Feedback Control Systems ( FCS )

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/

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)

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 (1) • Consider the state equation given below • Taking Laplace transform of the equation (1)

Solution of State Equations • Taking inverse Laplace State Transition Matrix

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

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(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

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

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 (1) • Consider the state equation with u(t) as forcing function • Taking Laplace transform of the equation (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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Feedback Control Systems ( FCS )