# Interactive Tuning Tool of Proportional-Integral Controllers for First Order Plus Time Delay Processes

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## Abstract

**:**

## 1. Introduction

#### 1.1. Background

#### 1.2. Formulation of the Problem of Interest for This Investigation

_{P}represents the proportional gain and K

_{I}the integral gain (thus, the integral time constant T

_{I}equals K

_{P}/K

_{I}).

_{P}and K

_{I}in the case of a PI controller) to the optimum values for a desired control response. One of the main objectives of this work is to analytically determine the set of K

_{P}–K

_{I}values for which the closed-loop system is stable. For this purpose, and for the analysis of the control response given specific K

_{P}and K

_{I}values, the feedback control system of Figure 1 is assumed. The controlled variable is given by y, sp represents the reference signal, u symbolizes the control signal, and load the input disturbance.

#### 1.3. Literature Survey

#### 1.4. Scope and Contribution of This Study

#### 1.5. Organization of the Paper

## 2. Control Problem

#### 2.1. The PI Stabilization Problem

_{P}–K

_{I}parameter space. Two cases are considered: open-loop stable processes (T > 0) and open-loop unstable processes (T < 0). The frequency response of the plant model is distinguished by the following magnitude and phase margin:

_{P}–K

_{I}space can be calculated using the following steps:

**Step 1**: First, find the frequency ω

_{max}verifying (5) in the interval (π/2, π) when T > 0 or the interval (0, π/2) when T < 0.

**Step 2**: According to tuning by critical gain margin (A

_{m}= 1) of [33], calculate the (K

_{P}, K

_{I}) values scanning the frequency range (0, ω

_{max}), as follows:

#### 2.2. PI Tuning Rules

_{P}–K

_{I}couple inside the corresponding stabilizing region in the K

_{P}–K

_{I}space.

- The “quarter decay ratio” criterion achieves a small settling time.
- The internal mode control (IMC) tuning rule of Rivera achieves an excellent set-point response; however, the load disturbance rejection is very slow. The Skogestad IMC tuning rule works well for both cases.
- The achievable performance specifications for unstable processes are usually worse than those obtained for stable systems. For the unstable case, PI controllers do not normally achieve very good responses with larger overshoot and settling times. Additionally, the robustness of the feedback system is very limited for small changes in the process parameters, which does not usually occur in stable systems.

#### 2.3. Stabilizing Regions for a Particular PI Controller

_{P}–K

_{I}space for a given FOPTD process, it is possible to calculate the set of FOPTD plants that can be stabilized by a particular PI controller. Assuming that the process stationary gain K remains constant, this set of processes can be depicted in the T–τ space. The greater the region, the more robust is the PI controller against changes in these model parameters. To calculate this region for a particular PI controller (a K

_{P}–K

_{I}couple), the following steps are proposed:

**Step 1**: Determine the possible range of the process time constant T ∈ [0, T

_{max}] and the desired incremental value to sweep the different T values in this range.

**Step 2**: For each possible T value, obtain the critical frequency ω

_{c}that achieves an open-loop module equal to one in the feedback system. The square of this frequency is given by (8)

**Step 3**: Given a specific T value and its corresponding critical frequency, the particular PI controller can stabilize a FOPTD process with a time delay τ ∈ [0, τ

_{max}], where the maximum time delay τ

_{max}is given by (9). The derivation of Expressions (8) and (9) is explained in Appendix B. When a negative time delay is obtained from (9), the current PI controller cannot stabilize any real FOPTD process with the same K.

_{P}= 1.5, K

_{I}= 0.1) for stable processes and (K

_{P}= −1.5, K

_{I}= −0.1) for unstable systems, Figure 3 shows the regions in the T–τ space containing the set of stable FOPTD plants (right) and unstable FOPTD processes (left) that can be stabilized by its corresponding PI controller. It is assumed that the process stationary gain remains constant and equal to one. Any process with a couple T–τ out of these zones cannot be stabilized by the corresponding PI controller. This region can give insight into the robustness of the controller. Although the regions are unbounded, the region for the open-loop unstable case is generally smaller than that for the open-loop stable case. Controller design for open-loop unstable systems with conventional PID/PI controllers cannot provide a good close-loop performance, especially for delay dominant unstable processes [38]. Performance specifications such as settling time and overshoot are larger compared with those obtained for open-loop stable systems.

#### 2.4. Discrete Implementation and Practical Considerations

_{max}and u

_{min}) is another important issue that should be considered. If the control signal reaches one of these limits, the undesired windup effect can appear. The tool allows one to configure these limits in the control signal and, thus, the possible windup effect can be analyzed. Furthermore, it is possible to enable an anti-windup method in the discrete PI controller to avoid the windup effect and verify the improvement in the system response [35]. This procedure is described in the anti-windup block of Table 1. If u(k) is initially out of range, I(k) is recalculated so that the new sum of the proportional and integral actions u

_{out}(k) are within the range of validity.

## 3. Developed Graphical User Interface

- Model parameter space (a). This plot shows the T–τ model parameter space. These two parameters can be jointly changed by dragging the blue point shown in the plot. The corresponding text fields located in the Model parameters panel (d) are automatically updated. In addition, the set of stable FOPTD plants is also plotted. This stability region is given for the PI controller specified in the tool (i.e., the K
_{P}and K_{I}values), and for a specific K. If any of these parameters are changed, the aforementioned stability region is automatically updated. - K
_{P}–K_{I}parameter space (b). This plot shows the K_{P}–K_{I}parameter space. The current K_{P}and K_{I}values are represented by a red point. These values are also shown in two text fields located on section f (see Figure 5). PI parameters can be modified either by editing these text fields or by dragging the aforesaid red point. In addition, the plot shows the set of PI controllers that stabilize the specified feedback system. If any of the model parameters are modified, the K_{P}–K_{I}region is automatically updated. - Time response simulation (c). This plot shows the temporal response of the specified feedback system. It is possible to show different temporal responses, which is useful to contrast different tuning rules. The simulation parameters window (Figure 6) allows the user to configure step changes in both the reference signal and load disturbance. The initial and final values of these step changes, as well as the times when they occur, must be specified.
- Model parameters (d). As mentioned before, the model parameters (steady-state gain, time constant, and time delay) can be changed by editing the text fields contained in this panel. In addition, the tool can work with one or two models by clicking the radio-button Mismatch models. By default, only the “nominal” model is used. However, when the second one, the “simulation” model, is enable, the two models are used and provide the corresponding stability regions, the requested tuning rules are applied on the “nominal” model, and simulations and analysis are performed using the “simulation” model. Therefore, enabling the Mismatch models button, the user can modify these model parameters and analyze the robustness of the closed-loop system. This feature can be also used to consider process model uncertainties due to non-linearities of the plant. Although there are control methodologies directly accounting for the non-linear nature of the system [39,40], a classical approach is linearizing the original system around the nominal operation point, and assuming the non-linearities as parameter uncertainties of the approximated linear model when the point of operation changes.
- Tuning rules (e). The Tuning Rules menu, located on the upper part of the main window, contains the tuning rules described in Table A1 and Table A2. These tuning rules are organized in two sub-menus: open-loop stable models and open-loop unstable models. When a tuning rule is selected (and its validity range is fulfilled by the nominal model parameters), the tool automatically generates a point or a set of points in the K
_{P}–K_{I}parameter space. Depending on the tuning rule, some extra information can be requested by means of a pop-up menu, such as gain margin, phase margin, or the closed-loop time constant. The generated points are tagged in the K_{P}–K_{I}parameter space with the name of the tuning rule and the design specification. All these points are also updated when model parameters are modified. - PI controller configuration (f). A set of control panels related with the PI controller are included in this section. The Control Design panel allows selecting between the continuous and discrete domains. As mentioned before, in this tool, the PI design is performed in the continuous domain. Thus, once the design is carried out (i.e., the PI parameters and the model parameters are defined), the performance and degradation of the time response can be analyzed by clicking on the discrete radio-button. A new simulation will then be executed according to the sampling period defined in the Period text field. In addition, when the user switches from the continuous to the discrete domain, the plot that displayed the K
_{P}–K_{I}parameter space region will display the closed-loop discrete pole localizations within the Z-plane. The saturation model can be specified in the Saturation model panel, where an anti-windup strategy can be enabled, and the minimum and maximum process input values can be specified. The saturation model is only applied when the discrete domain is selected. The tool can work with two types of PI structures: PI and I-P. In the I-P structure, only the measured output is introduced in the proportional term, and not the error, as in the PI structure. The K_{P}and K_{I}values are always visible in the text fields of the PI controller panel. As mentioned in point 2), they are also represented by the red point in the K_{P}–K_{I}parameter space. The Tuning point push button updates the control parameters with the K_{P}and K_{I}values shown in the text fields of pink color. These text fields show the obtained values by the last tuning rule selected by the user. In all cases, the temporal and frequency responses are updated. Finally, each time a simulation is carried out, the text fields contained in the Performance panel are updated. These field texts contain the integral of absolute error (IAE) and the total variation (TV). The first measure is useful to evaluate the output control performance, and the second to evaluate the “smoothness” of the control signal. - Secondary windows (g). The control signal, the Nyquist plot, and the frequency response features (gain margin, phase margin, and sensitivity and complementary sensitivity peaks) are only displayed when the user demands them by means of the respective push buttons placed on the center of the main window. After changing the model or controller parameters, these windows are quickly updated. These windows are shown in Figure 7.

_{m}allows plotting lines with different slopes τ/T. This feature is useful to test the validity range of different tuning rules. Similarly, using the edit text field named line τ/T

_{I}, lines with different slopes K

_{I}/K

_{P}can be plotted in the K

_{P}–K

_{I}parameter space, and pole cancellation methodologies can be applied. In addition, the user can zoom or pan the different axes, save and load sessions, or export the controller or the process model to the MATLAB workspace.

#### 3.1. Comparison with Similar Software and Main Novelties of the Proposed Tool

- Modification of the process model and the controller parameters.
- Simulation parameters configuration.
- Representation of the closed-loop temporal response (control signal and controlled variable).
- Representation of the system frequency response (Nyquist diagram).
- Performance indices computation (IAE, TV).
- Interactive zooming/panning.

- The tool gathers several rules of tuning of PI controllers for FOPTD systems to allow their comparative analysis. Although some applications, such as MATLAB PID Tuner tool, allows adjustment with tuning rules, it only has 4 implemented. The present tool contains 17 rules with the future possibility to increase that number or add custom-user tuning rules.
- K
_{P}–K_{I}controller parameter space and stability region. This special feature allows the user to choose the controller parameters interactively. Along with it we can also show the curves of different specifications, facilitating the previous task. Furthermore, several tuning rules can be compared by means of their associated points, which can be displayed in the mentioned region. - Stability region in the parameter space of the T–τ model. For a gain process K and a given PI controller, this region shows the set of T–τ combinations that the controller is able to stabilize. Thus, the two aforementioned stability regions are very useful to study the robustness of the system.
- In addition to the continuous time domain analysis, the tool allows checking the performance of the closed-loop system in the discrete domain, taking into consideration real aspects in the controller implementation, such as the selection of sampling period, definition of the control signal limits, or the activation of an anti-windup method.

#### 3.2. Training or Educational Uses

- Trial-and-error tuning, by means of the stabilizing region. The stabilizing region in the K
_{P}–K_{I}parameter space is provided by the tool. By moving the red point within this space, the system stability and closed-loop response can be studied as a function of the control parameters. - Comparison of PI tuning rules. The user can easily compare the performance achieved with different tuning rules and different specifications.
- Testing PI control limitations. For instance, in the case of an unstable FOPTD model, the user can check that it is not possible to achieve large gain and phase margins by moving the red point inside the corresponding stabilizing region (K
_{P}–K_{I}parameter space). Similarly, the user can observe the set of FOPTD processes that can be stabilized given a particular PI controller. - Analysis of PI controllers designed by pole cancellation. There are tuning rules that cancel the model pole with the controller zero doing T
_{I}= T. By specifying a line 1/T_{I}in the K_{P}–K_{I}parameter space and moving the red point along it, PI controllers designed by pole cancellation can be studied; where T_{I}remains constant, and only K_{P}is changing. - Robust analysis of the closed-loop system. As mentioned before, the tool can work simultaneously with the nominal and simulation models. Thus, when the Mismatch models radio-button is enabled, the user can analyze perturbations in the model parameter space and study the stabilizing regions or the frequency and temporal responses of the closed-loop system.
- Effects of practical considerations. As mentioned before, real considerations of PI controllers have been taken into account. In addition of the effects of a poor choice of the controller parameters, the degradation of the system closed-loop response when the sample time is not properly selected can be studied.

## 4. Illustrative Example

_{m}, T

_{m}, and τ. Immediately, the stabilizing region in the K

_{P}–K

_{I}space for this system is calculated. The corresponding one for this example is shown in Figure 5. For the simulations, the parameters provided in Figure 6 are configured. There are step changes from zero to one in the set-point, and the load disturbance at times t = 1 s and t = 150 s, respectively. The simulation time is 300 s.

_{P}–K

_{I}space in Figure 5. Their closed-loop responses are shown in section c) of Figure 5 (AMIGO in pink, O’Dwyer in blue, and Ziegler–Nichols in black), and several performance indices are gathered in Table 2. Using the AMIGO tuning rule, the response is the slowest of the three, and it has the higher IAE value. However, it does not show overshoot, and has better robustness with the higher gain and phase margins. The responses using the Ziegler–Nichols rule or O’Dwyer rule are faster, and have smaller IAE values. Nevertheless, they show an overshoot and lower gain and phase margins. Note that the design using O’Dwyer rule achieves the specified GM of 3.

_{P}–K

_{I}space is substituted by the stability area with the closed-loop poles in the discrete space. To determine a proper sampling time, the Nyquist-Shannon criterion states that the sampling rate must be twice the largest frequency contained in the signal spectral content. In control, a typical rule of thumb is to apply a sampling frequency of 5 to 10 times the bandwidth of the closed-loop system, to avoid degradation of the response. In this example, assuming the original design in the continuous domain, the closed-loop bandwidth is about 0.4 rad/s. Considering 10 times this value, a proper sample time h must be less than 2·π/4 = 1.57 s. In this example, two sample times are selected: one of 1 s, fulfilling the previous condition; and the other of 10 s, not satisfying the criterion. As shown in Figure 8, for a sample time of 1 s, the closed-loop poles lay inside the unit circle, which is necessary for stability. The corresponding closed-loop response is also shown in the bottom left corner (in black), together with the response obtained with a sample time of 10 s (in blue). In the first case, a sample time of 1 s is acceptable; however, the response is considerably deteriorated with a sample time of 10 s. It can be checked that the closed-loop response becomes unstable for sample times greater than 18 s (with closed-loop poles outside the unit circle).

_{max}of +0.25, and a minimum one u

_{min}of −1. The pink line represents the ideal closed-loop response without any constraints. The blue one is the response with the previous limitations, and the black one is the response with the control signal constraints and an anti-windup method in the PI controller. The corresponding control signals are also shown in the top of the figure, from left to right, respectively.

#### 4.1. Practical Laboratory Process

_{P}and K

_{I}parameters. The experimental and simulated step responses for each tuning rule are shown in Figure 11.

_{P}and K

_{I}parameters obtained with the tool. The last two columns show the IAE and TV performance indices computed with the tool and from real data, respectively. As can be noted, there is a good correspondence between the real and simulated indices.

## 5. Assessment and Evaluation

#### 5.1. Student Survey

- Theoretical ideas: The tool is mainly focused on the design of PI controllers for stable and unstable FOPTD systems and the application of different tuning rules. However, it has been developed to facilitate the learning other basic concepts related to process control, such as stability boundary, robustness, temporal response for reference tracking, disturbance rejection, frequency response (Nyquist plot), and different closed-loop performance indices.
- User-friendly interface: The graphical user interface (GUI) has been designed avoiding unnecessary elements. In addition, an introduction to the tool is given to the students based on the workflow shown in Figure 4.
- Real-life problem: The discrete implementation of PI controllers can be evaluated in the tool considering real aspects, such as the windup effect of process input constraints. This step can be carried out after a proper tuning is performed (see Figure 4), allowing one to study the possible performance degradations in the closed-loop system response.
- Visual sensation: The GUI is structured coherently and is user-friendly, trying to keep the interface simple, and ensuring that the possible actions that can be carried out are easy to understand.

- Introduction of PID/PI tuning rules, their classification, explanation of the most extended rules, and advantages and disadvantages.
- Description of the tool based on the workflow shown in Figure 4.
- Running the tool which is available to students, and resolution and discussion of several examples interactively.
- Brief introduction to the discrete implementation of controllers and practical aspects, such as sample time selection and input saturations and wind up effect. The fundamentals of these concepts are explained in detail in advanced control subjects. In this lesson, these problems are exemplified using the tool.

- Description of the process to be controlled and its operation.
- Obtaining a model of the system by identification and approximation to a FOPTD model.
- Design and simulation of different PI controllers using the proposed tool.
- Verification of the previous designs in the real process and comparison with simulation results.

- Learning value considers questions about the students’ perceptions of the effectiveness of the proposed tool in facilitating the learning of PI control theory.
- Value added evaluates the use of the tool as a complement for traditional lectures.
- Design usability and easy understanding of the tool is aimed to evaluate how the students perceive the clarity and ease to work with the GUI.

#### 5.2. Student Results

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

_{P}and T

_{I}controller parameters as a function of the plant model parameters (K, T and τ), and some design specification in some cases. Information about the tuning rule is provided in the last column: performance specification (minimum integral error, closed-loop time constant, gain margin, quarter decay ratio, etc.) and its validity range based on the model parameters. Although these tuning rules give the integral parameter of the PI controller as the integral time constant T

_{I}, the presented tool works directly with the integral gain K

_{I}= K

_{P}/T

_{I}.

Author Rule | K_{P} | T_{I} | Comment |
---|---|---|---|

Ziegler and Nichols (1942) | $\frac{0.9\text{}T}{K\text{}\tau}$ | $\frac{\tau}{0.3}$ | Quarter decay ratio, $0.1\le \frac{\tau}{T}\le 1$ |

AMIGO Aström and Hägglund (2005) | $\frac{0.15}{K}+\left(0.35-\frac{\tau \text{}T}{{\left(\tau \text{}+T\right)}^{2}}\right)\text{}\frac{T}{K\text{}\tau}$ | $0.35\text{}\tau +\frac{13\text{}\tau \text{}{T}^{2}}{{T}^{2}+12\text{}\tau \text{}T+7\text{}{\tau}^{2}}$ | I-P structure when $\text{}\frac{\tau}{\tau +T}\le 0.5$ otherwise PI structure |

MISE Murrill (1967) | $\frac{1.305}{K}\text{}{\left(\frac{T}{\tau}\right)}^{0.959}$ | $\frac{T}{0.492}\text{}{\left(\frac{\tau}{T}\right)}^{0.739}$ | Regulator tuning by minimum integral error $0.1\le \frac{\tau}{T}\le 1$ |

MIAE Murrill (1967) | $\frac{0.984}{K}\text{}{\left(\frac{T}{\tau}\right)}^{0.986}$ | $\frac{T}{0.608}\text{}{\left(\frac{\tau}{T}\right)}^{0.707}$ | |

MITAE Murrill (1967) | $\frac{1.305}{K}\text{}{\left(\frac{T}{\tau}\right)}^{0.959}$ | $\frac{T}{0.492}\text{}{\left(\frac{\tau}{T}\right)}^{0.739}$ | |

MIAE Rovira et al. (1969) | $\frac{0.758}{K}\text{}{\left(\frac{T}{\tau}\right)}^{0.861}$ | $\frac{T}{1.020-0.323\text{}\frac{\tau}{T}}$ | Servo tuning by minimum integral error $0.1\le \frac{\tau}{T}\le 1$ |

MITAE Rovira et al. (1969) | $\frac{0.586}{K}\text{}{\left(\frac{T}{\tau}\right)}^{0.916}$ | $\frac{T}{1.030-0.165\text{}\frac{\tau}{T}}$ | |

Cohen and Coon (1953) | $\frac{1}{K}\text{}\left(0.9\text{}\frac{T}{\tau}+0.083\right)$ | $T\text{}\left(\frac{3.33\text{}\frac{\tau}{T}+0.31\text{}{\left(\frac{\tau}{T}\right)}^{2}}{1+2.22\text{}\frac{\tau}{T}}\right)$ | Quarter decay ratio, $0<\frac{\tau}{T}\le 1$ |

O’Dwyer (2001) | $\frac{\pi \text{}T\text{}}{2\text{}{A}_{m}\text{}K\text{}\tau}$ | $T$ | A_{m}: Gain margin |

Skogestad (2003) | $\frac{T}{K\text{}\left({T}_{c}+\tau \right)}$ | $\mathrm{min}\left(T,4\left({T}_{c}+\tau \right)\right)$ | T_{c}: Closed-loop time constant${T}_{c}\le \left(T+\tau \right)$ |

IMC Rivera et al. (1986) | $\frac{\left(T+\frac{\tau}{2}\right)}{K\text{}{T}_{c}}$ | $T+\frac{\tau}{2}$ | T_{c}: Closed-loop time constant$1.7\text{}\tau \le {T}_{c}\le \left(T+\tau \right)$ |

Author Rule | K_{P} | T_{I} | Comment |
---|---|---|---|

Mahji and Atherton (2000) | $\frac{1}{K\text{}}\left(0.889+\frac{{e}^{\raisebox{1ex}{$-\tau $}\!\left/ \!\raisebox{-1ex}{$T$}\right.}-0.064}{{e}^{\raisebox{1ex}{$\tau $}\!\left/ \!\raisebox{-1ex}{$T$}\right.}-0.990}\right)$ | $\frac{2.6316T\left({e}^{\raisebox{1ex}{$\tau $}\!\left/ \!\raisebox{-1ex}{$T$}\right.}-0.966\right)}{{e}^{\raisebox{1ex}{$-\tau $}\!\left/ \!\raisebox{-1ex}{$T$}\right.}-0.377}$ | ISTE optimization criterion $0<\frac{\tau}{T}<0.693$ |

De Paor and O’Malley (1989) | $\frac{1}{K\text{}}\left(\mathrm{cos}\sqrt{\left(1-\frac{\tau}{T}\right)\frac{\tau}{T}}+\sqrt{\frac{T}{\tau}\left(1-\frac{\tau}{T}\right)}\text{}\mathrm{sin}\sqrt{\left(1-\frac{\tau}{T}\right)\frac{\tau}{T}}\right)$ | $\frac{T}{\left[\sqrt{\frac{T}{\tau}\left(1-\frac{\tau}{T}\right)}\right]\mathrm{tan}\left(0.5\phi \right)}$$\phi ={\mathrm{tan}}^{-1}\sqrt{\frac{T}{\tau}\left(1-\frac{\tau}{T}\right)}-\sqrt{\left(1-\frac{\tau}{T}\right)\frac{\tau}{T}}$ | Gain margin A_{m} = 2$\frac{\tau}{T}<1$ |

Venkatashankar and Chidambaram (1994) | $\frac{1}{K}\sqrt{0.98\sqrt{1+\frac{0.04{T}^{2}}{{\left(T-\tau \right)}^{2}}}\left(\frac{25}{\tau}\right)\beta \left(T-\tau \right)\sqrt{\frac{1+\frac{{\beta}^{2}{T}^{2}}{{\tau}^{2}}}{1+{\beta}^{2}\frac{625}{{\tau}^{2}}{\left(T-\tau \right)}^{2}}}}$ | $25\left(T-\tau \right)$ | $\beta =1.373\text{}\mathrm{when}\text{}\frac{\tau}{T}0.25$$\begin{array}{c}\beta =0.953\text{}\\ \mathrm{when}\text{}0.25\le \frac{\tau}{T}\text{}0.67\end{array}$ |

Chidambaram (1995) | $\frac{1}{K}\left(1+0.26\frac{\tau}{T}\right)$ | $25T-27\tau $ | $\frac{\tau}{T}<0.6$ |

Chidambaram (1997) | $\frac{1.678}{K}\mathrm{ln}\left(\frac{T}{\tau}\right)$ | $0.4015T{e}^{5.8\raisebox{1ex}{$\tau $}\!\left/ \!\raisebox{-1ex}{$T$}\right.}$ | |

Ho and Xu (1998) | $\frac{{\omega}_{p}T}{{A}_{m}K}$ ${\omega}_{p}=\frac{{A}_{m}{\Phi}_{m}+0.5\text{}\pi \text{}{A}_{m}\left({A}_{m}-1\right)}{\left({A}_{m}^{2}-1\right)\tau}$ | $\frac{1}{1.57{\omega}_{p}-{\omega}_{p}^{2}\tau -\frac{1}{T}}$ | Φ_{m}: Phase marginA _{m}: Gain margin$\frac{\tau}{T}<0.62$ |

## Appendix B

_{P}, K

_{I}, K, and T, the critical frequency ω

_{c}must fulfill the condition |C(jω

_{c})·G(jω

_{c})| = 1, obtaining Expression (A2). From this, the quadratic equation on ${\omega}_{c}^{2}$ and given by (A3) is achieved. Solving for ω

^{2}, Expression (8) is obtained.

_{max}, Expression (9) is achieved. Note that the first atan2 function can only take the values 0 or π rad, depending on the sign of the second argument. Equation (9) allows to determine the cases where a specific PI controller cannot stabilize the plant model G. When this situation occurs, Equation (9) provides a negative delay τ

_{max}.

_{max}value obtained from Equation (9) is the limit value before achieving a negative phase margin or enclosing clockwise the critical point, −1, in the Nyquist plot. In the case of open-loop unstable processes with a stabilizing PI controller, a positive τ

_{max}value indicates that the Nyquist plot will encircle once, in the anticlockwise direction, the critical point, if the process delay τ is below this maximum value. Thus, in this case, a positive value of τ

_{max}indicates the maximum delay that the plant model G can have, before leaving to encircle in the counterclockwise direction the critical point and, consequently, with instability arising in the closed-loop system. It is important to note that the developed expressions are only true for the specific system structure under study (FOPTD processes compensated with PI controllers).

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**Figure 2.**Stabilizing proportional–integral (PI) regions in the Kp–Ki space for the open-loop process (K = 1, T = −1, τ = 0.2) and the open-loop stable process (K = 1, T = 1, τ = 0.2).

**Figure 3.**Regions of the T–τ space including the process (with K = 1) that can be stabilized by the PI controller (K

_{P}= −1.5, K

_{I}= −0.1) for unstable plants, and by the PI controller (K

_{P}= 1.5, K

_{I}= 0.1) for stable plants.

% control block |

$P\left(k\right)={K}_{P}\xb7\left(a\xb7sp\left(k\right)-y\left(k\right)\right)$ |

$I\left(k\right)=I\left(k-1\right)+\frac{{K}_{P}\xb7h}{2\xb7{T}_{I}}\xb7\left(sp\left(k\right)-y\left(k\right)+sp\left(k-1\right)-y\left(k-1\right)\right)$ |

$u\left(k\right)=P\left(k\right)+I\left(k\right)$ |

% anti-windup block |

$ifu\left(k\right)umax$ |

$I\left(k\right)=umax-P\left(k\right)$ |

$elseifu\left(k\right)umin$ |

$I\left(k\right)=umin-P\left(k\right)$ |

$end$ |

${u}_{out}\left(k\right)=P\left(k\right)+I\left(k\right)$ |

Tuning Rule | K_{P} | K_{I} | IAE | TV | GM | PM |
---|---|---|---|---|---|---|

AMIGO | 0.021 | 0.022 | 67.87 | 1.32 | 7.6 | 75.7 |

O’Dwyer | 0.524 | 0.052 | 29.78 | 1.74 | 3 | 60 |

Ziegler–Nichols | 0.9 | 0.054 | 29.67 | 3.47 | 1.9 | 54.4 |

Tuning Rule | K_{P} | K_{I} | IAE | TV | IAE_{REAL} | TV_{REAL} |
---|---|---|---|---|---|---|

Murril (min. ITAE-servo) | 0.3304 | 0.1265 | 19.77 | 2.05 | 20.90 | 1.92 |

AMIGO | 0.1302 | 0.0573 | 41.78 | 1.80 | 46.30 | 1.78 |

Ziegler–Nichols | 0.5152 | 0.0793 | 29.97 | 3.65 | 29.81 | 3.62 |

Learning Value | |

Q1 | Did the tool facilitate you to understand new concepts of PI control and FOPTD processes? |

Q2 | Evaluate if the tool helps you to remember basic concepts about PI control theory. |

Q3 | Evaluate if you consider the tool useful as a complement of lecture explanations and if it motivates you to learn the explained control concepts. |

Value added | |

Q4 | Did you like the practical simulations carried out with the tool? |

Q5 | Rate if you think you have improved your theoretical knowledge about tuning rules for PI controllers. |

Q6 | Did the tool help you to understand the practical issues that can arise when implementing a PI control loop? |

Q7 | Rate the interactive capabilities of the tool. |

Design usability and easy understanding of the tool | |

Q8 | Is the graphical user interface of the tool user-friendly? |

Q9 | Rate if the tool is easy to understand and use. |

Q10 | Do you think the concepts explained with the tool were easy to follow? |

Group Items | Strongly Disagree | Disagree | Neutral | Agree | Strongly Agree |
---|---|---|---|---|---|

Learning value (Q1, Q2, Q3) | 0% | 7% | 8% | 47% | 38% |

Value added (Q4, Q5, Q6, Q7) | 1% | 4% | 24% | 41% | 29% |

Design usability and easy understanding of the tool (Q8, Q9, Q10) | 0% | 5% | 30% | 48% | 17% |

© 2018 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**

Ruz, M.L.; Garrido, J.; Vazquez, F.; Morilla, F.
Interactive Tuning Tool of Proportional-Integral Controllers for First Order Plus Time Delay Processes. *Symmetry* **2018**, *10*, 569.
https://doi.org/10.3390/sym10110569

**AMA Style**

Ruz ML, Garrido J, Vazquez F, Morilla F.
Interactive Tuning Tool of Proportional-Integral Controllers for First Order Plus Time Delay Processes. *Symmetry*. 2018; 10(11):569.
https://doi.org/10.3390/sym10110569

**Chicago/Turabian Style**

Ruz, Mario L., Juan Garrido, Francisco Vazquez, and Fernando Morilla.
2018. "Interactive Tuning Tool of Proportional-Integral Controllers for First Order Plus Time Delay Processes" *Symmetry* 10, no. 11: 569.
https://doi.org/10.3390/sym10110569