This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: ME 132
1 The statevariable 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 ...
View
Full
Document
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.
 Spring '08
 Tomizuka

Click to edit the document details