Implementation of a Generalized Predictive Controller with Fuzzy Suppression and Weighting Parameters for a Level Plant with Interconnected Concentric Tanks

Introduction

Optimal control plays a crucial role in modern engineering since it allows systems to operate efficiently, reliably, and with minimal resource usage. It provides a mathematical foundation for making real-time decisions that balance performance with constraints, making it essential in complex and dynamic environments. One key application is microgrids, where optimal control can be used to manage energy distribution effectively, especially when power loads are uncertain and the grid operates in islanded (i.e., isolated) mode ¹. Additionally, optimal control techniques are valuable for the optimal tuning of controllers, which is particularly beneficial in systems involving shape memory alloy (SMA) wire actuators, allowing for a precise and adaptive response ². Furthermore, fuzzy controllers, which are often guided by optimal control principles, have shown strong potential in managing nonlinear servo systems, offering a robust performance despite system uncertainties and nonlinearities ³. A survey that focuses on the fundamental aspects of fuzzy control for mechatronics applications can be consulted in ⁴.

Generalized predictive control (GPC) is a strategy belonging to the family of model predictive control (MPC), which seeks to minimize a cost function by predicting the controlled output of a system. GPC requires several tuning parameters, including weighting factors acting on the control effort and on the discrepancy between the output and the desired reference trajectory ⁵. The success of GPC has been reported in several processes and in different productive sectors, such as the chemical, aeronautics, energy efficiency, robotics, and mechanical industries, proving to be a solid strategy for controlling processes in different areas ⁶⁽⁹.

The weighting factors of GPC in conventional tuning are set to constant values, without considering that varying them can improve controller performance ¹⁰. Defining the point values for the weighting factors of a GPC is also a design problem. Sometimes, engineers regard the tuning of the weighting parameters as an optimization problem, proposing solutions that combine metaheuristic strategies with the controller. In ¹¹, for instance, a particle swarm optimizer (PSO) was employed to tune a GPC with constraints on the control signal and on the transient behavior of a bridge crane. The results demonstrated the robustness of the controller to variations in operating conditions. The effects of incorporating a genetic algorithm (GA) to tune the parameters of a GPC are shown in ¹². Here, an improvement in controller performance, including disturbance rejection, was achieved.

In general, there are no analytical, numerical, or theoretical methods to determine the values of the GPC weighting factors. Therefore, it is necessary to establish strategies for particular systems since the dynamics of the processes unquestionably exhibit variations reflected in the models that describe their behavior. Some research works have focused on varying GPC weighting parameters, achieving considerable improvements in controller performance.

In ¹³, an optimized weighting strategy was presented, varying the weighting factors for each operating point while implementing bilinear interpolation to cover the entire workspace. The results show that the proposed technique allows obtaining smaller deviations in the working region.

A case involving the variation of the weighting parameters of a tuned predictive controller for a power converter is reported in ¹⁴. Here, three types of cost functions with weighting factor variations were identified, selecting the ideal function for the objectives stated in the design of the model-based predictive controller. Simulation results demonstrated the effectiveness of the strategy for multi-objective control in power converters. The hybridization of a model-based predictive controller is shown in ¹⁵, which was carried out by implementing fuzzy logic to obtain the predictive model of the system and as a mechanism for establishing the conditions that ensured the closed-loop stability of a plant.

The combination of a GPC and fuzzy logic to switch controllers in the global model of a power plant is presented in ¹⁶. By means of fuzzy pattern clustering, the plant model was divided into local models, and the controllers were tuned based on each of them. Subsequently, controller planning was performed to jointly regulate the entire operating region of the plant. In discussing the results, the authors concluded that, by means of simulated schemes, this strategy exhibits a better performance than conventional proportional-integral-derivative (PID) and GPC controllers. Apart from modeling, fuzzy logic has been combined with GPC to tune controller parameters. ¹⁷ reported a strategy that enables the online tuning of GPC parameters through event triggering and fuzzy logic, demonstrating that it is an efficient combination that allows reducing the energy consumption of the system.

The proposed approach, which combines a GPC with fuzzy logic, offers several advantages over both general optimal and non-optimal control strategies. Compared to traditional optimal control approaches, due to its predictive nature, GPC provides enhanced adaptability to system dynamics and disturbances, allowing for a better handling of model uncertainties and external perturbations. The integration of fuzzy logic further improves the controller’s robustness by incorporating expert knowledge and linguistic rules, enabling effective decision-making even in the presence of nonlinearities and modeling inaccuracies ¹⁸.

In contrast to non-optimal methods such as PID or conventional rule-based controllers, the GPC-fuzzy combination offers improved performance in terms of setpoint tracking, disturbance rejection, and stability. While classical controllers typically require extensive tuning and may struggle with varying operating conditions, the predictive capability of GPC ensures proactive adjustments, and the fuzzy logic component enhances flexibility in dynamic environments ¹⁹.

This paper is structured as follows. It begins a description of a plant with interconnected concentric tanks, outlining the mathematical model that represents the system dynamics. The GPC is then tuned with static suppression and weighting parameters. Subsequently, fuzzy logic is implemented, generating a table of linguistic variables to tune the suppression and weighting parameters for the GPC. This is done to establish the block diagram and implement the controller. Afterwards, the performance of the proposed approach is compared against that of a fuzzy PID controller, both implemented on the aforementioned plant. Finally, the conclusions drawn from this work are presented.

Methodology

Water level control is a common challenge in industrial settings, and it has been extensively analyzed in numerous research works. For instance, in ²⁰, the authors focus on developing and implementing a water level control system based on a programmable logic controller (PLC), and ²¹ present the real-time implementation of a plug-and-play process monitoring and control system for an experimental three-tank setup. Furthermore, the control of interconnected concentric tanks, such as the one presented herein, is crucial in various industrial applications, including chemical processing, wastewater treatment, and pharmaceutical manufacturing. Practical implementations for controlling water levels have also been reported ^22,23.

Level plant with interconnected concentric tanks

Fig. 1 shows the studied level plant, which consists of two interconnected concentric tanks, a data acquisition card, two centrifugal pumps, an electrical circuit based on solid state relays, and an electronic circuit that allows for signal coupling. Together, the aforementionedelementsenabletheimplementation of control strategies for the level variable. As described in ²⁴, the plant was designed to implement controllers and process monitoring and supervision schemes.

Figure 1: Level plant with interconnected concentric tanks

The transfer function representing the dynamics of the plant is obtained from the open-loop excitation of the system, from which first-order with delay (FOD) and second-order with delay (SOD) models are derived. To select the model that best fits the real plant dynamics, two indicators are calculated: the mean squared error (MSE) and the correlation coefficient (r), as expressed in Eqs. (1)) and (2), respectively.

Table I shows the results of the two indicators calculated for each model, showing that the FOD model is the closest to the real dynamics of the interconnected concentric tank plant —see Eq. (3).

Table I: Indicators used for plant model selection

Then, the continuous-time FOD model is discretized, selecting the sampling period (T) via the system bandwidth criterion, obtaining T = 21 s. The discrete time model HGp(z) of the interconnected

concentric tank plant, which will be used for tuning the GPC with fuzzy weighting factors, is shown in Eq. (4).

Fuzzy logic

Fuzzy logic constitutes a flexible alternative that allows incorporating structures supported by linguistic rules into the solution of problems. This facilitates the handling of incomplete information and uncertainty. Since fuzzy logic seeks to simulate the human intellect, the experience of engineers and expert operators can be introduced into the design ^{27)- (929}.

The implementation of fuzzy logic involves three steps: 1) fuzzification, where a degree of membership is assigned to the inputs via membership functions in order to obtain the fuzzy set of variables; 2) linguistic rules, where the expert, by means of propositions, expresses their acquired knowledge of the process; and 3) defuzzification, where a quantifiable value is assigned to the process outputs. This value comes from the fuzzy inference performed to obtain the linguistic rules. The fuzzy inference system then obtains a conclusion based on input information in fuzzy terms ³⁰. In control systems, fuzzy logic is implemented in the following sequence:

• Determine the inputs and outputs of the fuzzy syste

• Elaborate the input and output variable characterization template

• Define the linguistic variables

• Define the universe of discourse for each input and output variable

• Define the fuzzy sets for each input and output variable

• Define the semantic rules for each linguistic variable

• Establish the rules according to an analysis of the behavior of the system’s input and output variables

Subsequently, the transfer function of the plant is rewritten in negative powers (z−1), as indicated in Eq. (9), and the discrete delay (d) is determined.

The studied system has a delay d = 1 and amaximumcoefficient m = 2.Fortheinitial tuning of the GPC, a control horizon (Hc = 5), a maximum prediction horizon (Hmaxc = 5), a minimum prediction horizon (N1 = 1), and T = 21 s are defined.

A, B, and ˆA are defined by the following polynomials:

The expressions used to obtain the prediction are given in Eqs. (13)-(15).

The polynomials Ej+1, Fj, and Gj, corresponding to each prediction, are obtained as indicated in Eqs. 16-30.

The prediction equation is given by(31).

Where:

Eq. (31) can be expressed as indicated below:

Fuzzy suppression and weighting parameters for the GPC

To design the system that enables the fuzzy variation of the GPC weighting factors, a linguistic characterization template is constructed for both the inputs and outputs of the system. This is shown in Tables II and III, along with the corresponding definitions, the universe of discourse, the set of terms for each variable, the definition of each term, and the semantic rules or membership functions for each term in each set. The inference and if-then rules are defined based on an analysis of the controlled output, the control error signal, and the control law. A total of 27 semantic rules are defined.

Table II: Template for characterizing the input variable

Table III: Template for characterizing the output variables

Results and discussion.

To implement the GPC with mobile fuzzy suppression and weighting factors, the block diagram shown in Fig. 2 was followed. The inputs and outputs of the system were configured according to the linguistic characterization templates. The parameters delivered by the fuzzy system correspond to the suppression and weighting factors, which were in turn configured as inputs to the GPC. The controller is in charge of generating the control law at each instant during the execution of the algorithm, while the plant delivers the controlled output to provide system feedback. The block diagram was programmed and implemented in the LabVIEW software.

Figure 2: Block diagram of the GPC with fuzzy weighting factors

Fig. 3 illustrates the responses of the final implementation of the GPC with fuzzymoving suppression and weighting factors in the studied level plant with interconnected concentric tanks. The responses were obtained for a set point value equal to 20%, determining the behavior of the control error, the set point, the controlled output, and the control law over time. The results show that the output follows the desired reference with an average error equal to ±0.0630%,. Moreover, the control law does not exhibit saturation and shows an under-damped behavior with a maximum output of 24%. After 50 s, it stabilizes without abrupt changes.

Figure 3: Response of the GPC with fuzzy moving suppression and weighting factors in the studied level plant with interconnected concentric tanks

As previously mentioned, a fuzzy PID controller was used for comparison, following the block diagram shown in Fig. 4. The responses of this controller are shown in Fig. 5, indicating a sustained oscillation in the output, caused by behavior of the control law. There is an evident steady-state error, as well as an overshoot of 21.2%.

Figure 4: Block diagram of the fuzzy PID controller

Figure 5: Response of the fuzzy PID controller in the studied level plant with interconnected concentric tanks

Toevaluate the controllers, we used metrics related to response time (Mp, ts(s), and ess), and metrics related to error integration criteria (IAE and ICE). The results are shown in Table IV. Note that the GPC with fuzzy weighting factors performs better than the fuzzy PID controller in all performance metrics.

Table IV: Performance comparison between the implemented controllers

Conclusions

In this work, aGPCstrategywascombinedwithfuzzylogicinordertotunethemovingsuppression and weighting factors of the controller, unlike conventional methods that use constant values. This strategy yielded satisfactory results, as confirmed through the graphical and numerical responses obtained and a comparison against a fuzzy PID controller. Both strategies were tuned for a level plant with interconnected concentric tanks. We started by selecting the plant model from two statistical metrics, wherein the FOD model was deemed to be the best fit for real plant dynamics. Then, the GPC was tuned until the cost function was obtained. Subsequently, fuzzy logic was used to establish the variable and fuzzy weighting factors until there was a control action for each instant during the execution of the algorithm. It was possible to demonstrate that varying the weighting factors in combination with fuzzy logic yields a solid control strategy. The behavior over time of the control law also stood out, characterized by a sustained performance without abrupt oscillations; there was constant current consumption in the plant under a stable regime, without current peaks in the recirculation pumps.