2018 05 03 midterm answers

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Control Systems (Prof. Casella) Midterm Exam – May 3th, 2018 Answer Sheet Question 1 Write down Lagrange’s formula for the motion of the output of a LTI system and discuss what important properties about the motion of LTI systems can be drawn from it. Lagrange’s formula shows that the superposition principle applies: 1. The motion of the system is the summation of the free motion (obtained by setting the input to zero) and of the forced motion (obtained by setting the initial state to zero). This is immediately apparent from the formula 2. The forced motion obtained from a linear combination of two inputs is the same linear combination of the forced motion obtained by applying each input separately; this stems from the linearity of the finite integral operator Question 2 Assess the stability of the systems described by the following transfer functions: Asym. stable Simply stable Unstable s2−1 s2 −10 s+ 20   x s+3 s2 +1  x  s s3 + s2  x  s ( s+1)3 x   s+1 s( s2 +1)  x  1 s2 −s   x 1 s( s+10)  x  1 ( s+1)(s2 +1)2   x y (t)=Ce At x(0)+C∫ 0 t eA(t−τ) Bu( τ)d τ+ Du(t) Question 3 Consider a cylindrical open tank of cross section A, containing a liquid of constant density r up to a level l, connected to a pump that feeds it with a prescribed mass flow rate wi and temperature Ti, and to a pump that discharges it with a prescribed flow rate wo. Assuming a constant specific heat capacity c and assuming that de = dh = cdT, where e and h are the specific energy and enthalpy of the liquid, the system is described as: 3.1 Write the system equations in standard state-space form, considering wi, wo, and Ti as inputs and l, T as states and outputs 3.2 Compute the equilibrium conditions of the system, assuming non-zero flow rate and level. 3.3 Write down the linearized…