Questions and Answers of Digital Control System Analysis And Design

Give a first-order time-invariant discrete system with a cost function N JN = Qx(k) + Ru(k) k = 0 Show that the optimal gains are a function of only the ratio Q R and not of Q and R singly. =
The system considered here is the classical system of Doyle and Stein [9] to illustrate robustness problems. The system of Doyle and Stein is analog; the discrete model of the system is used here
For the IH-LQG control system of Fig. 11-6, suppose that the plant is single-input single-output, such that B and C are vectors. To determine the system robustness, the open-loop transfer functions
Consider the robot-joint control system of Fig. P5.3-13. This system is described in Problem 1.5-4.Problem 1.5-4 T Controller D(z) Volts Power Zero-order hold Servomotor Ea 200 100 s (0.5 s + 1)
Consider the closed-loop system of Fig. P6.2-1.(a) Calculate and plot the unit-step response at the sampling instants, for the case that D(z) = 1 .(b) Calculate the system unit-step response of the
Consider the system of Fig. P6.2-1, with D(z) = 1 . Use the results of Problem 6.2-1 if available.(a) Find the system time constant τ for T = 2 s .(b) With the input a step function, find the time
In Example 6.1 the response of a sampled-data system between sample instants was expressed as a sum of delayed step responses.(a) Use this procedure to find the system output y(t) of Fig. P6.2-1 at
Shown in Fig. P6.2-4 is the block diagram of a temperature control system for a large test chamber. This system is described in Problem 1.6-1. Ignore the disturbance input for this problem.(a) With
Consider the temperature control system of Problem 6.2-4 and Fig. P6.2-4.(a) Let T = 6 s , and solve for the response to the input R(s) = 0.4 s−1 . Plot this response on the same graph with the
Consider the system of Fig. P6.2-4, with D(z) = 1 . Use the results of Problems 6.2-4 and 6.2-5 if available.(a) Find the system time constant τ for T = 0.6 s .(b) With the input a step function,
The block diagram of a control system of a joint in a robot arm is shown in Fig. P6.2-7. This system is discussed in Section 1.6. Let T = 0.1 s and D(z) = 1 .(a) Evaluate C(z) if the input is to
The block diagram of a control system of a joint in a robot arm is shown in Fig. P6.2-7. Let T = 0.1 s , K = 10 , and D(z) = 1 . The results of Problem 6.2-7 are useful in this problem if these
The block diagram of an attitude control system of a satellite is shown in Fig. P6.4-2. Let T = 1 s , K = 100 , J = 0.1 , Hk = 0.02 , and D(z) = 1.(a) Find the damping ratio ζ the natural frequency
Consider the system of Fig. P6.4-4. This system is called a regulator control system, in which it is desired to maintain the output, c(t) , at a value of zero in the presence of a disturbance, f (t)
Consider the sampled-data systems with the following characteristic equations: (1) 2- - 0.999 = 0 Z-0.99 = 0 (v) 21.85z + 0.854 = 0 (vi) 2-1=0 (vii) 22z+0.99 = 0 (11) (111) z - 0.9=0 (iv) z +0.9 = 0
Consider the system of Fig. P6.2-1. Suppose that an ideal time delay of 0.2 s is added to the plant, such that the plant transfer function is now given by(a) Find the time constant τ of the system
(a) Give the system type for the following systems, with D(z) = 1. It is not necessary to find the pulse transfer functions to find the system type. Why?(i) Fig. P6.2-1.(ii) Fig. P6.2-4.(iii) Fig.
Consider the system of Fig. P6.5-2. The digital filter is described by(a) Find the system type.(b) Find the steady-state response for a unit-step input, without finding C(z).(c) Find the
Consider the system of Fig. P6.5-2, with D(z) = 1 .(a) With the sampler and zero-order hold removed, write the system differential equation.(b) Using the rectangular rule for numerical integration
Consider the numerical integration of the differential equation (6-24) in Section 6.6. Applying the rectangular rule results in the difference equation (6-25), which can be expressed as(a) Use the
Consider the numerical integration of the differential equationusing the rectangular rule.(a) Develop the difference equation, as in (6-25), for the numerical integration of this differential
Consider the numerical integration of the differential equationusing the predictor-corrector of Section 6.6. The predictor method is the rectangular rule, and the corrector method is the trapezoidal
Consider the numerical integration of the differential equationusing the predictor-corrector of Section 6.6. The predictor method is the rectangular rule, and the corrector method is the trapezoidal
Consider the second-order analog system described byExtra \left or missing \rightwhere T(s)T(s) is the closed-loop transfer function and Gp(s)Gp(s) is the plant transfer function. Show that the phase
Consider the system of Fig. P8.2-2 with T=0.2T=0.2.(a) Show that the pulse transfer function of the plant is given byExtra \left or missing \right(b) The frequency response for G(z)G(z) is given in
Consider the system of Fig. P8.2-2 with T=1T=1 s. The plant frequency response Extra close brace or missing open brace is given in Table P8.4-1.(a) Let D(z)=1D(z)=1. If the system is stable, find its
This problem is based on the solution in Problem 8.4-3. Suppose that in Fig. P8.4-3, the sensor gain \(H=0.04\) is moved back to the feedback path. What effect does this have on the percent
Consider the system of Problem 8.4-1.(a) Design a unity-dc-gain phase-lead compensator that yields a phase margin of approximately \(45^{\circ}\).(b) Obtain the system step response of part (a) using
Repeat Problem 8.4-2 using a phase-lead controller. In part (c), the overshoot is approximately 26 percent.Problem 8.2-2 Consider the system of Fig. P8.2-2 with T = 0.2. (a) Show that the pulse
Three constraints are given on the choice of the phase-margin frequency, \(\omega_{w 1}\), in Section 8.6 for phase-lead design. Derive the three equivalent constraints on \(\omega_{w 1}\) for the
Consider the block diagram of a robot-arm control system shown in Fig. P8.6-4. This system is described in Problem 1.5-4. Let \(T=0.1\). It was shown in Problem 4.3-8 that\[G(z)=\frac{z-1}{z}
Consider the block diagram of an antenna control system shown in Fig. P8.6-5. Let T=0.05T=0.05 and the sensor gain be unity (H=1)(H=1).Extra \left or missing \rightThe frequency response for
Consider the block diagram of a satellite control system shown in Fig. P8.6-6. This system is described in Problem 1.4-1. Let \(T=0.1 \mathrm{~s}, K=1, J=0.1\), and \(H_{k}=0.02\). It was shown in
Write the difference equations required for the realization of the PID controller transfer function \(D(z)\) in (8-52). Let \({ }_{E(z)}\) be the controller input and \({ }_{M(z)}\) be the controller
If the rectangular integrator [see below (8-38)] is used in the PID transfer function in (8-52), the resulting controller is described by\[\frac{M(z)}{E(z)}=D(z)=K_{p}+K_{I}\left[\frac{T
Repeat Problem 8.6-2 using a proportional-plus-derivative (PD) filter.Problem 8.6-2Repeat Problem 8.4-2 using a phase-lead controller. In part (c), the overshoot is approximately 26 percent.Problem
Repeat Problem 8.4-3 for a proportional-plus-integral (PI) compensator. Note that in this case, the steady-state error for a constant input is zero.Problem 8.4-3 Shown in Fig. P8.4-3 is the block
Consider the system of Example 8.1.(a) Design a PI filter to achieve a phase margin of \(60^{\circ}\).(b) Obtain the system step response using MATLAB. Compare this response to that of the system of
Consider the chamber temperature control system of Problem 8.4-3. Suppose that a variable gain \(K\) is added to the plant. The pulse transfer function for this system is given in Problem 8.4-3.(a)
Consider the system of Fig. P8.2-2 with a gain factor KK added to the plant.(a) Show that the pulse transfer function is given by(b) Sketch the root locus for this system, and find the value of KK
Consider the system of Example 8.7.(a) Design a phase-lag compensator such that with \(K=0.5\), the system is critically damped with roots having a time constant of approximately \(2.03
The plant of Example 9.1 has the state equations given (9-1). Find the gain matrix KK required to realize the closed-loop characteristic equation with zeros which have a damping ratio ζζ of 0.46
Consider the plant of Problem 9.2-1, which has the transfer function, from Example
A satellite control system is modeled as shown in Fig. P9.2-4. This system is described in Problem 1.4-1. For this problem, let D(z)=1D(z)=1D(z)=1K=1,T=1 s,J=4. In addition, K=1,T=1 s,J=4Hk=1, and
Consider the pole-placement design of Problem 9.2-2.Problem 9.2-2Consider the plant of Problem 9.2-1, which has the transfer function, from Example 9.5,Problem 9.2-1The plant of Example 9.1 has the
Consider the satellite control system of Problem 9.2-4.Problem 9.2-4A satellite control system is modeled as shown in Fig. P9.2-4. This system is described in Problem 1.4-1.For this problem, let D(z)
Assume that equation (9-25), Ackermann's formula for pole-assignment design, is the solution of (9-15). Based on this result, show that (9-48), Ackermann's formula for observer design, is the
Given in (9-56) is the closed-loop state model for the pole-placement prediction-estimator design. Extend this model to include plant disturbances and sensor noise, as described in (9-44).Equation
Consider the control system of Problem 9.2-2.Problem 9.2-2Consider the plant of Problem 9.2-1, which has the transfer function, from Example 9.5,Problem 9.2-1The plant of Example 9.1 has the state
Consider the satellite control system of Problem 9.2-4.Problem 9.2-4A satellite control system is modeled as shown in Fig. P9.2-4. This system is described in Problem 1.4-1.For this problem, let D(z)
Consider the reduced-order observer designed in Problem 9.4-2. In this problem, velocity [dy/dt][dy/dt] is estimated, using position [y][y] plus other information. We could simply calculate velocity,
Consider that in Fig. 9-8, the observer is reduced order, and that the system is single-input single output. Show that the transfer function Dce(z)Dce(z) of the equivalent controller is given by.Fig.
Repeat all parts of Problem 9.3-1 using a current observer.Problem 9.3-1 (a) Design a predictor observer for this system, with the time constant equal to one-half the value of Problem 9.2-2(a) and
Repeat all parts of Problem 9.3-2, using a current observer.Problem 9.3-2 (a) Design a predictor observer for this system, with the time constant equal to one-half the value of Problem 9.2-3(b). (b)
(a) Find the closed-loop state equations for the system of Problem 9.3-2(a), of the form of (9-56).(b) Find the system characteristic equation using the results in part (a), and show that this is the
Consider the chamber temperature control system of Fig. P9.2-3. For this problem, replace the sensor gain \(H=0.04\) with the gain \(H=1\). The system is now a unity-gain feedback system.(a) Work
Repeat all parts of Problem 9.3-3, using a current observer.Problem 9.3-3 (a) Design a predictor observer for this system, with the time constant equal to one-half the value of Problem 9.2-4(b) and
(a) Show that for the current observer specified in Section 9.5, the transfer matrix from the input \(\mathbf{U}(z)\) to the estimated states \(\mathbf{Q}(z)\) is equal to that from the input to the
Derive the closed-loop state model for the pole-placement current-estimator design, given in (9-74). B] =[GCA x(k+ q(k+ 1). -BK A - GCA x(k) BKq(k) (9-74)
Consider a system described by (9-82).Equations 9-82For the case that u(k) is not zero, derive the conditions for observability. x(k+ 1) = Ax(k) + Bu(k) y(k) Cx(k) =
Consider the plant of Example 9.2, which is(a) Is this system observable?(b) Explain the reason for your answer in part (a) in terms of the physical aspects of the system.Example 9.2 x(k + 1) =
Consider the satellite control system of Problem 9.2-4. Suppose that the output is the measurement of angular velocity, such that(a) Is this system observable?(b) Explain your answer in part (a) in
Consider the first-order plant described byand a prediction observer described by (9-38).(a) Construct a single set of state equations for the plant-observer system, with the state vector
Problem 9.6-7 is to be repeated for the current observer, with the mode of Q(z)Q(z) equal to (A−GCA)k(A−GCA)k.Problem 9.6-7For the system of Problem 9.6-5, use a transfer function approach to
(a) Give the definition of the starred transform.(b) Give the definition of the z-transform.(c) For a function e(t) , derive a relationship between its starred transform E*(s) and its z-transform
A signal e(t) is sampled by the ideal sampler as specified by .(a) List the conditions under which e(t) can be completely recovered from e*(t) , that is, the conditions under which no loss of
A system is defined as linear if the principle of superposition applies. Is a sampler/zero-order-hold device linear? Prove your answer.
Use the residue method of (3-10) to find the starred transform of the following functions.Equation 3-10 (a) E(s) = (c) E(s) = (e) E(s) = 20 (s+2)(s+5) s+2 s(s+1) s + 5s +6 s(s+4) (s+5) (b) (d) 5 E(5)
Find E*(s) for each of the following functions. Express E*(s) in closed form. (a) e(t) = gat (c) e(t) = g(1-27)u(1-27) (b) E(s) = -2Ts s-a (d) e(t) = 8(t-1/2)u(t-T / 2)
For e(t) = ε−3t .(a) Express E*(s) as a series.(b) Express E*(s) in closed form.(c) Express E*(s) as a series which is different from that in part (a).
Express the starred transform of e(t − kT) u(t − kT), k an integer, in terms of E*(s) , the starred transform of e(t) . Base your derivation on (3-3).Equation 3-3 E*(s) = E(s) * AT(S)
Find E*(s) for E (s) 1-8-7s s(s+1)
(a) Find E*(s) , for T = 0.1 s , for the two functions below. Explain why the two transforms are equal, first from a time-function approach, and then from a pole–zero approach.(i) e1(t) =
Compare the pole–zero locations of E*(s) in the s-plane with those of E(s) , for the functions given in Problem 3.4-1(c).Problem 3.4-1(c). (c) E(s) = s+2 s(s+1)
Find E*(s) , with T = 0.5 s , for E(s) = (1-8-0.5) 0.5s (s+1)
Suppose that(a) Find e(kT) for all k.(b) Can e(t) be found? Justify your answer.(c) Sketch two different continuous-time functions that satisfy part (a).(d) Write the equations for the two
Suppose that the signalis applied to an ideal sampler and zero-order hold.(a) Show that the amplitude of the time function out of the zero-order hold is a function of the phase angle θ by sketching
(a) A sinusoid with a frequency of 2 Hz is applied to a sampler/zero-order hold combination. The sampling rate is 10 Hz. List all the frequencies present in the output that are less than 50
Given the signal e(t) = 3sin 4t + 2sin7t.(a) List all frequencies less than ω = 50 rad/s that are present in e(t) .(b) The signal e(t) is sampled at the frequency ωs = 22 rad/s . List all
A signal e(t) = 4sin7t is applied to a sampler/zero-order-hold device, with ω = 4 rad/s .(a) What is the frequency component in the output that has the largest amplitude?(b) Find the amplitude and
It is well known that the addition of phase lag to a closed-loop system is destabilizing. A sampler/data-hold device adds phase lag to a system, as described in Section 3.7. A certain analog control
A sinusoid is applied to a sampler/zero-order-hold device, with a distorted sine wave appearing at the output, as shown in Fig. 3-15.(a) With the sinusoid of unity amplitude and frequency 2 Hz, and
A polygonal data hold is a device that reconstructs the sampled signal by the straight-line approximation shown in Fig. P3.7-8. Show that the transfer function of this data hold isIs this data hold
A data hold is to be constructed that reconstructs the sampled signal by the straight-line approximation shown in Fig. P3.7-9. Note that this device is a polygonal data hold with a delay of T
Plot the ratio of the frequency responses (in decibels) and phase versus for the data holds of Problems 3.7-8 and 3.7-9. Note the effect on phase of making the data hold realizable.Problems 3.7-8A
Derive the transfer function of the fractional-order hold.
Shown in Fig. P3.7-12 is the output of a data hold that clamps the output to the input for the first half of the sampling period, and returns the output to a value of zero for the last half of the
(a) Show that a pole of E(s) in the left half-plane transforms into a pole of E(z) inside the unit circle.(b) Show that a pole of E(s) on the imaginary axis transforms into a pole of E(z) on the unit
Let T = 0.05 s and(a) Without calculating E(z), find its poles.(b) Give the rule that you used in part (a). (c) Verify the results of part (a) by calculating E(z).(d) Compare the zero of E(z)
Find the z-transform of the following functions, using z-transform tables. Compare the pole-zero locations of E(z) in the z-plane with those of E(s) and E * (s) in the s-plane. Let T = 0.1s- (a) (c)
Find the z-transforms of the following functions: (a) (b) E(s) = = E(s) = -1) es(s+1) T = 0.5 s (0.55+1)(1-8-0.25) 0.5s(s+ 0.25) T = 0.25 s
(a) Find the system response at the sampling instants to a unit step input for the system of Fig. P4.3-2. Plot c(nT) versus time.(b) Verify your results of (a) by determining the input to the
(a) Find the conditions on a transfer function G(z) such that its dc gain is zero. Prove your result.(b) Forfind the conditions of Gp (s) such that the de gain of G(z) is zero. Prove your
Find the system response at the sampling instants to a unit-step input for the system of Fig. P4.3-5. E(s) = -=-/ 1 3+0.5 T=1s Fig. P4.3-5 1-g-7's S 38 (s + 1)(s+2) C(s)
(a) Find the output c(kT) for the system of Fig. P4.3-6, for e(t) equal to a unit-step function.(b) What is the effect on c(kT) of the sampler and data hold in the upper path? Why?(c) Sketch the
(a) Express each C(s) and C(z) as functions of the input for the systems of Fig. P4.3-7.(b) List those transfer functions in Fig. P4.3-7 that contain the transfer function of a data hold. E(s) E(s)
Shown in Fig. P4.3-8 is the block diagram of one joint of a robot arm. This system is described in Problem 1.5-4. The signal M(s) is the sampler input, Ea (s) is the servomotor input voltage, Θm(s)
Fig. P4.3-9 illustrates a thermal stress chamber. This system is described in Problem 1.6-1. The system output c(t) is the chamber temperature in degrees Celsius, and the control-voltage input m(t)
Given in Fig. P4.3-10 is the block diagram of a rigid-body satellite. The control signal is the voltage e(t) . The zero-order hold output m(t) is converted into a torqueτ(t) by an amplifier and the
The antenna positioning system described in Section 1.5 and Problem 1.5-1 is depicted in Fig. P4.3-11. In this problem we consider the yaw angle control system, where θ(t) is the yaw angle.(a) Find

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