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(Solved): Table 2: Problem specific parameters. In the end, you have 4 states x=(x,x^(),\theta ,\theta ^() ...



Table 2: Problem specific parameters. In the end, you have 4 states x=(x,x^(?),\theta ,\theta ^(?)) and 1 input u=F. and assume that you can measure all the states. a\theta =\pi +\phi and derive the linearized equations in terms of \phi assuming that all the angular displacements and angular rates are sufficiently small such that (M+m)tilde(x)+bx^(?)-ml\phi ^(¨)=u, (I+ml^(2))\phi ^(¨)-mgl\phi (t)ml=0. bG_(1)(s)=\phi (s)/(u)(s) and G_(2)(s)=x(s)/(u)(s), draw the root locus (either manually or by using Matlab rlocus function) clearly for each of them. cG_(1)(s)=\phi (s)/(u)(s) in the form C(s)=K((s+a)(s+b))/(s) Adjust parameters a,b such that the root locus passes through points s=-1+-j0.5 (Hint: use angle condition). Now, determine the value of gain K to ensure conjugate poles placed on points s=-1+-j0.5 as well (Hint: use magnitude condition). d\Delta t=0.01{:x_(0),x_(0)^(?),\theta _(0),\theta _(0)^(?))=(0,0,(\pi )/(18),0). eG_(1)(s) you found. Use the same solver settings in part (d) and sufficiently large time span. Give a zero reference input to your closed-loop system and plot the rod angle. You should expect that the controller will drive rod angle from (\pi )/(18)10\deg


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