Did their performance change when applied to the real

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still meet the design specifications? Did their performance change when applied to the real nonlinear system? 8. (2 marks) Consider the case when the prosthesis actuator can only provide a maximum torque of τ max = 1Nm . Simulate this by introducing a saturation block to your control input. Comment on the performance of the controller. How does this change the behaviour and performance of the system? Is it possible to still satisfy the performance constraints while avoiding control input saturation? If so, how can this be achieved? 1.4 Design of feedback controllers (50 marks) 1.4.1 State Feedback Control (15 marks) When complete state information is available, an alternative controller can be designed to meet the desired specifications. Design a full state feedback controller to meet the following design constraints: Maximum overshoot less than 10 %, Zero steady state error to step references and step input disturbances. Please follow and answer the steps and questions below while developing your controller: 9. (10 marks) Using the linearised state space model, prove that it is possible for a full state feedback controller to be utilised to meet the given constraints. Design such a controller. 10. (5 marks) Discretise your controller by using u [ k ] = - Kx [ k ] over the interval t [ kΔ, ( k + 1 ) Δ ] . Simulate the performance that this controller has, when applied into the linearised system, to a step input disturbance. Comment on the performance of discretised controller? Does the controller still meet the design specifications? 1.4.2 Output Feedback Control (20 marks) Due to size and weight constraints of the prosthetic device, it is not possible to fit a sensor that measures the state ˙ q 2 . Therefore, an observer and output feedback controller need to be developed to meet the performance constraints. 11. (10 marks) Show that it is possible to design a state estimator to recapture the lost state information. Using the separation principle design an output feedback controller in continuous time for your linearised system model. 12. (10 marks) Simulate your full state feedback and output feedback controllers into the continuous-time linear plant and nonlinear plant. Discuss and compare the performance of the two controllers. 3
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1.4.3 Internal Model Principle (15 marks) Testing of your previously developed controller onto the physical platform shows that rather than a step input disturbance, there is a sinusoidal input disturbance of the form d o ( t ) = α sin ( 2t ) where α is unknown. 13. (10 marks) Augment your output feedback controller to reject this disturbance. 14. (5 marks) Discretise your augmented controller. Simulate your new controller with the sinusoidal input disturbance. Comment on the performance of your controller. 1.5 Optimal Control (40 marks) A widely accepted theory of human motor control is that the CNS seeks to optimize arm motion with respect to performance and the metabolic effort of the motion. This means that there is a trade-off between performance of the position tracking task and the control effort to achieve the task. This type of behavior can be emulated through Optimal Controllers. Investigate the use of optimal control in meeting the desired
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