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

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Unformatted text preview: EE236A (Fall 2007-08) Lecture 7 Applications in control optimal input design robust optimal input design pole placement (with low-authority control) 71 Linear dynamical system y ( t ) = h 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 < , 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 h 1 h h 2 h 1 h . . . . . . . . . . . . . . . h N h N- 1 h N- 2 h 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 72 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 73 example. single input/output, N = 200 step response 100 200 1 y des 100 200- 1 1 constraints on u : input horizon M = 150 amplitude constraint | u ( t ) | 1 . 1 slew rate constraint | u ( t ) u ( t 1) | . 25 Applications in control 74 output and desired output: y ( t ) , y des ( t ) 100 200- 1 1 optimal input sequence u : u ( t ) 100 200- 1 . 1 . 1 . 1 u ( t )- u ( t- 1) 100 200- . 25 . 00 . 25 Applications in control 75 Robust output tracking (1) impulse response is not exactly known; it can take two values: ( h (1) , h (1) 1 , . . . , h (1) k ) , ( h (2) , h (2) 1 , . . . , h (2) k ) design an input sequence that minimizes the worst-case peak tracking...
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control - EE236A (Fall 2007-08) Lecture 7 Applications in...

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