Linear Time Invariant systems

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Lavi Shpigelman, Dynamic Systems and control – 76929 – 1

Linear Time Invariant systems

 definitions,

 Laplace transform,

 solutions,

 stability

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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)

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Linearity and Impulse Response description of linear systems

 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)

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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

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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,)

 A 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

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LTI - State-Space Description

 Every (lumped, noise free) linear, time invariant (LTI) system can be described by a set of

equations of the form:

Linear, 1^st order ^ODEs

Linear algebraic equations

Controllable inputs u

State x Disturbance

(noise) w Measurement

Error (noise) n

Observations y

Plant Dynamic

Process

A B

+

Observation Process

C D x +

u 1/s

Fact: (instead of using the impulse response representation..)

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What About

n

^th

Order Linear ODEs?

 Can be transformed into n 1

^st

order ODEs

1. Define new variable:

2. Then:

Dx/dt = A x + B u y = [I 0 0  0] x

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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 ae

^t

(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!

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Some Laplace Transform Properties

 Linearity (superposition):

 Differentiation

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 Remember integration by parts:

 Using that and the transform definition:

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Some Laplace Transform Properties

 Linearity (superposition):

 Differentiation

 Convolution

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 Using definitions

 Integration over triangle 0 < < t

 Define  t, thend  = dt and region is t 

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Some Laplace Transform Properties

 Linearity (superposition):

 Differentiation

 Convolution

 Integration

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 By definition:

 Switch integration order

 Plug = t-

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Some specific Laplace

Transforms (good to know)

 Constant (or unit step)

 Impulse

 Exponential

 Time scaling

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Homogenous

(aka Autonomous / no input)

1

^st

order linear ODE

 Solve:

 Do the Laplace transform

 Do simple algebra

 Take inverse transform

Known as zero input response

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 Solve:

 Do the Laplace transform

 Do simple algebra

 Take inverse transform

1

^st

order linear ODE

with input (non-homogenous)

Known as the zero state response

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 Solve:

 Do the Laplace transform

 Do simple algebra

 Take inverse transform

Example: a 2

^nd

order system

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Using Laplace Transform to Analyze a 2

^nd

Order system

 Consider the autonomous (homogenous) 2^nd order system

 To find y(t), take the Laplace transform (to get an algebraic equation in s)

 Do some algebra

 Find y(t) by taking the inverse transform

characteristic polynomial determined by Initial condition

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2

^nd

Order system - Inverse Laplace

 Solution of inverse transform depends on 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

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Real & Distinct roots (b

²

>4ac)

 Some algebra helps fit the polynomial to Laplace tables.

 Use linearity, and a table entry To conclude:

• Sign{}  growth or decay

• ||  rate of growth/decay

p(s)=s²+3s+1 y(0)=1,y’(0)=0

₁=-2.62

₂=-0.38

y(t)=-0.17e^-2.62t+1.17e^-0.38t

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Real & Equal roots (b

²

=4ac)

 Some algebra helps fit the polynomial to Laplace tables.

 Use linearity, and a some table entries to conclude:

• Sign{}  growth or decay

• ||  rate of growth/decay

p(s)=s²+2s+1 y(0)=1,y’(0)=0

₁=-1

y(t)=-e^-t+te^-t

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Complex conjugate roots (b

²

<4ac)

 Some algebra helps fit the polynomial to Laplace tables.

 Use table entries (as before) to conclude:

 Reformulate y(t) in terms of  and 

Where:

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 E.g. p(s)=s²+0.35s+1 and initial condition y(0)=1 , y’(0)=0

 Roots are =+i=-0.175±i0.9846

 Solution has form:

with constants A=||=1.0157 r=0.5-i0.0889

=arctan(Im(r)/Re(r)) =-0.17591

 Solution is an exponentially

decaying oscillation

 Decay governed by  oscillation by 

Complex conjugate roots (b

²

<4ac)

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The “Roots” of a Response

Stable

Marginally Stable

Unstable

Re(s) Im(s)

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(Optional) Reading List

 LTI systems:

• Chen, 2.1-2.3

 Laplace:

• http://www.cs.huji.ac.il/~control/handouts/laplace_Boyd.pdf

• Also, Chen, 2.3

 2

^nd

order LTI system analysis:

• http://www.cs.huji.ac.il/~control/handouts/2nd_order_Boyd.pdf