Analysis of Control Systems in State Space

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

2. 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

3. 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

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

5. 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.

6. Example-1 • Consider RLC Circuit • Choosing vc and iL as state variables iL + + Vc Vo - -

7. Example-1 (cont...)

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

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

10. 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

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

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

13. Example-2 (cont...)

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