Process Dynamics and Control (419307), 7cr Kurt-Erik Häggblom 2. Basic control concepts

Process Dynamics and Control (419307), 7cr • Kurt-Erik Häggblom • 2. Basic control concepts • 2.1 Signals and systems • 2.2 Block diagrams • 2.3 From flow sheet to block diagram • 2.4 Control strategies • 2.5 Feedback control • Process • Control • Laboratory Process Dynamics and Control

2. Basic control concepts • 2.1 Signals and systems • A system can be defined as a combination of • components that act together to perform a • certain objective. Figure 2.1. A system. • A system interacts with its environmentthrough signals. • There are two main types of signals: • input signals(inputs) , which affect the system behavior in some way • output signals (outputs) , which give information about the system behavior There are two types of input signals: • control signals are inputs whose values we can adjust • disturbances are inputs whose values we cannot affect (in a rational way) Generally, signals are functions of timet , which we can indicate by u(t) and y(t). • Process • Control • Laboratory Process Dynamics and Control

2. Basic control concepts 2.1 Signals and systems A signal is (usually) a physical quantity or variable. Depending on the context, the term “signal” may refer to the • type of variable (e.g. a variable denoting a temperature) • value of a variable (e.g. a temperature expressed as a numerical value) In practice, this does not cause confusion. The value of a signal may be known if it is a measured variable. In particular, • some outputs are (nearly always) measured • somedisturbancesmight be measured • control signals are eithermeasured or knownbecausethey are given by the controller A system is a • static system if the outputs are completely determined by the inputs at the same time instant; such behavior can be described by algebraic equations • Dynamic(al) system if the outputs depend also on inputs at previous time instants; such behavior can be described by differential equations • Process • Control • Laboratory Process Dynamics and Control

2. Basic control concepts 2.1 Signals and systems • Example 2.1. Block diagram of a control valve. • Figure 2.2 illustrate a control valve. • The flowq through the control valve depends on the valve position x, primary pressure p1and secondary pressure p2 . • The valve characteristics give a relationship between the steady state values of the variables. In reality, the flow q depends on the other variables in a dynamic way. • The flow q is the output signalof the system, while x, p1and p2are the input signals. • From these input signals, x can be used as a control signal , while p1and p2are disturbances. • Process • Control • Laboratory Valve Process Dynamics and Control

2. Basic control concepts • 2.2 Block diagrams A block diagram is a • pictorial representation of cause-and-effect relationships between signals. • The signals are represented by • arrows, which show the direction of information flow. • In particular, a block with signal arrows denotes that • the outputs of a dynamical system depend on the inputs. • The simplest form a block diagram is a single block, illustrated by Fig. 2.1. • The interior of a block usually contains • adescriptionor the nameof the corresponding system, or • a symbol for the mathematical operation on the input to yield the output. • Figure 2.4. Examples of block labeling. • Process • Control • Laboratory Process Dynamics and Control

2. Basic control concepts 2.2 Block diagrams The blocks in a block diagram consisting of several blocks are connected via their signals. The following algebraic operations on signals of the same type are often needed: • addition • subtraction • branching • Process • Control • Laboratory Process Dynamics and Control

2. Basic control concepts 2.2 Block diagrams • Figure 2.5 shows the symbols for flow control in a process diagram. • - “FC” is the flow controller . • - “FT” is the flow transmitter. • The notations “FIC” and “FIT” are also used, where “I” indicates that the instrument is equipped with a “indicator” (analog or digital display of data). • Other common notations are • - “LC” for level controller • - “TC” for temperature controller • - “PC” for pressure controller • - “QC” for concentration controller • Process • Control • Laboratory Process Dynamics and Control

2. Basic control concepts • 2.3 From flow sheet to block diagram • Note the following input and output signals used in control engineering: • The input and output signals in a control system block diagram are not equivalent to the physical inlet and outlet currentsin a process flow diagram. • Theinput signals in a control system block diagram indicate which variables affect the system behavior while theoutput signals give information about the system behavior. • The input and output signals in control systems do not necessarily need to be currents in the literal sense of the word, and even if they are, these signals do not need to coincide with the corresponding course of their physical currents. For instance, a physical outlet current may well be a control input signal as shown in figure 2.2. • The output signals in a block diagram provide some information about the aim of the process, which cannot be directly understood from a process flow diagram. Usually the choice of the control signals and the presence of disturbance are not unambiguously apparent from the process flow diagram. In other words, the block diagram provides information of the process control in addition to the process flow diagram. • Process • Control • Laboratory Process Dynamics and Control

2. Basic control concepts 2.3 From flow sheet to block diagram • Example 2.2. Block diagram of a tank with continuous flow. • Process A. A container for liquids, where the fluid level hcan be controlled by the inflow F1 , while the outflow F2depends on h (discharged by gravity). • Block diagram: • Process B. A container for liquids, where the fluid level hcan be controlled by the outflow F2 , while the inflow F1is a disturbance variable. • Block diagram: • Process • Control • Laboratory level/inflow outflow/level disturbance control variable level/inflow disturbance level/outflow The block diagram illustrate also what is meant by positive and negative gain control variable Process Dynamics and Control

2. Basic control concepts 2.3 From flow sheet to block diagram • Exercise 2.1. • Design a block diagram for the following process, where a liquid flowing through a tube is heated and the temperature is controlled by the admission of steam into the tube. • Process • Control • Laboratory liquid steam Process Dynamics and Control

2. Basic control concepts • 2.4 Control strategies • 2.4.1 Open-loop control • In some simple applications, open-loop control without measurentscan be used. In this control strategy • the controller is tuned using a priori information (a “model”) about the process • after tuning, the control actions are a function of the setpoint only (setpoint = desired value of the controlled variable) • This control strategy has some advantages, but also clear disadvantages. Which? • Process • Control • Laboratory • Examples of open-loop control applications: • bread toaster • idle-speed control of (an old) car engine Figure 2.6. Open-loopcontrol. Process Dynamics and Control

2. Basic control concepts 2.4 Control strategies • 2.4.2 Feedforward control • Control is clearly needed to eliminate the effect of disturbances on the system output. Feedforward control is a type of open-loop control strategy, which can be used for disturbance elimination, if • disturbances can be measured • we know how the disturbances affect the output • we know how the control signal affects the output • Feedforward is an open-loop control strategy because the output, which we want to control, is not measured. • Obviously, this control strategy has advantages, but it also has some disadvantage. Which? • When feedforward control • is used, it is usually used in • combination with feedback • control. • Process • Control • Laboratory Figure 2.7. Feedforward control. Process Dynamics and Control

2. Basic control concepts 2.4 Control strategies • 2.4.3 Feedback control • Generally, successful control requires that an output variable is measured. In feedback control, this measurement is fed to the controller. Thus • the controller receives information about how a control action affects the output • usually the measured variable is the variable we want to control (in principle, it can also be some other variable) • Figure 2.8. Feedback control. • Process • Control • Laboratory Process Dynamics and Control

2. Basic control concepts 2.4 Control strategies • Example 2.3. Two different control strategies for house heating. • Figure 2.9 illustrates the heating of a house by (a) feedforward, (b) feedback. The following advantages and disadvantages can be noted: • Feedforward: rapid control because the controller acts before the effect of the disturbance (outdoor temperature) is seen in the output (indoor temperature), but requires good knowledge of the process model; do not consider other disturbances than the measured outside temperature, e.g. wind speed. • Feedback: slower control because the controller does not act before the effect of the disturbance (outside temperature) is seen in the output (inside temperature); insensitive to modeling errors an disturbances. • How would open-loop control of the indoor temperature look like? • (a) feedforward (b) feedback • Process • Control • Laboratory Temp. sensor Temp. sensor Controller Heater Controller Heater Process Dynamics and Control

2. Basic control concepts 2.4 Control strategies • Exercise 2.2. • Consider the two flow control diagrams below. Indicate the control strategies (feedback /feedforward) in each case and justify the answer. It can be assumed that the distance between the flow transmitter FT and the control valve is small. • Process • Control • Laboratory liquid liquid Process Dynamics and Control

2. Basic control concepts 2.4 Control strategies • Exercise 2.3. • The container for liquids to the right has an inflow F1and an outflow F2. The inflow is controlled so that F1 = 10 l/min. • The volume of the liquid is desired to remains constant at V = 1000 l. The volume of the liquid (or liquid level) is thus the systems output signal, while F1and F2 are the input signals. • Process • Control • Laboratory • The following control strategies are possible: • Open-loop control – outflow is measured and controlled so that • F2 = 10 l/min. • Feedback – liquid level h is measured and controlled by the outflow. • Feedforward – inflow is measured and the outflow is controlled so that F2 =F1. • Discuss the differences between these strategies and propose a suitable strategy. Process Dynamics and Control

2. Basic control concepts 2.4 Control strategies • a) b) • c) • Process • Control • Laboratory Process Dynamics and Control

Controller Controlled system Measuring device 2. Basic control concepts • 2.5 Feedback control • 2.5.1 The basic feedback structure • Figure 2.11 shows a block diagram of a simple closed-loop control. • The objective of this control system is to control the measured output signal y ( a single variable) of the controlled system to a desired level given by the desired value, also called setpoint or reference value. • Normally, the controller operates directly on the difference between the setpointr and the output signal measured value y, i.e. the control deviation or control error. • The output signal (at a certain instant) is sometimes called actual value. • Figure 2.9. Feedback control. • Process • Control • Laboratory Comparator v Disturbance Setpoint Control error Controlsignal Outputsignal u r + e y – Measuredvalue ym Process Dynamics and Control

2. Basic control concepts 2.5.1 The basic feedback structure • Two types of regulation are distinguished depending on whether the setpoint is constant or variable: • Regulatory control. The setpoint is usually constant and the main objective of the control system is to maintain the output signal at setpoint, despite the influence of the disturbance. This is sometimes called controller problems. • Tracking control. The setpoint varies and the main objective of the control system is to make the output signal to follow the setpoint with as little error as possible. This is sometimes called servo problems. • These two types of control may well be handled in parallel; the differences arise in the choice of the parameters for the controller (Chapter 7). • Process • Control • Laboratory Process Dynamics and Control

2. Basic control concepts 2.5 Feedback control • 2.5.2 An example of what can be achieved by feedback control • We shall illustrate some fundamental properties of feedback control by considering the control of the inside temperature of a house. • The temperature inside the house depends on the outside temperature and the heating power according to some dynamic relationship. If we assume that • depends linearly (or more accurately, affinity) on • the dynamics are of first order • the relationship between the variables can be written • (2.1) • where is the static gain and T is the time constant of the system. The system parameters have the following interpretations: • Kp denotes how strong the effect of a system input ( P ) is on the output ; a larger value means a stronger effect. • T denotes how fast the dynamics are; a larger value means a slower system. • Process • Control • Laboratory Process Dynamics and Control

2.5 Feedback control 2.5.2 An example of what can be achieved by feedback control • In this case . The equation shows that in the steady-state ( ) • if • an increase of increases • an increase of increases • Thus, the simple model (2.1) has the same basic properties as the true system. We want • the inside temperature to be equal to a desired temperature • in spite of variations in the outside temperature • even if the system gain and the time constant are not accurately known. A simple control law is to adjust the heating power in proportion to the difference between the desired and the actual inside temperature, i.e., (2.2) where Kcis the controller gain and P0is a constant initial power which we can set manually. This relationship describes a proportional controller, more commonly known as a P-controller. As we can see , the controller has the ability to increase the heating power when the inside temperature is below the desired temperature, if Kc> 0. • Process • Control • Laboratory Process Dynamics and Control

2.5 Feedback control 2.5.2 An example of what can be achieved by feedback control • By combining equation (2.1) and (2.2) we can get more explicit information about the controlled system behavior. The elimination of the control signal P gives • (2.3) • From this equation, for instance, we can deduce the following: • If the temperature control is turned off so that Kc= 0, we get i.e. the inside temperature is not dependent at all, as expected, on the desired temperature ϑr . • If, in addition to Kc= 0 the initial heating power is turned off so that P0= 0, the inside temperature will be equal to the outside temperature. • If we set the controller in automatic mode (Kc> 0), we get, for example, if we choose Kc= 1/Kp , , i.e. the inside temperature will be closer to the desired temperature than the outside temperature (if ). • Depending on how we tune P0 it will be even possible that we might get • Process • Control • Laboratory Process Dynamics and Control

2.5 Feedback control 2.5.2 An example of what can be achieved by feedback control • It is easy to see that • the higher Kcis, the more approaches to the reference value ϑr , independent of andP0 , i.e. if Kc→ , means that . • This illustrates a fundamental property of the feedback control • It can almost completely eliminate the effect of the disturbances (the outside temperature in this example) on the controlled system. • Normally, we do not need to know the characteristics of the system in detail (Kpin this example) to set the controller. • We can make the output signal to adopt or follow a desired value ( in this example). • Process • Control • Laboratory Process Dynamics and Control

2. Basic control concepts 2.5 Feedback control • 2.5.3 A counter-example: limiting factors • In the example above we neglected the system dynamics in order to illustrate in a simple way the advantages, that at least in principle, can be achieved with feedback control. • It is clear, for example, that in practice we might not have a controller gain that will approaches to infinity. • When the dynamic of the system is to be considered, an input power which approaches to infinity would be required according the dynamic counterpart of the equation (2.2) if the inside temperature deviates from the reference temperature. • In addition, the properties of the controlled system are generally limitations. This is best explained by the following example. • Process • Control • Laboratory Process Dynamics and Control

2.5 Feedback control 2.5.3 A counter-example: limitingfactors • Consider the process in Exercise 2.1, where the fluid flowing in a well insulated pipe is heated and the temperature is controlled by direct addition of steam. • The temperature of the liquid ϑ2 is measured 60 m after the mixing point, which means that the temperature measured at the mixing point ϑ1 reaches the temperature at the measuring point after 1 minute, considering the flow velocity v = 1 m/s. • If the temperature before the mixing point is denoted by ϑi and the mass flow of the added steam by ṁ, the following expression applies when the heat loss from the pipe is neglected, • ϑ2(t +1) = ϑ1(t) = ϑi(t) + Kpṁ(t) (2.4) • where t is the time expressed in minutes and Kpis a positive process gain, whose value we do not need to specify in this example. • If we use a P-controller for the control of ϑ2 with ṁ (we neglect the control valve) • ṁ(t) = Kc(ϑr – ϑ2(t)) + ṁ0 (2.5) • where Kcis the controller gain and ṁ0 is the normal value of the steam mass flow, which at the steady state gives . • Process • Control • Laboratory Process Dynamics and Control

2.5 Feedback control 2.5.3 A counter-example: limitingfactors • Combining equation (2.4) and (2.5) gives • ϑ2(t +1) = ϑi(t) + KpKc(ϑr – ϑ2(t)) + Kpṁ0 (2.6) • Consider a steady state . The following expression applies then, according equation (2.6) • (2.7) • Subtraction of eq. (2.7) from (2.6) gives with and • Δϑ2(t +1) = Δϑi(t) - KpKc(ϑr – Δϑ2(t)) + Kpṁ0 (2.8) • Assume that the steady state conditions prevail up to t = 0, and that a step change Δϑi,stepoccurs in the temperature ϑiat this time. According to eq. (2.8) we get Δϑ2(1) = Δϑi,step, Δϑ2(2) = Δϑi,step - KpKcΔϑ2(1) = (1 - KpKc)Δϑi,step , and for t = k a general expression is then • (2.9) • Process • Control • Laboratory Process Dynamics and Control

2.5 Feedback control 2.5.3 A counter-example: limitingfactors • Weseeimmediatly: • If |KpKc| > 1, every term on the right side of the absolute value is greater than the previous, i.e. the series diverges with instability. • If KpKc= 1, Δϑ2 will oscillate between the levels -Δϑi,stepand Δϑi,step“for ever”. • If |KpKc| < 1 , the sum of all the terms form a converging geometric series, an we get • when K→ ∞ , |KpKc| < 1 (2.10) • The expression (2.10) shows that the best control with a P-controller gives Δϑ2(k) ≈ 0,5Δϑi,step when K→ ∞ , although we would want to get the desired value Δϑ2 ≈ 0. • Process • Control • Laboratory Process Dynamics and Control

2.5 Feedback control 2.5.3 A counter-example: limitingfactors • In this example, we did not obtain very positive effects we did obtain in the previous example. • We can not say that the process is especially complicated, but it has a pure transport delay, or more generally, a time delay, also called dead time. • Such transport delays are of course very common in the process industry, but even other processes often include time delays. • In general, we can say that a time delay in a feedback control system can cause very harmful effect to the performance of the closed-loop control, and it can compromise the control loop stability. • Time delays are troublesome characteristics in a process, but some processes can also be difficult to control due to other factors. • For example, processes whose behavior is described by (linear) differential equations of third or higher order bring restrictions of similar type as those restrictions that time delays cause. • Process • Control • Laboratory Process Dynamics and Control

2.5 Feedback control • 2.5.4 PID controller • In the two examples above were used P controller, and we established the following: • A high gain controller is desirable for the elimination of the influence of external disturbanceson the controlled system, and also for the reduction in the sensitivity to uncertainty regarding the process parameters. • A high gain may cause instability, and the situation is aggravated by the process uncertainties; one can say that the risk is imminent when you rely too much on outdated information. • A stationary control deviation (a lastingcontrol error) is obtained after a load step change (i.e. a load disturbance); the smaller the controller gain is, the larger the error. • One can say that the first two items apply to feedback control in general. • Since they are mutually contradictory, they suggest that some compromises must be made in order to find an optimal controller settings. • It is also likely that a more complex controller than a P-controller is usually preferred. This is necessary, for example, for the elimination of the steady control deviation. • Process • Control • Laboratory Process Dynamics and Control

2.5 Feedback control 2.5.4 PID controller • The so-called PID controller is a "universal controller", which in addition to a pure gain, it also contains an integral and derivativecontroller. The control law of an ideal PID controller− in practice, however, modifications are often used− is given by • (2.11) • Where u(t) is the controller output signal and e(t) is the difference between the reference value and the measured value, i.e. the control error; see Figure 2.9. • The adjustable parameters of the controller are, in addition to the control signal initial value u0 (usually = 0), the control gainKc , the integral time Ti and the derivative time Td. • Process • Control • Laboratory Process Dynamics and Control

2.5 Feedback control 2.5.4 PID controller • By choosing appropriate controller parameters, those parts of the controller that are not needed can be disabled. • A so-called PI controller is obtained by setting Td = 0. • A P controller is obtained by choosing Ti = ∞ , in addition to Td = 0 (note that Ti ≠ 0 !). • Sometimes PD controllers are also used. • A P-effect is practically always required for controlling, and as it is written in the control law (2.11), it cannot be disabled without disabling the hole controller. This limitation can be eliminated by writing the control law in the following form • (2.12) • Process • Control • Laboratory Process Dynamics and Control

2.5 Feedback control 2.5.4 PID controller • The PI controller is without doubt the most common controller in the (process) industry, where it is specifically used for flow control. In conclusion, the PI controller has • good static properties, it eliminates the stationary control deviation; • the tendency to cause oscillatory behavior, which reduces the stability (the integral collect old data!). • The D-effects are often included (PD or PID) in the control of processes with slow dynamics, especially temperature and vapor pressure. The D-effect gives • good dynamic properties and good stability (the derivative “predicts” the future!); • sensitivity to measurement noise. • Process • Control • Laboratory Process Dynamics and Control

2.5 Feedback control 2.5.4 PID controller • Exercise 2.4. Consider a PI controller and assume that the steady state conditions prevail up to t = ts. This means that u(t) and e(t) are constant for t ≥ ts. Explain why this implies that e(ts) = 0 , i.e. that the control deviation must be zero at steady state. Exercise 2.5. Which stationary property has a double-integral controller (PII controller) , i.e. what can you say about e(t) and/or x(t) at the steady state? • Process • Control • Laboratory Process Dynamics and Control

2.5 Feedback control • 2.5.5 Negative and positive feedback • It is important to distinguish between negative feedback and positive feedback . • A negative feedback means that the control signal cancels the control error. • A positive feedback means that the control signal amplifies the control error. • Process • Control • Laboratory Process Dynamics and Control

2.5 Feedback control Exercise 2.6. • What type of feedback control would you use in a control system? • How do you know what kind of feedback control you have in a control system? • Is it always possible to choose the right type of feedback control? • What happens if the wrong type of feedback is chosen? Often other definitions of negative (and positive) feedback control are mentioned in control engineering, such as • A negative feedback control means that the control signal increases when the output signal decreases and vice versa. • A negative feedback control is obtained when the measured value of the output signal is subtracted from the setpoint. • Are these definitions in accordance with the definitions given in Section 2.5.5? • If not, what can be assumed from the process and/or the controller properties by using these definitions? • Process • Control • Laboratory Process Dynamics and Control

Process Dynamics and Control (419307), 7cr Kurt-Erik Häggblom 2. Basic control concepts