In this problem we will study PID controllers in a state space representation. Consider the linear...

 

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In this problem we will study PID controllers in a state space representation. Consider the linear control system: x 1 = x2, x 2 = 2x1 - 3x2 - u, y = x1. 1. Knowing that only y can be measured, design a stabilizing controller that only uses the measurement y. Relate the designed controller to a PD controller. 2. Assume now that the dynamics for x2 is affected by an unknown constant input c E R. i.e.: x 2 = 2x1 - 3x2 - u + c, and that the state x can be measured. Design a dynamic controller, i.e., a controller of the form: z = g(z, x), u = k(z, x), that guarantees limt → x(t) = 0. Does the variable z converge? If so, to which value does it converge? How is this value related to c? Relate the designed controller to a PI controller. In this problem we will study PID controllers in a state space representation. Consider the linear control system: x 1 = x2, x 2 = 2x1 - 3x2 - u, y = x1. 1. Knowing that only y can be measured, design a stabilizing controller that only uses the measurement y. Relate the designed controller to a PD controller. 2. Assume now that the dynamics for x2 is affected by an unknown constant input c E R. i.e.: x 2 = 2x1 - 3x2 - u + c, and that the state x can be measured. Design a dynamic controller, i.e., a controller of the form: z = g(z, x), u = k(z, x), that guarantees limt → x(t) = 0. Does the variable z converge? If so, to which value does it converge? How is this value related to c? Relate the designed controller to a PI controller.

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1 The stabilizing controller that only uses the measurement y is a PD cont... View the full answer