Process Dynamics and Control

Process Dynamics and Control
7. PID Controllers 7.0 Overview 7.1 PID controller variants 7.2 Choice of controller type 7.3 Specifications and performance criteria 7.4 Controller tuning based on frequency response 7.5 Controller tuning based on step response 7.6 Model-based controller tuning 7.7 Controller design by direct synthesis 7.8 Internal model control 7.9 Model simplification KEH Process Dynamics and Control

7.0 Overview PID controller (”pee-i-dee”) is a generic name for a controller containing a linear combination of proportional (P) integral (I) derivative (D) terms acting on a control error (or sometimes the process output). All parts need not be present. Frequently I and/or D action is missing, giving a controller like P, PI, or PD controller It has been estimated that of all controllers in the world 95 % are PID controllers KEH Process Dynamics and Control

7.1 PID controller variants
7.1.1 Ideal PID controller An ideal PID controller is described by the control law 𝑢 𝑡 = 𝐾 c 𝑒 𝑡 + 1 𝑇 i 0 𝑡 𝑒 𝜏 d𝜏 + 𝑇 d d𝑒(𝑡) d𝑡 + 𝑢 (7.1) 𝑢(𝑡) is the controller output 𝑒 𝑡 =𝑟 𝑡 −𝑦(𝑡) is the control error, which is the difference between the setpoint 𝑟(𝑡) and the measured process output 𝑦(𝑡) 𝐾 c is the proportional gain 𝑇 i is the integral time 𝑇 d is the derivative time 𝑢 0 is the “normal” value of the controller output The transfer function of the PID controller is 𝐺 PID = 𝑈(𝑠) 𝐸(𝑠) = 𝐾 c 𝑇 i 𝑠 + 𝑇 d 𝑠 = 𝐾 c 𝑇 i 𝑠 1+ 𝑇 i 𝑠+ 𝑇 i 𝑇 d 𝑠 (7.2) 𝑈(𝑠) is the Laplace transform of 𝑢 𝑡 − 𝑢 0 𝐸(𝑠) is the Laplace transform of the control error KEH Process Dynamics and Control

7.1.1 Ideal PID controller Depending on the values of 𝑇 i and 𝑇 d , the transfer function of the PID controller can have real or complex valued zeros Complex zeros might be useful for control of underdamped systems with complex poles. A PI controller is obtained from a PID controller by letting 𝑇 d =0. Its transfer function is 𝐺 PI = 𝐾 c 𝑇 i 𝑠 = 𝐾 c 𝑇 i 𝑠 1+ 𝑇 i 𝑠 (7.3) A PD controller is obtained from a PID controller by letting 𝑇 i =∞. Its transfer function is 𝐺 PD = 𝐾 c 1+ 𝑇 d 𝑠 (7.4) The ideal PID controller is sometimes referred to as the parallel form of a PID controller the (ISA) standard form KEH Process Dynamics and Control

7.1.2 The series form of a PID controller
7.1 PID controller variants 7.1.2 The series form of a PID controller In the pre-digital era it was convenient to implement an analog PID controller as a PI controller and a PD controller in series. This form of a PID controller is called the series form. Occasionally, the terms interactive form or classical form are used. The controller has the transfer function 𝐺 PIPD = 𝐾 c ′ 𝑇 i ′ 𝑠 1+ 𝑇 d ′ 𝑠 = 𝐾 c ′ 𝑇 i ′ 𝑠 1+ 𝑇 i ′ 𝑠 1+ 𝑇 d ′ 𝑠 (7.5) where ′ is used to distinguish the parameters from the parameters of the parallel form. The series form of a PID controller can only have real valued zeros. This means that the series form is less general than the parallel form. It is easy to find the controller parameters of the series form by frequency analytic methods by so-called lead-lag design. Exercise 7.1 Which is the control law in the time domain for a series form PID controller? KEH Process Dynamics and Control

7.1.3 A PID controller with derivative filter
7.1 PID controller variants 7.1.3 A PID controller with derivative filter A drawback with the ideal PID controller (7.1) is that the derivative part cannot be realized exactly in a real controller. For example, if the control error 𝑒(𝑡) changes as a step, the derivate in (7.1) becomes infinitely large. This problem can be remedied by filtering the signal to be differentiated. This also has the practical advantage that (high-frequency) noise is filtered before differentiation. The transfer function of a parallel form PID controller with a derivative filter is 𝐺 PIDf = 𝐾 c 𝑇 i 𝑠 + 𝑇 d 𝑠 𝑇 f 𝑠 (7.6) The transfer function of a series form PID controller with a derivative filter is 𝐺 PIPDf = 𝐾 c ′ 𝑇 i ′ 𝑠 𝑇 d ′ 𝑠 𝑇 f ′ 𝑠 (7.7) 𝑇 f and 𝑇 f ′ are filter constants, usually % of corresp. derivative time. KEH Process Dynamics and Control

7.1.3 A PID controller with derivative filter Relationships between parallel and series form If the parameters of the series form are known, the corresponding parameters of the parallel form can be calculated according to 𝑇 i = 𝑇 i ′ + 𝑇 d ′ − 𝑇 f ′ , 𝑇 d = 𝑇 d ′ 𝑇 i ′ 𝑇 i − 𝑇 f ′ , 𝑇 f = 𝑇 f ′ , 𝐾 c = 𝐾 c ′ 𝑇 i ′ 𝑇 i (7.8) For calculation of the parameters of the series form from the parameters of the parallel form, we define the parameter 𝛿=1− 4 𝑇 i ( 𝑇 d + 𝑇 f ) ( 𝑇 i + 𝑇 f ) 2 (7.9) If 𝛿≥0, the zeros of the parallel PID are real. Then, there exists a series form PID controller which is equivalent to the parallel form according to 𝑇 i ′ = ( 𝑇 i + 𝑇 f ) 2 1+ 𝛿 , 𝑇 d ′ = 𝑇 i + 𝑇 f − 𝑇 i ′ , 𝑇 f ′ = 𝑇 f , 𝐾 c ′ = 𝐾 c 𝑇 i ′ 𝑇 i (7.10) The condition for 𝛿≥0 in terms of the controller parameters is 𝑇 d ≤ ( 𝑇 i − 𝑇 f ) 2 4 𝑇 i (7.11) i.e., the derivative time has to be “small enough”. KEH Process Dynamics and Control

7.1.4 Differentiation of the measured output
7.1 PID controller variants 7.1.4 Differentiation of the measured output Even if we have a derivative filter, a step change in the setpoint 𝑟(𝑡) tends to affect the derivative part much more strongly than a disturbance in the output 𝑦(𝑡). A remedy to this is to differentiate the (filtered) output instead of the control error 𝑒(𝑡). The ideal control law (7.1) then becomes 𝑢 𝑡 = 𝐾 c 𝑒 𝑡 + 1 𝑇 i 0 𝑡 𝑒 𝜏 d𝜏 − 𝑇 d d 𝑦 f (𝑡) d𝑡 + 𝑢 (7.12a) 𝑇 f d 𝑦 f (𝑡) d𝑡 + 𝑦 f 𝑡 =𝑦(𝑡) (7.12b) In the Laplace domain we get 𝑈 𝑠 = 𝐾 c 𝑇 i 𝑠 𝑅 𝑠 − 𝐾 𝑐 𝑇 i 𝑠 + 𝑇 d 𝑠 𝑇 f 𝑠+1 𝑌(𝑠) (7.13) which is a combination of a PI controller and a PID controller 𝑈 𝑠 = 𝐺 PI 𝑅 𝑠 − 𝐺 PIDf 𝑌(𝑠) (7.14) This kind of 2-degrees-of-freedom (2DOF) controller can be tuned separately for setpoint tracking and disturbance rejection. KEH Process Dynamics and Control

7.1.4 Differentiation of the measured output Exercise 7.2 Which is the control law, both in the time domain and the Laplace domain, for the series form of a PID controller with differentiation of the filtered output measurement? A simple way of obtaining 2DOF PID controller is to use setpoint weighting. With the definitions 𝑒 p =𝑏𝑟−𝑦 , 𝑒=𝑟−𝑦 , 𝑒 d =𝑐𝑟− 𝑦 f (7.15) where 𝑏 and 𝑐 are setpoint weights, the control law becomes 𝑢 𝑡 = 𝐾 c 𝑒 p 𝑡 + 1 𝑇 i 0 𝑡 𝑒 𝜏 d𝜏 + 𝑇 d d 𝑒 d (𝑡) d𝑡 + 𝑢 0 (7.16a) 𝑇 f d 𝑦 f (𝑡) d𝑡 + 𝑦 f 𝑡 =𝑦(𝑡) (7.16b) 7.1.5 Setpoint weighting KEH Process Dynamics and Control

7.1.5 Setpoint weighting In the Laplace domain the control law with setpoint weighting is 𝑈 𝑠 = 𝐺 vPID 𝑅 𝑠 − 𝐺 PIDf 𝑌(𝑠) (7.17) where 𝐺 vPID = 𝐾 c 𝑏+ 1 𝑇 i 𝑠 +𝑐 𝑇 d 𝑠 (7.18) and 𝐺 PIDf is as in (7.6). With suitable choices of 𝑏 and 𝑐, all previously treated PID controllers on parallel form can be obtained. 𝑏 and 𝑐 do not affect the controller’s ability to reject disturbances in the output, only the ability to track setpoint changes. 𝐺 vPID can be tuned for setpoint tracking and 𝐺 PIDf for disturbance rejection (i.e., 𝐾 c , 𝑇 i and 𝑇 d need not have the same values in 𝐺 vPID and 𝐺 PIDf ). Exercise 7.3 Include setpoint weighting in the series form of a PID controller. KEH Process Dynamics and Control

7.1.6 Non-interactive form of a PID controller
7.1 PID controller variants 7.1.6 Non-interactive form of a PID controller In the control laws treated so far, the proportional part alone cannot be disconnected by letting 𝐾 c =0 because that would disconnect all parts; it would put the controller on “manual” with 𝑢 𝑡 = 𝑢 0 . Tuning the proportional part by adjusting 𝐾 c will affect all controller parts (however, this is often a desired feature); hence, it is an interactive controller form. The non-interactive form 𝑢 𝑡 = 𝐾 c 𝑒 𝑝 𝑡 + 𝐾 i 0 𝑡 𝑒 𝜏 d𝜏 + 𝐾 d d 𝑒 d (𝑡) d𝑡 + 𝑢 (7.19) is a more flexible control law. In the Laplace domain it can be written 𝑈 𝑠 = 𝐺 vP+I+D 𝑅 𝑠 − 𝐺 P+I+Df 𝑌(𝑠) (7.20) where 𝐺 vP+I+D = 𝐾 c 𝑏+ 𝐾 i 𝑠 −1 +𝑐 𝐾 d 𝑠 (7.21a) 𝐺 P+I+Df = 𝐾 c + 𝐾 i 𝑠 −1 + 𝐾 d 𝑠 ( 𝑇 f 𝑠+1) − (7.21b) Note: It is essential to know which form is used when tuning a controller! KEH Process Dynamics and Control

7.2 Choice of controller type
The choice between controller types such as P, PI, PD, PID is considered. In principle, the simplest controller that can do the job should be chosen. An on-off controller is the simplest type of controller, where the control signal has only two levels. If the variables are defined such that a positive control error 𝑒(𝑡) should be corrected by an increase of the control signal 𝑢(𝑡), the control law is 𝑢 𝑡 = 𝑢 max if 𝑒 𝑡 > 𝑒 hi 𝑢 0 or unchanged if 𝑒 lo ≤𝑒 𝑡 ≤ 𝑒 hi 𝑢 min if 𝑒 𝑡 < 𝑒 lo (7.23) where 𝑢 max , 𝑢 0 , 𝑢 min is the high, normal, low value of the control signal. The interval ( 𝑒 lo , 𝑒 hi ) is a dead zone. In the simplest case, 𝑒 lo = 𝑒 hi =0. The on-off controller is inexpensive, but it causes oscillations in the process. It is often used for temperature control in simple appliances such as ovens, irons, refrigerators and freezers, where oscillations are tolerated. 7.2.1 On-off controller KEH Process Dynamics and Control

7.2 Choice of controller type 7.2.2 P controller A P controller implements the simple control law 𝑢 𝑡 = 𝐾 c 𝑒 𝑡 + 𝑢 0 (7.24) where 𝐾 c is the adjustable controller gain and 𝑢 0 is the normal value of the control signal, which is also be adjustable. In principle, 𝑢 0 is selected to make the control error 𝑒 𝑡 =0 at the nominal operating point. If the output is changed by a disturbance or a setpoint change, the P controller is unable to bring the control error to zero, i.e., there will be a remaining control error. The higher the controller gain, the smaller the control error. Thus, P control is used when a (small) control error is allowed and a high controller gain can be used without risk of instability. A typical application for P control is level control in liquid tank. Another situation when P control is often sufficient is as an inner loop (a secon- dary loop) in so-called cascade control. KEH Process Dynamics and Control

7.2 Choice of controller type 7.2.3 PI controller A PI controller is by far the most common type of controller. The ideal PI controller implements the control law 𝑢 𝑡 = 𝐾 c 𝑒 𝑡 + 1 𝑇 i 0 𝑡 𝑒 𝜏 d𝜏 + 𝑢 0 (7.25) where the gain 𝐾 c and the integral time 𝑇 i are adjustable parameters; 𝑢 0 is less important due to the integral. The main advantage of the PI controller is that there will be no remaining control error after a setpoint change or a process disturbance. A dis- advantage is that there is a tendency for oscillations. PI control is used when no steady-state error is desired and there is no reason to use derivative action. Measurement noise is often a reason for not using derivative action. PI control is suitable for noisy processes, integrating processes and processes resembling first-order systems. The most typical application is flow control. PI control might also be preferable for processes with large time delays. KEH Process Dynamics and Control

7.2 Choice of controller type 7.2.4 PD controller The ideal form of a PD controller implements the control law 𝑢 𝑡 = 𝐾 c 𝑒 𝑡 + 𝑇 d d𝑒(𝑡) d𝑡 + 𝑢 0 (7.26) where the gain 𝐾 c and the derivative time 𝑇 d are adjustable parameters; 𝑢 0 is chosen as for a P controller. A PD controller is preferred when integral action is not needed, but the dynamics of the process are so slow that the predictive nature of derivative action is useful. Many thermal processes, where energy is stored with small heat losses (e.g., ovens), usually have slow dynamics, almost as integrating systems. A PD controller might then be suitable for temperature control. Another typical application for PD control is in servo mechanisms such as electrical motors, which usually behave as second-order integrating systems. KEH Process Dynamics and Control

7.2 Choice of controller type 7.2.5 PID controller As has been shown in Section 7.1, there are many variants of PID controllers. The ideal form and the classical series form have 3 adjustable parameters in addition to 𝑢 0 : the proportional gain, the integral time, and the derivative time. If a derivative filter is included, there are 4 adjustable parameters, but the filter time constant is usually selected as a given fraction (e.g., 10 %) of the derivative time. In addition, the setpoint can be weighted in the proportional part and the derivative part. If there is no reason to exclude integral action or derivative action, a PID controller is the natural choice. Typically PID control is used for underdamped processes, processes with slow dynamics and not very large time delays, and systems of second and higher order. Typical applications are control of temperature and chemical composition when the process is not close to an integrating system. KEH Process Dynamics and Control

7.3 Specifications and performance criteria
7.3.1 General performance criteria The task of a controller is to control a system to behave in a desired way despite unknown disturbances and an inaccurately known system. The controlled system should satisfy performance criteria such as: The controlled system must be stable; this is absolutely necessary. The effect of disturbances on the controlled output is minimized; this is especially important for regulatory control. The controlled output should follow setpoint changes fast and smoothly; this is especially important for setpoint tracking. The control error is minimized or kept within certain limits, The control signal variations should be moderate or at least not be excessively large; more variations wear out control equipment faster. The control system should be robust (insensitive) against moderate changes in system properties, which introduce model uncertainty. The importance of these criteria varies from case to case. Since many criteria are conflicting, compromises have to be made in the control design. KEH Process Dynamics and Control

7.3.2 Fundamental limitations
7.3 Specifications and performance criteria 7.3.2 Fundamental limitations One reason to the fact that there are usually good solutions to the conflicting control criteria is that feedback control is used. However, feedback also introduces limitations because a control error is required for the controller to take action. The fact that the available resources for control are always limited, also limit the achievable performance. In addition to the general limitations above, there are also limitations that depend on the process to be controlled, e.g., the dynamics of the process nonlinearities model and process uncertainty disturbances control signal limitations KEH Process Dynamics and Control

7.3.2 Fundamental limitations The process dynamics is often the performance limiting factor. Such factors are time delays as well as RHP (right-half plane) poles and zeros high-order dynamics In practice, all processes are nonlinear. Such a process cannot be described accurately at different operating points by a linear model with constant parameters; thus there is model/process uncertainty. Disturbances such as load disturbances and measurement noise limit how well a variable can be controlled. Efficient control of load disturbances often require derivative action, but measurement noise is bad for the derivative. Large load disturbances can cause the control variable to reach its (physical) maximum or minimum value. This is especially troublesome if the controller contains an integrator. Proportional band and integrator windup are two concepts that deal with this limitation. KEH Process Dynamics and Control

7.3.3 Proportional band and integrator windup
7.3 Specifications and performance criteria 7.3.3 Proportional band and integrator windup Proportional band A controller’s proportional band (PB) denotes the maximum control error the controller can handle with the available control signal. The PB is defined for a P controller, but it can be extended to a full PID controller. If the control signal is limited by 𝑢 min ≤𝑢(𝑡)≤ 𝑢 max , a P controller can according to (7.24) handle a control error that satisfies 𝑢 min − 𝑢 0 𝐾 c ≡ 𝑒 min ≤𝑒(𝑡)≤ 𝑒 max ≡ 𝑢 max − 𝑢 0 𝐾 c (7.27) The PB is equal to 𝑒 max − 𝑒 min = 𝑦 hi − 𝑦 lo , where 𝑦 hi is the highest output ( 𝑒 min =𝑟− 𝑦 hi ) and 𝑦 lo is the lowest output ( 𝑒 max =𝑟− 𝑦 lo ) the controller can handle. Usually, the PB is defined in percent of the total measurable output interval 𝑦 min , 𝑦 max . Then, the PB is 𝑃 b = 𝑦 hi − 𝑦 lo 𝑦 max − 𝑦 min 100%= 𝑢 max − 𝑢 min 𝑦 max − 𝑦 min ⋅ 100% 𝐾 c (7.28) KEH Process Dynamics and Control

Proportional band If the proportional band is known, the controller gain is given by 𝐾 c = 𝑦 hi − 𝑦 lo 𝑦 max − 𝑦 min 100%= 𝑢 max − 𝑢 min 𝑦 max − 𝑦 min ⋅ 100% 𝑃 b (7.29) In (old) automation systems, the signals are often expressed as a fraction or percentage of the total signal interval (0-1 or 0-100%). The PB is then 𝑃 b =100%/ 𝐾 c (7.30) Note that the controller gain here has to be expressed in terms of the normalized signals, which means that the controller gain is dimensionless. The practical usefulness of the PB is that it tells something about the size of control errors that can be handled without reaching an input signal constraint. If 𝑢 0 is in the middle of the interval 𝑢 min , 𝑢 max , a P controller with 𝑃 b =50 % can handle an instantaneous control error equal to ±25 % (i.e., 50 % in total) of the total output signal range. Note that the PB is an adjustable controller parameter — if it is to small, it can be increased (corresponding to a decrease of 𝐾 c ). KEH Process Dynamics and Control

7.3.3 PB and integrator windup Integrator windup Usually controllers are tuned for stability and performance, not for signal limits. Therefore, it is not uncommon that a control signal reaches a constraint. If the controller contains integral action, this can be very damaging to the control performance unless the situation is handled properly. Consider the figure, where the PI control law (7.25) has been used. A strong disturbance causes the process output to fall well below the setpoint. The controller is not able to elimi- nate the control error (A) because the control signal has reached a constraint. During this time, the positive control error will increase the integral in the controller. If the disturbance later disappears, the controller will still keep the control signal at the constraint due to the large value of the integral, even if the control error goes below zero. This will cause the output (B), which is entirely due to the controller. Illustration of integral windup. KEH Process Dynamics and Control

Integral windup The described phenomenon is called integral windup (also reset windup). There are sophisticated as well as simple methods for handling the problem. The term anti-windup is used for such arrangements. A simple solution is to stop integrating when a control signal reaches a constraint. This requires that it is known when the control signal reaches a constraint (e.g., through measurement) there is some built-in logic to interrupt the integration In the case of digital control, which nowadays is customary, automatic anti-windup can be built into the control law. KEH Process Dynamics and Control

7.3.4 Design specifications
7.3 Specifications and performance criteria 7.3.4 Design specifications Above, some general performance criteria and fundamental limitations to achievable control performance have been considered. Here, some ways of making more specific design specifications will introduced. If a process model is available, the specifications make it possible to calculate controller parameters. Step-response specifications It is of often desired that the closed-loop response to a step change in the setpoint resembles an underdamped second-order system. Therefore, parameters familiar from the step-response of such a system can be used to specify the desired behaviour. Such parameters are the maximum relative overshoot 𝑀 the rise time 𝑡 r the settling time 𝑡 𝛿 the relative damping 𝜁 the ratio between successive relative overshoots (or undershoots) 𝑀 R KEH Process Dynamics and Control

Step-response specifications According to the relationships in Section 5.3.3: The two parameters 𝑀 and 𝑡 r are sufficient to determine the transfer function of an underdamped second-order system with a given gain. The settling time 𝑡 𝛿 can be used instead of 𝑀 or 𝑡 r , but the relationships are then only approximate. The relative damping 𝜁 or the overshoot ratio 𝑀 R can be specified instead of 𝑀. Some classical tuning recommendations are based on the specification 𝑀 R =1/4. This may be acceptable for regulatory control, but not for setpoint tracking. 𝑀 R =1/4 corresponds to 𝑀=0.5 (i.e., a 50 % overshoot) and 𝜁=0.22 . For setpoint tracking, 𝑀≈0.1 (𝜁≈0.6) is usually more appropriate. If an overdamped closed-loop response is desired, this cannot be achieved with a specification 𝜁>1 , because the other parameters require an underdamped system. Instead, the closed-loop transfer function can be directly specified and controller parameters calculated by direct synthesis (Section 7.7), for example. KEH Process Dynamics and Control

7.3.4 Design specifications Error integrals In principle, a small overshoot, rise time and settling time are desired. In practice, the overshoot and settling time will increase with decreasing rise time, and vice versa. Therefore, compromises have to be made. One way of solving this problem in an optimal way is to specify some error integral to be minimized. Examples of such error integrals are 𝐽 IAE = 0 𝑡 s 𝑒(𝑡) d𝑡 , 𝐽 ISE = 0 𝑡 s 𝑒(𝑡) 2 d𝑡 𝐽 ITAE = 0 𝑡 s 𝑡 𝑒(𝑡) d𝑡 , 𝐽 ITSE = 0 𝑡 s 𝑡 𝑒(𝑡) 2 d𝑡 (7.31) where the acronyms are – IAE = “integrated absolute error” – ISE = “integrated square error” – ITAE = “integrated time-weighted absolute error” – ITSE = “integrated time-weighted square error” The weighting with time forces the control error towards zero as time in- creases. In principle, the integration time should be infinite, but because the minimization has to be done numerically, a finite 𝑡 s has to be used. KEH Process Dynamics and Control

Error integrals It is of interest to consider how the error integrals relate to step-response specifications when the controlled system is of second order, i.e., 𝐺 𝑠 = 𝜔 n 2 𝑠 2 +2𝜁 𝜔 n 𝑠+ 𝜔 n 2 (7.32) In the figure, IAE and ISE are normalized with 𝜔 n , ITAE and ITSE with 𝜔 n 2 . As can be seen, every normalized error integral has a minimum for a given relative damping 𝜁 . This damping as well as the corresponding relative overshoot 𝑀 are shown below. Table 7.1 Optimal relative damping for 2nd order system. Error integrals as function of 𝜁. KEH Process Dynamics and Control

7.4 Tuning based on frequency response
7.4.1 Experimental tuning An ideal PID controller of interactive form can be tuned experimentally by making closed-loop control experi- ments with the real process. The standard feedback structure is used. A P controller ( 𝐺 c = 𝐾 c ) is used for the first experiment. A low value is chosen for the gain 𝐾 c . Note that 𝐾 c must have the same sign as 𝐾 p . A change in the setpoint 𝑅 is introduced. (Some other disturbance could also be used.) The controller gain 𝐾 c is increased until the output 𝑌 starts to oscillate with a constant amplitude (see next slide). The value of the controller gain yielding constant oscillations is denoted 𝐾 c,max . The period of the oscillations is denoted 𝑃 c . The controller gain is changed to 𝐾 c = 0.5𝐾 c,max . If the intention was to tune a P controller, this is the final tuning. KEH Process Dynamics and Control

7.4.1 Experimental tuning To tune a controller with integral action (PI or PID), an experiment is done with a PI controller using 𝐾 c = 0.5𝐾 c,max . A large value is initially used for the integral time 𝑇 i . A change in the setpoint 𝑅 (or some other disturbance) is introduced. The integral time 𝑇 i is reduced until 𝑌 starts to oscillate with a constant amplitude. This occurs at 𝑇 i = 𝑇 i,min . The integral time for a PI or PID controller is chosen as 𝑇 i =3 𝑇 i,min . To tune the derivative part of a PID (or PD) controller, an experiment is done with such a controller using 𝐾 c = 0.5𝐾 c,max , 𝑇 i =3 𝑇 i,min (if a PID controller). The derivative time is initially set at 𝑇 d =0 . A change in the setpoint 𝑅 (or some other disturbance) is introduced. The derivative time 𝑇 d is increased until the output 𝑌 starts to oscillate with a constant amplitude. This occurs when 𝑇 d = 𝑇 d,max . The derivative time for a PD or PID controller is set at 𝑇 d = 𝑇 d,max . KEH Process Dynamics and Control

7.4.1 Experimental tuning If the control performance obtained by the above tunings turns out to be unsatisfactory, the controller parameters can be adjusted by “trial and error”. The next figure shows how changes of the controller gain 𝐾 c and the integral time 𝑇 i typically affect the control performance. The optimal performance is in this case obtained by 𝐾 c =3 and 𝑇 i =11 . 𝑇 i = 𝑇 i = 𝑇 i =20 𝐾 c =5 𝐾 c =3 𝐾 c =1 KEH Process Dynamics and Control

7.4.2 Ziegler-Nichols’s recommendations
7.4 Tuning based on frequency response 7.4.2 Ziegler-Nichols’s recommendations In 1942, Ziegler and Nichols suggested tunings for P, PI and PID controllers based on 𝐾 c,max and 𝑃 c only. To obtain this information, it is sufficient to do steps 1–3 in the experimental procedure. The tunings are primarily intended for regulatory control (i.e., disturbance rejection). For setpoint tracking, setpoint weighting is suggested, e.g. 𝑏=0.5. The controller tuning should Table 7.2. Ziegler-Nichols’s controller preferably not be used out- tuning recommendations based on side the range 0.1<𝜅<0.5, frequency response. where 𝜅= 𝐾 𝐾 c,max −1 . Here, 𝐾 is the process gain. The critical frequency 𝜔 c is often used instead of 𝑃 c : 𝜔 c =2𝜋/ 𝑃 c . KEH Process Dynamics and Control

7.4.3 Åström’s and Hägglund’s correlations
7.4 Tuning based on frequency response 7.4.3 Åström’s and Hägglund’s correlations In 2006, Åström and Hägglund showed that, in general, 𝐾 c,max and 𝑃 c alone do not provide sufficient information for good controller tuning. In addition to 𝐾 c,max and 𝑃 c , Åström and Hägglund also use the parameter 𝜅= 𝐾𝐾 c,max −1 in their controller tuning correlations. The tuning correlations are primarily intended for regulatory control; for setpoint tracking, setpoint weighting is suggested. The correlations should Table 7.3. Åström-Hägglund’s controller not be used below the tuning correlations based on frequency range 𝜅>0.1 . response. Large time delays are allowed, but clearly underdamped systems are less suitable. KEH Process Dynamics and Control

7.5 Tuning based on step response
A drawback with generating the frequency response is that it is quite cumbersome and time-consuming to generate oscillations with constant amplitude by adjusting a controller parameter. An alternative is to use a step response for the process. The figure illustrates how the needed parameters are obtained from a unit-step response, i.e., a step with size 𝑢 step =1 expressed in the units used for the control variable. The method is based on the (modified) tangent method, but here it is not necessary to wait for the new steady state; only the parameters 𝑎 and 𝐿 need Characteristic parameters from a to be determined. monotonous unit-step response. KEH Process Dynamics and Control

7.5 Tuning based on step response Instead of taking the 𝑎 parameter from the point, where the tangent through the inflexion point (i.e., the point where the slope is highest) of the step response crosses the vertical axis, it can be calculated when the coordinates ( 𝑡 i , 𝑦 i ) of the inflexion point are known. The calculation is valid for any size of 𝑢 step . The formula for 𝑎 is 𝑎= 𝐿 𝑦 i 𝑢 step ( 𝑡 i −𝐿) (7.34) Another useful parameter is 𝜃=𝐿/ 𝑇 eq , 𝑇 eq = 𝑡 63 −𝐿 (7.35) where 𝑇 eq is the equivalent time constant of the system and 𝑡 63 is the time it takes to reach 63% of the total output change. The step response of a purely integrating system is a ramp that changes linearly with time, i.e., it has a constant slope. Any point on the ramp can then be used as a pair of coordinates ( 𝑡 i , 𝑦 i ) for calculation of 𝑎 according to (7.34). KEH Process Dynamics and Control

7.5.1 Ziegler-Nichols’s recommendations
7.5 Tuning based on step response 7.5.1 Ziegler-Nichols’s recommendations In 1942, Ziegler and Nichols also suggested tunings for P, PI and PID controllers based on the information that can be obtained from a step test. Their recommendations for an ideal controller are given in Table 7.4. The method requires 𝐿>0 and preferably 0.1≤𝜃≤1. Table 7.4. Ziegler-Nichols’s controller tuning recommendations based on step response. Note that Ziegler-Nichols’s recommendations based on frequency response and step response do not necessarily give the same controller tuning for the same process. KEH Process Dynamics and Control

7.5 Tuning based on step response 7.5.2 The CHR method In 1952, Chien, Hrones and Reswick suggested improvements to Ziegler’s and Nichols’s recommendations based on a step response. The CHR- method gives different tunings for regulatory control and setpoint tracking tunings for aggressive control (with ~20 % overshoot) and cautious control (no overshoot) The method requires 𝐿>0 and preferably 0.1≤𝜃≤1. The CHR tunings (even the aggressive one) are less aggressive than the ZN tuning. Note that the different tunings for regulatory control and setpoint tracking can directly be used in a 2DOF controller. KEH Process Dynamics and Control

7.5.2 The CHR method Table 7.5. Controller tuning for regulatory control by the CHR method. Table 7.6. Controller tuning for setpoint tracking by the CHR method. KEH Process Dynamics and Control

7.5.3 Åström’s and Hägglund’s correlations
7.5 Tuning based on step response 7.5.3 Åström’s and Hägglund’s correlations In 2006, Åström and Hägglund presented improved controller tunings based on a step response. In addition to 𝑎 and 𝐿 , they use 𝜃 in their correlations, which can be used for all 𝜃≥0. However, for 𝜃<0.4 , they tend to be conservative. For an integrating process, 𝜃=0 is used. The tunings are primarily intended for regulatory control. For setpoint tracking, setpoint weighting can be used as follows: PI control: 𝑏=1 if 𝜃>0.4 , 𝑏<1 if 𝜃≤0.4 (optimal 𝑏 is unclear) PID control: 𝑏=1 if 𝜃>1 , 𝑏=0 if 𝜃≤1 Table Åström’s and Hägglund’s controller tuning correlations KEH Process Dynamics and Control

7.6 Model-based controller tuning
The controller tuning methods in Sections 7.4 and 7.5 employ parameters that can be determined from an experiment with an existing process. If a process model is known, the same parameters can be determined through a simulation experiment possibly by direct calculation from the process model For example, a first-order system with a time delay has the transfer function 𝐺 𝑠 = 𝐾 𝑇𝑠+1 e −𝐿𝑠 (7.36) from which the parameters 𝑎 and 𝜃 can be calculated according to 𝑎= 𝐾𝐿 𝑇 , 𝜃= 𝐿 𝑇 (7.37) The same tuning methods as in Sections 7.4 and 7.5 can then be used. However, the methods in Sections 7.4 and 7.5 are “general purpose” methods that are not optimized for any specific model type. For a given model, better controller tunings probably exist. KEH Process Dynamics and Control

7.6.1 First-order system with a time delay
7.6 Model-based controller tuning 7.6.1 First-order system with a time delay The transfer function is defined in (7.36) and the parameter 𝜃 in (7.37). Minimization of error integrals Controller tunings that minimize IAE and ITAE when 0.1≤𝜃≤1. Table 7.8. IAE and ITAE minimizing controller tunings for regulatory control. Table 7.9. IAE and ITAE minimizing controller tunings for setpoint tracking. KEH Process Dynamics and Control

7.6.1 First-order system with time delay Other optimality criteria The controller tunings for minimizing the error integrals IAE and ITAE in Tables 7.8 and 7.9 do not give any robustness guarantees. Thus, the control performance can be bad if the model contains errors. Cvejn (2009) has derived controller tunings that have a certain robustness even for systems with large time delays, i.e., for large 𝜃 values. Table Cvejn’s tunings for regulatory control and setpoint tracking. The PI controller tunings tend to give better robustness than the PID controller tunings, which tend to give better performance. KEH Process Dynamics and Control

7.6.2 Second-order no-zero system with a time delay
7.6 Model-based controller tuning 7.6.2 Second-order no-zero system with a time delay We shall consider second-order systems with a time delay but no zeros. Such a system has the transfer function 𝐺 𝑠 = 𝐾 𝜔 n 2 𝑠 2 +2𝜁 𝜔 n 𝑠+ 𝜔 n 2 e −𝐿𝑠 (7.40) In Cvejn’s method for tracking control, the controller 𝐺 c (𝑠) is tuned to give the loop transfer 𝐺 k (𝑠)=𝐺(𝑠) 𝐺 c (𝑠) such that 𝐺 k 𝑠 = 1 2𝐿𝑠 e −𝐿𝑠 (7.38) or 𝐺 k 𝑠 = 𝐿𝑠 e −𝐿𝑠 (7.39) Tuning by (7.38) gives better stability, (7.39) gives better performance. Exercise 7.3 Use Cvejn’s method for tracking control to tune a PID controller for the system (7.40). KEH Process Dynamics and Control

7.6.2 Second-order system with delay Overdamped system For an overdamped (or critically damped) second-order system, 𝜁≥1. In this case, (7.40) is more conveniently written as 𝐺 𝑠 = 𝐾 ( 𝑇 1 𝑠+1)( 𝑇 2 𝑠+1) e −𝐿𝑠 , 𝑇 1 ≥ 𝑇 2 (7.41) Cvejn’s method can be used also in this case, but Åström and Hägglund (2006) suggest the following tuning when the system is overdamped: 𝐾 𝐾 c = 𝜃 1 − 𝜃 2 − 𝜃 1 −1 𝜃 2 −1 𝐾 𝐾 c 𝐿/ 𝑇 i = 𝜃 1 −1 − 𝜃 2 − 𝜃 1 −1 𝜃 2 −1 (7.42) 𝐾 𝐾 c 𝑇 d /𝐿= 𝜃 1 −1 −0.2 𝜃 2 − 𝜃 1 −1 𝜃 2 −1 𝜃 1 + 𝜃 2 𝜃 1 + 𝜃 2 + 𝜃 1 𝜃 2 where 𝜃 1 =𝐿/ 𝑇 1 , 𝜃 2 =𝐿/ 𝑇 2 (7.43) KEH Process Dynamics and Control

Overdamped system Second-order system including integration A second-order no-zero system including an integrator has the transfer function 𝐺 𝑠 = 𝐾 𝑠 ( 𝑇 2 𝑠+1) e −𝐿𝑠 (7.44) For this kind of system, Åström and Hägglund (2006) suggest the tuning: 𝐾 𝐾 c 𝐿= 𝜃 2 −1 𝐾 𝐾 c 𝐿 2 / 𝑇 i = 𝜃 2 −1 (7.45) 𝐾 𝐾 c 𝑇 d = 𝜃 2 −1 If the system is a double integrator with the transfer function 𝐺 𝑠 = 𝐾 𝑠 2 e −𝐿𝑠 (7.46) the suggested tuning is 𝐾 𝐾 c 𝐿 2 =0.02 𝐾 𝐾 c 𝐿 3 / 𝑇 i = 𝐾 𝐾 c 𝑇 d 𝐿=0.28 KEH Process Dynamics and Control

7.7 Controller design by direct synthesis
In the previous sections, equations for controller tuning have been given for first- and second-order no-zero systems. The equations are usually the result of optimization of some criterion that is considered to imply “good control”. However, what is “good control” varies from case to case depending on the compromise between stability and performance. A drawback of the tuning equations is that the user cannot influence the tuning according to his/her opinion of “good control”. In this section, a method is introduced whereby the user can influence the controller tuning in a systematic way according to his/her opinion of “good control” more model types than in previous sections can be handled, e.g., systems with a zero KEH Process Dynamics and Control

7.7.1 Closed-loop transfer functions
7.7 Controller tuning by direct synthesis 7.7.1 Closed-loop transfer functions Consider the closed-loop system in the figure with the following transfer functions: – 𝐺 𝑠 process being controlled – 𝐺 c 𝑠 controller – 𝐺 d 𝑠 disturbance system Block diagram of closed-loop system Standard block diagram algebra gives 𝑌= 𝐺 𝐺 c 1+𝐺 𝐺 c 𝑅+ 𝐺 d 1+𝐺 𝐺 c 𝑉 (7.49) where 𝐺 r = 𝐺 𝐺 c 1+𝐺 𝐺 c , 𝐺 v = 𝐺 d 1+𝐺 𝐺 c (7.50,51) are the closed-loop transfer functions from the setpoint 𝑅 and the disturbance 𝑉 to the output 𝑌. The user can specify the desired 𝐺 r for setpoint tracking or 𝐺 v for regulatory control. For setpoint tracking, the required controller is given by 𝐺 c = 1 𝐺 𝐺 r (1− 𝐺 r ) (7.52) KEH Process Dynamics and Control

7.7.2 Low-order minimum-phase systems
7.7 Controller tuning by direct synthesis 7.7.2 Low-order minimum-phase systems First-order system A strictly proper first-order system without a time delay has the transfer function 𝐺= 𝐾 𝑇𝑠+1 (7.53) Assume that we want the controlled system to behave as a first-order system with the time constant 𝑇 r . Then, 𝐺 r = 1 𝑇 r 𝑠+1 , which gives 𝐺 r 1− 𝐺 r = 1 𝑇 r 𝑠 (7.54) Substitution of (7.53) and (7.54) into (7.52) gives 𝐺 c = 𝑇𝑠+1 𝐾 1 𝑇 r 𝑠 = 𝑇 𝐾 𝑇 r 1+ 1 𝑇𝑠 (7.55) which is a PI controller with the parameters 𝐾 c = 𝑇 𝐾 𝑇 r , 𝑇 i =𝑇 (7.56) Here, 𝑇 r is a design parameter, by which the performance of the control system can be affected. KEH Process Dynamics and Control

7.7.2 Low-order min-phase systems Second-order system with no zero A second-order system with no zero and no time delay has the transfer function 𝐺 𝑠 = 𝐾 𝜔 n 2 𝑠 2 +2𝜁 𝜔 n 𝑠+ 𝜔 n 2 (7.57) Even if the uncontrolled system is of second order, we can specify the controlled system to be of first order. Substitution of (7.54) and (7.57) into (7.52) then gives 𝐺 c = 𝑠 2 +2𝜁 𝜔 n 𝑠+ 𝜔 n 2 𝐾 𝜔 n 2 1 𝑇 r 𝑠 = 2𝜁 𝐾 𝜔 n 𝑇 r 1+ 𝜔 n 2𝜁𝑠 + 𝑠 2𝜁 𝜔 n (7.58) which is an ideal PID controller with the parameters 𝐾 c = 2𝜁 𝐾 𝜔 n 𝑇 r , 𝑇 i = 2𝜁 𝜔 n , 𝑇 d = 1 2𝜁 𝜔 n (7.59) Also here, 𝑇 r is a design parameter which only affects the controller gain. KEH Process Dynamics and Control

7.7.2 Low-order min-phase systems Overdamped second-order system with a LHP zero An overdamped second-order system with a zero in the left half of the complex plane (LHP) has the transfer function 𝐺 𝑠 = 𝐾( 𝑇 3 𝑠+1) ( 𝑇 1 𝑠+1)( 𝑇 2 𝑠+1) , 𝑇 𝑖 ≥0 (7.60) We can specify the controlled system to be of first order. Substitution of (7.54) and (7.60) into (7.52) gives 𝐺 c = ( 𝑇 1 𝑠+1)( 𝑇 2 𝑠+1) 𝐾( 𝑇 3 𝑠+1) 1 𝑇 r 𝑠 = 1 𝐾 𝑇 r 𝑠 𝑇 1 𝑇 2 𝑠 2 + 𝑇 1 + 𝑇 2 𝑠+1 𝑇 3 𝑠+1 = 1 𝐾 𝑇 r 𝑠 1+ 𝑇 1 + 𝑇 2 − 𝑇 3 𝑠+ 𝑇 1 𝑇 2 − 𝑇 1 + 𝑇 2 − 𝑇 3 𝑇 3 𝑇 3 𝑠+1 𝑠 2 or 𝐺 c = 𝐾 c 1+ 1 𝑇 i 𝑠 + 𝑇 d 𝑠 𝑇 f 𝑠+1 (7.61) where 𝐾 c = 𝑇 1 + 𝑇 2 − 𝑇 3 𝐾 𝑇 r , 𝑇 i = 𝑇 1 + 𝑇 2 − 𝑇 3 , 𝑇 d = 𝑇 1 𝑇 2 𝑇 1 + 𝑇 2 − 𝑇 3 − 𝑇 3 , 𝑇 f = 𝑇 3 (7.62) This is a PID controller with a derivative filter. KEH Process Dynamics and Control

7.7.3 High-order minimum-phase systems
7.7 Controller tuning by direct synthesis 7.7.3 High-order minimum-phase systems A high-order minimum-phase system with real poles and zeros, but with no time delay, has the transfer function 𝐺=𝐾 𝑗=𝑛+1 𝑛+𝑚 ( 𝑇 𝑗 𝑠+1) 𝑖=1 𝑛 ( 𝑇 𝑖 𝑠+1) , 𝑇 𝑖 >0 , 𝑇 𝑗 >0 , 𝑛> (7.63) If 𝑛=3 and 𝑚=0 or 1 , a closed-loop system of second order can be obtained by a full PID controller. If 𝑛>3, it is not possible to obtain a closed-loop system of lower order than 3 by a PID controller and an exact design by specifying 𝐺 r is thus not practical. In the case of 𝑛>3 , two possibilities are to specify a closed-loop system of first or second order and then to first calculate a 𝐺 c according to (7.52), then to approximate 𝐺 c by a PID controller; first approximate 𝐺 by a model of at most third order, then to calculate the PID controller according to (7.52). In Section 7.9, the latter approach will be described. KEH Process Dynamics and Control

7.7.4 Second-order system with RHP zero
7.7 Controller tuning by direct synthesis 7.7.4 Second-order system with RHP zero A second-order system with real poles and a right half plane (RHP) zero has the transfer function 𝐺 𝑠 = 𝐾( −𝑇 3 𝑠+1) ( 𝑇 1 𝑠+1)( 𝑇 2 𝑠+1) , 𝑇 𝑖 ≥ (7.71) Now division by 𝐺 in (7.52) will result in an unstable controller with a RHP pole if 𝐺 r is chosen as in the previous sections. One possible solution is to approximate the unstable controller by a stable controller. This tends to result in too aggressive control because the controller is then designed as if there were no RHP zero in 𝐺 . Another solution is to include the same RHP zero in 𝐺 r as in 𝐺 ; it will then be cancelled out in (7.52) and the controller will automatically be stable. This means that the choice of 𝐺 r is restricted, but otherwise the control performance tends to be as expected. In this section, the latter approach is used. KEH Process Dynamics and Control

7.7.4 System with RHP zero Closed-loop system of first order The closed-loop transfer function is chosen as 𝐺 r = − 𝑇 3 𝑠+1 𝑇 r 𝑠+1 , which gives 𝐺 r 1−𝐺 r = − 𝑇 3 𝑠+1 (𝑇 r + 𝑇 3 )𝑠 (7.72) Substitution of (7.71) and (7.72) into (7.52) gives 𝐺 c = ( 𝑇 1 𝑠+1)( 𝑇 2 𝑠+1) 𝐾 1 (𝑇 r + 𝑇 3 )𝑠 = 𝑇 1 + 𝑇 2 𝐾( 𝑇 r + 𝑇 3 ) 1+ 1 𝑇 1 + 𝑇 2 𝑠 + 𝑇 1 𝑇 2 𝑠 𝑇 1 + 𝑇 2 (7.73) which is a PID controller with the parameters 𝐾 c = 𝑇 1 + 𝑇 2 𝐾 (𝑇 r + 𝑇 3 ) , 𝑇 i = 𝑇 1 + 𝑇 2 , 𝑇 d = 𝑇 1 𝑇 2 𝑇 1 + 𝑇 2 (7.74) KEH Process Dynamics and Control

7.7.4 System with RHP zero Closed-loop system of second order A first-order system with a zero is proper, but not strictly proper. If a zero is present, a strictly proper system has to be at least second order. Hence, a more natural choice for 𝐺 r is 𝐺 r = (− 𝑇 3 𝑠+1) 𝜔 r 2 𝑠 2 +2 𝜁 r 𝜔 r 𝑠+ 𝜔 r 2 , which gives 𝐺 r 1− 𝐺 r = (− 𝑇 3 𝑠+1) 𝜔 r 2 𝑠(𝑠+2 𝜁 r 𝜔 r + 𝑇 3 𝜔 r 2 ) (7.75) To simplify the derivation of controller parameters, we define 𝑇 f =1/(2 𝜁 r 𝜔 r + 𝑇 3 𝜔 r 2 ) (7.76) Substitution of (7.71) and (7.75) into (7.52), gives with (7.76) 𝐺 c = ( 𝑇 1 𝑠+1)( 𝑇 2 𝑠+1) 𝑇 f 𝜔 r 2 𝐾 𝑇 f 𝑠+1 𝑠 = 𝑇 f 𝜔 r 2 𝐾𝑠 𝑇 1 𝑇 2 𝑠 2 + 𝑇 1 + 𝑇 2 𝑠+1 𝑇 f 𝑠+1 (7.77) Analogously with the derivation of (7.62), this gives the PID controller parameters (7.76) and 𝐾 c = 𝑇 f 𝜔 r 2 𝐾 ( 𝑇 1 + 𝑇 2 − 𝑇 f ), 𝑇 i = 𝑇 1 + 𝑇 2 − 𝑇 f , 𝑇 d = 𝑇 1 𝑇 2 𝑇 1 + 𝑇 2 − 𝑇 f − 𝑇 f (7.78) where 𝑇 f is the derivative filter time constant in a PID controller (7.61). KEH Process Dynamics and Control

Closed-loop system of 2nd order Choice of closed-loop system parameters In (7.75), there are two design parameters, the relative damping 𝜁 r , and the undamped natural frequency 𝜔 r . The meanings of these parameters are discussed in Section 5.3, especially Subsection The choice of design parameters can be simplified in the following two ways. Let 𝐺 r have two equally large real poles at −1/ 𝑇 r . This corresponds to 𝜁 r =1 and 𝜔 r =1/ 𝑇 r , which for (7.76) gives 𝑇 f = 𝑇 r 2 2 𝑇 r + 𝑇 3 Let 𝐺 r have real poles at −1/ 𝑇 r and −1/ 𝑇 3 . This corresponds to 𝜁 r =0.5( 𝑇 r + 𝑇 3 ) 𝜔 r and 𝜔 r =1/ 𝑇 r 𝑇 3 , which for (7.76) gives 𝑇 f = 𝑇 r 𝑇 3 𝑇 r + 2𝑇 3 KEH Process Dynamics and Control

7.7.5 First-order system with a time delay
7.7 Controller tuning by direct synthesis 7.7.5 First-order system with a time delay To illustrate how systems with a time delay can be handled by direct synthesis, a first-order system with a time delay will be studied. Such a system has the transfer function 𝐺 𝑠 = 𝐾 𝑇𝑠+1 e −𝐿𝑠 (7.79) Calculation of a controller by (7.52) will then result in a controller containing a time delay — there is no practical way to avoid this by the choice of 𝐺 r . There are methods to implement a controller resulting from (7.52) (see Section 7.8), but not by a regular PID controller. If a PID controller is desired, the time delay has to be approximated in some way. KEH Process Dynamics and Control

st order system with a delay Time-delay approximation in the model A standard way of approximating a time delay is to use a Padé approximation. I first-order Padé approximation e −𝐿𝑠 ≈ 1−0.5𝐿𝑠 1+0.5𝐿𝑠 (7.80) gives the model 𝐺 𝑠 = 𝐾(−0.5𝐿𝑠+1) (𝑇𝑠+1)(0.5𝐿𝑠+1) (7.81) A natural choice for 𝐺 r is then 𝐺 r = −0.5𝐿𝑠+1 (𝑇 r 𝑠+1)(0.5𝐿𝑠+1) , which gives 𝐺 r 1−𝐺 r = −0.5𝐿𝑠+1 𝑠(0.5 𝑇 r 𝐿𝑠+ 𝑇 r +𝐿) (7.82) Substitution of (7.81) and (7.82) into (7.52) gives a PID controller with the parameters 𝐾 c = 𝑇+0.5𝐿− 𝑇 f 𝐾 (𝑇 r +𝐿) , 𝑇 i =𝑇+0.5𝐿− 𝑇 f , 𝑇 d = 0.5𝐿𝑇 𝑇+0.5𝐿− 𝑇 f , 𝑇 f = 0.5𝐿 𝑇 r 𝑇 r +𝐿 (7.83) Here, 𝑇 f is the time constant of a derivative filter in the PID controller (7.61). KEH Process Dynamics and Control

st order system with a delay Time-delay approximation in the controller If e −𝐿𝑠 is retained in the model, it also has to be part of 𝐺 r , because it is impossible for the closed-loop system to have a shorter time-delay than the uncontrolled system. If 𝐺 r is chosen to be first order with a time delay 𝐺 r = 1 𝑇 r 𝑠+1 e −𝐿𝑠 , which gives 𝐺 r 1−𝐺 r = e −𝐿𝑠 𝑇 r 𝑠+1− e −𝐿𝑠 (7.84) Substitution of (7.79) and (7.84) into (7.52) gives 𝐺 c = 𝑇𝑠+1 𝐾( 𝑇 r 𝑠+1− e −𝐿𝑠 ) (7.85) Unfortunately, this controller cannot be implemented by a PID controller in a regular feedback loop. In order to do that, the time delay in (7.85) has to be approximated by a rational expression. If the approximation (7.80) is used, the controller parameters will be as in (7.83). The simpler approximation e −𝐿𝑠 ≈1−𝐿𝑠 gives a PI controller with 𝐾 c = 𝑇 𝐾 (𝑇 r +𝐿) , 𝑇 i =𝑇 (7.86) KEH Process Dynamics and Control

7.8 Internal model control
“Internal model control” (IMC) is closely related to “direct synthesis” (DS). As in DS, a model of the system to be controlled is explicitly built into the controller, but in a different way. An advantage with IMC is that it is easier to implement more complex control laws than regular PID controllers. For example, the controller transfer function (7.85) can easily be implemented exactly with IMC. Even if the controller design is based on IMC, it is often desirable to implement the controller as a regular PID controller. In such cases, the IMC approach offers better possibilities to deal with robustness issues than DS. KEH Process Dynamics and Control

7.8 Internal model control 7.8.1 The IMC structure Consider the closed-loop system in the figure with the following transfer functions: – 𝐺 𝑠 true process – 𝐺 𝑠 process model – 𝐺 IMC 𝑠 a controller – 𝐺 d 𝑠 disturbance system Standard block diagram algebra The IMC structure. gives 𝑈= 𝐺 IMC (𝐸+ 𝐺 𝑈) from which 𝑈 𝐸 = 𝐺 c = 𝐼− 𝐺 IMC 𝐺 −1 𝐺 IMC = 𝐺 IMC 𝐼− 𝐺 𝐺 IMC −1 = 𝐺 IMC 1− 𝐺 𝐺 IMC (7.87) Assume that 𝐺 IMC = 𝐺 −1 𝐺 f (7.88) where 𝐺 f is a “filter”. Substitution of (7.88) into (7.87) gives 𝐺 c = 𝐺 −1 𝐺 f 𝐼− 𝐺 f −1 = 1 𝐺 𝐺 f (1− 𝐺 f ) (7.89) If the filter is chosen as 𝐺 f = 𝐺 r (and 𝐺 =𝐺), this is the same as (7.52) ! KEH Process Dynamics and Control

7.8.2 Handling of time delays without approximation
7.8 Internal model control 7.8.2 Handling of time delays without approximation Consider a system modelled as a first-order system with a time delay, i.e., 𝐺 =𝐾 e −𝐿𝑠 /(𝑇𝑠+1). Choose the IMC filter as 𝐺 f = e −𝐿𝑠 /( 𝑇 r 𝑠+1) . Substitution into (7.88) now gives 𝐺 IMC = 1 𝐾 𝑇𝑠+1 𝑇 r 𝑠+1 = 1 𝐾 1+ 𝑇− 𝑇 r 𝑇 r 𝑠+1 𝑠 (7.90) which is a PD controller with a derivative filter having the parameters 𝐾 𝑐 =1/𝐾 , 𝑇 d =𝑇− 𝑇 r , 𝑇 f = 𝑇 r . Substitution of (7.90) and the model 𝐺 into (7.87) gives 𝐺 c = 𝑇𝑠+1 𝐾( 𝑇 r 𝑠+1− e −𝐿𝑠 ) (7.91) which identical with (7.85). The difference is that (7.91) can be implemented exactly with the IMC structure without time-delay approximation. Note that there is no integration in 𝐺 IMC , but the feedback of 𝐺 in the IMC structure introduces integration if 𝐺 IMC has been calculated using the same 𝐺 in (7.88); integration is achieved even if 𝐺 ≠𝐺 . Exercise. Calculate the closed-loop transfer function 𝐺 r when 𝐺 ≠𝐺 . Show that there will be no steady-state error, i.e., that 𝐺 r 0 =1 . KEH Process Dynamics and Control

7.8.3 The predictive character of the IMC structure
7.8 Internal model control 7.8.3 The predictive character of the IMC structure The previous block diagram of the IMC structure is drawn to emphasize how 𝐺 IMC combined with the feedback of 𝐺 is equivalent to 𝐺 c . The block diagram can also be drawn to emphasize the predictive character of the IMC structure, as shown below. (Note that the two diagrams are completely equivalent.) – The control signal is an input to the real system 𝐺 and the model 𝐺 . – 𝐺 predicts the output 𝑌 , which is compared with the true output 𝑌. – Only the prediction error 𝐸=𝑌− 𝑌 is fed back, not the entire 𝑌. The latter property is a clear advantage in controller design. If 𝐺 =𝐺 (i.e., 𝐸=0) 𝐺 r = 𝐺𝐺 IMC (7.93) which means that the closed- loop transfer function depends linearly on 𝐺 IMC making design of 𝐺 IMC easier than design of 𝐺 c . Predictive nature of IMC structure. KEH Process Dynamics and Control

7.8 Internal model control 7.8.4 Controller design The following conclusions can be drawn from (7.93): A stable closed-loop system 𝐺 r requires a stable IMC controller 𝐺 IMC ; in particular, the IMC controller may not contain integral action. Non-minimum phase properties (i.e., RHP zeros and time delays) in 𝐺 will also be present in 𝐺 r because they cannot be cancelled out by a stable and realizable 𝐺 IMC . From (7.88) it follows that the filter 𝐺 f has to be chosen to cancel out non-minimum phase properties of 𝐺 — this is equivalent to the choice of 𝐺 r in direct synthesis. In practice, the IMC design is done differently. Instead of guaranteeing the stability and realizability of 𝐺 IMC by the choice of 𝐺 f , it is handled by the choice of 𝐺 to be inverted: non-minimum phase parts of 𝐺 are not inverted. KEH Process Dynamics and Control

7.8.4 Controller design The process model 𝐺 can always be factorized as 𝐺 = 𝐺 ⊕ 𝐺 ⊖ (7.94) where 𝐺 ⊕ contains all non-minimum-phase elements of 𝐺 , but no minimum-phase elements, and normalized so that 𝐺 ⊕ 0 =1 (i.e., it has the static gain 1). This means that 𝐺 ⊕ contains all RHP zeros and time delays of 𝐺 ; if there are no such elements, 𝐺 ⊕ =1. When 𝐺 IMC is calculated according to (7.88), only 𝐺 ⊖ is inverted. Thus, 𝐺 IMC = 𝐺 ⊖ −1 𝐺 f (7.95) Note that the full 𝐺 should be used as internal model as illustrated by the IMC block diagrams — the use of 𝐺 ⊖ is only a technical aid for the calculation of 𝐺 IMC . The IMC filter 𝐺 f could be chosen as the desired closed-loop transfer function without any non-minimum phase elements (not even a time delay), but in practice a low-pass filter 𝐺 f = 1 ( 𝑇 r 𝑠+1) 𝑛 (7.96) is chosen. Here, 𝑛 is an integer, usually 𝑛=1, sometimes 𝑛>1. KEH Process Dynamics and Control

7.8.5 Implementation with a regular PID controller
7.8 Internal model control 7.8.5 Implementation with a regular PID controller An advantage of the IMC structure is that time delays can be handled exactly, but often a regular PID controller is preferred, because it is standard software in all automation systems. If an IMC controller 𝐺 IMC has been designed, the corresponding “regular” controller 𝐺 c can be calculated according to (7.87). If 𝐺 contains a time delay, it will also be present in 𝐺 c . In such cases, the time delay has to be approximated in a suitable way. Table 7.12 shows IMC-based tunings of regular PID controllers for some typical model structures. The tunings can also be used for models of lower degree or no time delay as long as 𝑇 1 >0 , 𝑇 2 ≥0 , 𝑇 3 ≥0 , 𝐿≥ (7.101) The tunings can be used for (underdamped) models expressed by the relative damping and the natural frequency by the substitutions 𝑇 1 + 𝑇 2 =2𝜁/ 𝜔 n , 𝑇 1 𝑇 2 =1/ 𝜔 n (7.103) Usually 𝑇 r is chosen such that 𝐿≤ 𝑇 r <𝑇 (but no clear consensus). KEH Process Dynamics and Control

7.8.5 Implementation with a PID controller Table IMC-based tuning of ideal PID controller. The desired time constant of the close-loop system is 𝑇 r . 𝜆 , which is used in the calculations, is closely related to 𝑇 r . Note that the calculated integral time 𝑇 i is used in several expressions. KEH Process Dynamics and Control

7.9 Model simplification Many controller tuning methods have been presented in the previous sections. Section 7.4: Controller tuning based on frequency-response parameters 𝐾 c,max , 𝑃 c (or 𝜔 c ) and 𝜅. These methods are “general- purpose methods” not optimized for any specific model type. Section 7.5: Controller tuning based on step-response parameters 𝑎 (or 𝑡 i , 𝑦 i ), 𝐿 and 𝜃. These methods are also general-purpose methods not optimized for any specific model type. Section 7.6: Model-based tuning optimized for given model structures and control criteria with no user interaction. Section 7.7: Direct synthesis for low-order models according to desired closed-loop response. Section 7.8: Internal model control mainly for low-order models according to desired closed-loop response. In this section, methods to reduce high-order models to first- or second- order models are presented. Any controller tuning method can be used. KEH Process Dynamics and Control

7.9 Model simplification 7.9.1 Skogestad’s method Skogestad and Grimholt (2012) have presented a method to simplify a high-order model with real poles and zeros to a first- or second-order model with a time delay but with no zeros. The transfer function to be simplified is factorized into a minimum-phase part 𝐺 ⊖ and a non-minimum-phase part 𝐺 ⊕ , i.e., 𝐺 𝑠 = 𝐺 ⊕ (𝑠) 𝐺 ⊖ (s) (7.106) Any left-half plane (LHP) zeros of 𝐺 ⊖ (s) and RHP zeros of 𝐺 ⊕ (𝑠) are eliminated by suitable approximations. Elimination of LHP zeros If the poles and zeros are real, the minimum-phase part has the form 𝐺 ⊖ 𝑠 = 𝐾 𝑇 𝑛+1 𝑠+1 𝑇 𝑛+2 𝑠+1 …( 𝑇 𝑛+𝑚 𝑠+1) 𝑇 1 𝑠+1 𝑇 2 𝑠+1 …( 𝑇 𝑛 𝑠+1) (7.107) where 𝑇 1 ≥ 𝑇 2 ≥…≥ 𝑇 𝑛 >0, 𝑇 𝑛+1 ≥ 𝑇 𝑛+2 ≥…≥ 𝑇 𝑛+𝑚 >0 , 𝑛>𝑚. The simplification procedure now goes as follows. The numerator time constants 𝑇 𝑛+1 , 𝑇 𝑛+2 , …, 𝑇 𝑛+𝑚 are considered in that order. Assume that 𝑇 𝑛+𝑗 is the one currently being considered. KEH Process Dynamics and Control

Elimination of LHP zeros Next, the smallest remaining denominator time constant 𝑇 𝑖 such that 𝑇 𝑖 ≥ 𝑇 𝑛+𝑗 is selected. If there is no such time constant, or if 𝑇 𝑖 ≫ 𝑇 𝑛+𝑗 , the smaller 𝑇 𝑖 closest to 𝑇 𝑛+𝑗 is chosen. It is considered that 𝑇 𝑖 ≫ 𝑇 𝑛+𝑗 if 𝑇 𝑖 > 𝑇 𝑛+𝑗 2 / 𝑇 𝑖+1 and 𝑇 𝑛+𝑗 / 𝑇 𝑖+1 <1.6 . The ratio ( 𝑇 𝑛+𝑗 𝑠+1)/( 𝑇 𝑖 𝑠+1) is now approximated as 𝑇 𝑛+𝑗 𝑠+1 𝑇 𝑖 𝑠+1 ≈ 𝑇 𝑛+𝑗 / 𝑇 𝑖 if 𝑇 𝑖 ≥ 𝑇 𝑛+𝑗 ≥5 𝑇 r a 𝑇 r / 𝑇 𝑖 5𝑇 r − 𝑇 𝑛+𝑗 𝑠+1 if 𝑇 𝑖 ≥5 𝑇 r ≥ 𝑇 𝑛+𝑗 b 𝑇 𝑖 − 𝑇 𝑛+𝑗 𝑠+1 if 𝑇 r ≥𝑇 𝑖 ≥ 𝑇 𝑛+𝑗 c 𝑇 𝑛+𝑗 / 𝑇 𝑖 if 𝑇 𝑛+𝑗 ≥ 𝑇 𝑖 ≥ 𝑇 r (d) 𝑇 𝑛+𝑗 / 𝑇 r if 𝑇 𝑛+𝑗 ≥ 𝑇 r ≥ 𝑇 𝑖 (e) if 𝑇 r ≥𝑇 𝑛+𝑗 ≥ 𝑇 𝑖 (f) (7.108) Here, 𝑇 r is the desired closed-loop time constant. If this is not known, the suggested value is 𝑇 r = 𝐿 , which is the time delay in the simplified model. Since this is not initially known, one may have to iterate (i.e., first guessing 𝐿 , then possibly correcting with the new 𝐿 ). KEH Process Dynamics and Control

Elimination of LHP zeros The above procedure gives an approximate minimum-phase part 𝐺 ⊖ of the form 𝐺 ⊖ 𝑠 = 𝐾 𝑇 1 𝑠+1 𝑇 2 𝑠+1 …( 𝑇 𝑛 𝑠+1) (7.109) Note that the gain as well as the values and number of denominator time constants may have changed from the original 𝐺 ⊖ . Elimination of RHP zeros and the half rule The transfer function 𝐺 𝑠 = 𝐺 ⊕ (𝑠) 𝐺 ⊖ (s) now has the form 𝐺 𝑠 = 𝐾 − 𝑇 𝑛+𝑚+1 𝑠+1 − 𝑇 𝑛+𝑚+2 𝑠+1 …(− 𝑇 𝑛+𝑚+𝑝 𝑠+1) 𝑇 1 𝑠+1 𝑇 2 𝑠+1 …( 𝑇 𝑛 𝑠+1) e −𝐿𝑠 (7.110) where 𝑇 1 ≥ 𝑇 2 ≥…≥ 𝑇 𝑛 >0, 𝑇 𝑛+𝑚+1 ≥ 𝑇 𝑛+𝑚+2 ≥…≥ 𝑇 𝑛+𝑚+𝑝 >0 . Skogestad’s half rule If an approximate model of order 𝑛 is desired, the 𝑛 largest denominator time constants are retained in the model with the modification that half of 𝑇 𝑛 +1 is added to 𝑇 𝑛 . Half of 𝑇 𝑛 +1 is also added to the time delay as well as all remaining smaller denominator time constants. In addition, all negative numerator time constants are subtracted from the time delay. KEH Process Dynamics and Control

Elimination of RHP zeros and the half rule Approximation by first-order system If a first-order model is desired, the half rule gives 𝐺 𝑠 = 𝐾 𝑇 𝑠+1 e − 𝐿 𝑠 (7.111a) 𝑇 = 𝑇 𝑇 2 , 𝐿 =𝐿+ 1 2 𝑇 2 + 𝑖=3 𝑛 𝑇 𝑖 + 𝑗=𝑛+𝑚+1 𝑛+𝑚+𝑝 𝑇 𝑗 (7.111b) Approximation by second-order system If a second-order model is desired, the half rule gives 𝐺 𝑠 = 𝐾 ( 𝑇 1 𝑠+1)( 𝑇 2 𝑠+1) e − 𝐿 𝑠 (7.112a) 𝑇 2 = 𝑇 𝑇 3 , 𝐿 =𝐿+ 1 2 𝑇 3 + 𝑖=4 𝑛 𝑇 𝑖 + 𝑗=𝑛+𝑚+1 𝑛+𝑚+𝑝 𝑇 𝑗 (7.112b) KEH Process Dynamics and Control

7.9.1 Skogestad’s method Example 7.2. IMC via model reduction by Skogestad’s method. Simplify the model 𝐺 𝑠 = (16𝑠+1)(4𝑠+1)(−8𝑠+1) e −2𝑠 (50𝑠+1)(20𝑠+1)(12𝑠+1)(6𝑠+1)(3𝑠+1)(𝑠+1) to a second-order model by Skogestad’s method and determine the parameters of a PID controller by IMC-based tuning for this model. Use a first-order filter time constant 𝑇 r =10. Here 𝐺 ⊖ 𝑠 = (16𝑠+1)(4𝑠+1) (50𝑠+1)(20𝑠+1)(12𝑠+1)(6𝑠+1)(3𝑠+1)(𝑠+1) . According to (7.108c), 16𝑠+1 20𝑠+1 ≈ 1 4𝑠+1 . The numerator factor (4𝑠+1) can now be cancelled out against the new denominator factor, which gives 𝐺 ⊖ 𝑠 = 1 (50𝑠+1)(12𝑠+1)(6𝑠+1)(3𝑠+1)(𝑠+1) and 𝐺 𝑠 = (−8𝑠+1) e −2𝑠 (50𝑠+1)(12𝑠+1)(6𝑠+1)(3𝑠+1)(𝑠+1) . KEH Process Dynamics and Control

Example 7.2 The resulting second-order model is 𝐺 𝑠 = 𝐾 ( 𝑇 1 𝑠+1)( 𝑇 2 𝑠+1) e − 𝐿 𝑠 with 𝑇 1 =50 , 𝑇 2 = ⋅6=15 , 𝐿 = ⋅ =17. Thus 𝐺 𝑠 = 1 (50𝑠+1)(15𝑠+1) e −17𝑠 . According to Table 7.12 for IMC-based tuning of second-order model: – 𝜆= 𝑇 r + 𝐿 =10+17=27 – 𝑇 i = 𝑇 1 + 𝑇 2 =50+12=62 – 𝐾 c = 𝑇 i /( 𝐾 𝜆)=62/(1⋅27)=2.3 – 𝑇 d = 𝑇 1 𝑇 2 / 𝑇 i =50⋅15/62 =12.1 KEH Process Dynamics and Control

7.9.2 Isaksson’s and Graebe’s method
7.9 Model simplification 7.9.2 Isaksson’s and Graebe’s method Isaksson and Graebe (1999) have presented a method to simplify a high- order model, where the fast and slow dynamics are combined to yield a model with a desired number of poles and zeros. If the original model contains a time delay, it is either left intact or substituted by a Padé approximation. To describe the method, both factorized and polynomial forms of the original transfer function are employed. If the numerator order is 𝑚 and the denominator order is 𝑛 , the transfer function is 𝐺 𝑠 =𝐾 𝑇 𝑛+1 𝑠+1 𝑇 𝑛+2 𝑠+1 …( 𝑇 𝑛+𝑚 𝑠+1) 𝑇 1 𝑠+1 𝑇 2 𝑠+1 …( 𝑇 𝑛 𝑠+1) (7.113a) =𝐾 𝑏 0 𝑠 𝑚 +…+ 𝑏 𝑚−2 𝑠 2 + 𝑏 𝑚−1 𝑠+1 𝑎 0 𝑠 𝑛 +…+ 𝑎 𝑛−2 𝑠 2 + 𝑎 𝑛−1 𝑠+1 (7.113b) where 𝑇 1 ≥ 𝑇 2 ≥…≥ 𝑇 𝑛 >0 (i.e., a stable system) and | 𝑇 𝑛+1 |≥ |𝑇 𝑛+2 |≥…≥ |𝑇 𝑛+𝑚 | . The numerator time constants can be positive or negative. KEH Process Dynamics and Control

7.9.2 Isakssons’s and Graebe’s method If a model with the numerator order 𝑚 and the denominator order 𝑛 is desired, the simplified model is 𝐺 𝑠 =𝐾 𝑇 𝑛+1 𝑠+1 …( 𝑇 𝑛+ 𝑚 𝑠+1) + 𝑏 𝑚− 𝑚 𝑠 𝑚 +…+ 𝑏 𝑚−1 𝑠+1 𝑇 1 𝑠+1 …( 𝑇 𝑛 𝑠+1) + 𝑎 𝑛− 𝑛 𝑠 𝑛 +…+ 𝑎 𝑛−1 𝑠+1 (7.114) Complex-conjugated poles or zeros is no problem, except if they occur as poles number 𝑛 and 𝑛 +1 or zeros number 𝑛+ 𝑚 and 𝑛+ 𝑚 +1. One solution is then to use the real part of the complex conjugate as 𝑇 𝑛 or 𝑇 𝑛+ 𝑚 . If the model is to be used for controller tuning, a strictly proper first- or second-order model, possibly with a time delay, is usually desired. Then 𝐺 𝑠 = 𝐾 1 2 𝑇 1 + 𝑎 𝑛−1 𝑠+1 (1st order) (7.115a) 𝐺 𝑠 = 𝐾 1 2 𝑇 𝑛+1 + 𝑏 𝑚−1 𝑠 𝑇 1 𝑇 2 + 𝑎 𝑛−2 𝑠 𝑇 1 + 𝑇 2 + 𝑎 𝑛−1 𝑠+1 (2nd order) (7.115b) where 𝑏 𝑚−1 = 𝑗=1 𝑚 𝑇 𝑛+𝑗 , 𝑎 𝑛−1 = 𝑖=1 𝑛 𝑇 𝑖 , 𝑎 𝑛−2 = 1 2 𝑖=1 𝑛 𝑇 𝑖 2 − 𝑖=1 𝑛 𝑇 𝑖 2 (7.116) KEH Process Dynamics and Control

7.9.2 Isakssons’s and Graebe’s method Example 7.3. IMC via model reduction by Isaksson–Graebe’s method. Solve the same problem as in Example 7.2 by Isaksson’s and Graebe’s model reduction method. The model gives 𝑏 𝑚−1 =16+4−8=12 , 𝑎 𝑛−1 = =92 𝑎 𝑛−2 = −( ) =2687 from which 𝐺 𝑠 = 𝑠 𝑠 𝑠+1 e −2𝑠 = (14𝑠+1) e −2𝑠 𝑠 2 +81𝑠+1 This model has complex-conjugated poles, but according to (7.103), 𝑇 1 + 𝑇 2 =81 and 𝑇 1 𝑇 2 = can be used in the controller calcula-tions. Table 7.12 for IMC-based tuning of second-order model then gives – 𝜆= 𝑇 r +𝐿=10+2=12 – 𝑇 i = 𝑇 1 + 𝑇 2 − 𝑇 3 =81−14=67 – 𝐾 c = 𝑇 i /(𝐾𝜆)=67/(1⋅12)=5.6 (much bigger than in Ex. 7.2!) – 𝑇 d = 𝑇 1 𝑇 2 / 𝑇 i − 𝑇 3 =1843.5/67 −14=13.5 KEH Process Dynamics and Control

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