individual loops [5], but cannot strictly be generalised to larger-dimensional systems (without making the interaction measure dependent on the controller). The Niederlinsk Niederlinskii index applies applies also to systems systems of large dimension, dimension, but it is this authors (subjective) opinion that it is of less use than the RGA.
5
Tun uning ing of dec decen entra tralize lized d con controller trollers s
5.1 5.1
Intr In trodu oduct ctio ion n
In this section, we consider the case when the control con figuration is fixed, and focus focus on fully fully decen decentr tral aliz ized ed contr control ol.. That That is, it is assum assumed ed that the overall controller consists of multiple single-input, single-output controllers, and the pairing of manipulated and controlled variables has been determined. Despite Despite the prevalence prevalence of decentraliz decentralized ed controllers controllers in industry industry,, the tuning (determination of controller parameters) of decentralized controllers is not a solved solved problem problem in mathem mathemati atical cal terms. terms. The well well establi establishe shed d controller synthesis synthesis methodologies, methodologies, like like H 2− or H −optimal optimal control, control, cannot handle a pre-specified struct structur uree for the cont contro roll ller. er. In fact, fact, a trul truly y H 2 − or H −optimal decentralized controller would have an infinite number of states[4 states[40]. 0]. This This follow follows, s, since since these controll controller er synthesis synthesis procedure proceduress result result in controll controllers ers which which have have the same number number of states states as the ’plant ’plant’. ’. When When synthesizing one decentralized controller element, all the other decentralized controllers would become a part of the ’plant’ as seen from the controller to be synthesized, and this controller element would therefore have a large numb number er of states states.. No Now, w, with this this new control controller ler in operation, operation, it becomes becomes a part of the ’plant’ as seen from the other controllers, and the other controllers may therefore be improved - thereby introducing yet more states. Sourlas et al. have looked at l1 -optimal 13 decentrali decentralized zed control control [47, 46], and have developed a method for calculating the best achievable decentralized performance, both for decentralized control in general and for fixed order decentraliz decentralized ed controller controllers. s. Ho Howev wever, er, the computations computations invol involved ved are rather rather complex, and may well become hard to solve even for problems of moderate dimension. dimension. In the absence of any decentraliz decentralized ed controller controller synthesis synthesis method that that has both b oth solid theoreti theoretical cal foundatio foundation n and is easily easily applicabl applicable, e, a few practical approaches have been developed: ∞
∞
13 In
l1 -optimal control, the ratio ky(t)k / kd(t)k ∞
47
∞
is minimized.
• Independent design. The individual decentralized controller elements are designed independently, but bouds on the controller designs are sought which ensure that the overall system will behave acceptably. • Sequential design. The controller elements are designed sequentially, and the controllers that have been designed are assumed to be in operation when the next controller element is designed. • Simultaneous design. Optimization is used to simultaneously optimize the controller parameters in all decentralized controller elements. A particular controller parametrization (e.g. PI-controllers) have to be chose a priori. In the following, these three tuning approaches will be described in some detail, but first some methods for tuning conventional single-loop controllers will be reviewed.
5.2
Tuning of single-loop controllers
There are a number of methods for tuning single-loop controllers, and no attempt will be made here at providing a comprehensive review of such tuning methods. Instead, a few methods will be described, which all are based on simple experiments or simple models, and do not require any frequencydomain analysis (although such analysis may enhance understanding of the resulting closed loop behaviour).
5.2.1
Ziegler-Nichols closed-loop tuning method
This tuning method can be found in many introductory textbooks, and is probably the most well-known tuning method. It is based on a simple closed loop experiment, using proportional control only. The proportional gain is increased until a sustained oscillation of the output occurs (which neither grows nor decays significantly with time). The proportional gain giving the sustained oscillation, K u , and the oscillation period (time), T u , are recorded. The proposed tuning parameters can then be found in Table 1. In most cases, increasing the proportional gain will provide a sufficient disturbance to initiate the oscillation - measurement noise may also do the trick. Only if the output is very close to the setpoint will it be necessary to introduce a setpoint change after increasing the gain, in order to initiate an oscillation. Note that for controllers giving positive output signals, i.e., controllers giving 48
output signals scaled in the range 0−1 or 0%−100%, a constant bias must be included in the controller in addition to the proportional term, thus allowing a negative proportional term to have an eff ect. Otherwise, the negative part of the oscillation in the plant input will be cut o ff , which would also eff ect the oscillation of the output - both the input and output of the plant may still oscillate, but would show a more complex behaviour than the singlefrequency sinusoids that the experiment should produce. Table 1. Tuning parameters for the closed loop Ziegler-Nichols method
Controller type Gain, K P Integral time, T I Derivative time, T D P 0.5 · K u PI 0.45 · K u 0.85 · T u PID 0.6 · K u 0.5 · T u 0.12 · T u Essentially, the tuning method works by identifying the frequency for which there is a phase lag of 180 . In order for the tuning method to work, the system to be controlled must therefore have a phase lag of 180 in a reasonable frequency range, and with a gain that is large enough such that the proportional controller is able to achieve a loop gain of 1 (0 dB). These assumptions are fulfilled for many systems. The tuning method can also lead to ambigous results for systems with a phase lag of 180 at more than one frequency. This would apply for instance to a system with one slow, unstable time constant, and some faster, but stable time constants. Such a system would have a phase lag of 180 both at steady state and at some higher frequency. It would then be essential to find the higher of these two frequencies. Furthermore, the system would be unstable for low proportional gains, which could definitely lead to practical problems in the experiment, since it is common to start the experiment with a low gain. Despite its popularity, the Ziegler-Nicols closed loop tuning rule is often (particularly in the rather conservative chemical processing industries) considered to give somewhat aggressive controllers. ◦
◦
◦
◦
5.2.2
Tuning based on the process reaction curve
In process control, the term ’reaction curve’ is sometimes used as a synonym for a step response curve. Many chemical processes are stable and well damped, and for such systems the step response curve can be approximated by a first-order-plus-deadtime model, i.e., Ke θs (25) y(s) = u(s) 1 + Ts and it is relatively straight forward to fit the model parameters to the observed step response. This is illustrated in Figure 10. Assuming that −
49
Output
A
0
Time q
T
Figure 10: Estimating model parameters from the process reaction curve. the response in Fig. 10 is the result of a step of size B at time 0 in the manipulated variable, the model parameters are found as follows: 1. Locate the in fl ection point , i.e., the point where the curve stops curving upwards and starts to curve downwards. 2. Draw a straight line through the inflection point, with the same gradient as the gradient of the reaction curve at that point. 3. The point where this line crosses the initial value of the output (in Fig.10 this is assumed to be zero) gives the apparent time delay θ. 4. The straight line reaches the steady state value of the output at time T + θ. 5. The gain K is given by A/B.
Ziegler and Nichols [51] propose the tuning rules in Table 2 based on the model in Eq. (25). Ziegler-Nichols open loop tuning
Table 2. Tuning parameters for the open loop Ziegler-Nichols method
Controller type Gain, K P Integral time, T I Derivative time, T D T P K θ θ 0.9T PI K θ 0.3 θ 4T PID 0.5θ 3K θ 0.5 50
Corresponding conventional controller, K Reference _
IMC controller, Q
Manipulated variable
Plant, G
Plant model, Gm
Controlled variable
+ _
Figure 11: An internal model controller. Cohen and Coon [10] have modified the ZieglerNichols open loop tuning rules. The modifications are quite insignificant when the deadtime θ small relative to the time constant T , but can be important for large θ. The Cohen-Coon tuning parameters are given in Table 3. Cohen-Coon tuning
Table 3. Tuning parameters for Cohen-Coon method
Controller type P PI PID
5.2.3
Gain, K P Integral time, T I Derivative time, T D T θ (1 + 3T ) K θ 30+3θ/T θ T θ ( 9+20θ/T ) K θ (0.9 + 12T ) 32+6θ/T θ T 4 4 θ ( 13+8θ/T ) θ 11+2θ/T K θ ( 3 + 4T )
IMC-PID tuning
In internal model control (IMC), the controller essentially includes a process model operating in ”parallel” with the process, as illustrated in Figure 1. The IMC controller Q and the corresponding conventional feedback controller K are related by K = Q(1 − GmQ)
1
−
(26)
Note that if the model is perfect, Gm = G, IMC control essentially results in an open loop control system. This means that it is not straight forward to use it for unstable systems, but for stable systems (and a perfect model) any stable IMC controller Q results in a stable closed loop system - this holds also for non-linear systems. In the following discussion on IMC controllers we 51