Feedback Control Systems (FCS) Dr. Imtiaz Hussain URL : Lecture Transfer Matrix and solution of state equations

Transfer Matrix (State Space to T.F) 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) (1) (2) (3) (4) (5)

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

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

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

Example#1

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

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

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

Solution of State Equations Taking inverse Laplace State Transition Matrix

Example-3 Consider RLC Circuit obtain the state transition matrix ɸ(t). VcVc VoVo iLiL

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

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

Example#4 Compute the state transition matrix if Solution

State Space Trajectories The unforced response of a system released from any initial point x(t o ) 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

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( x 1, x 2 ) t0t0 t1t1 t2t2 t3t3 t4t4 t5t5 t6t6

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

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

Example-5 (cont...)

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

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

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

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

Example#6 Obtain the state transition equation of the system

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