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computer science
digital control system analysis and design

Questions and Answers of Digital Control System Analysis And Design

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
Example 4.3 calculates the step response of the system in Fig. 4-2. Example 4.4 calculates the step response of the same system preceded by a digital filter with the transfer function D(z) = (2 −
Consider the hardware depicted in Fig. P4.4-2. The transfer function of the digital controller implemented in the computer is given by D(2) = 4.5(z - 0.90) z - 0.85 The input voltage rating of the
For the system of Fig. P4.4-3, the filter solves the difference equation m(k) = 0.9m(k −1)+ 0.2e(k)The sampling rate is 1 Hz and the plant transfer function is given by E(s) e(t) T using E*(s)
Consider the system of Fig. P4.4-5. The filter transfer function is D(z). E(s) G(s) A/D Digital filter DIA Fig. P4.4-5 Express c(z) as a function of E G(s) C(s) (a) (b) A discrete state model of this
Find the modified z-transform of the following functions: (a) (c) (e) E(s) = 20 (s+2)(s+5) s+2 E(s) = s(s+1) E(s) = s +55+6 s(s+4) (s+5) (b) (d) (f) 5 E(s) = s(s+1) E(s) = E(s) = s+2 s (s+1) 2 s + 25
For Problem 4.4-6, assume that the total computational delay in D(z) plus the plant delay totals 3.4T seconds. Repeat the problem under these new conditions.Problem 4.4-6Consider again the system of
Find a discrete-state-variable representation for the system shown in Fig. P4.8-1. A discrete-state-variable description of the continuous-time system is given by E(s) T Digital filter 0.1z z-1 0.9 1
(a) The model of a continuous-time system with algebraic loops is given as x(t)=2x(t) + 0.5x(t) + 3u(t) y(t) = x(t) + 4u(t) Derive the state equations for this system. (b) Repeat part (a) for the
Consider the system of Fig. P4.10-1. The plant is described by the first-order differential equation Let T = 2 s. E(s) dy(t) + 0,05y (t) = 0.1m(t) dt G(s) A/D Digital filter DIA G(s) G (s) C(s) Fig.
Consider the robot arm system of Fig. P4.3-8. Let T = 0.1 s .(a) Find the system transfer function Θa (z) M(z).(b) Draw a discrete simulation diagram, using the results of part (a), and give the
Consider the thermal stress chamber depicted in Fig. P4.3-9. Let T = 0.6 s . Ignore the disturbance input d(t) for this problem.(a) Find the system transfer function C(z) E(z) .(b) Draw a discrete
Repeat Problem 4.10-4 for the satellite system of Fig. P4.3-10, with T = 1 s . In part (a), the required transfer function is Θ(z) E(z).Problem 4.10-4Consider the thermal stress chamber depicted in
Repeat Problem 4.10-4 for the antenna system of Fig. P4.3-11, with T = 0.05 s . In part (a), the required transfer function is Θ(z) E(z).Problem 4.10-4Consider the thermal stress chamber depicted in
Consider a proportional-integral (PI) digital controller with the transfer function D(z) = 1.2 + 0.1z 2-1 This controller is placed in the designated system between the sampler and data hold. Find a
For each of the systems of Fig. P5.3-1, express Error! Objects cannot be created from editing field codes. as a function of the input and the transfer functions shown. R(s) R(8) R(s) R(s) T T T (a) T
For each of the systems of Fig. P5.3-2, express C (z) as a function of the input and the transfer functions shown. R R(s) R R T T -G(s) G(s) H(s) H(s) (a) G(s) H(s) (b) H(s) 6 G(s) H(8) (d) -G(s)
(a) Derive the transfer function C(z) R(z) for the system of Fig. P5.3-1(b).(b) Derive the transfer function C(z) R(z) for the system of Fig. P5.3-1(c).(c) Even though the two systems are different,
Consider the system of Fig. P5.3-4.(a) Calculate the system output C(z) for the signal fromG2 (s) disconnected from the middlesumming junction.(b) Calculate the system output C(z) for the signal
For the system of Fig. P5.3-1(a), suppose that the sampler in the forward path samples at t = 0, T, 2T, …, and the sampler in the feedback path samples at t = T 2, 3T 2, 5T 2 , . . . . (a) Find the
The system of Fig. P5.3-6 contains a digital filter with the transfer function D(z). Express φm(z) as a function of the input. The roll-axis control system of the Pershing missile is of this
Consider the two-loop system of Fig. P5.3-7. The gain K is used to give the inner loop certain specified characteristics. Then the controller D(z) is designed to compensate the entire system. The
The system of Fig. P5.3-8 is the same as that of Example 5.1, except that the sampler has been moved to the feedback path. This may occur for two reasons: (1) the sensor output is in sampled form and
For Fig. P5.3-1, list, for each system, all transfer functions that contain the transfer function of azero-order hold.Fig. P5.3-1 R(s) R(s) R(s) + T T T T (a) (b) G(s) H(s) G(s) H(s) G(s) H(s) T C(s)
For Fig. P5.3-2, list, for each system, all transfer functions that contain the transfer function of azero-order hold. R + R(s) + R + T T G(s) G(s) H(s) G(s) H(s) (b) + H(s) (c) T H(s) G(s) G(s) T
In the system of Fig. P5.3-11, the ideal time delay represents the time required to complete the computations in the computer.(a) Derive the output function C(z) for this system.(b) Suppose that the
Consider the satellite control system of Fig. P5.3-14. The units of the attitude angle θ(t) is degrees, and the range is 0 to 360° . The sensor gain is Hk = 0.02 .(a) Suppose that the input ranges
Consider the antenna control system of Fig. P5.3-15. The units of the antenna angle θ(t) is degrees, and the range is ±45° .(a) The input signalr (k) is generated in the computer. Find the
Given a closed-loop system described by the transfer function(a) Express c(k) as a function of r(k), as a single difference equation.(b) Find a set of state equations for this system.(c) Calculate
For the system of Fig. P5.3-1(a), let T = 0.1 s and(a) Calculate G(z) and H(z) .(b) Draw simulation diagrams for G(z) and H(z) , and interconnect these diagrams to form the control system of Fig.
Let T = 0.1 s for the system of Fig. P5.4-4. Derive a set of discrete state equations for the close dloopsystem, for the plant described by each of the differential equations. U(s) + M(s) T 1-8-7's S
Suppose that the plant in Fig. P5.4-4 has the discrete state modelDerive the state model for the closed-loop system, in terms of A, B, C, and D. 1 (k+ 1) = Ax(k) + Bm(k) y (k) = Cx(k) + Dm(k)
Find a discrete state variable model of the closed-loop system shown in Fig. P5.4-4 if the discretestate model of the plant is given by:Fig. P5.4-4 (a) x(k+1)= 0.7x(k)+0.3m (k) y (k) = 0.2x(k) +
Suppose that, for the system of Fig. 5.4-5, equation (5-34) isFig. 5.4-5 y(k) = Cv (k) + dm (k) (a) Derive the state model of (5-37) for this case. (b) This system has an algebraic loop. Identify
Assume that for the system of Fig. 7-1, the system closed-loop transfer-function pole p1 is repeatedsuch that the system characteristic equation is given byFig. 7-1 (z P) (2 Pr + 1)( Pr + 2)(2
The system of Example 7.1 and Fig. 7-3 has two samplers. The system characteristic equation is derived in Example 7.1 asShow that the same characteristic equation is obtained by opening the system at
(a) The unit-step response of a discrete system is the system response c(k) with the input r(k) = 1 for k ≥ 0 . Show that if the discrete system is stable, the unit-step response, c(k), approaches
Consider a sampled-data system with T = 0.5 s and the characteristic equation given by(a) Find the terms in the system natural response.(b) A discrete LTI system is stable, unstable, or marginally
Consider the system of Fig. P7.2-5 with T = 1 s . Let the digital controller be a variable gain K such that D(z) = K . Hence m(kT) = Ke(kT).(a) Write the closed-loop system characteristic
Consider the system of Fig. P7.2-5, and let the digital controller be a variable gain K such that D(z) = K. Hence m(kT) = Ke(kT).(a) Write the closed-loop system characteristic equation as a
Consider the general bilinear transformationwhere a is real and nonzero.(a) Show that this function transforms the stability boundary of the z-plane into the imaginary axis in the w-plane.(b) Find
Given below are the characteristic equations of certain discrete systems.(a) Use the Jury test to determine the stability of each of the systems.(b) List the natural-response terms for each of the
Consider the system of Fig. P7.2-5 with T = 1 s. Let the digital controller be a variable gain K such that D(z) = K. Hence m(kT) = Ke(kT).(a) Write the closed-loop system characteristic
Consider the temperature control system of Fig. P7.5-3. This system is described in Problem 1.6-1. For this problem, ignore the disturbance input, let T = 0.6 s, and let the digital controller be a
Consider the robot arm joint control system of Fig. P7.5-4. This system is described in Problem 1.5-4. For this problem, T = 0.1 s and D(z) = 1. It was shown in Problem 6-7 that(a) Write the
For the system of Fig. P7.5-7, T = 2 s and(a) Determine the range of K for stability using the Routh–Hurwitz criterion.(b) Determine the range of K for stability using the Jury test.(c) Show
Given the pulse transfer function G(z) of a plant. For w = 2 rad/s, G (EJWT) is equal to the
(a) Find the transfer function Θ(s) / Θi (s) for the antenna pointing system of Problem 1.5-1(b). This transfer function yields the angle θ(t) in degrees.(b) Modify the transfer function in part
From Table 2-3, *[cos akT] = z(z - cos aT) z - 2z cos aT +1 2
Solve the given difference equation for x(k) using: x(k)-3x(k-1)+2x(k-2)= e(k), e(k) = (1, k = 0, 1 0, k2 x(-2) = x(-1) = 0 (a) The sequential technique. (b) The Z-transform. (c) Will the final-value
Given the difference equation y(k+ 2) / v(k + 1) + v(k) = e(k) where y(0) = y(1) = 0, e(0) = 0, and e(k) = 1, k = 1, 2,... . (a) Solve for y(k) as a function of k, and give the numerical values of
Given the difference equation where e(k)= x(k+ 2) + 3x (k+1) + 2x(k)= e(k) 1, k=0 0, otherwise x(0) = 1 x(1) = -1 (a) Solve for x(x) as a function of k. (b)Evaluate x(0), x(1), x(2), and x(3) in part
Given the difference equation x(k+ 3) - 2.2x(k+ 2) +1.57x(k+1) 0.36x(k) = e(k) where e(k) = 1 for all k 0, and x(0) = x(1) = x(2) = 0 . (a) Write a digital computer program that will calculate x(k).
(a) Find e(0) , e(1) , and e(10) for E (2) = 0.1 z(z - 0.9) using the inversion formula. (b)Check the value of e(0) using the initial-value property. (c) Check the values calculated in part (a)
Find the inverse z-transform of each E(z) below by the four methods given in the text. Compare the values of e(z) , for k = 0, 1, 2, and 3, obtained by the four methods. (a) E(z) = 0.5z (2-1)(z-0.6)
Given in Fig. P2.8-1 are two digital-filter structures, or realizations, for second-order filters.Example 2.10It is desired to find m(k) for the equation B T B T Bo (a) y(k) T 7 3 (x1 CLO T y(k)
Shown in Fig. P2.8-2 is the second-order digital-filter structure 1X. e(k) 83 84 fi(k) f(k) b 81 T 82 82 T 81 FIGURE P2.8-2 Digital-filter structure 1X. y(k)
Given the second-order digital-filter transfer functionFigure p2.8-2Figure p2. 8-1 2z-2.4z+0.72 z - 1.4z +0.98 (a) Find the coefficients of the 3D structure of Fig. P2.8-1 such that D(z) is realized.
Find two different state-variable formulations that model the system whose difference equation is given by: (a) y(k+ 2) + 6y(k+1) + 5y (k)= 2e(k) (b) y(k + 2) + 6y(k+1) + 5y(k) = e(k+1) + 2e(k) (c)
Find a state-variable formulation for the system described by the coupled second-order difference equations given. The system output is y(k) , and e1(k) and e2 (k) are the system inputs. y(k) = v(k +
Consider 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 transfer function of the
Consider a system with the transfer functionEquation 2-84 Y(z) G() = 2(3) = 2 (2-1) U (a) Find three different state-variable models of this system. (b) Verify the transfer function of each state
Consider a system described by the coupled difference equationEquation 2-84 y (k+ 2) - v (k)= 0 v(k + 1) + y (k + 1) = u(k) where u(k) is the system input.
Section 2.9 gives some standard forms for state equations (simulation diagrams for the control canonical and observer canonical forms). The MATLAB statementgenerates a standard set of state equations
The system described by the equations(a) Use (2-89) to solve for x(k), k ≥0.(b) Find the output y(z) .(c) Show that Φ(k) in (a) satisfies the property Φ(0) = I.(d) Show that the solution in
(a) Show that the transfer function of two systems in parallel, as shown in Fig. P1.1-l(a), is equal to the sum of the transfer functions.(b) Show that the transfer function of two systems in series
By writing algebraic equations and eliminating variables, calculate the transfer function C(s) R(s) for the system of:(a) Figure P1.1-2(a).(b) Figure P1.1-2(b).(c) Figure P1.1-2(c). R(s) + L R(s)
Use Mason’s gain formula of Appendix II to verify the results of Problem 1.1-2 for the system of:(a) Figure P1.1-2(a).(b) Figure P1.1-2(b).(c) Figure P1.1-2(c). R(s) R(s) R(s) E(s) G(s) E(s) Ge(s)
A feedback control system is illustrated in Fig. P1.1-4. The plant transfer function is given by(a) Write the differential equation of the plant. This equation relates c(t) and m(t) .(b) Modify the
Repeat Problem 1.1-4 with the transfer functionsProblem 1.1-4A feedback control system is illustrated in Fig. P1.1-4. The plant transfer function is given by(a) Write the differential equation of the
Repeat Problem 1.1-4 with the transfer functionsProblem 1.1-4A feedback control system is illustrated in Fig. P1.1-4. The plant transfer function is given by(a) Write the differential equation of the
The satellite of Section 1.4 is connected in the closed-loop control system shown in Fig. P1.4-1.The torque is directly proportional to the error signal. Thrusters (s) (a) Amplifiers and thrusters K
(a) In the system of Problem 7, J = 0.4 and K = 14.4, in appropriate units. The attitude of the satellite is initially at 0°. At t = 0, the attitude is commanded to 20°; that is, a 20° step is
The input to the satellite system of Fig. P1.4-1 is a step function θc (t) = 5u(t) in degrees. As a result, the satellite angle θ(t) varies sinusoidally at a frequency of 10 cycles per minute. Find
The satellite control system of Fig. P1.4-1 is not usable, since the response to any excitation includes an undamped sinusoid. The usual compensation for this system involves measuring the angular
The antenna positioning system described in Section 1.5 is shown in Fig. P1.5–1. In this problem we consider the yaw angle control system, whereθ(t) is the yaw angle. Suppose that the gain of the
The state-variable model of a servomotor is given in Section 1.5. Expand these state equations to model the antenna pointing system of Problem 1.5-1(b).Problem 1.5-1(b)The system block diagram is
Shown in Fig. P1.5-4 is the block diagram of one joint of a robot arm. This system is described in Section 1.5. The input M(s) is the controlling signal, Ea (s) is the servomotor input voltage, is
Consider the robot arm depicted in Fig. P1.5-4.Fig. P1.5-4. M Power amplifier K Ea Servomotor 200 0.5 s +1 Fig. P1.5-4 S I Gears 100 a
A thermal test chamber is illustrated in Fig. P1.6-1(a). This chamber, which is a large room, is used to test large devices under various thermal stresses. The chamber is heated with steam, which is
The rectangular rules for numerical integration are illustrated in Fig. P2.2-1. The left-side rule is depicted in Fig. P2.2-1(a), and the right-side rule is depicted in Fig. P2.2-1(b). The integral
The trapezoidal rule (modified Euler method) for numerical integration approximates the integral of a function x(t) by summing trapezoid areas as shown in Fig. P2.2-2. Let y(t) be the integral of
Find the z-transform of the number sequence generated by sampling the time function e(t) = t every T seconds, beginning at t = 0 . Can you express this transform in closed form?
(a) Write, as a series, the z-transform of the number sequence generated by sampling the time function e(t) = ε−t every T seconds, beginning at t = 0 . Can you express this transform in closed
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) e(t) = -at 8 (b)
A function e(t) is sampled, and the resultant sequence has the z-transform z - 2z E (2)=4 0.92 +0.8 Solve this problem using E(z) and the properties of the z-transform. (a) Find the z-transform of
A function e(t) is sampled, and the resultant sequence has the z-transform E (2) = 2-b 2-cz + d Find the z-transform of gate(kr). Solve this problem using E(z) and the properties of the z- transform.
Find the z-transform, in closed form, of the number sequence generated by sampling the time function e(t) every T seconds beginning at t = 0 . The function e(t) is specified by its Laplace transform,
Given the difference equation x(k) - x(k-1) + x(k 2) = e(k) - where e(k)= 1 for k 0. (a) Solve for x(*) as a function of k, using the z-transform. Give the values of x(0), x(1), and x(2). (b)
For the number sequence{e(k)}, Z E B () = (= + 1) (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) Repeat parts (a) and (b)
Write the state equations for the observer canonical form of a system, shown in Fig. 2-10, which has the transfer function given in (2-51) and (2-61)Figure 2-10 G(z) = + 2"+an-12"-1 +byz + bo *** + +
Given the system described by the state equations(a) Calculate the transfer function Y(z)/U(z), using (2-84).(b) Draw a simulation diagram for this system, from the state equations given.(c) Use
Let Φ(k) be the state transition matrix for the equations x (k+ 1) = Ax(k) Show that (k) satisfies the difference equation (k+ 1) = A (k)

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