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HW13_solution

# HW13_solution - ME 132 1 The state-variable equations for...

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ME 132 Solutions # 13 1 The state-variable equations for the system are ˙ x 1 = x 2 ˙ x 2 = g - c m u 2 x 2 1 y = x 1 (a) The control u needed to maintain y = ζ is found from evaluating the system at equilib- rium. This gives the equilibrium parenleftBiggbracketleftBigg x 1 x 2 bracketrightBigg , u, y parenrightBigg = parenleftBiggbracketleftBigg ζ 0 bracketrightBigg , ζ radicalbigg mg c , ζ parenrightBigg The linear system governing small deviations away from the operating point listed above is ˙ δx 1 ˙ δx 2 δy = 0 1 0 2 g ζ 0 - 2 ζ radicalBig gc m 1 0 0 δx 1 δx 2 δu (b) The observer and controller gains can be found using the standard observer/controller design methods K = bracketleftBigg - 20 - 100 - 2 g ζ bracketrightBigg , L = bracketleftBig - 4 - 2 g ζ - 4 bracketrightBig parenleftbigg - ζ 2 radicalbigg m gc parenrightbigg For the controller equation given (which is not in terms of the deviation variables), F can be found by letting y be equal to r in the steady state u = L ˆ x + ζ radicalbigg mg c = L 1 ζ + F = radicalbigg

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HW13_solution - ME 132 1 The state-variable equations for...

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