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Analysis of Control Systems in State Space

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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

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