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engineering
modern control systems
Modern Control Systems 14th Global Edition Richard Dorf, Robert Bishop - Solutions
AP12.1 To minimize vibrational effects, a telescope is magnetically levitated. This method also eliminates friction in the azimuth magnetic drive system. The photodetectors for the sensing system require electrical connections. The system block diagram is shown in Figure AP12.1. Design a PID
P12.12 Consider a control system with the plant’s model == x(t) = Ax(t) + Bu(t) y(t) = Cx(t), 0 1 0 where A = 0 0 1-8- -9-6-4 C=[010] B=0 and The system uses output feedback u(t) = r(t) - Ky(t), where r(t) is the reference input. Plot the root locus. Derive the transfer function of the
P12.11 Consider the three dimensional cam shown in Figure P12.11 [18]. The control of x may be achieved with a DC motor and position feedback of the form shown in Figure P12.11. Assume 1 5 ≤ ≤ K and 2 5 ≤ ≤p . NormallyK = 1 and p = 3. Design a PID controller so that the settling time
P12.10 A satellite system can be modeled as a double integrator with a plant transfer function G s s 10.2( ) =We want to use a PID controller and a prefilter with unity feedback for the system to achieve the requirements of P.O. = 2% and settling time Ts = 2 sec. The desired characteristic
P12.9 Future astronauts may drive on the Moon in a pressurized vehicle, shown in Figure P12.9(a), that would have a long range and could be used for missions of up to six months. Engineers first analyzed the Apolloera Lunar Roving Vehicle, then designed the new vehicle, incorporating improvements
P12.8 A unity feedback system has a nominal characteristic equation q(s) =s3+2s2+4s +5 = 0. The coefficients vary as follows: 1
P12.7 A motor and load with negligible friction and a voltage-to-current amplifier Ka is used in the feedback control system, shown in Figure P12.7. A designer selects a PID controllerwhere K K P I = = 5, 500, and KD = 0.0475.(a) Determine the appropriate value of Ka so that the phase margin of
P12.6 The function of a steel plate mill is to roll reheated slabs into plates of scheduled thickness and dimension [5, 10]. The final products are of rectangular plane view shapes having a width of up to 3300 mm and a thickness of 180 mm.A schematic layout of the mill is shown in Figure P12.6(a).
P12.5 A roll-wrapping machine (RWM) receives, wraps,and labels large paper rolls produced in a paper mill[9, 16]. The RWM consists of several major stations:positioning station, waiting station, wrapping station, and so forth. We will focus on the positioning station shown in Figure P12.5(a). The
P12.4 An automatically guided vehicle is shown in Figure P12.4(a) and its control system is shown in Figure P12.4(b). The goal is to track the guide wire accurately, to be insensitive to changes in the gain K1, and to reduce the effect of the disturbance [15, 22]. The gainK1 is normally equal to 1
P12.3 Magnetic levitation (maglev) trains may replace airplanes on routes shorter than 200 miles. The maglev train developed by a German firm uses electromagnetic attraction to propel and levitate heavy vehicles, carrying up to 400 passengers at 300-mph speeds. But the 1 4 -inch gap between car and
P12.2 Consider the control system is shown in Figure P12.2, where τ1 = 10 ms and τ2 = 1 ms.(a) Select K so that Mpω = 1.39.(b) Plot 20 log T j ( ) ω and 20 log S j T K ( ) ω on one Bode plot.(c) Evaluate S j T K ( ) ω at ω ω B B , /2, and ωB/4. (d) LetR s( ) = 0, and determine the
P12.1 Consider the uncrewed underwater vehicle(UUV) problem. The control system is shown in Figure P12.1, where R s( ) = 0, the desired roll angle, and T s d ( ) = 1 . s (a) Plot 20 log T j ( ) ω and 20 log S j T K ( ) ω . (b) Evaluate S j T K ( ) ω at ω ω B B , , /2 and ωB/4.
E12.12 Consider the second-order system 1 *0-240+20 x(1) -a -b y(t)=[10]x(t)+[0]u(t). The parameters a, b, c, and c are unknown a pri- ori. Under what conditions is the system completely controllable? Select valid values of a, b, c, and c to ensure controllability and plot the step response.
E12.11 Consider a second-order system with the following state space representationFor a nominal value of p = 20, what is the range of the system damping ratio? Plot the root locus with variation of k. If the system is required to have a percent overshoot of less than 10%, a controller is added to
E12.10 A system has the form shown in Figure E12.6 with K G(s) = s(s+2)(s+7) where K 1. Design a PI controller so that the dominant roots have a damping ratio = 0.65. Determine the step response of the system. Predict the effect of a change in K of +50% on the percent overshoot. Estimate the step
E12.9 A system has the form shown in Figure E12.6 withwhere K = 1. Design a PD controller such that the dominant closed-loop poles possess a damping ratio of ζ = 0.6. Determine the step response of the system. Predict the effect of a change in K of ±50%, on the percent overshoot. Estimate the
E12.8 Repeat Exercise 12.6, striving to achieve a minimum settling time while adding the constraint that u(t) 0 for a unit step input, r(t)=1, 11.
E12.7 For the control system of Figure E12.6 withG s( ) = + 1 6 ( ) s 2 , select a PID controller to achieve a settling time (with a 2% criterion) of less than 1.0 second for an ITAE step response. Plot y t( ) for a step input r t( ) with and without a prefilter. Determine and plot y t( ) for a
E12.6 Consider the control system shown in Figure E12.6 when G s( ) = + 2 3 ( ) s 2 , and select a PID controller so that the settling time (with a 2% criterion) is less than 1.5 second for an ITAE step response.Plot y t( ) for a step input r t( ) with and without a prefilter. Determine and plot
E12.5 A system has a process function K G(s) = s(s+4)(s+7) with K 50 and unity feedback with a PD compensator G(s) = Kp + KDs. The objective is to design G. (s) so that the percent overshoot to a step is P.O.
E12.4 A PID controller is used in a unity feedback system where G(s) = 1 (s+ 10)(s +25) The gain Kp of the controller Gd(s) = Kp + Kps+ K S is limited to 500. Select a set of compensator zeros so that the pair of closed-loop roots is approximately equal to the zeros. Find the step response for the
E12.3 A closed-loop unity feedback system has the loop transfer function 22 L = G(s)G(s) = s(s+b) where b is normally equal to 4. Determine ST, and plot 20log10 T(jw) and 20log10|S(jw) on a Bode plot.
E12.2 For the ITAE design obtained in Exercise E12.1, determine the response due to a disturbanceT s d ( ) = 0.5 s. FIGURE E12.1 Closed-loop control system. (a) Signal flow graph. (b) block diagram. Td(s) G(s) G(s) Y(s) R(s) T (a) N(s) T(s) Controller Process + E(s) R(s) Ge(s) G(s) Y(s) N(s) (b)
E12.1 Consider a system of the form shown in Figure E12.1,whereTable 5.6. Determine the step response with and without a prefilter G s p( ). 5 G(s) = (s+5) Using the ITAE performance method for a step input, determine the required Ge(s). Assume w = 25 for
15. A feedback control system has the nominal characteristic equation 9(s)=3+as+as+ao = $ +38+2s+3 = 0. The process varies such that 2
14. Consider the feedback control system in Figure 12.45 withThe nominal value of J = 5, but it is known to change with time. It is thus necessary to design controller with sufficient phase margin to retain stability as J changes. A suitable PID controller such that the phase margin is greater
13. Consider a unity negative feedback system with a loop transfer function (with nominal values) K L(s) = Ge(s)G(s)= 4.5 s(s+a)(s+b) s(s+1)(s+2) Using the Routh-Hurwitz stability analysis, it can be shown that the closed-loop system is nominally stable. However, if the system has uncertain
12. Consider the feedback control system in Figure 12.45 with plant G(s) = s+2 A proportional-plus-integral (PI) controller and prefilter pair that results in a settling time T,
11. Consider the system in Figure 12.45 with G s p( ) = 1 and loop transfer function K L(s) =G(s)G(s) = s(s+5)* The sensitivity of the closed-loop system with respect to variations in the parameter K is s(s+3) s+3s+ K a. Sk = b. S 8+5 = c. Sk d. Sk = = $2+5s+K S s+5s+K s(s+5) s +58 + K
10. Consider the system in Figure 12.45 with G s p( ) = 1, b G(s)= = s2+as+b and 1 < a
9. Consider the closed-loop system block-diagram in Figure 12.45, where 1 G(s)= and Gp(s) = 1. s(s+8s) Determine which of the following PID controllers results in a closed-loop system possessing two pairs of equal roots. 22.5(s+1.11) a. Ge(s) = S b. Ge(s)= 10.5 (s +1.11) S 2.5(s+2.3) c. Ge(s)= S d.
8. Considering the same PI controller as in Problem 7, a suitable prefilter, Gp(s), which provides optimum ITAE response to a step input is: Prefilter Controller Process R(s) Gp(s) Ge(s) G(s) Y(s) FIGURE 12.45 Block diagram for the Skills Check.
7. Assume that the prefilter is G s p( ) = 1. The proportional-plus-integral (PI) controller,G s c ( ), that provides optimum coefficients of the characteristic equation for ITAE(assuming ωn = 12 and a step input) is which of the following a. Ge(s) = 72+ b. Ge(s) = 6.9 + c. Ge(s)=1+1 S 6.9 S
6. A closed-loop feedback system has the third-order characteristic equationwhere the nominal values of the coefficients are a2 = 3, a1 = 6, and a0 = 11. The uncertainty in the coefficients is such that the actual values of the coefficients can lie in the intervalsConsidering all possible
5. Control system designers seek small loop gain L s( ) in order to minimize the sensitivity S s( ).True or False
4. A plant model will always be an inaccurate representation of the actual physical system.True or False
3. The PID controller consists of three terms in which the output is the sum of a proportional term, an integrating term, and a differentiating term, with an adjustable gain for each term.True or False
2. For physically realizable systems, the loop gain L s( ) = G s c ( )G s( )must be large for high frequencies.True or False
1. A robust control system exhibits the desired performance in the presence of significant plant uncertainty.True or False
CP11.13 Consider the system in state variable formDesign a full-state feedback gain matrix and an observer gain matrix to place the closed-loop system poles at s j 1,2 3 = −2 2 ± = , s j ,4 − ±5 and the observer poles s j 1,2 3 = −9 2 ± = , s ,4 −15.Construct the state variable
Implement the system shown in Figure CP11.12 in an m-file. Obtain the step response of the system. U(s) 513 20 20 511 12 10 10 + Y(s) FIGURE CP11.12 Control system for m-file implementation. 15
CP11.11 Consider the third-order system x(t)= 01 0 0 0 0 1 x(t)+ -4.3 -1.7 -6.7 0 u(t) 0.35 y(t)=[010]x(t) +[0]u(t). (a) Using the acker function, determine a full- state feedback gain matrix and an observer gain matrix to place the closed-loop system poles at $1,2 -1.4j1. 4, 83-2 and the observer
CP11.10 Consider the system represented in state variable form x(t)=Ax(t)+ Bu(t) y(t)=Cx(t) + Du(t), where A = A=| 0 8.8 B= -25.5 -17.5 19.1 C=[10] and D= [0]. Using the acker function, determine a full-state feedback gain matrix and an observer gain matrix to place the closed-loop system poles at
A first-order system is given by x(t)=-x(t)+u(t) with the initial condition x(0) = x0. We want to design a feedback controller u(t) = -kx(t) such that the performance index is minimized. J = f (x (1) + Au (1)) dt (a) Let = 1. Develop a formula for J in terms of k, valid for any xo, and use an
CP11.8 Consider the system x(t) = Ax (t) + Bu(t)
CP11.7 Consider the system 0 1 0 x(1) = 0 0 1 x (1), -2-4 -4-6 y(t) = [100] x(t). (CP11.1) Suppose that we are given three observations y(),i=1, 2, 3, as follows: y(1) 1 at 11=0 y(12)=-0.0256 at 1 = 2 y(13) -0.2522 at 13 = 4. (a) Using the three observations, develop a method to determine the
CP11.6 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, is
CP11.5 A linearized model of a vertical takeoff and landing (VTOL) aircraft is [24]The state vector components are: (i) x t 1 ( ) is the horizontal velocity (knots), (ii) x t 2 ( ) is the vertical velocity (knots), (iii) x t 3 ( ) is the pitch rate(degrees/second), and (iv) x t 4 ( ) is the
CP11.4 The following model has been proposed to describe the motion of a constant-velocity rocket: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 the transfer function
CP11.3 Find a gain matrix K so that the closed-loop poles of the system, x(1) = 0 -6-5 y(t) = [1 1 ]|x(t), 25. are s =-2 and $2 -5. Use state feedback $1 u(t) = -Kx(t).
CP11.2 Consider the systemDetermine if the system is controllable and observable. Compute the transfer function from u t( ) to y(t). x(1) =| -15-23 23 |x() + | 180 | u(t), 16 y(t) = [ 10 ]x(t).
CP11.1 Consider the systemUsing the ctrb and obsv functions, show that the system is controllable and observable. 5 -20 12 x(t) = 3 08 x(t)+ 0 u(t), -9 13 3 -5 y(t)= [1 5 1 ]x(t). [15
DP11.7 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 by x(t) = Ax (t) + Bu(t) y(t) = Cx(t) where 0 1 0 A = 0 0 1 B C=[100] -4 -8-10 1
DP11.6 A coupled-drive apparatus is shown in Figure DP11.6. The coupled drives consist of two pulleys connected via an elastic belt, which is tensioned by a third pulley mounted on springs providing an underdamped dynamic mode. One of the main pulleys, pulley A, is driven by an electric DC motor.
DP11.5 The headbox 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
DP11.4 A high-performance helicopter has a model shown in Figure DP11.4. The goal is to control the pitch angle θ( )t of the helicopter by adjusting the rotor thrust angle δ( )t . The equations of motion of the helicopter are (t)=(t)-ax (1) + n(1) (t) = g(t)-a20(1) - 02x(1) + 88(1), Body fixed
DP11.3 Consider the feedback system depicted in Figure DP11.3. The system model is given by x(t)=Ax(t)+ Bu(t) y(t) = Cx(t)
DP11.2 The control of the fuel-to-air ratio in an automobile of prime importance as automakers work 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 used as an input to a
DP11.1 Consider the device for the magnetic levitation of a steel ball, as shown in Figures DP11.1(a) and (b).Design a feedback controller i = −k1x1 − k2x2 + βr where x1(t) = y(t), x2(t) = y˙(t), and β is selected to produce a zero steady-state error to a unit step. The goal for y(t) is
CDP11.1 We wish to obtain a state variable feedback system for the capstan-slide the state variable model developed in CDP3.1 and determine the feedback system.The step response should have a percent overshoot ofP O. . ≤ 2% and a settling time of Ts ≤ 250 ms.
AP11.15 Consider the system depicted in Figure AP11.15. Design a full-state observer for the system.Determine the observer gain matrix L to place the observer poles at s j 1, 2 = −10 ± 10. U(s) 2 7 4 113 FIGURE AP11.15 A second-order system block diagram. Y(s)
AP11.14 Consider the third-order systemVerify that the system is observable and controllable. Then, design a full-state feedback law and an observer by placing the closed-loop system poles ats j 1, 2 = −4 1 ± = , s3 −5 and the observer poles ats j 1, 2 = −10 ± = 5, s3 −25. 0 1 0 x(t) =
AP11.13 Consider the system represented in state variable form x(t) = Ax(t) + Bu(t) y(t)=Cx(t)+Du(t), where 1 7 A B = -6 -3 C = [81], and D =[0]. Verify that the system is observable and control- lable. If so, design a full-state feedback law and an observer by placing the closed-loop system poles
AP11.12 A fourth-order system has the model.What are the poles of the open-loop system? By using the state variable feedback control u(t) = −Kx(t), determine matrix K to place the closed-loop poles at−1, −2, −4, and −8. x(t) = Ax(t)+Bu(t) y(t) = Cx(t), where A = -2.4762 3.3755 -0.0225
AP11.11 Determine an internal model controller G s c( )for the system shown in Figure AP11.11. 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 Ts ≤ 5 s. FIGURE AP11.11 Internal model control. R(s) Ge(s) Process (s+1)(s+2) K
AP11.10 Consider the inverted pendulum mounted to a motor, as shown in Figure AP11.10. 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
AP11.9 The motion control of a lightweight hospital transport vehicle can be represented by a system of two masses, as shown in Figure AP11.9, wherem m 1 2 = = 1 and k k 1 2 = = 1 [21]. (a) Determine the state vector differential equation. (b) Find the roots of the characteristic equation. (c) We
AP11.8 Consider the system x(t) = Ax(t) + Bu(t) where -1 1.6 0 0 A = 0 0 1 and B = 0 00-11.8 8333.0 (a) Design a state variable controller using only x (1) as the feedback variable, so that the step response has a percent overshoot of P.O.
AP11.7 The Radisson Diamond uses pontoons and stabilizers to damp out the effect of waves hitting the ship, as shown in Figure AP11.7(a). The block diagram of the ship roll control system is shown in Figure AP11.7(b)Determine the feedback gains K2 and K3 so that the characteristic roots are s =
AP11.6 A system is represented by the differential equation y(t) + 4y (t) + 4y (t) = u(t) + 2(1), where y(t) = output and u(t) = input. (a) Define the state variables as x1(t) = y(t) and x2(1) =(t). Develop a state variable representation and show that it is a controllable system. (b) Define the
AP11.5 An automobile suspension system has three physical state variables, as shown in Figure AP11.5[13]. 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
AP11.4 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 j = −5 2 ± −, 3, and −3. x(t)= 00 0 1 0 0 00-1 -1 0 0 1 x(1) +
AP11.3 A system has a matrix differential equationWhat values for b1 and b2 are required so that the system is controllable? 0 1 b *(1) = x(1) + u(t). -2-3 2b2
AP11.2 A system has the model -5-2-1 16 x(t) = -1 0 0 x(t)+0u(t) 0 1 0 y(t) = [0010]x(t). 0 Add state variable feedback so that the closed-loop poles are s-2 +2j and s=-20.
AP11.1 A DC motor control system has the form shown in Figure AP11.1 [6]. The three state variables are available for measurement; the output position isx t 1 ( ). 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
P11.29 The block diagram shown in Figure P11.29 is an example of an interacting system. Determine a state variable representation of the system in the form + 2 x(1) = Ax(t) + Bu(t) y(t) = Cx(t) + Du(t) 3 FIGURE P11.29 Interacting feedback system. +
P11.28 Consider the single-input, single-output system is described bya. 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( ) = − u t( ) y t( ).b. Plot the response to a unit step
P11.27 Consider the second-order system *(1)= *0-310+20 -5 -9 y(t)= [21]x(t)[0]u(t). Determine the observer gain matrix required to place the observer poles at S12 = 1 + j2.
P11.26 Consider the second-order system 0 1 0 0 *(1)= 0 0 1x(t)+0u(t) -1 -2-6 7 y(t) =[540]x(t) +[0]u(t). Verify that the system is observable. If so, determine the observer gain matrix required to place the ob- server poles at $1,2 -2j3 and 83 = -6.
P11.25 Consider the system represented in state variable formVerify that the system is observable. Then design a full-state observer by placing the observer poles ats1,2 = −1. Plot the response of the estimation errore x ( )t t = − ( ) xˆ( )t with an initial estimation error ofe( ) 0 1 = [ ]
P11.24 Letand D = [0]. Then design a controller using internal model methods so that the steady-state error for a ramp input is zero and the roots of the characteristic equation are s j = −2 2 ± , s s = −2, and 1 = − . B 00 C = [10]
P11.23 Letand D = [0]. Then design a controller using internal model methods so that the steady-state error for a step input is zero and the desired roots of the characteristic equation are s j = −2 2 ± = and s −20. -1 2 A B= C=[10], 0 1 1
P11.22 The speed control system of an electric car has a plant transfer function of G(s) = 1. s(0.28 + 1) in the open loop and a negative feedback transfer function of H(s) = k with a reference input of r(t) and an output y(t). a. Represent the system in the block diagram for unity feedback systems
P11.21 An op-amp circuit is shown in Figure P11.21 with input u(t) = vi (t), and output y t( ) = v t 0 ( ).a. Determine the system transfer function.b. Select the state variables, and write the state equations for this system.c. Determine its observability, and find the condition when the
P11.20 Consider the automatic ship-steering system. The state variable form of the system differential equation is -0.06 -5 0 0. -0.1 -0.01 -0.2 -0.2 0 0 0.05 x (1) = x(1) + 8(1), 1 0 0 10 0 0 1 0 0 0 y(t)= [0 0 10 0] x(t)
P11.19 The block diagram of a system is shown in Figure P11.19. Determine whether the system is controllable and observable.
P11.18 A system has a plant Y(s) G(s) = _ U(s) 1 (s+2)21 (a) Find the matrix differential equation to represent this system. Identify the state variables on a block dia- gram model. (b) Select a state variable feedback struc- ture using u(t), and select the feedback gains so that the response y(t)
P11.17 A system has a transfer function Y(s) s+as+b R(s) s +123 + 48s + 72s +52* Determine real values of a and b so that the system is either uncontrollable or unobservable.
P11.16 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 byWe want to place the closed-loop poles at s j = −5 .
P11.15 A telerobot system has the matrix equations [16] -5 1 0 x (1)= 0-1 1 x(t) + 1 (1) 0 0-6 0 y(t)= [310]x(t). (a) Determine the transfer function, G(s) = Y(s)/ U(s). (b) Draw the block diagram indicating the state variables. (c) Determine whether the system is control- lable. (d) Determine
P11.14 A process has the transfer function x(t) = -8 0 1 8 x 10 + | | 40 10 y(t)=-11]x(t)+[0]u(t). Determine the state variable feedback gains to achieve a settling time (with a 2% criterion) of T = 1.5s and a percent overshoot of P.O. = 8%. Assume the complete state vector is available for
P11.13 A feedback system has a plant transfer function Y(s) 45.78 G(s) = R(s) s(s+50) We want the percent overshoot to a step to be P.O. 10% and the settling time (with a 2% criterion) T1s. Design an appropriate state variable feedback system for r(t)=-kx(1) - k2x2 (1).
P11.12 A voice-coil actuator-driven electromechanical system has the following state-space model: where x(t) = Ax (t) + Bu(t) y(t) = Cx(t), -1.4890 -0.7681 -0.0945 -0.0424 1 0 0 0 0 1 0 0 0 0 1 0 0 B 0 and C = [0001] 0 Compute the transfer function and the poles of this system. Determine whether it
P11.11 The state variable model of a plant to be controlled is -7 0.2 *(1)= 4 0 x(1) + 0 Ju(t) y(t) =[01]x(t)+[0]u(t). Use state variable feedback, and incorporate a com- mand input u(t) = -Kx(t) + ar(t). Select the gains K and a so that the system has a rapid response with a percent overshoot of
P11.10 The dynamics of a rocket are represented by 1 0 y(t) = [01]x(t), and state variable feedback is used, where u(t)=7x1 (1) 12x2 (1)+r(t). 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 (0)=-1.
P11.9 An interesting mechanical system with a challenging control problem is the ball and beam, shown in Figure P11.9(a) [10]. 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
P11.8 For the system of P11.7, determine the optimum value fork2 when k = 0.25, and x7(0)= [1,0
P11.7 A system has the vector differential equation where x(t) = Ax(t) + Bu(t) A = and B = 0 0
P11.6 For the solutions of Problems P11.3, P11.4, and P11.5, determine the roots of the closed-loop optimal control system. Note that the resulting closed-loop roots depend on the performance index selected.
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