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# lec8_print - Analysis of Linear System Dynamics Srinivas...

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Analysis of Linear System Dynamics Srinivas Palanki University of South Alabama Srinivas Palanki (USA) Analysis of Linear System Dynamics 1 / 16

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Dynamics of a Linear System Recap A linear dynamical system in deviation form is represented as dX dt = AX + BU X (0) = X 0 (1) where X = x - x s U = u - u s (2) Srinivas Palanki (USA) Analysis of Linear System Dynamics 2 / 16
Dynamics of a Linear System We saw in the previous lecture that the solution of the above diﬀerential equation is given by X ( t ) = e At . X (0) + e At . Z t 0 e - At BU ( t ) dt (3) It is clear from the above equation that X ( t ) is aﬀected by: the initial conditions , X (0) the input vector , U ( t ) We study these eﬀects separately Srinivas Palanki (USA) Analysis of Linear System Dynamics 3 / 16

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Unforced Dynamics Unforced Dynamics Consider the system represented by eq. (1) in the previous slide. Suppose that: U ( t ) = 0 (the inputs are at their steady state values u s ) X (0) 6 = 0 (the initial conditions are not at their steady state values) How does X ( t ) change with time? Do the system states go to their steady state values, x s ? If so, how long does this take? Srinivas Palanki (USA) Analysis of Linear System Dynamics 4 / 16
Unforced Dynamics Since X ( t ) = e At . X (0) + e At . Z

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lec8_print - Analysis of Linear System Dynamics Srinivas...

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