Problem 4 (23 points). Motorized cart. Consider a pendulum of mass m and length 1 on...

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Problem 4 (23 points). Motorized cart. Consider a pendulum of mass m and length 1 on a motorized cart of mass M, as shown in Figure Suppose that an external force u is being applied horizontally on the cart. Let s denote the cart's position and the pendulum's angle as depicted in Figure 1. The (nonlinear) equations of motion for this system are given by where g denotes the standard gravitational acceleration on the surface of the Earth. Linearizing the equations of motion around 0 (for which sin 0) and rearranging terms, we find that the linearized model of this system is given by S (M +m)s - mle cos 0 + m 6 sin 0 = u 10 g sin = = $ cos 0, M m mg 1 -0 + U M M = (M + m)gg + MU S = 1. (3 points) Determine the state equations of this system with state x = 1 MU U. (external force) Figure 1: Inverted pendulum on a motorized cart. 2. (3 points) Is the system controllable? 3. (3 points) Is the system observable when we only measure the cart's position? What about if we only measure the pendulum's angle? 4. (16 points) Consider the following three closed-loop schemes: (i) Full-state feedback. (ii) An observer-based controller measuring both cart position and pendulum angle. (iii) An observer-based controller measuring only cart position. Let m= 1kg, M = 5 kg, l = 2m, and g = 9.8 ms-2. (a) (5 points) Design the schemes (i), (ii), (iii) by solving a suitable LQR problem for each scheme, so that they each stabilize the closed-loop system. For (ii) and (iii), design only the observer gain and reuse the feedback gain designed for (i). Choose the LQR parameters as simple as possible, while still ensuring stabilization. In Matlab, you can use the lqr (A,B,C,Q,R, N) function from the control toolbox, and in Python you can use control. lqr(A, B,Q,R,N) from the control library. Hint: Use Problem 4f and recall the separation principle. (b) (1 point) Compute the poles of the closed-loop systems. (c) (6 points) Simulate the three linearized closed-loop systems and generate 3 figures: one for the position, one for the angle, and one for the external force, where each figure overlays the results of the three schemes. For initial conditions, assume that the system is initially at rest (initial velocities at zero), in position s(0) = 2m and angle 0(0) = 30 rad. For the state observer, use (0) = [0, 0, 0, 0]T. Plot over a time interval of 20 seconds. You can use ode45 in Matlab or scipy.integrate. solve_ivp in Python (from the scipy library). Alternatively, you can discretize the system via sampling as learned in class, but make the sampling period no larger than T, = 0.1 s. (d) (2 points) Comment on the results of your simulations and its relationship with the poles. Problem 4 (23 points). Motorized cart. Consider a pendulum of mass m and length 1 on a motorized cart of mass M, as shown in Figure Suppose that an external force u is being applied horizontally on the cart. Let s denote the cart's position and the pendulum's angle as depicted in Figure 1. The (nonlinear) equations of motion for this system are given by where g denotes the standard gravitational acceleration on the surface of the Earth. Linearizing the equations of motion around 0 (for which sin 0) and rearranging terms, we find that the linearized model of this system is given by S (M +m)s - mle cos 0 + m 6 sin 0 = u 10 g sin = = $ cos 0, M m mg 1 -0 + U M M = (M + m)gg + MU S = 1. (3 points) Determine the state equations of this system with state x = 1 MU U. (external force) Figure 1: Inverted pendulum on a motorized cart. 2. (3 points) Is the system controllable? 3. (3 points) Is the system observable when we only measure the cart's position? What about if we only measure the pendulum's angle? 4. (16 points) Consider the following three closed-loop schemes: (i) Full-state feedback. (ii) An observer-based controller measuring both cart position and pendulum angle. (iii) An observer-based controller measuring only cart position. Let m= 1kg, M = 5 kg, l = 2m, and g = 9.8 ms-2. (a) (5 points) Design the schemes (i), (ii), (iii) by solving a suitable LQR problem for each scheme, so that they each stabilize the closed-loop system. For (ii) and (iii), design only the observer gain and reuse the feedback gain designed for (i). Choose the LQR parameters as simple as possible, while still ensuring stabilization. In Matlab, you can use the lqr (A,B,C,Q,R, N) function from the control toolbox, and in Python you can use control. lqr(A, B,Q,R,N) from the control library. Hint: Use Problem 4f and recall the separation principle. (b) (1 point) Compute the poles of the closed-loop systems. (c) (6 points) Simulate the three linearized closed-loop systems and generate 3 figures: one for the position, one for the angle, and one for the external force, where each figure overlays the results of the three schemes. For initial conditions, assume that the system is initially at rest (initial velocities at zero), in position s(0) = 2m and angle 0(0) = 30 rad. For the state observer, use (0) = [0, 0, 0, 0]T. Plot over a time interval of 20 seconds. You can use ode45 in Matlab or scipy.integrate. solve_ivp in Python (from the scipy library). Alternatively, you can discretize the system via sampling as learned in class, but make the sampling period no larger than T, = 0.1 s. (d) (2 points) Comment on the results of your simulations and its relationship with the poles.