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**Analysis of Control Systems in State Space**

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**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

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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

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**Forced and Unforced Response**

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

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**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.

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

+ - iL Vo

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Example-1 (cont...)

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**Example-1 (cont...) State transition matrix can be obtained as**

Which is further simplified as

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

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**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

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

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

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Example-2 (cont...)

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

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