EE 380 Linear Control Systems Lecture...
Transcript of EE 380 Linear Control Systems Lecture...
EE 380 Fall 2014Lecture 12.
EE 380
Linear Control Systems
Lecture 12
Professor Jeffrey SchianoDepartment of Electrical Engineering
1
EE 380 Fall 2014Lecture 12.
Lecture 12 Topics
• Nonlinear System Analysis– Small-signal Analysis– Chapter 9, Sections 9.1 and 9.2
2
EE 380 Fall 2014Lecture 12.
Fundamental Definitions• A system is said to be linear if the following are true
1. The response can be represented as the sum a zero-input and zero-state response
2. The zero-input response obeys the principle of superposition with respect to the initial state
3. The zero-state response obeys the principle of superposition with respect to the input
• If a system does not satisfy all three properties, then it is nonlinear
3
EE 380 Fall 2014Lecture 12.
Example 1
• Determine if the following systems are zero-state linear
4
2
2
2
2
1
1
(1)
(2) sin(
(
)
)
( )
o
o
d y dya dtdtd y dya ydtd
u
a tt
a y t
u
EE 380 Fall 2014Lecture 12.
Control Design for Nonlinear Plants• Most physical systems are inherently nonlinear
• Control design techniques that account for nonlinear plant behavior are beyond the scope of EE 380
• In many applications it is possible to develop a linear model that approximates the behavior of the nonlinear system for small variations about an operating point
• Given a linear small signal-plant model, one can design a linear feedback system using methods from EE 380
6
EE 380 Fall 2014Lecture 12.
Nonlinear Plant Representation• From Lecture 4, an arbitrary time-invariant nonlinear
system can be represented by the state space model
7
,
,
x f x u
y g x u
1 1 1 1 ,, , , ( , )
,n n m n
x x u f x ux x u f x u
x x u f x u
1 1 ,, ( , )
,r r
y g x uy g x u
y g x u
EE 380 Fall 2014Lecture 12.
Example 2
• Determine a state-space representation for a pendulum with damping at the pivot point. Is the system linear?
8
M
B
EE 380 Fall 2014Lecture 12.
Static Equilibrium States• Consider the time-invariant nonlinear system
• Drive the system with a constant input
• The constant state vector(s) xe satisfying
are called static equilibrium states
• In many cases, the constant input uo and equilibrium state xe represent a desired operating point
10
( ) ou t u
,x f x u
10 ,n e of x u
EE 380 Fall 2014Lecture 12.
Example 3
• Determine the static equilibrium state(s) of the pendulum system considered in example 2
11
M
B
EE 380 Fall 2014Lecture 12.
Stability of Static Equilibrium States
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• Once in a static equilibrium state, if left unperturbed, the system remains in that state because dx/dt = 0
• The equilibrium state xe is said to be stable if for small perturbations from xe, the system returns to the equilibrium state
• The equilibrium state xe is said to be unstable if for small perturbations from xe, the system diverges from xe
EE 380 Fall 2014Lecture 12.
Example 4
• Determine the stability of the static equilibrium state(s) found in example 3
14
M
B
EE 380 Fall 2014Lecture 12.
EE 380
Linear Control Systems
Lecture 12
Professor Jeffrey SchianoDepartment of Electrical Engineering
1
EE 380 Fall 2014Lecture 12.
Lecture 12 Topics
• Nonlinear System Analysis– Small-signal Analysis– Chapter 9, Sections 9.1 and 9.2
2
EE 380 Fall 2014Lecture 12.
Fundamental Definitions• A system is said to be linear if the following are true
1. The response can be represented as the sum a zero-input and zero-state response
2. The zero-input response obeys the principle of superposition with respect to the initial state
3. The zero-state response obeys the principle of superposition with respect to the input
• If a system does not satisfy all three properties, then it is nonlinear
3
EE 380 Fall 2014Lecture 12.
Control Design for Nonlinear Plants• Most physical systems are inherently nonlinear
• Control design techniques that account for nonlinear plant behavior are beyond the scope of EE 380
• In many applications it is possible to develop a linear model that approximates the behavior of the nonlinear system for small variations about an operating point
• Given a linear small signal-plant model, one can design a linear feedback system using methods from EE 380
6
EE 380 Fall 2014Lecture 12.
Nonlinear Plant Representation• From Lecture 4, an arbitrary time-invariant nonlinear
system can be represented by the state space model
7
EE 380 Fall 2014Lecture 12.
Example 2
• Determine a state-space representation for a pendulum with damping at the pivot point. Is the system linear?
8
EE 380 Fall 2014Lecture 12.
Static Equilibrium States• Consider the time-invariant nonlinear system
• Drive the system with a constant input
• The constant state vector(s) xe satisfying
are called static equilibrium states
• In many cases, the constant input uo and equilibrium state xe represent a desired operating point
10
EE 380 Fall 2014Lecture 12.
Example 3
• Determine the static equilibrium state(s) of the pendulum system considered in example 2
11
EE 380 Fall 2014Lecture 12.
Stability of Static Equilibrium States
13
• Once in a static equilibrium state, if left unperturbed, the system remains in that state because dx/dt = 0
• The equilibrium state xe is said to be stable if for small perturbations from xe, the system returns to the equilibrium state
• The equilibrium state xe is said to be unstable if for small perturbations from xe, the system diverges from xe
EE 380 Fall 2014Lecture 12.
Example 4
• Determine the stability of the static equilibrium state(s) found in example 3
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