Questions and Answers of Digital Control System Analysis And Design

Consider an open-loop system having a transfer function \(G(s)=(s+2)^{-1}\). Find its output \(c(t)\) in block pulse function (BPF) domain as well as in non-optimal block pulse function (NOBPF)
Consider the first-order system of Problem 9.1. Identify the system using "deconvolution" process for the same value of \(m, T\) and similar input in NOBPF and optimal block pulse function (OBPF)
Consider an output of a closed-loop system as \(c(t)=\exp (-t) \sin (3 t)\), having the feedback element \(h(t)=2 u(t)\). Identify the system \(g(t)\) in non-optimal and optimal BPF domain for a step
Consider an open loop system having a transfer function \(G(s)=(s+1)^{-1}\). Find its output \(c(t)\) in LPWM-GBPF domain for a step input \(u(t)\) using the convolution matrix. Consider \(m=8\) and
Consider the system of Problem 10.1. For the same values of \(m\) and \(T\), find its output \(c(t)\) in conventional BPF domain using the convolution matrix. Finally, compare the convolution result
Consider a closed loop system having the forward path transfer function \(G(s)=(s+1)^{-1}\) and the feedback path transfer function \(H(s)=\frac{4}{s}\). Find its output \(c(t)\) in LPWM-GBPF domain
Consider the feedback system of Problem 10.3. Double the value of \(m\) for the same interval \(T\) and find its output \(c(t)\) in BPF domain using the convolution matrix. Finally, comment on the
Consider the open loop system of Problem 10.1. Knowing the input and output, identify the system both in BPF domain and in LPWMGBPF domain using the "deconvolution" matrix, for \(m=8, T=1
Consider the closed loop system of Problem 10.3. Knowing the input output and feedback path transfer function, identify the system both in BPF domain and in LPWM-GBPF domain using the convolution
Consider the closed loop system of Problem 10.3. Identify the system in LPWM-GBPF domain for double the value of \(m\) keeping the time interval \(T\) same and new choices for \(h_{0}\) and
State the properties of an orthogonal function set.
What are piecewise constant orthogonal functions? What are the advantages of using these functions instead of sine-cosine functions?
Given the plant of Problem 11.4-4, with the cost function(a) Calculate the feedback gains required to minimize the cost function, using the difference-equation approach of Section 11.4.(b) Find the
A satellite control system is modeled as shown in Fig. P11.4-6. This system is described in Problem 1.4-1.For this problem, ignore the sensor gain and let D(z) = 1. In addition, K = 1, T = 1 s, and J
(a) Repeat Problem 11.4-6 for minimizing the cost functionPlot the gains on the graph from Problem 11.4-6(c)(b) Explain the differences in the gains in part (a) and those in Problem 11.4-6.Problem
A chamber temperature control system is modeled as shown in Fig. P11.4-8. This system is described in Problem 1.6-1. For this problem, ignore the disturbance input, T = 0.6 s and let D(z) = 1. It
Given the plant of Problem 11.4-9, with the cost function(a) Calculate the feedback gains required to minimize the cost function, using the difference-equation approach of Section 11.4.(b) Find the
Consider the system of Problem 11.4-4.(a) Find the feedback gain required to minimize the given cost function for the infinite-time problem.(b) Find the closed-loop system characteristic
Consider the system of Problem 11.4-5.(a) Find the feedback gain required to minimize the given cost function for the infinite-time problem.(b) Find the closed-loop system characteristic
Consider the satellite control system of Problem 11.4-6.(a) By computer, find the feedback gains required to minimize the given cost function for the infinite-time problem.(b) Find the closed-loop
Consider the satellite control system of Problem 11.4-7.(a) By computer, find the feedback gains required to minimize the given cost function for the infinite-time problem.(b) Find the closed-loop
Consider the system of Problem 11.4-9.(a) Find the feedback gain required to minimize the given cost function for the infinite-time problem.(b) Find the closed-loop system characteristic
Consider the system of Problem 11.4-10.(a) Find the feedback gain required to minimize the given cost function for the infinite-time problem.(b) Find the closed-loop system characteristic
Suppose that a plant is described by(a) Design a Kalman filter for this system. Continue the gain calculations until the gain is approximately constant. Use M(0) = 2.(b) In part (a), we specified
The Kalman filter design in Problem 11.7-1 resulted in the steady-state filter equations(a) Consider the plant and filter to be open-loop; that is, the state estimate is not fed back for control
Consider the system of Problem 11.7-1.(a) Suppose that all specifications are the same, except E[w( j)w(k)] = 0; that is, there are no random plant disturbances. Design a Kalman filter and estimate
Consider the satellite described in Problem 11.4-8. The plant model is given asThe plant disturbances are caused by random variations in the Earth’s gravity field. Suppose that Rw = 1 and Rv =
It is shown in Example 6.6 that the system of Fig. 6.2-2 has the parameters ζ = 0.250 , ωn = 0.9191 , and τ = 4.36 s .Example 6.6For this example we will consider the system of Example 6.4. For
Consider the system of Fig. P6.5-2, with the digital filter as described in Problem 6.5-2.(a) Find the system type.(b) Find the steady-state response for a unit ramp input, without finding
Consider the system of Problem 6.5-2. Let the plant transfer function be given byHence the dc gain of the plant is zero. The digital filter is as given in Problem 6.5-2.(a) Find the system
Repeat Problem 6.6-1, but with the process transfer function given byProblem 6.6-1 G(s) = 20 s(3s +1) Solve for the unit-step responses for 0 t 1.0s.
Consider the system of Problem 8.2-2. It is desired that the steady-state error constant for a unitramp input, \(K_{v}\) of (6-20), be 4, resulting in a steady-state error of 0.25.(a) Repeat Problem
8.2-4. Consider the system of Problem 8.2-2. The plant frequency response \(G\left(j \omega_{w}ight)\) is given in Table P8.2-2.(a) Let the digital controller realize a gain \(K\), that is,
8.4-2. To satisfy the steady-state constraints for the system of Problem 8.2-3, the dc gain of the digital controller must be equal 4(a) Design a phase-lag controller with this dc gain that will
8.4-3. Shown in Fig. P8.4-3 is the block diagram of the temperature control system described in Problem 1.6-1. For this problem, ignore the disturbance input. In Fig. P8.4-3 the sensor gain of
Repeat Problem 8.4-1, using a proportional-plus-integral digital filter.Problem 8.4-1 Consider the system of Fig. P8.2-2 with T = 1 s. The plant frequency response G(jo) is given in Table P8.4-1. (a)
Repeat Problem 8.6-1, using a proportional-plus-derivative filter.Problem 8.6-1Consider the system of Problem 8.4-1.(a) Design a unity-dc-gain phase-lead compensator that yields a phase margin of
Consider Problem 8.4-2 again. In this problem a proportional-plus-integral (PI) filter is to be designed.(a) The gain added in Problem 8.4-6 to reduce steady-state errors is no longer necessary.
In this problem we consider the effects of the disturbance input in Fig. P8.4-3.(a) Suppose that the disturbance input is a unit step, which models the door to the chamber remaining open. Find the
(a) Design a PI controller for Problem 8.6-5(b).(b) Design a PD controller for Problem 8.6-5(e).(c) Use the results of parts (a) and (b) to repeat Problem 8.6-5(d).Problem 8.6-5(b) (d) (e)(b) To
(a) Repeat Problem 8.6-6(a) and (b).(b) Design a PD controller for Problem 8.6-6(c). Use \(\omega_{w 1}=1\).(c) Use the results of part (b) to repeat Problem 8.6-6(d).Problem 8.6-6 Consider the block
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 Problem 8.10-10.(a) Plot the root locus for this system, and find the value of \(K>0\) for which the system is stable.(b) Find the time constant for the system with \(K=1\).(c)
A chamber temperature control system is modeled as shown in Fig. P9.2-3. This system is described in Problem 1.6-1. For this problem, ignore the disturbance input, T=0.6 sT=0.6 s, and let
Consider the chamber temperature control system of Problem 9.2-3.(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) To check
Consider the temperature control system of Problem 9.2-3.(a) Determine if this system is controllable.(b) An observer is added to this system in Problem 9.3-2, with the equation [see (9-38)]Construct
Repeat Problem 9.6-5 for the current observer.Problem 9.6-5Consider the first-order plant described byand a prediction observer described by (9-38). x(k + 1) = Px(k) + Qu(k) y(k) = Rx(k)
For the system of Problem 9.6-5, use a transfer function approach to show that the transfer function Q(z)/U(z)Q(z)/U(z) is first order (even though the system is second order) and is equal to
In Problem 9.2-4, a pole-placement design for a satellite control system results in the gain matrix K = [0.194 0.88]. It is desired to have an input signal r(t) applied to the system, so as to
Repeat Problem 9.7-1 for the temperature control system of Problem 9.2-3, where the state equations are given byProblem 9.7-1In Problem 9.2-4, a pole-placement design for a satellite control system
Repeat Problem 4.3-2 for the case that T = 0.1 s and the plant transfer function is given by:Problem 4.3-2(a) Find the system response at the sampling instants to a unit step input for the system of
Repeat Problem 4.4-3 for the case that the filter solves the difference equationProblem 4.4-3 m(k+ 1) = 0.5e(k+1) - (0.5)(0.98)e(k) +0.995m(k)* the sampling rate is 10 Hz, and the plant transfer
Find the z-transform of the following functions. The results of Problem 4.5-1 may be useful. (a) (c) (e) E(s) = E(s) = E(s) = 20 -0.37s (s+2)(s+5) -1.17s 5+267 s(s+1) (s +55 +6) e-0.37 s(s+4)(s+5)
Generally, a temperature control system is modeled more accurately if an ideal time delay is added to the plant. Suppose that in the thermal test chamber of Problem 4.3-9, the plant transfer function
For the thermal test chamber of Problems 4.3-9 and 4.6-1, let T = 0.6 s . Suppose that a proportional-integral (PI) digital controller with the transfer functionProblems 4.3-9Problems 4.6-1Generally,
Repeat Problem 4.10-1 for the plant described by the second-order differential equationProblem 4.10-1 dy(t) dt 15 dy(t) + 0.005y(t) = 0.1m(t) dt +0.15-
Shown in Fig. P5.3-12 is the block diagram for the temperature control system for a large test chamber. This system is described in Problem 1.6-1. The disturbance shown is the model of the effects of
Repeat Problem 5.4-1 for each of the transfer functionsProblem 5.4-1Given a closed-loop system described by the transfer function(a) Express c(k) as a function of r(k), as a single difference
Consider the temperature control system of Problem 5.3-12 and Fig. P5.3-12. Suppose that the digital filter transfer function is given bywhich is a PI (proportional-integral) controller. Suppose that
Consider the satellite control system of Problem 5.3-14 and Fig. P5.3-14. Let D(z) = 1 , T = 1 s, K = 2 , J = 0.1 , and Hk = 0.02 .(a) Using the closed-loop transfer function, derive a discrete state
Consider the antenna control system of Problem 5.3-15 and Fig. P5.3-15. Let D(z) = 1, T = 0.05 s , and K = 20 .(a) Using the closed-loop transfer function, derive a discrete state model for the
Consider the robot joint control system of Problem 5.3-13 and Fig. P5.3-13. Let D(z) = 1 ,T = 0.1 s, and K = 2.4 .(a) Using the closed-loop transfer function, derive a discrete state model for the
Consider the antenna control system of Fig. P7.5-5. This system is described in Problem 1.5-1. For this problem, T = 0.05 s and D(z) = 1. It was shown in Problem 5.3-15 that(a) Write the
Consider the satellite control system of Fig. P7.5-6. This system is described in Problem 1.4-1. For this problem, T = 0.1 s, J = 0.1, Hk = 0.02, and D(z) = 1. From the z-transform table,(a) Write
For the system of Problem 7.2-5 and Fig. P7.2-5:(a) Plot the z-plane root locus.(b) Plot the w-plane root locus.(c) Determine the range of K for stability using the results of part
For the chamber temperature control system of Problem 7.5-3 and Fig. P7.5-3:(a) Plot the z-plane root locus.(b) Plot the w-plane root locus.(c) Determine the range of K for stability using the
For the robot arm joint control system of Problem 7.5-4 and Fig. P7.5-4:(a) Plot the z-plane root locus.(b) Plot the w-plane root locus.(c) Determine the range of K for stability using the results
For the satellite control system of Problem 7.5-6 and Fig. P7.5-6:(a) Plot the z-plane root locus.(b) Plot the w-plane root locus.(c) Determine the range of K for stability using the results of
For the robot arm joint control system of Problem 7.5-4 and Fig. P7.5-4, let K = 1 .(a) The frequency response for G(z) was calculated by computer and is given in Table P7.7-1.Sketch the Nyquist
For the antenna control system of Problem 7.5-5 and Fig. P7.5-5, let K = 1 .Problem 7.5-5 (a) The frequency response for was calculated by computer and is given in Table P7.7-2. G(z) Sketch the
For the satellite control system of Problem 7.5-6, the frequency response for G(z) was calculated by computer and is given in Table P7.7-3.(a) Sketch the Nyquist diagram for the open-loop function
For the antenna control system of Problem 7.5-5 and Fig. P7.5-5:(a) Plot the z-plane root locus.(b) Plot the w-plane root locus.(c) Determine the range of K for stability using the results of part
For the system of Problem 7.2-5 and Fig. P7.2-5, let K = 1 .(a) Determine the stability of the system.(b) Sketch the Bode diagram, and use this diagram to sketch the Nyquist diagram.(c) If the
For the temperature control system of Problem 7.5-3 and Fig. P7.5-3, let K = 1 .(a) Determine the stability of the system.(b) Plot the Bode diagram, and the Nyquist diagram.(c) If the system is
The thermal chamber transfer function C(s) E(s) = 2 (s + 0.5) of Problem 1.6-1 is based on the units of time being minutes.(a) Modify this transfer function to yield the chamber temperature c(t)
(a) The transfer function for the right-side rectangular-rule integrator was found in Problem 2.2-1 to be Y(z)/X(z) = Tz/(z −1) . We would suspect that the reciprocal of this transfer function
Consider the system of Problem 2.10-2. A similarity transformation on these equations yields d w(k+ 1) = y (k) = C(k) 22 w(k)+ Bu(k) d (a) Find d, and d. (b) Find a similarity transformation that
Repeat Problem 2.10-2 for the system described by(a) Find the transfer function Y(z)/U(z).(b) Using any similarity transformation, find a different state model for this system.(c) Find the
Consider the system described in Problem 2.10-2.(a) Find the transfer function of this system.(b) Let u(k) = 1, k ≥0 (a unit step function) and x(0) = 0 . Use the transfer function of
Suppose that a square-law circuit has the input–output relationshipwhere x is the input voltage and y is the output voltage.(a) Derive a least-squares procedure for calculating
Equation (10-5) gives a least-squares estimate for curve fitting. This equation was derived by finding the point at which the slope of the cost function is zero. The maximum of a function also occurs
Consider the sinusoidal function y(k) = sin (10 k) for k = 0, 1, . . . , 9. Find the minimum degree of the polynomial which fits this function with a maximum least-squares error of 10-3.
Using the data given in Problem 10.5-1, derive the state-variable model using the algorithm described in Section 10.4.Problem 10.5-1(a) A first-order system yields the following input–output
Derive the SISO black-box identification algorithm described in Section 10.4 when the excitation input u(k) is a unit step signal.SISO black-box 10.4 BLACK-BOX IDENTIFICATION For simplicity of
Consider the SISO continuous-time model:Assume u(k) to be an impulse input. Sample this system with T = 10 s, 1 s, and 0.1 s, and compare how accurately the estimated impulse responses, computed
(a) A first-order system yields the following input–output measurements:Find the system transfer function by the least-squares batch procedure, using all the data.(b) An additional data pair for
For a third-order discrete system, suppose that the data pairs [u(k), y(k)], k = 0, 1,g5, are available.In these data, u(k) is the input and y(k) is the output. Write the complete expression for
Derive (10-47), the weighted least-squares estimate.Equation 10-47 WLS = [FT(N)W(N)F(N)]F(N)W(N)y(N) (10-47)
Consider that b1, b2,g, bm in (10-30) for some given positive integer m are all known constants. Using the method proposed in Section 10.6 re-derive (10.43) to compute the least-squares estimate of
Using (10-47), (10-52), (10-56), and (10-57), derive the equations for the recursive least-squares estimation, (10-59)–(10-61). OWLS = [FT(N)W(N)F(N)JFT(N)W(N)y(N) (10-47)
Using the MATLAB function awgn, repeat the recursive least-squares estimation of Example 10.4 when the output is corrupted by a white Gaussian noise with 10.0 dB SNR. Compare the estimation results
Consider the values of y(k) in Problem 10.6-3 at k = 0, 2, 4, 6, and 8. Using these values and assuming T = 0.2 s, repeat the least-squares estimation of the transfer function. Next, consider the
Consider the transfer functionExcite this transfer function with the following inputs, and collect the respective outputs in a vector y(k), k = 0, 1, . . . , 20. Assume the sampling time T = 0.1
Given that (11-21) and (11-22) are valid, derive (11-23). S S + x(N-1)Q(N-1)x(N-1) + u(N-1)R(N 1)u(N 1) = (11-21)
Show that (11-35) can also be expressed as P(Nm) = A[P(Nm +1) P(N m + 1)BDB/P(N m + 1)]A + Q - where D= [B'P(Nm + 1)B + RJ-. -
Given the discrete systemshow that the optimal gains which minimize JN are unchanged if the elements both in Q and in R are multiplied by the positive scalar β. with the cost function x(k+1) = Ax(k)
Given the first-order plant described by(a) Calculate the feedback gains required to minimize the cost function, using the partial-differentiation procedure of Section 11.3.(b) Repeat part
Consider the third-order continuous-time LTI system with A = 208 0 0 0 3 0 -8 -6 x = Ax + Bu y = Cx 0 B = 0, and C= [1 0 0]. Using = 8 0 0 0 6 0 0 4 3 R = 1.5 (a) First design a LQ controller for
Given a general first-order plant described by(a) Show that for R = 0, the optimal gain K(k) is constant for all k ≥ 0.(b) Give the input sequence u(k), k ≥ 0, for part (a), where u(k) is a
It is shown in [6] that given the partitioned matrix H = D E F G where each partition is n X n, the determinant of H is given by |H| = |G||D EGF| = |D||G FDE| Show that the determinant of H in

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