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Dr.Y.NARASIMHA MURTHY Ph.D yayavaram@yahoo.com CONTROL SYSTEMSINTRODUCTION :A control system is one which can control any quantity of interest in a machine, mechanism orother equipment in order to achieve the desired performance or output.(or) A control system is aninterconnection of components connected or related in such a manner as to command, direct, orregulate itself or another system. For example consider, the driving system of an automobile.Speed of the automobile is a function of the position of its accelerator. The desired speed can bemaintained (or a desired change in speed can be achieved) by controlling pressure on theaccelerator pedal. This automobile driving system (accelerator, carburetor and engine-vehicle)constitutes a control system.Control systems find numerous and widespread applications from everyday to extraordinary inscience, industry, and home. Here are a few examples:(a) Home heating and air-conditioning systems controlled by a thermostat(b) The cruise (speed) control of an automobile(c) Manual control (i) Opening or closing of a window for regulating air temperature or air quality (ii) Activation of a light switch to regulate the illumination in a room (iii) Human controlling the speed of an automobile by regulating the gas supply to the engine(d) Automatic traffic control (signal) system at roadway intersections(e) Control system which automatically turns on a room lamp at night, and turns it off in Day lightThe general block diagram of a control system is shown below.. Fig.1 .Block diagram of a control system.1
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Dr.Y.NARASIMHA MURTHY Ph.D yayavaram@yahoo.comThe above method of representation of a control system is known as block diagramrepresentation where in each block represents an element, a plant, mechanism, device etc.,whose inner details are not indicated. Each block has an input and output signal which are linkedby a relationship characterizing the block. It may be noted that the signal flow through the blockis unidirectionalBasic Control System components : The basic control system components are objectives i.einputs or actuating signals to the system and the Output signals or controlled variables etc.Thecontrol system will control the outputs in accordance with the input signals.The relation beteenthese components is shown in the block diagram.The components of the control system changes as we move from openloop control system toclosed loop control systems. In a closed loop control system ,the feedback control network playan important role in getting the correct output.The general block diagram of a control system with feed back is shown below. The error detectorcompares a signal obtained through feedback elements, which is a function of the outputresponse, with the reference input. Any difference between these two signals gives an error oractuating signal, which actuates the control elements. The control elements in turn alter theconditions in the plant in such a manner as to reduce the original error.2
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Dr.Y.NARASIMHA MURTHY Ph.D yayavaram@yahoo.com Fig.2 .General block diagram of an automatic control system.Types of control systems :There are basically two types of control systems (i) the open loop system and the (ii) closed loopsystem. They can both be represented by block diagrams. A block diagram uses blocks torepresent processes, while arrows are used to connect different input, process and output parts.Open loop Control System : Asystem which do not possess any feed back network ,andcontains only the input and output relationship is known as a open loop control system.Examples of the open loop control systems are washing machines, light switches, gas ovens,burglar alarm system etc.The drawback of an open loop control system is that it is incapable of making automaticadjustments. Even when the magnitude of the output is too big or too small, the system can‟tmake the necessary adjustments. For this reason, an open loop control system is not suitable foruse as a complex control system.3
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Dr.Y.NARASIMHA MURTHY Ph.D yayavaram@yahoo.comClosed loop control system : A closed loop system is one which uses a feed back controlbetween input and output. A closed loop control system compares the output with the expectedresult or command status, then it takes appropriate control actions to adjust the input signal.Therefore, a closed loop system is always equipped with a sensor, which is used to monitor theoutput and compare it with the expected result.The output signal is fed back to the input to produce a new output. A well-designed feedbacksystem can often increase the accuracy of the output.Examples for closed loop systems are air conditioners, refrigerators, automatic rice cookers,automatic ticketing machines, etc. For example An air conditioner, uses a thermostat to detectthe temperature and control the operation of its electrical parts to keep the room temperature at apreset constant.One advantage of using the closed loop control system is that it is able to adjust its outputautomatically by feeding the output signal back to the input. When the load changes, the errorsignals generated by the system will adjust the output suitably. The limitation of a closed loopcontrol systems is they are generally more complicated and thus also more more expensive todesign.Linear versus Nonlinear Control Systems : Linear feedback control systems are idealizedmodels fabricated by the analyst purely for the simplicity of analysis and designWhen the magnitudes of signals in a control system are limited to ranges in which systemcomponents exhibit linear characteristics (i.e., the principle of superposition applies), the systemis essentially linear.But when the magnitudes of signals are extended beyond the range of the linear operation,depending on the severity of the nonlinearity, the system should no longer be considered linear.For instance, amplifiers used in control systems often exhibit a saturation effect when their input4
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Dr.Y.NARASIMHA MURTHY Ph.D yayavaram@yahoo.comsignals become large; the magnetic field of a motor usually has saturation properties. Othercommon nonlinear effects found in control systems are the backlash or dead play betweencoupled gear members, nonlinear spring characteristics, nonlinear friction force or torquebetween moving members, and so on. Quite often, nonlinear characteristics are intentionallyintroduced in a control system to improve its performance or provide more effective control.Also the analysis of linear systems is easy and lot of mathematical solutions are available fortheir simplification.Nonlinear systems, on the other hand, are usually difficult to treat mathematically, and there areno general methods available for solving a wide class of nonlinear systems. In practice , first alinear-system is modeled by neglecting the nonlinearities of the system and the designedcontroller is then applied to the nonlinear system model for evaluation or redesign by computersimulation.Distinguish between Openloop and Closed loop control systemsTime-Invariant control Systems :When the parameters of a control system do not change with respect to time during theoperation of the system, the system is called a time-invariant system. In practice, most physicalsystems contain elements that drift or vary with time. For example, the winding resistance of anelectric motor will vary when the motor is first being excited and its temperature is rising.Another example of a time-varying system is a guided-missile control system in which the mass5
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Dr.Y.NARASIMHA MURTHY Ph.D yayavaram@yahoo.comof the missile decreases as the fuel on board is being consumed during flight. Although a time-varying system without nonlinearity is still a linear system, the analysis and design of this classof systems are usually much more complex than that of the linear time-invariant systems.Continuous and Discrete Data Control Systems : A continuous-data system is one in whichthe signals at various parts of the system are all functions of the continuous time variable t. Thesignals in continuous-data systems may be further classified as ac or dc. In control systems theac control system, means that the signals in the system are modulated by some form ofmodulation scheme. A dc control system, on the other hand, simply implies that the signals areunmodulated, but they are still ac signals according to the conventional definition. The schematicdiagram of a closed loop dc control system is shown below. Typical waveforms of the signals inresponseto a step-function input are shown in the figure. Typical components of a dc controlsystem are potentiometers, dc amplifiers, dc motors, dc tachometers, and so on.In ac control systems , the signals are modulated i.e the information is transmitted by an accarrier signal. Here the output controlled variable behaves similarly to that of the dc system. Inthis case, the modulated signals are demodulated by the low-pass characteristics of the ac motor.Ac control systems are used extensively in aircraft and missile control systems in which noise6
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Dr.Y.NARASIMHA MURTHY Ph.D yayavaram@yahoo.comand disturbance often create problems. By using modulated ac control systems with carrierfrequencies of 400 Hz or higher, the system will be less susceptible to low-frequency noise.Typical components of an ac control system are synchros, ac amplifiers, ac motors, gyroscopes,accelerometers etc.Discrete Data Control SystemIf the signal is not continuously varying with time but it is in the form of pulses then the controlsystem is called Discrete Data Control System.If the signal is in the form of pulse data, then thesystem is called Sampled Data Control System. Here the information supplied intermittently atspecific instants of time. This has the advantage of Time sharing system. On the other hand, ifthe signal is in the form of digital code, the system is called Digital Coded System. Here use ofDigital computers, micro processors or microcontrollers are made use of such systems and areanalyzed by the Z- transform theory.Block Diagrammatic Representation :It is a representation of the control system giving the inter-relation between the transfer functionof various components. The block diagram is obtained after obtaining the differential equation &Transfer function of all components of a control system. The arrow head pointing towards theblock indicates the i/p & pointing away from the block indicates the o/p.Suppose G(S) is the Transfer function then G(S) = C(S) / R(S)After obtaining the block diagram for each & every component, all blocks are combined to get acomplete representation. It is then reduced to a simple form with the help of block diagramalgebra.Basic elements of a block diagram Blocks Transfer functions of elements inside the blocks Summing points Take off points Arrow7
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Dr.Y.NARASIMHA MURTHY Ph.D yayavaram@yahoo.comA control system may consist of a number of components. A block diagram of a system is apictorial representation of the functions performed by each component and of the flow of signals.The elements of a block diagram are block, branch point and summing point.Block :In a block diagram all system variables are linked to each other through functional blocks. Thefunctional block or simply block is a symbol for the mathematical operation on the input signalto the block that produces the output.Summing point :The blocks are used to identify many types of mathematical operations, like addition andsubtraction and represented by a circle, called a summing point. As shown belowdiagram asumming point may have one or several inputs. Each input has its own appropriate plus or minussign. A summing point has only one output and is equal to the algebraic sum of the inputs8
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Dr.Y.NARASIMHA MURTHY Ph.D yayavaram@yahoo.comA takeoff point is used to allow a signal to be used by more than one block or summing pointArrow – associated with each branch to indicate the direction of flow of signalAdvantages of Block Diagram Representation : It is always easy to construct the block diagram even for a complicated system Function of individual element can be visualized Individual & Overall performance can be studied Over all transfer function can be calculated easilyLimitations of a Block Diagram Representation No information can be obtained about the physical construction Source of energy is not shownBlock diagram reduction technique: Because of the simplicity and versatility, the blockdiagrams are often used by control engineers to describe all types of systems. A block diagramcan be used simply to represent the composition and interconnection of a system. Also, it can beused, together with transfer functions, to represent the cause-and-effect relationships throughoutthe system. Transfer Function is defined as the relationship between an input signal and anoutput signal to a device.Procedure to solve Block Diagram Reduction s :Step 1: Reduce the blocks connected in seriesStep 2: Reduce the blocks connected in parallelStep 3: Reduce the minor feedback loopsStep 4: Try to shift take off points towards right and Summing point towards leftStep 5: Repeat steps 1 to 4 till simple form is obtainedStep 6: Obtain the Transfer Function of Overall SystemBlock diagram rules(1) Blocks in Cascade [Series] : When two blocks are connected in series ,their resultant transferfunction is the product of two individual transfer functions.9
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Dr.Y.NARASIMHA MURTHY Ph.D yayavaram@yahoo.com(2) Combining blocks in Parallel: When two blocks are connected parallel as shown below ,theresultant transfer function is equal to the algebraic sum (or difference) of the two transferfunctions.This is shown in the diagram below.(3) Eliminating a feed back loop:The following diagram shows how to eliminate the feed backloop in the resultant control system(4) Moving a take-off point beyond a block: The effect of moving the takeoff point beyond ablock is shown below.(5) Moving a Take-off point ahead of a block: The effect of moving the takeoff point ahead of ablock is shown below.10
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Dr.Y.NARASIMHA MURTHY Ph.D yayavaram@yahoo.comApplications of the control systems : There are various applications of control systems whichinclude biological propulsion; locomotion; robotics; material handling; biomedical, surgical,and endoscopic ; aeronautics; marine and the defense and space industries. There are also manyhousehold and industrial application examples of the control systems, such as washing machine,air conditioner, security alarm system and automatic ticket selling machine, etc.(i) Washing machine :The most commonly used house hold application is the washing machine.It comes underautomatic control system ,where the machine automatically starts to pour water, add washingpowder, spin and wash clothes, discharge wastewater, etc. After the completion of all theprocedures, the washing machine will stop the operation.However, this kind of machine only operates according to the preset time to complete thewhole washing process. It ignores the cleanness of the clothes and does not generate feedback.Therefore, this kind of washing machine is of open loop control system.(ii). Air conditionerThe air conditioner is used to automatically control the temperature of the room.In the airconditioner the coolant circulated in the machine will absorb heat indoor, then it will betransported from the vaporization device to cooling device. The hot air is then blown to outdoorby a fan. There is an adjustable temperature device equipped in the air conditioner for the usersto adjust the extent of cooling. When the temperature of the cool air is lower than the preset one,the controller of the air conditioner will stop the operation of the compressor to cease thecirculation of the coolant. The temperature sensor installed near the vaporization device willcontinuously measure the indoor temperature, and send the results to the controller for furtherprocessing.This operation will come under closed loop control system.Thesimple block diagramof air conditioner system is shown below.11
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Dr.Y.NARASIMHA MURTHY Ph.D yayavaram@yahoo.comFeed back Control System : The feed back control system is represented by the followingblock diagram .In the diagram feed back signal is denoted by B(S) and the output is C(S).Theinput function is denoted by R(S).The open loop gain of the system is G(S) and the feed back loop gain H(S). Then the feedback signal b(s) is given by B(S) = H(S). C(S)Transfer function : The input- output relationship in a linear time invariant system is given bythe transfer function.For a time invariant system it is defined as the ratio of Laplace transformof the out to the Lapalce transform of the inputThe important features of the transfer functions are, The transfer function of a system is the mathematical model expressing the differential equation that relates the output to input of the system. The transfer function is the property of a system independent of magnitude and the nature of the input . The transfer function includes the transfer functions of the individual elements. But at the same time, it does not provide any information regarding physical structure of the system If the transfer function of the system is known, the output response can be studied for various types of inputs to understand the nature of the system It is applicable to Linear Time Invariant system. It is assumed that initial conditions are zero. It is independent of i/p excitation. It is used to obtain systems o/p response. If the transfer function is unknown, it may be found out experimentally by applying known inputs to the device and studying the output of the system12
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Dr.Y.NARASIMHA MURTHY Ph.D yayavaram@yahoo.comFrom the above block diagram G(S) = C(S) / E(S) & E(S) = R(S) – B(S) So, C(S) = G(S) .E(S) = G(S)[ R(S)- B(S) = G(S) [ R(S) – H(S).C(S)ThereforeThis is the transfer function of the closed loop control systemProperties of Systems : For any control system to understand its performance the followingproperties are very important.(i).Linearity : A system is said to be linear if it follows both the law of addtivity and law ofhomogeneity. The system which do not follow the law of homogeneity and additivity is called anon-linear system.If input x1(t) produces response y1(t) and input x2(t) produces response y2(t) then the scaledand summed input a1x1(t) +b1x2(t) produces the scaled and summed response a1y1(t) +b1y2(t)where a1 and a2 are real scalars. It follows that this can be extended to an arbitrary number ofterms, and so for real numbers .(ii) Time Invariance : A system with input x(t) and output y(t) is time-invariant if x(t- t0) iscreates output y ( t – t0) for all inputs x and shifts t0.(iii). Causality : A system is causal, if the output y(t) at time t is not a function of future inputsand it depends only on the present and past inputs . All analog systems are causal and allmemeoryless systems are causal .If the system is causal, then this implies h(t) = 0, t < 0. Alternatively, h[n]=0, n < 0.(iv).Stability : A system is said to be a stable if for every bounded-input there exists a boundedoutput .Transfer Function: For a open loop control system shown below the transfer function is theratio of Laplace transform of the out-put to the Laplace transform of the input.13
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Dr.Y.NARASIMHA MURTHY Ph.D yayavaram@yahoo.com The Laplace transform of the Input is R(S) and the Laplace transform of the outputis C(S) . So,the Transfer function of the system is G(S) = C(S) / R(S)Example : Find the transfer function of the following RC circuitBy definition ,the transfer function is G(S) =14
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Dr.Y.NARASIMHA MURTHY Ph.D yayavaram@yahoo.comThe Laplace transformed Network is shown above.From the circuit we can write thatPoles and Zeros : In the transfer function of a control system both numerator anddenominator will be polynomials. If these numerator and denominator are solved , the roots ofthe numerator are called Zeros and the roots of the denominator are called the Poles. Thesepoles of the transfer function decides the stability of the control system.The transfer function provides a basis for determining important system responsecharacteristics without solving the complete differential equation. As defined, the transferfunction is given by the following expression with variable s = σ + jω,It is always easy to factor the polynomials in the numerator and denominator, and to write15
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Dr.Y.NARASIMHA MURTHY Ph.D yayavaram@yahoo.comthe transfer function in terms of those factors .then the above transfer function can be writtenaswhere the numerator and denominator polynomials, N(s) and D(s), have real coefficientsdefined by the system’s differential equation and K = bm/an.As written in the above equation the Zi s are the roots of the equation N(s) = 0, and aredefined to be the system Zeros, and the pi s are the roots of the equation D(s) = 0, and aredefined to be the system poles.All of the coefficients of polynomials N(s) and D(s) are real, therefore the poles and zeros mustbe either purely real, or appear in complex conjugate pairs.The stability of a linear system may be determined directly from its transfer function. An nthorder linear system is asymptotically stable only if all of the components in the homogeneousresponse from a finite set of initial conditions decay to zero as time increases, orwhere the pi are the system poles. In a stable system all components of the homogeneousresponse must decay to zero as time increases. If any pole has a positive real part there is acomponent in the output that increases without bound, causing the system to be unstable.16
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Dr.Y.NARASIMHA MURTHY Ph.D yayavaram@yahoo.comExample : A linear system is described by the differential equationFind the system poles and zeros.Solution: From the differential equation the transfer function is So,the system has a single real zero at s = −1/2, and a pair of real poles at s = −3 and s = −2.SIMULATION DIAGRAMS: The simulation diagram is similar to the diagram used to representthe system on an analog computer.The basic elements used are ideal integrators, idealamplifiers, and ideal summers, shown in below diagram. Additional elements such asmultipliers and dividers may be used for nonlinear systems.17
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Dr.Y.NARASIMHA MURTHY Ph.D yayavaram@yahoo.comTo obtain the simulation diagram ,the following steps are to be followed.1. Start with differential equation.2. On the left side of the equation put the highest-order derivative of the dependent variable. Afirst-order or higher-order derivative of the input may appear in the equation. In this case thehighest-order derivative of the input is also placed on the left side of the equation. All otherterms are put on the right side.3. Start the diagram by assuming that the signal, represented by the terms on the left side ofthe equation, is available. Then integrate it as many times as needed to obtain all the lower-order derivatives. It may be necessary to add a summer in the simulation diagram to obtain thedependent variable explicitly.4. Complete the diagram by feeding back the approximate outputs of the integrators to asummer to generate the original signal of step 2.Include the input function if it is required.Example: Draw the simulation diagram for the series RLCcircuit of below figure in which theoutput is the voltage across the capacitor.SIGNAL FLOW GRAPHS(SFG) : The block diagram method is a useful tool for simplifying therepresentation of a control system. But when there are more than two feed back loops and ifthere exists inter-coupling between feedback loops, and when a system has more than oneinput and one output, the block diagram approach is very complex. Hence an alternate methodis proposed by S.J. Mason. This method is called signal flow graphs. In these graphs each node18
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Dr.Y.NARASIMHA MURTHY Ph.D yayavaram@yahoo.comrepresents a system variable & each branch connected between two nodes acts as SignalMultiplier. The direction of signal flow is indicated by an arrow.A signal flow graph is a diagram that represents a set of simultaneous equations. It consists of agraph in which nodes are connected by directed branches. The nodes represent each of thesystem variables. A branch connected between two nodes acts as a one-way signal multiplier: thedirection of signal flow is indicated by an arrow placed on the branch, and the multiplicationfactor (transmittance or transfer function) is indicated by a letter placed near the arrow.So,in the figure above , the branch transmits the signal x1 from left to right and multiplies it bythe quantity a in the process. The quantity a is the transmittance, or transfer function.Flow-Graph Definitions : A node performs two functions: Addition of the signals on all incoming branches and Transmission of the total node signal (the sum of all incoming signals) to all outgoing branchesThere are three types of nodes .They are Source nodes , Sink nodes and Mixed nodesSource nodes (independent nodes) : These represent independent variables and have onlyoutgoing branches. In Fig. 5.21, nodes u and v are source nodes.Sink nodes (dependent nodes): These represent dependent variables and have only incomingbranches. In Fig (a), nodes x and y are sink nodes.Mixed nodes (general nodes): These have both incoming and outgoing branches. In Fig. (a),node w is a mixed node. A mixed node may be treated as a sink node by adding an out goingbranch of unity transmittance, as shown in Fig (b), for the equation x = au +bvand w = cx = cau + cbv19
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Dr.Y.NARASIMHA MURTHY Ph.D yayavaram@yahoo.com Fig(a) Fig (b)A path is any connected sequence of branches whose arrows are in the same direction and Aforward path between two nodes is one that follows the arrows of successive branches and inwhich a node appears only once. In Fig.(a) the path uwx is a forward path between the nodesu and x.Flow-Graph Algebra : The following rules are useful for simplifying a signal flow graph:Series paths (cascade nodes). Series paths can be combined into a single path by multiplyingthe transmittances as shown in Fig ( A ).Path gain. The product of the transmittances in a series path.Parallel paths. Parallel paths can be combined by adding the transmittances as shown in Fig(B).Node absorption. A node representing a variable other than a source or sink can be eliminatedas shown in Fig (C).Feedback loop. Aclosed path that starts at a node and ends at the same node.Loop gain. The product of the transmittances of a feedback loop.These results are shown diagrammatically in the following figures (A) ,(B) and C) where theoriginal diagram and equivalent diagrams are shown.20
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Dr.Y.NARASIMHA MURTHY Ph.D yayavaram@yahoo.comMasons gain formula: The relationship between an input variable and an output variable of asignal flow graphis given by the net gain between input and output nodes and is known as overallgain ofthe system. Masons gain formula is used to obtain the over all gain (transfer function) ofsignal flow graphs. According to Mason‟s gain formula Gain is given byWhere, Pk is gain of k th forward path and Δ is determinant of graph. Here the Δ is given byΔ = 1-(sum of all individual loop gains)+(sum of gain products of all possible combinations oftwo non touching loops –sum of gain products of all possible combination of three non touchingloops)Δk is cofactor of kth forward path determinant of graph with loops touching k th forward path.It is obtained from Δ by removing the loops touching the path Pk.Finding transfer function from the system flow graphs is explained below by example.Example1 : Obtain the transfer function of the system whose signal flow graph is shown below.21
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Dr.Y.NARASIMHA MURTHY Ph.D yayavaram@yahoo.comThere are two forward paths: One is Gain of path 1 : P1=G1 and the other is Gain of path 2:P2=G2There are four loops with loop gains :L1=-G1G3, L2=G1G4, L3= -G2G3, L4= G2G4There are no non-touching loops.Δ = 1+G1G3-G1G4+G2G3-G2G4Forward paths 1 and 2 touch all the loops. Therefore, Δ1= 1, Δ2= 1So,the transfer function T is given by22
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Dr.Y.NARASIMHA MURTHY Ph.D yayavaram@yahoo.comExample 2 : Obtain the transfer function of C(s) /R(s) of the system whose signal flow graphshown below.From the system flow graph it is clear thatThere is one forward path, whose gain is: P1=G1G2G3There are three loops with loop gains:L1=-G1G2H1, L2=G2G3H2, L3= -G1G2G3There are no non-touching loops: Δ = 1-G1G2H1+G2G3H2+G1G2G3Forward path 1 touches all the loops. Therefore, Δ1= 1.The transfer function T is given bySystem Stability:The study of stability of a control system is very important to understand the performance .This means that the system must be stable at all times during operation. Stability may be used todefine the usefulness of the system. Stability studies include absolute & relative stability.Absolute stability is the quality of stable or unstable performance. Relative Stability is thequantitative study of stability.23
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Dr.Y.NARASIMHA MURTHY Ph.D yayavaram@yahoo.comThe stability study is based on the properties of the Transfer Function. In the analysis, thecharacteristic equation is very important ,which describe the transient response of the system.From the roots of the characteristic equation, following conclusions about the stability can bedrawn.(1) When all the roots of the characteristic equation lie in the left half of the S-plane, the systemresponse due to initial condition will decrease to zero at time Thus the system will be termed asa stable system.(2) When one or more roots lie on the imaginary axis & there are no roots on the RHS of S-plane, the response will be oscillatory without damping. Such a system will be termed ascritically stable.(3) When one or more roots lie on the RHS of S-plane, the response will exponentially increasein magnitude and there by the system will be Unstable.Stability – Definitions : A system is stable, if its o/p is bounded for any bounded i/p . or A system is stable, if it‟s response to a bounded disturbing signal vanishes ultimately as time ‘ t ‘ approaches infinity. A system is un stable, if it’s response to a bounded disturbing signal results in an o/p of infinite amplitude or an Oscillatory signal. If the o/p response to a bounded i/p signal results in constant amplitude or constant amplitude oscillations, then the system may be stable or unstable under some limited constraints. Such a system is called Limitedly Stable system. If a system response is stable for a limited range of variation of its parameters, it is called Conditionally Stable System. If a system response is stable for all variation of its parameters, it is called Absolutely Stable system.Routh-Hurwitz Stability Criterion : This criterion is derived from the theory of equationsand is an algebraic method to determine the number of roots of a given equation with positivereal part.The Routh-Hurwitz criterion is a method of finding whether a linear system is stable or not byexamining the locations of the roots of the characteristic equation of the system. In fact, the24
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Dr.Y.NARASIMHA MURTHY Ph.D yayavaram@yahoo.commethod determines only if there are roots that lie outside of the left half plane; it does notactually compute the roots.To determine whether this system is stable or not, check the following conditionsTwo necessary but not sufficient conditions that all the roots have negative real parts area) All the polynomial coefficients must have the same sign.b) All the polynomial coefficients must be nonzero.A sufficient condition for a system to be stable is that each & every term of the column ofthe Routh array must be positive or should have the same sign. Routh array can be obtained asfollowsConsider the Characteristic equation of the form,Similarly rest of the elements, can be evaluated.The limitations of the Routh-Hurwitz stability criteria are(1) It is valid only if the Characteristic equation is algebraic.(2) If any co-efficient of the Characteristic equation is complex or contains power of „e‟ thiscriterion can not be applied.(3) It gives information about how many roots are lying in the RHS of S-plane; values of theroots are not available. Also it cannot distinguish between real & complex roots.25
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Dr.Y.NARASIMHA MURTHY Ph.D yayavaram@yahoo.comSpecial cases in Routh-Hurwitz criteria :(1) When the 1st term in a row is zero, but all other terms is non-zeroes then substitute a smallpositive number ε for zero & proceed to evaluate the rest of the elements. When the 1 st columnterm is zero, it means that there is an imaginary root.(2) All zero row: In the case, write auxiliary equation from preceding row, differentiate thisequation & substitute all zero row by the co-efficient obtained by differentiating the auxiliaryequation. This case occurs when the roots are in pairs. The system is said to be limitedly stable.Application of Routh’s Criteria : Routh‟s criterion can be applied to determine range of certainparameters of a system to ensure stability. For example, it is usually of interest to find the rangeof the open loop gain K for closed loop stability.Example 1 : Find the stability of a system whose characteristic equation is given below.From the above table it is clear that the no. of sign changes in the 1st column = zero. No roots arelying in the RHS of S-plane. So, the given System is Absolutely Stable.26
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Dr.Y.NARASIMHA MURTHY Ph.D yayavaram@yahoo.comExample 2: Find the stability of a system whose characteristic equation is given below .From the above table it is clear that the no. of sign changes in the 1st column = 2 and two rootsare lying in the RHS of S-plane. So , the given System is unstableROOT LOCUS :Root locus is the plot of the loci of the root of the complementary equation when one or moreparameters of the open-loop Transfer function are varied, mostly the only one variable availableis the gain „K‟ The negative gain has no physical significance hence varying „K‟ from „0‟ to „∞‟ ,the plot is obtained called the “Root Locus Point”.Root locus gives the complete dynamic response of the system. It provides a measure ofsensitivity of roots to the variation in the parameter being considered. It is applied for single aswell as multiple loop systemRules for the Construction of Root Locus :(1) The root locus is symmetrical about the real axis.(2) The no. of branches terminating on „∞‟ equals the no. of open-loop pole-zeroes.27
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Dr.Y.NARASIMHA MURTHY Ph.D yayavaram@yahoo.com(3) Each branch of the root locus originates from an open-loop pole at „K = 0‟ & terminates atopen-loop zero corresponding to „K = ∞‟.(4) A point on the real axis lies on the locus, if the no. of open-loop poles & zeroes on the realaxis to the right of this point is odd.(5) The root locus branches that tend to „∞‟, do so along the straight line.Asymptotes making angle with the real axis is given by Where, n=1,3,5,…………………P is the number of poles and z is the number of zeros(6) The asymptotes cross the real axis at a point known as Centroid(7) The break away or the break in points [Saddle points] of the root locus are determined fromthe roots of the equation dk /ds = 0.(8) The intersection of the root locus branches with the imaginary axis can be determined by theuse of Routh-Hurwitz criteria or by putting s= jω in the characteristic equation & equating thereal part and imaginary to zero. To solve for ω and K i.e., the value of „ω‟ is intersection pointon the imaginary axis & „K‟ is the value of gain at the intersection point.9) The angle of departure from a complex open-loop pole θd is given by θd = 1800 + ∟GHNyquist Criteria:A stability test for time invariant linear systems can also be derived in the frequency domain. It isknown as Nyquist stability criterion. It is based on the complex analysis result known asCauchy‟s principle of argument.The Nyquist stability criterion relates the location of the roots of the characteristic equation tothe open-loop frequency response of the system. In this, the computation of closed-loop poles isnot necessary to determine the stability of the system and the stability study can be carried out28
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Dr.Y.NARASIMHA MURTHY Ph.D yayavaram@yahoo.comgraphically from the open-loop frequency response. Therefore experimentally determined open-loop frequency response can be used directly for the study of stability. When the feedback path isclosed. The Nyquist criterion provides the information on absolute and relative stability of thesystem.Let us suppose that the system transfer function is a complex function. By applying Cauchy‟sprinciple of argument to the open-loop system transfer function, we will get information aboutstability of the closed-loop system transfer function and arrive at the Nyquist stability criterionThe importance of Nyquist stability lies in the fact that it can also be used to determine therelative degree of system stability by producing the so-called phase and gain stability margins.These stability margins are needed for frequency domain controller design techniquesThe Nyquist plot is a polar plot of the function D(s) = 1 = G(s).H(s)29
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Dr.Y.NARASIMHA MURTHY Ph.D yayavaram@yahoo.comThe Nyquist criterion states that the number of unstable closed-loop poles is equal to thenumber of unstable open-loop poles plus the number of encirclements of the origin of theNyquist plot of the complex function D(s).Continuous Time Feedback Control Systems : If the signals in all parts of a control system arecontinuous functions of time, the system is classified as continuous time feedback controlsystem. Typically all control signals are of low frequency and if these signals are un modulated,the system is known as a d.c. control system. These systems use potentiometers as errordetectors, d.c amplifiers to amplify the error signal, d.c. servo motor as actuating device and d.ctachometers or potentiometers as feedback elements. If the control signal is modulated by ana.c carrier wave, the resulting system is usually referred to as an a.c control system. Thesesystems frequently use synchros as error detectors and modulators of error signal, a.camplifiers to amplify the error signal and a.c servo motors as actuators. These motors also serveas demodulators and produce an un modulated output signal.Discrete Data Feedback Control SystemsDiscrete data control systems are those systems in which at one or more pans of the feedbackcontrol system, the signal is in the form of pulses. Usually, the error in such system is sampledat uniform rate and the resulting pulses are fed to the control system. In most sampled datacontrol systems, the signal is reconstructed as a continuous signal, using a device called holddevice. Holds of different orders are employed, but the most common hold device is a zeroorder hold. It holds the signal value constant, at a value equal to the amplitude of the inputtime function at that sampling instant, until the next sampling instant .These systems are alsoknown as sampled data control systems.30
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Dr.Y.NARASIMHA MURTHY Ph.D yayavaram@yahoo.com Fig : Discrete Data Feedback Control SystemsDiscreet data control systems, in which a digital computer is used as one of the elements, areknown as digital control systems. The input and output to the digital computer must be binarynumbers and hence these systems require the use of digital to analog and analog to digitalconvertersTime response analysis : It is an equation or a plot that describes the behavior of a system andgives information about it with respect to time response specification as overshooting, settlingtime, peak time, rise time and steady state error. Time response is formed by the transientresponse and the steady state response. Time response = Transient response + Steady state response.Transient time response or Natural response describes the behavior of the system in its first shorttime until arrives the steady state value. If the input is step function then the output or theresponse is called step time response and if the input is ramp, the response is called ramp timeresponse .. etc.Transient Response: The transient response is defined as the part of the time response thatgoes to zero as time becomes very large. Thus yt(t) has the property Lim y(t) = 0 t -->∞The time required to achieve the final value is called transient period. The transient response maybe exponential or oscillatory in nature. Output response consists of the sum of forced response(form the input) and natural response (from the nature of the system).The transient response isthe change in output response from the beginning of the response to the final state of theresponse and the steady state response is the output response as time is approaching infinity (orno more changes at the output). The behavior of a system in transient state is shown below.31
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Dr.Y.NARASIMHA MURTHY Ph.D yayavaram@yahoo.comSteady State Response: The steady state response is the part of the total response that remainsafter the transient has died out. For a position control system, the steady state response whencompared to with the desired reference position gives an indication of the final accuracy of thesystem. If the steady state response of the output does not agree with the desired referenceexactly, the system is said to have steady state error.Response to a Unit Step Input –First OrderConsider a feedback system with G)s) = 1/τs as shown belowThe closed loop transfer function of the system is given by32
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Dr.Y.NARASIMHA MURTHY Ph.D yayavaram@yahoo.comFor a unit step input R (s) = 1 / s and the output is given byInverse Laplace transformation yieldsThe plot of c(t) Vs t is shown belowThe response is an exponentially increasing function and it approaches a value of unity ast --- > ∞At t = τ the response reaches a value,which is 63.2 percent of the steady value. This time, τ is known as the time constant of thesystem. One of the important characteristics about the system is its speed of response or howfast the response is approaching the final value. The time constant τ is indicative of this measureand the speed of response is inversely proportional to the time constant of the system.Anotherimportant characteristic of the system is the error between the desired value and the actual valueunder steady state conditions. This quantity is known as the steady state error of the - system andis denoted by ess.The error E(s) for a unity feedback system is given by33
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Dr.Y.NARASIMHA MURTHY Ph.D yayavaram@yahoo.com For the system under consideration G(s) = 1 / τs and R(s) = 1 /s ThereforeAs t ~ ∞ e (t) ~ 0 . Thus the output of the first order system approaches the reference input,which is the desired output, without any error. In other words, we say a first order system tracksthe step input without any steady state error.Response to a Unit Ramp Input :To study the response of a unit ramp let us consider a feedback system with G)s) = 1/τs as shown belowThe response of the system for which , R(s) = 1 / s2 is given byThe time response is obtained by taking inverse Laplace transform of above equation34
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Dr.Y.NARASIMHA MURTHY Ph.D yayavaram@yahoo.comDifferentiating the above equation we getThis is the response of the system to a step input. Thus no additional information about thespeed of response is obtained by considering a ramp input.So, no additional information aboutthe speed of response is obtained by considering a ramp input.But the effect on the steady state error is given byThus the steady state error is equal to the time constant of the system. The first order system,therefore, can not track the ramp input without a finite steady state error. If the time constant isreduced not only the speed of response increases but also the steady state error for ramp inputdecreases. Hence the ramp input is important to the extent that it produces a finite steady stateerror.The response of a first order system for unit ramp input is shown below.35
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Dr.Y.NARASIMHA MURTHY Ph.D yayavaram@yahoo.comResponse to a Unit Step Input - Second Order SystemLet us consider a type 1, second order system as shown in Fig.below. Since G(s) has one pole atthe origin, it is a type one system.The closed loop transfer function is give by,The transient response of any system depends on the poles of the transfer function T(s). Theroots of the denominator polynomial in s of T(s) are the poles of the transfer function. Thus thedenominator polynomial of T(s), given by D(s) = τs2 + S + K36
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Dr.Y.NARASIMHA MURTHY Ph.D yayavaram@yahoo.comis known as the characteristic polynomial of the system and D(s) = 0 is known as thecharacteristic equation of the system. The above Eqn. is normally put in standard from, givenby,The poles of T( s), or, the roots of the characteristic equation S2 + 2δ ωn s + ωn 2 = 0are given by,Where is known as the damped natural frequency of the system. If δ > 1,the two roots s1, s2 are real and we have an over damped system. If δ = 1, the system is known asa critically damped system. The more common case of δ < 1 is known as the under dampedsystem.Steady State Errors : One of the important design specifications for a control system is thesteady state error. The steady state output of any system should be as close to desired output aspossible. If it deviates from this desired output, the performance of the system is not satisfactory37
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Dr.Y.NARASIMHA MURTHY Ph.D yayavaram@yahoo.comunder steady state conditions. The steady state error reflects the accuracy of the system. Amongmany reasons for these errors, the most important ones are the type of input, the type of thesystem and the nonlinearities present in the system. Since the actual input in a physical system isoften a random signal, the steady state errors are obtained for the standard test signals, namely,step, ramp and parabolic signals.Error Constants : Let us consider a feedback control system as shown below.The error signal E (s) is given by E (s) = R (s) - H (s) C (s)But C (s) = G (s) E (s)From the above equations we haveApplying final value theorem, we can get the steady state error ess as,The above equation shows that the steady state error is a function of the input R(s) and the openloop transfer function G(s). Let us consider various standard test signals and obtain the steadystate error for these inputs.Proportional, Integral and Derivative Controller (PID Control) :A Proportional–Integral–Derivative (PID) controller is a three-term controller which isconsidered as a standard controller in industrial settings. . It can be found in virtually all kinds of38
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Dr.Y.NARASIMHA MURTHY Ph.D yayavaram@yahoo.comcontrol equipments, either as a stand-alone (single-station)controller or as a functional block inProgrammable Logic Controllers (PLCs)and Distributed Control Systems (DCSs).An integral control eliminates steady state error due to a velocity input, but its effect on dynamicresponse is difficult to predict as the system order increases to three. It is known that aderivative term in the forward path improves the damping in the system. One of the best-knowncontrollers used in practice is the PID controller, where the letters stand for proportional,integral,and derivative. The integral and derivative components of the PID controller haveindividual performance implications, and their applications require an understanding ofthe basicsof these elements. Hence a suitable combination of integral and derivative controls results in aproportional, integral and derivate control, usually called PID control. The transfer function ofthe PID controller is given by,The diagram below gives the ideal PID controller.The overall forward path transfer function is given by,39
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Dr.Y.NARASIMHA MURTHY Ph.D yayavaram@yahoo.comand the overall transfer function is given by,Proper choice of Kp, KD and Kr results in satisfactory transient and steadystate responses. Theprocess of choosing proper Kp, KD, at Kr for a given system is known as tuning of a P IDcontroller.Bode plots : Bode plots are graphs of the steady state response of stable continuous time Lineartime invariant systems for sinusoidal inputs,plotted as change in magnitude and phase versusfrequency on logarithmic scale.Bode plots are a visual description of the system.So,these Bodeplots are used for the representation of sinusoidal transfer function . In this representation themagnitude of G(jω) in db, i.e, 20 log G(jω) is plotted against log ω.Similarly phase angle ofG(jω) is plotted against log ω .Hence the abscissa is logarithm of the frequency and hence theplots are known as logarithmic plots. The plots are named after the famous mathematician H. W.Bode.The transfer function G(jω) can be written as G(jω) = G(jω)∟Ф( ω) Where Ф( ω) is the angleG(jω) .Since G(jω) consists of many multiplicative factors in the numerator and denominator it isconvenient to take logarithm of G(jω) to convert these factors into additions and subtractions,which can be carried out easily.To plot the magnitude plot , magnitude is plotted against input frequency on a logarithmicscale.It can be approximated by two lines and it forms the asymptotic (approximate) magnitudeBode plot of the transfer function:40
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Dr.Y.NARASIMHA MURTHY Ph.D yayavaram@yahoo.com(i) for angular frequencies below ωc it is a horizontal line at 0 dB since at low frequencies theterm ω / ωc is small and can be neglected, making the decibel gain equation above equal tozero,(ii) for angular frequencies above ωc it is a line with a slope of −20 dB per decade since athigh frequencies the term ω / ωc dominates and the decibel gain expression above simplifies to-20 log (ω / ωc ) which is a straight line with a slope of −20 dB per decade.These two lines meet at the corner frequency. From the Bode plot, it can be seen that forfrequencies well below the corner frequency, the circuit has an attenuation of 0 dB,corresponding to a unity pass band gain, i.e. the amplitude of the filter output equals theamplitude of the input. Frequencies above the corner frequency are attenuated – the higher thefrequency, the higher the attenuation.Phase plotThe phase Bode plot is obtained by plotting the phase angle of the transfer function given by –Φ = - tan (ω/ωc ) Versus ω , where ω and ωc are the input and cutoff angular frequenciesrespectively.For input frequencies much lower than corner, the ratio ω/ωc is small and therefore the phaseangle is close to zero. As the ratio increases the absolute value of the phase increases andbecomes –45 degrees when ω= ωc As the ratio increases for input frequencies much greaterthan the corner frequency, the phase angle asymptotically approaches −90 degrees. Thefrequency scale for the phase plot is logarithmic.The main advantage of using the Bode plots is that multiplication of magnitudes can beconverted into addition. Also a simple method of plotting an approximate log-magnitude curve is41
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Dr.Y.NARASIMHA MURTHY Ph.D yayavaram@yahoo.comobtained. It is based on asymptotic approximations. Such approximation by straight lineasymptotes is sufficient if only rough information on the frequency response characteristics is isneeded.The phase angle curves can be drawn easily if a template for the phase angle curve of1+jω is available. Expanding the low frequency range ,by use of a logarithmic scale for thefrequency is very advantageous ,since characteristics at low frequencies are most important inpractical systems. The only limitation is ,it is not possible to draw the curve ,right down to zerofrequency because of the logarithmic frequency (log 0 = ∞) .But this do not give any seriousproblem.Polar plot : This plot is drawn to study the frequency response in polar co-ordinates. The polarplot of a sinusoidal transfer function G(jω) is a plot of the magnitude of G(jω) Vs the phase ofG(jω) on polar co-ordinates as ω is varied from 0 to ∞. Polar graph sheet has concentric circlesand radial lines. Concentric circles represent the magnitude. Radial lines represents the phaseangles. In polar sheet +ve phase angle is measured in ACW from 00 and -ve phase angle ismeasured in CW from 00 .To sketch the polar plot of G(jω) for the entire range of frequency ω, i.e., from 0 to infinity, thereare four key points that usually need to be known: The start of plot where ω = 0, The end of plot where ω = ∞, where the plot crosses the real axis, i.e., Im(G(jω)) = 0, and where the plot crosses the imaginary axis, i.e., Re(G(jω)) = 0.Procedure: The following steps are involved in the plotting of Polar graphs Express the given expression of OLTF in (1+sT) form. Get the expressions for | G(jω)H(jω)| & G(jω)H(jω). Tabulate various values of magnitude and phase angles for different values of ω ranging from 0 to ∞. Usually the choice of frequencies will be the corner frequency and around corner frequencies. Choose proper scale for the magnitude circles. Fix all the points in the polar graph sheet and join the points by a smooth curve. Write the frequency corresponding to each of the point of the plot.42
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Dr.Y.NARASIMHA MURTHY Ph.D yayavaram@yahoo.comGain Margin: Gain Margin is defined as “the factor by which the system gain can be increasedto drive the system to the verge of instability”.For stable systems, ωgc < ωpc │G(j)H(j)│ at ω= ωpc < 1 GM = in positive dB .More positive the Gain Margin , more stable is the systemFor marginally stable systems, ωgc = ωpc │G(j)H(j)│ at ω=ωpc = 1 GM = 0 dBFor Unstable systems, ωgc > ωpc │G(j)H(j)│ at ω=ωpc > 1 GM = in negative dBGain is to be reduced to make the system stableThe important conclusions are : If the gain is high, the GM is low and the system‟s step response shows high overshoots and long settling time. On the contrary, very low gains give high GM and PM, but also causes higher ess(Error), higher values of rise time and settling time and in general give sluggish response. Thus we should keep the gain as high as possible to reduce ess and obtain acceptable response speed and yet maintain adequate GM and PM. As a thumb rule an adequate GM of 2 ( ie 6 dB ) and a PM of about 30 is generally considered sufficient enough. If the gain of the system is increased by a factor 1/β, then the G(j)H(j) at ω=ωpc becomes β(1/β) = 1 and hence the │G(j)H(j)│ locus pass through -1+j0 point driving the system to the verge of instabilityPHASE MARGIN: Phase Margin is defined as “ the additional phase lag that can be introducedbefore the system becomes unstable”.43
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Dr.Y.NARASIMHA MURTHY Ph.D yayavaram@yahoo.com Let „A‟ be the point of intersection of │G(j)H(j)│ plot and a unit circle centered at the origin. Draw a line connecting the points „O‟ & „A‟ and measure the phase angle between the line OA and +ve real axis. This angle is the phase angle of the system at the gain cross over frequency. G(jωgc)H(jgc) = Фgc If an additional phase lag of PM is introduced at this frequency, then the phase angle G(jωgc)H(jωgc) will become 180 and the point „A„ coincides with (-1+j0) driving the system to the verge of instability. This additional phase lag is known as the Phase Margin. For a stable system, the phase margin is positive. A Phase margin close to zero corresponds to highly oscillatory system A polar plot may be constructed from experimental data or from a system transfer function If values of ω are marked along the contour, a polar plot gives the same information as a Bode plot Usually, the shape of a polar plot is very important as it gives much information about the system.Relative and absolute stability: A stable system is a dynamic system with a bounded responseto a bounded input. A necessary and sufficient condition for a feedback system to be stable isthat all the poles of the system transfer function have negative real parts44
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Dr.Y.NARASIMHA MURTHY Ph.D yayavaram@yahoo.comA system is considered marginally stable if only certain bounded inputs will result in a boundedoutput.In practical systems, it is not sufficient to know that the system is stable but a stable system mustmeet the specifications of relative stability which is a quantitative measure of how fast thetransients die out in the system.Relative stability of a system is usually defined in terms of two design parameters-phase marginand gain margin.The relative stability of a system can be defined as the property that is measured by the relativereal part of each root or pair of roots. The relative stability of a system can also be defined interms of the relative damping coefficients of each complex root pair and, therefore, in terms ofthe speed of response and overshoot instead of settling time.Time Domain Analysis Vs Frequency Domain Analysis Variable frequency, sinusoidal signal generators are readily available and precision measuring instruments are available for measurement of magnitude and phase angle. The time response for a step input is more difficult to measure with accuracy. It is easier to obtain the transfer function of a system by a simple frequency domain test.Obtaining transfer function from the step response is more tedious. If the system has large time constants, it makes more time to reach steady state at each frequency of the sinusoidal input. Hence time domain method is preferred over frequency domain method in such systems. In order to do a frequency response test on a system, the system has to be isolated and the sinusoidal signal has to be applied to the system. This may not be possible in systems which can not be interrupted. In such cases, a step signal or an impulse signal may be given to the system to find its transfer function. Hence for systems which cannot be interrupted, time domain method is more suitable. The design of a controller is easily done in the frequency domain method than in time domain method. For a given set of performance measures in frequency domain , the parameters of the open loop transfer function can be adjusted easily. The effect of noise signals can be assessed easily in frequency domain rather than time domain. The most important advantage of frequency domain analysis is the ability to obtain the relative stability of feedback control systems. The Routh Hurwitz criterion is essentially a time do main method which determines the absolute stability of a system.45
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Dr.Y.NARASIMHA MURTHY Ph.D yayavaram@yahoo.com Since the time response and frequency response of a system are related through Fourier transform , the time response can be easily obtained from the frequency response. The correlation between time and frequency response can be easily established so that the time domain performance measures can be obtained from the frequency domain specifications and vice versa.Frequency Response of a Control System : To study the frequency response of a controlsystem let us consider second order system with the transfer function,The steady state sinusoidal response is obtained by substituting s = j ω in the above equationNormalising the frequency ω, with respect to the natural frequency ωn by defining a variable We have,From the above equation the magnitude and angle of the frequency response is obtained as, and The time response for a unit sinusoidal input with frequency OJ is given by,46
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Dr.Y.NARASIMHA MURTHY Ph.D yayavaram@yahoo.comThe magnitude and phase of steadystate sinusoidal response for variable frequency can beplotted and are shown in the Fig. (a) and (b).An important performance measure, in frequency domain, is the bandwidth of the system.From Fig. , we observe that for δ < 0.707 and U > Ur the magnitude decreases monotonically.The frequency Ub where the magnitude becomes 0.707 is known as the cut off frequency. Atthis frequency, the magnitude will beControl system Compensators:A control system is usually required to meet three time response specifications, namely, steadystate accuracy, damping factor and settling time. To get the desired design with minimum errors47
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Dr.Y.NARASIMHA MURTHY Ph.D yayavaram@yahoo.comand to adjust the parameters of the overall system to satisfy the design criterion an additionalsubsystem called Compensator „ must be used.This compensator may be used in series with the plant in the forward path or in the feedbackpath shown in Fig. below. The compensation in the forward path is known as series or cascadecompensation and the later is known as feedback compensation. The compensator may be apassive network or an active network. Series compensation Feedback compensationIn general the series compensation is much simpler than the feed back compensation.But theseries compensation frequently require additional amplifiers to increase the gain and to provideisolation.i.e to avoid power dissipation the series compensator is inserted in the lowest energypoint in the feed forward path.In general the number of components required in feed backcompensation will be less than the number of components in series compensation.There are three important types of compensators.(i) Lead Compensator (ii) Lag compensator and(iii) Lag-Lead compensator.In a network ,when a sinusoidal input signal ei is applied at its input and if the steady stateoutput eo has a phase lead then the network is called a lead network. Similarly In the network fora sinusoidal input ei ,if the steady state output has a phase lag ,then the network is called a lagnetwork.Similarly in a network for a sinusoidal input .if the steady state out put has both phase lag andlead ,but different frequency regions ,then the network is called lag-lead network.Generally thephase lag occurs at low frequency regions and the phase lead occurs at higher frequencyregion.A compensator having the characteristics of a lead network ,lag network or lead-lagnetwork is called a lead compensator ,lag compensator or lag-lead compensator.The lead compensators, lag compensators and lag-lead compensators are be designed eitherusing electronic components like operational amplifiers or using Root Locus methods. The first48
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Dr.Y.NARASIMHA MURTHY Ph.D yayavaram@yahoo.comtype are called electronic lag compensators .The second type are Root locus compensators. TheRoot locus compensators are have many advantages over electronic type.Lead compensation basically speeds up the response of the control system and increase thestability of the system. Lag compensation improves the steady –state accuracy of the system butreduces the speed of the response. If improvements in both transient response and steady stateresponse is required ,both lead compensator and lag compensator are used simultaneously .Ingeneral using a single lag-lead compensator is always economical.Lag-lead compensation combines the advantage of lag and lead compensation. Since, the lag –lead compensator possesses two poles and two zeros ,such a compensation increases the order ofthe system by 2 ,unless the cancellation of the poles and zeros occurs in the compensated system.Lead Compensator : It has a zero and a pole with zero closer to the origin. The general form ofthe transfer function of the load compensator is49
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Dr.Y.NARASIMHA MURTHY Ph.D yayavaram@yahoo.comSubsistingTransfer function s50
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Dr.Y.NARASIMHA MURTHY Ph.D yayavaram@yahoo.comLag Compensator :It has a zero and a pole with the zero situated on the left of the pole on the negative real axis. Thegeneral form of the transfer function of the lag compensator isTherefore, the frequency response of the above transfer function will be51
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Dr.Y.NARASIMHA MURTHY Ph.D yayavaram@yahoo.comNow comparing withTherefore,the transfer function isLag-Lead CompensatorThe lag-lead compensator is the combination of a lag compensator and a lead compensator. Thelag-section is provided with one real pole and one real zero, the pole being to the right of zero,52
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Dr.Y.NARASIMHA MURTHY Ph.D yayavaram@yahoo.comwhereas the lead section has one real pole and one real zero with the zero being to the right ofthe pole.The transfer function of the lag-lead compensator will beThe figure below shows the lag lead compensator53
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Dr.Y.NARASIMHA MURTHY Ph.D yayavaram@yahoo.comThe above transfer functions are comparing with54
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Dr.Y.NARASIMHA MURTHY Ph.D yayavaram@yahoo.comThenTherefore, the transfer function is given bystate variable analysis :Analysis and the design of feedback control systems by the classical design methods (root locusand frequency domain methods) based on the transfer function approach are inadequate and notconvenient. So, the development of the state-variable approach, took place .This methos hasthe following advantages over the classical approach.55
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Dr.Y.NARASIMHA MURTHY Ph.D yayavaram@yahoo.com1. It is a direct time-domain approach. Thus, this approach is suitable for digital computercomputations2. It is a very powerful technique for the design and analysis of linear or nonlinear, time-variantor time-invariant, and SISO or MIMO systems.3. In this technique, the nth order differential equations can be expressed as „n‟ equations of firstorder. Thus, making the solutions easier.4. Using this approach, the system can be designed for optimal conditions with respect to given performance indices.Definitions:State : The state of a dynamical system is a minimal set of variables x1(t), x2(t)x3(t) …xn(t)such that the knowledge of these variables at t = t0 (initial condition), together with theKnowledge of inputs u1(t), u2(t), u3(t)… um (t) for t ≥0, completely determines the behavior ofthe system for t < t0.State-Variables :The variables x1(t),x2(t),x3(t)… xm(t) such that the knowledge of these variables at t = t0 (initialcondition), together with the knowledge of inputs u1(t), u2(t), u3(t)… um(t) for t ≥ t0, completelydetermines the behavior of the system for t < t0 ; are called state-variables. In other words, thevariables that determine the state of a dynamical system, are called state-variables.State-Vector :If n state variables x1(t), x2(t), x3(t)… xn(t) are necessary to determine the behavior of adynamical system, then these n state-variables can be considered as n components of a vectorx(t), called state-vector.State-Space : The n dimensional space, whose elements are the n state-variables, is called state-space. Any state can be represented by a point in the state-space.State equation of a linear time-invariant system :For a general system of the Figure shown below the state representation can be arranged in theform of n first-order differential equations as56
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Dr.Y.NARASIMHA MURTHY Ph.D yayavaram@yahoo.com ……(1)Integrating equation 1 , we getWhere I = 1,2,3……..nThus, the n state-variables and, the state of the system can uniquely be determined at any t < tn,provided each state-variable is known at t = tn and all the m control forces are known throughoutthe interval t 0 to t.The n differential equations of 1 may be written in vector form as ………… (2)57
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Dr.Y.NARASIMHA MURTHY Ph.D yayavaram@yahoo.comEquation 2 is the state equation for time-invariant systems. However, for time-varying systems, the function vector f(.) is dependent on time as well, and the vector equation may be given as ……………….(3)Equation (3) is the state equation for time-varying systems.The output y(t) can, in general, be expressed in terms of the state vector x(t) and input vector u(t)asFor time-invariant systems : ------------------------ -----(4)For time-varying systems: ----------------(5)STATE MODEL OF A LINEAR SINGLE-INPUT-SINGLE-OUTPUT SYSTEM:The state model of a linear single-input-single-output system can be written as (1) (2)Where58
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Dr.Y.NARASIMHA MURTHY Ph.D yayavaram@yahoo.comd = Transmission Constant , u(t) = Input or Control Variable (scalar)and, y(t) = Output Variable (scalar).The block-diagram representation of the state model of linear single-input-single-output systemis shown below.A very important conclusion is that the derivatives of all the state-variables are zero at theequilibrium point. Thus, the system continues to lie at the equilibrium point unless otherwisedisturbed.59
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Dr.Y.NARASIMHA MURTHY Ph.D yayavaram@yahoo.comSOLUTION OF STATE EQUATIONS FOR LINEAR TIME-INVARIANT SYSTEMS :The state equation of a linear time-invariant system is given byFor a homogeneous (unforced) systemSo , we have ………..(1)Take the Laplace Transform on both sides …………… (2)The above may also be written asTaking the inverse Laplace transform --------------(3)This equation (3) gives the solution of the LTI homogeneous state Equation.It can be observedthat the initial state x(0) at t = 0, is driven to a state x(t) at time t. This transition in state is carriedout by the matrix exponential eAt. Because of this property, eAt is termed as the State TransitionMatrix and is denoted by ø(t).Determination of transfer function using SVA:Let us consider the following state space equation ,Now, take the Laplace Transform (with zero initial conditions)60
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Dr.Y.NARASIMHA MURTHY Ph.D yayavaram@yahoo.comWe want to solve for the ratio of Y(s) to U(s), which gives the transfer function.So, we need toremove Q(s) from the output equation. We solve the state equation for Q(s)The matrix Φ(s) is called the state transition matrix. Now , put this into the output equationNow solving for transfer function ,we getThis is the method of determining the transfer function from state variable analysis (SVA). --------------------xxxxxxxxxxxxxx-------------------------Acknowledgment: Dear Reader, I don’t claim any ownership for this material, as it is acollection from various books and websites and other articles. I have simply added myexperience and try to provide this material for those people who are preparing for NET/UGCand other Engineering exams.Please download this material for your preparation .But nevermake it commercial. It is purely meant for students who are in need of this material.References: 1:Automatic control systems - B.C. Kuo, Prentice-Hall of India. 2. Modern Control Engineering- K. Ogata, Prenticd-Hall of India. 3. Control Systems Engineering - I.G. Nagrath, M. Gopal; Wiley Eastern Ltd 4. Control Systems -Second Edition - Dr. N.C. JaganAnd many more people‟s study material on the net.61