s
s
1,2
1,2
1
1
s j
s j
Characteristic equation
Center Point
II001
I001
xxx
xxx
1
2
I 1
II 1e
e
x
x
Singular point
PolesCenter Point
Switch line — The boundary line to divide different linear area.
Equilibrium line (Singular line) — Generated by the interaction between phase locus in different area.
Example 6 Consider the system . Analyze its free response.
0sign xxx
Summary
7.1 Introduction to Nonlinear Control Systems
7.1.1 Nonlinearities in Physics Systems
7.1.2 Typical Nonlinear Factors in Control Systems
7.1.3 Characteristics of Nonlinear Control System
7.1.4 Methods to Analyze Nonlinear Control System
7.2 Phase Plane Method 7.2.1 Phase Plane
(1) Phase plane and phase locus
(2) Features of phase locus (The direction of movement,
Singular point, Singular line, Switched line )
(3) Phase locus of the second order linear system (Analyze the free response a class of nonlinear systems)
Automatic Control Theory
Exercises (36)
7 — 1,4
