EE472 Linear Systems: Control Design and Analysis: Homework 2, Spring 2019 Instructor: Dr. Oğuzhan Çifdalöz

Problem 1: Consider the DC motor control system with rate feedback shown in Figure below.

(a) Find values for K´ and kt´ so that the system of Figure 1(b) has the same transfer function as the system

of Figure 1(a). Your answers should be in terms of Kp, K, kt, Km, and k.

(b) Suppose τm = 0.25 sec. For the case with no rate feedback (i.e., kt´ = 0), use root-locus techniques to select

the proportional gain K´ to achieve a closed-loop damping ratio of ζ = 0.2.

(c) Using the value of K´ found in part (b), sketch the locus of closed-loop pole locations for kt´ > 0.

(d) With respect to tracking θr, and compute the corresponding error constant in terms of parameters K´ and kt´. What happens to the steady state error if K´ is increased? If kt´ is increased?

Hint: Your tracking error should take the form e(t) = θr(t) – θ(t).

Problem 2: Simplify the following block diagram into a single block

Problem 3: A unity feedback system has a loop transfer function given by

(a) For a unit step input, the closed-loop system must have peak overshoot of 16%. Determine the

required value of K. (b) If K = 2, what is the approximate rise time of the output response to a step input? (c) If K = 2 and the input is a unit ramp, what I the steady state error?

Problem 4: The attitude-control system of a space booster is shown in Figure 2. The attitude angle θ is controlled by commanding the engine angle δ, which is then the angle of the applied thrust, FT. The vehicle velocity is denoted by v. A typical rigid-body transfer function for such a booster might take the form

(This is very simplified, ignoring fuel slosh, aeroelasticity, motor dynamics, etc.) The rigid-body vehicle can be stabilized by the addition of rate feedback (Figure 2(b)).

(a) With KD = 0, plot the root locus and state the different types of responses possible (relate the response with the possible pole locations). Why is KP alone not sufficient to stabilize the dynamics?

(b) Design the compensator shown (which is PD) to place the closed-loop poles at s = –0.2±j0.3 (the resulting time constant is 5 sec).

(c) Plot the root locus of the compensated system, with KP variable and KD set to the value found in (b). Compare with your answer in (a).

(d) For some value of KP, use Matlab to compute the closed-loop response to an impulse for θc. Problem 5: Suppose that you are to design a unity gain feedback controller for a first order plant. The plant and controller respectively take the form

where K > 0, p, z are parameters to be specified.

(a) Using root-locus methods, specify some p and z for which it is possible to make the closed-loop system strictly stable. Include a sketch of the closed-loop root locus, as well as the corresponding range of gains K for which the system is strictly stable. (b) Suppose p and z are fixed to the values chosen in (a). Design K to meet the following specifications:

 The closed-loop system must be strictly stable.  The damping ratio ζ must be between 0.4 and 0.6.  Given these constraints, minimize the natural frequency ωn.

Problem 6: Analyze the stability of the unity gain negative feedback systems described by the following open-loop transfer functions, using the (i) root locus method and (ii) asymptotic Bode plot:

Plots should be drawn by hand for each transfer function (though you may check your answers in Matlab). You should note when a particular method cannot be used to assess stability. Problem 7: Match the step response to the appropriate pole-zero plot in the figure below:

Problem 8: A simplified model of a glider is

where γ is the flight path angle in radians, v is the airspeed in m/sec, n =L/mg is the load factor, L is the lift in Newtons, m is the mass in kg, and k1 = 61.6594 and k2 = 4.8747×10-5

are constants for the glider.

(a) Given that γ = -0.15 rad, and the airspeed is 50.8691 m/sec, find the necessary load factor to maintain equilibrium. (b) Let the state vector be [γ v]T

, let the input be n, and let the output of interest be v. Derive the linearized

system about the equilibrium point obtained from above. (c) From the linearized system dynamics, derive the transfer function from input n to output v, and the transfer function from input n to output γ.(Hint: Use tf and ss in Matlab to check your answers.)

(d) Plot the unity gain negative feedback root locus of the linearized system using Matlab. Give the locations of the open loop poles, and the range of gains for which the closed-loop system is stable.

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