Abstract
An inverse problem is considered for the determination of the parameters, involved in the right-hand side of the system of nonlinear ordinary differential equations by given initial and final conditions. The solution of the problem is reduced to the minimization of the quadratic functional, which indeed is a deviation of the value of the solution from the given values at the end points. Using the quasilinearization method a calculation method is proposed to the solution of the considered problem. The application of this method is demonstrated on the example of the determination of the hydraulic resistance in the tubes.
1. Introduction
As is known, different classes of inverse and identification problems play an important role in the solition of many applied problems from physics, hydrodynamics, and industry [1–5]. There exist various methods to solve these problems, as well as optimization methods [6–10]. One of the principle steps of these methods is a choice of suitable functional. Since many applied problems are described by the nonlinear systems, the choice of such functional and further solution of the corresponding optimization problem is problematic [2]. These difficulties may be avoid, for example, by the iterational quasilinearization method, convergence of which is in detail studied in [11, 12].
In the present work a multidimensional identification problem is considered for the determination of the parameters involving the right-hand side of the system of nonlinear differential equations by given initial and final conditions. Solution of the problem is reduced to the optimization problem, in which the functional under minimization is constructed as a quadratic deviation of the solition of the system from the given data at the end points. Since solution of the problem in the stated nonlinear formulation presents certain difficulties [13, 14], the considered problem is reduced to the linear case with respect to phase coordinates and vector of parameters by the help of quasilinearization method. A quadratic functional is constructed and an expression for its gradient is derived. Using Gram-Schmidt orthogonalization method a calculation algorithm is proposed, which allows one to define sought parameters. This algorithm is applied to the example, describing the flow in the pipes.
2. Problem Statement
Let the movement of the object be described by the system of differential equations:where is -dimensional phase vector, is -dimensional differentiable function continuous in the interval , and is -dimensional constant vector to be found.
Let the following initial conditions be given:where , are given natural numbers and is an independent variable , given -dimensional vector. The problem consists in the finding of the vector , by which the solution of the Cauchy problem (1)-(2) in the point satisfies the given condition
Such problems are often met in applications [1, 3, 6, 15], when initial data (2) are given and final ones are statistically measured. In these cases it is required to find the vector such that the solution of the problem by initial data (2) is maximally close to the measured data at the end points. As an example the problem in oil-gas production may be shown when it needs to define the coefficient of the hydraulic resistance.
3. Solution Method
Since the function is nonlinear, to solve the problem (1)–(3) it is expedient to use any numerical method, as well as quasilinearization method [11, 13]. So in the first step we linearize the problem (1)–(3). For this purpose some nominal trajectory and parameter are chosen and it is assumed that th iteration is already hold. If we linearize (1) with respect to these data we obtain After integration of the linear differential equation (4) with condition (2) we get the representation [16] where is a fundamental matrix of the system of homogeneous equationsand the matrices , are defined as in [16]
To provide that the solution of the linearized differential equation (4) with initial data (2) coincides with the values of the measurements in the point we construct the following quadratic functional in the th iteration:where the sign stands for transpore, is a constant symmetre -dimensional weight matrix, that is chosen in each iteration, considering the specifics of the concrete problem; is -dimensional vector of observation; is -dimensional vector defined by (5). Then the solution of the stated problem is reduced to the problem: Find a constant vector , by which the solution of (1) with conditions (2) minimizes the functional (8).
Various algorithms exist for the minimization of the functional (8). However in the solution of the concrete problem, as well as problem arising in the oil production, these algorithms met some difficulties [14] (e.g., to reach the necessary accuracy and speed of convergence). Therefore in [6] the use of Gram-Schmidt orthogonalization method is proposed.
4. Algorithm for the Minimization of the Functional (8)
Here we consider the minimization of the functional (8) by the help of the relation (4) with conditions (2). Putting from (5) into (8) we get Considering the symmetricity of the matrix the relation (9) may be written as and gradient of the functional (9) has a formSince the terms , , , , , , , , do not depend on the parameter , we haveBased on the formulasfrom [17, 18], for the gradients of , , , , , , , we get the formulas
Finally if we consider these results, then the gradient of the functional (9) will be defined by the formula
Then, for the gradient of the functional relatively to the parameter we get the expression
Taking equal to zero the expression (16) we getSolution of (17) with respect to giveswhere it is assumed that exists.
Value of the parameter , defined by (18) is a solution of the multiparameter optimization problem for the functional (9) that gives minimum to the cost functional.
Considering the above the following algorithm may be proposed to the solution of the identification problem (1), (2).
Algorithm 1. Construct the function from (1), initial and final data and from (2) and (3) correspondingly.
Calculate the derivatives , taking as initial approaches and .
Calculate the fundamental matrix from (6); reconstruct , from (7) and functional from (8).
Solving the system of algebraic equations (14) relatively find the value of the -dimensional vector in the th iteration.
Check the condition where is a given small enough number. If the condition (16) is satisfied the process stops; otherwise go to Step . The convergence of this algorithm may be proved similar to [13, 14].
Now we discuss the realization of this algorithm.
5. Calculational Algorithm
As one can see, in the realization of the above algorithm the main step is a calculation of the fundamental matrix and the matrices , . Note that, as is mentioned in [16], construction of these matrices is an enough difficult procedure. So for simplicity we try (4) and find the corresponding derivatives by using the Euler method. Really and everywhere are replaced by and correspondingly, where and are small enough numbers.
To calculate the fundamental matrix and the matrices , it is proper to replace (4) by the following discrete one:where is unit matrix of proper dimension.
Then from (20) we get that is a fundamental matrix for the system of homogeneous equations:
Therefore, similar to the nongradient methods we propose an algorithm, based on the orthogonalization of gradient directions using the Gram-Schmidt procedure.
Step 1. Using Gram-Schmidt orthogonalization, vectors , ,, are calculated and the set of vectorsis found which form an orthogonal basis in .
If we apply the orthogonalization algorithm, then a linearly independent system should be formed, that is, orthogonal system and each vector should be linearly expressed through . Here and are upper triangular matrices. Thus it is possible to ensure that the systems were orthonormal, where the diagonal elements of the transition matrix are positive; by these conditions the system and the transition matrix are uniquely determined.
The algorithm considers , if the vectors, , are constructed. Then where is the sign of the scalar product of vectors.
Step 2. For the orthogonalization of the gradient directions we compute in the form Here is any small parameter.
Step 3. The orthogonal gradient directions are chosen in the form Replacing by in (17) the nongradient iterative minimization procedure will bewhere is a scalar, which is determined by golden section method.
Now we apply the above proposed technique to the example of 15 production by gas-lift method.
Example 2. It is known that nonstable motion of gas in tubes and gas liquid mixture (GLM) in vertical tubes, that is, in the lift pipe of the gas-lift well with constant across profile, is described by the following system of linear partial differential equations of hyperbolic type (see Figure 1):where , , is an additional pressure on its stationary value and averaged over across section speed of motion of GLM; : time and coordinate; : speed of sound in gas and GLM; : gas, oil, and GLM in correspondence with coordinates; ; , : free fall accel ; : interval effective diameter of the tube [19].
The partial differential equation of gas and GLM motion by are averaging over time may be reduced to the following ordinary differential equation [20]where and all quantities are assumed constant and is area of across section of the pump-compressor tubes and is constant relative to axes.
It is assumed that the passing from the end of tube through the layer to the beginning of the lift () is described by the following difference equations:where and , , are constant numbers to be found. For the sake of simplicity we suppose that the parameters , , , are known and it is required to reconstruct involved in (19) due to .
Then some nominal trajectory and parameter are chosen assuming that th iteration is already held. Let us linearize (29) among these datawhere Note that by the help of the relations (20) and (21) the matrices , , are calculated as follows: where is small enough number.
Let some statistical data be given.
Let us assume that some observation points for and are given (see Table 1).
We give in Table 2 the values of obtained by using MATLAB by given input parameters.
As we see from Table 2 by gets value 0.2299, with error estimation 10−3.
Here is the dependence of or (see Figure 2).
The above algorithm reaches given accuracy after 4 iterations and gives .
Note that the inequality holds for any that shows the stability of the proposed quasilinearization algorithm.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
This work is supported by the joint grant of ANAS and SOCAR 17, 2013–2015, Baku State University “” Grant.