FINALS FEEDBACK PDF

1. A system that uses a measurement of the output and compares it with the desired output. Answer: Close-loop 2. To produce the desired transient response, we ____ parameters or _____components. Answer: adjust, design 3. A measure of the output of the system used for feedback to control the system. Answer: Feedback signal 4. The result of making judgment about how much compromise must be made between conflicting criteria. Answer: Trade-Off 5. A gap between the complex physical system and the design model intrinsic to the progression from the initial concept to the final product. Answer: Design Gap 6. A system configuration in which the control action is somehow dependent on the output. Answer: Control System 7. It is a shorthand, graphical representation of a physical system. illustrating the functional relationships among its components. Answer: Block Diagram 8. The ratio of physical output to physical input of an industrial process. Answer: Productivity 9. Uncertainties embodied in unintended consequences of a design. Answer: Risk 10. It is a measure of how quickly a first-order system responds to a unit step input. Answer: Time or Time constant of the system 11. It is sometimes called as reference Answer: Input 12. To reduce the steady state error, we___ and corrective actions, Answer: analyze 13. It is the simplest form of the block diagram Answer: single block 14. Control of an industrial process by automatic rather than manual means is called Answer: Automation 15. A system configuration in which the control action is independent of the output. Answer: Open-loop system 16. Elements of the systems are also called Answer: System components 17. It measures the output response and converts it into the form used by the controller Answer: Output transducer 18. A system with more than one input variable or more than one output variable. Answer: Multivariable control system 19. This response resembles the input and is usually what remains after the transients have decayed to zero Answer: Steady-state Response 20. It consists of subsystems and processes assembled to obtain a desired output with desired performance, given a specific unit Answer: Control System II. a. Given the mass- spring -damper system below: 1. Write the differential equation of the system 2 Write the equivalent Laplace Transform of the ODE 𝑥(𝑠) 3. Solve for the system's transfer function 𝐹(𝑥) b. Refer to question 3.a. If the system has a damping coefficient of 10 N/m and spring constant of 20 N/m. If the block weighs 1 kg. Solve for the following: 𝑥(𝑠) 1 = 2 𝐹(𝑥) 𝑠 + 10𝑠 + 20 1. Open-loop process gain or steady state output value 2 Closed-loop process gain or steady state output value. 3 Open-loop steady-state error 4. Closed-loop steady-state error 1 – 0.0476 = 0.9524 Larger Steady State error c. Refer to question 3.b. Using a P controller to the system, solve for the following: 𝑥(𝑠) 1. Closed-loop system's transfer function 𝐹(𝑥) 2 Closed-loop system's process gain if kp = 300 3. New closed-loop steady-state error value. d. Refer to question 3.b. Using a PD controller to the system, solve for the following: 1. Closed-loop system's transfer function 2. Closed-loop system's process gain if kp = 300 & kd = 10 3. New closed-loop steady-state error value e. Refer to question 3 b. Using a Pl controller to the system solve for the following: 𝑥(𝑠) 1. Closed-loop system's transfer function 𝐹(𝑠) 2. Closed-loop system's process gain if kp = 300 & ki = 160 3. New closed-loop steady-state error value f. Refer to question 3 b. Using a PID controller to the system, solve for the following: 𝑥(𝑠) 1. Closed-loop system's transfer function 𝐹(𝑠) 2 Closed-loop system's process gain if kp=300, ki 160 & kd = 10 3. New closed - loop steady - state error value. g. [10 points] Determine the inverse Laplace transform of the function below s2 8𝑠+4 𝑠2 +6𝑠+13 FORMULA: ------------------------------------------------------------------------------------------------------------------ 𝑠+4 4 𝑠2 +3𝑠+2 4 4 3 4 4 -2 3 2 3 2 LESSON 1 EST7 FEEDBACK CONTROL SYSTEMS LECTURE TUPM-21-1726 BET-ET-NS-4A Student ID No. Section CIFRA, MA. ASHLEY S. 10/03/2024 Student’s Name Date of Submission EST7 – Feedback and Control System Control systems are a basic part of modern society. Various applications are all around us, such as the rockets fire, and the space transport lifts off to earth orbit, sprinkling cooling water, a metallic part is consequently machined, an independent vehicle conveying material to workstations in an aerospace plant glides along the floor seeking for its destination. These are just a few examples of the automatically controlled systems that we can make. Thus this module will introduce feedback and control systems. LESSON 1 – Introduction to Feedback & Control System Terminology OBJECTIVES Define a control system and describe some applications. Describe historical developments leading to modern-day control theory. Describe the basic features and configurations of control systems. Describe control systems analysis and design objectives. Describe a control system's design process. Describe the benefit of studying control systems. PRE-TEST True/False. Write true if the statement is correct and write false otherwise. 1. The flyball governor is generally agreed to be the first automatic feedback controller used in an industrial process. ANS: TRUE 2. A closed-loop control system uses a measurement of the output and feedback of the signal to compare it with the desired input. ANS: TRUE 3. Engineering synthesis and engineering analysis are the same. ANS: FALSE 4. The block diagram shown below is an example of a closed-loop feedback system. ANS: FALSE 5. A multivariate system is a system with more than one input and/or more than one output. ANS: TRUE EST7 – Feedback and Control System DISCUSSION Control System Definition A control system consists of subsystems and processes (or plants) assembled to obtain a desired output with desired performance, given a specified input. Figure 1.0 Simplified description of a control system Example: Elevator Figure 1.1 Elevator response Advantages of Control Systems We build control systems for four primary reasons: 1. Power Amplification 2. Remote Control 3. Convenience of input form 4. Compensation for disturbances Page | 7 EST7 – Feedback and Control System Figure 1.2 a. Early elevators were controlled by hand ropes or an elevator operator. b. One of two modern Duo-lift elevators makes its way up the Grande Arche in Paris System Configuration Open-loop system – One in which the control action is independent of the output. Closed-loop system – One in which the control action is somehow dependent on the output. Figure 1.3 Block diagrams of control systems: a. open-loop system; b. closed-loop system Page | 8 EST7 – Feedback and Control System Input – sometimes called as reference. Input transducer – converts the form of the input to that used by the controller. Controller – drives a process or plant. Process/Plant – driven by the controller Output – can be called the controlled variable Disturbances – are signals added to the controller and process outputs via summing junctions, which yields to the algebraic sum of their input signals using associated signs. Output transducer – measures the output response and converts it into the form used by the controller. Actuating signal – is the result when the output signal is subtracted from the input signal. Analysis and Design Objectives Analysis is the process by which a system's performance is determined. Design is the process by which a system's performance is created or changed. A control system is dynamic: It responds to an input by undergoing a transient response before reaching a steady-state response that generally resembles the input. Three major objectives of Systems Analysis and Design: ✓ Producing the desired transient response ✓ Reducing Steady State Error ✓ Achieving Stability Transient Response In the case of elevator, a slow transient response makes passengers impatient, whereas an excessively rapid response makes them uncomfortable. Too fast a transient response could cause permanent physical damage. To produce the desired transient response, we adjust parameters or design components. Steady-State Response This response resembles the input and is usually what remains after the transients have decayed to zero. (Accuracy of the steady-state response) Page | 9 EST7 – Feedback and Control System To reduce steady-state error, we analyze and then design corrective action. Stability Discussion of transient response and steady-state error is moot if the system does not have stability. The total response of a system is the sum of the natural response and the forced response. Total Response = Natural Response + Forced Response Natural Responses – describes the way the system dissipates or acquires energy. - dependent only on the system, not the input. Forced Response – dependent on the input For a Control System to be useful: Natural Response – must eventually approach zero thus leaving the forced response, or oscillate. If NR >> FR || The system is no longer controlled. Other considerations: Factors affecting hardware selection Examples: motors – fulfill power requirements sensors – for accuracy Must be considered early in the design Finances Robust Design - the engineer wants to create a robust design so that the system will not be sensitive to parameter changes. Page | 10 EST7 – Feedback and Control System The Design Process Step 1: Example: Antenna Azimuth Position Control System − Weight and Physical Dimensions Step 2: Qualitative Description of the system into a functional block diagram that describes the parts of the system (i.e., function and/or hardware and shows their interconnection). Step 3: Consists of electrical, mechanical, and electromechanical components. Step 4: The designer uses physical laws, such as Kirchhoff’s laws for electrical networks and Newton’s Law for mechanical systems along with simplifying assumptions, to model the system mathematically. − Kirchhoff’s Voltage Law − Kirchhoff’s Current Law − Newton’s Laws. The sum of forces on a body equals zero − Kirchhoff’s and Newton’s Laws (Relationship between output and input of dynamic systems) − Linear, Time-invariant Differential Equation − Transfer Function – is another way of mathematically modeling a system. − State-Space Representation – they can also be used for systems that cannot be described by linear differential equations. Page | 11 EST7 – Feedback and Control System Step 5: Reduce the block diagram. Step 6: Analyze and Design. Analyze and design the system to meet specified requirements and specifications that include stability, transient response, and steady -state performance. Page | 12 EST7 – Feedback and Control System CHECKPOINT #1 Multiple Choice. Choose the letter that corresponds to the best and correct answer. C 1. A system with more than one input variable or more than one output variable is known by what name? a. Closed-loop feedback system b. Open-loop feedback system c. Multivariable control system d. Robust control system D 2. Closed-loop control systems should have which of the following properties: a. Good regulation against disturbances b. Desirable responses to commands c. Low sensitivity to changes in the plant parameters d. All of the above B 3. Complete the following sentence: Control of an industrial process by automatic rather than manual means is often called a. negative feedback b. automation c a design gap d. a specification A 4. Complete the following sentence: Control engineers are concerned with understanding and controlling segments of their environments, often called a. systems b. design synthesis c. trade-offs d. risk A 5. Complete the following sentence: An open-loop control system utilizes an actuating device to control a process a. without using feedback b. using feedback c. in engineering design d. in engineering synthesis EST7 – Feedback and Control System POST-TEST A. Review Questions 1. Research a brief history of human-designed control systems leading to modern- day control theory. Ancient Foundations The history of control systems begins with ancient Greeks, notably Ctesibius, who invented the clepsydra (water clock) around 270 BC, using a float mechanism to maintain a constant water level. The machine used a float to maintain a constant water level in the container. When the level dropped, a valve opened to refill it, ensuring a consistent outflow of water. This water then flowed into a second container, filling it proportionally to the elapsed time. The Industrial Revolution The Industrial Revolution marked a significant leap in control technology. James Watt's centrifugal governor (1769) revolutionized steam engine operation by regulating speed based on engine performance. This era saw the introduction of automatic control for temperature, pressure, and speed across various industrial applications, setting the stage for modern control principles. Mathematical Foundations Following the Industrial Revolution, the focus shifted to the mathematical description of control systems. By the mid-20th century, engineers began to analyze and synthesize control loops using mathematical models, enabling more precise system behavior predictions and designs. Technological Advances and WWII The invention of the telephone and the development of feedback operational amplifiers opened new avenues for control systems, emphasizing signal accuracy. During World War II, precision control systems were extensively developed for applications such as automated air traffic control and radar calibration, further enhancing the practical aspects of control engineering. The Computer Era The advent of computers in the latter half of the 20th century transformed control systems, allowing for real-time integration and complex computations within control loops. This integration led to the design of industrial robots capable of performing precision tasks and adapting to various environments, marking a new phase in automation. Modern Developments Today, control theory emphasizes robust and reliable systems, focusing on adaptability and non-linear dynamics. The emergence of distributed control systems (DCS) enables coordinated operations across multiple locations, while advancements in connectivity and intelligent systems continue to redefine the field, positioning control engineering at the forefront of technological innovation. 2. Give some examples of open-loop systems. Washing Machine: Operates on a preset cycle time regardless of the cleanliness of the clothes Bread Toaster: Toasts bread for a predetermined time set by the user, irrespective of the toast's actual readiness Traffic Control System: Operates on a fixed schedule for traffic lights without adjusting to real-time traffic conditions Electric Bulb: Lights up when switched on, independent of its temperature or any other conditions Sprinkler System: Watering occurs based on a timer rather than soil moisture levels Electric Clothes Dryer: Runs for a set time based on user input, regardless of whether clothes are dry or damp 3. Physically, what happens to an unstable system? In an unstable system, when you give it an input like a unit step, the output keeps increasing without stopping, which can lead to saturation at maximum limits. This can cause parts to overheat or fail. The system may also experience positive feedback loops that amplify responses, making it highly sensitive to disturbances and unpredictable in nature, ultimately compromising stability and safety. In short, if the physical system becomes unstable, then it would destroy itself. 4. Instability is attributable to what part of the total response? Instability comes mainly from the transient response of a system. This is how the system reacts to changes before it settles down. In unstable systems, this response can keep growing or fluctuate wildly, while stable systems eventually calm down and reach a steady state. 5. Adjustments of the forward path gain can cause changes in the transient response. True or false? True. Changing the forward path gain can affect how a system responds over time. If you increase the gain, the system may react faster but might overshoot its target. If you decrease it, the response will be slower and more stable. So, adjusting the gain can change how the system behaves when it gets a new input. 6. Name three approaches to the mathematical modeling of control systems. a. Differential Equation Model It is a time domain mathematical model of control system which describe the relationship between input and output of a control system in terms of the rates of change of system variables.These models can be used to analyse the stability of the system and its transient response. b. Transfer Function Model It is an s-domain mathematical model which describe the relationship between input and output of a control system in terms of complex frequency domain (s-domain) functions. These models can be used to analyse the frequency response of the system and design controllers. c. State Space Model It is a mathematical representation of a physical system as a set of input, output and state variables related by first-order differential equations. Hence, we can say that state space models describe the behavior of a control system in terms of a set of first-order differential equations. B. In the following Word Match problems, match the term with the definition by writing the correct letter per item. a. Optimization The output signal is fed back so that it subtracts from the input signal. P b. Risk A system that uses a measurement of the output and compares it with the desired output. F c. Complexity of design A set of prescribed performance criteria. H d. System A measure of the output of the system used for feedback to control the system. K e. Design A system with more than one input variable or more than one output variable. M f. Closed-loop feedback The result of making a judgment about how much control system compromise must be made between conflicting criteria. Q g. Flyball governor An interconnection of elements and devices for a desired purpose. D h. Specifications A reprogrammable, multifunctional manipulator is used for a variety of tasks. L i. Synthesis A gap between the complex physical system and the design model intrinsic to the progression from the initial N concept to the final product. EST7 – Feedback and Control System j. Open-loop The intricate pattern of interwoven parts and knowledge control system are required. C k. Feedback signal The ratio of physical output to physical input of an industrial process. R 1. Robot The process of designing a technical system. S m. Multivariable A system that utilizes a device to control the process control system without using feedback. J n. Design gap Uncertainties embodied in the unintended consequences of a design. B o. Positive feedback The process of conceiving or inventing the forms, parts, and details of a system to achieve a specified purpose. E p. Negative feedback The device, plant, or system under control. T q. Trade-off The output signal is fed back so that it adds to the input signal. O r. Productivity An interconnection of components forming a system configuration that will provide a desired response. U s. Engineering design The control of a process by automatic means. V t. Process The adjustment of the parameters to achieve the most favorable or advantageous design. A u. Control system The process by which new physical configurations are created. I v. Automation A mechanical device for controlling the speed of a steam engine. G C. Control Problems 1. In the past, control systems used a human operator as part of a closed-loop control system. Sketch the block diagram of the valve control system shown in the figure below. Page | 15 EST7 – Feedback and Control System BLOCK DIAGRAM OF THE VALVE CONTROL SYSTEM 2. The potential of employing two or more helicopters for transporting payloads that are too heavy for a single helicopter is a well-addressed issue in the civil and military rotorcraft design arenas. Overall requirements can be satisfied more efficiently with a smaller aircraft by using multi lift for infrequent peak demands. Hence the principal motivation for using multi lift can be attributed to the promise of obtaining increased productivity without having to manufacture larger and more expensive helicopters. A specific case of a multi lift arrangement, where two helicopters jointly transport payloads has been named twin lift. shows a typical "two-point pendant" twin lift configuration in the lateral/vertical plane. Develop the block diagram describing the pilots' action, the position of each helicopter, and the position of the load. BLOCK DIAGRAM OF TWIN LIFT HELICOPTER CONTROL SYSTEM Page | 16 Introduction to control system: What is Control System? In most systems there will be an input and an output as shown in the following block diagram. (Control system designers and engineers use block diagrams to represent systems). Signals flow from the input, through the system and produce an output.  The input will usually be an ideal form of the output. In other words the input is really what we want the output to be. It's the desired output.  The output of the system has to be measured. In the figure below, we show the system we are trying to control -the "plant"- and a sensor that measures what the controlled system is doing.  The input to the plant is usually called the control effort, and the output of the sensor is usually called the measured output, as shown below in the figure. If we want to control the output, we first need to measure the output. Within the whole system is the system we want to control - the plant - along with a sensor that measures what the output actually is.  In our block diagram representation, we show the output signal being fed to the sensor which produces another signal that is dependent upon the output.  A sensor, which produces a voltage proportional to temperature - if the output signal is a temperature. We need the sensor in the system to measure what the system is doing. To control the system we need to use the information provided by the sensor.  Usually, the output, as measured by the sensor is subtracted from the input (which is the desired output) as shown below. That forms an error signal that the controller can use to control the plant.  The device which performs the subtraction to compute the error, E, is a comparator. Finally, the last part of this system is the controller. Definitions System – An interconnection of elements and devices for a desired purpose. Control System – An interconnection of components forming a system configuration that will provide a desired response. Process – The device, plant, or system under control. The input and output relationship represents the cause-and-effect relationship of the process. Input Process Output Controlled Variable– It is the quantity or condition that is measured and Controlled. Normally controlled variable is the output of the control system. Manipulated Variable– It is the quantity of the condition that is varied by the controller so as to affect the value of controlled variable. Control – Control means measuring the value of controlled variable of the system and applying the manipulated variable to the system to correct or limit the deviation of the measured value from a desired value. Manipulated Variable Input or Output Set point Controller Process Or or Controlled reference Variable Disturbances– A disturbance is a signal that tends to adversely affect the value of the system. It is an unwanted input of the system. If a disturbance is generated within the system, it is called internal disturbance. While an external disturbance is generated outside the system. Types of Control System Manual Control Systems – Room Temperature regulation Via Electric Fan – Water Level Control Automatic Control System – Room Temperature regulation Via A.C – Human Body Temperature Control Types of Control System Open-Loop Control Systems utilize a controller or control actuator to obtain the desired response. Output has no effect on the control action. In other words output is neither measured nor fed back. Input Output Controller Process Open-Loop Control Systems Practical Examples of Open Loop Control System: 1.Automatic Washing Machine – This machine runs according to the pre-set time irrespective of washing is completed or not. 2. Bread Toaster - This machine runs as per adjusted time irrespective of toasting is completed or not. 3. Automatic Tea/Coffee Maker – These machines also function for pre adjusted time only. Advantages of Open Loop Control System: 1. Simple in construction and design. 2. Economical. 3. Easy to maintain. 4. Generally stable. 5. Convenient to use when output is difficult to measure. Disadvantages of Open Loop Control System: 1. They are inaccurate. 2. They are unreliable. 3. Any change in output cannot be corrected automatically. Closed-Loop Control Systems utilizes feedback to compare the actual output to the desired output response. Input Output Comparator Controller Process Measurement Closed-Loop Control Systems Practical Examples of Closed Loop Control System: 1. Automatic Electric Iron – Heating elements are controlled by output temperature of the iron. 2. Servo Voltage Stabilizer – Voltage controller operates depending upon output voltage of the system. 3. Water Level Controller– Input water is controlled by water level of the reservoir. 4. Missile Launched & Auto Tracked by Radar – The direction of missile is controlled by comparing the target and position of the missile. Advantages of Closed Loop Control System: 1. Closed loop control systems are more accurate even in the presence of non-linearity. 2. Highly accurate as any error arising is corrected due to presence of feedback signal. 3. Bandwidth range is large. 4. Facilitates automation. 5. The sensitivity of system may be made small to make system more stable. 6. This system is less affected by noise. Disadvantages of Closed Loop Control System: 1. They are costlier. 2. They are complicated to design. 3. Required more maintenance. 4. Feedback leads to oscillatory response. 5. Overall gain is reduced due to presence of feedback. 6. Stability is the major problem and more care is needed to design a stable closed loop system. Comparison of Closed Loop and Open Loop Control System: No. Open loop control system Closed loop control system 1 The feedback element is absent. The feedback element is always present. 2 An error detector is not present. An error detector is always present. 3 It is stable one. It may become unstable. 4 Easy to construct. Complicated construction. 5 It is an economical. It is costly. 6 Having small bandwidth. Having large bandwidth. 7 It is inaccurate. It is accurate. 8 Less maintenance. More maintenance. 9 It is unreliable. It is reliable. 10 Examples: Hand drier, tea maker Examples: Servo voltage stabilizer, perspiration Feedback Control System: A system that maintains a prescribed relationship between the output and some reference input by comparing them and using the difference (i.e. error) as a means of control is called a feedback control system. Feedback can be positive or negative. Input error +- Controller Process Output Feedback Effect of Feedback: As can be seen in figure below, which represents feedback system where: R = Input signal, E = Error signal, G = forward path gain H = Feedback gain, C = Output signal, B = Feedback signal 1. Error between system input and system output is reduced. 2. System gain is reduced by a factor 1/(1 ± GH). 3. Improvement in sensitivity. 4. Stability may be affected. 5. Improve the speed of response. LESSON 2 Introduction Block diagram is a shorthand, graphical representation of a physical system, illustrating the functional relationships among its components. A Block Diagram is a shorthand pictorial representation of the cause-and-effect relationship of a system. The simplest form of the block diagram is the single block, with one input and one output. The interior of the rectangle representing the block usually contains a description of or the name of the element, or the symbol for the mathematical operation to be performed on the input to yield the output. The arrows represent the direction of information or signal flow. ▸ The operations of addition and subtraction have a special representation. ▸ The block becomes a small circle, called a summing point, with the appropriate plus or minus sign associated with the arrows entering the circle. ▸ Any number of inputs may enter a summing point. The output is the algebraic sum of the inputs. ▸ Some books put a cross in the circle. Components of a Block Diagram for a Linear Time Invariant System ‣ System components are alternatively called elements of the system. ▸ Block diagram has four components: a. Signals b. System/block c. Summing junction d. Pick-off/ Take-off point ‣ In order to have the same signal or variable be an input to more than one block or summing point, a takeoff point is used. ▸ Distributes the input signal, undiminished, to several output points. ▸ This permits the signal to proceed unaltered along several different paths to several destinations. Example-1 ‣ Consider the following equations in which X1, X2, X3, are variables, and a1, a2 are general coefficients or mathematical operators. X3 = а1 x1 + а2 x2 – 5 Example-2 ‣ Consider the following equations in which X1, X2,. Xn, are variables, and a¡, a1⁄2,..., an, coefficients or mathematical operators. xn=a1x1 + a2x2 + an-1Xn-1 CASCADE Any finite number of blocks in series may be algebraically combined by multiplication of transfer functions. That is, n components or blocks with transfer functions G1, G2,..., Gn connected in cascade are equivalent to a single element G with a transfer function given by Topologies We will now examine some common topologies for interconnecting subsystems and derive the single transfer function representation for each of them. These common topologies will form the basis for reducing more complicated systems to a single block. Parallel Form: ▸ Parallel subsystems have a common input and an output formed by the algebraic sum of the outputs from all of the subsystems. Feedback Form: ▸ The third topology is the feedback form. Let us derive the transfer function that represents the system from its input to its output. The typical feedback system, shown in figure: The system is said to have negative feedback if the sign at the summing junction is negative and positive feedback if the sign is positive. Characteristic Equation ▸The control ratio is the closed loop transfer function of the system. The denominator of closed loop transfer function determines the characteristic equation of the system. Which is usually determined as: Canonical Form of a Feedback Control System G = direct transfer function = forward transfer function H = feedback transfer function GH = loop transfer function = open-loop transfer function C/R = closed-loop transfer function = control ratio E/R = actuating signal ratio = error ratio B/R primary feedback ratio REDUCTION TECHNIQUES 1 2 3 4 5 6 7 Another Example: Given: LESSON 3 SPLITTING THE CONTROLLER THE PRESENT This part of the controller is only concerned with what the error is now Let's take a simple law: let the control signal be proportional to the error: u = kp × e PROPORTIONAL CONTROL This is what is referred to as proportional control. The control action at any instant is the same as a constant times the error at the same instant The constant kp, is the Proportional Gain, and is the first of our controller's parameters IS PROPORTIONAL CONTROL ENOUGH? Intuitively it seems like it should be fine on its own: when the error is big, the control input is big to correct it. As the error reduced so does the control input. But there are problems... STEADY STATE ERROR (2ND PROBLEM) This problem is normally called steady state error It's a confusing name. The issue is just that the controller can't produce any output when the error is zero. Easiest to see for tank level control: If there is a constant flow out of the tank, the controller must provide the same flow in, while the level is at the setpoint. P CONTROL PROBLEM SUMMARY Problem 1: oscillations o P control will give us oscillations in some processes, regardless of the value of the gain parameter. Problem 2: steady state error o P control cannot give us a nonzero value of the control at zero error for some types of process SOLVING P CONTROL'S PROBLEM How to get rid of steady state error? Let's ignore the present for the moment and concentrate on what has happened in the past. SOLVING THE STEADY STATE ERROR We can examine how the controller error has evolved in the past If we sum up the past values of the error, we can get a value that increases when there is a constant error. INTEGRAL ACTION We can let the control be given by the sum of the past values of the error, scaled by some gain. In continuous time the sum is the integral: PROPORTIONAL AND INTEGRAL CONTROLLER We solved the steady state error by adding integral action (summing the past) How can we solve the oscillation problem? Let's look at the future! HOW DO WE PREDICT THE FUTURE OF THE ERROR? Look at its gradient! If the gradient (the time derivative) of the error is in a direction that makes the error smaller, we can reduce the control input. DAMPING It can be easier to think of this as damping, something that resists velocity Think of the wheel on your car… The spring is a proportional controller for the wheel position. The damper adds a derivative action by opposing the velocity of the wheel. DERIVATIVE ACTION Let’s let the control be dependent on the derivative of the error: Here kd is the derivative gain. Let’s again split this into KpTd, where Td is the derivative time. DERIVATIVE TIME Why do we want Td as a parameter? We can think of it as how far ahead we want to predict. PID CONTROLLER TECHNIQUES IN SOLVING SYSTEM EQUATIONS General Form of the 1st Order System 3 Transient Response Specifications o Time Constant ✓ Time it takes for the step response to rise up to 63% of its final value. ✓ Referred to as t=1/a ✓ The reciprocal of time constant is frequency or unit/second o Rise Time ✓ Time for the waveform to go from 10% to 90% of its final value. o Settling Time ✓ Time for the response to reach and stay within 2% of its final value. FIRST ORDER CONTROL SYSTEM Open Loop Step Response o If our step response looks something like this, we have a stable system with a first order response. Process Gain o This is the ratio of the change in MV to the change in control input. Time Constant o This is the ‘speed’ of the process. o To read it from the trend, look for the time it takes for the MV to rise 63% of its final value o Ok why 63% ? Time Delay o This is simpler, it is just the time it takes from the start of the step until the MV starts to move. FIRST ORDER PROCESS MODEL We have 3 parameters that determine behavior K determines how big the output change will be T determines how long the process takes to get there 𝜏 detemines how long it takes before the process starts doing anything at all IMPLICATIONS FOR CONTROL The key feature is the relation between T and 𝜏 If 𝜏 is small and T is big, control PI is fine ( 𝜏 < T) If 𝜏 is similar in size to T, we may need a more complex controller like PID (𝜏 = T) If 𝜏 is big relative to T, PID will struggle it will integrate during the delay and then see a large jump! (𝜏>T) FIRST ORDER CONTROL SYSTEM Example: Where: K is the DC Gain T is the time constant of the system (the time constant is a measure of how quickly a first-order system responds to a unit step input) GENERALIZED CLOSED LOOP CONTROL Reference = R Error = E Controlled Variable = C Measured Variable = Cm LESSON 4 Online Class Feedback Control Systems, EST7 Engr. Rommel Aunario Feedback Control Systems 202110-20 Laplace Transform Steps involved in using the Laplace Transform Differential Equation  Transform differential equation to algebraic equation. L[ f (t )] F ( s ) F ( s )  f (t )e dt  st 0 Solve equation by algebra. 1 L [ F ( s )]  f (t ) Determine inverse transform. Solution Feedback Control Systems 202110-20 Laplace Transform Basic Theorems of Linearity L[ Kf (t )] KL[ f (t )] KF ( s ) L[ f1 (t )  f 2 (t )] L[ f1 (t )]  L[ f 2 (t )] F1 ( s )  F2 ( s ) The Laplace transform of a product is not the product of the transforms. L[ f1 (t ) f 2 (t )]  F1 ( s ) F2 ( s ) Feedback Control Systems 202110-20 Laplace Transform Illustration of Unit Step Function u (t ) 1 0 t Feedback Control Systems 202110-20 Table 10-1. Common transform pairs. f (t ) F ( s )  L [ f ( t )] 1 or u ( t ) 1 T-1 ut is s e t 1 T-2 e-at 1SA s  sin  t sinat  T-3 wisiw s  2 2 cos  t cosot s T-4 sisiw s2   2 e   t sin  t e-at si nat  T-5* ( s   ) 2   2 wa saw e   t cos  t e-at cosot s  T-6* sa saw (s   )2   2 t te 1 T-7 is-s s2 tn tin n! T-8 nisinil s n 1 e tt n n! T-9 eat tin nisanil ( s   ) n 1  (t ) delta 1 one T-10 *Use when roots are complex. 5 Laplace Transform Example: Derive the Laplace transform of the unit step function.  F ( s )  (1)e dt  st 0  st  0 e  e  1 F ( s)   0    s 0  s s Feedback Control Systems 202110-20 Laplace Transform Example: Derive the Laplace transform of the exponential function.  t f (t ) e   F ( s )  e e dt  e   t  st  (  s ) t dt 0 0  ( s  ) t  0 e  e   0   (s   )  0  (s   ) 1  s  Feedback Control Systems 202110-20 Laplace Transform Common Transform Pairs Example. A force in newtons (N) is given below. Determine the Laplace transform. 50 f (t ) 50u (t ) F (s)  s Feedback Control Systems 202110-20 Laplace Transform Common Transform Pairs Example. A voltage in volts (V) starting at t = 0 is given below. Determine the Laplace transform.  2t v(t ) 5e sin 4t 4 V ( s ) L[v(t )] 5  ( s  2) 2  (4) 2 20 20  2  2 s  4s  4  16 s  4 s  20 Feedback Control Systems 202110-20 Laplace Transform Common Transform Pairs Example. A pressure in pascals (p) starting at t = 0 is given below. Determine the Laplace transform.  4t p (t ) 5cos 2t  3e s 1 P ( s ) L[ p (t )] 5  2 2  3 s  (2) s4 5s 3  2  s 4 s4 Feedback Control Systems 202110-20 Inverse Laplace Transform by Identification When a differential equation is solved by Laplace transforms, the solution is obtained as a function of the variable s. The inverse transform must be formed in order to determine the time response. The simplest forms are those that can be recognized within the tables and a few of those will now be considered. Feedback Control Systems, 202110-20 Laplace Transform Common Transform Pairs Example. Determine the inverse transform of the function below 5 12 8 F (s)   2  s s s 3  3t f (t ) 5  12t  8e Feedback Control Systems 202110-20 Laplace Transform Common Transform Pairs Example. Determine the inverse transform of the function below 200 V ( s)  2 s  100  10  V ( s ) 20  2 2   s  (10)  v(t ) 20sin10t Feedback Control Systems 202110-20 Laplace Transform Example. Determine the inverse transform of the function below 8s  4 When the denominator contains a V (s)  2 quadratic, check the roots. s  6s  13 In this case, the roots are: If they are real, a partial fraction s1,2  3 2i Complex expansion will be required. number If they are complex, the table may be used. Feedback Control Systems 202110-20 Laplace Transform Example. Determine the inverse transform of the function below 8s  4 2 V (s)  2 s  6 s  13 s  6s  13 2 2 2 s  6 s  (3)  13  (3) 2 s  6 s  9  4 2 2 ( s  3)  (2) Feedback Control Systems 202110-20 Laplace Transform Example. Determine the inverse transform of the function below 8( s  3) 4  24 V (s)  2 2  2 2 ( s  3)  (2) ( s  3)  (2) 8( s  3) 10(2)  2 2  2 2 ( s  3)  (2) ( s  3)  (2)  3t  3t v(t ) 8e cos 2t  10e sin 2t Feedback Control Systems 202110-20 Laplace Transform Example. Determine the inverse transform of the function below s 6 s 6 When the denominator F ( s)  2  contains a quadratic, s  3s  2 ( s  1)( s  2) check the roots. In this case, the roots are: If they are real, a real number partial fraction expansion will be required. If they are complex, the table may be used. Feedback Control Systems 202110-20 Laplace Transform Example. Determine the inverse transform of the function below s 6 A1 A2 F ( s)    ( s  1)( s  2) s  1 s  2 s 6  1 6 A1 ( s  1) F ( s )  s  1   5 s  2  s  1  1  2 Feedback Control Systems 202110-20 Laplace Transform Example. Determine the inverse transform of the function below s 6 A1 A2 F ( s)    ( s  1)( s  2) s  1 s  2 s 6  26 A2 ( s  2) F ( s )  s  2    4 s  1  s  2  2  1 Feedback Control Systems 202110-20 Laplace Transform Example. Determine the inverse transform of the function below s 6 A1 A2 5 4 F ( s)    F ( s)   ( s  1)( s  2) s  1 s  2 s 1 s  2 t  2t f (t ) 5e  4e Feedback Control Systems 202110-20 Common Transform Pairs f (t ) F ( s )  L [ f ( t )] 1 or u ( t ) 1 T-1 s e t 1 T-2 s  sin  t  T-3 s  2 2 cos  t s T-4 s2   2 e   t sin  t  T-5* (s   )2   2 e   t cos  t s  T-6* (s   )2   2 t 1 T-7 s2 tn n! T-8 s n 1 e tt n n! T-9 ( s   ) n 1  (t ) 1 T-10 *Use when roots are complex. Feedback Control Systems 202110-20 FEEDBACK CONTROL Feedback Control seeks to bring the measured quantity to its desired value or set- point (also called reference trajectory) automatically. There are two possible causes of difference between measured value and desired value: a) Disturbance and noise b) Change of set point, where the control system must act to bring the measured quantity to the new set point DIFFERENT TYPES OF FEEDBACK CONTROL On-Off Control o This is the simplest form of control Proportional Control o A proportional control attempts to perform better than on-off type by applying power in proportion to the difference in temperature between the measured and the set point. Proportional Derivative Control o The stability and overshoot problems that arise when a proportional controller is used at high gain can be mitigated by adding a term proportional to the time derivative of the error signal. Proportional Integral Derivative Control o Cures the steady state error. PID control eliminates this while using relatively low gain by adding an integral term to the control function. Tips for Designing a PID Controller Obtain an open loop response and determine what needs to be improved. Add a proportional control to improve the rise time. Add a derivative control to control the overshoot. Add an integral control to eliminate the steady state error. Adjust each Kp, Ki, and Kd until you obtained a desired overall response. The Three Term Controller: PID Controller The PID Controller is the most popular feedback control algorithm used in process control and industries. It is robust yet simple, easy to understand, and can provide excellent control performance despite the variation of dynamic characteristics of the process. As the name suggests, the PID Controller consists of three basic terms: the Proportional term, the Integral term, and the Derivative term. Each term can be activated separately (P, I, or D) or combined to obtain the desired controller. Depending on the characteristics of the process, the following controller are generally used: P, PI, PD, or PID. PROPORTIONAL TERM INTERGRAL TERM DERIVATIVE TERM PID CONTROL Effect of Each Term Problem Example Problem Example Problem Example: Open-Loop Problem Example: Open-Loop Problem Example: Closed-Loop Problem Example: Closed-Loop Problem Example: Using P-Controller Problem Example: Using P-Controller Problem Example: Using PD-Controller Problem Example: Using PD-Controller Problem Example: Using PI-Controller Problem Example: Using PI-Controller Problem Example: Using PID-Controller Problem Example: Using PID-Controller Problem Example Conclusion Control of 1st Order System Using P-Controller Control of 1st Order System Using P-Controller Control of 1st Order System Using PI-Controller Control of 1st Order System Using PI-Controller Control of 2nd Order System Using P-Controller Control of 2nd Order System Using P-Controller Control of 2nd Order System Using P-Controller Control of 2nd Order System Using P-Controller Control of 2nd Order System Using PD-Controller Control of 2nd Order System Using PD-Controller

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