# Deterministic Performance Assessment and Retuning of Industrial Controllers Based on Routine Operating Data: Applications

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## Abstract

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## 1. Introduction

## 2. Self-Regulating Processes

#### 2.1. Control Scheme

_{10}> τ

_{20}> ⋯). Then, Model (3) can be approximated by a second order plus dead time (SOPDT) transfer function:

_{0}the sum of the time constants and of the dead time of the process, it results:

#### 2.2. Set-Point Following Performance Assessment

^{−}

^{θs}= 1 − θs, the closed-loop transfer function is:

_{s}is the amplitude of the set-point step.

_{m}of the apparent dead time of the system can be performed by considering the time interval from the application of the step signal to the set-point and the time instant when the process output attains 2% of the new set-point value A

_{s}, namely, when the condition y > 0.02A

_{s}occurs. In this context, in order to cope with the measurement noise, a simple sensible solution is to define a noise band NB [19] (whose amplitude should be equal to the amplitude of the measurement noise) and to rewrite the condition as y > NB. Then, Expression (13) can be rewritten as:

_{0}[20]. The process gain µ can be determined by considering the following trivial relations, which involve the final steady-state value of the control variable u:

_{s}of the step.

_{PID}= 1. From a practical point of view, however, the controller is considered to be well tuned if the index is greater than a given threshold, which can be taken equal to 0.6. Note that this last value has been selected by considering the SIMC tuning rule applied to many different processes [18], but in any case, another value of the threshold can be selected by the user depending on how tight its control specifications are. If the controller results in being badly tuned, then two cases might occur. If the value of SI is less than one, this means that the current tuning is based on an underestimation of the process lags and/or of the dead time. Thus, an improvement of the performance can be achieved by decreasing the value of K

_{p}and/or by increasing the value of T

_{i}and/or T

_{d}(see Equation (10)). On the contrary, if the value of SI is greater than one, this means that the current tuning is based on an overestimation of the process lags and/or of the dead time. Thus, an improvement of the performance can be achieved by increasing the value of K

_{p}and/or by decreasing the value of T

_{i}and/or T

_{d}.

#### 2.3. Retuning Algorithm

_{m}and T

_{0}have to be calculated according to the technique described in Section 2.2 (see Equation (22)). Then, if an overshoot occurs, the proportional gain K

_{p}is decreased, and the integral time constant T

_{i}and the derivative time constant T

_{d}are subsequently calculated by means of Expressions (9) and (10). If the resulting value of T

_{i}is less than the value of T

_{d}, then T

_{i}has to be decreased, and the values of K

_{p}and T

_{d}are updated again according to Equations (9) and (10). The procedure is iterated until a value T

_{d}> 0 is obtained. Conversely, if the closed-loop step response is overdamped (namely, there is no overshoot), the procedure starts by increasing the proportional gain K

_{p}, and then, the same steps of the case with overshoot are applied. By taking into account the considerations made in Section 2.2, a sound measure on how much to initially increase or decrease the parameters is given by the sigma index SI.

_{p}, T

_{i}and T

_{d}, the algorithm for retuning the PID controller is described as follows [20].

- Evaluate a closed-loop set-point step response y(t) and the corresponding control variable u(t), and determine µ, θ
_{m}and T_{0}. - Calculate SI according to Equation (23).
- Set K
_{pi}= K_{p}(store the initial value of the proportional gain). - If max
_{t}y(t) > A_{s}(namely, if there is overshoot), then:- Set K
_{p}= K_{pi}· min{SI, 1/SI}. - Set T
_{i}= 2µK_{p}θ_{m}. - Set ${T}_{d}={T}_{0}-{T}_{i}-\frac{{T}_{i}}{2\mu {K}_{p}}$.
- If T
_{i}< T_{d}, then:- Set T
_{i}= T_{i}· max{SI, 1/SI}. - Set ${K}_{p}=\frac{{T}_{i}}{2\mu {\theta}_{m}}$.
- Set ${T}_{d}={T}_{0}-{T}_{i}-\frac{{T}_{i}}{2\mu {K}_{p}}$.

- If T
_{d}< 0, then:- If SI > 1, then set SI = 2SI; else set SI = SI/2.
- Go to 4(a).

- If max
_{t}y(t) ≤ A_{s}(namely, if there is no overshoot), then:- Set K
_{p}= K_{pi}· max{SI, 1/SI}. - Set T
_{i}= 2µK_{p}θ_{m}. - Set ${T}_{d}={T}_{0}-{T}_{i}-\frac{{T}_{i}}{2\mu {K}_{p}}$.
- If T
_{i}< T_{d}, then:- Set T
_{i}= T_{i}· max{SI, 1/SI}. - Set ${K}_{p}=\frac{{T}_{i}}{2\mu {\theta}_{m}}$.
- Set ${T}_{d}={T}_{0}-{T}_{i}-\frac{{T}_{i}}{2\mu {K}_{p}}$.

- If T
_{d}< 0, then:- If SI < 1, then set SI = 2SI; else set SI = SI/2.
- Go to 5(a).

- End.

_{m}and τ = T

_{0}− θ

_{m}:

#### 2.4. Experimental Results

_{p}) and an integral time constant equal to T

_{i}= 70 (s) (T

_{d}= 0). After the application of the step signal to the set-point, the process parameters have been determined as θ

_{m}= 94 (s), µ = 0.285 and T

_{0}= 184.3 (s). The corresponding values of SI and SFPI

_{PID}have been determined as 0.84 and 0.545, respectively, indicating the need for a retuning. By applying the retuning algorithm, the new values of the PID controller have been determined as PB = 82.78%, T

_{i}= 64.82 (s) and T

_{d}= 24.45 (s), with a corresponding value of SFPI

_{PID}= 0.973 (obviously, it is SI = 1). The set-point step responses before and after the retuning procedure are shown in Figures 3 and 4, respectively, where a clear improvement in the performance appears (note the different time range in the two figures). In particular, the settling time has been considerably reduced, which is obviously appreciated in the start-up phase.

_{i}= 60 (s), and the resulting response to a set-point step change from 1500 (kg/h) to 1750 (kg/h) was the one shown in Figure 6. The corresponding value of the integrated absolute error (IAE) was 7.585 (kg) (the flow is measured in (kg/h), but the values are stored at each second). When the system attained its steady-state value, the following values of the process parameters were estimated: θ

_{m}= 19 (s), µ = 3.125 and T

_{0}= 58 (s). The value of the sigma index SI resulted in being 1.86, while the value of the closed-loop index SFPI

_{PID}resulted in being equal to 0.35, which means that both indices indicate that the controller had to be retuned. The PI controller has then to be retuned by applying Formula (25), which yielded PB = 304.5 and T

_{i}= 39 (s). These values were applied to the PI controller before changing the set-point value from 1750 (kg/h) back to the value it had previously (1,500 (kg/h)). The step response obtained is shown in Figure 7. It can be seen that, although a small overshoot appeared, the settling time was shorter. The value of the integrated absolute error IAE decreased to 4.01 (kg), and the set-point following performance index increased to 0.66.

#### 2.5. Load Disturbance Rejection Performance Assessment

_{d}is assumed to enter into the process at a known time instant, which is taken as t = 0 without loss of generality. The disturbance amplitude A

_{d}can be estimated by considering the final value of the integral of the control error:

_{d}) as:

_{0}as:

_{m}of the system can be estimated by considering the time interval from the occurrence of the load disturbance and the time instant when the absolute value of the difference between the process output and its previous steady-state value exceed the noise band threshold.

_{c}is a design parameter, which can be selected as τ

_{c}= θ in order to provide a satisfactory robustness.

_{0}− θ

_{m}, the PID controller parameters can be calculated as:

_{p}and T

_{i}in Equation (32) with the values in Equation (31), it results:

_{c}= θ to achieve a satisfactory robustness. Correspondingly, by considering the estimated apparent dead time, the tuning formulas are:

_{p}= 1, T

_{i}= 10 and T

_{d}= 1. After the load disturbance occurs on the process, the gain µ is correctly determined by means of Equation (72), while the computation (73) gives exactly T

_{0}= 20; finally, the dead time is estimated to be θ

_{m}= 6.27. The performance indices related to this initial manual tuning are LRPI

_{PID}= 0.582 and SFPI

_{PID}= 0.642, which indicates a quite poor behavior of the controller. Hence, the PID parameters are retuned by using Equation (36), resulting in K

_{p}= 1.785, T

_{i}= 14.06, T

_{d}= 2.087. Figure 8 shows the performance improvement that results from the retuning; of course, when the set-point filter of Equation (42) is applied, the typical aggressiveness in the load disturbance rejection task does not have to be paid by a high overshoot in the set-point following, like happens if the filter is not used. In fact, in both cases, it is LRPI

_{PID}= 0.877, but by filtering the set-point, it is SFPI

_{PID}= 0.849.

## 3. Distributed Lag Processes

#### 3.1. Modeling

_{0}, is equal to 0.5τ. The sum of all time constants can be estimated easily, by means of an open-loop step response, as the time the process variable takes to attain 63.2% of its steady-state value, and the process gain can be determined by considering the steady-state value of the process output and the amplitude of the step input.

_{0}and µ can be estimated with a closed-loop experiment, namely by employing a PID controller with any values of the parameters (provided that the closed-loop system is asymptotically stable).

#### 3.2. Performance Assessment and Retuning for Load Disturbance Rejection

_{p}= 5/µ and T

_{i}= 0.54T

_{0}, while, in the case of a PID controller, it is K

_{p}= 100/15/µ, T

_{i}= 0.25T

_{0}and T

_{d}= 0.10T

_{0}. The benchmark performance, in terms of integrated absolute error, can be determined by applying these tuning rules to many (simulated) distributed lag processes (45) with different values of µ and τ and different process order n. Results for µ = 1 are shown in Figures 9 and 10 for PI and PID controllers, respectively. By interpolating the results, the following values of IAE are obtained (for PI and PID controllers, respectively):

_{PI}= 1 or LRPI

_{PID}= 1, the controller can be considered to be well tuned if the index is greater than 0.8.

#### 3.3. Simulation Results

_{p}= 3.3, T

_{i}= 1.9, T

_{d}= 0.25. Then, a unit step load disturbance is applied to the process (see the dashed line in Figure 11), and the amplitude of the disturbance, the gain of the process and the sum of the time constants are estimated as A

_{d}= 1, µ = 1.0 and T

_{0}= 4.97. Based on these values, the PID parameters are retuned, as K

_{p}= 6.66, T

_{i}= 1.24, T

_{d}= 0.5. The load disturbance step response provided by the new values of the PID controller parameters is shown as a solid line in Figure 11. By retuning the controller, the performance index is improved from LRPI

_{PID}= 0.32 to LRPI

_{PID}= 1.03, while the integrated absolute error decreases from IAE = 0.62 to IAE = 0.19. It is worth noting that the same result is achieved if a set-point step response is employed for estimating the process parameters.

_{p}= 7 and T

_{i}= 2. By evaluating a unit step load disturbance, the gain of the process and the sum of the time constants are estimated as A

_{d}= 1, µ = 0.99 and T

_{0}= 4.97. Based on these values, the PI parameters are retuned as K

_{p}= 5.06 and T

_{i}= 2.68. The load disturbance step responses provided by the initial and new values of the PI controller parameters are shown in Figure 12. The performance index is improved from LRPI

_{PID}= 0.64 to LRPI

_{PID}= 1.01, while the integrated absolute error decreases from IAE = 0.88 to IAE = 0.56.

## 4. Integral Processes

#### 4.1. Modeling

_{10}> τ

_{20}>⋯). Then, by applying the “half rule”, a second order plus dead time (SOPDT) transfer function:

#### 4.2. Performance Assessment and Retuning

^{−}

^{θs}≅ = 1 − θs, yields a PD controller whose parameters are selected according to the following tuning rule:

_{s}applied to the set-point results as in (13). Thus, as in (24), the controller performance with respect to the set-point following task can be assessed by considering the following performance index:

_{s}is applied to the set-point can be calculated as [28]:

_{PID}to assess the controller performance when a step signal is applied to the set-point can be defined as:

_{d}occurs, as there are no complex poles in the transfer function T(s) between the load disturbance and the process output, the integrated absolute error can be calculated as:

_{PID}to assess the controller performance when a step load disturbance occurs can be defined as:

_{PID}is defined as:

_{0}and of the dead time θ, as well as the value of the amplitude of the load step disturbance A

_{d}for the load disturbance case.

_{m}of the system can be determined as described in Section 2. Then, if a set-point step response is evaluated, by following again a procedure involving the final value theorem, the determination of the sum of the lags and of the dead time of the process can be obtained as:

_{d}of the step load disturbance can be determined as:

_{1}can be calculated as τ

_{1}= T

_{0}− θ

_{m}.

#### 4.3. Experimental Results

_{i}(t) − q

_{0}(t) is the difference between the input and output flow rates. An apparent dead time is present in the process because of the neglected dynamics and because of the length of the pipe. A load disturbance is added to the process by applying an additional input flow rate by means of another pump.

_{p}= 1 and T

_{i}= 100 (s). The results obtained by applying a set-point step from 1.5 to 2.3 V (namely, A

_{s}= 0.8) at time t = 0 and a load disturbance at t = 200 are shown in Figure 13. By using the data of the set-point step response, the values θ

_{m}= 2 (s), T

_{0}= 16.02 (s) and µ = 0.06 have been estimated. The resulting integrated absolute error is IAE = 9.04, and the resulting set-point following performance index is SFPI

_{PID}= 0.61. By considering the load disturbance response, the resulting integrated absolute error is IAE = 107.9, and the resulting load disturbance performance index is LRPI

_{PID}= 0.05. This latter result indicates that it is worth retuning the controller. By applying the tuning rule (57), the PID parameters have been calculated as K

_{p}= 3.70, T

_{i}= 16 (s) and T

_{d}= 0.02 (s). The corresponding results are shown in Figure 14, where the resulting integrated absolute errors are IAE = 4.92 and IAE = 5.93 for the set-point and load disturbance response, respectively. The corresponding performance indices are SFPI

_{PID}= 0.84 and LRPI

_{PID}= 0.85.

## 5. Conclusions

## Author Contributions

## Conflicts of Interest

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**Figure 4.**Set-point step response after the PID controller retuning in the temperature control loop.

**Figure 6.**Initial set-point step response in the flow control loop. Top: process variable. Bottom: control variable.

**Figure 7.**Set-point step response after the PID controller retuning in the flow control loop. Top: process variable. Bottom: control variable.

**Figure 8.**Results of the illustrative example for disturbance rejection performance assessment. Dotted line: initial tuning; dashed line: simple PID retuned; solid line: retuned PID plus set-point filter.

**Figure 9.**Values of the integrated absolute error (IAE) for different values of τ and n (process order) with a PI controller tuned according to the benchmark tuning rules.

**Figure 10.**Values of IAE for different values of τ and n (process order) with a PID controller tuned according to the benchmark tuning rules.

**Figure 11.**Load disturbance step response for PID control for a distributed lag process. Dashed line: initial tuning; solid line: after retuning.

**Figure 12.**Load disturbance step response for PI control for a distributed lag process. Dashed line: initial tuning; solid line: after retuning.

Process Type | Controller | Control Task | Performance Index |
---|---|---|---|

Self-regulating | PI(D) | set-point following | (24) |

Self-regulating | PID | load disturbance rejection | (34) |

Self-regulating | PI | load disturbance rejection | (37) |

Self-regulating | PI(D) with set-point filter | set-point following (and load disturbance rejection) | (24) |

Distributed-lag | PI | load disturbance rejection | (49) |

Distributed-lag | PID | load disturbance rejection | (50) |

Non-self-regulating | PD | set-point following | (56) |

Non-self-regulating | PID | set-point following (and load disturbance rejection) | (61) |

Non-self-regulating | PID | load disturbance rejection (and set-point following) | (63) |

Non-self-regulating | PID | load disturbance rejection | (66) |

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**MDPI and ACS Style**

Veronesi, M.; Visioli, A.
Deterministic Performance Assessment and Retuning of Industrial Controllers Based on Routine Operating Data: Applications. *Processes* **2015**, *3*, 113-137.
https://doi.org/10.3390/pr3010113

**AMA Style**

Veronesi M, Visioli A.
Deterministic Performance Assessment and Retuning of Industrial Controllers Based on Routine Operating Data: Applications. *Processes*. 2015; 3(1):113-137.
https://doi.org/10.3390/pr3010113

**Chicago/Turabian Style**

Veronesi, Massimiliano, and Antonio Visioli.
2015. "Deterministic Performance Assessment and Retuning of Industrial Controllers Based on Routine Operating Data: Applications" *Processes* 3, no. 1: 113-137.
https://doi.org/10.3390/pr3010113