ECE121_2010FALL_EXAM1__[0]

# ECE121_2010FALL_EXAM1__[0] - Electrical Engineering...

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Unformatted text preview: Electrical Engineering Department The Cooper Union ECE 121A Mid-Term Examination Control Systems ‘ Fall 2010 1. The closed-loop transfer function M(s) deﬁned by Y(s) := M(s)X(s) for the unity—feedback system (G(s), 1) has the s—plane pole-zero distribution shown below. In addition, the steady—state step response limp m{y(t) ixm = u(t)} = 1' w +.. 3) Determine the differential equation relating the output y(t) and the input x(t) of the feedback system (G(s), l). b) Determine the system transfer function G(s). c) Determine the expression for the response y(t) due to the input x(t) = 8.55 (t - 1), where 5(t) is the unit impulse function. 2. The linear time—invariant system 21 with input x(t) and output. y(t) is described by the following differential equation x(t) 2: x(t) ———>- y(t) y(t) (Yb/(2‘) (iii/(t) a’y(t)=,~ a'XU) dtsx +6‘—#——df2 +8 dt 2 dt +10X(t) with initial conditions y”(0) = O, y’(0) = 1, y(0) = -1 and x(0) = 1.5. ' 3) Obtain an appropriate simulation diagram for this system , using integrators, scalar multipliers and summing units. b) Let the state variable q1(t) := y(t). Then, use the simulation diagram in part (a) to obtain the state variable model (A, b, c') for the system 21. ' c) Determine the eigenvalues for the system matrix A. (1) Determine the state transition matrix @(t) for the system (A, b, c‘). ECE 121A Control Systems Mid-Term Examination Fall 2010 3. A linear time-invariant feedback control system is described by the block diagram shown below G(S) = 5/(S + 2) P Y(s) 3) Determine the characteristic equation for the given feedback system. b) Determine the expression for the step response of the feedback system with K = 0.6. c) Determine the expression for the steady-state error esr(K) due to a unit ramp input, as a function of the system parameter K. (1) Give a sketch for esr(K) versus K, and identify on this sketch the regions corresponding to overdamped, critically damped and underdamped feedback systems. 4. A unity—feedback system with disturbance input D(s) is described by the block diagram in the figure below. I)(s) Y(s) G(S) = 2/(S + 4) 3) Determine the sensitivity function S(MIG; s) of M(s) with respect to G(s) Where M(s) is deﬁned by Y(s) := M(s)X(s) with the disturbance D(s) = O. b) Determine the expression for the response Y(s) in terms of the input X(s) and the disturbance D(s). c) Determine the expression for the tracking error E(s) := X(s) — Y(s), in terms of the sensitivity function S(MlG; s). (1) Evaluate l T(jw) | and ] S(M |G; joo) | for the system with w = 2 rad/s. T(s) is the complementary sensitivity function of the system. ...
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## This note was uploaded on 02/27/2012 for the course CHEMISTRY/ CH/ECE/PH/ taught by Professor Faculty during the Spring '08 term at Cooper Union.

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ECE121_2010FALL_EXAM1__[0] - Electrical Engineering...

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