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Transfer Matrix and Solutions of State Equations: FCS Lecture 30-31

In this lecture, Dr. Imtiaz Hussain delves into the concepts of transfer matrices and the solution of state equations. Topics include converting state space models to transfer function models using Laplace transforms, understanding forced and unforced responses, and deriving the state transition matrix for RLC circuits. By exploring practical examples, students will gain insights into the dynamic behavior of systems in state space, emphasizing the significance of initial conditions and trajectories. Further resources and detailed lecture notes are available online.

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Transfer Matrix and Solutions of State Equations: FCS Lecture 30-31

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  1. Feedback Control Systems (FCS) Lecture-30-31 Transfer Matrix and solution of state equations Dr. Imtiaz Hussain email: imtiaz.hussain@faculty.muet.edu.pk URL :http://imtiazhussainkalwar.weebly.com/

  2. Transfer Matrix (State Space to T.F) (1) (3) • Now Let us convert a space model to a transfer function model. • Taking Laplace transform of equation (1) and (2) considering initial conditions to zero. • From equation (3) (2) (4) (5)

  3. Transfer Matrix (State Space to T.F) • Substituting equation (5) into equation (4) yields

  4. Example#1 • Convert the following State Space Model to Transfer Function Model if K=3, B=1 and M=10;

  5. Example#1 • Substitute the given values and obtain A, B, C and D matrices.

  6. Example#1

  7. Example#1

  8. Example#1

  9. Example#1

  10. Example#1

  11. Example#2 • Obtain the transfer function T(s) from following state space representation. Answer

  12. Forced and Unforced Response • Forced Response, with u(t) as forcing function • Unforced Response (response due to initial conditions)

  13. Solution of State Equations (1) • Consider the state equation given below • Taking Laplace transform of the equation (1)

  14. Solution of State Equations • Taking inverse Laplace State Transition Matrix

  15. Example-3 • Consider RLC Circuit obtain the state transition matrix ɸ(t). Vo iL + + Vc - -

  16. Example-3 (cont...) • State transition matrix can be obtained as • Which is further simplified as

  17. Example-3 (cont...) • Taking the inverse Laplace transform of each element

  18. Example#4 • Compute the state transition matrix if Solution

  19. State Space Trajectories • The unforced response of a system released from any initial point x(to)traces a curve or trajectory in state space, with time t as an implicit function along the trajectory. • Unforced system’s response depend upon initial conditions. • Response due to initial conditions can be obtained as

  20. State Transition • Any point P in state space represents the state of the system at a specific time t. • State transitions provide complete picture of the system P(x1,x2) t0 t1 t6 t2 t3 t5 t4

  21. Example-5 • For the RLC circuit of example-3 draw the state space trajectory with following initial conditions. • Solution

  22. Example-5 (cont...) • Following trajectory is obtained

  23. Example-5 (cont...)

  24. Equilibrium Point • The equilibrium or stationary state of the system is when

  25. Solution of State Equations (1) • Consider the state equation with u(t) as forcing function • Taking Laplace transform of the equation (1)

  26. Solution of State Equations • Taking the inverse Laplace transform of above equation. Natural Response Forced Response

  27. Example#6 • Obtain the time response of the following system: • Where u(t) is unit step function occurring at t=0. consider x(0)=0. Solution • Calculate the state transition matrix

  28. Example#6 • Obtain the state transition equation of the system

  29. To download this lecture visit http://imtiazhussainkalwar.weebly.com/ End of Lectures-30-31

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