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systems analysis and design

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

A high-speed train is under development in Texas [21] with a design based on the French Train a Grande Vitesse (TGV). Train speeds of 186 miles per hour are foreseen. To achieve these speeds on tight curves, the train may use independent axles combined with the ability to tilt the train. Hydraulic
High-performance tape transport systems are designed with a small capstan to pull the tape past the read/write heads and with take-up reels turned by DC motors. The tape is to be controlled at speeds up to 200 inches per second, with start-up as fast as possible, while preventing permanent
The past several years have witnessed a significant engine model-building activity in the automotive industry in a category referred to as "control-oriented" or "control design" models. These models contain representations of the throttle body, engine pumping phenomena, induction process dynamics,
A high-performance jet airplane is shown, and the roll-angle control system is shown. Design a controller Gc(s) so that the step response is well behaved and the steady-state error is zero.
A simple closed-loop control system has been proposed to demonstrate proportional-integral (PI) control of a windmill radiometer [27]. The windmill radiometer is shown in Figure DP10.8(a) and the control system is shown in Figure DPI0.8(b). The variable to be controlled is the angular velocity ta
The feedback control system shown in Figure DPI 0.9 has the transfer functionDesign a PID compensator Gcl (s) and a lead-lag compensator GC2 (s) such that, in each case, the closed-loop system is stable in the presence of a time-delay T = 0.1 s. Discuss the capability of each compensator to insure
Consider the control system, whereDevelop an m-file to show that the phase margin is approximately 50° and that the percent overshoot to a unit step input is 18%.
The feedback control system shown in FigureCP10.10 has the transfer functionThe time delay is T = 0.2 s. Plot the phase margin for the system versus the gain in the range 0.1 s K
A negative feedback control system is shown. Design the proportional controller Gc(s) = K so that the system has a 40° phase margin. Develop an m-file to obtain a Bode plot and verify that the design specification is satisfied.
Consider the system, whereDesign a compensator Gc(s) so that the steady-state tracking error to a ramp input is zero and the settling time (with a 2% criterion) is less than 5 seconds. Obtain the response of the closed-loop system to the input R(s) = l/.s2 and verify that the settling time
A fighter aircraft has the transfer functionWhere Î¸ is the pitch rate (rad/s) and 5 is the elevator deflection (rad). The four poles represent the phugoid and short-period modes. The phugoid mode has a natural frequency of 0.1 rad/s, and the short period mode is 1.4 rad/s. The block diagram is
The pitch attitude motion of a rigid spacecraft is described bywhere J is the principal moment of inertia, and u is the input torque on the vehicle [7]. Consider the PD controller Gc{s) = KP + KDs. (a) Obtain a block diagram of the control system. Design a control system to meet the following
Consider the control system shown in Figure CP10.6. Design a lag compensator using root locus methods to meet the following specifications: (1) steady state error less than 10% for a step input, (2) phase margin greater than 45°, and (3) settling time (with a 2% criterion) less than 5 seconds for
A lateral beam guidance system has an inner loop as shown in Figure CP10.7, where the transfer function for the coordinated aircraft is [26]Consider the PI controller (a) Design a control system to meet the following specifications: (1) settling time (with a 2% criterion) to a unit step input of
Consider again the system and the lead compensator designed in Example 10.3. The actual overshoot of the compensated system will be 46%. We want to reduce the overshoot to 32%. Using a m-file script, determine an appropriate value for the zero of Gc(s).
Plot the frequency response of the circuit of AP10.10.
The capstan-slide system of Figure CDP4.1uses a PD controller. Determine the necessary values of the gain constants of the PD controller so that the deadbeat response is achieved. Also, we want the settling time (with a 2% criterion) to be less than 250 ms. Verify the results.
The ability to balance actively is a key ingredient in the mobility of a device that hops and runs on one springy leg. The control of the attitude of the device uses a gyroscope and a feedback such that u = Kx, whereandwhereDetermine a value for k so that the response of each hop is critically
Consider the block diagram model. Write the corresponding state variable model in the formy = Cx + Dµ.
Consider the system shown in block diagram form in Figure E ll . ll . Obtain a state variable representation of the system. Determine if the system is controllable and observable.FIGURE E11.11State variable block diagram with a feed forward term.
Consider the single-input, single-output system is described byx(f) = Ax(0 + Bu(f)y(0 = Cx(r)whereCompute the corresponding transfer function representation of the system. If the initial conditions are zero (i.e., Xy(0) = 0 and x2(0) = 0), determine the response when u(t) is a unit step input for t
A magnetically suspended steel ball can be described by the linear equationThe state variables are x1 = position and x2 = velocity, and both are measurable. Select a feedback so that the system is critically damped and the settling time (with a 2% criterion) is 4 seconds. Choose the feedback in the
A system is described by the matrix equationsv = [0 2]x.Determine whether the system is controllable and observable.
A system is described by the matrix equationsy = [1 0) xDetermine whether the system is controllable and observable.
A system is described by the matrix equationsDetermine whether the system is controllable and observable.
A system is described by the matrix equationsy = [l 0]x.Determine whether the system is controllable and observable.
Consider the system represented in state variable formy = Cx + DM, where C = [2 -2], and D = [0]. Sketch a block diagram model of the system.
Consider the third-order systemy = [2 8 10]x + [l]µ. Sketch a block diagram model of the system.
Consider the second-order system y = [1 0]x + [0]u.For what values of k1 and k2 is the system completely controllable?
A first-order system is represented by the time domain differential equationA feedback controller is to be designed such thatu(t) = -kx,and the desired equilibrium condition is x(t) = 0 as t -> oo. The performance integral is defined asand the initial value of the state variable is x(0) = ˆš2
The dynamics of a rocket are represented byand state variable feedback is used, where µ = - x1 - 25x2 + r.. Determine the roots of the characteristic equation of this system and the response of the system when the initial conditions are x1(0) = 1 and x2(Q) = - 1. Assume the reference input r(t) -
The state variable model of a plant to be controlled is y = [0 l]x + ~[0]u.Use state variable feedback and incorporate a command input u = -Kx + ar. Select the gains K and a so that the system has a rapid response with an overshoot of approximately 1 %, a settling time (with a 2% criterion)
A DC motor has the state variable modely = [00002.75]x.Det ermine whether this system is controllable and observable.
A feedback system has a plant transfer function
A process has the transfer function y = [0 l]x + [0]µ.Determine the state variable feedback gains t achieve a settling time (with a 2% criterion) of 1 second and an overshoot of about 10%. Also sketch the block diagram of the resulting system. Assume the complete state vector is available
A tele robot system has the matrix equations [16]andy = [1 0 2]x.(a) Determine the transfer function, G(s) = Y(s)/U(s). (b) Draw the block diagram indicatingthe state variables, (c) Determine whether the system is controllable, (d) Determine whether r the system is observable.
Hydraulic power actuators were used to drive the dinosaurs of the movie Jurassic Park [20]. The motions of the large monsters required high-power actuators requiring 1200 watts. One specific limb motion has dynamics represented byy = [0 l]x + [0]u.We want to place the closed-loop poles at s = -1 ±
A system has a transfer functionDetermine a real value of a so that the system is either uncontrollable or unobservable.
A system has a plant(a) Find the matrix differential equation to represent this system. Identify the state variables on a block diagram model, (b) Select a state variable feedback structure using u(t), and select the feedback gains so that the response y(t) of the unforced system is critically
The block diagram of a system is shown. Determine whether the system is controllable and observable.
To account for the expenditure of energy and resources, the control signal is often included in the performance integral. Then the operation will not involve an unlimited control signal u{t). One suitable performance index, which includes the effect of the magnitude of the control signal, is(a)
Consider the automatic ship-steering system discussed in Problems P8.ll and P9.15.The state variable form of the system differential equation is(a) Determine whether the system is stable, (b) Feedback can be added so that Determine whether this system is stable for suitable values of k1 and k3.
An RL circuit is shown in Figure PI 1.21. (a) Select the two stable variables and obtain the vector differential equation where the output is v0 (t). (b) Determine whether the state variables are observable when R1/L1 - R2/L2. (c) Find the conditions when the system has two equal roots.FIGURE
A manipulator control system has a loop transfer function ofand negative unity feedback [15]. Represent this system by a state variable signal-flow graph or block diagram and a vector differential equation, (a) Plot the response of the closed-loop system to a step input.(b) Use state variable
Consider again the system of Example 11.7 when we desire that the steady-state error for a step input be zero and the desired roots of the characteristic equation be s = -2 ± jl and .s = -10.
Consider again the system of Example 11.7 when we desire that the steady-state error for a ramp input be zero and the roots of the characteristic equation be s = -2 ± j2 and s = -20.
Consider the system represented in state variable formy = Cx + Du,whereC = [1 -4], and D = [0].
Consider the third-order systemy = [2 -4 0]x + [0]u.Verify that the system is observable. If so, determine the observer gain matrix required to place the observer poles at s1,2 = - 1 ± j and s3 = -5.
Consider the second-order systemy = [1 0]x + [0]«.Determine the observer gain matrix required to place the observer poles at s1,2 = - 1 ± j.
Consider the single-input, single-output system is described by y(t) = Cx(t) where (a) Determine the value of K resulting in a zero steady-state tracking error when u(t) is a unit step input for t ‰¥ 0 . The tracking error is defined here as e(t) = tt(t) - y(t). (b) Plot the response
The block diagram shown in Figure PI 1.29 is an example of an interacting system. Determine a state variable representation of the system in the formx(t) = Ax(t) + Ba(t)y(t) - Cx(t) + Du(t)
An unstable robot system is described by the vector differential equation [9]Both state variables are measurable, and so the control signal is set as u(t) = -k(xi + x2). Following the method of Section 11.7, design gain k so that the performance index is minimized. Evaluate the minimum value of the
Determine the feedback gain k of Example that minimizes the performance index 11.12when xT (0) = [1 -1], Plot the performance index j versus the gain k.
Determine the feedback gain k of Example 11.13 that minimizes the performance indexwhen xT (0) = [1 1]. Plot the performance index j versus the gain k.
For the solutions of Problems PI 1.3, P11.4, and Pll.5, determine the roots of the closed-loop optimal control system. The resulting closed-loop roots depend on the performance index selected.
A system has the vector differential equation as given in Equation (11.42). We want both state variables to be used in the feedback so that u(t) = -ki{Xi - k2 x2. Also, we desire to have a natural frequency wn for this system equal to 2. Find a set of gains k1 and k2 in order to achieve an optimal
For the system of Example 11.11 determine the optimum value for k2 when k1] - 1 and xT (0) = [1 0].
An interesting mechanical system with a challenging control problem is the ball and beam. It consists of a rigid beam that is free to rotate in the plane of the paper around a center pivot, with a solid ball rolling along a groove in the top of the beam. The control problem is to position the ball
A DC motor control system has the form. The three state variables are available for measurement; the output position is Xi(t). Select the feedback gains so that the system has a steady-state error equal to zero for a step input and a response with a percent overshoot less than 3%.
Consider the inverted pendulum mounted to a motor. The motor and load are assumed to have no friction damping. The pendulum to be balanced is attached to the horizontal shaft of a servomotor. The servomotor carries a tachogenerator, so that a velocity signal is available, but there is no position
Determine an internal model controller Gc(s) for the system. We want the steady-state error to a step input to be zero. We also want the settling time (with a 2% criterion) to be less than 5 seconds.
Repeat Advanced Problem APll.ll when we want the steady-state error to a ramp input to be zero and the settling time (with a 2% criterion) of the ramp response to be less than 6 seconds.
Consider the system represented in state variable formy = Cx + Duwhere C = [4 -3] and D = [0]Verify that the system is observable and controllable. If so, design a full-state feedback law and an observer by placing the closed-loop system poles at s1,2 - -1 ± / and the observer poles at
Consider the third-order system y = [2 -9 2]x + [0]µ.Verify that the system is observable and controllable. Then, design a full-state feedback law and an observer by placing the closed-loop system poles at.$1,2 = -1 ± j.s3 = -3 and the observer poles at $1,2 = -12 ± j2,s3 = -30.
Consider the system depicted. Design a full-stale observer for the system. Determine the observer gain matrix L to place the observer poles at s1,2 = -10 ± j10.
A system has the modelAdd state variable feedback so that the closed-loop poles are s = -4, -5, and -6.
A system has a matrix differential equationWhat values for h1 and b2 are required so that the system is controllable?
The vector differential equation describing the inverted pendulum of Example 3.3 isAssume that all state variables are available for measurement and use state variable feedback. Place the system characteristic roots at s = -2 ± , -5 , and - 5 .
An automobile suspension system has three physical state variables. The state variable feedback structure is shown in the figure, with K1 = 1. Select K2 and K3 so that the roots of the characteristic equation are three real roots lying between s = -3 and s = -6. Also, select Kp so that the
A system is represented by the differential equationwhere y = output and u = input.(a) Develop a state variable representation and show that it is a controllable system, (b) Define the state variables as x1 = y and x2 = dy/dt -µ, and determine whether the system is controllable. The
The Radisson Diamond uses pontoons and stabilizers to damp out the effect of waves hitting the ship. The block diagram of the ship's roll control system is shown. Determine the feedback gains K2 and K^ so that the characteristic roots are s = -15 and s = -2 ± j2. Plot the roll output ϕ(t) for a
Consider again the liquid-level control system described in Problem P3.36.(a) Design a state variable controller using only h(t) as the feedback variable, so that the step response has an overshoot less than 10% and a settling time (with a 2% criterion) less than or equal to 5 seconds.(b) Design a
The motion control of a lightweight hospital transport vehicle can be represented by a system of two masses, where m1 - m2 = 1 and k1= k2 = 1 [21]. (a) Determine the state vector differential equation, (b) Find the roots of the characteristic equation, (c) We wish to stabilize the system by letting
Consider the device for the magnetic levitation of a steel ball. Obtain a design that will provide a stable response where the ball will remain within 10% of its desired position. Assume that y and dy / dt are measurable.
The control of the fuel-to-air ratio in an automobile carburetor became of prime importance in the 1980s as automakers worked to reduce exhaust-pollution emissions. Thus, auto engine designers turned to the feedback control of the fuel-to-air ratio. A sensor was placed in the exhaust stream and
Consider the feedback system depicted. The system model is given byy{t) = Cx (t)where
A high-performance helicopter has a model. The goal is to control the pitch angle 0 of the helicopter by adjusting the rotor thrust angle 8.The equations of motion of the helicopter arewhere x is the translation in the horizontal direction. For a military high-performance helicopter, we find
The head box process is used in the manufacture of paper to transform the pulp slurry flow into a jet of 2 cm and then spread it onto a mesh belt [22]. To achieve desirable paper quality, the pulp slurry must be distributed as evenly as possible on the belt, and the relationship between the
A coupled-drive apparatus is shown. The coupled drives consist of two pulleys connected via an elastic belt, which is tensioned by a third pulley mounted on springs providing an under damped dynamic mode. One of the main pulleys, pulley A, is driven by an electric DC motor. Both pulleys A and B are
A closed-loop feedback system is to be designed to track a reference input. The desired feedback block diagram is shown in Figure DP11.3.The system model is given bym = cx(t)whereDesign the observer and the control law to meet the following specifications:1. The steady-state error of the
Consider the system y = [1 2 1]x. Using the ctrb and obsv functions, show that the system is controllable and observable.
Consider the system represented in state variable formy = Cx + Du C = [1 0] and D - [0]. Using the function, determine a full-state feedback gain matrix and an observer gain matrix to place the closed-loop system poles at sl,2 = -2 and the observer poles at s1,2 = -20 ± j4.
Consider the third-order system y = [0 1 0]x + [0]µ.
Implement the system shown in Figure CP11.12 in an m-file. Obtain the step response of the svstem.
Consider the system in state variable formy = [1 0 0 0]x + [0)u.Design a full-state feedback gain matrix and an observer gain matrix to place the closed-loop system poles at $i2 - -1.4 ± / 1 . 4 , ^ - -2 ± j and the observer poles s^2 = -18 ± fi*s34 = ~20. Construct the state variable
Consider the systemy = [1 0]x Determine if the system is controllable and observable. Compute the transfer function from u to y.
Find a gain matrix K so that the closed-loop poles of the systemy = [l -l]xare S1 = - 1 and s2 = -2 = -2. Use state feedback µ = -kx.
The following model has been proposed to describe the motion of a constant-velocity guided missile: y - [0 0 0 1 0]x.(a) Verify that the system is not controllable by analyzing the controllability matrix using the ctrb function.(b) Develop a controllable state variable model by first computing
A linearized model of a vertical takeoff and landing (VTOL) aircraft is [24]where and The state vector components are (i) x1 is the horizontal velocity (knots), (ii) x2 is the vertical velocity (knots), (iii) .x3 is the pitch rale (degrees/second), and (iv) ,v4 is the pitch angle (degrees). The
In an effort to open up the far side of the Moon to exploration, studies have been conducted to determine the feasibility of operating a communication satellite around the translunar equilibrium point in the Earth-Sun-Moon system. The desired satellite orbit, known as a halo orbit. The objective of
Consider the systemy (t) = [1 0 0]x(t).Suppose that we are given three observations y(t1), i - 1,2,3, as follows:y(t1) = 1 at t1 = 0y(t2) = -0.0256 at t2 = 2y(t3) = -0.2522 at t3 = 4.(a) Using the three observations, develop a method to determine the initial value of the state vector x(r0) for the
A system is described by a single-input state equation withUsing the method of Section 11.7 (Equation 11.40) and a negative unity feedback, determine the optimal system when xT(0) = [1 0].
A first-order system is given bywith the initial condition x(0) = .v0. We want to design a feedback controller u = -kx such that the performance index is minimized. (a) Let A = 1. Develop a formula for / in terms of k, valid for any x0, and use an m-file to plot J / x20 versus k. From the plot,
We wish to obtain a state variable feedback sysf- tem for the capstan-slide the state variable model de- C VJ veloped in CDP3.1 and determine the feedback system. The step response should have an over shoot less than 2% and a settling time less than 250 ms.
Consider a system of the form shown, whereUsing the ITAE performance method for a step input, determine the required Gc(s), Assume wn = 30 for Table 5.6. Determine the step response with and without a prefilter Gp(s).
A system has the form shown withwhere K = 1. Design a PI controller so that the dominant roots have a damping ratio £ - 0.70. Determine the step response of the system. Predict the effect of a change in K of ±50% on the percent overshoot. Estimate the step response of the worst-case
Consider the closed-loop system represented in state variable formX = Ax + BrY = Cx + Dr,whereThe nominal value of k -2. However, the value of k can vary in the range 0.1 ‰¤ k ‰¤ 4, Plot the percent overshoot to a unit step input as k varies from 0.1 to 4.
Consider the second-order systemy = [1 0]x + [0]«.The parameters a, b, c1, and c2 are unknown a priori. Under what conditions is the system completely controllable? Select valid values of a, b, cl and c2 to ensure controllability and plot the step response.
For the ITAE design obtained in Exercise E12.1, determine the response due to a disturbance Td(s) = 1/s,
A closed-loop unity feedback system has the loop transfer functionwhere b is normally equal to 8. Determine Si and plot |r(;'a>)| and S(ja)) on a Bode plot.

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