2023 04 20 midterm answers

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Control Systems (Prof. Casella) Midterm Exam – Apr 20th, 2023 ANSWERS SHEET Question 1 Consider two asymptotically stable LTI dynamical systems. Discuss the stability of their feedback connection. Barring pole/zero cancellations, the transfer function from input to output of the feedback connection of two LTI’s with transfer functions A(s) = nA(s)/dA(s) and B(s) = nB(s)/dB(s) has a denominator nA(s)nB(s) + dA(s)dB(s). Its poles, which are the roots of such polynomials, are in general different from the poles of A(s) and B(s), which are the roots dA(s) and dB(s). Hence, the feedback connection of two asymptotically stable LTI systems can be either stable or unstable in general. Question 2 Give the most general definition of state variable for a generic dynamical system. Then, define what are the state variables for a continuous-time state-space model of a continuous-time dynamical system, explaining why it fits the general definition. A dynamical system is such that the present value of the output also depends on the past values of the input. The state variables are those values whose knowledge at a certain time t0 allows to determine the value of the output without knowing the input values for t < t0. In the specific case of a state-space model, the knowledge of the initial state x0 allows to solve the Cauchy problem for the state equation for values of time t > t0 without the need of knowing u(t) for t < t0, and then to solve the output equations also without the need of knowing u(t) for t < t0. Hence, for state-space systems, the state variables are the ones that appear under derivative sign. dx dt = f ( x(t) , u(t) , t) y=g ( x(t ) , u(t) ,t) x(t0)= x0 Question 3 For each of the following four diagrams indicate what is the transfer function, among those listed…