# Simulation Tool for Tuning and Performance Analysis of Robust, Tracking, Disturbance Rejection and Aggressiveness Controller

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

_{R}, θ

_{T}, θ

_{D}and θ

_{A}). The tuning of RTD-A controller is more transparent than classic PID controllers. The RTD-A tuning parameters values lies between ZERO and ONE. Availability of a tool to design optimal parameters for this controller and evaluating the performance on a given system is necessary for the researchers. In this paper, the new simulation tool is presented to deal with the RTD-A control scheme. There are four graphical user interface tools included in the proposed tool and working of each tool is explained in detail. To demonstrate the proposed tool, two examples, which involve a liquid level control application and an air pressure control application, are presented in this work. The performance of the RTD-A controller is compared with PID controller. RTD-A controllers are tuned using optimization algorithms and their performances are observed and analyzed in both cases under deterministic and uncertain conditions.

## 1. Introduction

_{P}), integral gain (K

_{I}) and derivative gain (K

_{D}) [4,5]. The PID controller parameters are difficult to tune in achieving more robust performance with the existence of process uncertainties [6]. Even after choosing the tuning values user will be facing difficulties in achieving the disturbance rejection and set point tracking simultaneously within the restricted time. Most of the industrial processes need a controller with the combo of set point tracking and disturbance rejection ability, which is challenging with the classic PID controller. One may get some unsatisfactory results using tuning rules such as Ziegler–Nichols (ZN) and Cohen–Coon (CC) [7]. It may be debated that contemporary optimization techniques have made the classical tuning methods needless. These techniques mandate a reliable plant model, which is to be controlled effectively [8]. The user can solve the problem of optimization by considering some advanced version of control scheme such as the model predictive controller (MPC). MPC can be effective in the typical industry situations, but still, they have certain limitations, which are generally not mentioned in advance process control applications. The limitations are difficulty in industry operations, high maintenance cost and less flexibility towards industrial functioning. All the mentioned limits may lead to insubstantial controllers that are not cost-effective. There are many discrepancies while calculating the tuning parameters [9]. The Smith predictor algorithm can be implemented across the controller transfer function for reducing the effect of dead time on the process output. Internal model control (IMC) has the advantage of robust tuning rules, but it has the disadvantage of a large process time constant. In the IMC scheme the user needs to choose integral time equal to the process time constant [10]. Therefore, it is inevitable to design a novel alternate control scheme that has all the combined advantages of PID and MPC.

## 2. RTD-A Control Scheme

- Predicting the process output;
- Updating the model prediction;
- Computing the control action.

- a = e
^{−Δt/τ}, Δt is the sampling time; - b = K(1 − e
^{−Δt/τ}) and m = round(α/Δt).

- $\mu \left(k,m\right)={\displaystyle \sum}_{i=1}^{m}{a}^{i}u\left(k-i\right)$.

_{m}(k) and unmeasured disturbance e

_{D}(k). Therefore, the total error is (7),

_{D}(k) is given as (8).

_{R}lies between 0 and 1. The controller has the robustness ability to handle the plant uncertainties. The future disturbance effect is predicted according to (9).

_{D}is the disturbance rejection controller tuning parameter. Finally, by solving the (5) and (9) the updated N-step model output prediction is given by (10).

_{d}(k), then the desired trajectory to follow y*(k) is given by (11).

_{T}is introduced, which varies from 0 to 1. The value of N will be calculated by the controller aggressiveness tuning parameter θ

_{A}given in (12).

#### 2.1. Stability Analysis

#### 2.1.1. State Variable Form

_{d}, d and d

_{o}are the state vector, setpoint, load disturbance and output disturbance respectively.

#### 2.1.2. Polynomial Form

^{th}order polynomial $\sum _{i=0}^{n}{c}_{0}{z}^{i}$ to lie inside the unit circle is that:

_{0}and c

_{n}can be calculated using (21) and (22).

_{R}can also be determined. The necessary condition is given in (25). The right-side term in (25) is defined as θ

_{ref}for simplicity.

## 3. Tool Design

#### 3.1. Tool for Model Order Reduction (Tool-I)

- Skogestad method;
- Two point method;
- Fraction incomplete method.

#### 3.2. Tool for Tuning RTD-A Parameter (Tool-II)

- The Mukati–Ogunnaike method;
- The Kariwala method.

#### 3.3. Tool for Analyzing the Effect of the RTD-A Parameter (Tool-III)

- Servo response (performance of the controller without disturbances);
- Regulatory response (performance of the controller considering only disturbances).

_{R}, θ

_{T}, θ

_{D}and θ

_{A}) on the plant output is clearly explained by Ogunnaike and Srinivasan [13,31].

#### 3.3.1. Effect of Robustness Parameter

_{R}parameter determines the controller robustness on the process output. The high open loop process gain causes for significant robustness in the face of process parameter uncertainty. It is desirable to have a negligible effect of parameter changes on the process output. The controller action is expected to be swift when ${\theta}_{R}$ is closer to unity and vice versa when it is near zero.

#### 3.3.2. Effect of the Setpoint Tracking Parameter

_{T}is the parameter that deals with the controller setpoint tracking ability. It does not affect the closed loop stability. θ

_{T}is chosen by considering the closed loop conditions. The error in the process output is determined by the tracking parameter. If the value of ${\theta}_{T}$ is close to 1, it leads to a sluggish response and vice versa when it is zero.

#### 3.3.3. Effect of the Disturbance Rejection Parameter

_{D}is the parameter that deals with future conditions. Here, the disturbances are deliberated as a composite disturbance. It includes both process model mismatch and unmeasured disturbances. The controller will be more aggressive when θ

_{D}is close to zero. If θ

_{D}is closer to unity the disturbance effect will be changing into the future. Conversely, when θ

_{D}is zero then the disturbance effect will remain constant throughout the system response.

#### 3.3.4. Effect of the Aggressiveness Parameter

_{A}is the parameter associated with the prediction horizon N. The prediction horizon value provides information about how much the future output can be predicted. The controller will be more conservative when the tuning value is high. The RTD-A will be more conservative when compared to the PID controller. An N value is large when the tuning value is near one and it has a short prediction horizon when θ

_{A}is close to zero [14].

#### 3.4. Tool for Optimizing the RTD-A Controller Parameter (Tool-IV)

_{R}, θ

_{T}, θ

_{D}and θ

_{A}) with the help of four different optimization algorithms. The algorithms are chosen arbitrarily. A cost function required for optimization can be chosen by selecting the “Get Cost” function. The main aim of the tool is to determine the optimal tuning parameters by minimizing the error between the actual process output and set-point. In order to minimize the cost function, the tool implements the intuitively chosen five heuristic algorithms such as:

- Galactic swarm optimization;
- Particle swarm optimization;
- Genetic algorithm;
- Firefly algorithm;
- Grey wolf algorithm.

#### 3.4.1. Galactic Swarm Optimization

#### 3.4.2. Particle Swarm Optimization

#### 3.4.3. Genetic Algorithm

#### 3.4.4. Firefly Algorithm

#### 3.4.5. Grey Wolf Algorithm (GWA)

^{0}= K(1 + λ), τ

^{0}= τ(1 + λ) and α

^{0}= α(1 + λ).

^{0}= e

^{−Δt/τ0}, b

^{0}= K

^{0}(1 − a

^{0}) and m

^{0}= round(α

^{0}/Δt).

_{p}(k) = a

^{0}y

_{p}(k − 1) + b

^{0}(u(k − m

^{0}− 1) + d(k − m

^{0}− 1)).

## 4. Tool Description

_{R}, θ

_{T}, θ

_{D}and θ

_{A}) on the process output the user has to change the value manually. The tool is made in such a way that the researcher has the facility to tune the controller manually with a slider option. The representation of Tool-III from the proposed tool is shown in Figure 5. The designed Tool-III has the facility to give a different set point at a different time instant and the user can perform a time domain analysis from the step response of the process.

## 5. Evaluation

#### 5.1. Example 1: Interacting a Two Tank Liquid Level System

_{1}and the outlet valve resistance is R

_{2}.

_{2}(s) and inlet flow rate Q(s) is derived and given in (33).

_{1}= R

_{2}=1; C

_{1}= C

_{2}= 1. Now the resulting transfer function is given in (34):

_{2}(t) by manipulating the inflow rate q(t). Since it is required to control h

_{2}(t) using the RTD-A controller, the actual system in Equation (34) is reduced into the FOPDT system. The three methods in Tool-I were used to obtain the FOPDT model of the process. The result of each method is presented in Table 3. The accuracy of the reduced model in terms of the integral square error (ISE) performance measure was also compared. It was found that the fraction incomplete method based model was more accurate for this interacting liquid level system. Hence, in this section, the remaining analysis was carried out with the obtained accurate reduced model in (35).

_{2}(t) without uncertainties and disturbance. The structure of the PID controller transfer function used is given in (36). The PID tuning parameters proportional gain (K

_{P}), integral gain (K

_{I}), derivative gain (K

_{D}) and filter co-efficient (T

_{f}) used in the simulation were determined using the auto tune function in MATLAB.

_{P}= 0.496, K

_{I}= 0.116, K

_{D}= −0.857 and T

_{f}= 0.128. The setpoint for h

_{2}(t) was considered as one meter. The servo response with the PID and RTD-A for zero uncertainty and zero disturbance is shown in Figure 10. Similarly, the servo response with RTD-A tuned using the optimization algorithm is shown in Figure 11.

_{R}, θ

_{T}and θ

_{A}values were changed and the output of the plant with respect to time was plotted. The output of Tool-III for Example 1 with sequential changes in setpoints (SP = 1 m, SP = 5 m and SP = 2 m) is shown in Figure 16, Figure 17 and Figure 18.

_{R}approaches unity. The response was robust as the θ

_{R}value was small. The effect of the setpoint tracking parameter on the process output is shown in Figure 17. The response became sluggish and the settling time increased as the θ

_{T}approached unity. In Figure 18, as the θ

_{A}value approached 1, the response was moving away from the desired set point. Therefore, for the lesser value of θ

_{A}, the process output was closer to the desired level. Similarly, the process output reached the set point very quickly without overshoot as the parameter θ

_{A}was small.

_{d}) and actual process output (y

_{p}). Therefore, the integral absolute error (IAE) cost function in (37) was chosen for both example 1 and example 2. During this optimal tuning process, an output disturbance was also considered with ±10% random variation.

#### 5.1.1. Example 1: Closed Loop Stability Analysis

_{0}and c

_{n}were calculated using (21) and (22). The value of the polynomial P(z) when z = 1 and z = −1 was calculated according to (18). The θ

_{ref}value also was calculated to check the stability in terms of θ

_{R}using (25). The calculated values are tabulated in Table 10. The value of c

_{n}= 51,164, θ

_{ref}= 0.07 and the order of P(z) was an even number. The necessary conditions in (20), (23), (24) and (25) were checked for stability. It was found that all the conditions were satisfied for closed loop stability for the calculated tuning values. Therefore, the designed RTD-A controller ensured closed loop stability in liquid level control. A similar analysis was done for pressure control in Section 5.2.1.

#### 5.2. Example 2: Pressure Control System

_{out}(t). All four parameters were tuned using the Tool-II and -IV. The tuned values are listed in Table 11 and Table 12 respectively. The control objective of the optimal control problem was to reduce the error between the desired pressure and actual pressure. The IAE cost function given in (37) was used to determine the optimal tuning values of RTD-A.

_{P}= 1.373, K

_{I}= 0.143, K

_{D}= −0.178 and the filter co-efficient (T

_{f}) = 0.115.

#### 5.2.1. Example 2: Closed Loop Stability Analysis

_{ref}, coefficients c

_{0}and c

_{n}were calculated using (18), (21), (22) and (25). The values are tabulated in Table 14. The calculated values were checked for stability necessary conditions in (20) and (23)–(25). The value of c

_{n}= 1.5929e + 07, θ

_{R}

_{ef}= 0.3295 and the order of P(z) was an odd number.

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**General closed loop diagram with a robust, tracking, disturbance rejection and aggressiveness (RTD-A) controller.

**Figure 7.**Two tank liquid-level system [4].

**Figure 13.**Example 1: Servo response with the RTD-A controller with uncertainty using optimization algorithms.

**Figure 14.**Example 1: Regulatory response with the RTD-A controller using tuning rules and proportional integral derivative (PID).

**Figure 20.**Pressure control station [38].

**Figure 23.**Comparison of the servo response for the pressure control station with two tuning rules and PID.

**Figure 24.**Comparison of the servo response for the pressure control station with optimization algorithms.

**Figure 25.**Example 2: Regulatory response for the pressure control station with tuning rules and PID.

**Figure 26.**Example 2: Regulatory response for the pressure control station with optimization algorithms.

**Table 1.**Mukati–Ogunnaike tuning rules [4].

Composite Parameter | ${\mathit{\theta}}_{\mathit{R}}$ | ${\mathit{\theta}}_{\mathit{T}}$ | ${\mathit{\theta}}_{\mathit{D}}$ | ${\mathit{\theta}}_{\mathit{A}}$ |
---|---|---|---|---|

$\rho <1$ | 0.5 | 0.8 | $\ge 0.5$ | $0.9\rho $ |

$1\le \rho \le 1.6$ | 0.9 | 0.8 | $\ge $0.1 | $1.25\left(1-{e}^{-\rho}\right)$ |

$\rho \ge 1.6$ | $>0.95$ | 0.8 | $\ge 1-{\theta}_{R}$ | 0.995 |

θ_{R} | θ_{T} | θ_{D} | θ_{A} |
---|---|---|---|

$\mathrm{min}\left[\frac{m}{m+1},0.95\right]$ | ${a}^{2}$ | $1-a$ | $1-\left(1/ae\right)$ |

Methods | Skogestad | Two Point | Fraction Incomplete |
---|---|---|---|

Reduced Model G(s) | $\frac{{e}^{-4.19s}}{2.809s+1}$ | $\frac{{e}^{-4.9s}}{2.68s+1}$ | $\frac{{e}^{-4.82s}}{2.69s+1}$ |

ISE | 0.066 | 0.0518 | 0.0444 |

**Table 4.**Example 1: Tuning parameters with two tuning methods [4].

Parameter | Ogunnaike | Kariwala |
---|---|---|

θ_{R} | 0.5 | 0.95 |

θ_{T} | 0.9180 | 0.9207 |

θ_{D} | 0.5 | 0.0405 |

θ_{A} | 0.1971 | 1 |

Specifications | PID | Ogunnaike | Kariwala | GSO | GA | PSO | Fire-Fly | GWA |
---|---|---|---|---|---|---|---|---|

IAE | 9.52 | 9.079 | 7.61 | 6.89 | 6.88 | 7.427 | 6.85 | 6.932 |

Rise Time (s) | 5.87 | 1.641 | 3.578 | 0.22 | 0.1408 | 0.1626 | 0.1407 | 0.14 |

Settling Time (s) | 38.05 | 21.1 | 13.78 | 20.49 | 20.36 | 22.81 | 20.12 | 20.24 |

Overshoot (%) | 20.78 | 33.57 | 12.98 | 29.67 | 31.94 | 56.92 | 29.17 | 35.81 |

Steady State Error (e_{ss}) | 2.46 × 10^{−5} | 0.02 | 0 | 0 | 0 | 0 | 0 | 0 |

Parameter | GSO | GA | PSO | Firefly | GWA |
---|---|---|---|---|---|

Maximum Iteration | 125 | 30 | 30 | 50 | 70 |

Swarm Size | 100 | 50 | 70 | 60 | 50 |

Algorithm | Parameter |
---|---|

PSO | Inertia coefficient = 1 Damping coefficient = 0.99 Acceleration coefficients c1 = 1.5, c2 = 2 |

Firefly | α = 0.2, β = 2, γ = 1 |

GSO | Acceleration coefficients c1 = c2 = c3 = c4 = 2.05 |

Algorithm | ${\mathit{\theta}}_{\mathit{R}}$ | ${\mathit{\theta}}_{\mathit{T}}$ | ${\mathit{\theta}}_{\mathit{D}}$ | ${\mathit{\theta}}_{\mathit{A}}$ |
---|---|---|---|---|

GSO | 0.9597 | 0.4782 | 0.8880 | 0.0356 |

GA | 0.9581 | 0.00103 | 0.67 | 0.0211 |

PSO | 0.8931 | 0.2532 | 0.5945 | 0.03 |

Firefly | 0.9663 | 0.0001 | 0.519 | 0.033 |

GWA | 0.9470 | 0.0015 | 0.914 | 0.0016 |

Tuning Method | Ogunnaike | Kariwala | GSO | GA | PSO | FF | GWA |
---|---|---|---|---|---|---|---|

Closed Loop Roots | 0.8300 | 0.0793 | 0.5170 | 0.4389 | 0.5962 | 0.6730 | 0.6754 |

0.3348 | 0.0500 | 0.4072 | 0.5590 | 0.5705 | 0.7277 | 0.7287 | |

0.0704 | 0.0407 | 0.0330 | 0.0327 | 0.0909 | 0.0432 | 0.0531 | |

0.0377 | 0.0364 | 0.0458 | 0.0056 | 0.0371 | 0.0307 | 0.0353 | |

0.0365 | 0.0365 | 0.0365 | 0.0054 | 0.0365 | 0.0365 | 0.0365 |

**Table 10.**Example 1: Jury’s stability test for liquid level control (c

_{n}= 5.1164e + 04 and θ

_{ref}= 0.07).

Tuning Method | Ogunnaike | Kariwala | GSO | GA | PSO | FF | GWA |
---|---|---|---|---|---|---|---|

θ_{R} | 0.5 | 0.95 | 0.9597 | 0.9581 | 0.8931 | 0.9663 | 0.947 |

$\left|{c}_{0}\right|$ | 2.7507 × 10^{4} | 2.75 × 10^{3} | 2.217 × 10^{3} | 2.305 × 10^{3} | 5.881 × 10^{3} | 1.854 × 10^{3} | 2.9158 × 10^{3} |

P(z) when z = 1 | 40.5457 | 4.0546 | 3.2680 | 3.3977 | 8.6687 | 2.7328 | 4.2978 |

P(z) when z = −1 | 1.1878 × 10^{7} | 1.54 × 10^{6} | 1.318 × 10^{6} | 1.3548 × 10^{6} | 2.8480 × 10^{6} | 1.1665 × 10^{6} | 1.6098 × 10^{6} |

Algorithm | ${\mathit{\theta}}_{\mathit{R}}$ | ${\mathit{\theta}}_{\mathit{T}}$ | ${\mathit{\theta}}_{\mathit{D}}$ | ${\mathit{\theta}}_{\mathit{A}}$ |
---|---|---|---|---|

Ogunnaike | 0.5 | 0.9876 | 0.5 | 0.003 |

Kariwala | 0.66 | 0.98 | 0.866 | 0.006 |

Algorithm | ${\mathit{\theta}}_{\mathit{R}}$ | ${\mathit{\theta}}_{\mathit{T}}$ | ${\mathit{\theta}}_{\mathit{D}}$ | ${\mathit{\theta}}_{\mathit{A}}$ |
---|---|---|---|---|

GSO | 0.5121 | 0.0036 | 0.176 | 0.0352 |

GA | 0.4620 | 0.00101 | 0.2450 | 0.0255 |

PSO | 0.4097 | 0.8845 | 0.8346 | 0.0196 |

Firefly | 0.7398 | 0.7424 | 0.3032 | 0.0044 |

GWA | 0.4356 | 0.0083 | 0.3270 | 0.0163 |

Tuning Method | Ogunnaike | Kariwala | GSO | GA | PSO | FF | GWA |
---|---|---|---|---|---|---|---|

Closed Loop Roots | 0.6032 | 0.7360 | 0.2247 | 0.3681 | 0.1985 | 0.5930 | 0.4940 |

0.3840 | 0.2621 | 0.6452 | 0.7198 | 0.4132 | 0.2589 | 0.7538 | |

0.0108 | 0.0164 | 0.2302 | 0.2725 | 0.0837 | 0.1515 | 0.2980 | |

0.0056 | 0.0056 | 0.0056 | 0.0056 | 0.0056 | 0.0056 | 0.0056 | |

0.0054 | 0.0054 | 0.0054 | 0.0054 | 0.0054 | 0.0054 | 0.0054 |

**Table 14.**Example 2: Jury’s stability test for pressure control (c

_{n}= 1.5929 × 10

^{7}and θ

_{Ref}= 0.3295).

Tuning Method | Ogunnaike | Kariwala | GSO | GA | PSO | FF | GWA |
---|---|---|---|---|---|---|---|

θ_{R} | 0.5 | 0.66 | 0.5121 | 0.462 | 0.4097 | 0.7398 | 0.4356 |

$\left|{c}_{0}\right|$ | 1.1878 × 10^{7} | 8.0769 × 10^{6} | 1.159 × 10^{7} | 1.278 × 10^{7} | 1.4023 × 10^{7} | 6.1812 × 10^{6} | 1.34 × 10^{7} |

P(z) when z = 1 | 277.8394 | 188.9308 | 271.1157 | 298.9552 | 328.0172 | 144.5876 | 313.6251 |

P(z) when z = −1 | −2.092 × 10^{10} | −1.418 × 10^{10} | −2.04 × 10^{10} | −2.25 × 10^{10} | −2.472 × 10^{10} | −1.082 × 10^{10} | −2.36 × 10^{10} |

Specifications | PID | Ogunnaike | Kariwala | GSO | GA | PSO | Firefly | GWA |
---|---|---|---|---|---|---|---|---|

IAE | 570.6 | 452.1788 | 327.21 | 291.1297 | 291.0047 | 291.6691 | 291.2998 | 291.0940 |

Rise Time (s) | 15.47 | 17.53 | 10.4164 | 8.4854 | 8.4855 | 8.4845 | 8.4855 | 8.4854 |

Settling Time (s) | 57.78 | 32.09 | 47.77 | 11.0437 | 11.0425 | 11.0412 | 11.0424 | 11.0424 |

Overshoot (%) | 7.9623 | 0 | 0 | 0 | 0.3746 | 0.7189 | 1.0824 | 0.5502 |

Steady State Error (e_{ss}) | 0.0756 | 0.0016 | 0.0015 | 0.0182 | 0.0175 | 0.0220 | 0.0178 | 0 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Bagyaveereswaran, V.; Umashankar, S.; Arulmozhivarman, P.
Simulation Tool for Tuning and Performance Analysis of Robust, Tracking, Disturbance Rejection and Aggressiveness Controller. *Algorithms* **2019**, *12*, 144.
https://doi.org/10.3390/a12070144

**AMA Style**

Bagyaveereswaran V, Umashankar S, Arulmozhivarman P.
Simulation Tool for Tuning and Performance Analysis of Robust, Tracking, Disturbance Rejection and Aggressiveness Controller. *Algorithms*. 2019; 12(7):144.
https://doi.org/10.3390/a12070144

**Chicago/Turabian Style**

Bagyaveereswaran, Veeramani, Subramaniam Umashankar, and Pachiyappan Arulmozhivarman.
2019. "Simulation Tool for Tuning and Performance Analysis of Robust, Tracking, Disturbance Rejection and Aggressiveness Controller" *Algorithms* 12, no. 7: 144.
https://doi.org/10.3390/a12070144