On the Modelling and Control of a Laboratory Prototype of a Hydraulic Canal Based on a TITO Fractional-Order Model

Abstract

In this paper a two-input, two-output (TITO) fractional order mathematical model of a laboratory prototype of a hydraulic canal is proposed. This canal is made up of two pools that have a strong interaction between them. The inputs of the TITO model are the pump flow and the opening of an intermediate gate, and the two outputs are the water levels in the two pools. Based on the experiments developed in a laboratory prototype the parameters of the mathematical models have been identified. Then, considering the TITO model, a first control loop of the pump is closed to reproduce real-world conditions in which the water level of the first pool is not dependent on the opening of the upstream gate, thus leading to an equivalent single input, single output (SISO) system. The comparison of the resulting system with the classical first order systems typically utilized to model hydraulic canals shows that the proposed model has significantly lower error: about 50%, and, therefore, higher accuracy in capturing the canal dynamics. This model has also been utilized to optimize the design of the controller of the pump of the canal, thus achieving a faster response to step commands and thus minimizing the interaction between the two pools of the experimental platform.

A Fractional-Order Partially Non-Linear Model of a Laboratory Prototype of Hydraulic Canal System

“…Previous papers on identifying the dynamics of this system [25,[27][28][29] reported linear models-of either integer or fractional order natures-that were fitted to responses of the canal to step commands. These models were identified around a specific operating point and badly reproduced the canal behaviour when the operating point diverged from that specific one.…”

“…Such model significantly improved the description of the canal dynamics, reducing in more than 30% the ISE (integral squared error) between the model response and the recorded experimental data. The above multivariable model was used later in [28] to improve the control system which is in charge of the water level of the first pool y up (t). By closing this loop, the two input/two output system of [27] becomes a single input/single output system whose input is u(t) and its output y dwe (t).…”

“…Later, the two pools that constitute the canal were identified and a TITO (two input/two output) fractional-order model was obtained in [27] that took into account the coupling existing between the two pools. In [28] the combination of the TITO model with a water level closed-loop control of the upstream pool of our canal allowed to reproduce the experimental results with a higher fidelity (up to a 50% of improvement) than the linearized integer-order MI MO models traditionally proposed in the scientific literature. Based on these fractional-order models, a fractional order Wiener-Hopf optimal control system was designed and tested in [29], that efficiently adjusted the downstream end water level of the main canal pool.…”

Fractional-Order System Modeling and its Applications

“…In particular with fractional-orders, oscillators [74], supercapacitors [75,76] thermo-mechanical systems [78], atmospheric dispersion [79], lithium-ion batteries [80,81,82,83], acid-lead battery [84], electrochemical cell [85], polymer electrolyte membrane fuel cell (PEMFC) [86], thermal systems [87,88,89,90], heat transfer [91], solid-oxide fuel cells [92], viscoelastic material [93], diffusion modeling [94], solid-core magnetic bearing [95], Zener diode [49], permanent magnet synchronous motor [96], electrical circuits [97,98], magnetic levitation [99,100,101] were shown with a remarkable result. Recently, laboratory prototype of a hydraulic canal modeling and control strategy have been discussed by [102] using fractional-order time delay TITO models. Modeling of two-pool laboratory hydraulic canal was described by [103].…”

Analytical Solutions for Multi-Time Scale Fractional Stochastic Differential Equations Driven by Fractional Brownian Motion and Their Applications

“…In the last few years, the interest of the scientific community towards fractional calculus has experienced an exceptional boost, and so its applications can now be found in a great variety of scientific fields-for example, anomalous diffusion [1][2][3], medicine [4], solute transport [5], random and disordered media [6][7][8], information theory [9], electrical circuits [10], and so on. The reason for the success of fractional calculus in modeling natural phenomena is that the operators are nonlocal, which makes them suitable to describe the long memory or nonlocal effects characterizing most physical phenomena.…”