1. Consider a robotic cart that tries to balance a pendulum at the vertical position by...

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1. Consider a robotic cart that tries to balance a pendulum at the vertical position by appropriate control u as shown in Figure 1. Let H and V be, respectively, the horizontal and vertical forces exerted by the cart to the pendulum. H-> M tv OO Figure 1. Inverted Pendulum on a Cart (a) Derive the dynamical equations of this system and show that they are nonlinear. (b) Under the assumption that is small, show that the dynamical equations of the system can be represented by two coupled linear differential equations as (M+m) ÿ+ m lö=u mfö+mly-mg10 (c) Find the transfer functions of the pendulum 0(s)/U(s) and Y(s)/U(s). Obtain the poles of the system with respect to each transfer function and show that the system is open loop unstable. (d) Write the state equations of the system using the coupled differential equations from part (b) by defining the state variables 0=x1, 6=x2, y=x3, y=x4. (e) Use a state feedback control law u=v+kx to stabilize the system with desired eigenvalues -1, -2, -3, -4 where M-2 kg, m=0.1 kg, 1=0.5 m. Use command "place" from MATLAB to perform the task. Note that u=v-k x is used in MATLAB to obtain k. 1. Consider a robotic cart that tries to balance a pendulum at the vertical position by appropriate control u as shown in Figure 1. Let H and V be, respectively, the horizontal and vertical forces exerted by the cart to the pendulum. H-> M tv OO Figure 1. Inverted Pendulum on a Cart (a) Derive the dynamical equations of this system and show that they are nonlinear. (b) Under the assumption that is small, show that the dynamical equations of the system can be represented by two coupled linear differential equations as (M+m) ÿ+ m lö=u mfö+mly-mg10 (c) Find the transfer functions of the pendulum 0(s)/U(s) and Y(s)/U(s). Obtain the poles of the system with respect to each transfer function and show that the system is open loop unstable. (d) Write the state equations of the system using the coupled differential equations from part (b) by defining the state variables 0=x1, 6=x2, y=x3, y=x4. (e) Use a state feedback control law u=v+kx to stabilize the system with desired eigenvalues -1, -2, -3, -4 where M-2 kg, m=0.1 kg, 1=0.5 m. Use command "place" from MATLAB to perform the task. Note that u=v-k x is used in MATLAB to obtain k.