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CONTROL SYSTEMS-EE2253PART A & PART BUNIT I - SYSTEMS AND THEIR REPRESENTATIONPART A (2 MARKS)What is control system? (April 2009)A system consists of a number of components connected together to perform a specific function. In a system when the output quantity is controlled by varying the input quantity then the system is called control system. What are the two major types of control system? (May 2007)The two major types of control systems areOpen Loop Control SystemClosed Loop Control SystemDefine open loop control system. (April 2005)The control system in which the output quantity has no effect upon the input quantity is called open loop control system. This means that the output is not feedback to the input for correction. Define closed loop control system. (May 2006)The control system in which the output has an effect upon the input quantity so as to maintain the desired output value is called closed loop control system. What are the components of feedback control system? (Dec 2006)The components of feedback control system are plant, feedback path elements, error detector and controller. Define transfer function. (April 2009)The T.F of a system is defined as the ratio of the Laplace transform of output to Laplace transform of input with zero initial conditions. Laplace Transform of OutputTransfer Function = -----------------------------------------Laplace Transform of Input with zero initial conditions The transfer function can be obtained by taking Laplace transform of the differential equations governing the system with zero initial conditions and rearranging the resulting algebraic equations to get the ratio of output to input.7. What are the basic elements used for modeling mechanical translational system?(May 2007)The model of mechanical translational systems can be obtained by using three basic elements mass, spring, and dash -pot. These three elements represent three essential phenomena which occur in various ways in mechanical systems.The weight of the mechanical system is represented by the element mass and it is assumed to be concentrated at the centre of the body. The elastic deformation of the body can be represented by a spring. The friction existing in mechanical system can be represented by the dash – pot. The dash – pot is a piston moving inside a cylinder filled with viscous fluid.8. What are the basic elements used for modeling mechanical rotational system? (Dec 2009)The model of mechanical rotational systems can be obtained by using three basic elements moment of inertia J, dashpot with rotational frictional coefficient, B, and torsional spring with stiffness, K. The weight of the rotational mechanical system is represented by the moment of inertia of the mass. The moment of inertia of the system or body is considered to be concentrated at the centre of gravity of the body. The elastic deformation of the body can be represented by a spring (torsional spring). The friction excising in rotational mechanical system can be represented by a dash – pot. The dash – pot is a piston rotating inside a cylinder filled with viscous fluid. 9. Name two types of electrical analogous for mechanical system. (May 2008)The two types of analogies for the mechanical system are:Force voltage analogyForce current analogy 10. What is block diagram? A block diagram of a system is a pictorial representation of the functions performed by each component of the system and shows the flow of signals. The basic elements of block diagram are blocks, branch point and summing point. 11. What are the advantages of closed loop systems? (April 2009)In these systems accuracy is very high due to correctiosn of any arising errorSince these systems sense environmental changes as well as internal disturbances, the errors are modified.There is reduced effect of nonlinearity in these systemsThese systems have high bandwidth,i.e high operating frequency zone.There are facilities of automation in these systems12. What is the basis for framing the rules of block diagram reduction technique? (May 2006)The rules for block diagram reduction technique are framed such that any modification made on the diagram does not alter the input - output relation. 13. What is a signal flow graph? (May 2008)A signal flow graph is a diagram that represents a set of simultaneous algebraic equations. By taking L.T the time domain differential equations governing a control system can be transferred to a set of algebraic equations in s-domain. 14. What is transmittance? (Dec 2007)The transmittance is the gain acquired by the signal when it travels from one node to another node in signal flow graph. The transmittance can be real or complex.16. Define non - touching loop. (Dec 2013)The loops are said to be non - touching if they do not have common nodes. 17. Write Masons Gain formula. (May 2011)Masons Gain formula states that the overall gain of the system is 1Overall gain, T = ---------- ∑ Pk Δk Δ kT = T (s) = Transfer function of the systemPk- Forward path gain of kth forward pathΔ= 1- [sum of individual loop gains] + [sum of gain products of all possible combinations of two non touching loops] - [sum of gain products of all possible combinations of three non touching loops] + ……………………..Δk= Δ for that part of the graph which is not touching kth forward path.18. Write the analogous electrical elements in force voltage analogy for the elements of mechanical translational system. (May 2009)Force – Voltage, e Velocity, v – Current, i Displacement, x – Charge, q Frictional Coefficient, B – Resistance, R Mass, M – Inductance, L Stiffness, K - Inverse of capacitance, 1/C 19. Write the analogous electrical elements in force current analogy for the elements ofmechanical translational system. (May 2007)Force, F-Current, i Velocity v-Voltage, V Displacement x-flux, Ф Frictional coefficient, B -Conductance 1/R Mass, M-Capacitance, C Stiffness, K-Inverse of inductance, 1/L 20. Write the force balance equation of the following ideal element?(April 2010)Mass: F = M d2x /dt2Dashpot: F = B dx /dt Spring: F = Kx 21. Distinguish between open loop and closed loop system.(April 2009)Sl. No:OPEN LOOPCLOSED LOOP1Inaccurate Accurate 2Simple and economical Complex and costlier 3The changes in output due to external disturbance are not corrected. The changes in output due to external disturbances are corrected automatically.4Unstable They are generally stable Great efforts are needed to design a stable system.22. What is servomechanism? (May 2005)The servomechanism is a feedback control system in which the output is mechanical Position, i.e., when the objective of the system is to control the position of an object then the system is called servomechanism.23. Why is negative feedback system is invariably preferred in closed loop system?(Dec 2013) The negative feedback results in better stability in steady state and rejects any disturbance signals. electrical signal (control voltage) applied to them into an angular displacement of the shaft. They can either operate in a continuous duty or step duty depending on construction.24. What is synchro? (April 2012)Synchros are a device used to convert an angular motion to an electrical signal or vice versa. It works on the principle of a rotating transformer (induction motor). The term Synchros is a generic name for a family of inductive devices which works on the principle of a rotating transformer (Induction motor). The trade names for Synchros are Selsyn, Autosyn and Telesyn.25.Mention applications of synchros(Dec 2013)To control the angular position of load from a remote place/long distance.For automatic correction of changes due to ditrubance in the angular position of the load.PART B (16 MARKS)1. For a given mechanical or electrical or electromechanical system derive the transfer function. Procedure: Free body diagram, Differential equation, Transfer function (May 2009)Control systems, A. Nagoor Kani, Page: 9 – 28, 37 – 692. Determination of the overall transfer function using Block diagram reduction technique. Procedure: Reduction of block diagram, Determination of Transfer function Control systems, A. Nagoor Kani, Page: 70 – 1003. Determination of the overall transfer function from the given signal flow graph using Mason’s gain formula. Procedure: Reduction of signal flow graph, Apply masons gain formula Control systems, A. Nagoor Kani, Page: 100 – 1294. Write the procedure for constructing the block diagram of a given mechanical or electrical or electromechanical system. Procedure: Write the differential equation, Develop block diagram Control systems, A. Nagoor Kani, Page: 11 - 12, 23 – 24, 37 – 40, 58–605. Find the electrical analogical variables of a mechanical rotational system. Procedure: Write the differential equation, Draw the equivalent electrical networkControl systems, A. Nagoor Kani, Page: 58 – 616. Explain the working of Synchros transmitter and receiver.Procedure: Diagram, Working principleControl systems, A. Nagoor Kani, Page: 171 – 1787. Use Mason’s gain formula for determining the overall transfer function of the system shown in fig.Control systems, A. Nagoor Kani, Page: 106 – 1198. Obtain the transfer function of the mechanical system shown in fig: and draw its analogous electrical circuits.Control systems, A. Nagoor Kani, Page: 41 – 699. a. Explain how the sensitivity to parameter variations in a system is reduced using Feedback. (8) b. Develop signal flow graph of the closed loop controlled drive of a dc motor and find the transfer function. (10)Control systems, A. Nagoor Kani, Page: 2, 31 – 3610. Reduce the block diagram shown in the figure and obtain the overall transfer function.UNIT II - TIME RESPONSEPART A (2 MARKS)1. What is transient response? (May2009)The transient response is the response of the system when the system changes from one state to another. The transient response of a system to a unit step input depends on the initial conditions. Therefore to compare the time response the time response of various systems it is necessary to start with standard initial conditions. The most practical standard is to start with the system at rest and so output and all time derivates before t = 0 will be zero.2. What is steady state response? (April 2004)The steady state response is the response of the system when it approaches infinity. 3. What is an order of a system? (Dec 2012)The order of a system is the order of the differential equation governing the system. The order of the system can be obtained from the transfer function of the given system. 4. Define Damping ratio. (Dec 2013)Damping ratio is defined as the ratio of actual damping to critical damping. 5. List the time domain specifications.(April 2008) The time domain specifications are Delay time Rise time Peak time Peak overshoot 6. Define Delay time. (Dec 2007)The time taken for response to reach 50% of final value for the very first time is called delay time. 7. Define Rise time. (Dec 2004) It is the time for response to rise from 0 to 100% of its final value for the very first time. For under damped system, the rise time is calculated from 0 to 100%. But, for over damped system it is the time taken by the response to raise form 10% to 90%. For critically damped system, it is the time taken for response to rise from 5% to 95%.Rise time, tr for a step input, Π – cos -1 δRise time, tr =-------------------------- ωn √ (1 – δ2)8. Define peak time. (Dec 2005)It is the time required for the response to reach the peak value for the very first time. Or it is the time taken for the response to reach the peak overshoot, Mp. πPeak time, tp = -------------------------- ωn √ (1 – δ2)9. Define peak overshoot. (April 20060Peak overshoot is defined as the ratio of maximum peak value measured from themaximum value to final value. The peak overshoot is defined as c (tp) - c(∞)Mp=------------------------c (∞)For step input, c (∞) = css = 1; - π δMp =exp ----------------- √ (1 – δ2)10. Define Settling time. (Dec 2006)It is defined as the time taken by the response to reach and stay within a specified tolerance band ±Δ. It is usually expressed as % of final value. This tolerance band or error is either ±2% or ±5% of the final steady state value. 4For 2% tolerance, settling time, ts = ------------δωn 3For 5% tolerance, settling time, ts = ------------δωn11. What is the need for a controller? The controller is provided in the feedback control systems to modify the error signal and to achieve better control action. The introduction of controllers will modify the transient response and the steady state error of the system. 12. What are the different types of controllers? (May 2006)The different types of controllers are:ON – OFF ControllerProportional Controller PI Controller PD Controller PID Controller 13. What is proportional controller? (April 2006)It is device that produces a control signal which is proportional to the input error signal. The transfer function of proportional controller is Kp. The term Kp is called the gain of the controller. Hence the proportional controller amplifies the error signal and increases the loop gain of the system. The following aspects of system behaviour are improved by increasing loop gain.Steady state tracking accuracyDisturbance signal rejectionRelative stabilityIn addition to increase the loop gain it decreases the sensitivity of the system toparameter variations. The drawback in proportional control action is that it produces a constant steady state error.14. What is PI controller? (Dec 2005)It is device that produces a control signal consisting of two terms: one is proportional to error signal and the other is proportional to integral of error signal. Transfer Function of PI controller = Kp (1+1/Ti S) where, Kp is equal to proportional gain and Ti is equal to integral time.15. What is the significance of integral controller and derivative controller in a PID controller? (Dec 2007)The proportional controller stabilizes the gain but produces a steady state error. The integral control reduces or eliminates the steady state error. The derivative controller reduces the rate of change of error. The combined effect of all the three cannot be judged from the parameters Kp, Ki and Kd.16. Why derivative controller is not used in control systems? (May 2011)The derivative controller produces a control action based on the rate of change of error signal and it does not produce corrective measures for any constant error. Hence derivative controller is not used in control system 17. Define Steady state error. (May 2012)The steady state error is defined as the value of error signal e (t), when t tends to infinity. The steady state error is a measure of system accuracy. These errors arise from the nature of inputs, type of system and from non – linearity of system components.18. What is the drawback of static coefficients? (May 2013)The main draw back of static coefficient is that it does not show the variation of error with time and input should be standard input. 19. What is step signal? (May 2004)The step signal is a signal whose value changes from zero to A at t= 0 and remains constant at A for t > 0. r (t) r (t) = A u (t) A where, u (t) = 1; t ≥ 0 u (t) = 0; t < 0 0t20. What is ramp signal? (April 2007)The ramp signal is a signal whose value increases linearly with time from an initial value of zero at t=0.the ramp signal resembles a constant velocity. r (t)r (t) = At; t ≥ 0 = 0; t < 0 0 t21. What is a parabolic signal? (May 2005)The parabolic signal is a signal whose value varies as a square of time from an initial value of zero at t = 0.This parabolic signal represents constant acceleration input to the signal. r (t) r (t) = At2 ; t ≥ 0 2 = 0; t < 0 0 t22. What are the three constants associated with a steady state error? (April 2007)Positional error constant Velocity error constant Acceleration error constant 23. What are the main advantages of generalized error co-efficients? (Dec 2006)Steady state is function of time.Steady state can be determined from any type of input24. What is steady state error and what are static error constants? (April 2008)The steady state error is the value of error signal e (t) when t tends to infinity and the constants Kp, Kv and Ka are called static error constants. 25. What is the effect of PD controller on system performance? (May 2009)The effect of PD controller is to increase the damping ratio of the system and so the peak overshoot is reduced. 26 . What is meant by ‘corner frequency’ in frequency response analysis?(Dec 2013)The magnitude plot can be approximated by asymptotic straight lines. The frequencies corresponding to the meeting point of asymptotes are called corner frequency. The slope of the magnitude plot changes at every corner frequencies27. What is meant by state state error?(Dec 2009)The stady state error is the value of error signal e(t),when t tends to infinity.The steady state error is a measure of system accuracy.These errors arise from the nature of inputs, type of system and from non-linearity of systems componentsPART B (16 MARKS)1:The unity feedback system is characterized by an open loop transfer function G(s)=k/s(s+10).determine the gain k,so that the system will have a damping ratio of0.5.for this value of k,determine settling time,peak overshoot and time to peak overshoot for unit step input.(April 2009)ii) a unity feedback system has the forward transfer function G(S)=k1(2s+1)/s(5s+1)(1+s)^2. The input r(t)=(1+6t) is applied to the system. determine the minimum value of k1,if the steady error is to be less than 0.1.(Dec 2012)2)with suitable block diagram and equations,explain the following types of controllers employed in control systems(May 2012)i)propotional controllerii)propotional-plus-integratal controlleriii)PID controlleriv)integral controller.(nov/dec2012)3)explain in detail PI,PID,PD controllers.(nov/dec2010)4)the open loop transfer function of a unity feedback system is given by G(S)=k/s(sT+1), where k and T are positive constant.by which factor should the amplifier gain k be reduced,so that the peak overshoot of unit step response of the systemis reduced from 75% to 25%.(nov/dec 2010)5)i)determine k to limit the error error of a system for input 1+8t+18/2 t^2 to 0.8 having G(S)H(S)=K/s^2(s+1)(s+4)ii)the forward path transfer function of an unity feedback control system is given by G(S)=2/s(s+3). Obtain an expression for unit step response of a system.(nov/dec 2010)6)a)the open loop transfer function of servosystem with unity feedback is G(S)=10/s(0.1s+1). Obtain the steady state error when subjected to an input given by the polynomial r(t)=a0+a1 t+a2 t^2.also find the three error constants for the system(nov/dec2010 eie)7)consider a unity feedback system with open loop transfer function G(S)=75/(s+1)(s+3)(s+8). Design a PID controller to satisfy the following specifications:i)steady state error for unit ramp input should be less than 0.08.ii)damping ratio=0.8iii)natural frequency of oscillation=2.5rad/sec.state the expressions for the transfer function of the PID controller and for the open looptransfer functionof the compensated system(nov/dec2011 ec)8)i)derive an expression to find steady state error of a closed loop control system.ii)the closed loop transfer function of a second order system is given by T(S)=100/s^2+10s+100. Determine the damping ratio,natural frequency of oscillation, rise time,settling time and peak overshoot.(nov/dec2011ec)9)Determine all the time domain specifications.ii)derive the expression for rise time and peak overshoot.(nov/dec 2011 eie).10) Discuss the effect of PI,P,PID,PD controllers.(nov/dec 2011 eie)UNIT IIIFREQUENCY RESPONSEPART A1.What is frequency response?(April 2009)A frequency responses the steady state response of a system when the input to the systemis a sinusoidal signal.2.List out the different frequency domain specifications?(May 2009)The frequency domain specification are i)Resonant peak. ii)Resonant frequency.3.Define –resonant Peak?(Dec 2009)The maximum value of the magnitude of closed loop transfer function is called resonantpeak.4.Define –Resonant frequency?(Dec 2006)The frequency at which resonant peak occurs is called resonant frequency.5.What is bandwidth?The bandwidth is the range of frequencies for which the system gain is more than 3 dbB. The bandwidth is a measure of the ability of a feedback system to reproduce the input signal,noise rejection characteristics and rise time.6.Define Cut-off rate?(April 2008)The slope of the log-magnitude curve near the cut-off is called cut-off rate. The cut-off rate indicates the ability to distinguish the signal from noise.7.Define –Gain Margin?(Dec 2008)The gain margin, kg is defined as the reciprocal of the magnitude of the open loop transfer function at phase cross over frequency. Gain margin kg = 1 / | G(jw pc) |.8.Define Phase cross over?(Dec 2012)The frequency at which, the phase of open loop transfer functions is called phase cross over frequency wpc.9. What is phase margin?(Dec 2009)The phase margin ,g is the amount of phase lag at the gain cross over frequency required to bring system to the verge of instability.11.Define Gain cross over?(May 2006)The gain cross over frequency wgc is the frequency at which the magnitude of the open looptransfer function is unity.12.What is Bode plot?(Dec 2006)The Bode plot is the frequency response plot of the transfer function of a system. A Bode plot consists of two graphs. One is the plot of magnitude of sinusoidal transfer function versus log w.The other is a plot of the phase angle of a sinusoidal function versus log w.13.What are the main advantages of Bode plot?(April 2008)The main advantages are:i) Multiplication of magnitude can be in to addition.ii) A simple method for sketching an approximate log curve is available.iii) It is based on asymptotic approximation. Such approximation is sufficient ifrough information on the frequency response characteristic is needed.iv) The phase angle curves can be easily drawn if a template for the phase anglecurve of 1+ jw is available.14.Define Corner frequency?(Dec 2011)The frequency at which the two asymptotic meet in a magnitude plot is called cornerfrequency.15.What is Polar plot?(April 2004)The Polar plot of a Sinusoidal transfer function G(j_) is a plot of the magnitude of G(j_)versus the phase angle/argument of G(j_) on polar or rectangular co-ordinates as _ is variedfrom zero to infinity.16.What is minimum phase system?(April 2007)The minimum phase systems are systems with minimum phase transfer functions. Inminimum phase transfer functions, all poles and zeros will lie on the left half of s-plane.17.What is non-minimum phase transfer function?A transfer function which has one or more zeros in the right half of s-plane is known as nonminimumphase transfer function.18.What are M circles?The magnitude of closed loop transfer function with unit feed back can be shown to be in the for every value if M.These circles are called M circles.19.What is Nichols chart?The chart consisting if M & N loci in the log magnitude versus phase diagram is calledNichols chart.20.What are two contours of Nichols chart?(Dec2007)Nichols chart of M and N contours, superimposed on ordinary graph. The M contours are the magnitude of closed loop system in decibels and the N contours are the phase angle locus ofclosed loop system.21.How is the Resonant Peak, resonant frequency, and bandwidth determined fromNichols chart?(April 2007)i) The resonant peak is given by the value of _.contour which is tangent to G(jw ) locus.ii) The resonant frequency is given by the frequency of G(jw ) at the tangency point.iii) The bandwidth is given by frequency corresponding to the intersection point of G(jw ) and–3dB M-contour.22.What are the advantages of Nichols chart?(Dec 2009)The advantages are:i) It is used to find the closed loop frequency response from open loop frequency response.ii) Frequency domain specifications can be determined from Nichols chart.iii) The gain of the system can be adjusted to satisfy the given specification23.What are N circles?( Dec 2005)If the phase of closed loop transfer function with unity feedback is _, then tan _ will be in the form of circles for every value of _. These circles are called N circles.PART B1.given G(S)=Ke-0.2s/s(S+2)(s+8),find K for the following two cases:(|) Gain margin equal to 6 db(||)phase margin equal to 45(nov/dec 2012)2.Draw the pole-zero diagram of a lead compensator.propose lead compensation using electrical network.derive the transfer function.draw the bode plots.(nov/dec 2012)3.sketch bode plot for the following transfer function and determine the system gain K for thegain cross over frequency to be rad/sec G(s)=KS2/(1+0.2S)(1+0.02S)(NOV/DEC 2010)4.The open loop transter function of a unity feed back system by G(S)=1/S2(1+s)(1+2S)sketch the polar plot and determine the gain margin and phase margin.(nov/dec 2010)5.A unity feed back control system has G(s)=K/S(s+4)(s+10). draw the bode plot. find K when phase margin=30.(apr/may 2010)6.A Unity feed back control system has G(s)= 10/S(S+1)(S+2).Draw the nyquist plot and comment on closed loop stability.(apr/may 2010)7.sketch the bode magnitude plot for the transfer function G(s)=100(1+0.1s)/(1+0.01s)(1+S)(nov/dec 2011)8.draw the polar plot for the the following transfer function G(S)=10(s+2)/S(s+1)(s+3).(NOV/DEC 2011)9.Skecth the bode plot for the following transfer function and determine the phase margin and gain margin of the system.G(S)=10/s(1+0.5s)(1+0.1s)(nov/dec 2011)10. sketch the polar plot for the following transfer function G(s)H(s)=10(s+4)(s+4)/S(S2-3S+10).(NOV/DEC 2011) UNIT-4STA BILITY OF CONTROL SYSTEM PART AState the advantages of Nyquist stability criterion over that of Routh’s criterion. (Dec 2008) We can predit the closed loop stability of a given system from open loop data.It can be modified for non-linear control system.What is meant by BIBO stability.(April 2005)Ans: A linear relaxed system is said to have BIBO stability if every bounded(finite)input results in a bounded(finite) output.Using Routh criterion, determine the stability of the system represented by the characteristic equation S4 + 8S3 +18S2 + 16s + 5=ment on the location of the roots of characteristics equ.(Dec 2012)Ans:The given characteris equ is 4th order equ and so it has 4 roots.Since the highest power of s is even no,form the frst row of routh array using the coefficients of even powers of s and form the second row using the using the coefficients of odd powers of s. s4 : 1 18 5 …..row1s3 : 8 16 …..row2The elements of s3 row can be divided bu 8 to simplify the computations.s4 : 1 18 5 …..row1s3 : 1 2 …..row2s2 : 16 2 …..row3s1 : 1.7 …..row4s0 : 5 …..row5On examining the elements of 1st column of routh array it is observed that all the elements are positive and there is no sign change.Hence all the roots are lying on the left half of s-plane and the system is stable. What are the effects of adding open loop pole to root locus and the system?(Dec 2012)Ans: The addition of a zero to open loop t.f of a system will reduce the steady state error. The closer the pole to origin lesser will be the steady state error. Thus the steady state performance of the system is improved.Also the addition of pole will increase the order of the system,which in turn makes the system less stable than the original system.State Nyquist stability criterion (Dec 2008)Ans: If G(s)H(s) contour in the G(s)H(s) plane corresponding to Nyquist contour in s-plane encircles the point -1+j0 in the anti-clockwise direction as many times as the umber of right half s-plane poles of G(s)H(s).Then the closed loop system is stable.How is the value of gain “K” determined from root locus plot? (April 2007) Ans:The gain K at a point s=sa on root locus is given by,K= product of length of vector from open loop poles to the point sa/ product of length of vector from open loop zeros to the point sa.State the necessary and sufficient conditions of Routh-Hurwitz criterion.(April 2009)Ans: Routh criterion states that the necessary and sufficient condition for stability is that all of the elements in the 1st column of the routh array be positive. If this condition is not met, the system is unstable and the number of sign changes in the elements of the 1st column ot routh array corresponds to the number of roots of characteristic equ in the right half of the s-plane.Define stability of a system.(Dec 2009)Ans: The term stability refers to the stable working condition of a control system.Every working system is designed to be stable.What is centroid?Ans: The meeting point of asymptotes with real axis is called centroid. sum of poles – Sum of zerosCentroid = -----------------------------------------n - m Write the expressions for gain margin and phase margin.(April 2008) Ans: Gain Margin,Kg=1/|G9j)| ω=ωpcPhase Margin,γ=180 + φgc How is pole locations and stability related?(Dec 2010)Ans; The closed loop transfer function,M(s) can be expressed as a ratio of 2 polynomials in s. The denominator polynomial of closed loop transfer function is called characteristic equ.The roots of charac equ are poles of closed loop transfer function. 2 limitation of routh criterion.(April 2011)It gives only information about the absolute stability of the system.The degree of stability of a stable system can not be obtained.The criterion can be applied if the characteristics equation has constant coefficients and can not be applied if they they are not real or contains exponential terms as in the case of systems with dead time What are the two segments of Nyquist contour? (May 2012)The nyquist contour is directed clockwise and comprises of two segments;A finite line segment C1 along the imaginary axis. An arc C2 of infinite radius. Along C1, s = jω with s varying from –j∞ to +j∞Along C2, s = Lt R e jθ with θ varying from +π /2 to - π /2 R ∞What are root loci? (April 2004)The path taken by the roots of the open loop transfer function when the loop gain is varied from 0 to ∞ are called root loci. What is a dominant pole? (May 2011)The dominant pole is pair of complex conjugate pair which decides the transient response of the system. In higher order systems the dominant poles are very close to origin and all other poles of the system are widely separated and so they have less effect on transient response of the system. What are the main significances of root locus? (april 2010)The main root locus technique is used for stability analysis. Using root locus technique the range of values of K, for a stable system can be determined What are the effects of adding a zero to a system? (April 2009)Adding a zero to a system increases peak overshoot appreciably. What is limitedly stable system? (Dec2013)For a bounded input signal if the output has constant amplitude oscillations. Then the system may be stable or unstable under some limited constraints such a system is called limitedly stable system. What is break away and break in point? How to determine them?(Dec 2013)At break away point the root locus breaks from the real axis to enter in to the complex plane. At break in point the root locus enters the real axis from the complex plane.To find the breakaway and breakin points, form an equation for K from the characteristics equation and differentiate the equation of K with respect to s. Then find the roots of equation, dK /ds = 0. The roots of dk / ds = 0 are breakaway or breakin points provided for this value of root the gain K should be positive and real.PART B(i) Determine the range of K for stability of unity feedback system whose open loop transfer function is G(s)= K/s(s+1)(s+2) using Routh stability criterion.(ii) Draw the approximate root locus diagram for a closed loop system whose loop transfer function is given by G(s)H(s)=k/s(s+5)(s+10). Comment on the stability. (April 2009) Sketch the Nyquist plot for a system with open loop transfer function G(s)H(s)=K(1+0.4s(s+1)/(1+8s)(s-1) and determine the range of K for which the system is stable. (Dec2013) Sketch the root locus of the system whose open loop transfer function is G(s)= K/s(s+2)(s+4).find the value of K,so that the damping ratio of the closed loop system is 0.5.(Dec 2013)Construct the Nyquist plot for a system whose open loop transfer function is given by G(s) H(s)=K(1+ s)2 /s3 . Find the range of K for stability.(Dec 2009)(i) For a system with,F(s)=s4 + 22s3 + 10s2 +s+k=0, obtain the marginal value of K, and the frequency of oscillations at that value of K.(ii) Explain how gain margin and phase margin are obtained using polar plot.(Dec2009)(i) construct the root locus for the function, G(s)H(s)= K(s+2)/(s+1)2 and discuss about the stability of the system.(ii) For a system with, G(s) H(s)= 40/(s+4)(s2 +2s+2), obtain the gain margin and stability using Nyquist plot.(April 2010)Draw the root locus for the open-loop transfer function of a unity feedback control system give below and determine.(Dec 2010)the value of K for G=0.5.the value of K for marginal stabilitythe value of K at S=-4 G(s)=K/s(s+1)(s+3)(i) using Routh criterion,investigate the stability of a unity feedback control system whose open-loop transfer function is givem by G(s)=e-st /s(s+2)(ii) A Closed loop control system has the characteristic equation given by s3 + 4.5s2 +3.5s+1.5=0.(Dec 2009)Consider the open-loop transfer function of a unity feedback control system; G(s)= K(s+2)/s(s+4)(s+6).(April 2012)Using Routh criterion,find the range of values of k that correspond to a stable system. Note that K is a positive real constant. A unity feedback control system has an open-loop transfer functionG(s)=k/s(s2 +4s+13), sketch the root locus plot of the system.Draw the circuit of a lag lead compensator nad derive its transfer function. What are the effects?(April 2012)Draw and explain a typical circuit to demonstrate the action of proportional and differential controller. Derive the transfer function.(April 2011)The open loop T.F of a unity feed back system is given by G(s)=k/(s+2)(s+4)(s2 +6s+25), by applying the Routh criterion, discuss the stability of the closed loop system as a function of K.Determine the value of K which will cause sustained oscillations in the closed-loop system.what are the corresponding oscillation frequencies.(Dec 2008)Sketch the root locus of the system having G(s)H(s)=K(s+2)/(s+1)(s+3+j2)(s+3+j2) for positive value of K.9Dec 2008)(i) A certain unity negative feedback control system has the following open loop transfer function G(s)H(s)=K/(s(s+2)( s2 +2s+5). Find the break away points and draw Root Locus for 0≤w≤∞.(April 2008) (ii) List the advantages pf Routh’s array method of examining stability of a control system. (i)Construct Routh array and determine the stability of the system whose characteristic equ is s6 + 2s5 + 8s4 +12s3 + 20s2 +16s+16=0. Also determine the number of roots lying on right-half of s plane,left-half of s plane and on imaginary axis. (ii) Discuss the difficulties encountered in the Routh stability test.(Dec 2008)Plot the root-locus for a unity feedback closed loop system whose open loop T.F is G(s)=1/s(s+4)(s2 +2s+2).(Dec 2008_The open loop transfer function of a unity negative feedback system is given by G(s)=k(s+3)/s(s2 +2s+2). Using the Nyquist criterion or otherwise find the value of K for which the closed loop system just stable.(May 2009)UNIT V - COMPENSATOR DESIGNPART A (2 MARKS)What are the two types of compensation? (May 2004)The two different types of compensation techniques are:Cascade or series compensation Feedback compensation or parallel compensation 2. What are the three types of compensators? (May 2005)The three different types of compensators are:Lag compensator Lead compensator Lag-Lead compensator 3. What are the uses of lead compensator? (Dec 2009)Speeds up the transient response Increases the margin of stability of a system Increases the system error constant to a limited extent. 4. What is the use of lag compensator? (April 2008)Improve the steady state behavior of a system, while nearly preserving its transient response. 5. When lag - lead compensator is required? (May 2003)The lag lead compensator is required when both the transient and steady state response of a system has to be improved 6. What is a compensator? (Dec 2007)A device inserted into the system for the purpose of satisfying the specifications is called as a compensator. 7. What are the advantages of frequency domain design?(Dec 2013)Effects of disturbances, sensor noise plant uncertainties can be easily visualized.Experimental information can be used for design purposes.8. State the characteristics of lead compensator?(2013)The lead compensator transfer function is represented as, Ts + 1 s + 1/T Gc (s) = Kc α -------------- = Kc ----------------. α Ts + 1 s + 1/ αTThe range of values of α lies between 0 and 1. The compensator Gc (s) has a zero at – 1 / T and a pole at – 1 / αT. The zero – 1 /T always located at the right of the pole in the complex plane since α lies between 0 and 1. The geometric mean of corner frequencies, ωm = ω1 ω2/2.9. State design procedure of lead compensator.(Dec 2009)Write the general transfer function of the lead compensator, Gc (s)Draw the block diagram of CL uncompensated and compensated system.Write down the forward path transfer function, Gc (s) * G (s).Here, K is the OL gain to be adjusted, Find the value of K’.Draw the bode plot for the system.Find the phase lead angle.Find the new phase margin & gain cross over frequencyFind α & TFind Gc (S), Go(s)Verify Phase marginWhat are the formulae to find the maximum phase angle, new gain corner frequency, magnitude to the compensated system? (Dec 2009)Maximum phase angle, sin Фm = 1 – α / 1 + αNew gain corner frequency, ωm = 1 / √α TMagnitude to the compensated system, Δm = 20 log (1 / √α) dB. 12. State the characteristics of lag compensator?(Dec 2012)The lag compensator transfer function is represented as, Ts + 1 s + 1/T Gc (s) = Kc α -------------- = Kc ----------------. Where α > 1.α Ts + 1 s + 1/ αTThe compensator Gc (s) has a zero at – 1 / T and a pole at – 1 / αT. The pole is close to the origin than the zero. 13. State the magnitude lag characteristic of a lag compensator.(SMay 2011)The magnitude of the lag compensator has 20dB.At low frequencies and 0dB at higher frequencies. Lag compensator acts as a low pass filter. It attenuates high frequencies and allows low frequencies. So, it acts as high frequency noise filter. Lag compensator increase the order of the system by one so the transient response becomes sluggish.14. State the phase lag characteristic of a lag compensator.(April 2011)This type is of no use for compensation purposes.It has a high gain at low frequencies and low gain at high frequencies.High gain at low frequencies improves steady state performance and low gain at high frequencies improves phase margin. 15. State design procedure of lag compensator.(Dec 2012)Write the general transfer function of the lag compensator, Gc (s)Draw the block diagram of CL uncompensated and compensated system.Write down the forward path transfer function, Gc (s) * G (s).Here, K is the OL gain to be adjusted, Find the value of K’.Draw the bode plot for the system.Find the phase angle.Find the new phase margin & gain cross over frequencyFind α & TFind Gc (S), Go(s)Verify Phase margin16. State the selection criteria of compensator.(April 2006)The lead compensation becomes ineffective when the phase angle of the uncompensated system deceases rapidly near the gain crossover frequency because he shift in the gain crossover frequency to the right make it difficult to provide enough phase angle poles close to the gain crossover frequency. Under this condition lag compensator is used.The combination of increased open loop gain and lead compensator improves the state error and phase margin. But it is boosting the noise level. Therefore we have to choose lag compensator whenever we want to reduce noise level.Selection of compensator is the compromise between sluggish response due to lag compensator and noise amplification property of lead compensator.17. What is the need of lag – lead compensator?(April 2006)Situation may arise, For an unstable uncompensated system lead compensator provides fast response but does not provide enough phase margin.For an unstable uncompensated system lag compensator stabilize the system but does not provide enough bandwidth.For a slightly unstable system faster response is needed with nearly critical damping.18. State the characteristics of lag – lead compensator.(May 2007)The transfer function of lag – lead compensator, s + 1 / T1 s + 1 / T2 Gc (s) = Kc ------------- -------------- s + 1 / αT1 s + 1 / βT2 , where α < 1 and β > 1. The term s + 1 / T1 works as lead network and the term s + 1 / T2 works as a lag network. s + 1 / αT1 s + 1 / βT2 19. Write down the formula for new gain crossover frequency?(Dec 2009)New gain crossover frequency, ω’g = frequency at which G1 (s) = (- 180 + given phase margin + ε). 5? < ε < 12? is to compensate for the phase lag of lag compensator.20. What is a lag-lead compensator?(Dec 2007) Ans: A compensator having the characteristics of lag-lead network is called lad lead compensator.If a sinusoidal signal is applied to a lag lead compensator then the output will have both phase lag and lead with respect to input, but in different frequency regions20. Write down the procedure for lag – lead compensator?(May 2009)Write down the transfer function, (s + 1 / T1) (s + 1 / T2) Gc (s) = Kc (s + 1 / αT1) (s + 1 / βT2) Condition, α < 1, β > 1. α =1/ β PART-B 1.Design a suitable compensator such that a phase margin of 45? is achieved without sacrificing system velocity error constant.(Derc 2009)2. Consider the system as shown.Design a lead compensator for the system to meet the following specifications: Damping ratio ζ=0.7,Settling time ts=1.4 sec,Velocity error constant Kv=2 sec-1 (Dec2009)3. Derive the steady state error for a rate servomechanism. Draw and explain the block diagram of the servo.(April 2011)4.Design a phase lag compensator so the system G(s)H(s)=100/s(s+1) will have phas5.unity feedback system has an open loop transfer function G(s)H(s)=k/s(s+1)(1+0.2s)6. Design a phase lag compensator to achieve the following specofications.(May 2006)Velocity error constants Kv=5, Phase margin 45? .To open loop T.F of a system is given below G(s)=k/s(s=1)(s+4)7. Design a suitable lag compensator to meet the following specifications.(Dec 2012)Phase Margin=43, Bandwidth=1.02 rad/sec, Velocity error constant Kv ≥ 5 sec-1 15. State the characteristics of lead compensator and write the procedure for designing the lead compensator. Control Systems Engineering, M. Gopal, Page: 6. 2 – 6.5 58. The open loop transfer function of the uncompensated system is G (s) = ------------. s (s + 2)Design a suitable compensator for the system so that the static velocity error constant kv is 20 sec -1, the phase margin is atleast 55? and the gain margin is atleast 12 dB.(Dec2013)Procedure:Find the value of K, phase margin of uncompensated systemFind the new phase margin & gain cross over frequencyCalculate α & TFind Gc (S), Go(s)Verify Phase marginControl Systems Engineering, M. Gopal, Page: 6.5 - 6.89. State the characteristics of lead compensator and write the procedure for designing the lead compensator.Control Systems Engineering, M. Gopal, Page: 6. 9 – 6.11 510. The open loop transfer function of the uncompensated system is G (s) = ------------. s (s + 2)Design a suitable lag compensator for the system so that the static velocity error constant kv is 20 sec -1, the phase margin is atleast 55? and the gain margin is atleast 12 dB.Procedure: Find the value of K, the phase margin of uncompensated systemFind the new phase margin & gain cross over frequencyCalculate α & TFind Gc(S), Go(s)Verify Phase marginControl Systems Engineering, M. Gopal, Page: 6. 11 – 6.1411. State the characteristics of lag - lead compensator and write the procedure for designing the lead compensator.(May 2007)Control Systems Engineering, M. Gopal, Page: 6. 14 – 6. 15 112. Open loop transfer function of the uncompensated system is G (s) = -------------------. (s + 1) (s + 2)Compensate the system by cascading suitable lag – lead compensator so that the compensated system has the static velocity error constant kv is 10 sec -1, the phase margin is atleast 45? and the gain margin of 10 dB or more.Procedure:Find the value of K, phase margin of uncompensated systemFind the new phase margin, new gain cross over frequency Calculate α, β , T1 & T2 Find Gc(S), Go(s) Verify Phase margin Control Systems Engineering, M. Gopal, Page: 6. 15 – 6. 18 K13. Consider a type 1 unity feedback system with an OLTF G (s) = ---------------. It is s (s + 1)specified that Kv = 12 sec -1 and ФPM = 40?. Design lead compensator to meet the specifications.Procedure:Find the value of K, phase margin of uncompensated systemFind the new phase margin, new gain cross over frequency Calculate α, β , T1 & T2 Find Gc(S), Go(s) Verify Phase margin Control Systems Engineering, M. Gopal, Page: 6. 18 – 6. 22 K14. Consider a type 1 unity feedback system with an OLTF G (s) = ------------------. The s (s + 1) (s + 4)System is to be compensated to meet the following specifications: damping ratio, δ = 0.4, settling time, ts = 10sec. Velocity error constant, kv = 5sec -1(Dec2010)Procedure:Convert the time domain specifications in to frequency domain specification.Find the value of ts, K, phase margin of uncompensated systemFind the new phase margin, new gain cross over frequency Calculate α, T1 & T2 Find Gc(s), Go(s) Verify Phase margin Control Systems Engineering, M. Gopal, Page: 6. 29 – 6. 3215. An unity feedback type 0 system with transportation lag has a forward loop transfer function of 10e –j0.02ω G (s) = ------------------------------------------------- (1 + j 0.5 ω) (1 + j 0.1 ω) (1 + j0.05 ω). Design a suitable compensation scheme so that the system acquires a damping factor, 0.4 without loss of steady state accuracy. Estimate the bandwidth and settling time of compensating system.Find the value of K, phase margin of uncompensated systemFind the new phase margin, new gain cross over frequency Calculate α, T1 & T2 Find Gc(s), Go(s) Verify Phase margin Control Systems Engineering, M. Gopal, Page: 6. 34 – 6. 39K16. The open loop transfer function of the given system is G (s) = ------------------------------- s (2s + 1) (0.5s + 1). It is desired to design a compensator to obtain a phase margin of 35? and velocity error constant = 10sec -1. (Dec 2008)Find the value of K.,Find the phase margin of uncompensated systemFind the new phase margin, new gain cross over frequency Calculate α, T Find Gc(s), Go(s) Verify Phase margin Control Systems Engineering, M. Gopal, Page: 6. 22– 6. 26 ................
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