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In this paper, we present a system identification (SI) procedure that enables building linear time-dependent fractional-order differential equation (FDE) models able to accurately describe time-dependent behavior of complex systems. The parameters in the models are the order of the equation, the coefficients in it, and, when necessary, the initial conditions. The Caputo definition of the fractional derivative, and the Mittag-Leffler function, is used to obtain the corresponding solutions. Since the set of parameters for the model and its initial conditions are nonunique, and there are small but significant differences in the predictions from the possible models thus obtained, the SI operation is carried out via global regression of an error-cost function by a simulated annealing optimization algorithm. The SI approach is assessed by considering previously published experimental data from a shell-and-tube heat exchanger and a recently constructed multiroom building test bed. The results show that the proposed model is reliable within the interpolation domain but cannot be used with confidence for predictions outside this region. However, the proposed system identification methodology is robust and can be used to derive accurate and compact models from experimental data. In addition, given a functional form of a fractional-order differential equation model, as new data become available, the SI technique can be used to expand the region of reliability of the resulting model.

Complex systems are common in mechanical engineering applications, for example, automobiles, washing machines, and thermal power plants. As illustrated schematically in Figure

Schematic representation of a complex system with subsystems

One approach to modeling is from the ground up, i.e., to use first principles to model each component and their interactions to create a so-called white box model. If we think of each subsystem as being governed by a single ordinary differential equation (ODE), then the overall system is governed by a large set of coupled ODEs. If, however, there is a governing partial differential equation (PDE), then that may be considered to be equivalent to an infinite set of ODEs. Modeling is possible only when there is physical and mathematical understanding of the behavior of each subsystem. In any case, for control of the overall system one ends up working with a large set of ODEs that must be solved in real time. This may be computationally undesirable for control purposes.

In many cases it is advantageous to use existing or continually acquired experimental data to make predictions [

Two new aspects will be explored here in relation to SI.

In the present investigation, we propose a methodology of analysis to derive accurate models – based on fractional-order differential equations – that approximate relaxation processes of complex systems. To this end, the paper presents first a brief description of the background information about fractional calculus. Next, the system identification technique, based on global regression of an error-cost function, is introduced in detail. Application to previously published experimental data derived from (1) a heat exchanger and (2) a recently constructed building test facility is later carried out with special emphasis on the accuracy of the model within the region of interest. A vital conclusion is the nonuniqueness in the resulting parameter set that defines both the fractional-order model and the corresponding initial conditions. Although the proposed model is accurate within the interpolation region, it is unreliable outside it.

Fractional (noninteger) calculus, which can be thought of as a generalization of the well-known integer calculus, has a long history dating back to Leibniz [

In principle, the field of fractional calculus entails the generalization of the concept of a derivative beyond that of the common integer order. It is concerned with the meaning of the “in-between” derivatives. For instance, the 5/4-derivative of a function

Let us start with the Cauchy formula for repeated integrals of integer order

Regardless of the definition used, the Laplace transform is a powerful technique to solve equations involving fractional derivatives. For instance, the Riemann-Liouville definition in the Laplace domain is

The first step in the SI process is to propose the model and then to determine the parameters therein that will approximate – as closely as possible – its output to the target data. For our purposes, the proposed model is the linear fractional differential equation, given as

To find the unknowns

It is to be noted that other global optimization algorithms can also be used in the search for the set of parameters

SI using the fractional model in (

Details about the experimental setup in the thermal systems laboratory at the University of Notre Dame, and the corresponding data obtained with it, have been reported in Mayes [

Schematic of a shell-and-tube heat exchanger.

For purposes of analysis, we take the approach of [

Multiplicity of solutions in the parameter set

Experimental data and model approximation

Section of surface

In this case, the variance of the error is given as

Initial conditions provide the set of values of the system at the point of departure towards a new set of states. Mathematically, these conditions can be easily set up so that the model can be solved. Experimentally, however, this may not be the case since it is difficult to measure the value of the function (e.g., temperature) at a specific time (e.g.,

As an example of this situation, let us focus on the experimental measurements of the shell-and-tube heat exchanger described before. The system was analyzed in [

An example is the case of a sensor measuring temperature that either is not appropriately calibrated or that has a slow response time-constant, then the initial states for the temperature or its rate of change will not be correctly established. In other cases, the system may be so complex that establishing actual initial conditions is virtually impossible. A car comprising a large number of subsystems may have initial states for each of those components that are different than those of the overall system, and deciding which initial state is the most appropriate (and thus, the one that should be used) may be extremely difficult. A similar situation occurs in a fluid, where initial conditions must be specified at all locations. From this perspective, the inclusion of the initial conditions in the unknown parameter set may be advantageous since the optimal values of these states can be directly computed from the data via SI. On the other hand, mathematically, by increasing the number of adjustable parameters it may be possible to obtain a better approximation from the model. This is what we have done within the context of the two thermal systems considered here.

In what follows, we apply the global-regression-based FOSI to the heat exchanger experimental data to obtain the parameters in (

The results from the two fractional-order models are shown in Figure

Model parameters from heat exchanger data.

Model | | | | | |
---|---|---|---|---|---|

A | 1 | -2.4903 | 0.8055 | | 20 |

B | 1.6708 | 0.3313 | 1.8067 | -0.1093 | 6.94 |

Figure

By using the heat exchanger data described before, and relaxing the restriction on the number of parameters in

Model parameters from shifted heat exchanger data.

| | | | | | |
---|---|---|---|---|---|---|

| 1.5061 | 0.2717 | 1.3858 | 0.005 | -0.8368 | 6.6471 |

| 1.5061 | 0.2621 | 1.3373 | 0.0313 | -0.8366 | 6.6470 |

| 1.5061 | 0.2532 | 1.2919 | 0.0576 | -0.8364 | 6.6471 |

Comparison of prediction of

The use of fractional-order approximation models stems from the fact that as the complexity of a system increases, so does its required mathematical representation. Though this can be achieved starting from conservation equations, for purposes of control it is important that the plant model is compact, efficient, and sufficiently accurate. For instance, a model based on the Navier-Stokes equations may be more accurate than an approximate model, but would be computationally intensive for control (for design and prediction, on the other hand, the mathematical model needs to be accurate). An approximate model, however, may also serve the purpose of bringing the output close to the desired state from where some other form of nonmodel control (e.g., PID) can take over. From this perspective, the fractional-order model given in (

This data-driven model interpolation/extrapolation issue is particularly problematic for the case of complex systems, an example of which is heat exchangers, and the phenomena associated with them. In these thermal systems, complexity arises from geometrical configurations, the large number of parameters involved in their operation, and the nonlinear nature of the system. In the latter case, for example, there is nonlinearity due to variation of properties (e.g., density, viscosity, and thermal conductivity), with temperature. Nonlinearity in the phenomena then generates the possibility of bifurcations, including instability and transition to turbulence. Finally, the conservation equations themselves, which provide the most accurate description of the system, are nonlinear (e.g., in the advective terms). It is apparent that nonlinearities associated with the system may be a main reason for constraining predictions from data-driven models to within the interpolation region only.

The issue of interpolation vs. extrapolation is prevalent in models derived from experimental data. For instance, techniques such as artificial neural networks and correlation equations, among others, are unreliable for extrapolation; yet, they are extremely useful for predictions

Comparison in predictions from fractional- and integer-order fitted models.

Predictions in interpolated region

Predictions in extrapolated region

The issue of reliability of mathematical models is an active research topic [

In practice, experimental uncertainty may be present due to nonideal sensor measurement, thus it may be necessary to establish the applicability of the resulting model for the system under analysis. In the present case, we use the experimental data, along with a definition based on the root-mean-square (RMS) error, to establish the range of applicability for the five-parameter fractional-order model, i.e., (

The first step is to define the RMS error as

Range of applicability of five-parameter set

The second test deals with a more complex multiroom building system. A subscaled two-floor building facility, designed specifically to be a scaled model of an actual building for energy-related studies, is located in the thermofluids laboratory at the California State University, Los Angeles, and is used to conduct experiments for thermal analysis and control [

Subscaled building test facility.

Schematic view

Photograph

The average air temperature measurements for the eight rooms of the building are illustrated in Figure

Time evolution of average room temperatures in multiroom building device.

For the analysis that follows, room 1 is selected because of its smooth transition from a maximum temperature,

The FOSI methodology is now applied to the experimental temperature measurements of room 1 of the test facility. Again, the variance of the error is

Model parameters for the relaxation process in room 1 of a building test facility.

Model | | | | | | |
---|---|---|---|---|---|---|

SA | 1.2319 | 3.4428 | 10 | 1.0223 | -0.1043 | 2.04 |

LOA | 1.0266 | -0.0213 | -0.1052 | 0.9577 | 47.5124 | 2.86 |

Prediction of time evolution of average room temperature for room 1 from fractional-order models.

These results are confirmed quantitatively in Table

Finally, it is important to note that, though commonly done with integer-order calculus, a physical interpretation of the fractional derivative in the modeling of the two thermal systems analyzed here is not necessary since the corresponding fractional-order model – in each case – is merely a stand-in for the true mathematical model. Some efforts have been made in the physical interpretation of fractional-order derivatives (see [

Complex systems, of which heat exchangers and multiroom buildings are examples, can be found in widespread applications. For their performance control it is necessary to obtain an accurate description of their dynamic behavior. In this regard, fractional-order differential equations, derived via SI, may provide compact and efficient models of complex systems. In this work, a fractional-order-based system SI procedure grounded on global regression has been proposed to build accurate models from data. The SI methodology seeks the optimum values of a parameter set that includes the fractional order of the differential equation, its parametric constants, and, when necessary, the initial conditions. This last case arises when accurate knowledge of the initial conditions, whether the value of the function or its rate of change, is not possible, and such conditions have to be included as part of the search. It is to be noted, however, that there may be cases when some values of these conditions may not be physically feasible.

The application of the SI methodology to experimental measurements from a shell-and-tube heat exchanger and a multiroom building has confirmed that the approach is accurate and robust. The fractional-based model obtained via global regression provides better approximations to the data than those obtained by SI with local optimization algorithms. The results show that not only the apparent multiplicity in the parameter set that includes the order of the proposed model equation along with its constants and the initial conditions, as provided by the results from the global regression analysis, is due to the fact that the initial condition(s) is(are) included in the search, rather than in the model itself, but also the proposed fractional-order differential equation model is reliable within the interpolation region, but cannot be used with confidence for predictions outside this region. The SI methodology proposed here is robust and can be used to derive accurate and compact models from experimental data. In addition, given a functional form of a fractional-order differential equation model, as new data become available the SI technique can be used to expand the region of reliability (interpolation) of the resulting model. The methodology described here can be extended to model other complex physical systems.

Fractional integral operator of order

Coefficient for the linear fractional differential equation

Riemann-Liouville fractional derivative of order

Caputo fractional derivative of order

Two-parameter Mittag-Leffler function

Mass flow rate

Fractional order

Variance of the error

Number of experimental data points

Vector of search parameters

Fluid bulk temperature

Time

Input variable

Output variable

rms error

Gamma function

Dimensionless temperature

Independent variable

Dimensionless time

Cold fluid

Experimental value

Hot fluid

Inlet

Outlet

Predicted value

Data acquisition system

Fractional-order differential equation

Fractional-order system identification

Local optimization algorithm

Ordinary differential equation

Simulated annealing algorithm

System identification

Partial differential equation.

The data used in this study can be made available, after publication, via a hyperlink to the corresponding author’s personal website.

The authors declare that they have no conflicts of interest.

Kin M. Li was the recipient of a CREST-CEaS fellowship, for which we are grateful. The comments from an anonymous referee have greatly influenced the final version of this paper and are very much appreciated. This work has been supported by an NSF HRD-1547723 grant.