Mathematical Problems in Engineering

Volume 2015, Article ID 989542, 14 pages

http://dx.doi.org/10.1155/2015/989542

## Optimal Design of Stochastic Distributed Order Linear SISO Systems Using Hybrid Spectral Method

^{1}School of Chemical Engineering, Yeungnam University, Gyeongsan 712-749, Republic of Korea^{2}Chemical & Biological Engineering, University of British Columbia, Vancouver, Canada

Received 20 May 2015; Revised 18 August 2015; Accepted 2 September 2015

Academic Editor: Son Nguyen

Copyright © 2015 Pham Luu Trung Duong et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The distributed order concept, which is a parallel connection of fractional order integrals and derivatives taken to the infinitesimal limit in delta order, has been the main focus in many engineering areas recently. On the other hand, there are few numerical methods available for analyzing distributed order systems, particularly under stochastic forcing. This paper proposes a novel numerical scheme for analyzing the behavior of a distributed order linear single input single output control system under random forcing. The method is based on the operational matrix technique to handle stochastic distributed order systems. The existing Monte Carlo, polynomial chaos, and frequency methods were first adapted to the stochastic distributed order system for comparison. Numerical examples were used to illustrate the accuracy and computational efficiency of the proposed method for the analysis of stochastic distributed order systems. The stability of the systems under stochastic perturbations can also be inferred easily from the moment of random output obtained using the proposed method. Based on the hybrid spectral framework, the optimal design was elaborated on by minimizing the suitably defined constrained-optimization problem.

#### 1. Introduction

Fractional/distributed order calculus is applied widely across a range of disciplines, such as physics, biology, chemistry, finance, physiology, and control engineering [1–6]. The memory property of fractional order calculus provides a novel tool to model real-world plants better than integer order ones such as diffusion plants [5]. Fractional calculus has been used for modeling of turbulence in [2]. In [3], the concept of fractional calculus is used for interpreting the underlying mechanism of dielectric relaxation. A method for design fractional order controllers for deterministic systems is proposed in [6].

The distributed order (DO) equation, which is a generalized concept fractional order, was first proposed by Caputo in 1969 [7] and solved by him in 1995 [8]. The general solution of linear DO was then discussed systematically [9]. Later, the DO concept was used to examine the diffusion equation [10], the rheological properties of composite materials, and other real complex physical phenomena [11–14]. Several different methods for the time domain analysis of DO systems have been reported [15–18]. On the other hand, a numerical method for the analysis of a DO operator is still immature and requires further development. In particular, there are few methods to analyze DO systems under the excitation of random processes. This motivated the theme of this study: the development of a computational scheme for the analysis basic of a DO system with stochastic settings. The operational matrix (OP) has attracted considerable attention for the analysis of a range of dynamic systems [19–21]. The main characteristic of this technique is that different analysis problems can be reduced to a system of algebraic equations using different types of orthogonal functions, which greatly simplifies the problem [19]. On the other hand, to the best of the author’s knowledge, there are no reports on the analysis of stochastic DO systems using an OP. Many natural systems often suffer from stochastic noise that causes fluctuations in their behavior, making them deviate from deterministic models. Therefore, it is important to examine the statistical characteristics of states (mean, variance) for those stochastic systems. This problem is often called statistical analysis (or uncertainty quantification) of a system [22–24]. This paper proposes a numerical scheme based on the OP technique for the statistical analysis of DO systems.

The Monte Carlo (MC) method is commonly used to simulate a stochastic model [25, 26]. The method relies on the sampling of independent realizations of random inputs according to their prescribed probability distribution. The data is fixed for each realization and the problem becomes deterministic. Solving the multiple deterministic realizations builds an ensemble of solutions, that is, the realization of random solutions, from which statistical information can be extracted, for example, the mean and variance. Nevertheless, this method typically reveals slow convergence and has a large computational demand. For example, the mean values typically converge as , where is the number of samples.

Generalized polynomial chaos (gPC) [27–32] represents a more recent tool for quantifying the uncertainty within system models. The approach involves expressing stochastic quantities as the orthogonal polynomials of random input parameters. This method is actually a spectral representation in random space and converges rapidly when the expanded function depends smoothly on random parameters. On the other hand, the stochastic inputs of many systems involve random processes parameterized by truncated Karhunen-Loeve (KL) expansions, and the dimensionality of the KL expansions depends on the correlation lengths of these processes. For input processes with low correlation lengths, the number of dimensions required for an accurate representation can be extremely large.

The OP method [29], where a system is described by a stochastic operator (operational matrix), is an alternative approach for the simulation of stochastic integer order systems. This method involves the inverse of the stochastic operators as Neumann series and is most effective for systems with inputs with low correlation lengths. On the other hand, it is restricted to small random parametric uncertainty.

In a recent study [33], the authors introduced a hybrid spectral method, which combines the advantages of both the OP and polynomial chaos (PC), to simulate single input single output (SISO) stochastic fractional order systems. In the present study, the method reported in [33] was extended to the statistical analysis of DO systems affected by stochastic fluctuations. Here, the stochastic operator was approximated using PC instead of a Neumann series. This method provides the algebraic relationships between the first- and second-order stochastic moments of the input and output of a system, hence bypassing the KL expansions that can require large dimensions for accurate results. In contrast to the traditional OP method, the proposed method is not limited by the magnitude of the uncertainty.

Section 2 briefly introduces a DO system and the OP technique for uncertainty quantification in this system, leading to computation of the moments of random matrices. Section 3 summarizes the process of calculating the moments of the random matrices using a stochastic collocation. Section 4 defines the suitable performance objectives coupled with the spectral method for the design of a stochastic linear DO system. Section 5 provides examples to demonstrate the use of the proposed method. The results of the proposed deterministic system with a DO were compared with those of other existing numerical and analytical methods. To assess a stochastic DO system, the MC, gPC, and frequency methods were first adopted to the stochastic DO system for comparison because the analytical results were unavailable. The results from the proposed method were then compared with the numerical results from the MC, gPC, and frequency methods.

#### 2. Preliminary of Fractional and Distributed Order System

In this section, we give some necessary definitions and preliminaries of the fractional calculus theory which will be used in this paper.

##### 2.1. Governing Equation for System Dynamics with Fractional Order Dynamics

Fractional calculus considers the generalization of the integration and differentiation operator to a noninteger order [34, 35]:where is the order of the operator.

Among many formulations of the generalized derivative, the Riemann-Liouville (RL) definition is used most often:where denotes the gamma function and is an integer satisfying .

The RL fractional integral of a function is defined as follows:Another popular definition of a fractional order derivative is the Caputo () definition [36],The Laplace transform for a fractional order derivative under zero initial conditions can be defined as .

Note that, under a zero initial condition, the two Riemann-Liouville and Caputo definitions are equivalent.

Therefore, a fractional order single input single output (SISO) system can be described by a fractional order differential equation as or by a transfer function:where and are the arbitrary real positive numbers and and are the input and output of the system, respectively.

##### 2.2. Distributed Order Systems

The DO differential operation is defined as follows [17]:where denotes the distribution function of order .

Therefore, the general form of the DO differential equation isFor time domain analysis of the DO system, the integral in (7) is discretized using the quadrature formula as follows [16, 17]:where , are the node and weight from the quadrature formula, respectively. In other words, the DO equation is approximated as a multiterm fractional order equation and can be rearranged as (5).

##### 2.3. Operational Matrices of Block Pulse Function for the Analysis of Distributed Order Systems

Block pulse functions (BPFs) are a complete set of orthogonal functions that are defined over the time interval, ,where is the number of block pulse functions.

Therefore, any function that can be absolutely integrated on the time interval can be expanded to a series from the block pulse basis as follows:where constitutes the block pulse basis. From here, the subscript of is dropped out for the convenience of notation.

The expansion coefficients (or spectral characteristics) can be evaluated as follows:Furthermore, any function that is absolutely integrable on the time interval can be expanded as follows:with expansion coefficients (or spectral characteristics) ofEquation (3) can be expressed in terms of the OP [19],where the generalized OP integration of the block pulse function, , isThe elements of the generalized OP integration can be given byThe generalized OP of a derivative of order iswhere is the identity matrix.

The generalized OP of derivative can be used to approximate (2) as follows:Therefore, using the OP, the discretization of DO can be expressed asThe DO system in (7) can be rewritten in terms of the OP, , as follows:The input and output are related by the following equation:

##### 2.4. Stochastic Analysis of Distributed Order Systems

Consider the system described by (7), which has the spectral characteristics of input and output linked by (21). Assume that the system is excited by random forcing with a given mean and covariance function as follows:where denotes the expectation operator and the spectral characteristics of the mean and covariance function of the input are calculated in (11) and (13).

Using the OP, the spectral characteristics of the mean and covariance of the output are given by the following [33] (the details are available in Appendices):Therefore,The random parameters , result in the random OP in (23) and (24), and its (OP ) moment can be estimated using a stochastic collocation method, which is described in the next section.

When the parameters, , , are deterministic, (23) and (24) become

*Remarks*. The relationship in (25) is invariant with respect to the orthogonal polynomial used to construct the OP of the fractional order integral and derivative. The relationships in (24) and (25) are only available for a linear system.

#### 3. Stochastic Collocation for the Operational Matrix

A stochastic collocation method, which is described briefly below, is based on the gPC and can easily estimate the means and variances of complex dynamics. Therefore, it has been used to estimate the moment of the random matrix in (24).(i)Assume that a random OP has the form, , where is a vector of independent random parameters with the probability density functions . Vector has the joint probability density function, , with the support, .(ii)Choose a suitable quadrature set for each random parameter according to the probability density so that a one-dimensional integration can be approximated as accurately as possible by , where is the th node and is the corresponding weight.(iii)Construct a multidimensional cubature set by tensorization of the one-dimensional quadrature set over all the combined multi-index . Because manipulation of the multi-index is cumbersome in practice, a single index is preferable for manipulating these equations. The multi-index is often replaced by a graded lexicographic order index, [27]. Because the weighting functions of the cubature are the same as the probability density functions, the moment of the random matrix can be approximated byThe Matlab suite, OPQ, can be used to obtain the one-dimensional quadrature sets and their corresponding orthogonal polynomials (polynomial chaos) with respect to the different density weights [36].

The algorithm of the proposed method for the analysis of a stochastic system can be summarized as follows:(a)Calculate the coefficients , of the expansions of the mean and covariance of the input as shown in (11) and (13).(b)Rewrite the DO differential equation in (7) in terms of OP as (20).(c)The coefficients of expansions of the mean and covariance function of the output are obtained from (23). In (23), the moments of several random matrices need to be calculated. The moment of a random matrix is calculated by the stochastic collocation method as (26).(d)Finally, the mean and covariance of the output are obtained as (24).For a clearer understanding of the algorithm, a similar algorithm is depicted graphically in [33] for the analysis of stochastic linear fractional order systems.

#### 4. Optimal D Controller Design

Assume that the system is described by (7), where coefficients , are independent random variables with given distributions. The set point input is a random process with a given mean and covariance function as follows:The system is in the closed loop configuration, as shown in Figure 1, with a fractional order controller [35]