2023 06 16 exam answers

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Control Systems (Prof. Casella) Written Exam – June 16th, 2023 ANSWERS SHEET Question 1 Consider a generic time-invariant nonlinear dynamical system, given in state-space form. Explain how to compute the stability of its equilibria. The system must first be linearized around the equilibrium. Then, the transfer function between the variations of the input and the variations of the output must be computed. If all the poles of such transfer function have negative real part, the equilibrium is asymptotically stable. If there exists at least one pole with positive real part, the equilibrium is unstable. Otherwise, nothing can be said, as the stability depends on higher-order terms, which are neglected by the linearization process. Question 2 Describe the phenomenon of integral wind-up in linear controllers and why it should be avoided by properly implemented control systems. See the course notes. ˙x(t)= f ( x(t) ,u(t)) y(t)= g ( x(t) , u(t)) Question 3 The equations shown on the right describe the dynamic response of the power output in a steam Rankine cycle with reheat, operated at constant boiler pressure. wh and wl are the steam flow rates through the high and low pressure tur - bines, pr the reheater steam pressure, Ph, Pl, and P the high-pressure, low- pressure, and total turbine power outputs, Mr is the mass of vapour contained in the reheater tubes (assumed as a function of pr), and q is the partial arc opening degree of the high-pressure turbine in per-unit. The design parameters w0, pr0, Ph0, and Pl0 are all constants. 3.1 Write down the system equations in standard state-space form, consider - ing q as input and P as output. 3.2 Compute the equilibrium conditions at full load, q = 1 3.3 Write down the system's linearized equations around such an equilibrium (Hint:…