Linear Time Invariant systems definitions, Laplace transform, solutions,
Download
Report
Transcript Linear Time Invariant systems definitions, Laplace transform, solutions,
Lavi Shpigelman, Dynamic Systems and control – 76929 –
Linear Time Invariant systems
definitions,
Laplace transform,
solutions,
stability
1
Lavi Shpigelman, Dynamic Systems and control – 67522 – Practical Exercise 1
Ready to GO !?
Disclaimer: the following slides are a quick review of
Linear, Time Invariant systems
a.
b.
c.
If you feel a bit disoriented think:
I can easily read up on it (references will be given)
Its VERY simple if you’ve seen it before (i.e. intuitive).
This part will be over soon.
2
Lavi Shpigelman, Dynamic Systems and control – 76929 –
Lumpedness and causality
Definition: a system is lumped if it can be
described by a state vector of finite
dimension. Otherwise it is called distributed.
Examples:
• distributed system: y(t)=u(t- t)
• lumped system (mass and spring with friction)
Definition: a system is causal if its current
state is not a function of future events (all
‘real’ physical systems are causal)
3
Linearity and Impulse Response
description of linear systems
Lavi Shpigelman, Dynamic Systems and control – 76929 –
Definition: a function f(x) is linear if
(this is known as the superposition property)
Impulse response:
Suppose we have a SISO (Single Input Single Output) system
system as follows:
where:
y(t) is the system’s response (i.e. the observed output) to the
control signal, u(t) .
The system is linear in x(t) (the system’s state) and in u(t)
4
Lavi Shpigelman, Dynamic Systems and control – 76929 –
Linearity and Impulse Response
description of linear systems
Define the system’s impulse response, g(t,), to be the
response, y(t) of the system at time t, to a delta function
control signal at time (i.e. u(t)=t,) given that the
system state at time is zero (i.e. x()=0 )
Then the system response to any u(t) can be found by
solving:
Thus, the impulse response contains all the information
on the linear system
5
Lavi Shpigelman, Dynamic Systems and control – 76929 –
Time Invariance
A system is said to be time invariant if its response to an
initial state x(t0) and a control signal u is independent of
the value of t0.
So g(t,) can be simply described as g(t)=g(t,0)
A time linear time invariant system is said to be causal if
A system is said to be relaxed at time 0 if x(0) =0
A linear, causal, time invariant (SISO) system that is
relaxed at time 0 can be described by
causal
relaxed
Time invariant
Convolution
6
LTI - State-Space Description
Lavi Shpigelman, Dynamic Systems and control – 76929 –
Fact: (instead of using the impulse response representation..)
Every (lumped, noise free) linear, time invariant
(LTI) system can be described by a set of
equations of the form:
Linear, 1st order ODEs
Linear algebraic equations
Dynamic
Process
Observation
Process
Controllable
inputs u
u
D
B
+
1/s
x
Observations
y
+
C
A
State x
Disturbance
(noise) w
Measurement
Error (noise) n
Plant
7
What About
nth Order Linear ODEs?
Lavi Shpigelman, Dynamic Systems and control – 76929 –
Can be transformed into n 1st order ODEs
1. Define new variable:
2. Then:
Dx/dt
=
y = [I 0 0 0] x
A
x
+
B
u
8
Lavi Shpigelman, Dynamic Systems and control – 76929 –
Using Laplace Transform to
Solve ODEs
The Laplace transform is a very useful tool in the
solution of linear ODEs
(i.e. LTI systems).
Definition: the Laplace transform of f(t)
It exists for any function that can be bounded by
aet (and s>a ) and it is unique
The inverse exists as well
Laplace transform pairs are known for many
useful functions (in the form of tables and Matlab
functions)
Will be useful in solving differential equations!
9
Some Laplace Transform
Properties
Lavi Shpigelman, Dynamic Systems and control – 76929 –
Linearity (superposition):
Differentiation
Convolution
Integration
10
Some specific Laplace
Transforms (good to know)
Lavi Shpigelman, Dynamic Systems and control – 76929 –
Constant (or unit step)
Impulse
Exponential
Time scaling
11
Using Laplace Transform to
Analyze a 2nd Order system
Lavi Shpigelman, Dynamic Systems and control – 76929 –
Consider the unforced (homogenous) 2nd order system
To find y(t):
Take the Laplace transform
(to get an algebraic equation in s)
Do some algebra
characteristic polynomial
determined by Initial condition
Find y(t) by taking the inverse transform
12
Lavi Shpigelman, Dynamic Systems and control – 76929 –
2nd Order system Inverse Laplace
The solution of the inverse depends on the
nature of the roots 1,2 of the
characteristic polynomial p(s)=as2+bs+c:
• real & distinct, b2>4ac
• real & equal, b2=4ac
• complex conjugates b2<4ac
In shock absorber example:
a=m, b=damping coeff., c=spring coeff.
We will see:
Re{} exponential effect
Im{} Oscillatory effect
13
Real & Distinct roots (b2>4ac)
Lavi Shpigelman, Dynamic Systems and control – 76929 –
Some algebra helps fit the polynomial to Laplace
tables.
Use linearity, and a table entry
To conclude:
p(s)=s2+3s+1
y(0)=1,y’(0)=0
1=-2.62
2=-0.38
• Sign{} growth or decay
• || rate of growth/decay
y(t)=-0.17e-2.62t+1.17e-0.38t
14
Real & Equal roots (b2=4ac)
Lavi Shpigelman, Dynamic Systems and control – 76929 –
Some algebra helps fit the polynomial to Laplace
tables.
Use linearity, and a some table entries to
conclude:
p(s)=s2+2s+1
y(0)=1,y’(0)=0
1=-1
• Sign{} growth or decay
• || rate of growth/decay
y(t)=-e-t+te-t
15
Complex conjugate roots (b2<4ac)
Lavi Shpigelman, Dynamic Systems and control – 76929 –
Some algebra helps fit the polynomial to Laplace
tables.
Use table entries (as before) to conclude:
Reformulate y(t) in terms of and
Where:
16
Lavi Shpigelman, Dynamic Systems and control – 76929 –
Complex roots (b2<4ac)
For p(s)=s2+0.35s+1 and initial condition y(0)=1,y’(0)=0
The roots are =+i=-0.175±i0.9846
The solution has form:
and the constants are
A=||=1.0157
r=0.5-i0.0889
=arctan(Im(r)/Re(r))
=-0.17591
We see the solution
is an exponentially
decaying oscillation
where the decay is
governed by and
the oscillation by .
17
The “Roots” of a Response
Lavi Shpigelman, Dynamic Systems and control – 76929 –
Im(s)
Marginally
Stable
Stable
Unstable
Re(s)
18
(Optional) Reading List
LTI systems:
Lavi Shpigelman, Dynamic Systems and control – 76929 –
• Chen, 2.1-2.3
Laplace:
• http://www.cs.huji.ac.il/~control/handouts/laplace_Boyd.pdf
• Also, Chen, 2.3
2nd order LTI system analysis:
• http://www.cs.huji.ac.il/~control/handouts/2nd_order_Boyd.pdf
Linear algebra (matrix identities and
eigenstuff)
• Chen, chp. 3
• Stengel, 2.1,2.2
19