This project is to model and simulate a Cart-Ball control system, as shown in Figure 1 The detailed procedure is described as follows: 1. Derive a dynamic equation of this nonlinear system through the Lagrange Equation: whereq), and the inertial matrix and the derivative matrix OP The torque Tedue to gravity is , where the potential energy is Pm2g(R+r)(1- cose). The horizontal push/pull force u on the cart is.considered to.be a single input of the system. Then, the entire second term becomes -m2(R + r)g sin 2. Locally linearize the state-space equation from .x =f(&W tox = Ax + Bu, where x is the 4-dimensional state vector to be predefined and the horizontal force on the cart is considered to be the only input u of the system: 3. If the cart displacementy is the only output to be monitored, test the system to see if it is controllable and observable: 4. Design a state-feedback control u-Kxto place the eigenvalues of the linearized matrix A into (-1,-0.6,-0.4 2); 5. All the parameters of the system are given as follows: M = 5 Kg.. m-2.5 Kg., r = 0.015 m., R = 0.25 m., and g = 9.8 m/sec- 6. The initial condition will be specified in the class 7. Using the above results that you have just derived, program your state-feedback controllaw and the nonlinear real-plant into MatlabT to run the simulation and animation by setting the sampling interval 0.025 sec. This project is to model and simulate a Cart-Ball control system, as shown in Figure 1 The detailed procedure is described as follows: 1. Derive a dynamic equation of this nonlinear system through the Lagrange Equation: whereq), and the inertial matrix and the derivative matrix OP The torque Tedue to gravity is , where the potential energy is Pm2g(R+r)(1- cose). The horizontal push/pull force u on the cart is.considered to.be a single input of the system. Then, the entire second term becomes -m2(R + r)g sin 2. Locally linearize the state-space equation from .x =f(&W tox = Ax + Bu, where x is the 4-dimensional state vector to be predefined and the horizontal force on the cart is considered to be the only input u of the system: 3. If the cart displacementy is the only output to be monitored, test the system to see if it is controllable and observable: 4. Design a state-feedback control u-Kxto place the eigenvalues of the linearized matrix A into (-1,-0.6,-0.4 2); 5. All the parameters of the system are given as follows: M = 5 Kg.. m-2.5 Kg., r = 0.015 m., R = 0.25 m., and g = 9.8 m/sec- 6. The initial condition will be specified in the class 7. Using the above results that you have just derived, program your state-feedback controllaw and the nonlinear real-plant into MatlabT to run the simulation and animation by setting the sampling interval 0.025 sec