1. Linear Time Invariant systems
definitions,
Laplace transform,
solutions,
stability
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2. 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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We assume that the state vector contains all the information needed to describe the future of the system given the future control signal and disturbances.
Systems that are not lumped may be rather simple as in the above example but do not yield to the tools made for the state space description to be introduced later.

3. 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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The linearity of a system allows us to treat it’s response to control inputs at every instant separately (as if the control input is a continuous series of impulses) and then simply sum them up in the integral.

4. 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,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
Lavi Shpigelman, Dynamic Systems and control – –
The linearity of a system allows us to treat it’s response to control inputs at every instant separately (as if the control input is a continuous series of impulses) and then simply sum them up in the integral.
If a system is causal (as we will assume in a minute) then its current state is not a function of future inputs, thus for t<tau g(t,tau)=0.

5. 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 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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In practice, the difference between time invariant to time varying systems is the time constants of system changes. For example a new car in its “running in” period is time varying on the scale of days (till its parts settle down) but is time invariant on a scale of minutes. After the running in period it is relatively time invariant on the scale of years for the next 2-5 years and then it begins to show time dependency again as the rate it which it breaks down accelerates.
The changes from the previous integral are independent. Each can be undone if it’s corresponding assumption doesn’t hold.
causal
Time invariant
Convolution
relaxed

6. 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:
Linear, 1st order ODEs
Linear algebraic equations
Lavi Shpigelman, Dynamic Systems and control – –
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
If the system is not lumped, the state vector is infinitely long (assuming it holds all the information about the system) and therefore can not generally be described by a finite set of equations.
Later we will deal with additive Gaussian white noise in this LTI setting in the context of estimation and control.

7. What About nth Order Linear ODEs?
Can be transformed into n 1st order ODEs
Define new variable:
Then:
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Any linear nth order ODE can similarly be transformed into a set of n 1st order ODEs as seen in the example.
Dx/dt = A x B u y = [I 0 0  0] x

8. 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!
Lavi Shpigelman, Dynamic Systems and control – –

9. Some Laplace Transform Properties
Linearity (superposition):
Differentiation
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10. Remember integration by parts:
Using that and the transform definition:
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11. Some Laplace Transform Properties
Linearity (superposition):
Differentiation
Convolution
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12. Integration over triangle 0 <  < t
Using definitions
Integration over triangle 0 <  < t
Define  = t-t, then ds = dt and region is  > 0, t > 0
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13. Some Laplace Transform Properties
Linearity (superposition):
Differentiation
Convolution
Integration
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14. Switch integration order Plug  = t-
By definition:
Switch integration order
Plug  = t-
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15. Some specific Laplace Transforms (good to know)
Constant (or unit step)
Impulse
Exponential
Time scaling
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16. Homogenous (aka Autonomous / no input) 1st order linear ODE
Solve:
Do the Laplace transform
Do simple algebra
Take inverse transform
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Known as zero input response

17. 1st order linear ODE with input (non-homogenous)
Solve:
Do the Laplace transform
Do simple algebra
Take inverse transform
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Known as the zero state response

18. Example: a 2nd order system
Solve:
Do the Laplace transform
Do simple algebra
Take inverse transform
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19. 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
Lavi Shpigelman, Dynamic Systems and control – –
characteristic polynomial
determined by Initial condition

20. 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 effect Im{} Oscillatory effect
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21. 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
Lavi Shpigelman, Dynamic Systems and control – –
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

22. 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
Lavi Shpigelman, Dynamic Systems and control – –
p(s)=s2+2s+1 y(0)=1,y’(0)=0 1=-1
y(t)=-e-t+te-t

23. 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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24. Complex conjugate roots (b2<4ac)
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=||= r=0.5-i =arctan(Im(r)/Re(r)) =
Solution is an exponentially decaying oscillation
Decay governed by  oscillation by .
Lavi Shpigelman, Dynamic Systems and control – –

25. The “Roots” of a Response
Stable
Marginally Stable
Unstable
Re(s)
Im(s)
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26. (Optional) Reading List
LTI systems:
Chen,
Laplace:
Also, Chen, 2.3
2nd order LTI system analysis:
Linear algebra (matrix identities and eigenstuff)
Chen, chp. 3
Stengel, 2.1,2.2
Lavi Shpigelman, Dynamic Systems and control – –