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

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Unformatted text preview: ME 132 1 The state-variable equations for the system are x1 = x2 ˙ x2 = g − ˙ y = x1 c u2 m x2 1 Solutions # 13 (a) The control u needed to maintain y = ζ is found from evaluating the system at equilibrium. This gives the equilibrium x1 x2 , u, y = ζ 0 ,ζ mg ,ζ c The linear system governing small deviations away from the operating point listed above is ˙ 0 01 δx1 δx1 ˙ 2g gc 2 δx2 = ζ 0 − ζ m δx2 δy δu 10 0 (b) The observer and controller gains can be found using the standard observer/controller design methods K= −20 −100 − 2g ζ , L= −4 − 2g ζ −4 − ζ 2 m gc 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 = Lx + F ζ ˆ ζ mg = L1 ζ + F ζ c mg − 2ζ ⇒F = c m F = −2ζ gc m − gc mg c The Simulink block diagrams are given below. The second block diagram is a subsystem for the actual plant and the third block diagram is a subsystem for the observer. The Matlab Fcn blocks contain expressions for the dynamics and controller/observer gains. 1 y In1 Out1 To Workspace1 Plant In1 Out1 Obs MATLAB Function MATLAB Fcn ref u To Workspace From Workspace r To Workspace2 x2 To Workspace 1 In 1 Mux Mux MATLAB Function MATLAB Fcn 1 s Integrator 1 s Integrator 1 1 Out 1 2 1 In 1 y_hat To Workspace1 MATLAB Function MATLAB Fcn 1 s x1 1 s x2 1 Out1 delta _x1_hat To Workspace2 x2_hat To Workspace The response of the system is given below. 1.6 Ball height Reference 1.4 1.2 meters 1 0.8 0.6 0.4 0.2 0 10 20 Time (sec) 30 40 50 3 ...
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This note was uploaded on 10/20/2010 for the course ME 132 taught by Professor Tomizuka during the Spring '08 term at University of California, Berkeley.

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

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