Multi-linear Systems and Invariant Theory in the Context of Computer Vision and Graphics
Linear Time Invariant systems
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Transcript of Linear Time Invariant systems
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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