Lecture notes on Control system Engineering

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IC- 6501 CONTROL SYSTEMS A Course Material on IC – 6501 CONTROL SYSTEMS By Mr. S.SRIRAM HEAD & ASSISTANT PROFESSOR DEPARTMENT OF ELECTRICAL AND ELECTRONICS ENGINEERING SASURIE COLLEGE OF ENGINEERING VIJAYAMANGALAM – 638 056 Page 1 of 116 SCE ELECTRICAL AND ELECTRONICS ENGINEERING IC- 6501 CONTROL SYSTEMS QUALITY CERTIFICATE This is to certify that the e-course material Subject Code : IC- 6501 Subject : Control Systems Class : III Year EEE Being prepared by me and it meets the knowledge requirement of the university curriculum. Signature of the Author Name: Designation: This is to certify that the course material being prepared by Mr.S.Sriram is of adequate quality. He has referred more than five books among them minimum one is from aboard author. Signature of HD Name: S.SRIRAM SEAL Page 2 of 116 SCE ELECTRICAL AND ELECTRONICS ENGINEERING IC- 6501 CONTROL SYSTEMS S. No TOPICS PAGE No. UNIT –I SYSTEMS AND THEIR REPRESENTATION 1 Basic Elements of Control System 6 2 Open loop and Closed loop systems 6 3 Electrical analogy of mechanical and thermal systems 6 4 Transfer function 14 Synchros 15 5 6 AC and DC servomotors 17 7 Block diagram reduction Techniques 21 8 Signal flow graph 29 UNIT II TIME RESPONSE Time response 31 9 10 Time domain specifications 40 11 Types of test input 32 12 First Order Systems 33 13 Impulse Response analysis of second order systems 34 Step Response analysis of second order systems 34 14 15 Steady state errors 37 16 Root locus construction 41 17 P, PI, PD and PID Compensation 38 UNIT III FREQUENCY RESPONSE Frequency Response 44 18 19 Bode Plot 45 20 Polar Plot 48 Determination of closed loop response from open loop 21 49 response Correlation between frequency domain and time domain 22 50 specifications Effect of Lag, lead and lag-lead compensation on 23 51 frequency response 24 Analysis. 55 UNIT IV STABILITY ANALYSIS 25 Characteristics equation 58 26 Routh-Hurwitz Criterion 58 27 Nyquist Stability Criterion 60 28 Performance criteria 50 Lag, lead and lag-lead networks 51 29 30 Lag/Lead compensator design using bode plots. 54 UNIT V STATE VARIABLE ANALYSIS 31 Concept of state variables 66 32 State models for linear and time invariant Systems 67 33 Solution of state and output equation in controllable 69 canonical form Concepts of controllability and observability 70 34 35 Effect of state feedback 72 TUTORIAL PROBLEMS 73 Page 3 of 116 SCE ELECTRICAL AND ELECTRONICS ENGINEERING IC- 6501 CONTROL SYSTEMS QUESTION BANK 86 UNIVERSITY QUESTION PAPERS GLOSSARY 107 IC6501 CONTROL SYSTEMS L T P C 3 1 0 4 OBJECTIVES: Page 4 of 116 SCE ELECTRICAL AND ELECTRONICS ENGINEERING IC- 6501 CONTROL SYSTEMS To understand the use of transfer function models for analysis physical systems and introduce the control system components. To provide adequate knowledge in the time response of systems and steady state error analysis. To accord basic knowledge in obtaining the open loop and closed–loop frequency responses of systems. To introduce stability analysis and design of compensators To introduce state variable representation of physical systems and study the effect of state feedback UNIT I SYSTEMS AND THEIR REPRESENTATION 9 Basic elements in control systems – Open and closed loop systems – Electrical analogy of mechanical and thermal systems – Transfer function – Synchros – AC and DC servomotors – Block diagram reduction techniques – Signal flow graphs. UNIT II TIME RESPONSE 9 Time response – Time domain specifications – Types of test input – I and II order system response – Error coefficients – Generalized error series – Steady state error – Root locus construction- Effects of P, PI, PID modes of feedback control –Time response analysis. UNIT III FREQUENCY RESPONSE 9 Frequency response – Bode plot – Polar plot – Determination of closed loop response from open loop response - Correlation between frequency domain and time domain specifications- Effect of Lag, lead and lag-lead compensation on frequency response- Analysis. UNIT IV STABILITY AND COMPENSATOR DESIGN 9 Characteristics equation – Routh Hurwitz criterion – Nyquist stability criterion- Performance criteria – Lag, lead and lag-lead networks – Lag/Lead compensator design using bode plots. UNIT V STATE VARIABLE ANALYSIS 9 Concept of state variables – State models for linear and time invariant Systems – Solution of state and output equation in controllable canonical form – Concepts of controllability and observability – Effect of state feedback. TOTAL (L:45+T:15): 60 PERIODS OUTCOMES: Ability to understand and apply basic science, circuit theory, theory control theory Signal processing and apply them to electrical engineering problems. TEXT BOOKS: 1. M. Gopal, ‘Control Systems, Principles and Design’, 4th Edition, Tata McGraw Hill, New Delhi, 2012 2. S.K.Bhattacharya, Control System Engineering, 3rd Edition, Pearson, 2013. 3. Dhanesh. N. Manik, Control System, Cengage Learning, 2012. REFERENCES: 1. Arthur, G.O.Mutambara, Design and Analysis of Control; Systems, CRC Press, 2009. 2. Richard C. Dorf and Robert H. Bishop, “ Modern Control Systems”, Pearson Prentice Hall, 2012. 3. Benjamin C. Kuo, Automatic Control systems, 7th Edition, PHI, 2010. 4. K. Ogata, ‘Modern Control Engineering’, 5th edition, PHI, 2012. CHAPTER 1 SYSTEMS AND THEIR REPRESENTATION Page 5 of 116 SCE ELECTRICAL AND ELECTRONICS ENGINEERING IC- 6501 CONTROL SYSTEMS 1.1 Basic elements of control system In recent years, control systems have gained an increasingly importance in the development and advancement of the modern civilization and technology. Figure shows the basic components of a control system. Disregard the complexity of the system; it consists of an input (objective), the control system and its output (result). Practically our day-to-day activities are affected by some type of control systems. There are two main branches of control systems: 1) Open-loop systems and 2) Closed-loop systems. Basic Components of Control System 1.2 Open-loop systems: The open-loop system is also called the non-feedback system. This is the simpler of the two systems. A simple example is illustrated by the speed control of an automobile as shown in Figure 1-2. In this open-loop system, there is no way to ensure the actual speed is close to the desired speed automatically. The actual speed might be way off the desired speed because of the wind speed and/or road conditions, such as uphill or downhill etc. Basic Open Loop System Closed-loop systems: The closed-loop system is also called the feedback system. A simple closed-system is shown in Figure 1-3. It has a mechanism to ensure the actual speed is close to the desired speed automatically. 1.3 Mechanical Translational systems The model of mechanical translational systems can obtain by using three basic elements mass, spring and dashpot. When a force is applied to a translational mechanical system, it is Page 6 of 116 SCE ELECTRICAL AND ELECTRONICS ENGINEERING IC- 6501 CONTROL SYSTEMS opposed by opposing forces due to mass, friction and elasticity of the system. The force acting on a mechanical body is governed by Newton‗s second law of motion. For translational systems it states that the sum of forces acting on a body is zero. Force balance equations of idealized elements: Consider an ideal mass element shown in fig. which has negligible friction and elasticity. Let a force be applied on it. The mass will offer an opposing force which is proportional to acceleration of a body. Let f = applied force fm =opposing force due to mass 2 2 Here fm α M d x / dt 2 2 By Newton‗s second law, f = f m= M d x / dt Consider an ideal frictional element dash-pot shown in fig. which has negligible mass and elasticity. Let a force be applied on it. The dashpot will be offer an opposing force which is proportional to velocity of the body. Let f = applied force f b = opposing force due to friction Here, f b α B dx / dt By Newton‗s second law, f = f = M d x / dt b Consider an ideal elastic element spring is shown in fig. This has negligible mass and friction. Let f = applied force f k = opposing force due to elasticity Page 7 of 116 SCE ELECTRICAL AND ELECTRONICS ENGINEERING IC- 6501 CONTROL SYSTEMS Here, f k α x By Newton‗s second law, f = f k = x Mechanical Rotational Systems: The model of rotational mechanical systems can be obtained by using three elements, moment of inertia J of mass, dash pot with rotational frictional coefficient B and torsional spring with stiffnessk. When a torque is applied to a rotational mechanical system, it is opposed by opposing torques due to moment of inertia, friction and elasticity of the system. The torque acting on rotational mechanical bodies is governed by Newton‗s second law of motion for rotational systems. Torque balance equations of idealized elements Consider an ideal mass element shown in fig. which has negligible friction and elasticity. The opposing torque due to moment of inertia is proportional to the angular acceleration. Let T = applied torque Tj =opposing torque due to moment of inertia of the body 2 2 Here Tj= α J d θ / dt By Newton‗s law 2 2 T= Tj = J d θ / dt Consider an ideal frictional element dash pot shown in fig. which has negligible moment of inertia and elasticity. Let a torque be applied on it. The dash pot will offer an opposing torque is proportional to angular velocity of the body. Let T = applied torque Tb =opposing torque due to friction Here Tb = α B d / dt (θ - θ ) 1 2 By Newton‗s law T= Tb = B d / dt (θ - θ ) 1 2 . Consider an ideal elastic element, torsional spring as shown in fig. which has negligible moment of inertia and friction. Let a torque be applied on it. The torsional spring will offer an opposing torque which is proportional to angular displacement of the body Let T = applied torque Page 8 of 116 SCE ELECTRICAL AND ELECTRONICS ENGINEERING IC- 6501 CONTROL SYSTEMS Tk =opposing torque due to friction Here Tk α K (θ - θ ) 1 2 By Newton‗s law T = Tk = K (θ - θ ) 1 2 Modeling of electrical system Electrical circuits involving resistors, capacitors and inductors are considered. The behaviour of such systems is governed by Ohm‗s law and Kirchhoff‗s laws Resistor: Consider a resistance of ‗R‗ Ω carrying current ‗i‗ Amps as shown in Fig (a), then the voltage drop across it is v = R I Inductor: Consider an inductor ―L‗ H carrying current ‗i‗ Amps as shown in Fig (a), then the voltage drop across it can be written as v = L di/dt Capacitor: Consider a capacitor ‗C‗ F carrying current ‗i‗ Amps as shown in Fig (a), then the voltage drop across it can be written as v = (1/C)∫ i dt Steps for modeling of electrical system Apply Kirchhoff‗s voltage law or Kirchhoff‗s current law to form the differential equations describing electrical circuits comprising of resistors, capacitors, and inductors. Form Transfer Functions from the describing differential equations. Then simulate the model. Example R i(t) + R i(t) + 1/ C ∫ i(t) dt = V (t) 1 2 1 R i(t) + 1/ C ∫ i(t) dt = V (t) 2 2 Page 9 of 116 SCE ELECTRICAL AND ELECTRONICS ENGINEERING IC- 6501 CONTROL SYSTEMS Electrical systems LRC circuit. Applying Kirchhoff‗s voltage law to the system shown. We obtain the following equation; Resistance circuit L(di /dt) + Ri + 1/ C ∫ i(t) dt =e i …………………….. (1) 1/ C ∫ i(t) dt =e 0 ……………………………………….. (2) Equation (1) & (2) give a mathematical model of the circuit. Taking the L.T. of equations (1)&(2), assuming zero initial conditions, we obtain Armature-Controlled dc motors The dc motors have separately excited fields. They are either armature-controlled with fixed field or field-controlled with fixed armature current. For example, dc motors used in instruments employ a fixed permanent-magnet field, and the controlled signal is applied to the armature terminals. Consider the armature-controlled dc motor shown in the following figure. Ra = armature-winding resistance, ohms La = armature-winding inductance, henrys ia = armature-winding current, amperes if = field current, a-pares ea = applied armature voltage, volt eb = back emf, volts θ = angular displacement of the motor shaft, radians T = torque delivered by the motor, Newtonmeter J = equivalent moment of inertia of the motor and load referred to the motor shaft kg.m2 Page 10 of 116 SCE ELECTRICAL AND ELECTRONICS ENGINEERING IC- 6501 CONTROL SYSTEMS f = equivalent viscous-friction coefficient of the motor and load referred to the motor shaft. Newtonm/rad/s T = k1 ia ψ where ψ is the air gap flux, ψ = kf if , k1 is constant For the constant flux Where Kb is a back emf constant (1) The differential equation for the armature circuit The armature current produces the torque which is applied to the inertia and friction; hence Assuming that all initial conditions are condition are zero/and taking the L.T. of equations (1), (2) & (3), we obtain K s θ (s) = E (s) p b 2 (L s+Ra ) I (s) + E (s) = E (s) (Js +fs) a a b a θ (s) = T(s) = K I (s) a The T.F can be obtained is Analogous Systems Let us consider a mechanical (both translational and rotational) and electrical system as shown in the fig. From the fig (a) 2 2 We get M d x / dt + D d x / dt + K x = f From the fig (b) 2 2 We get M d θ / dt + D d θ / dt + K θ = T From the fig (c) 2 2 We get L d q / dt + R d q / dt + (1/C) q = V(t) Page 11 of 116 SCE ELECTRICAL AND ELECTRONICS ENGINEERING IC- 6501 CONTROL SYSTEMS Where q = ∫i dt They are two methods to get analogous system. These are (i) force- voltage (f-v) analogy and (ii) force-current (f-c) analogy Force –Voltage Analogy Force – Current Analog Problem 1. Find the system equation for system shown in the fig. And also determine f-v and f-i analogies Page 12 of 116 SCE ELECTRICAL AND ELECTRONICS ENGINEERING IC- 6501 CONTROL SYSTEMS For free body diagram M1 For free body diagram M2 (2) Force –voltage analogy From eq (1) we get From eq (2) we get …..(4) From eq (3) and (4) we can draw f-v analogy Force–current analogy Page 13 of 116 SCE ELECTRICAL AND ELECTRONICS ENGINEERING IC- 6501 CONTROL SYSTEMS From eq (1) we get ……..(5) From eq (2) we get …………(6) From eq (5) and (6) we can draw force-current analogy The system can be represented in two forms: Block diagram representation Signal flow graph 1.4 Transfer Function A simpler system or element maybe governed by first order or second order differential equation. When several elements are connected in sequence, say ―n‖ elements, each one with first order, the total order of the system will be nth order In general, a collection of components or system shall be represented by nth order differential equation. In control systems, transfer function characterizes the input output relationship of components or systems that can be described by Liner Time Invariant Differential Equation In the earlier period, the input output relationship of a device was represented graphically. In a system having two or more components in sequence, it is very difficult to find graphical relation between the input of the first element and the output of the last element. This problem is solved by transfer function Definition of Transfer Function: Page 14 of 116 SCE ELECTRICAL AND ELECTRONICS ENGINEERING IC- 6501 CONTROL SYSTEMS Transfer function of a LTIV system is defined as the ratio of the Laplace Transform of the output variable to the Laplace Transform of the input variable assuming all the initial condition as zero. Properties of Transfer Function: 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. The transfer functions of many physically different systems shall be identical. 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. 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 system. How you can obtain the transfer function (T. F.): Write the differential equation of the system. Take the L. T. of the differential equation, assuming all initial condition to be zero. Take the ratio of the output to the input. This ratio is the T. F. Mathematical Model of control systems A control system is a collection of physical object connected together to serve an objective. The mathematical model of a control system constitutes a set of differential equation. 1.5 Synchros A commonly used error detector of mechanical positions of rotating shafts in AC control systems is the Synchro. It consists of two electro mechanical devices. Synchro transmitter Synchro receiver or control transformer. The principle of operation of these two devices is sarne but they differ slightly in their construction. The construction of a Synchro transmitter is similar to a phase alternator. The stator consists of a balanced three phase winding and is star connected. The rotor is of dumbbell type construction and is wound with a coil to produce a magnetic field. When a no voltage is applied to the winding of the rotor, a magnetic field is produced. The coils in the stator link with this sinusoidal distributed magnetic flux and voltages are induced in the three coils due to transformer action. Than the three voltages are in time phase with each other and the rotor voltage. The magnitudes of the voltages are proportional to the cosine of the angle between the rotor position and the respective coil axis. The position of the rotor and the coils are shown in Fig. Page 15 of 116 SCE ELECTRICAL AND ELECTRONICS ENGINEERING IC- 6501 CONTROL SYSTEMS When 90 the axis of the magnetic field coincides with the axis of coil S2 and maximum voltage is induced in it as seen. For this position of the rotor, the voltage c, is zero, this position of the rotor is known as the 'Electrical Zero' of die transmitter and is taken as reference for specifying the rotor position. In summary, it can be seen that the input to the transmitter is the angular position of the rotor and the set of three single phase voltages is the output. The magnitudes of these voltages depend on the angular position of the rotor as given Hence Now consider these three voltages to he applied to the stator of a similar device called control transformer or synchro receiver. Page 16 of 116 SCE ELECTRICAL AND ELECTRONICS ENGINEERING IC- 6501 CONTROL SYSTEMS The construction of a control transformer is similar to that of the transmitter except that the rotor is made cylindrical in shape whereas the rotor of transmitter is dumbbell in shape. Since the rotor is cylindrical, the air gap is uniform and the reluctance of the magnetic path is constant. This makes the output impedance of rotor to be a constant. Usually the rotor winding of control transformer is connected teas amplifier which requires signal with constant impedance for better performance. A synchro transmitter is usually required to supply several control transformers and hence the stator winding of control transformer is wound with higher impedance per phase. Since the some currents flow through the stators of the synchro transmitter and receiver, the same pattern of flux distribution will be produced in the air gap of the control transformer. The control transformer flux axis is in the same position as that of the synchro transmitter. Thus the voltage induced in the rotor coil of control transformer is proportional to the cosine of the angle between the two rotors. 1.6 AC Servo Motors An AC servo motor is essentially a two phase induction motor with modified constructional features to suit servo applications. The schematic of a two phase or servo motor is shown o It has two windings displaced by 90 on the stator One winding, called as reference winding, is supplied with a constant sinusoidal voltage. The second winding, called control winding, is supplied with a variable control voltage o which is displaced by 90 out of phase from the reference voltage. The major differences between the normal induction motor and an AC servo motor are The rotor winding of an ac servo motor has high resistance (R) compared to its inductive reactance (X) so that its X / R ratio is very low. For a normal induction motor, X / R ratio is high so that the maximum torque is obtained in normal operating region which is around 5% of slip. The torque speed characteristics of a normal induction motor and an ac servo motor are shown in fig Page 17 of 116 SCE ELECTRICAL AND ELECTRONICS ENGINEERING IC- 6501 CONTROL SYSTEMS The Torque speed characteristic of a normal induction motor is highly nonlinear and has a positive slope for some portion of the curve. This is not desirable for control applications. as the positive slope makes the systems unstable. The torque speed characteristic of an ac servo motor is fairly linear and has negative slope throughout. The rotor construction is usually squirrel cage or drag cup type for an ac servo motor. The diameter is small compared to the length of the rotor which reduces inertia of the moving parts. Thus it has good accelerating characteristic and good dynamic response. The supplies to the two windings of ac servo motor are not balanced as in the case of a normal induction motor. The control voltage varies both in magnitude and phase with respect to the constant reference vulture applied to the reference winding. The direction of rotation of the motor depends on the phase (± 90°) of the control voltage with respect to the reference voltage. For different rms values of control voltage the torque speed characteristics are shown in Fig. The torque varies approximately linearly with respect to speed and also controls voltage. The torque speed characteristics can be linearised at the operating point and the transfer function of the motor can be obtained. DC Servo Motor A DC servo motor is used as an actuator to drive a load. It is usually a DC motor of low power rating. DC servo motors have a high ratio of starting torque to inertia and therefore they have a faster dynamic response. DC motors are constructed using rare earth permanent magnets which have high residual flux density and high coercively. As no field winding is used, the field copper losses am zero and hence, the overall efficiency of the motor is high. Page 18 of 116 SCE ELECTRICAL AND ELECTRONICS ENGINEERING IC- 6501 CONTROL SYSTEMS The speed torque characteristic of this motor is flat over a wide range, as the armature reaction is negligible. Moreover speed in directly proportional to the armature voltage for a given torque. Armature of a DC servo motor is specially designed to have low inertia. In some application DC servo motors are used with magnetic flux produced by field windings. The speed of PMDC motors can be controlled by applying variable armature voltage. These are called armature voltage controlled DC servo motors. Wound field DC motors can be controlled by either controlling the armature voltage or controlling rho field current. Let us now consider modelling of these two types or DC servo motors. (a) Armature controlled DC servo motor The physical model of an armature controlled DC servo motor is given in The armature winding has a resistance R a and inductance La. The field is produced either by a permanent magnet or the field winding is separately excited and supplied with constant voltage so that the field current If is a constant. When the armature is supplied with a DC voltage of e a volts, the armature rotates and produces a back e.m.f eb. The armature current ia depends on the difference of eb and en. The armature has a permanent of inertia J, frictional coefficient B0 The angular displacement of the motor is 8. The torque produced by the motor is given by Where K T is the motor torque constant. The back emf is proportional to the speed of the motor and hence The differential equation representing the electrical system is given by Taking Laplace transform of equation from above equation The mathematical model of the mechanical system is given by Page 19 of 116 SCE ELECTRICAL AND ELECTRONICS ENGINEERING IC- 6501 CONTROL SYSTEMS Taking Laplace transform Solving for (s),we get The block diagram representation of the armature controlled DC servo motor is developed in Steps Combining these blocks we have Usually the inductance of the armature winding is small and hence neglected Where Field Controlled Dc Servo Motor The field servo motor Page 20 of 116 SCE ELECTRICAL AND ELECTRONICS ENGINEERING

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