Linear%20Time%20Invariant%20systems

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Linear Time Invariant systems

• definitions,
• Laplace transform,
• solutions,
• stability

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Ready to GO !?

• Disclaimer the following slides are a quick
  review of Linear, Time Invariant systems
• If you feel a bit disoriented think
• I can easily read up on it (references will be
  given)
• Its VERY simple if youve seen it before (i.e.
  intuitive).
• This part will be over soon.

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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 systems response (i.e. the observed
  output) to the control signal, u(t) .
• The system is linear in x(t) (the systems state)
  and in u(t)

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Linearity and Impulse Response description of
linear systems

• Define the systems 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,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,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

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LTI - State-Space Description
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

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What About nth Order Linear ODEs?

• Can be transformed into n 1st order ODEs
• Define new variable
• 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 sgta ) 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
• Convolution
• Integration

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Some specific Laplace Transforms (good to know)

• Constant (or unit step)
• Impulse
• Exponential
• Time scaling

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Using Laplace Transform to Analyze a 2nd Order
system

• 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
• Find y(t) by taking the inverse transform

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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)as2bsc
• real distinct, b2gt4ac
• real equal, b24ac
• complex conjugates b2lt4ac
• In shock absorber exampleam, bdamping coeff.,
  cspring coeff.
• We will see Re?? exponential effectIm??
  Oscillatory effect

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Real Distinct roots (b2gt4ac)

• 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

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Real Equal roots (b24ac)

• 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

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Complex conjugate roots (b2lt4ac)

• 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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Complex roots (b2lt4ac)

• For p(s)s20.35s1 and initial condition
  y(0)1,y(0)0
• The roots are ??i?-0.175i0.9846
• The solution has formand the constants
  areA?1.0157r0.5-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 ?.

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The Roots of a Response
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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_ord
  er_Boyd.pdf
• Linear algebra (matrix identities and eigenstuff)
• Chen, chp. 3
• Stengel, 2.1,2.2