CONTROL SYSTEM (EE0704) STUDY NOTES

Transcription

1 CONTROL SYSTEM (EE0704) STUDY NOTES M.S.Narkhede Lecturer in Electrical Engineering GOVERNMENT POLYTECHNIC, NASHIK (An Autonomous Institute of Government of Maharashtra) Samangaon Road, Nashik Road, Nashik Phone : (0253) , Fax : (0253) Website: info@gpnashik.com

2 Preface Dear students, It gives me immense pleasure in bringing these study notes for you. In fact this is my first attempt to provide computerized notes to students in this format. I have also given you Switchgear & Protection Notes, but those were in power point presentation form. These notes are prepared by referring following books. Control System Engineering By: I.J.Nagrath, M.Gopal Automatic Control Systems By Benjamin C.Kuo Control Systems Engineering By S.N.Sivanandam These notes cover your first three modules i.e. Basis of Control system, Time response of Control system & Concept of stability in all for 55 marks. For next modules you can refer my OHP transparencies. I must thank our honorable Principal Dr. R.S.Naidu for his constant encouragement in providing latest study material to students in good readable form. Thanks are also due to my colleagues Prof. Mrs. S.S.Umare & Prof.Mrs.D.R.Kirtane for their kind support. Thanks are due to Prof. D.D.Lulekar, HOD of Electrical Engg. Department for providing the infrastructure required for preparing these notes. Hope these notes will be useful to you in studying Control System. M.S.Narkhede LEE, GP Nashik Thursday, 12 th February

3 CONTROL SYSTEMS An average human being is capable of performing a wide range of-tasks, including decision making. Some of these tasks, such as picking up objects, or walking from one point to another, are normally carried out in a routine fashion. Under certain conditions, some of these tasks are to be performed in the best possible way. For instance, a marathon runner, not only must run the distance as quickly as possible, but in doing so, he or she must control the consumption of energy so that the best result can be achieved. Therefore, we can state in general that in life there are numerous objectives that need to be accomplished, and the means of achieving the objectives usually involve the need for control systems. In recent years control systems have assumed an increasingly important role in the development and advancement of modern civilization and technology. Practically every aspect of our day-to-day activities is affected by some type of control system. For example, in the domestic domain, automatic controls in heating and air-conditioning systems regulate the temperature and humidity of homes and buildings for comfortable living. To achieve maximum efficiency in energy consumption, many modern heating and air-conditioning systems in large office and factory buildings are computer controlled. Control systems are found in abundance in all sectors of industry, such as quality control of manufactured products, automatic assembly line, machine-tool control, space technology and weapon systems, computer control, transportation systems, power systems, robotics, and many others. Even such problems as inventory control, and social and economic systems control, may be approached from the theory of automatic controls. Regardless of what type of control system we have, the basic ingredients of the system can be described by I. Objectives of control 2. Control system components 3. Results In block diagram form, the basic relationship between these three basic ingredients is illustrated in above Fig. In more scientific terms, these three basic ingredients can be identified with inputs, system components, and outputs, respectively, as shown in following Fig. In general, the objective of the control system is to control the outputs c in some prescribed manner by the inputs u through the elements of the control system. The inputs of the system are also called the actuating signals, and the outputs are known as the controlled variables. 3

4 OPEN LOOP SYSFEMS Above figure illustrates the block diagram of an open-loop control system. A small input signal r( t) is amplified by an amplifier and the output of this amplifier actuates the power actuator. The output of this power actuator drives the controlled variable c(t). From the block diagram it can be seen that the actuating signal or the control action does not change the output. An automatic toaster is an example of open-loop control system because it is controlled by a timer. The time required to make a good toast must be estimated by the user, who is not a part of the system. The quality of toast (output) is determined by the time set by the user. CLOSED-LOOP SYSTEMS Above figure illustrates the block diagram of a basic closed-loop control system in which the control action (input) is dependent upon the controlled variable (output). The output van-able c(t) is compared with the reference input r(t). This comparison is done by an element called error detector. The output of the error detector is the actuating signal e(t). If the actuating signal is weak, it should be amplified by an amplifier before it is applied to the power actuator. An auto pilot mechanism and the air-plane is an example of a closed-loop system. Its purpose is to maintain a specified air-plane heading despite atmospheric changes. It performs this task continuously by measuring the actual air-plane heading and automatically adjusting the air-plane control surfaces, so as to bring the actual air-plane heading into correspondence with the specified heading. The human pilot is not a part of the control system. 4

5 DISTINCTION BETWEEN OPEN-LOOP AND CLOSED-LOOP CONTROL SYSTEMS Merits of Open-loop Systems : 1. Simplest and most economical. 2. Easier to build. 3. Generally not disturbed with unstable operations. Demerits of Open-loop Systems: 1. Usually inaccurate and unreliable. Hence they are not preferred. 2. Do not adapt to environmental changes or to external disturbances. Merits of Closed-loop Systems : 1. Accurate due to feedback, leading to faithful reproduction of the input. 2. Reduced sensitivity of the ratio of output to input for variations in system characteristics. 3. Reduced effects on non-linearities and distortion. 4. Increased bandwidth (The bandwidth of a system is that range of input frequencies over which the arguments of the system will respond satisfactorily). Demerits of Closed-loop Systems : 1. Tendency towards oscillation (unstable operation). 2. Reduction in overall gain. CLASSIFICATION OF CONTROL SYSTEMS: Feedback control systems are classified as follows. 1) According to the method of analysis and design These are further classified as linear and nonlinear, time varying or time invariant. 2) According to the types of signal found in the system These are further classified as continuous-data and discrete-data systems, or modulated and un modulated systems. 3) According to the main purpose of the system. These may be classified as positional control system, speed control system, temperature control system etc. 5

6 EFFECTS OF FEEDBACK: Feedback has effects on system performance characteristics as stability, bandwidth, overall gain, impedance, and sensitivity. Effect of Feedback on Overall Gain : Feedback could increase the gain of the system in one frequency range but decrease it in another. Effect of Feedback on Stability: Stability is a notion that describes whether the system will be able to follow the input command. A system is said to be unstable if its output is out of control or increases without bound. Feedback can cause a system that is originally stable to become unstable. Certainly, feedback is a two-edged sword; when it is improperly used, it can be harmful. Effect of Feedback on Sensitivity: Sensitivity of a system can be decreased by providing a feedback. Effect of Feedback on External Disturbance or Noise: All physical control systems are subject to some types of extraneous signals or noise during operation. Examples of these signals are thermal noise voltage in electronic amplifiers and brush or commutator noise in electric motors. The effect of feedback on noise depends greatly on where the noise is introduced into the system; no general conclusions can be made. However, in many situations, feedback can reduce the effect of noise on system performance. 6

7 MATHEMATICAL MODELLING OF LINEAR SYSTEMS Translational Systems : Let us consider the above mechanical system. It is simply a mass M attached to a spring (stiffness k) and a dashpot (viscous friction coefficient f) on which the force F acts. Displacement x is positive in the direction shown. The zero position is taken to be at the point where the spring and mass are in static equilibrium. Let k be the spring constant called as stiffness in N/m. Let x be the deformation of the spring in meters which is also a displacement of mass M. When force F in Newtons is applied to a mass M in kg. a reactive force is produced which acts in a direction opposite to that of acceleration & this force is given by Newton s third law. i.e. F =Ma = M (d 2 x)/(dt 2 ) Where a is the acceleration in m/s 2 X is the displacement of mass in meters. The spring provides a retarding force which is given by F s = kx The viscous force also acts in opposite direction of applied force. F r = f.dx/dt Where, F r is the viscous friction force in N F is the coefficient of viscous friction in N/m/s Therefore, F= M (d 2 x)/(dt 2 ) + f.dx/dt + kx (1) 7

8 Rotational Systems : Consider a rotational mechanical system consisting of a rotatable disc of moment of inertia J and a shaft of stiffness k. The disc rotates in a viscous medium with viscous friction coefficient f. Let T be the applied torque which tends to rotate the disc. The torque equation obtained from free body diagram is T- f.(dθ/dt) - kθ = J (d 2 θ/dt 2 ) T = J (d 2 θ/dt 2 ) + f.(dθ/dt) + kθ (2) Here θ is the angular displacement in rad. Electrical systems : RLC series Circuit : The resistor, inductor and capacitor are the three basic elements of electrical circuits. These circuits are analyzed by the application of Kirchhoff s voltage and current laws. Let us analyze the L-R-C series circuit shown in above figure. by using Kirchhoff s voltage law. Applied voltage = sum of voltage drops in the circuit. e = V R + V L + V C e = Ri + L (di/dt) + (1/c) i.dt Now, q = i.dt Therfore, E = R.(dq/dt) + L (d 2 q/dt 2 )+ (1/c).q (3) 8

9 RLC Parallel Circuit : Applying Kirchoff s current law C(de/dt) + (1/L) e.dt + e/r = i In terms of magnetic flux linkage, Φ = e.dt C.(d 2 Φ/dt 2 ) + (1/R).(dΦ/dt) + 1/L Φ = i i = C.(d 2 Φ/dt 2 ) + (1/R).(dΦ/dt) + 1/L Φ (4) So we have got the equations (1), (2), (3) & (4) as follows. F= M (d 2 x)/(dt 2 ) + f.dx/dt + kx (1) T = J (d 2 θ/dt 2 ) + f.(dθ/dt) + kθ (2) E = R.(dq/dt) + L (d 2 q/dt 2 )+ (1/c).q (3) i = C.(d 2 Φ/dt 2 ) + (1/R).(dΦ/dt) + 1/L Φ (4) Comparing equations (1), (2) & (3) we will get Force- Torque- Voltage analogy. Similarly comparing equations (1), (2) & (4) we will get Force- Torque- Current analogy as follows. Force- Torque- Voltage analogy. Mechanical translational Mechanical rotational system. Electrical Systems systems Force F Torque T Voltage e Mass M Moment of inertia J Inductance L Viscous friction coefficient f Viscous friction coefficient f Resistance R Spring stiffness K Torsional spring stiffness K Reciprocal of capacitance 1/C Displacement x Angular displacement θ Charge q Velocity dx/dt Angular velocity dθ/dt Current i Force- Torque- Current analogy. Mechanical translational systems Mechanical rotational systems Electrical systems Force F Torque T Current i Mass M Moment of inertia J Capacitance C Viscous friction coefficient f Viscous friction coefficient f Reciprocal of resistance 1/R Spring stiffness K Torsional spring stiffness K Reciprocal of inductance 1/L Displacement x Angular displacement θ Magnetic flux linkage Φ Velocity dx/dt Angular velocity dθ/dt Voltage e 9

10 BLOCK DIAGRAMS : Because of its simplicity and versatility, block diagram is often used by control engineers to portray systems of all types. A block diagram can be used simply to represent the composition and interconnection of a system. Or, it can be used, together with transfer functions, to represent the cause-and-effect relationships throughout the system Block Diagrams of Control Systems : We shall now define some block diagram elements used frequently in control systems and the block diagram algebra. The important block diagram components are as follows. Block Diagram Algebra : We can use following rules for reduction of Block Diagrams. 10

11 Transfer Function : Transfer function of a system is defined as a ratio of Laplace of output variable to Laplace of input variable. Derivation of Transfer function of closed loop system : Consider a block diagram of a linear feedback control system as shown below. Let r(t), R(s) be the reference input. c(t),c(s) be the output signal (controlled variable) b(t), B(s) be the feedback signal ε(t), ε(s) be the actuating signal e(t),e(s) = R(s) C(s) = error signal G(s) = C(s)/ ε(s) = open -loop transfer function or forward path transfer function M( s) = C(s)/R(s) = closed-loop transfer function 11

12 H(s) feedback-path transfer function G(s)H(s) = loop transfer function The closed-loop transfer function, M(s) = C(s)/R(s), can be expressed as a function of G(s) and H(s). From above figure we can write C(s) = G(s) ε(s) (1) and B(s) = H(s) C(s) (2) The actuating signal is written ε(s) = R(s) - B(s) (3) Substituting Eq. (3) into Eq. (1) yields C(s) = G(s)R(s) - G(s)B(s) (4) Substituting Eq. (2) into Eq. (4) gives C(s) = G(s)R(s) - G(s)H(s)C(s) (5) Solving C(s) from the last equation, the closed-loop transfer function of the system is given by M(s) = C(s)/ R(s) = G(s)/[1+G(s)H(s)] (6) SIGNAL FLOW GRAPHS: A signal flow graph may be regarded as a simplified notation for a block diagram. It was originally introduced by S. J. Mason as a cause-and-effect representation of linear systems. In general, besides the difference in the physical appearances of the signal flow graph and the block diagram, we may regard the signal flow graph to be constrained by more rigid mathematical relationships, whereas the rules of using the block diagram notation are far more flexible and less stringent. A signal flow graph may be defined as a graphical means of portraying the input-output relationships between the variables of a set of linear algebraic equations. e.g. Here y 1 & y 2 are input & output nodes respectively. a 12 is the gain. BASIC PROPERTIES OF SIGNAL FLOW GRAPHS Following are the important properties of the signal flow graph. 1. A signal flow graph applies only to linear systems. 2. The equations based on which a signal flow graph is drawn must be algebraic equations in the form of effects as functions of causes. 3. Nodes are used to represent variables. Normally, the nodes are arranged from left to right, following a succession of causes and effects through the system. 4. Signals travel along branches only in the direction described by the arrows of the branches. 5. The branch directing from node y k to y j, represents the dependence of the variable y j upon y k but not the reverse. 12

13 6. A signal y k traveling along a branch between nodes y k and y j, is multiplied by the gain of the branch, a kj so that a signal a kj y k is delivered at node. DEFINITIONS FOR SIGNAL FLOW GRAPHS Input Node (Source) : An input node is a node that has only outgoing branches. (Example: node y 1 in above figure. Output Node (Sink) : An output node is a node which has only incoming branches. (Example: node y 2 in above figure. Path : A path is any collection of a continuous succession of branches traversed in the same direction. Forward Path : A forward path is a path that starts at an input node and ends at an output node and along which no node is traversed more than once. Loop: A loop is a path that originates and terminates on the same node and along which no other node is encountered more than once. Path Gain / Transmittance: The product of the branch gains encountered in traversing a path is called the path gain. 13

14 SIGNAL-FLOW-GRAPH ALGEBRA Based on the properties of the signal flow graph. we can state the following manipulation and algebra of the signal flow graph. Consider the following figures 1. The value of the variable represented by a node is equal to the sum of all the signals entering the node. Therefore, for the signal flow graph of first figure, the value of y 1 is equal to the sum of the signals transmitted through all the incoming branches; that is, y 1 = a 21 y 2 +a 31 y 3 +a 41 y 4 +a 51 y 5 2. The value of the variable represented by a node is transmitted through all branches leaving the node. In the signal flow graph of first figure, we have y 6 = a 16 y 1 y 7 =a 17 y 1 y 8 =a 18 y 1 3. Parallel branches in the same direction connecting two nodes can be replaced by a single branch with gain equal to the sum of the gains of the parallel branches. An example of this case is illustrated in Fig

15 4. A series connection of unidirectional branches, as shown last figure, can be replaced by a single branch with lain equal to the product of the branch gains. 5. Signal flow graph of a feedback control system. Figure 1 shows the signal flow graph of a feedback control system whose block diagram is given in Figure 2. Figure (1) Figure (2) Therefore, the signal flow graph may be regarded as a simplified notation for the block diagram. Writing the equations for the signals at the nodes E(s) and C(s) we have ε(s) = R(s) H(s)C(s) and C(s) = G(s) ε(s) The closed-loop transfer function is obtained from these two equations, C(s)/R(s) = G(s) / [1+G(s)H(s)] MASON S GAIN FORMULA FOR SIGNAL FLOW GRAPHS Given a signal flow graph or a block diagram, it is usually a tedious task to solve for its inputoutput relationships by analytical means. Fortunately, there is a general Mason s gain formula available which allows the determination of the input-output relationship of a signal flow graph by mere inspection. The general gain formula is, 15

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17 TIME RESPONSE OF CONTROL SYSTEM Introduction : Since time is used as an independent variable in most control systems, it is usually of interest to evaluate the state and output responses with respect to time, or simply, the time response. In the analysis problem, a reference input signal is applied to a system, and the performance of the system is evaluated by studying the system response in the time domain. For instance, if the objective of the control system is to have the output variable track the input signal, starting at some initial condition, it is necessary to compare the input and the output response as functions of time. Therefore, in most control system problems the final evaluation of the performance of the system is based on the time responses. The time response of a control system is usually divided into two parts: the transient response and the steady-state response. Let c(t) denote a time response; then, in general, it may be written as c(t) = c t (t) + Css(t) Where, c t (t) = transient response c ss (t) = steady state response In network analysis it is sometimes useful to define steady state as a condition when the response has reached a constant value with respect to the independent variable. In control systems studies, however, it is more appropriate to define steady-state as the fixed response when time reaches infinity. Therefore, a sine wave is considered as a steady-state response because its behavior is fixed for any time interval, as when time approaches infinity. Similarly, the ramp function c( t) = t is a steady-state response, although it increases with time. Transient response is defined as the part of the response that goes to zero as time becomes very large. All control systems exhibit transient phenomenon to some extent before a steady state is reached. Since inertia, mass, and inductance cannot be avoided entirely in physical systems, the responses of a typical control system cannot follow sudden changes in the input instantaneously, and transients are usually observed. Therefore, the control of the transient response is necessarily important. The steady-state response of a control system is also very important, since when compared with the input, it gives an indication of the final accuracy of the system. When the steady-state response of the output does not agree with the steady-state of the input exactly, the system is said to have a steady-state error. The study of a control system in the time domain essentially involves the evaluation of the transient and the steady-state responses of the system. TYPICAL TEST SIGNALS FOR THE TIME RESPONSE OF CONTROL SYSTEMS To facilitate the time-domain analysis, the following deterministic test signals are often used. Step Input Function : The step input function represents an instantaneous change in the reference input variables For example, if the input is the angular position of a mechanical shaft, the step input represents the sudden rotation of the shaft. The mathematical representation of a step function is 17

18 where R is a constant. Or r(t) =Ru s (t) where, u s (t) is the unit step function. The step function is not defined at t=0. It shown graphically as below. Ramp Input Function : In the case of the ramp function, the signal is considered to have a constant change in value with respect to time. Mathematically, a ramp function is represented by, or simply, The ramp function is shown in following figure. If the input variable is of the form of the angular displacement of a shaft, the ramp- input represents the constant-speed rotation of the shaft. 18

19 Parabolic Input Function : The mathematical representation of a parabolic input function is or simply The graphical representation of the parabolic function is shown in following figure. 19

20 TlME-DOMAlN PERNORMANCE OF CONTROL SYATEMS- TRANSIENT RESPONSE The transient portion of the time response is that part which goes to zero as time becomes large. The transient performance of a control system is usually characterized by the use of a unit step input. Typical performance criteria that are used to characterize the transient response to a unit step input include overshoot, delay time, rise time and settling time. Following figure illustrates a typical unit step response of a linear control system. The abovementioned criteria are defined with respect to the step response: 1. Maximum overshoot : The maximum overshoot is defined as the largest deviation of the output over the step input during the transient state. The amount of maximum overshoot is also used as a measure of the relative stability of the system. The maximum overshoot is often represented as a percentage of the final value of the step response; that is, 2. Delay time : The delay time T d is defined as the time required for the step response to reach 50 percent of its final value. 3. Rise time : The rise time T r is defined as the time required for the step response to rise from 10 percent to 90 percent of its final value 4. Settling time :The settling time T d is defined as the time required for the step response to decrease and stay within a specified percentage of its final value. A frequently used figure is 5 percent. 20

21 STABILITY OF CONTROL SYSTEM Roughly speaking, stability in a system implies that small changes in the system input, in initial conditions or in system parameters, do not result in large changes in system output. Stability is a very important characteristic of the transient performance of a system. Almost every working system is designed to be stable. A linear time-invariant system is stable if the following two notions of system stability at satisfied: (i) When the system is excited by a bounded input, the output is bounded. (ii) In the absence of the input, the output tends towards zero (the equilibrium state of the system) irrespective of initial conditions. The stability of a system is governed by roots of the characteristic equations. Following figure shows the various locations of roots of characteristic equation & corresponding impulse responses. 21

22 From above it is clear that if the roots of characteristic equation lie in left half of the s plane the system is stable. This can be shown by following figure. From above we can summarize the relation between the transient response and the characteristic equation roots as follows. 1. When all the roots of the characteristic equation are found in the left of the s plane, the system responses due to the initial conditions will decrease to zero as time approaches infinity. 2. If one or more pairs of simple roots are located on the imaginary axis of the s plane, but there are no roots in the right half of the s plane, the responses due to initial conditions will be undamped sinusoidal oscillations. 3. If one or more roots are found in the right half of the s plane, the responses will increase in magnitude as time increases. For analysis and design purposes stability is classified into absolute stability and relative stability. Absolute stability refers top the condition of stable or unstable. It is a yes or no condition. Once the system is stable then it is determined how stable it is? This degree of stability gives relative stability. METHODS OF DETERMINING STABILITY UNEAR CONTROL SYSTEMS Although the stability of linear time-invariant systems may be checked by investigating the impulse response, or by finding the roots of the characteristic equation, these criteria are difficult to implement in practice. For instance, the impulse response is obtained by taking the inverse Laplace transform of the transfer function, which is not always a simple task. The solving of the roots of a high-order polynomial can only be carried out by a digital computer. The methods outlined below are frequently used for the stability studies of linear time-invariant systems. 1) Routh - Hurwitz criterion: It is an algebraic method that provides information on the absolute stability of a linear time-invariant system. The criterion tests whether any roots of the characteristic equation lie in the right half of the s-plane. The number of roots that lie on the imaginary axis and in the right half of the s-plane are also indicated. 22

23 2) Nyquist criterion: It is a semi graphical method that gives information on the difference between the number of poles and zeros of the closed-loop transfer function by observing the behavior of the Nyquist plot of the loop transfer function. The poles of the closed-loop transfer function are the roots of the characteristic equation. This method requires that we know the relative location of the zeros of the closed-loop transfer function. 3) Root locus plot: It represents a diagram of loci of the characteristic equation roots when a certain system parameter varies. When the root loci lie in the right half of the s-plane, the closedloop system is unstable. 4)Bode diagram: The Bode plot of the loop transfer function G(s)H(s) may be used to determine the stability of the closed-loop system. However, the method can be used only if G(s)H(s) has no poles and zeros in the right-half s-plane. 5)Lyapunov s stability criterion: It is a method of determining the stability of nonlinear systems, although it can also be applied to linear systems. The stability of the system is determined by checking on the properties of the Lyapunov function of the system. HURWITZ CRITERION: Consider that the characteristic equation of a linear time-invariant system is of the form where all the coefficients are real numbers. The Hurwitz determinants of above equation are given by, Hurwitz criterion states that, all roots of above characteristic equation lie in left half of the s plane if Hurwitz determinants, D k where k= 1,2,3.n are all positive. 23

24 ROUTH HURWITZ CRITERION: The first step in the simplification of the Routh-Hurwitz criterion is to arrange the polynomial coefficients into two rows. The first row consists of the first, third, fifth,... coefficients, and the second row consists of the second, the fourth, sixth,... coefficients, as shown in the following tabulation: The next step is to form the following array of numbers by the indicated operations (the example shown is for a sixth-order system): The array of numbers and operations given above is known as The Routh tabulation or the Routh array. The first column of a 0 & a 1 on the left side is used for identification purpose. Once the Routh tabulation has been comp1eted, the last step in the Routh Hurwitz criterion is to investigate the signs of the numbers in the above first column of the tabulation. The following conclusions are drawn. The roots of the polynomial are all in the left half of the s-plane if all the elements of the first column of the Routh tabulation are of the same sign. If there are changes of signs in the elements of the first column, the number of sign changes indicates the number of roots with positive real parts i.e lying in right hand side of s plane. It means if there is a sign change in first column the system is unstable & the number of sign changes indicates the number of roots lying on the right hand side of the s plane. 24

25 Examples on Routh- Hurwitz criterion: 1) Comment on the stability of the system, whose characteristic equation is given as, by applying Routh -Hurwitz criterion. Solution : Applying Routh -Hurwitz criterion we get Rouths tabulation as, Since there are two sign changes in the first column of the tabulation, the system has two roots located in the right half of the s plane & is unstable. 2) Comment on the stability of the system, whose characteristic equation is given as, by applying Routh -Hurwitz criterion. Solution : Applying Routh -Hurwitz criterion we get Rouths tabulation as, Since there are two changes in sign in the first column, the equation has two roots in the right half of the s-plane & is unstable. 25

26 ROOT LOCUS TECHNIQUE : Consider a design problem in which the designer is required to achieve the desired performance for a system by adjusting the location of its closed-loop poles in the s-plane by varying one or more system parameters. The Routh s criterion, obviously does not help much in such problems. For determining the location of the closed-loop poles, we may use technique of factoring the characteristic polynomial and determining its roots, since the closed-loop poles are the roots of the characteristic equation. This technique is very laborious when the degree of the characteristic polynomial is three or higher. Furthermore, repeated calculations are required as a system parameter is varied for adjustments. A simple technique, known as the root locus techniques, for finding the roots of the characteristic equation, introduced by W.R. Evans, is extensively used in control engineering practice. This technique provides a graphical method of plotting the locus of the roots in the s-plane as a given system parameter is varied over the complete range of values ( may be from zero to infinity). The roots corresponding to a particular value of the system parameter can then be located on the locus or the value of the parameter for a desired root location can be determined from the locus. The root locus is a powerful technique as it brings into focus the complete dynamic response of the system and further, being a graphical technique, an approximate root locus sketch can be made quickly and the designer can easily visualize the effects of varying various system parameters on root locations. Also in the design of control systems it is often necessary to investigate the performance of a system when one or more parameters of the system varies over a given range. This technique is useful in such case. A characteristic equation can be written as, where, K is the parameter considered to vary between - and +. The coefficients a 1,a 2.a n,b 1,b 2..b m are assumed to be fixed. These coefficients can be real or complex, although our main interest here is in real coefficients. Now based on variation of K following categories are defined. 1) Root Loci : The portion of the root loci when K assumes positive values that is 2) Complimentary root loci : The portion of the root loci when K assumes negative values, that is, 3) Root contours : loci of roots when more than one parameter varies. 4) Complete root loci : It refers to the combination of the root loci and the complementary root loci. ROOT LOCUS CONCEPT: Consider a transfer function of a system as Here the characteristic equation of the system is This second order system under consideration is always stable for positive values of a and K but its dynamic behavior is controlled by the roots of above equation. 26

27 The roots are given by, From above it is seen that as any of the parameter (a or K) varies, the roots of the characteristic equation change. Let us vary K by keeping a constant. Following conditions are obtained. the roots are real and distinct, when K=0, the two roots are s 1 =0, s 2 = -a i.e. they coincide with the open loop pole of the system. (2) K=a 2 /4, the roots are real and equal in value. i.e.s 1 =s 2 = - a/2 (3) a 2 /4<K<, the roots are complex with real part = - a/2 i.e. unvarying real part. The root locus with varying K is plotted in following figure. 1) The root locus plot has two branches starting at the two open-loop poles (s = 0, and s =-a) for K = 0, 2) As K is increased from 0 to a 2 /4, the roots move towards the point (- a/2, 0) from opposite directions. Both the roots lie on the negative real axis which corresponds to an over damped system. The two roots meet at s = -a/2 for K=a 2 /4. This point corresponds to a critically damped system. As K is increased further (K> a 2 /4), the roots break away from the real axis, become complex conjugate and since the real part of both the roots remains fixed at - a/2, the roots move along the line σ =- a/2 and the system becomes under damped. 3)For K> a 2 /4, the real parts of the roots are fixed, therefore the settling time is nearly constant Example on Steady state error: 1) If a 10 volts reference supply is used to regulate a 100 volts supply & if H is a constant equal to 0.1,calculate the error signal when output voltage is exactly 100volts. Solution: The error signal ε(t) is given by following formula. ε(t) = r(t)-b(t) i.e. ε(s) = R(s) H(s)C(s) = 10-(0.1)x(100) =0 Best of Luck 27