1. 1 Lavi Shpigelman, Dynamic Systems and control – 76929 – Linear Time Invariant systems  definitions,  Laplace transform,  solutions,  stability

2. Lavi Shpigelman, Dynamic Systems and control – 67522 – Practical Exercise 1 2 Ready to GO !? Disclaimer: the following slides are a quick review of Linear, Time Invariant systems If you feel a bit disoriented think: a.I can easily read up on it (references will be given) b.Its VERY simple if you’ve seen it before (i.e. intuitive). c.This part will be over soon.

3. 3 Lavi Shpigelman, Dynamic Systems and control – 76929 – 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)

4. 4 Lavi Shpigelman, Dynamic Systems and control – 76929 – 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)

5. 5 Lavi Shpigelman, Dynamic Systems and control – 76929 – 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

6. 6 Lavi Shpigelman, Dynamic Systems and control – 76929 – Time Invariance  A system is said to be time invariant if its response to an initial state x(t 0 ) and a control signal u is independent of the value of t 0. 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 ConvolutionTime invariant

7. 7 Lavi Shpigelman, Dynamic Systems and control – 76929 – 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..)

8. 8 Lavi Shpigelman, Dynamic Systems and control – 76929 – 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

9. 9 Lavi Shpigelman, Dynamic Systems and control – 76929 – 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 i t 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!

10. 10 Lavi Shpigelman, Dynamic Systems and control – 76929 – Some Laplace Transform Properties  Linearity (superposition):  Differentiation

11. 11 Lavi Shpigelman, Dynamic Systems and control – 76929 –  Remember integration by parts:  Using that and the transform definition:

12. 12 Lavi Shpigelman, Dynamic Systems and control – 76929 – Some Laplace Transform Properties  Linearity (superposition):  Differentiation  Convolution

13. 13 Lavi Shpigelman, Dynamic Systems and control – 76929 –  Using definitions  Integration over triangle 0 <  < t  Define  t , then  d  = dt and region is  t 

14. 14 Lavi Shpigelman, Dynamic Systems and control – 76929 – Some Laplace Transform Properties  Linearity (superposition):  Differentiation  Convolution  Integration

15. 15 Lavi Shpigelman, Dynamic Systems and control – 76929 –  By definition:  Switch integration order  Plug  = t- 

16. 16 Lavi Shpigelman, Dynamic Systems and control – 76929 – Some specific Laplace Transforms (good to know)  Constant (or unit step)  Impulse  Exponential  Time scaling

17. 17 Lavi Shpigelman, Dynamic Systems and control – 76929 – 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

18. 18 Lavi Shpigelman, Dynamic Systems and control – 76929 –  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

19. 19 Lavi Shpigelman, Dynamic Systems and control – 76929 –  Solve:  Do the Laplace transform  Do simple algebra  Take inverse transform Example: a 2 nd order system

20. 20 Lavi Shpigelman, Dynamic Systems and control – 76929 – 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

21. 21 Lavi Shpigelman, Dynamic Systems and control – 76929 – 2 nd Order system - Inverse Laplace  Solution of inverse transform depends on nature of the roots 1, 2 of the characteristic polynomial p(s)=as 2 +bs+c : real & distinct, b 2 >4ac real & equal, b 2 =4ac complex conjugates b 2 <4ac  In shock absorber example: a=m, b= damping coeff., c= spring coeff.  We will see: Re{ }  exponential effect Im{ }  Oscillatory effect

22. 22 Lavi Shpigelman, Dynamic Systems and control – 76929 – 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)=s 2 +3s+1 y(0)=1,y’(0)=0 1 =-2.62 2 =-0.38 y(t)=-0.17e -2.62t +1.17e -0.38t

23. 23 Lavi Shpigelman, Dynamic Systems and control – 76929 – 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)=s 2 +2s+1 y(0)=1,y’(0)=0 1 =-1 y(t)=-e -t +te -t

24. 24 Lavi Shpigelman, Dynamic Systems and control – 76929 – 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:

25. 25 Lavi Shpigelman, Dynamic Systems and control – 76929 –  E.g. p(s)=s 2 +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 )

26. 26 Lavi Shpigelman, Dynamic Systems and control – 76929 – The “Roots” of a Response Stable Marginally Stable Unstable Re(s) Im(s)

27. 27 Lavi Shpigelman, Dynamic Systems and control – 76929 – (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  Linear algebra (matrix identities and eigenstuff) Chen, chp. 3 Stengel, 2.1,2.2