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lec11_print

# lec11_print - d dt ± X 1 X 2 ² = ± 1 1 0 1 ²± X 1 X 2...

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Process Outputs Srinivas Palanki University of South Alabama Srinivas Palanki (USA) Process Outputs 1 / 7

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Process Outputs Definition Very often, we are not interested in calculating the response for the entire state vector, x ( t ); we may care about only a few variables that are critical to the operation of the process. Such variables are called Outputs . In general, the outputs are algebraic functions of the states, x and the inputs, u , and are represented by the vector, y . y = g ( x , u ) (1) At steady state y s = g ( x s , u s ) (2) We define deviation variables for the outputs as Y = y - y s (3) Srinivas Palanki (USA) Process Outputs 2 / 7
Process Outputs Using the linearization formula, a linearized representation of the outputs is given by Y = CX + DU (4) where C = g x x s , u s D = g u x s , u s (5) Srinivas Palanki (USA) Process Outputs 3 / 7

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Process Outputs Illustrative Example 1 Linearize the following nonlinear system around the steady state (0 , 0) dx 1 dt = x 2 1 + x 1 + x 2 + u dx 2 dt = x 2 1 x 2 + x 2 + u y = x 2 (6) Srinivas Palanki (USA) Process Outputs 4 / 7
Process Outputs Solution A = 1 1 0 1 B = 1 1 C = 0 1 D

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Unformatted text preview: d dt ± X 1 X 2 ² = ± 1 1 0 1 ²± X 1 X 2 ² + ± 1 1 ² U Y = ³ 0 1 ´ ± X 1 X 2 ² + 0 . U (7) Srinivas Palanki (USA) Process Outputs 5 / 7 Process Outputs Dynamics of a Linear System with Process Outputs A linear dynamical system in deviation form is represented as dX dt = AX + BU Y = CX + DU X (0) = X (8) where X = x-x s U = u-u s Y = y-y s (9) Srinivas Palanki (USA) Process Outputs 6 / 7 Process Outputs We saw in the previously that the dynamics of the states, X ( t ), is given by X ( t ) = e At . X (0) + e At . Z t e-At BU ( t ) dt (10) The dynamics of the outputs is given by Y ( t ) = C . e At . X (0) + C . e At . Z t e-At BU ( t ) dt + D . U (11) Unforced Dynamics: Y ( t ) = C . e At . X (0) (12) Forced Dynamics: Y ( t ) = C . e At . Z t e-At BU ( t ) dt + D . U (13) Srinivas Palanki (USA) Process Outputs 7 / 7...
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lec11_print - d dt ± X 1 X 2 ² = ± 1 1 0 1 ²± X 1 X 2...

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