1. Analysis of Control Systems in State Space

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 t2 t3 t4 t5 t6

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 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