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

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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 2 nd order system state space is 2-dimensional space with x 1 and x 2 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( x 1, x 2 ) t0t0 t1t1 t2t2 t3t3 t4t4 t5t5 t6t6

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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 v c and i L as state variables VcVc + - + - VoVo iLiL

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Example-1 (cont...)

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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(t o ) 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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