2024 06 14 exam answers

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Control Systems (Prof. Casella) Written Exam – June 14th, 2024 ANSWER SHEET Question 1 Define the transfer function G(s) of a LTI system described by state-space equations. Then, explain what is the relationship between G(s) and the impulse response of the system. Given the LTI system by applying the Laplace transform to the equations and solving them assuming zero initial conditions, one gets where G(s) is defined as the transfer function. Since the Laplace transform of the impulse function is one, from the first equation one concludes that the transfer function can be interpreted as the Laplace transform of the impulse response. Question 2 (Respect the unstable!) The block diagram on the right represents a control system, where G(s) is an unstable LTI plant, while the linear controller C(s) is designed to stabilize the closed-loop system equilibria when the actuator is not saturated. Assume that the system is at one of those equilibrium points, and that at t = 0 the set point y° is increased to a value that would require a value of u larger than the maximum actuator output to get y = y° in steady-state. What will happen to the system output y in that case? (Hint: consider the saturation block output values ) When the set-point is increased, the controller output u will also increase, until eventually its value will become larger than the maximum actuator value, in the vain attempt by controller to drive the output to the new set point value. In fact, the actuator output m will remain constant from that point onward, so that G(s) is driven by a fixed input and is no longer at equilibrium. Since G(s) is unstable, its output y will then increase without bounds. In other words, the actuator saturation opens the feedback loop, which then does no longer stabilize the…