DOCSLIB.ORG

Basics of Automation and Control I

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Basics of Automation and Control I Three-term controller Ziegler-Nichols tuning rules Summary Basics of Automation and Control I Lecture 13: PID controllers Paweł Malczyk Division of Theory of Machines and Robots Institute of Aeronautics and Applied Mechanics Faculty of Power and Aeronautical Engineering Warsaw University of Technology © Paweł Malczyk. Basics of Automation and Control I Lecture 13: PID controllers 1 / 35 Three-term controller Ziegler-Nichols tuning rules Summary Outline 1 Three-term controller 2 Ziegler-Nichols tuning rules 3 Summary © Paweł Malczyk. Basics of Automation and Control I Lecture 13: PID controllers 2 / 35 Three-term controller Ziegler-Nichols tuning rules Summary Three-term controller 1 Three-term controller Introduction Basic control functions P controller PI controller PD controller PID controller 2 Ziegler-Nichols tuning rules 3 Summary © Paweł Malczyk. Basics of Automation and Control I Lecture 13: PID controllers 3 / 35 Three-term controller Ziegler-Nichols tuning rules Summary Introduction Fig. 1: Block diagram of a process with feedback controller 1 The three-term controller, i.e. proportional-integral-derivative (PID) controller, is a control loop feedback mechanism widely used in many industrial control systems. 2 PID controllers appear in many different forms: as stand-alone controllers, as part of hierarchical, distributed control systems and built into embedded components. 3 By tuning the three parameters in the PID controller algorithm, the controller can provide control action designed for specific process requirements (benign process dynamics, moderate performance). © Paweł Malczyk. Basics of Automation and Control I Lecture 13: PID controllers 4 / 35 Three-term controller Ziegler-Nichols tuning rules Summary Basic control functions Fig. 2: Block diagram of feedback control system with ideal PID controller The input/output relation for an ideal PID controller*: [ ∫ ] 1 t de(t) u(t) = k e(t) + e(τ)dτ + T = p T d dt ∫i 0 (1) t de(t) = kpe(t) + ki e(τ)dτ + kd 0 dt e(t) = r(t) − y(t) – error signal, u(t) – control signal, kp – proportional gain, Ti – integral time, Td – derivative time, k = kp – integral gain, k = k T – derivative gain. i Ti d p d * Modified versions of the PID controller are used in practical applications. © Paweł Malczyk. Basics of Automation and Control I Lecture 13: PID controllers 5 / 35 Three-term controller Ziegler-Nichols tuning rules Summary Basic control functions Ideal PID controller: [ ∫ ] t 1 de(t) u(t) = kp e(t) + e(τ)dτ + T T d dt i 0 ( ) (2) ( ) → U s 1 C(s) = = kp 1 + + Tds E(s) Tis Fig. 3: Block diagram of ideal PID controller • P controller (Ti = ∞, Td = 0) C(s) = kp (3) • PI controller (Td = 0)( ) 1 C(s) = kp 1 + (4) Tis • PD controller (Ti = ∞) C(s) = kp(1 + Tds) (5) Fig. 4: PID interpretation → Gentle introduction to PID controllers: https://www.youtube.com/watch?v=4Y7zG48uHRo. © Paweł Malczyk. Basics of Automation and Control I Lecture 13: PID controllers 6 / 35 Three-term controller Ziegler-Nichols tuning rules Summary Mechanical system Ns N Fig. 5: Układ mechaniczny. Stałe fizyczne: m = 1 kg, b = 2 m , k = 5 m Goal: choose a control force f(t) (in N) such that the mass will stop at the desired position (say 1 m from equilibrium). Y(s) 1 Plant tranfer function G(s) = F(s) = ms2+bs+k . Reference signal (desired position): r(t) = yd · 1(t) (yd = 1 m). In steady-state the force f(t) need to balance the spring force, i.e.: f = k · yd · 1(t), then 1 yss = lim y(t) = lim sY(s) = lim sG(s)F(s) = lim sG(s)kyd = G(0)kyd = yd t→∞ s→0 s→0 s→0 s © Paweł Malczyk. Basics of Automation and Control I Lecture 13: PID controllers 7 / 35 Three-term controller Ziegler-Nichols tuning rules Summary Mechanical system with disturbance Fig. 6: Mechanical system. Physical constants: m = 1 kg, b = 2 Ns , k = 5 N , and g = 10 m m m s2 Y(s) 1 Plant transfer function G(s) = F(s) = ms2+bs+k . Disturbance – gravity force: d(t) = −mg · 1(t). Reference signal (desired position): r(t) = yd · 1(t) (yd = 1 m). In steady-state the force f(t) need to balance the sum of spring force and the gravity force, i.e.: u = (k · yd + mg) · 1(t). Assumption: perfect knowledge of parameters and disturbance... © Paweł Malczyk. Basics of Automation and Control I Lecture 13: PID controllers 8 / 35 Three-term controller Ziegler-Nichols tuning rules Summary Mechanical system with parametric uncertainty True values: m, b, k. Estimated params: m, b, k. Real parameters and disturbance signals are usually unknown. Fig. 7: Mechanical system with parametric uncertainty Control signal: u = (k · yd + mg) · 1(t). Input signal: f = (kyd + (m − m)g) · 1(t). Steady-state response: − 1 1 − yss = lim y(t) = lim sY(s) = lim sG(s)(kyd+(m m)g) = (kyd+(m m)g) t→∞ s→0 s→0 s k Error signal: 1 1 e = y − y = y − (ky + (m − m)g) = ((k − k)y + (m − m)g) ss ss ss d k d k d The greater the uncertainty is, the greater the steady-state error is. Transient-response is not controlled. Open-loop control is sensitive to parametric uncertainty. © Paweł Malczyk. Basics of Automation and Control I Lecture 13: PID controllers 9 / 35 Three-term controller Ziegler-Nichols tuning rules Summary Proportional control Fig. 8: Feedback control system Assumption: u(t) = kp(r(t) − y(t)) – force proportional to the error. Open-loop transfer function: L(s) = C(s)G(s). Y(s) L(s) kp Closed-loop transfer function: T(s) = = = 2 . R(s) 1+L(s) ms +bs+k+kp kp – virtual spring coefficient. Let us calculate the response y(t) and steady-state error e(t). © Paweł Malczyk. Basics of Automation and Control I Lecture 13: PID controllers 10 / 35 Three-term controller Ziegler-Nichols tuning rules Summary Proportional control Fig. 9: Feedback control system The response: Y(s) = T(s)R(s) + G(s)S(s)D(s). Sensitivity function: 1 ms2 + bs + k ( ) = = S s 2 1 + L(s) ms + bs + k + kp Load disturbance sensitivity function: 1 ( ) = ( ) ( ) = W s G s S s 2 ms + bs + k + kp · → yd Reference signal: r(t) = yd 1(t) R(s) = s − · → − mg Disturbance: d(t) = mg 1(t) D(s) = s . © Paweł Malczyk. Basics of Automation and Control I Lecture 13: PID controllers 11 / 35 Three-term controller Ziegler-Nichols tuning rules Summary Proportional control Fig. 10: Feedback control system Steady-state response: yd mg yss = lim y(t) = lim sY(s) = lim s(T(s) + W(s)(− )) = t→∞ s→0 s→0 s s − kp − 1 = ydT(0) mgW(0) = yd mg k + kp k + kp If k → ∞, then kp → 1 and 1 → 0. In consequence: y = y . p k+kp k+kp ss d Conclusion: → disturbance rejection (even unknown) and trajectory tracking. © Paweł Malczyk. Basics of Automation and Control I Lecture 13: PID controllers 12 / 35 Three-term controller Ziegler-Nichols tuning rules Summary P controller Fig. 11: P controller, C(s) = kp Effects of P controller Proportional term, P, causes a corrective control actuation proportional to the error. The system with P controller will usually have nonzero steady-state errors. As kp increases, then the static position error decreases. As kp increases, the stability margins decrease and the system may become unstable. As kp increases, the BW (as well as overshoot and settling time) increases. It is difficult to ensure both good transient response accuracy and steady- state performance. © Paweł Malczyk. Basics of Automation and Control I Lecture 13: PID controllers 13 / 35 Three-term controller Ziegler-Nichols tuning rules Summary P controller kp=30 40 1.4 30 1.2 20 10 1 r(t) 0 0.8 -10 -20 y(t) 0.6 kp=30 kp=25 -30 0.4 kp=20 Magnitude (dB) -40 0.2 kp=15 -50 -60 0 10-2 10-1 100 101 102 0 1 2 3 4 5 6 7 8 9 10 Frequency (rad/sec) t kp=30 0 30 25 -30 20 -60 15 10 -90 G C u(t) 5 kp=15 -120 L 0 kp=20 Phase (deg) S -5 kp=25 -150 T -10 kp=30 -180 -15 10-2 10-1 100 101 102 0 1 2 3 4 5 6 7 8 9 10 Frequency (rad/sec) t Fig. 12: P controller characteristics Comment 1 If kp ↗, then Mp ↗, tp ↘, ts ↗, ess ↘, stability issues. © Paweł Malczyk. Basics of Automation and Control I Lecture 13: PID controllers 14 / 35 Three-term controller Ziegler-Nichols tuning rules Summary P controller kp=30 40 1.4 30 1.2 20 10 1 r(t) 0 0.8 -10 y(t) 0.6 -20 kp=30 kp=25 Magnitude (dB) G -30 0.4 kp=20 C -40 L kp=15 S 0.2 -50 T -60 0 10-2 10-1 100 101 102 0 1 2 3 4 5 6 7 8 9 10 Frequency (rad/sec) t Fig. 13: P controller characteristics © Paweł Malczyk. Basics of Automation and Control I Lecture 13: PID controllers 15 / 35 Three-term controller Ziegler-Nichols tuning rules Summary PI controller ( ) 1 Fig. 14: PI controller, C(s) = kp 1 + Tis Effects of PI controller The integral term, I, gives a correction proportional to the integral of the error. PI largely reduces the steady-state errors (compared to P controller). It may cause the closed loop system less stable (or even unstable). PI may slow down the transient response.
Load more

Recommended publications
  • 53188593.Pdf
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by ethesis@nitr COMPARISION OF PERFORMANCE ANALYSIS OF DIFFERENT CONTROL STRUCTURES A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FORTHE DEGREE OF Bachelor of Technology in Electronics & Communication Engineering By SOMJIT SWAIN (108EI029) DEVENDRA SINGH MANDAVI (10607022) ARUP AVISHEK BEHERA (108EC039) Department of Electronics & Communication Engineering National Institute of Technology Rourkela 2012 COMPARISION OF PERFORMANCE ANALYSIS OF DIFFERENT CONTROL STRUCTURES A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FORTHE DEGREE OF Bachelor of Technology in Electronics & Communication Engineering By SOMJIT SWAIN (108EI029) DEVENDRA SINGH MANDAVI (10607022) ARUP AVISHEK BEHERA (108EC039) Under the guidance of Prof. T.K.Dan Department of Electronics & Communication Engineering National Institute of Technology Rourkela 2012 CERTIFICATE ===================================================================== This is to certify that the work in this thesis entitled “Comparison of Performance Analysis of Different Control Structures” by Somjit Swain, Devendra Singh Mandavi and Arup Avishek Behera has been carried out under my supervision in partial fulfillment of the requirements for the degree of Bachelor of Technology in „Electronics & Instrumentation‟ and „Electronics & Communication‟ during session 2008-2012 in the Department of Electronics and Communication Engineering, National Institute of Technology, Rourkela. Place: Dated: Prof. T.K.Dan Dept. of ECE National Institute of Technology, Rourkela Acknowledgement We place on record and warmly acknowledge the continuous encouragement, invaluable supervision, timely suggestions and inspired guidance offered by our guide Prof. T. K. DAN Professor, Department of Electronics and Communication Engineering, National Institute of Technology, Rourkela, in bringing this report to a successful completion.
    [Show full text]
  • Introduction to Building Automation Systems (BAS)
    3/11/2013 Introduction to Building Automation Systems (BAS) Ryan R. Hoger, LEED AP 708.670.6383 [email protected] Building Automation Systems z Centralized controls z Change scheduling for multiple HVAC units at same time z Monitor “health” of equipment z Internet accessible z Alarming via text msg or email z Collect/trend data z Integrate to lighting control or security system 1 3/11/2013 z DDC - Direct Digital Control of an HVAC system z A method of monitoring and controlling HVAC system performance by collecting, processing, and sending information using sensors, actuators, and microprocessors. What is DDC? z DDC is the concept or theory of HVAC system control that uses digital controls z Physically, DDC encompasses all the devices used to implement this control method: a whole group of DDC controllers/microprocessors, actuators, sensors, and other devices. What is DDC? 2 3/11/2013 DDC - the Control Theory input-process-output cycle z A point is ANY input or output device used to control the overall or specific performance of equipment or output devices related to the equipment. What Is a Point? 3 3/11/2013 AI z Analog input -a sensor that monitors physical data, such as temperature, flow, or pressure. DI z Discrete input -a sensor that monitors status. Momentary and maintained switches, ON-OFF equipment status, and digital pulses from flow and electric power meters are discrete inputs. AOz Analog output - a physical action of a proportional device in the controlled equipment - e.g., actuator opens air damper from 20% to 40%, other dampers, valves, inlet guide vanes, etc.
    [Show full text]
  • Control Theory
    Control theory This article is about control theory in engineering. For 1 Overview control theory in linguistics, see control (linguistics). For control theory in psychology and sociology, see control theory (sociology) and Perceptual control theory. Control theory is an interdisciplinary branch of engi- Smooth nonlinear trajectory planning with linear quadratic Gaussian feedback (LQR) control on a dual pendula system. Control theory is The concept of the feedback loop to control the dynamic behavior of the system: this is negative feedback, because the sensed value is subtracted from the desired value to create the error signal, • a theory that deals with influencing the behavior of which is amplified by the controller. dynamical systems • an interdisciplinary subfield of science, which orig- inated in engineering and mathematics, and evolved neering and mathematics that deals with the behavior of into use by the social sciences, such as economics, dynamical systems with inputs, and how their behavior psychology, sociology, criminology and in the is modified by feedback. The usual objective of con- financial system. trol theory is to control a system, often called the plant, so its output follows a desired control signal, called the Control systems may be thought of as having four func- reference, which may be a fixed or changing value. To do tions: measure, compare, compute and correct. These this a controller is designed, which monitors the output four functions are completed by five elements: detector, and compares it with the reference. The difference be- transducer, transmitter, controller and final control ele- tween actual and desired output, called the error signal, ment.
    [Show full text]
  • Control Theory Tutorial Basic Concepts Illustrated
    SPRINGER BRIEFS IN APPLIED SCIENCES AND TECHNOLOGY Steven A. Frank Control Theory Tutorial Basic Concepts Illustrated by Software Examples SpringerBriefs in Applied Sciences and Technology SpringerBriefs present concise summaries of cutting-edge research and practical applications across a wide spectrum of fields. Featuring compact volumes of 50– 125 pages, the series covers a range of content from professional to academic. Typical publications can be: • A timely report of state-of-the art methods • An introduction to or a manual for the application of mathematical or computer techniques • A bridge between new research results, as published in journal articles • A snapshot of a hot or emerging topic • An in-depth case study • A presentation of core concepts that students must understand in order to make independent contributions SpringerBriefs are characterized by fast, global electronic dissemination, standard publishing contracts, standardized manuscript preparation and formatting guidelines, and expedited production schedules. On the one hand, SpringerBriefs in Applied Sciences and Technology are devoted to the publication of fundamentals and applications within the different classical engineering disciplines as well as in interdisciplinary fields that recently emerged between these areas. On the other hand, as the boundary separating fundamental research and applied technology is more and more dissolving, this series is particularly open to trans-disciplinary topics between fundamental science and engineering. Indexed by EI-Compendex,
    [Show full text]
  • APJ15 L8.Pdf
    Controllers 1 PID controllers The most widespread 85% are PIDs, 15% are others. • easy of use (easy to tune) • robustness (works even if badly tuned) r e u R e = r − y u(t) = F [e(t)] − c Terms in industry: SV Set Value (r) SP Set Point PV Process/ Present Value(с) CV Control variable (u) MV Manipulated Variable Z de u(t) = K · e + K · e dt + K · (1) c i d dt Parameters Kc;Ki;Kd Components P, I, D 1 Lecture 8 ISS0080 Automation and Process Control 1.1 Controller’s components Proportional control control law u = Kc · e control signal proportional to the error, where Kc is a proportional gain/v˜oimendustegur/ коэффициент усиления Controller static characteristic is u(e) with respect to error e. Output value is limited 0 ::: 100% u 100% three parts: 8 >0% − min <> u = Kc · e − linear > :>100% − max 0% e Output signals types: (4 − 20) mA, 0-4000, 0-1, 0% ::: 100%. ◦ The gain Kc units are (if we deal with a thermal process) [%= C]. Proportional action responds only to a change in the magnitude of the error. The controller has an error between PV and SP called proportional-only offset / staatiline viga/ статическая ошибка. Proportional action will not return the PV to set point (PV 6= SP ). It will, however, return the PV to a value that is within a defined span around the PV [6]. To minimize the proportional-only offset we need to increase the gain of the controller gain (decreas- ing its proportional band). That makes controller more "aggressive".
    [Show full text]
  • EEEN20030 Control Systems I William Heath Revised January 2012
    EEEN20030 Control Systems I William Heath Revised January 2012 Contents Sections marked * are not examinable 1 Introduction 6 1.1 Feedbacksystems........................... 6 1.2 Historyoffeedbackcontrollers . 6 1.3 Example:automotivecontrol . 8 1.4 Scopeofthecourse.......................... 9 1.5 Limitationsofthecourse....................... 9 1.6 Furtherreading............................ 10 2 Feedback 12 2.1 Negativefeedbackamplifier . 12 2.2 Standardcontrolloop ........................ 12 2.3 Standardcontrolloopwithoutdynamics . 13 2.4 Stability and dynamics . 14 2.5 Casestudy:edibleoilrefining. 15 2.6 Keypoints .............................. 18 3 Differential equations 19 3.1 Lineardifferentialequations . 19 3.2 Example: electrical circuit . 20 3.3 Example:mechanicalsystem . 20 3.4 Example:flowsystem ........................ 21 3.5 Discussion............................... 22 3.6 Keypoint ............................... 23 4 Laplace transforms and transfer functions 24 4.1 Laplacetransformdefinition. 24 4.2 Laplacetransformexamples . 24 4.2.1 Unitstep ........................... 24 4.2.2 Exponentialfunction. 25 4.2.3 Sinusoid............................ 25 4.2.4 Dirac delta function . 26 4.3 Laplacetransformproperties . 26 4.3.1 Linearity ........................... 26 4.3.2 Differentiation . 26 4.3.3 Integration .......................... 27 4.3.4 Delay ............................. 27 4.3.5 Finalvaluetheorem ..................... 28 4.4 Transferfunctions .......................... 28 4.4.1 Definitions .......................... 28 4.4.2
    [Show full text]
  • PID Control Theory Tutorial the P Stands for Proportional Control, I for Integral Control and D for Derivative Control
    PID Control Theory Tutorial The P stands for proportional control, I for integral control and D for derivative control. This is also what is called a three term controller. The basic function of a controller is to execute an algorithm (electronic controller) based on the control engineer's input (tuning constants), the operators desired operating value (setpoint) and the current plant process value. In most cases, the requirement is for the controller to act so that the process value is as close to the setpoint as possible. In a basic process control loop, the control engineer utilises the PID algorithms to achieve this. The PID control algorithm is used for the control of almost all loops in the process industries, and is also the basis for many advanced control algorithms and strategies. In order for control loops to work properly, the PID loop must be properly tuned. Standard methods for tuning loops and criteria for judging the loop tuning have been used for many years, but should be reevaluated for use on modern digital control systems. While the basic algorithm has been unchanged for many years and is used in all distributed control systems, the actual digital implementation of the algorithm has changed and differs from one system to another. How a PID Controller Works The PID controllers job is to maintain the output at a level so that there is no difference (error) between the process variable (PV) and the setpoint (SP). In the diagram shown above the valve could be controlling the gas going to a heater, the chilling of a cooler, the pressure in a pipe, the flow through a pipe, the level in a tank, or any other process control system.
    [Show full text]
  • Chapter 6. PID Control
    6 PID Control 6.1 Introduction The PID controller is the most common form of feedback. It was an es- sential element of early governors and it became the standard tool when process control emerged in the 1940s. In process control today, more than 95% of the control loops are of PID type, most loops are actually PI con- trol. PID controllers are today found in all areas where control is used. The controllers come in many different forms. There are stand-alone sys- tems in boxes for one or a few loops, which are manufactured by the hundred thousands yearly. PID control is an important ingredient of a distributed control system. The controllers are also embedded in many special-purpose control systems. PID control is often combined with logic, sequential functions, selectors, and simple function blocks to build the complicated automation systems used for energy production, transporta- tion, and manufacturing. Many sophisticated control strategies, such as model predictive control, are also organized hierarchically. PID control is used at the lowest level; the multivariable controller gives the setpoints to the controllers at the lower level. The PID controller can thus be said to be the “bread and buttert’t’ of control engineering. It is an important component in every control engineer’s tool box. PID controllers have survived many changes in technology, from me- chanics and pneumatics to microprocessors via electronic tubes, transis- tors, integrated circuits. The microprocessor has had a dramatic influence on the PID controller. Practically all PID controllers made today are based on microprocessors. This has given opportunities to provide additional fea- tures like automatic tuning, gain scheduling, and continuous adaptation.
    [Show full text]
  • PID Control Theory
    9 PID Control Theory Kambiz Arab Tehrani1 and Augustin Mpanda2,3 1University of Nancy, Teaching and Research at the University of Picardie, INSSET, Saint-Quentin, Director of Power Electronic Society IPDRP, 2Tshwane University of Technology/FSATI 3ESIEE-Amiens 1,3France 2South Africa 1. Introduction Feedback control is a control mechanism that uses information from measurements. In a feedback control system, the output is sensed. There are two main types of feedback control systems: 1) positive feedback 2) negative feedback. The positive feedback is used to increase the size of the input but in a negative feedback, the feedback is used to decrease the size of the input. The negative systems are usually stable. A PID is widely used in feedback control of industrial processes on the market in 1939 and has remained the most widely used controller in process control until today. Thus, the PID controller can be understood as a controller that takes the present, the past, and the future of the error into consideration. After digital implementation was introduced, a certain change of the structure of the control system was proposed and has been adopted in many applications. But that change does not influence the essential part of the analysis and design of PID controllers. A proportional– integral–derivative controller (PID controller) is a method of the control loop feedback. This method is composing of three controllers [1]: 1. Proportional controller (PC) 2. Integral controller (IC) 3. Derivative controller (DC) 1.1 Role of a Proportional Controller (PC) The role of a proportional depends on the present error, I on the accumulation of past error and D on prediction of future error.
    [Show full text]
  • Controllers and Controlled Systems 1
    Technical Information Controllers and Controlled Systems 1 Generator Add PID PT1 PT1PT2 Time + y-t A AEAEAEA1 _ 2 3 PT1 AE Y t Part 1 Fundamentals Technical Information Part 1: Fundamentals Part 2: Self-operated Regulators Part 3: Control Valves Part 4: Communication Part 5: Building Automation Part 6: Process Automation Should you have any further questions or suggestions, please do not hesitate to contact us: SAMSON AG Phone (+49 69) 4 00 94 67 V74 / Schulung Telefax (+49 69) 4 00 97 16 Weismüllerstraße 3 E-Mail: [email protected] D-60314 Frankfurt Internet: http://www.samson.de Part 1 ⋅ L102EN Controller and Controlled Systems Controller and Controlled Systems . 3 Introduction . 5 Controlled Systems . 7 P controlled system . 8 I controlled system . 9 Controlled system with dead time . 11 Controlled system with energy storing components . 12 Characterizing Controlled Systems . 18 System response . 18 Proportional-action coefficient . 18 Nonlinear response. 19 Operating point (OP). 20 Controllability of systems with self-regulation . 21 Controllers and Control Elements . 23 Classification . 23 Continuous and discontinuous controllers . 24 Auxiliary energy . 24 Determining the dynamic behavior . 25 Continuous Controllers. 27 Proportional controller (P controller) . 27 99/10 Proportional-action coefficient . 27 ⋅ System deviation . 29 SAMSON AG CONTENTS 3 Fundamentals ⋅ Controllers and Controlled Systems Adjusting the operating point . 30 Integral controller (I controller) . 35 Derivative controller (D controller) . 38 PI controllers . 40 PID controller . 42 Discontinuous Controllers . 45 Two-position controller . 45 Two-position feedback controller . 47 Three-position controller and three-position stepping controller . 48 Selecting a Controller . 50 Selection criteria . 50 Adjusting the control parameters.
    [Show full text]