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# control - EE236A(Fall 2007-08 Lecture 7 Applications in...

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EE236A (Fall 2007-08) Lecture 7 Applications in control optimal input design robust optimal input design pole placement (with low-authority control) 7–1 Linear dynamical system y ( t ) = h 0 u ( t ) + h 1 u ( t 1) + h 2 u ( t 2) + · · · single input/single output: input u ( t ) R , output y ( t ) R h i are called impulse response coefficients finite impulse response (FIR) system of order k : h i = 0 for i > k if u ( t ) = 0 for t < 0 , 2 6 6 6 6 6 4 y (0) y (1) y (2) . . . y ( N ) 3 7 7 7 7 7 5 = 2 6 6 6 6 6 4 h 0 0 0 · · · 0 h 1 h 0 0 · · · 0 h 2 h 1 h 0 · · · 0 . . . . . . . . . . . . . . . h N h N - 1 h N - 2 · · · h 0 3 7 7 7 7 7 5 2 6 6 6 6 6 4 u (0) u (1) u (2) . . . u ( N ) 3 7 7 7 7 7 5 a linear mapping from input to output sequence Applications in control 7–2

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Output tracking problem choose inputs u ( t ) , t = 0 , . . . , M ( M < N ) that minimize peak deviation between y ( t ) and a desired output y des ( t ) , t = 0 , . . . , N , max t =0 ,...,N | y ( t ) y des ( t ) | satisfy amplitude and slew rate constraints: | u ( t ) | ≤ U, | u ( t + 1) u ( t ) | ≤ S as a linear program (variables: w , u (0) , . . . , u ( N ) ): minimize. w subject to w t i =0 h i u ( t i ) y des ( t ) w, t = 0 , . . . , N u ( t ) = 0 , t = M + 1 , . . . , N U u ( t ) U, t = 0 , . . . , M S u ( t + 1) u ( t ) S, t = 0 , . . . , M + 1 Applications in control 7–3 example. single input/output, N = 200 step response 0 100 200 0 1 y des 0 100 200 - 1 0 1 constraints on u : input horizon M = 150 amplitude constraint | u ( t ) | ≤ 1 . 1 slew rate constraint | u ( t ) u ( t 1) | ≤ 0 . 25 Applications in control 7–4
output and desired output: y ( t ) , y des ( t ) 0 100 200 - 1 0 1 optimal input sequence u : u ( t ) 0 100 200 - 1 . 1 0 . 0 1 . 1 u ( t ) - u ( t - 1) 0 100 200 - 0 . 25 0 . 00 0 . 25 Applications in control 7–5 Robust output tracking (1) impulse response is not exactly known; it can take two values: ( h (1) 0 , h (1) 1 , . . . , h (1) k ) , ( h (2) 0 , h (2) 1 , . . . , h (2) k ) design an input sequence that minimizes the worst-case peak tracking error minimize w subject to w t i =0 h (1) i u ( t i ) y des ( t ) w, t = 0 , . . . , N w t i =0 h (2) i u ( t i ) y des ( t ) w, t = 0 , . . . , N u ( t ) = 0 , t = M + 1 , . . . , N U u ( t ) U, t = 0 , . . . , M S u ( t + 1) u ( t ) S, t = 0 , . . . , M + 1 an LP in the variables w , u (0) , . . . , u ( N )

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control - EE236A(Fall 2007-08 Lecture 7 Applications in...

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