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Modern Control Systems 12th edition Richard C. Dorf, Robert H. Bishop - Solutions

A lamp's intensity stays constant when monitored by an opto transistor-controlled feedback loop. When the voltage drops, the lamp's output also drops, and opto transistor Q1 draws less current. As a result, a power transistor conducts more heavily and charges a capacitor more rapidly [24]. The
A control engineer, N. Minorsky, designed an innovative ship steering system in the 1930s for the U.S. Navy. The system is represented by the block diagram shown in Figure E2.8, where Y(s) is the ship's course, R(s) is the desired course, and A(s) is the rudder angle [16]. Find the transfer
A four-wheel antilock automobile braking system uses electronic feedback to control automatically the brake force on each wheel [15]. A block diagram model of a brake control system is shown in Figure E2.9, where Ff(s) and FR(s) are the braking force of the front and rear wheels, respectively, and
An electric circuit is shown in Figure P2.1. Obtain a set of simultaneous integro differential equations representing the network.
Determine the transfer function Y1(s)/F(s) for the vibration absorber system of Problem P2.2. Determine the necessary parameters M2 and k12 so that the mass M1 does not vibrate in the steady state when F(t) = a sin(wot).
For electromechanical systems that require large power amplification, rotary amplifiers are often used [8, 19]. An amplidyne is a power amplifying rotary amplifier. An amplidyne and a servomotor are shown in Figure P2.11. Obtain the transfer function Î¸(s)/Vc(s), and draw the block
For the open-loop control system described by the block diagram shown in Figure P2.12, determine the value of K such that y(t) †’ 1 as t †’ ˆž when r(t) is a unit step input. Assume zero initial conditions.
An electromechanical open-loop control system is shown in Figure P2.13. The generator, driven at a constant speed, provides the field voltage for the motor. The motor has an inertia Jm and bearing friction bm. Obtain the transfer function Î¸L(s)/Vf(s) and draw a block diagram of the
A rotating load is connected to a field-controlled DC electric motor through a gear system. The motor is assumed to be linear. A test results in the output load reaching a speed of 1 rad/s within 0.5 s when a constant 80 V is applied to the motor terminals. The output steady-state speed is 2.4
Consider the spring-mass system depicted in Figure P2.15. Determine a differential equation to describe the motion of the mass m. Obtain the system response x(t) with the initial conditions x(0) = x0 and x(0) = 0.
Obtain a signal-flow graph to represent the following set of algebraic equations where x1 and x2 are to be considered the dependent variables and 6 and 11 are the inputs: x1 + 1.5x2 = 6, 2x1 + 4x2 = 11.
A mechanical system is shown in Figure P2.17, which is subjected to a known displacement x3(t) with respect to the reference,(a) Determine the two independent equations of motion.(b) Obtain the equations of motion in terms of the Laplace transform, assuming that the initial conditions are zero.(c)
An LC ladder network is shown in Figure P2.18. One may write the equations describing the network as follows:I1 = (V1 - Va)Y1, Va = (I1 - Ia)Z2.Ia = (Va - V2)Y3, V2 = IaZ4.Construct a flow graph from the equations and determine the transfer function V2(s)/V1(s).
A voltage follower (buffer amplifier) is shown in Figure P2.19. Show that T = v0/vin = 1. Assume an ideal op-amp.
A dynamic vibration absorber is shown in Figure P2.2. This system is representative of many situations involving the vibration of machines containing unbalanced components. The parameters M2 and kl2 may be chosen so that the main mass M1 does not vibrate in the steady state when F(t) = a sin(w0t).
The source follower amplifier provides lower output impedance and essentially unity gain. The circuit diagram is shown in Figure P2.20(a), and the small-signal model is shown in Figure P2.20(b).This circuit uses an FET and provides a gain of approximately unity. Assume that R2 >> R1 for biasing
A hydraulic servomechanism with mechanical feedback is shown in Figure P2.21 [18]. The power piston has an area equal to A. When the valve is moved a small amount ˆ†z, the oil will flow through to the cylinder at a rate p ˆ™ ˆ†z, where p is the port coefficient. The
Figure P2.22 shows two pendulums suspended from frictionless pivots and connected at their midpoints by a spring [1]. Assume that each pendulum can be represented by a mass M at the end of a mass less bar of length L. Also assume that the displacement is small and linear approximations can be used
The small-signal circuit equivalent to a common-emitter transistor amplifier is shown in Figure P2.23. The transistor amplifier includes a feedback resistor Rf. Determine the input-output ratio vce/vin.
A two-transistor series voltage feedback amplifier is shown in Figure P2.24(a). This AC equivalent circuit neglects the bias resistors and the shunt capacitors. A block diagram representing the circuit is shown in Figure P2.24(b).This block diagram neglects the effect of hie, which is usually an
H. S. Black is noted for developing a negative feedback amplifier in 1927. Often overlooked is the fact that three years earlier he had invented a circuit design technique known as feed forward correction [19]. Recent experiments have shown that this technique offers the potential for yielding
A robot includes significant flexibility in the arm members with a heavy load in the gripper [6, 20]. A two-mass model of the robot is shown in Figure. P2.26. Find the transfer function Y(s)/F(s).
Magnetic levitation trains provide a high-speed, very low friction alternative to steel wheels on steel rails. The train floats on an air gap as shown in Figure P2.27 [25]. The levitation force FL is controlled by the coil current i in the levitation coils and may be approximated byFL = k
A multiple-loop model of an urban ecological system might include the following variables: number of people in the city (P), modernization (M), migration into the city (C), sanitation facilities (S), number of diseases (D), bacteria/area (B), and amount of garbage/area (G), where the symbol for the
We desire to balance a rolling ball on a tilting beam as shown in Figure P2.29. We will assume the motor input current i controls the torque with negligible friction. Assume the beam may be balanced near the horizontal (Ï• = 0); therefore, we have a small deviation of Ï•. Find
A coupled spring-mass system is shown in Figure P2.3. The masses and springs are assumed to be equal. Obtain the differential equations describing the system.
The measurement or sensor element in a feedback system is important to the accuracy of the system [6]. The dynamic response of the sensor is important. Most sensor elements possess a transfer function Hs = k / τs + 1 Suppose that a position-sensing photo detector has τ = 4μs and 0.999 < k <
An interacting control system with two inputs and two outputs is shown in Figure P2.31. Solve for Y1(s)/R1(s) and Y2(s)/R1(s) when R2 = 0.
A system consists of two electric motors that are coupled by a continuous flexible belt. The belt also passes over a swinging arm that is instrumented to allow measurement of the belt speed and tension. The basic control problem is to regulate the belt speed and tension by varying the motor
Find the transfer function for Y(s)/R(s) for the idle-speed control system for a fuel-injected engine as shown in Figure P2.33.
The suspension system for one wheel of an old-fashioned pickup truck is illustrated in Figure P2.34. The mass of the vehicle is m1 and the mass of the wheel is m2. The suspension spring has a spring constant k1 and the tire has a spring constant k2. The damping constant of the shock absorber is b.
A feedback control system has the structure shown in Figure P2.35. Determine the closed-loop transfer function Y(s)/R(s)(a) By block diagram manipulation and(b) By using a signal-flow graph and Mason's signal-flow gain formula.(c) Select the gains K1, and K2 so that the closed-loop response to a
A system is represented by Figure P2.36.(a) Determine the partial fraction expansion and y(t) for a ramp input, r(t) = t, t > 0.(b) Obtain a plot of y(t) for part (a), and find y(t) for t = 1.0 s.(c) Determine the impulse response of the system y(t) for t > 0.(d) Obtain a plot of y(t) for
A two-mass system is shown in Figure P2.37 with an input force u(t). When m1 = m2 = 1 and K1 = K2 = 1, find the set of differential equations describing the system.
A winding oscillator consists of two steel spheres on each end of a long slender rod, as shown in Figure P2.38. The rod is hung on a thin wire that can be twisted many revolutions without breaking. The device will be wound up 4000 degrees. How long will it take until the motion decays to a swing of
For the circuit of Figure P2.39, determine the transform of the output voltage V0(s). Assume that the circuit is in steady state when t Figure P2.39Model of an electronic circuit.
A nonlinear amplifier can be described by the following characteristic:The amplifier will be operated over a range of ±0.5 volts around the operating point for vin. Describe the amplifier by a linear approximation (a) When the operating point is vin = 0 and (b) When the operating point is
A damping device is used to reduce the undesired vibrations of machines. A viscous fluid, such as a heavy oil, is placed between the wheels, as shown in Figure P2.40. When vibration becomes excessive, the relative motion of the two wheels creates damping. When the device is rotating without
The lateral control of a rocket with a gimbaled engine is shown in Figure P2.41. The lateral deviation from the desired trajectory is h and the forward rocket speed is V. The control torque of the engine is Tc and the disturbance torque is Td. Derive the describing equations of a linear model of
In many applications, such as reading product codes in supermarkets and in printing and manufacturing, an optical scanner is utilized to read codes, as shown in Figure P2.42. As the mirror rotates, a friction force is developed that is proportional to its angular speed. The friction constant is
An ideal set of gears is shown in Table 2.5, item 10. Neglect the inertia and friction of the gears and assume that the work done by one gear is equal to that of the other. Derive the relationships given in item 10 of Table 2.5. Also, determine the relationship between the torques Tm and TL.
An ideal set of gears is connected to a solid cylinder load as shown in Figure P2.44. The inertia of the motor shaft and gear G2 is Jm. Determine(a) The inertia of the load JL and(b) The torque T at the motor shaft.Assume the friction at the load is bL and the friction at the motor shaft is bm.
To exploit the strength advantage of robot manipulators and the intellectual advantage of humans, a class of manipulators called extenders has been examined [22]. The extender is defined as an active manipulator worn by a human to augment the human's strength. The human provides an input U(s), as
A load added to a truck results in a force F on the support spring, and the tire flexes as shown in Figure P2.46(a). The model for the tire movement is shown in Figure P2.46(b). Determine the transfer function X1(s)/F(s).Figure P2.46Truck support model.
The water level h(t) in a tank is controlled by an open-loop system, as shown in Figure P2.47. A DC motor controlled by an armature current ia turns a shaft, opening a valve. The inductance of the DC motor is negligible, that is, La = 0. Also, the rotational friction of the motor shaft and valve is
The circuit shown in Figure P2.48 is called a lead-lag filter.(a) Find the transfer function V2(s)/V1(s). Assume an ideal op-amp.(b) Determine V2(s)/V1(s) when R1 = 100 kÎ©, R2 = 200 kÎ©, C1 = 1 Î¼F, and C2 = 0.1 Î¼F.
A closed-loop control system is shown in Figure P2.49.(a) Determine the transfer functionT(s) = Y(s)/R(s).(b) Determine the poles and zeros of T(s).(c) Use a unit step input, R(s) = 1/s, and obtain the partial fraction expansion for Y(s) and the value of the residues.(d) Plot y(t) and discuss the
Fluid flowing through an orifice can be represented by the nonlinear equationQ = K(P1 - P2)1/2,where the variables are shown in Figure P2.5 and K is a constant [2].(a) Determine a linear approximation for the fluid-flow equation,(b) What happens to the approximation obtained in part (a) if the
A closed-loop control system is shown in Figure P2.50.(a) Determine the transfer function T(s) = Y(s)/R(s).(b) Determine the poles and zeros of T(s).(c) Use a unit step input, R(s) = l/s, and obtain the partial fraction expansion for Y(s) and the value of the residues.(d) Plot y(t) and discuss the
Consider the two-mass system in Figure P2.51. Find the set of differential equations describing the system.
Using the Laplace transformation, obtain the current I2(s) of Problem P2.1. Assume that all the initial currents are zero, the initial voltage across capacitor C1 is zero, v(t) is zero, and the initial voltage across C2 is 10 volts.
Obtain the transfer function of the differentiating circuit shown in Figure P2.7.
A bridged-T network is often used in AC control systems as a filter network [8]. The circuit of one bridged-T network is shown in Figure P2.8. Show that the transfer function of the network isSketch the pole-zero diagram when R1 = 0.5, R2 = 1, and C = 0.5.
Determine the transfer function X1(s)/F(s) for the coupled spring-mass system of Problem P2.3. Sketch the s-plane pole-zero diagram for low damping when M = l, b/k = l, and
Determine the transfer function X1(s)/F(s) for the coupled spring-mass system of Problem P2.3. Sketch the s-plane pole-zero diagram for low damping when M = l, b/k = l, andDiscuss.
An armature-controlled DC motor is driving a load. The input voltage is 5 V. The speed at t = 2 seconds is 30 rad/s, and the steady speed is 70 rad/s when t → ∞. Determine the transfer function ω(s)/V(s).
A system has a block diagram as shown in Figure AP2.2. Determine the transfer functionT(s) = Y2(s) / R1(s)It is desired to decouple Y2(s) from R1(s) by obtaining T(s) = 0. Select G5(s) in terms of the other G1(s) to achieve decoupling.
Consider the feedback control system in Figure AP2.3. Define the tracking error asE(s) = R(s) - Y(s).(a) Determine a suitable H(s) such that the tracking error is zero for any input R(s) in the absence of a disturbance input (that is, when Td(s) = 0).(b) Using H{s) determined in part (a), determine
Consider a thermal heating system given bywhere the output Ï„(s) is the temperature difference due to the thermal process, the input q(s) is the rate of heat flow of the heating element. The system parameters are Ct, Q, S, and Rt. The thermal heating system is illustrated in Table 2.5.(a)
For the three-cart system illustrated in Figure AP2.5, obtain the equations of motion. The system has three inputs u1, u2, and u3 and three outputs x1, x2, and x3. Obtain three second-order ordinary differential equations with constant coefficients. If possible, write the equations of motion in
Consider the hanging crane structure in Figure AP2.6. Write the equations of motion describing the motion of the cart and the payload. The mass of the cart is M, the mass of the payload is m, the mass less rigid connector has length L, and the friction is modeled as Fb = -bx where x is the distance
Consider the unity feedback system described in the block diagram in Figure AP2.7. Compute analytically the response of the system to an impulse disturbance. Determine a relationship between the gain K and the minimum time it takes the impulse disturbance response of the system to reach y(t) 0.
Consider the cable reel control system given in Figure AP2.8. Find the value of A and K such that the percent overshoot is P.O. ‰¤ 10% and a desired velocity of 50 m/s in the steady state is achieved. Compute the closed-loop response y(t) analytically and confirm that the steady-state
Consider the inverting operational amplifier in Figure AP2.9. Find the transfer function V0(s)/V1(s), Show that the transfer function can be expressed aswhere the gains KP, KI, and KD are functions of C1, C2, R1, and R2, This circuit is a proportional-integral-derivative (PID) controller.
We want to accurately position a table for a machine as shown in Figure CDP2.1. A traction-drive motor with a capstan roller possesses several desirable characteristics compared to the more popular ball screw. The traction drive exhibits low friction and no backlash. However, it is susceptible to
A control system is shown in Figure DP2.1. The transfer functions G2(s) and H2(s) are fixed. Determine the transfer functions G1(s) and H1(s) so that the closed-loop transfer function Y(s)/R(s) is exactly equal to 1.Figure DP2.1Selection of transfer functions.
The television beam circuit of a television is represented by the model in Figure DP2.2. Select the unknown conductance G so that the voltage v is 24 V. Each conductance is given in Siemens (S).
An input r(t) = t.t ‰¥ 0, is applied to a black box with a transfer function G(s). The resulting output response, when the initial conditions are zero, isDetermine G(s) for this system.
An operational amplifier circuit that can serve as a filter circuit is shown in Figure DP2.4. Determine the transfer function of the circuit, assuming an ideal op-amp. Find v0(t) when the input is v1(t) = At, t ‰¥ 0 .
Consider the clock shown in Figure DP2.5. The pendulum rod of length L supports a pendulum disk. Assume that the pendulum rod is a mass less rigid thin rod and the pendulum disc has mass m. Design the length of the pendulum, L, so that the period of motion is 2 seconds. Note that with a period of 2
Consider the two polynomials p(s) = s2 + 7s + 10 and q(s) = s + 2. Compute the following (a) p(s)q(s) (b) Poles and zeros of G(s) = q(s) / p(s) (c) p(-1)
Consider the block diagram in Figure CP2.10. Create an m-file to complete the following tasks:(a) Compute the step response of the closed-loop system (that is, R(s) = 1/s and Td(s) = 0) and plot the steady-state value of the output Y(s) as a function of the controller gain 0 (b) Compute the
Consider the feedback system depicted in Figure CP2.2.(a) Compute the closed-loop transfer function using the series and feedback functions.(b) Obtain the closed-loop system unit step response with the step function, and verify that final value of the output is 2/5.
Consider the differential equation
Consider the mechanical system depicted in Figure CP2.4.The input is given by f(t), and the output is y(t). Determine the transfer function from f(t) to y(t) and, using an m-file, plot the system response to a unit step input. Let m = 10, k = 1, and b = 0.5. Show that the peak amplitude of the
A satellite single-axis attitude control system can be represented by the block diagram in Figure CP2.5. The variables k, a, and b are controller parameters, and J is the spacecraft moment of inertia. Suppose the nominal moment of inertia is J = 10.8E8 (slug ft2), and the controller parameters are
Consider the block diagram in Figure CP2.6.(a) Use an m-file to reduce the block diagram in Figure CP2.6, and compute the closed-loop transfer function.(b) Generate a pole-zero map of the closed-loop transfer function in graphical form using the pzmap function.(c) Determine explicitly the poles and
For the simple pendulum shown in Figure CP2.7, the nonlinear equation of motion is given bywhere L = 0.5 m, m = 1 kg, and g = 9.8 m/s2. When the nonlinear equation is linearized about the equilibrium point Î¸ = 0, we obtain the linear time-invariant model, Create an m-file to plot both
A system has a transfer functionPlot the response of the system when R(s) is a unit step for the parameter z = 5, 10, and 15.
Consider the feedback control system in Figure CP2.9, whereG(s) = s + 1/s + 2 and H(s) = 1/s + 1(a) Using an m-file, determine the closed-loop transfer function.(b) Obtain the pole-zero map using the pzmap function. Where are the closed-loop system poles and zeros?(c) Are there any pole-zero
For the circuit shown in Figure E3.1 identify a set of state variables.
A hovering vehicle control system is represented by two state variables, and [13](a) Find the roots of the characteristic equation. (b) Find the state transition matrix Î¦(t).
Determine a state variable representation for the system described by the transfer function
Use a state variable model to describe the circuit of Figure E3.12. Obtain the response to an input unit step when the initial current is zero and the initial capacitor voltage is zero.
A system is described by the two differential equations dy/dt + y - 2u + aw = 0, and dw/dt - by + 4u = 0, where w and y are functions of time, and u is an input u(t). (a) Select a set of state variables, (b) Write the matrix differential equation and specify the elements of the matrices, (c) Find
Develop the state-space representation of a radioactive material of mass M to which additional radioactive material is added at the rate r(t) = Ku(t), where K is a constant. Identify the state variables.
Consider the case of the two masses connected as shown in Figure E3.15. The sliding friction of each mass has the constant b. Determine a state variable matrix differential equation.
Two carts with negligible rolling friction are connected as shown in Figure E3.16. An input force is u(t). The output is the position of cart 2, that is, y(t) = q(t). Determine a state space representation of the system.
Determine a state variable differential matrix equation for the circuit shown in Figure E3.17:
Consider a system represented by the following differential equations:where R, L1, L2 and C are given constants, and va and vb are inputs. Let the state variables be defined as x1 = i1, x2 = i2, and x3 = v. Obtain a state variable representation of the system where the output is x3.
A single-input, single-output system has the matrix equationsand y = [10 0]x. Determine the transfer function G(s) = Y(s)/U(s).
A robot-arm drive system for one joint can be represented by the differential equation [8]where v(t) = velocity, y(t) = position, and i(t) is the control-motor current. Put the equations in state variable form and set up the matrix form for k1 = k2 = 1.
For the simple pendulum shown in Figure E3.20, the nonlinear equations of motion are given bywhere g is gravity, L is the length of the pendulum, m is the mass attached at the end of the pendulum (we assume the rod is mass less), and k is the coefficient of friction at the pivot point. (a)
A single-input, single-output system is described byy(t) = [0 1]x(t) Obtain the transfer function G(s) = Y(s)/U(s) and determine the response of the system to a unit step input.
Consider the system in state variable formx = Ax + Buy = Cx + Duwith(a) Compute the transfer function G(s) = Y(s)/U(s). (b) Determine the poles and zeros of the system. (c) If possible, represent the system as a first-order system. x = ax + bu y = cx + du where a, b, c, and d are scalars such that
Consider a system modeled via the third-order differential equationDevelop a state variable representation and obtain a block diagram of the system assuming the output is x(t) and the input is u(t).
A system can be represented by the state vector differential equation of Equation (3.16), whereFind the characteristic roots of the system.
Obtain a state variable matrix for a system with a differential equation
A system is represented by a block diagram as shown in Figure E3.5. Write the state equations in the form of Equations (3.16) and (3.17).
A system is represented by Equation (3.16), where(a) Find the matrix Î¦(t). (b) For the initial conditions x1(0) = x2(0) = l, find x(t).

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