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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 bycausal

relaxed ConvolutionTime invariant

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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, 1st order ODEs

Linear algebraic equations

Controllable inputs u

State xDisturbance

(noise) wMeasurement Error (noise) n

Observationsy

Plant

Dynamic Process

A

B+

ObservationProcess

C

D+x

u

1/s

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

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What About nth Order Linear ODEs? Can be transformed into n 1st 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 aet (and s>a ) and it is unique

The inverse exists as wellLaplace 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, thend= 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) 1st 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

1st 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 2nd order system

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Using Laplace Transform to Analyze a 2nd Order system Consider the autonomous (homogenous) 2nd 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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2nd 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 effectIm{} Oscillatory effect

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Real & Distinct roots (b2>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+1y(0)=1,y’(0)=01=-2.622=-0.38

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

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Real & Equal roots (b2=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+1y(0)=1,y’(0)=01=-1

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

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Complex conjugate roots (b2<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 constantsA=||=1.0157r=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 (b2<4ac)

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The “Roots” of a Response

Stable

MarginallyStable

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 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