10.450 Process Dynamics, Operations, and Control Lecture Notes - 21 Lesson 21. Controllers 21.0 Context Once upon a time, a controller was a box that was installed near the sensor and valve. Later it was a card and display device inserted in a rack in a central control room. Now it can be a computer code that runs on a processor. This lesson describes some of the ways in which the ideal PID algorithm is actually deployed in operations. 21.1 Hardware and software The first automatic controllers were mechanical devices that behaved like the PID equation. Design of the device, and tuning during operation, depended on mechanical properties of materials (e.g., spring constant, elastic modulus) and geometric arrangement (e.g., spacing, clearance, position of a set screw). For example, the float and lever arrangement in the toilet tank is a proportional controller; the set point is always chosen such that the valve closes trying to reach it. One can still find pneumatic controllers in the process industries. Nozzles, baffles, and bellows operate upon a compressed air supply to regulate the compressed air signal given to the control valve. The next generation of controllers used electronic elements to represent the controller algorithms. These devices are classified as analog controllers - they process their signals continuously, and are expressed in dedicated hardware. Availability of powerful digital computers has allowed the “controller” to become a program executing on a computer. These digital controllers receive their input signals and deliver their output instructions intermittently. By virtue of being a code, rather than hardware, the algorithms may be more diverse, intricate, and flexible. 21.2 PID algorithm modifications We defined the ideal PID algorithm as an intuitive response to the presence of an error. * Gc (s) = xCO (s) = Kc  1 + T 1 is + Tds  (21.2.1)*ε (s) Some of the older controllers, constrained by hardware, executed the derivative and integral modes in series, rather than parallel as implied by (21.2.1) * Gc ′ (s) = xCO (s) = Kc ′ 1+ 1 1+ Td ′ s  (21.2.2)ε*(s)  Ti ′ s  110.450 Process Dynamics, Operations, and Control Lecture Notes - 21 Because a change to, e.g., the derivative time Td’ will affect the integral action, this PID algorithm is called interacting. Controllers that implement something more like the ideal (21.2.1) are termed non-interacting. The mode interaction can be seen by expressing (21.2.2) in the form of (21.2.1).  1 ′ Gc ′ (s) = Kc ′1+ Ti ′ s + Td ′ s + TTdi ′ ss   (21.2.3)′ ′ = Kc ′1+ Td ′′  1+ 1 + TiTd s   Ti  Ti ′+ Td ′s Ti ′+ Td ′   Thus an interacting controller set at Kc’, Ti’, Td’ will behave like a non-interacting controller set at K = K ′1+ Td ′  c c  Ti ′  ′ ′ Ti = Ti + Td (21.2.4) ′ ′ Td = T ′ i Td Ti + Td ′ Notice that TiTd = Ti’Td’. While (21.2.4) shows that a given interacting setting has an equivalent non-interacting setting, the converse is not necessarily true. That is, one cannot always take a given tuning recommendation (Kc, Ti, Td) and find a setting (Kc’, Ti’, Td’) to execute it on an interacting controller. To see this, invert (21.2.4) to find ′ Td = 42d ii i TTTT− ± (21.2.5)2 2 Equation (21.2.5) shows that real values of Td’ are available only if Td ≤ Ti (21.2.6)4 If Td = Td’ = 0, then the interacting and non-interacting algorithms are equivalent PI controllers, with Kc = Kc’ and Ti = Ti’. For any Td between this lower limit and the constraint of (21.2.6), equivalent performance can be obtained on both algorithms. For larger Td, however, the integrating 210.450 Process Dynamics, Operations, and Control Lecture Notes - 21 controller cannot duplicate the non-interacting controller. Shinskey (1996) recommends that interacting controllers be tuned so that ′ Td = 1 (21.2.7)′ Ti 3 This implies 3Kc = 1.33Kc ′ and TTdi = 16 (21.2.8) 21.3 P-mode terminology, units, and output limits On some controllers, the gain is increased by decreasing a quantity known as the proportional band P, where 100Kc = P (21.3.1) In keeping with the notion that PID is an intuitive response to error, irrespective of the nature of the controlled variable, useful units of gain would be Kc () %output (21.3.2)= %input In many cases it would be appropriate to have response capability equivalent to expected deviation, so that the magnitude of Kc would be around unity. If, however, the controller were programmed to have ‘engineering units’ (e.g., gpm/ºC), the magnitude of Kc set on the controller could be considerably different from 1. When working with a particular controller, be sure to understand the units of the gain, so that the magnitude of the setting can be properly interpreted. Even though the PID algorithm (21.1.1) can call for arbitrarily large values of output xCO, the valve can only range between 0 and 100% open. Hence, computations of xCO will normally be confined to the extreme values should they exceed this range. (As a practical matter, this limitation would occur anyway in the physical valve, but applying the limit in computations would be very important when using a simulation to tune a controller.) 310.450 Process Dynamics, Operations, and Control Lecture Notes - 21 21.4 I-mode terminology and reset windup On some controllers, the integral effect is increased by increasing a quantity known as the reset rate R, where 1R = Ti (21.4.1) The units of R are () repeats (21.4.2)R = time referring to the integral mode ‘repeating’ the effect of the proportional mode for a persistent constant error. That is, if the initial controller output is Kcε, after a duration of Ti with no change in error, the controller output is 2Kcε, or one “repeat”. The assumption of the integral mode is that the controller output gets stronger, as needed, until the error is driven to zero. In normal circumstances, the manipulated variable does influence the controlled variable, and the integral mode is effective in countering error. However, if something is wrong - the valve stuck, the disturbance too large for the manipulated variable to correct, some other fault in the process - the controlled variable