Solution of Time-Varying Index-2 Linear Differential Algebraic Control Systems Via A Variational Formulation Technique

This paper deals with finding an approximate solution to the index-2 time-varying linear differential algebraic control system based on the theory of variational formulation. The solution of index-2 time-varying differential algebraic equations (DAEs) is the critical point of the equivalent variational formulation. In addition, the variational problem is transformed from the indirect into direct method by using a generalized Ritz bases approach. The approximate solution is found by solving an explicit linear algebraic equation, which makes the proposed technique reliable and efficient for many physical problems. From the numerical results, it can be implied that very good efficiency, accuracy, and simplicity of the present approach are obtained.


Introduction
Many real life problems can be modelled as a differential algebraic (control) system. Finding a novel reliable and efficient technique for solving differential algebraic equations ISSN: 0067-2904

Zaboon and Abd
Iraqi Journal of Science, 2021, Vol. 62, No. 10, pp: 3656-3671 3657 has become an interesting aim for mathematicians and engineers. Numerical methods that solve higher index differential algebraic equations can be found in literature [1][2][3][4][5][6][7]. Many of these methods were based on the index reduction technique to avoid the difficulties in the higher index differential algebraic equations. Time-varying linear differential algebraic equations is a subject of many real life problems and has been the subject of many researchers in recent years [8,9,10]. An efficient and easily implemented technique to solve some classes of DAEs (index-1), approximately using non-classical variational formulation approach, was developed [11,12,13]. The aim of this work is to extend and develop the results of the latter three studies to solve higher index time-varying linear differential algebraic control equations, especially for index-2 problems, without using the reducing technique which is not applicable for many real life problems. Since the proposed DAEs problem has the non-symmetrical time-derivative linear operator with respect to the classical bilinear form, a new bilinear form, based on the old one, is taken to ensure the necessary requirements for the existences of the variational problem corresponding to the given constrained problem.

Basic Concepts
Hence,, the semi explicit system is transformed into differential algebraic systems: is invertible matrix with or is not invertible matrix with , then the system (1) and (2) are index two linear DAEs with control .
From the Jacobean of the algebraic constraint with respect to one can use the implicit function theorem [5,17,18]  where the class of consistent initial condition at is defined according to the given algebraic constraint (2), as follows: …(4) Note that, if there is an interest in finding the explicit expression for ̇ to obtain the statespace ̇ ̇ then one has to derive (3) with respect to t, as follows: Then systems (1) and (2) are equivalent to state-space differential equation defined on manifold: where ̇ subject to the manifold …(6) As one can see, the terms ̇ ̈ are not appropriate for an application point of view, and the usage of the implicit function theorem to reduce the number of variables and estimate by (3) is better than solving problems (5) and (6).

Index-2 Time-Varying DAEs and their Variational Formulations
The main theme of this section is to discuss the solvability of index-2 time-varying DAEs using the variational formulation approach. We are looking for a suitable function, such that its critical points lead to a solution to the proposed problem and vice-versa.
) is invertible matrix, then (9) is solvable and gives that This leads to ̇ , , , are given and real numbers with . The selection of a consistent initial condition is based on the nature of equations (8), as: Since the operator is appeared in , the linear operator is not symmetric with the given usual bilinear form basic concept. Hence, no variational formulation exists unless one can redefine the linear operator or its bilinear form [11,19]. To create a functional (variational) equivalent to a linear problem , where is not symmetric with respect to the chosen bilinear form, by the functional ( ̇ ̇) we have: We define the first variation, due to the linear part of the increment of the functional as: is an arbitrary selection from the class of consistency initial condition . Otherwise, one can assume it as fixed numbered and set , It is noticed that, if there is satisfying the operator equations (10)-(12), uniquely over the class , then these equations will be identically satisfied. Since the aim is to fix , then this variational problem is well defined. Also, since from the linearity in , for arbitrary , then 3. may also be assumed as fixed to produce that and this will not affect the previous results.
Then, if , with is the solution of the proposed problem, then 〈 〉 . The other direction is clearly understood and the solution is a critical point of variable formulation (10). From practical point of view, one has to evaluate the functional in order to find its critical points. Moreover, critical points of a functional are equivalent to solve the necessary Euler equation corresponding to the given problem, which is difficult too. Thus, a direct method of variational problem is adapted to approximate the solution by a finite number of bases functions of separable Banach space as: is linearly independent bases function of time t. By substituting (14) and (15) in (13), we have ( ̇ ̇) ... (16) where is the total number of unknown variables. The critical point of variational formulation (13) is then equivalent to find the derivative of the functional (16) with respect to .
i.e. … (17) Since the varitional formulation is of quadratic type, the linear system of algebraic equation was obtained from equation (14), with the class of consistency initial condition where the given functions ̇ ̇ are selected from the class of admissible functions. Once this system (17) is being solved for , approximate solutions are obtained according to equations (14) and (15) and hence the original solution of (7) and (8) is obtained approximately. . The class of consistency initial condition is , -Then, the index-2 semi explicit system will be as

Illustrations
And, as we mentioned in algorithm 5 for finding variational formulation, The numerical results of the unknown coefficients of linear algebraic system were found to be: . And The exact solution that taken from [1] is for a given Then one can shows the comparison between the proposed solution and the exact solution in tables 5.1 and 5.2.   , where . This system is equivalent to the following differential-algebraic system ̇ ) is nonsingular, then it is possible to rewrite our system as .
/ defined with the class .
The class of consistency initial condition is

( )
The variational formulation with the class of consistent initial condition is defined as where the exact answer is obtained as: The numerical results used in the proposed technique and the comparisons with given exact solutions are shown in following tables.   To test the accuracy of the solution presented by the present method, the -norm is examined. For equality constraint, we have This criterion represents a good test to show the extent to which the approximate solution matches the exact solution, and thus indicates the accuracy of the method used. The differential-equality states are illustrated in Figure 5.2. where the matrices in the table below represent the respective models. Matrix Represent in mechanical model the displacement vector the vector of lagrangian multiplier the known input force the inertial matrix the damping matrix the stiffness matrix matrix of force distribution , the coefficient matrices All these matrices are known with appropriate dimensions. For more detail about this mechanical model, see [14]. Now, based on these matrices the semi explicit descriptor system can be rewritten as Under the transformation the mechanical system will be differential algebraic control system as: ̇ with 0 1 0 1 0 1 which is an index-2 system. By differentiating the equality constraint with respect to to estimate and since invertible matrix, then is defined with the class And the class of consistency initial condition is {( ) } The variational function with the class of consistent initial condition is defined as

Results and discussion
The illustrated examples 5.1-5.3 are ranked from simple to more complex. The examples 5.1-5.2 are of semi explicit index-2, time-varying differential algebraic system, with known exact solutions. While the last example 5.3 is a descriptor system which firstly needs transformation, using singular value decomposition, to semi explicit DAEs, and needs to be taken from real life application without knowing its exact form solution. These examples are taken as a test for the proposed method. By step by step implementation, the approximate solution is parameterized via polynomial base function, which is dense in . Even with reasonable small number of these polynomials with unknown coefficient, the obtained solutions are shown to be very accurate and efficient.  show the excellent matching between the approximate solution, using the present method, and the given exact solution. The overall error value, using norm of the linear operators, showed very good results on [ ] , for each example. The pointwise error in tables 5.1-5.4 demonstrated the good accuracy. For example 5.3, due to the absence of the exact form solution, the norm errors for all the constraints (the differential and the algebraic) are adapted ‖ ‖ ‖ ‖ to test the convergence of the solution. As an overall evaluation, the method has very good accuracy, being simple and effective as a tool to solve index 2, time-varying control DAEs.

Conclusions
As one can see, the present method is suitably applicable for an efficient class of index two DAEs with input or even with semi-explicit index two, descriptor system. The method is easily implemented and a very good accuracy has been obtained, even for simple types of polynomial bases functions. This approach is reliable and efficient for this class of functions and can be extended to higher index DAEs (index greater than two).