ENGG2440 — Modelling and Control Viva Voce Major Assignment — Part 2 For Extra Help: E-mail: XXXXXXXXXX HelpDesk: ES204, Fridays, 11am-1pm Learning Outcomes The Viva Voce major assignment will assess...

hi it's a assignment based viva i need everything like how to presenting it what type of question they can ask me and need run file of simulink please read it twice time before starting the assignment i need hand written point everything


ENGG2440 — Modelling and Control Viva Voce Major Assignment — Part 2 For Extra Help: E-mail: [email protected] HelpDesk: ES204, Fridays, 11am-1pm Learning Outcomes The Viva Voce major assignment will assess your ability to: 1. Design controllers for the cart-pendulum using a linear approximation. 2. Simulate the closed-loop system using Matlab-Simulink. 3. Evaluate the performance of the controllers on the nonlinear model of the cart- pendulum system. Key Point: 1 Inverted pendulum on a cart We use the cart-pendulum system as a benchmark to design linear controllers. Figure 1 shows the idealised model of the system that consists of a pendulum of mass m and length ` attached to a cart of mass Mc. The pendulum moves under the action of the gravity (g) and the cart moves on the horizontal direction and is actuated by the control force F . The state-space model can be written in the form1 ẋ1 ẋ2 ẋ3 ẋ4  = f(x1, x2, x3, x4, F ), (1) where the states are: • x1: the position of the cart, • x2: the angle of the pendulum, • x3: the velocity of the cart, • x4: the angular velocity of the pendulum. 1.1 Problem formulation. In this lab assignment, we consider the stabilisation problem of two equilibriums of the cart-pendulum system. The equilibriums are x̄a =  x̄1a x̄2a x̄3a x̄4a  =  0 0 0 0  ; x̄b =  x̄1b x̄2b x̄3b x̄4b  =  0 π 0 0  . (2) We consider the control structure given in Figure 2. The control objective is to design the controllers C1(s) and C2(s) that generate the input force to stabilise the equilibriums x̄a and x̄b. 1The full model of the cart-pendulum system and the values of the model parameters are explicitly given in the part I of the VIVA assignment. 1 mailto:[email protected] Figure 1: Cart-pendulum system. Controller System Cart-pendulum Figure 2: Control system for the cart-pendulum. 1.2 Stabilisation of the cart-pendulum about the equilibrium x̄a. The task in this section is to design a controller to stabilise the equilibrium x̄a. To do that, consider the linearised model of the cart-pendulum about x̄a: ˙̃xa = Aax̃a +BaF, (3) y = Cax̃a +DaF, (4) where the matrices Aa, Ba, Ca and Da can be computed using the linearisation of the nonlinear model of the cart-pendulum about the equilibrium x̄a. The control structure for the linearised model of the cart-pendulum is shown in Figure 3, where the system transfer functions are H1(s) = X1(s) F (s) , (5) H2(s) = X2(s) F (s) , (6) with X1(s) = L[x1(t)], X2(s) = L[x2(t)] and F (s) = L[F (t)], and L is the Laplace trasform operator. The signals x̄1 and x̄2 are the desired equilibrium points. Control design. Consider the PD controllers C1(s) and C2(s) given in the form C1(s) = Ka1 +Ka3s, (7) C2(s) = Ka2 +Ka4s, (8) where Ka1, Ka2, Ka3 and Ka4 are the constant gains of the controllers. Design the controller following the next steps 1. Compute the matrices Aa, Ba, Ca and Da of the linearised model of the cart-pendulum about x̄a. 2. Obtain the transfer functions H1(s) and H2(s) of the linearised model of the cart- pendulum about x̄a. 2 Controller Linearised system Figure 3: Control system for the linearised model of the cart-pendulum. 3. Derive the closed-loop transfer function HCL(s) = X1(s) X̄1(s) , (9) where X̄1(s) = L[x̄1(t)]. 4. Pole placement. Compute the gains Ka1, Ka2, Ka3 and Ka4 such that the poles of of the closed-loop transfer function HCL(s) are λ1 = −3, λ2 = −4, λ3 = −5, λ4 = −6. 1.3 Simulation of the cart-pendulum control system about the equi- librium x̄a. Write a script that performs the following tasks: 1. Define the parameters of the model. 2. Define the matrices of the linearised model. 3. Define the gains of the controller. 4. Simulate the linearised and nonlinear closed-loop systems: i) Create a Simulink model of the linearised system in closed loop with the PD controllers. Save your model as CP_Control_Lin_a_yourstudentnumber.slx. An example of the Simulink model is shown in Figure 4 (see Appendix for details). ii) Create a Simulink model of the nonlinear system in closed loop with the PD controllers (use the controller gains obtained for the linearised model). Save your model as CP_Control_NLin_yourstudentnumber.slx. An example of the Simulink model is shown in Figure 5. iii) Export the states, the control input and the simulation time from Simulink to Matlab. iv) Simulate both the linearised and the nonlinear closed-loop models with the cart stating at 0.2m and the pendulum at 20deg. Set the initial conditions of the velocities to zero. That is x(0) = [ 0.2 20π/180 0 0 ]> . Suggestion for the simulation: use the fixed-step solver ode4 and select 0.02 as step time. 5. Plot the results of the simulations in a figure that shows the time histories of the position of the cart, the angle of the pendulum, the velocity of the cart, the angular velocity of the pendulum and the control forces for both the linearised and nonlinear control systems. An example of the simulation results is shown in Figure 6. 6. Save the script as MainFile_Control_Comparison_a_yourstudentnumber.m. 7. Run the script using different initial conditions for the simulation and analyse the results. The initial conditions should be defined in the script. 8. (Optional) Use the function Cart_Pendulum_Animation.m to create an animation of your control system. Important: The plots and (optional) animation should be produced automatically when the script is executed without further intervention of the user. 3 Figure 4: Control system for the linearised model. Figure 5: Control system for the nonlinear model. 4 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time [s] -0.2 0 0.2 0.4 C ar t p os iti on [m ] Time histories of the states using the nonlinear model and linearised model about EPa x 1 Nonlinear x 1 Linearised 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time [s] -20 0 20 P en du lu m a ng le [d eg ] x 2 Nonlinear x 2 Linearised 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time [s] -2 0 2 C ar t v el oc ity [m /s ] x 3 Nonlinear x 3 Linearised 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time [s] -600 -400 -200 0 P en du lu m r at e [d eg /s ] x 4 Nonlinear x 4 Linearised 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time [s] -50 0 50 100 150 In pu t F or ce [N ] F Nonlinear F Linearised Figure 6: Time histories of the states and input. 5 1.4 Stabilisation of the cart-pendulum about the equilibrium x̄b. The task in this section is to design a controller to stabilise the equilibrium x̄b. To do that, consider the linearised model of the cart-pendulum about x̄b: ˙̃xb = Abx̃b +BbF, (10) y = Cax̃b +DbF, (11) where the matrices Ab, Bb, Cb and Db can be computed using the linearisation of the nonlinear model of the cart-pendulum about the equilibrium x̄b. Consider the control structure for the linearised model of the cart-pendulum shown in Figure 3. Control design. Consider the PD controllers C1(s) and C2(s) given in the form C1(s) = Kb1 +Kb3s, (12) C2(s) = Kb2 +Kb4s, (13) where Kb1, Kb2, Kb3 and Kb4 are the constant gains of the controllers. Design the controller following the next steps 1. Compute the matrices Ab, Bb, Cb and Db of the linearised model of the cart-pendulum about x̄b. 2. Obtain the transfer functions H1(s) and H2(s) of the linearised model of the cart- pendulum about x̄b. 3. Derive the closed-loop transfer function HCL(s) = X1(s) X̄1(s) , (14) where X̄1(s) = L[x̄1(t)]. 4. Pole placement. Compute the gainsKb1, Kb2, Kb3 and Kb4 such that the poles of of the closed-loop transfer function HCL(s) are λ1 = −3, λ2 = −4, λ3 = −5, λ4 = −6. 1.5 Simulation of the cart-pendulum control system about the equi- librium x̄b. Write a script that performs the following tasks: 1. Define the parameters of the model. 2. Define the matrices of the linearised model. 3. Define the gains of the controller. 4. Simulate the linearised and nonlinear closed-loop systems: i) Create a Simulink model of the linearised system in closed loop with the PD controllers. Save your model as CP_Control_Lin_b_yourstudentnumber.slx. ii) Create a Simulink model of the nonlinear system in closed loop with the PD controllers (use the controller gains obtained for the linearised model). Save your model as CP_Control_NLin_yourstudentnumber.slx. iii) Export the states, the control input and the simulation time from Simulink to Matlab. iv) Simulate both the linearised and the nonlinear closed-loop models with the cart stating at 0.2m and the pendulum at 200deg. Set the initial conditions of the velocities to zero. That is x(0) = [ 0.2 200π/180 0 0 ]> . Note that the ini- tial condition is given for the states of the nonlinear model. Suggestion for the simulation: use the fixed-step solver ode4 and select 0.02 as step time. f) Plot the results of the simulations in a figure that shows the time histories of the position of the cart, the angle of the pendulum, the velocity of the cart, the angular velocity of the pendulum and the control forces for both the linearised and nonlinear control systems. An example of the simulation results is shown in Figure 7. 6 5. Save the script as MainFile_Control_Comparison_b_yourstudentnumber.m. 6. Run the script using different initial conditions for the simulation and analyse the results. The initial conditions should be defined in the script. i) (Optional) Use the function Cart_Pendulum_Animation.m to create an animation of your control system. Important: The plots and (optional) animation should be produced automatically when the script is executed without further intervention of the user. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time [s] -0.1 0 0.1 0.2 C ar t p os iti on [m ] Time histories of the states using the nonlinear model and linearised model about EPb x 1 Nonlinear x 1 Linearised 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time [s] 160 180 200 P en du lu m a ng le [d eg ] x 2 Nonlinear x 2 Linearised 0 0.5 1 1.5