Analysis of Control Systems in State Space imtiaz.hussain@faculty.muet.edu.pk

Introduction to State Space The state space is defined as the n-dimensional space in which the components of the state vector represents its coordinate axes. In case of 2nd order system state space is 2-dimensional space with x1 and x2 as its coordinates (Fig-1). Fig-1: Two Dimensional State space

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 t2 t3 t4 t5 t6

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

Solution of State Equations & State Transition Matrix Consider the state space model Solution of this state equation is given as Where is state transition matrix.

Example-1 Consider RLC Circuit Choosing vc and iL as state variables + - iL Vo

Example-1 (cont...)

Example-1 (cont...) State transition matrix can be obtained as Which is further simplified as

Example-1 (cont...) Taking the inverse Laplace transform of each element

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

Example-2 For the RLC circuit of example-1 draw the state space trajectory with following initial conditions. Solution

Example-2 (cont...) Following trajectory is obtained

Example-2 (cont...)

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