Design And Analysis Of Control Systems 1st Edition Arthur G O Mutambara - Solutions

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Determine the equation of the phase plane trajectory of the differential equation d2x dx dt 2 + dt = o, with the initial conditions x(O) = 0 and ±(0) = 1.
Show that the phase plane trajectory of the solution of the differential equation d2x dt2 +X= 0, with initial conditions x(O) = 0 and ±(0) = 1, is a circle of unit radius centered at the origin.Problem
Show that equation ~:~ = f ( x, ~~) can be equivalently described by a pair of first-order differential equations.Problem
Show that the describing f7mction for the sat·amtion element in the previous problem is given by Problem
Sketch the isoclines for the equations:(a)(b)(c)dy ( ') - = xy y -2 dx Problem 9. 7 The equations describing the motion of an earth satellite in the orbit plane are d2 r_r(d8) 2 dt2 dt A satellite is nearly in a circular orbit determined by r and ~~ = w. An exactly circular orbit is defined by r =
For the following system., determine the equilibrium points and discuss the stability of each equil·ibrium using Lyapunov 's (indirect)linearization method.. 2 1 XI =XI- XI - 2X1X2' 2 1 X2 = X2 - X 2 + 2X1X2.Sketch the flow of trajectories in the (xi, xz) phase plane and classify each
A nonlinear system is described by. 3 2 :1:1 = XI + Xz - (xi + XIXz)Xz = Xz -XI - (x~ +XIX~)if r 2 = :d + x§, show that the system has a limit cycle at T = 1.Problem
Construct a Lyapv.nov function for the system±1 = xi + x~ - XI±z = xi - x~ - x2 and use it to investigate the stability at the origin. State the domain of attraction if the system is asymptotically stable.Problem
Investigate whether the following functions V ( x) are poS'iiive definite.(a) V(x) = 2xi + 2xiX2 + 4x~- 4xi:I:;3 + 5:r~(b) V(x) xi+ 4xixz + 5x~- 6xiX3- 8xz:r:3 + 8x§Problem
A nonlinear system is described by the following eq'uations, XI = 2xi + 3xi - 2xz + u±z = xz - 0.25(sin XI +cos x2) + u.(a) With u = 0 plot the state-space trajectories in the neighborhood of the origin defined by lxii ~ 1, lxzl ~ 2.(b) Using Lyapunov's first method, linearize the above system
Consider the digital control system x[(k + 1)T] = Ax(kT) + Bu(kT)where [ 0 -1] A= -1-1 ' B= [n.The state feedback control is described by u(kT) = -Kx(kT), where Find the values of k1 and k2 so that the mots of the characteristic equation of the closed-loop system are at 0.5 and 0. 'l.Problem
Determine which of the following digital transfer functions are physically realizable.Gc(z) = 10(1 + 0.2z- 1 + 0.5z- 2 )· z- 1 + z- 2 + 1.5z-3 1 5 -1 -2 G() .z -z c z = · 1 + z- 1 + 2z- 2 G (z)- z+l.5 c - z 3 + z2 + z + 1 G ( ) = z- 1 + 2z- 2 + 0.5z- 3 c z 1 2 z- +zGc(z) = 0.1z + 1 + z- 1 Gc(z) =
A closed-loop system has a hold circuit, as shown in Problem 8.16. Determine G(z) when T = 1 sec and 2Gp(s)=s+2'Problem
A unit ramp r(t) = t, t > 0, is used as an input to a pmcess where 1G(s) = (s + 1)'as shown in the diagram below.r(t) --~/~y(t)Sampling System Determine the output y(kT) for the first four sampling instants.Problem
The characteristic equation of a sampled system is z2 + (K- 1.5)z + 0.5= 0.Find the range of K so that the system is stable. (Answer: 0 < K < 3}Problem
Consider the following discrete-time system in state-space form:[ xl(k + 1)] = [0 -1] [x1(k)] + [ 0] u(k). X2(k + 1) 0 -1 X2(k) lQ Use state feedback to relocate all of the system's poles to 0.5.Problem
Write a computer program to compute and r from A, B, and the sample period T. Use the program to compute and r when(a)[ -1 0 ] A= 0 -2 ' B = [i]' T = 0.2sec.(b)A= [ ~3 ~2], B = [~]' T = 0.2sec.Problem
to lOradf sec.Problem
(a) The following transfer function is a lag network designed to introduce a gain attenuation of 10 (-20dB) at w = 3 radjsec, lOs+ 1 H ( s) = lOOs + 1 .Assume a sampling period of T = 0.25 sec, and compute and plot in the z-plane the pole and zero locations of the digital implementations of
A unity feedback system has an open-loop transfer function given by G(s) _ 250 - s[(s/10) + 1]'The following lag compensator added in series with the plant yields a phase margin of 50°, Dc(s) = (s/1.25) + 1.50s+ 1(a) Using the matched pole-zero approximation, determine an equivalent digital
Use the Z-transform to solve the difference equation where y(k)- 3y(k- 1) + 2y(k- 2) = 2u(k- 1)- 2u(k- 2)u(k)=k, k::O:O= 0, k < 0 y(k) = 0, k > 0.Problem
(a) Transf01m G _ (z + 0.3)(z- 0.3)(z)- (z- 0.1)(z- 0.5+ 0.5i)(z- 0.5- 0.5i)into a continuous-time equivalent system by means of ZOH method of sampling intervals ranging fr-om T8 = 0.01sec toTs = 1sec.{b) Determine the poles and zeros of the resulting continuous-time systems.(c) Plot the root locus
(a) Comp1de the poles and zeros of the discrete-time systems that result from discretizing the continuous-time system with transfer function G s = 10(s2 +0.2s+2)() (s 2 +0.5s+l)(s+10)'for sampling intervals ranging from Ts = lsec to T8 = O.Olsec.(b) Create a root locus plot showing the evolution of
Suppose that the following continuous-time lead compensator D(s) = lO(s + 1)s + 10 is to be used in controlling the plant G(s) = 20 s(s + l)(s + 10) ·If this system is to be implemented using a digital computer at a sampling interval O.lsec, determine the transfer function D(z) of the controller
Find the time function y(kT) corresponding to the following closed form z-transforms Y{z):(a){b)(c)Y(z)=~ z 2 -1 Y(z) = 2z z 2 - 0.5z - 0.5 Y(z) = 0.522z2 + 0.36lz- 0.203 z3- 2.347z2 + 1.797z- 0.449 For each of these, use the following methods:(i) Inverse transform method {ii) Partial fraction
A unit feedback digital control system at a sampling inteTval of 0.058ec. If the plant transfer function is given by z+0.8 G(z) = (z- 1)(z- 0.8) ·Design the controlleT D(s) using root locus methods so that the closed-loop system satisfies the following time domain specification:settling time :::;
The following open-loop continuous-time transfer function represents a second-ordeT 8ystem that is to be controlled U8ing a digital computer with ZOH.1 G(s) = (s + 1)(s + 10) ·Choose a suitable 8ampling interval T and design a digital PID contmlleT that ensures that the settling time t 8 :S 1sec,
Represent the SISO system with the following transfeT function using 8tate-space Y(z) z + 0.3 U(z) z2 - 0.6z- 0.16·Problem
Assess the controllability, observability, and stability of the following digital system.[ -1 0.8] [ 1] x(k + 1) = 0.5-1.6 x(k) + 2 u(k)y(k) = [ 1 0] x(k) +[OJ u(k).Problem
A linear time-invar-iant system is descr-ibed by the differ--ential equation.d3y(t) 3d2 y. (t) dy(t) ()- ·()d 3 + d2 +3d +yt-rt. t t .t(a) Let the state var-iables be defined as x1 = y, Xz = dyjdt, x 3 =d2 y / dt 2 . Wr-ite the state equations of the system in vector-matrix for-m.(b) Find the
The following differential equations represent linear timeinvariant systems. Write the dynamic equations (state equations and output equations) in vector- form.(a) d2 y(t) d1') (t) --+ 4 _. - + (t) = 5 (t) dt2 dt y T(b) d3 (t) d2 (t) d (t) 2-y- + 3-y- + 5-y- + 2y(t) = r(t) dt3 dt 2 dt(c) y(T)dT =
A feedback system has a plant tmnsfer- function G s _ Y(s) _ K( ) - R(s) - s(s + 70)It is desir-ed that the velocity er-mT constant, Kv be 35 and the 011ershoot to a step be approximately 4% so that~ is 1/J2. The settling time (2%cr-iter-ion) desir-ed is 0.11 sec. Design an appropr-iate state
An unstable robot system is described by the vector differential equation!!__ [X1] = [ 1 OJ [X1] + [1] u(t). dt Xz -1 2 xz 1 Both state VaT"iables are measurable, and so the control signal is set as u(t) = -K(x1 + xz).(a) Design gain K so tho.t the performance inde:r; is minimized.(b) Eval1wte the
The linearized equations of motion for a satellite are where:X= Ax+Bu y = Cx+Du A= r 3~~ 2 ~ ~ 2~ 1 r ~ ~ 1 [ 1 0 0 0 J 0 01 'B= 00 'C= 0010'-2w 0 0 0 1 D= [OJ The inputs u 1 and u 2 are the radial and tangential thrusts, the state variables x1 and x2 are the radial and angular deviations from the
For each of the tnmsfer functions below, wr-ite the state equations in both contml and observer- canonical for-m. In each case, dmw a block diagram and give the appmpriate expressions for A, B, and C(a)(b)s2 + 1 s2(s2 - 1)3s+4 s2 + 2s + 2 Problem
Using the indicated state variables, obtain the stale equations .for the two systems shown below. Find the transfer- function for- each system using both block-diagram manipulation and matr-ix algebm.1 s+4 51------'y Problem
Consider the control i!ystem shown below.(a) Find the transfer function fmm U(s) to Y(s).(b) Write the state equations for the system using the state variables indicated.Problem
A certain system with state x is described by the state matrices A= [ =~ ~] , B = [ ~]C = [ 1 0] , D = [OJ .Find the transformation T so the if x = Tz, the state matrices describing the dynamics of z are in control canonical form. Compute the new matrices Az, Bz, , and.Problem
(a) Check the controllability and observability of the following two systems.[ 1 4 3] [-1] x = 0 2 16 x+ 0 u, 0 -25 -20 0 y=[-130]x x = [ ~ ~ ~ ] x+ [ ~ ] u,-2 -4-3 -1 y = [1 o o] x.(b) For each of the systems of part (a), find the contmllable modes, uncontrollable modes, obscTvable modes, and
Obtain a state-space representation for a system whose differential equation is given by·:r· + 3x + 3x + x = u + u, where the output is y = x.(a) Use this result to determine the system transition matrix ¢(t) and¢(s).(b) Use Ackermann's formula to determine the controller K that places the
Optimization theory is a wide subject that was introduced in this chapter as a way towaTd optimal control theory. Use the given introduction to establish the minimurn of subject to the conditions where f(x) = x1,(x1 - 1)2 + :r;~ = 1(:r1 + 1) 2 Problem
The state transfer function G(s) is given by G(s)=. s 2 +7s+10 .· s 3 + 8s2 + 19s + 122(a) Find the controllable and observable canonical forms of the system.(b) Draw the signal flow graph for the canonical form in (a).Problem
A linear system is described by x= [~2 ~2 i] x+ [~] u 0 -3 -2 2 y = [ -11 0) X.Using standard notat·ion used in this chapter(a) Find the transfer function of this system and establish its controllability, observability and stability.(b) Find a non-singular transformation T such that r- 1 AT is
For the linear system x=Ax+Bu y = Cx+Du,[0 0 -1] A= 1 0-3 , 0 1 -3[0 1 OJ C= 001 ' D=[O].(a) Design a full-order state estimator for the system.(b) Find an optimal control that minimizes the cost function where [3 0 0] R =I, Q = 0 2 0 0 0 1 Problem
A linear system X:= Ax+Bu y = Cx+Du has D = 0, where the plant and input matrices (A, B) are given by([ ~i -20] [1]) (A, B) = ~ 3 ~ , ~(a) Discuss the controllability and stability of this system(b) Determine the feedback matrix K that will place the poles of the system at -1,-1 ±j.Problem
Use MATLAB to find the resonant peak, Mpeak, resonant frequency Wr, and bandwidth B W of the following unity feedback contml systems. Make sure that the systems are stable.(a) 5 G(s) = s(1 + 0.5s)(1 + 0.1s)(b) G(s) _ lO(s + 1)- s(s + 2)(s + 10)(c) G( ) 0.5 s = s(s2 + s + 1){d) G woe-s(s)= s(s2
The forward-path transfer function of a unity feedback control system is K G ( s) - ---,-------,- - s(s+6.54)Analytically, find the resonant peak Mpeak! resonant frequency Wr, and bandwidth BW of the closed-loop system for the following values of K:(a) K = 5 (b) K = 21.:39 (c) K = 100.Problem
Draw a Nyquist d·iagram for each of the following systems, and compare the result with that obtained using the MATLAB command Nyquist.(a)(b)(c)KG(s) = K(s + 1)s+2 KKG(s) = (s + 1)(s + 2)2 KG(s) = K(s + 4)(s + 1)s(s + 2)4 Problem
Draw a Nyquist diagram for each of the following systems, and compare the result with that obtained using the MATLAB command Nyquist.(a)(b)(c)KG(s) = K(s + 2) s + 10 KKG ( s) = ..,---.,...,---:-:c-(s + 10)(s + 2) 2 KG(s) = K(s + 10)(s + 1)(s + lOO)(s + 2)3 Using the plots, estimate the range of K
Prove that the magnitude slope of -1 in the Bode plot corresponds to - 20dB per decade.(a) Sketch the polar plot for an open-loop system with transfer function 1/ s 2 , that is, sketch where C1 is a contour enclosing the entire RHP. (Hint: assume C1 takes a small detour around the poles at s =
A certain system is represented by the asymptotic Bode diagram shown below. Find and sketch the response of this system to a unit step input (assuming zero initial conditions).10 ~1 ~0. 1 1 10 100 0)1000 Problem
(a) Calculate the magnitude and phase of 1G(s)= s+1'for w = 0.1, 0.2, 0.5, 1, 2, 5, and 10rad/sec.(b) Sketch the asymptotes for G ( s), and compare these with the cmnpv.ted results from part (a).Problem 6. 7 Draw the Bode plots for each of the following systems. Compare the sketches with the plots
Autonomozts, self-driven veh·icles can be used in warehouses, airports, and many other applications. These vehicles follow a wire embedded in the floor and adjust the steerable front wheel in order to maintain proper direction. The sensing coils, mounted on the fmnl wheel assembly, detect an
A control system for controlling the pressv.re m a closed chamber is shown below, together with its flow graph model.Desire~ Controller pressure Irrfmite pressure r---Y'source 1Pressure Controller Controller De (s)Valve-H(s)Measurement Flow Graph Model The transfer function for the measuring
A rejection network that can be utilized instead of the tmin-T net•work is the bridged- T network: shown below.Bridged T-network The tmnsfcr function of this network is given by s2 +w2 G(s) = s2 + 2(wns/Q) + w;,'where wn = LC, and Q = --, and R 2 is adjusted so that R 2 = -'-------'-Rl 4Rl Show
Sketch the polar plot of the frequency response for the following transfer functions:{a){b){e){d)1 KG(s) = (1 + 0.5s)(1 + 2s)KG(s) = (1 + 0.5s)s2 KG(s) = s + 10 s2 + 6s + 10 KG(s) = 30(s + 8)s(s + 2)(s + 4)Problem
IJmw Bode, Nyquist plots for a sixth-onie1· Fade approximation to a p11,re delay of 1 second.Problem
The block diagram of a control system with feedback is shown in following rhagram.L E(s) 0---+®-- 1 Y(s)s2 (s+5) -I~ s I I I(a) Construct the root loci of the characteristic equation forK 2 0 when Kt = 0.(b) Set K = 10, construct the root loci of the characteristic equation for Kt 2: 0.Problem
The characteristic equation of the liquid level control system is written as 0.006s(s + 12.5)(As + K) + 250N = 0.(a) For A = K = 5, constrztct the root loci of the characteristic equation as N vaTics from 0 to oo.(b) FoT N = 10 and K = const.ruct the mot loci of the characteTistic equation joT A 2:
The forward path transfer functions of unity feedback control systems are given below.{a) G(s) _ K(s + 4) - s(s2 + 4s + 4)(s + 5)(s + 6){b) K G(s) = s(s+2)(s'+5)(s+10){c) G(s) _ K(s2 + 2s + 10)- s(s + 5)(s + 10){d) G(s) - K(s2 + 4)- (s + 2)2(s + 5)(s + 6)Construct the root loci for K :2: 0 .
Determine the step and ramp error constants of unity feedback systems whose forward path transfer functions are given by:( ) G(s) _ 1000 a - (1 + O.ls)(l +lOs){b)(c)G(s) = K(l + 2s)(l + 4s)s2 (s2 + s + 1)G(s) - 100 - s(s2 +lOs+ 100)Problem
Determine the system type for following unity feedback systems whose forward path transfer functions are given below.(a) K G(s) = (1 + s)(l + lOs)(l + 20s){b) G(s) _ lO(s + 1)- s3 (s2 +5s+5){c) G(s) _ 100 - s3 (s + 2) 2 Problem
Consider a system with the following plant transfer function(where the controller gain K = 1).G(s)- s2+s+a- s(s + l)(s2 + s + 1.25)(s2 + s + 4.25){a) Plot the root locus for the plant as a varies from 1.25 to 4.25.{b) Find the exact value of a for which the pair of poles which asymptotically
Two systems have the following plant gains 3s- 4 C(s) = -=----82- 2s + 2 G(s) =3s- 4(a) Obtain the root loci for the two systems.(b) Is there any value of contmller gain K that will stabilize either of these plants?Problem
Stability can be improved by addition of a zero and worsened by the addition of a pole Problem
The addition of poles and zeros (compensation) to the open-loop transfer function in order to influence the root locus, thus satisfying design specifications (compensator design by the root locus method)
Determination of control system stability from the intersection (or lack of intersection) of the root loci with the imaginary axis
Root locus design when two or more parameters are varying, e.g., PI]) where there are three adjustable parameter::;
Use of computers (MATLAB) to generate root loci
Use of the root locus to design and analyze feedback control systems
Identifying the locations of the closed-loop pole in the s-plane as the parameter changes (this produces the root locus).
How changes in the system's feedback characteristics and other parameters influence the pole locations
Welding Contml: Large welding mbots are used in today 's auto plants. The welding head ·is moued to different positions on the av.to body, and rapid, accurate response is requ·ired. A block diagram of a welding head positioning system is shown in the following diagram.~~~-~R(s) +~~Desired
Robot control: Let us consider the control of a robot arm.It is pre1hcted that there w'ill be about 1 DO, DOD robots in service thToughout the world by 2000. The robot is a six-legged micro robot system highly flexible legs with high gain controllers that may become unstable and oscillate. Under
Given the system in state equation form,~dx(t)= Ax(t) + Bu(t)where[ (a) A= 0 1
Using the Routh-Hurwitz criteT"ion, deterrn.ine the stability of the closed-loop system that has the following characteristic equations.Determine the nurnber of roots of each equation that are in the right half s-plane and on the jw-axis.(a) s3 + 25s2 + lOs + 450 = 0{b) s3 + 25s2 + lOs + 50 = 0(c)
Without using the Routh-Hur-witz criterion, determine if the following systems are asymptoi'ically stable, marginally stable, or unstable. In each case, the closed-loop transfer .function is given.(a){b)(c){d)M 8 = 10(s + 2)( ) s3 + 3s2 + 5s K M(s) = .....,3,----s + 5s + 5 100]\If ( s) =
Consider the system x=Ax where A is given by[ A= -b3 0 0 1 0 1 l 0 -b2 -bl(A is called the Schwar-z matn::r.) Show that the fir-st column of the Routh's ar-my of the characteristic equation lsi- AI = 0 consists of 1, b1 , b2 , and blb3.Problem
Consider the unity-feedback control system with the following open-loop transfer function:G(s)- 10 - s(s- 1)(2s + 3)Is this system stable?Problem
Determine the mnge of K joT stability of a unity-feedback control system whose open-loop tmnsfer function is KG ( s) = ---,s (,--s -+--:1 )--,-( s-+---,-2)Problem
Consider the position control syste·rn. shown in the following fig·u·re.Control System: Multiple Loop Control Write a MATLAB progmm to obtain a unit-step response and a unitmmp Tesponse of the system. Plot C1LTves x1 (t) vs. t, x2 (t) vs. t, and e(t)vs. t (where e( t) = T( t) - x1 ( t )} joT
The following diagram shows three systems: System I is a positional servo system. System II is a positional servo system with P D control. System III is a positional servo system with velocity feedback.CompaTe the unit-step, unit-impulse, and unit-mmp responses of the thTee systems. VVhich systeTn
Consider a unity-feedback control system whose open-loop transfer function is G(s) - -:-::-K---== -s(Js+B)Discuss the effects that varying the values of K and B has on the steady state error in unit-ramp response. Sketch typical unit-ramp response curves for a small value, medium value, and large
A traffic control signal is designed to control the distance between vehicles as shown in the following diagram Throttle, engine, Controller and automobile K 1 Y(s) L -- R(s) +f"?\_~-Desired ~ f distance - s s2+ 10s+20 Actual Sensor dis tan A Traffic Control Signal Control(a) Determine the range
An automatically g?tided vehicle on Mars is represented by the following figure.R(s) +®---I 10 I ~Y(s) Steering_ Ll 1 S+iO . S2 Direction command oftravel'----------~~-Mars Vehicle Control Sys/.em The system has a steerable wheel in both the fmnt and back of the vehicle, and the design requires
The control of the spark ignition of an automotive engine requ·ires constant peTjormance over a wide range of parameters. The control system is shown ·in the following diagram, where a controller gain K is to be selected.R(s)Spark Ignition Control System The paran1.cter- p is equal to 2 for many
(5%).(a) Determine the range of a that satisfies the ermr requirement.(b) Determine the range of a that satisfies the stability required.(c) Select an a that meets both requirements.341 Problem
A chem.ical control system is shown in the following diagram. It is des·ired that the system be stable and the steady st.ate error for a v.nit step input be less than or equal to
Consider the case of a navy pilot landing an aircraft on an carrier. The pilot has three basic tasks, the first of which is to t.he aircraft's approach to the ship along the extended centerline of !.he nLnway. The second task is maintaining tht: aircraft on the correct glideslope, the third task is
On July 16, 1993, the elevator in Yokohama's 70-story Landmark ToweT, operating at a peak speed of 45 km/hr {28 mph), was inaugurated as /.he fastest super-fast elevator. To Teach such a speed without leaving passengers' stomachs on the ground flo err, the lift accelemtes f oT longer periods,
A unity feedback control system is shown in the follow·ing diagram.Unity Feedback Control Sysl.em Determine the relative stability of the system with the following transfer function by locating the complex roots in the s-plane.(a) .(b)(c)~ 65 + 33s(,.(s) = s2 (s + 9)24 G ( s) =
A feedback control system is shown in the following diagram.The pmcess transfer function is G(s) = K(s + 40), s(s + 10)and the feedback transfer function is H(s) = 1/(s + 20).(a) DeleTmine the limiting value of gain K for a stable system.339 (b) FoT the gain that resv.lts in ·marginal stabildy,
Arc welding is one of the most irnpoT'tant areas of application for indnstTial robots. In most manufacturing welding sitnations, WI certainties in dimensions of the part, geometry of the joint, and the welding process itself TequiTe the use of sensors for maintaining the weld quality.Se·ueTcLl
Utilizing Routh-Hurwitz criterion, determine the stability of the following polynomials:(a) s2 + 5s + 2(b) s4 + s3 + 3s2 + 2s + K(c) s5 + s4 + 2s3 + s + 5(d) s5 + s4 + 2s3 + s 2 + s + K Problem
A system is Tepresented by equation x =Ax where A.ii == [ 0 1 0 l ~1 ~c ~2 Find the range of c where the system is stable.Problem
Designers have developed a srnall, fast, vertical-takeoff fighter airaaft that is invisible to radar (stealth aircraft). This aircmjt concept uses quickly turning jet nozzles to steer the airplane. The control system for the heading or direction contml is sho·wn in Lhe following diagram.Controller
A control system has the structure shown in the following diagram..L ·@I]-_!®- K Y(s)I s+ 1 - r s(s+2)Control System Determine ihe gain at which Lhe system wilt become unstable. ( Ans:0 < K < 1 Problem
Cons'ider the following control system where the feedback gain {3 is subject to var"iations.10 r +ff\ ~-~() a--------~~- (s+l)(s+lO)-1 []]--------'Control System with Variations in {3 y ~ -u The objective is to design a controller for this system so that the output y(t) accv.mtely tracks the
Consider the following unit feedback control system.o-r _ __,;; ... ® L ---·K{as+b} y s(s+ 1)Control System Problem (a) Disc11ss the effect of different values of (a,b, K) on the shape of the step response of the system.(b) Determine the system type with respect to r( t).Problem
A DC motor speed control is described by the differential equation y + 60y = 600va - 1500w, where y( t) is the motor speed, Va ( t) is the armature voltage, and w ( t) is the load torque. Assume the armature voltage is computed using the PI control law Kp [r(t)- y(t)] + K 1 lt [r(t)- y(t)] dt,where

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Design And Analysis Of Control Systems 1st Edition Arthur G O Mutambara - Solutions

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