Title ⋆

First A. Author^{∗} Second B. Author, Jr.^{∗∗}

Third C. Author^{∗∗∗}

∗ National Institute of Standards and Technology, Boulder, CO 80305 USA (e-mail: author@ boulder.nist.gov).

∗∗ Colorado State University, Fort Collins, CO 80523 USA (e-mail:

author@lamar. colostate.edu)

∗∗∗ Electrical Engineering Department, Seoul National University, Seoul, Korea, (e-mail: author@snu.ac.kr)

Abstract: abs

Keywords: Five to ten keywords, preferably chosen from the IFAC keyword list.

Fig. 1. Block diagram of feedback control system.

In this chapter we consider an input (“load”) disturbance (g_{d} = g).

1. INTRODUCTION

Notation. The notation is summarized in Figure 1. Here u is the manipulated input (controller output), d the disturbance, y the controlled output, and y_{s} the setpoint (reference) for the controlled output. g(s) = \frac{Δy}{Δu} denotes the process transfer function and c(s) is the feedback part of the controller. Note that all the variables u, d and y are deviations from the initial steady state, but the Δ used to indicate deviation variables is usually omitted. Similarly, the Laplace variable s is often omitted to simplify notation. The settings given in this chapter are for the series (cascade, “interacting”) form PID controller:

Series PID : c(s) = K_{c} · (\frac{τ_{I}s + 1}{τ_{I}s}) · (τ_{D}s + 1) = \frac{K_{c}}{τ_{I}s} (τ_{I}τ_{D}s^{2} + (τ_{I} + τ_{D})s + 1)(1)

where K_{c} is the controller gain, τ_{I} the integral time, and τ_{D} the derivative time. The reason for using the series form is that the PID rules with derivative action are then much simpler. The corresponding settings for the ideal (parallel form) PID controller are easily obtained using (??).

2. SIMC RULE FOR PI CONTROL

For a first-order model

g_{1}(s) = \frac{k}{(τ_{1}s + 1)}e^{−θs} (2)

the SIMC method results in a PI controller with settings

K_{c} = \frac{1}{k} \frac{τ_{1}}{τ_{c} + θ} = \frac{1}{k′} \frac{1}{τ_{c} + θ} (3)

τ_{I} = min{τ_{1}, 4(τ_{c} + θ)} (4)

The desired first-order closed-loop time constant τ_{c} is the only tuning parameter.

3. OPTIMAL PI TUNING

How good are the SIMC PI rules, that is, how much room is there for improvements? To study this, we compare the SIMC PI performance, with τ_{c} as a parameter, to the “Pareto-optimal” PI-controller. Pareto-optimality applies to multiobjective problems, and means that no further improvement can be made in objective 1 (output performance in our case) without sacrificing objective 2 (robustness and input usage in our case).

3.1 Evaluation of performance and robustness

Output performance: We choose to quantify performance in terms of the integrated absolute error in response to a setpoint change (IAE_{ys}) and to an input “load” disturbance (IAE_{d}). The setpoint performance is often referred to as the “servo” behavior and the disturbance (in this case the input “load” disturbance) performance is often referred to as “regulator” behavior.

IAE = ∫_{0}^{∞}|y(t) − y_{s}(t)|dt

⋆ Sponsor and financial support acknowledgment goes here. Paper titles should be written in uppercase and lowercase letters, not all uppercase.

Input performance:

TV = ∫_{0}^{∞}|\frac{du}{dt}|dt

Robustness: We choose to quantify robustness and input usage in terms of the sensitivity peak M_{s}. We also considered other “robustness” measures, for example, the relative delay margin as suggested by ?, but we choose to use M_{s}. One reason is that we found that the M_{s}- value correlates well with the input usage as given by its total variation (TV), which agrees with the findings of ?. Such a correlation is reasonable since a large M_{s}- value corresponds to an oscillatory system with large input variations.

IAE, TV and Ms + (figures TV vs MS and ISE vs MS)

3.2 Cost Function

It may be argued that a two-degree of freedom controller (“feedforward action”) may be used to improve the response for setpoints, but note that a setpoint change is equivalent to an output disturbance (with g_{d} = 1 in Figure 1) which can only be counteracted by feedback. Thus, both setpoint changes (output disturbances) and input disturbances should be included when evaluating performance, and to get a good balance between the two, we weigh them about equally by defining the following performance cost

J(c) = 0.5 [\frac{IAE_{ys}(c)}{IAE_{ys}^{∘}} + \frac{IAE_{d}(c)}{IAE_{d}^{∘}}] (5)

where the reference values, IAE_{ys}^{∘} and IAE_{d}^{∘}, are for IAEoptimal PI-controllers (with M_{s} = 1.59) for a setpoint change and input disturbance, respectively. We could have used the truly optimal IAE-value as the reference when computing J (without the restriction M_{s} = 1.59), but this would not have changed the results much because the IAEvalue is anyway quite close to its minimum at M_{s} = 1.59.

Table ?? gives the tunings and reference values obtained using IAE-optimal PI-controllers (with M_{s} = 1.59) for four different processes, and Table ?? gives the tunings, costs J and M_{s}-values for the SIMC PI-controller (with τ_{c} = θ). Importantly, the weighted cost J is independent of the process gain k and the disturbance magnitude, and also of the unit used for time. Note that two different optimal PI-controllers are used to obtain the two reference values, whereas a single controller c is used to find IAE_{ys}(c) and IAE_{d}(c) when evaluating the weighted IAE-cost J(c).

3.3 Pareto-Optimal Tuning Curve

intro to Pareto-Optimal, Main Trade-offs(high gain-low gain): performance vs robustness, and tracking vs. regulation. Point out that outp.perf and robustness nota always competing (figure pareto-optimal curve for FOPTD proses, with shaded area of unwanted region)

3.4 Effect of Weighting on Integrating Process

Why is the optimal to have IAEd closed to IAEd0 (show table that illustrates sensitivity)

4. OPTIMALITY OF SIMC RULE

Figure 2 shows the trade-off between performance (J) and robustness (M_{s}) for the SIMC PI-controller (blue solid curve) and the Pareto-optimal controller (dashed black curve) for four different processes: pure time delay (τ_{1}/θ = 0), small time constant (τ_{1}/θ = 1), intermediate time constant (τ_{1}/θ = 8), and integrating process (τ_{1}/θ = ∞). The curve for the SIMC controller was generated by varying the tuning parameter τ_{c} from a large to a small value. The controllers corresponding to the choices τ_{c} = 1.5θ (smoother), τ_{c} = θ (recommended) and τ_{c} = 0.5θ (aggressive) are shown by circles. The Pareto-optimal curve was generated by finding for each value of M_{s}, the optimal PI-controller c, with the smallest IAE-value J(c). Except for the pure time delay process, the differences between the J-values for SIMC (blue solid curve) and optimal (dashed black curve) are small (within 10%), which shows that the SIMC PI-rules are close to optimal.

Note that we have a real trade-off between performance (J) and robustness (M_{s}) only when there is a negative slope between these variables (in the left region in the figures in Figure 2). We never want to be in the region with a zero or positive slope (to the right in the figures), because here we can improve both performance (J) and robustness (M_{s}) at the same time with another choice for the tuning parameter (using a larger value for τ_{c}). Another important observation from Figure 2 is then that the SIMC-recommendation τ_{c} = θ for “tight” control (as given by middle of the three circles) in all cases is located in the desired trade-off region with a negative slope, well before we reach the minimum. Also, the recommended choice give a fairly constant M_{s}-value in the region 1.59 to 1.7. From this we conclude that, except for the time delay process, there is little room to improve on the SIMC PI rules, at least when performance and robustness are as defined above (J and M_{s}).

The IAE-cost J in (5) is based on equal weighting of servo (output disturbance) and regulator (input disturbance) performance. The existence of a trade-off between servo and regulator performance, can be quantified by considering how much larger the (Pareto) optimal cost J_{opt} (dashed black line) is than 1 at the reference robustness, M_{s} = 1.59, see also Table ??. For a pure time delay-process, we have that J_{opt} = 1 for M_{s} = 1.59 and there is no trade-off. The reason is that the setpoint and output disturbance responses are the same. On the other hand, for the other extreme of an integrating process, we have a clear trade off since the optimal PI-controller has J_{opt} = 1.51 (the SIMC PI-controller with M_{s} = 1.59 is close to this with J about 1.6). The existence of the servo/regulator trade-off for an integrating process, implies that one for a given robustness (M_{s}-value) can find PI-settings with significantly better regulator (load disturbance) performance or better servo (setpoint) performance, but not both at the same time. To be able to shift the trade-off, one may introduce an extra parameter in the PID rules ?, in addition to τ_{c}. For the SIMC method, this extra servo/regulator tradeoff parameter could be c in the following expression for the integral time,

τ_{I} = min(τ_{1}, c(τ_{c} + θ)) (6)

Table 1. Optimal PI-controllers (M_{s} = 1.59) and corresponding IAE-values for four processes.

Setpoint  Input disturbance  Optimal combined (minimize J)
Process K_{c}  τ_{I}  IAE_{ys}^{∘}  K_{c}  τ_{I}  IAE_{d}^{∘}  K_{c}  τ_{I}  IAE_{ys} IAE_{d}  J  M_{s}
e^{−s}  0.20 0.32 1.607  0.20 0.32 1.607  0.20 0.32 1.607 1.607  1  1.59
\frac{e^{−s}}{s+1}  0.54 1.10 2.083  0.50 1.0 2.036  0.54 1.10 2.083 2.041 1.00 1.59
\frac{e^{−s}}{8s+1}  4.0  8  2.169  3.34 3.7 1.135  3.46 4.0  3.111 1.158 1.23 1.59
\frac{e^{−s}}{s}  0.50 ∞  2.169  0.40 5.8 15.09  0.41 6.3  4.314  15.4 1.51 1.59

IAE_{ys} is for a unit setpoint change. IAE_{d} is for a unit input disturbance.

Fig. 2. Check of optimality of SIMC PI tuning rules for four processes.

where c = 4 gives the original SIMC-rule. A larger value if c improves the setpoint performance, and a smaller value, e.g. c = 2, improves the input disturbance performance ?.

However, introducing an extra parameter adds complexity and the potential benefit does not seem sufficiently large. Nevertheless, one may consider choosing another (lower) fixed value for c. There are two reasons why we recommend keeping the SIMC-value of c = 4. First, it is close to the Pareto-optimal PI controller (as seen from Figure 2), so we cannot get a significant improvement with our performance objective J. Second, with a smaller value for c, say c = 2.5, the recommended choice τ_{c} = θ becomes less robust (with a higher M_{s}), so one would need to recommend a different value for τ_{c} for an integrating process, say τ_{c} = 1.5θ, which would add complexity. In summary, we find that the value c = 4 in the original SIMC rule provides a well-balanced servo/regulator tradeoff.

5. IMPROVED SIMC RULE

For a pure time delay process, we see from Figure 2 that the IAE-value (J) for the SIMC controller is about 40% higher than the minimum with the same robustness (M_{s}). This is further illustrated by the closed-loop simulations in Figure 3 where we see that the SIMC PI-controller (denoted SIMC-original in the figure) gives a nice and smooth response. However, the response is somewhat sluggish initially, because it is actually a pure I-controller (with K_{c} = 0, τ_{I} = 0 and K_{I} = K_{c}/τ_{I} = 0.5). On the other hand, the IAE-optimal PI-controller (with minimum J for M_{s} = 1.59) has K_{c} about 0.2 and τ_{I} about 0.32 (and K_{I} = 0.62). In fact, the optimal PI-controller for a pure time delay process (dashed black line in Figure 2), has an almost fixed integral time of approximately θ/3 for all values of M_{s} between 1.4 and 1.7.

Based on this fact, we propose a simple change to the SIMC-rules, namely to replace τ_{1} by τ_{1} + θ/3 in the rules (PI control), which markedly improved the responses for a pure time delay process. It is important that the change is simple because “simplicity” was one of the main objectives when originally deriving the SIMC rules.

A similar change, but with θ/2 rather than θ/3, was originally proposed by ? for their “improved PI” tuning rule, and the effectiveness of this modification is also clear from the paper of ?foley:05.

However, as seen in Figure 3, the response with this IMC PI controller also settles rather slowly towards the setpoint, indicating that the integral time θ/2 is too large. The proposed value θ/3 gives a faster settling and is also closer to the original SIMC-rule (which is zero for a time delay process). The conclusion is that we recommend to replace τ_{1} by τ_{1}+θ/3 in the SIMC rules to get the improved SIMC rules:

K_{c} = \frac{1}{k} \frac{τ_{1} + \frac{θ}{3}}{τ_{c} + θ} (7)

τ_{I} = min\{τ_{1} + \frac{θ}{3}, 4(τ_{c} + θ)\} (8)

The improvement of this rule for a pure time delay processes is clear from the red curves in Figures 3 and 2 (upper left); for small M_{s}-values the improved SIMCcontroller is almost identical to the Pareto-optimal, which confirms that τ_{I} = θ/3 is close to optimal for a pure time delay process. For the process with a small time constant (τ_{1} = θ), the improved SIMC rule (red curve in upper right plot in Figure 2) is slightly better than the “original”

SIMC rule (blue curve) for higher M_{s}-values (where we get better performance) but slightly worse for lower M_{s}- values. For the two processes with a large time constant (τ_{1} = 8θ and τ_{1} = ∞) there are, as expected, almost no difference between the original and improved SIMC rules.

Fig. 3. Closed-loop setpoint responses for pure time delay process (θ = 1, k = 1, τ_{1} = 0) with PI-control. All three controllers have the same robustness (M_{s} = 1.59).

For a pure time delay process, the setpoint and disturbance responses are identical, and the input and output are identical. IMC PI: K_{c} = 0.29 and τ_{I} = 0.5 (K_{I} = K_{c}/τ_{I} = 0.58). SIMC PI original (τ_{c} = θ): K_{c} = 0 and τ_{I} = 0 (K_{I} = 0.5). SIMC PI improved (τ_{c} = 0.61θ): K_{c} = 0.207 and τ_{I} = 0.333 (K_{I} = 0.62).

6. CONCLUSION

REFERENCES

Rivera, D.E., Morarl, M., and Skogestad, S. (1986). Internal model control. 4. pid controller design. Ind. Eng. Chem. Process Des. Dev., 25, 252–256.

Appendix A. MODEL APPROXIMATION FROM CLOSED-LOOP SETPOINT RESPONSE