Questions and Answers of Digital Control System Analysis And Design

Repeat Problem 4.10-1 for the plant described by the second-order differential equation for \(T=2\)\[\frac{d^{2} y(t)}{d t^{2}}+0.25 \frac{d y(t)}{d t}+0.005 y(t)=0.2 m(t)\]
Repeat Problem 5.4-1 for each of the transfer functions(a) \(\frac{C(z)}{R(z)}=\frac{0.5 z}{z-1.8}\)(b) \(\frac{C(z)}{R(z)}=\frac{0.4}{z-0.16}\)(c) \(\frac{C(z)}{R(z)}=\frac{0.1 z-0.05}{z-0.9}\)(d)
Let \(T=0.5 \mathrm{~s}\) for the system of Fig. P5.4-4. Derive a set of discrete state equations for the closed-loop system, for the plant described by each of the differential equations.(a)
Consider the satellite control system of Problem 5.3-14 and Fig. P5.3-14. Let \(D(z)=2, T=2 \mathrm{~s}, K=4\), \(J=0.2\), and \(H_{k}=0.04\).(a) Using the closed-loop transfer function, derive a
Suppose that, for the system of Fig. 5-20, equation (5-34) is\[y(k)=\mathbf{C v}(k)+d_{2} m(k)\](a) Derive the state model of (5-37) for this case.(b) This system has an algebraic loop. Identify this
(a) In the system of Problem \(1.4-1, J=0.6\) and \(K=12.4\), in appropriate units. The attitude of the satellite is initially at \(0^{\circ}\). At \(t=0\), the attitude is commanded to
Consider the robot arm depicted in Fig. P1.5-4.(a) Suppose that the units of \(e_{a}(t)\) are volts, that the units of both \(\theta_{m}(t)\) and \(\theta_{a}(t)\) are degrees, and that the units of
The thermal chamber transfer function \(C(s) / E(s)=2.5 /(s+1)\) is based on the units of time being minutes.(a) Modify this transfer function to yield the chamber temperature \(c(t)\) based on
Consider the single-machine infinite bus power system of Fig. 1-13 with \(M=0.5, d=0.1\), and \(k=10\). Find the steady-state gain of the closed-loop transfer function when:(a) \(C(s)=1\)(b)
Consider the single-machine infinite bus power system of Fig. 1-13 with \(M=0.5, d=0.1, k=10\), and \(C(s)=1\). Simulate the unit step response for this system, and compute the rise time of the
Consider a slightly different block-diagram for the closed-loop single-machine infinite-bus power system of Fig. 1-13 as shown below in Fig. P1.7-3. In this block diagram the controller \(C(s)\) is
Find the \(z\)-transforms of the number sequences generated by sampling the following time functions every \(T\) seconds, beginning at \(t=0\). Express these transforms in closed form.(a)
From Table 2-3,\[y[\cos a k T]=\frac{z(z-\cos a T)}{z^{2}-2 z \cos a T+1}\](a) Find the conditions on the parameter \(a\) such that \({ }_{z}[\cos a k T]\) is first order (pole-zero cancellation
Given the difference equation\[x(k+3)-2.2 x(k+2)+1.57 x(k+1)-0.36 x(k)=e(k)\]where \(e(k)=1\) for all \(k \geq 0\), and \(x(0)=x(1)=x(2)=0\).(a) Write a digital computer program that will calculate
Given the MATLAB programthat solves the difference equation of a digital controller.(a) Find the transfer function of the controller.(b) Find the \(z\)-transform of the controller input.(c) Use the
(a) Find \(e(0), e(1)\), and \(e(10)\) for\[E(z)=\frac{0.8}{z(z-0.6)} \]using the inversion formula.(b) Check the value of \(e(0)\) using the initial-value property.(c) Check the values calculated in
For the number sequence \(\{e(k)\}\),\[E(z)=\frac{z}{(z-1)^{2}}\](a) Apply the final-value theorem to \(E(z)\).(b) Check your result in part (a) by finding the inverse \(z\)-transform of \(E(z)\).(c)
Find the inverse \(z\)-transform of each \(E(z)\) below by any method.(a) \(E(z)=\frac{0.5 z^{2}}{(z-1)^{2}(z-0.6)}\)(b) \(E(z)=\frac{0.5}{(z-1)(z-0.6)^{2}}\)(c) \(E(z)=\frac{0.5
Given in Fig. P2.8-1 are two digital-filter structures, or realizations, for second-order filters.(a) Write the difference equation for the 3D structure of Fig. P2.8-1(a), expressing \(y(k)\) as a
Shown in Fig. P2.8-2 is the second-order digital-filter structure 1X. This structure realizes the filter transfer function\[D(z)=b_{2}+\frac{A}{z-p}+\frac{A^{*}}{z-p^{*}}\]where \(p\) and \(p^{*}\)
Given the second-order digital-filter transfer function\[ D(z)=\frac{2 z^{2}-2.4 z+0.72}{z^{2}-1.4 z+0.98} \](a) Find the coefficients of the 3D structure of Fig. P2.8-1 such that \(D(z)\) is
Find two different state-variable formulations that model the system whose difference equation is given by:(a) \(y(k+2)+6 y(k+1)+5 y(k)=2 e(k)\)(b) \(y(k+2)+6 y(k+1)+5 y(k)=e(k+1)+2 e(k)\)(c)
Note the relationship between the control canonical form in Fig. 2-9 and the observer canonical form in Fig. 2-10. The diagram of Fig. 2-9 (b) can be converted to that of Fig. 9-10 by changing all
Consider a system described by the coupled difference equation\[\begin{aligned}y(k+2)-v(k) & =0 \\v(k+1)+y(k+1) & =u(k)\end{aligned}\]where \(u(k)\) is the system input.(a) Find a state-variable
Show that for the similarity transformation of (2-71),\[\mathbf{C}[z \mathbf{I}-\mathbf{A}]^{-1} \mathbf{B}+\mathbf{D}=\mathbf{C}_{w}\left[z \mathbf{I}-\mathbf{A}_{w}ight]^{-1}
Find \(E^{*}(s)\) for\[E(s)=\frac{1-\varepsilon^{-T s}}{s(s+1)}\]
Find \(E^{*}(s)\), with \(T=0.2 \mathrm{~s}\), for\[E(s)=\frac{\varepsilon^{-2 s}}{(s+1)(s+2)}\]
Derive the Walsh coefficients for the following functions for \(m=4\) :a. \(f(t)=\left\{\begin{array}{l}0,0 \leq t
Derive the block pulse function (BPF) coefficients for the functions given in Problem 2.3.Data From Problem 2.3Derive the Walsh coefficients for the following functions for \(m=4\) :a.
Compare the results of Problem 4.5 with the direct BPF expansion of the exact time domain output by estimating the mean integral square error (MISE) and comment on the effectiveness of the BPOTF
Consider a closed loop system having a loop transfer function \(G(s) H(s)=\left(s^{2}+s+4ight)^{-1}\). Find its output \(c(t)\) in BPF domain for a step input \(u(t)\), using block pulse operational
Consider an open loop system having a transfer function \(G(s)=\) \(\left(s^{2}+s+4ight)^{-1}\) and an S/H device at the input. For a unit step input \(u(t)\), determine its output \(c(t)\) by the
Determine the output in z-domain for the following systems when a step input is applied through a sampler:i. \(\quad G(s)=\frac{1}{(s+4)}\)ii. \(G(s)=\frac{1}{(s+3)^{2}}\)iii. \(G(s)=\frac{1}{\left(2
Consider an open loop system having a transfer function \(G(s)=\) \(\left(s^{2}+4ight)^{-1}\) and a sampler placed before \(G(s)\). Find its output \(c(t)\) by the conventional \(z\)-transform
Repeat Problem 8.5 considering \(G(s)=\left(s^{2}+s+4ight)^{-1}\).Data From Problem 8.5Consider an open loop system having a transfer function \(G(s)=\) \(\left(s^{2}+4ight)^{-1}\) and a sampler
Consider an undamped system having a transfer function \(G(s)=\) \(\left(s^{2}+2ight)^{-1}\). Find its output \(c(t)\) in BPF domain as well as in NOBPF domain, for a ramp input \(r(t)\) using the
Consider the second-order undamped system of Problem 9.3. Identify the system using "deconvolution" process for the same value of \(m, T\) and similar input in NOBPF and OBPF approaches. Compared to
Consider an underdamped system having a transfer function \(G(s)=\) \(\left(s^{2}+2 s+2ight)^{-1}\). Find its output \(c(t)\) in BPF domain as well as in NOBPF domain, for a step input \(u(t)\) using
Consider the second-order underdamped system of Problem 9.5. Identify the system using "deconvolution" process for the same value of \(m, T\) and similar input in NOBPF and OBPF approaches. Compared
In Equation (10-7), show that if N 6 n then the solution of β is non-unique. For the example in Problem 10.2-3, calculate the error bound when the polynomial order n is exactly equal to the number
Consider a variant of the matrix A used in Example 10-2. In this example, we used A(2,1) = -0.6.(a) Changing this to A(2,1) = -0.9, and assuming exact same values for B and C as in the example, what
Consider the following three data sets, which are the same data set given in Example 10.2 but corrupted with varying levels of white Gaussian noise. Three different SNRs are used to generate the
Derive the transfer function for the LTI model in Examples 10-3 and 10-4 using weighted least squares.Manipulate the weights and test whether the transfer function can be identified with lesser
Consider the data set given in Example 10.2. Assume that the system has three poles and two zeros. One of the poles is at z = 0.1 and one of the zeros is at z = 0.10002. Identify the system transfer
Consider the following unit step response of a discrete-time LTI system:It is known that the transfer function of the system has three poles and two zeros. All three poles are atz = 1. Assuming the
Use recursive least squares to estimate the transfer function for the LTI model given in Example 10-2 using the impulse response data y(k). Verify that the poles of this transfer function match the
Write down the Haar matrix for \(m=4\).
Write down the Walsh matrix for \(m=4\).
Define MISE. Show that the selection of \(c_{j}\) as\[c_{j}=\frac{1}{T} \int_{0}^{T} f(t) \varphi_{j}(t) \mathrm{d} t\]reduces MISE to a minimum.
Show how Rademacher functions are related to Walsh functions. Then, find \(\varphi_{17}\) of a Walsh function set using Rademacher functions.
Show with a neat sketch a block pulse function (BPF) set of order 6 . Show mathematically, how a BPF set is normalized.
Compare the major piecewise constant basis orthogonal functions, e.g., Walsh, Haar, block pulse function (BPF) and sampleand-hold function (SHF) with respect to their attributes and properties.
What are the basic differences between the optimal block pulse function set and the non-optimal block pulse function set?
Chyi Hwang, Solution of functional differential equation via delayed unit step functions, Int. J. Syst. Sci., vol. 14, no. 9, pp. 1065-1073, 1983.
Anish Deb, Gautam Sarkar, Manabrata Bhattacharjee, and Sunit K. Sen, A new set of piecewise constant orthogonal functions for the analysis of linear SISO systems with sample-and-hold, J. Franklin
G. P. Rao, Piecewise constant orthogonal functions and their application to systems and control, LNC1S, vol. 55, Springer-Verlag, Berlin, 1983.
G. P. Rao and T. Srinivasan, Analysis and synthesis of dynamic systems containing time delays via block-pulse functions, Proc. IEE, vol. 125, no. 9, pp. 1064-1068, 1978.
Derive the relation \(\boldsymbol{\Phi}=\mathbf{W} \Psi\) where \(\mathbf{W}\) is the Walsh matrix, \(\boldsymbol{\Phi}\) is the Walsh function vector, \(\Psi\) is the block pulse function vector of
Show that the Walsh matrix \(\mathbf{W}\) has the property \(\mathbf{W}_{(m)}^{2}=m \mathbf{I}_{(m)}\), where each of the matrices is of order \(m\).
Find the BPF coefficients of the following functions for \(m=6\) :a. \(f(t)=\exp (-2 t), 0 \leq t
Derive the NOBPF coefficients of the following functions for \(m=7\) :a. \(f(t)=t^{2}, 0 \leq t
Derive the delayed unit step function (DUSF) coefficients for the functions of Problem 2.6 with m = 10. Also show that the results in DUSF domain are same as that obtained using BPF domain
In BPF domain, using the operational matrix \(\mathbf{P}\), we can integrate any time function expanded via block pulse functions. Why this matrix \(\mathbf{P}\) is called an 'Operational Matrix'?To
Decompose the \((i+1)\) th block pulse function component using two delayed unit step functions, and graphically prove that the integration of these component functions finally result in the figure
Explain why the operational matrix for integration \(\mathbf{P}\), is of an upper triangular form. What are the mathematical advantages of an upper triangular matrix?
Integrate a function \(f(t)=\sin (\pi t)\) in BPF domain, considering \(m=\) 10 and \(T=1 \mathrm{~s}\), and estimate the MISE.
Integrate a function \(f(t)=\exp (-t)\) in GBPF domain, considering \(h_{0}=0.55 \mathrm{~s}, \delta=0.1 \mathrm{~s}, m=10\) and \(T=10 \mathrm{~s}\). Then estimate the MISE.
Integrate a function \(f(t)=t\) in BPF domain, using the operational matrix for integration and its improved versions \(\mathbf{P 1}\) and \(\mathbf{P 2}\), considering \(m=10\) and \(T=1
Integrate the function \(f(t)=t\) twice using the operational matrix \(\mathbf{P}\) and the one-shot operational matrix \(\mathbf{P ( 2 )}\) in BPF domain, considering \(m=10\) and \(T=1
Differentiate the function \(f(t)=t^{3}\) in BPF domain using the \(\mathbf{D}\) matrix considering \(m=10\) and \(T=1 \mathrm{~s}\). Then find out the percentage error of the BPF coefficients
Differentiate the function \(f(t)=t^{4}\) twice in BPF domain using the first-order operational matrix \(\mathbf{D}\) twice and the one-shot operational matrix \(\mathbf{D}(2)\) considering \(m=10\)
What are the advantages of using block pulse operational transfer function (BPOTF) in control system analysis?
Consider an open loop system having a transfer function \(G(s)=\) \((s+4)^{-1}\). Find its output \(c(t)\) in BPF domain for a step input \(u(t)\) using block pulse operational transfer function
Compare the results of Problem 4.2 with direct BPF expansion of the exact time domain output by estimating the mean integral square error (MISE). Plot the exact solution along with the two BPF
Solve Problem 4.2 using Walsh Operational Transfer Function (WOTF) and prove that the results are identical with that obtained via BPOTF.Data From Exercise 4.2Consider an open loop system having a
Consider an open loop system having a transfer function \(G(s)=\) \(\left(s^{2}+4ight)^{-1}\). Find its output \(c(t)\) in BPF domain for a step input \(u(t)\) using block pulse operational transfer
Consider two block pulse functions of different heights \(c_{0}\) and \(c_{1}\) having the same width of \(h\) seconds. Then graphically determine the convolution result in time domain and
Consider an open loop system having a transfer function \(G(s)=\) \((s+1)^{-1}\). Find its output \(c(t)\) in BPF domain for a step input \(u(t)\) using the convolution matrix. Consider \(m=4\) and
Repeat Problem 5.2 for \(m=8\) and \(T=1 \mathrm{~s}\). Compare the two results and discuss.Data From Problem 5.2Consider an open loop system having a transfer function \(G(s)=\) \((s+1)^{-1}\). Find
Consider a closed loop system having the forward path transfer function \(G(s)=(s+1)^{-1}\) and the feedback path transfer function \(H(s)=\frac{2}{s}\). Find its output \(c(t)\) in BPF domain for a
Consider the feedback system of Problem 5.4. Double the value of mm for the same interval TT and find its output c(t)c(t) using the convolution matrix. Finally, comment on the convolution result.Data
Consider the open loop system of Problem 5.2. Knowing the input and output, identify the system using (a) the "deconvolution" matrix and (b) the recursive approach, for m=8m=8 and T=1 sT=1 s.
Consider the open loop system of Problem 5.6. After identifying the system using the "deconvolution" approach for \(m=8\) and \(T=1 \mathrm{~s}\), compare the result with the direct BPF expansion of
Consider the open loop system of Problem 5.6. Identify the system for double the value of \(m\) keeping the time interval \(T\) same. Compute percentage error for each segment and make your
Consider the closed loop system of Problem 5.4. Knowing the input, output, and feedback path transfer function, identify the system using (a) the "deconvolution" matrix and (b) the recursive approach
Derive the delayed unit step function (DUSF) coefficients for the function \(f(t)=\sin (\pi t), 0 \leq t
Compare qualitatively the basic properties of the two piecewise constant basis function sets: BPF and DUSF.
Derive the operational matrix for integration \(\mathbf{P}_{D}\) in DUSF domain. Explain why the integration matrix is identical with \(\mathbf{P}_{B}\), the operational matrix for integration in BPF
Expand the function \(f(t)=1-\exp (-t)\) in BPF domain with \(m=8\) and \(T=1 \mathrm{~s}\) and plot the result along with the exact curve of \(f(t)\).Also, obtain the approximation of the scaled
Expand the function \(f(t)=\cos (\pi t)\) in BPF domain for \(m=10\) and \(T=1\) \(\mathrm{s}\) and then integrate it using the operational matrix for integration \(\mathbf{P}_{\mathrm{B}}\).Also,
Consider the equation \(\dot{x}(t)=-x(t / \lambda), x(0)=1\). For \(m=16, T=2 \mathrm{~s}\), and \(\lambda=2\), determine the solution in DUSF domain. Convert the DUSF result into BPF domain as well
Consider the equation \(\dot{x}(t)=-x(0.99 t)-0.95 x(t), x(0)=1\). For \(m=8\) and \(T=1 \mathrm{~s}\) determine the solution in DUSF domain. Convert the DUSF result into BPF domain as well as Walsh
What are the advantages of using sample-and-hold operational transfer function (SHOTF) over the block pulse operational transfer function (BPOTF) in analyzing control systems involving
To implement an S/H device in practice what components do we need?
Consider an open loop system having a transfer function \(G(s)=\) \((s+4)^{-1}\) with an S/H device at the input. For a unit step input \(u(t)\), find the output \(c(t)\) of the system by the
Compare the results of Problem 7.3 obtained using BPOTF, SHOTF, and conventional z-transform analysis by estimating the percentage errors at the sampling instants and discuss the effectiveness of the
Compare the results of Problem 7.5 obtained using BPOTF, SHOTF, and conventional z-transform analysis by estimating the percentage errors at the sampling instants and discuss the effectiveness of the
Consider an open loop system having a transfer function G(s) = (s + 4)–1 and a sampler at the input. Find its output c(t) by the conventional z-transform analysis, for a unit step input u(t) and
Consider an open loop system having a transfer function G(s) = (s + 4)–1 and a sampler placed before G(s). Find its output c(t) by the conventional z-transform analysis for a sampling period h = 1
Repeat Problem 8.3 considering G(s) = (s + 2)–2.Data From Problem 8.3Consider an open loop system having a transfer function \(G(s)=\) \((s+4)^{-1}\) and a sampler placed before \(G(s)\). Find its

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