A characteristic feature of modern control systems is the widespread introduction of various types of computing systems that implement methods and algorithms of control and modeling. Therefore, the quality of functioning of control systems is largely determined by the characteristics of the used computational tools. The problem of ensuring the quality of functioning of control systems with a computer in the control loop is very complex both technically and mathematically. In solving this problem, practice has put forward one of the first places the scientific direction, associated with the development of methods and means to ensure the reliability of computational processes and information processing. In real control systems or modeling processes of calculation are inevitably accompanied by various kinds of noises and errors. Thus, in the case of numerical analysis of the equations of dynamics on a digital machine, the solution errors, in addition to all, can be caused by failures and failures in the work of the hardware. Under failures we understand distortions of processed and controlling information under the influence of primary self-eliminating reasons (incorrect operation of binary hardware elements, interferences, power supply voltage fluctuations and so on). Moreover, most failures have an accidental character. Therefore, an effective current control is needed, which allows detecting and eliminating an error before it spreads in the control circuit, which determines the operability as one of the main characteristics of error control means.
Currently, both software and hardware methods of control of simulation processes are used, as
well as their combinations [
In the case under consideration, it is necessary to control the computational process in the
working mode of operation of the control system or simulation. And the main object of control are
hardware failures. More rational are program-logical methods of control, allowing you to control in
the process of solving the problem. Sufficiently complete review of existing methods of
programlogic control is given in a number of works [
The aim of the work is to develop constructive algorithms to control the solution of the equations of dynamics of objects in the problems of synthesis of control actions.
Numerical methods for solving the dynamics equations are step by step procedures for obtaining the desired function. In this case, the computational process is naturally divided into relatively constant in time parts corresponding to a certain desired function at the integration step. Besides, the computational process is divided inside the integration step into relatively constant in time parts corresponding to determination of values of the right part of the solved equation system Y = f (Y, U,t ), Y(t0 ) = Y0 (1) where Y = ( y1, y2 ,..., yn )T – vector of state variables, U = (u1 , u2 ,..., um )T – vector of control signals, f = ( f1 , f 2 , ..., f n )T – vector-function, t – independent time parameter.
Let us consider the possibilities of numerical solution procedures (1) on organization of their control (self-control). To organize control (self-control) it is necessary to have reference values to compare the results of calculations. Controlled results of previous calculations or results with low probability of failure can be used as reference values. For any method of numerical solution of equations of dynamics of a type (1) failure control can be organized by comparing the results yi and yi+1, obtained at the current and subsequent steps of integration, respectively. The presence or absence of failures is determined by checking the condition
= (yi , yi+1 ) D (2) where – is a measure of proximity of values yi and yi+1, D – is the region of admissible values of this measure.
The sizes of the region D depend on solution changes from step to step and determine the accuracy of control. If the boundaries of the area D correspond to the maximum possible difference between the values of yi and yi+1 at the whole step of integration in the absence of failures, the error that can be missed in the control is equal to twice the value of the area D, which follows from (2), since in the latter the value yi+1 should be replaced with yi. In this case, the accuracy of control may be unsatisfactory.
To control errors caused by failures and malfunctions, methods of controlling the methodological error of numerical solution on the integration step or peculiarities of construction of computational algorithms can be used.
In prediction and correction methods [
Let us show it on the example of the improved Euler-Cauchy method. Let the prediction be carried out by the formula:
yi+1 = yi +hfi , (3) where h – is a step of integration, fi=f(Y, U, ti), and the correction is carried out according to expression
~ ~ where f i+1 = f (Y, U, ti+1 ).
As can be seen from (3) and (4), at one step of integration the function f(Y, U, ti) is calculated twice at the points ti and ti+1. Since this process, as a rule, takes the main part of computational work, let us assume that a failure has occurred at the moment of the calculation f(Y, U, ti) and the function value fi has been obtained with error ∆fi. Then the function value calculated by formula (3) will be ~ yi+1 = yi + h( f i + f i ) = yi + h f i + hf i (5)
Here the value is the ∆fi+1=h∆fi – error in the prediction value. Next, we will propose methods of algorithmic control, suitable for control of calculation process at any methods of numerical solution of equations of dynamics. In this case we will consider the control, based on application of additional (control) algorithm on each step of numerical solution of the equations of dynamics of the form (1).
Let us consider the problem of obtaining algorithm Ak, which we will formulate as follows: for a certain class of problems Q to obtain an algorithm of solution Ak, the result of which Y in relation to the problem solution qQ coincides with the accuracy ε of the result obtained by the main algorithm Ak. In this case the time and memory costs for the algorithm's realization must be within the specified limits.
When computing by the basic algorithm Ab for the class of problems under consideration for modeling the dynamics of dynamic objects, most of the time is spent on computing the right side of the system of equations (1). The computation time by the algorithm Ak can be significantly reduced in comparison with the algorithm if it uses the values of the function f(Y, U, t) obtained by the algorithm Ab. Since the computing process according to the algorithm Ak is also subject to failures, the refusal to calculate the function f(Y, U, t) will greatly reduce the probability of a first-order error α (i.e., the correct initial hypothesis will be rejected). In this case the accuracy of control at the controlled step is improved, if the controlled algorithm uses the information controlled at previous steps, in other words: use the idea of extrapolation for control. Then control of each component of vector equation (1) using extrapolation method can be done separately. Therefore, the construction of the control algorithm Ak can be considered for some component of the vector Y and, at the same time, not to lose generality of the statement.
In this case, the control value of the function at the (i+1)-th step of integration can be defined by the formula
p1 s1 yi+1 = a p y p−1 + h bs f (Y,U , ti−s )
p=0 s=1 h=ti+1-ti – step of integration, ap, bs, – coefficients.
In (6), only values of function and its derivative, controlled during previous steps of integration, are used. Upper limits of sums p1 and s1 are selected from the conditions of required accuracy of the controlled algorithm. Since the controlled algorithm is not always required to ensure stability, the coefficients ap, bs, can be selected from the condition of minimal local error (i.e., equality to zero of the residual term ri=0 between the exact and numerical solution (6)).
Many algorithms for the numerical solution of ordinary differential equations determine estimates of the derivative function between integration nodes. Since these estimates provide information on the behavior of the solution at a point closer to the controlled value of the function than the value f(Y, U, ti-s) in (6) at s1=1, it is advisable to take this information into account when extrapolating. If, in extrapolation, we take into account one estimate of the derivative obtained by the basic algorithm Ab, then one term will be added to expression (6)
p1 s1 yi+1 = a p y p−1 + hbs f (Y,U , ti−s ) + b0 f (Y,U , ti+l )
p=0 s=1 where f(Y, U, ti-s) – is the estimate of the derivative, and b0 – is the coefficient. (7)
For example, a fourth-order precision control formula in which the derivative estimate is used is yi+1 = a1 y1 + a2 y2 + h(b0 y i−1+ + b1 y i−1 + b2 yi−2 ).
Obviously, in the general case, any information about the solution obtained by the basic algorithm Ab and checked at the previous steps of integration can be used for extrapolation.
It should be noted that extrapolation uses finite differences of the necessary order. For example, in the linear method of extrapolation, the control value of the function at a point ti+1 is defined as (8) yi+1 = yi + yi − yi−1 (ti+1 − ti )
ti − ti−1
Application of expression (8) for organization of control of numerical solution of equation of dynamics of a kind (1) does not allow taking into account all available information about controlled solution. Indeed, in numerical solution of equation of dynamics (1), as a rule, not only the values of function in integration nodes are known, but also the values of derivatives and their estimates which are used in (8). In addition, the use of finite differences to determine derivatives increases the error of the control value of the function yi+1.
Thus, expression (6), in contrast to formulas of the type (8), allows taking into account the results of calculations using the main algorithm more completely Ab, which leads to more accurate results in extrapolations.
As an example, Fig. 1 and Fig. 2 show the dependencies of the mathematical expectation and the standard deviation of the value from the integration step h in the equation of the form (1), which describe the dynamics of an airplane flight.
When using the extrapolation method for control, as already noted, it is possible to correct the solution for errors not by the repeated method, but by replacing yi+1 by yi+1 , which is essential in real-time control and modeling. However, the control accuracy depends on the accuracy of extrapolation of the solution using expression (6) and may not be satisfactory.
From the point of view of increasing control accuracy, a method based on the idea of interpolation may be considered preferable to the extrapolation method. The interpolation control formulas, in contrast to extrapolation formulas, take into account changes in the solution at the controlled calculation step, which leads to a decrease in the difference of values and, obtained by algorithms Ab and Ak. Increasing the accuracy of interpolation can be obtained by attracting the results of calculations by the main algorithm Ab before the controlled calculation step. This leads to multi-step interpolation formulas. To increase the control sensitivity to errors that have been classified as failures, use the control interpolation formula to determine the function value obtained in the calculation step closest to the controlled calculation step. p1 s1 (9) yi = a p yi− p + h bs f (Y,U , ti−s )
p=1 s=1 where ap, bs – are real coefficients; p 0 (otherwise a0=1, and all other coefficients are zero).
As in the case of the extrapolation formula (6) in (9), the upper limits of summation have to be chosen depending on the required order of control accuracy, and the coefficients ap and bs – from the condition of the accuracy of representation of polynomials of appropriate degree.
The principal difference between the interpolation expression (8) and the extrapolation expression (6) as control algorithms is as follows. In the case of extrapolation formulas, the directly controlled value is the value of the function obtained by the basic algorithm Ab at the controlled step of integration, and the control value yi is obtained by expression (6). In case of interpolation formulas the controlled value is obtained by expression (8), and the reference value is the value of the function obtained by the main algorithm Ab on the previous (i-p) step of the calculation, i.e., in fact, indirect control is carried out. The error of calculating the value yi+1 according to the basic algorithm Ab is transformed into the error yi of determination yi by the formula (8). The deviation yi in this case, in the framework of the linear theory of accuracy, will be defined as yi = L where
L = yi yi+1 As applied to expression (8), L is determined by
yi+1 where a* and b* are the coefficients, at yi+1 in (9).
L = a* − b*h
f (Y,U , ti+1 )
The interpolation method of control allows you to increase the accuracy of obtaining reference values. However, when organizing control of the computational process it should be borne in mind that when using the interpolation method of control there is no possibility to replace the erroneous value obtained by the basic algorithm Ab with the value obtained by the control algorithm Ak (which is possible by the extrapolation method).
The coefficients of control algorithms of the form (6) and (8) are determined from the condition of obtaining the exact result by the control algorithm Ak for the solution of yj(t), j = 1, n , representing a polynomial of the highest possible degree. Taking into account the above said, the choice of coefficients ap and bs allows to get the given order of control error at relatively small volume of calculations by the control algorithm Ak in the course of calculation. In fact, the control accuracy defined by the residual of the Taylor series for control formulas varies from step to step, depending on the smoothness of the solution yj(t), j = 1, n .
Let us propose a method for organizing the control of the solution of the system of equations (1) with the help of the control algorithm Ak, which reconstructs itself depending on the behavior of the controlled solution yj(t), j = 1, n . The peculiarity of this approach consists in the fact that the results controlled at the next step are used to determine the parameters of the control algorithm Ak. In this case, the parameters are determined from the condition of a minimum of some measure of proximity of results obtained by the control algorithm Ak and the basic algorithm Ab at the section of the solution including the last step of calculations under control. The obtained algorithm, denoted as Aa, is used to control the next calculation step. After that the parameters are refined again. Obviously, the algorithm Aa, in view of the above, can be regarded as adaptive.
The method under consideration allows the following interpretation: on the basis of the
controlled information about a physical process (i.e. its dynamics described by system (1) and the
assumption that behavior of the solution yj(t), j = 1, n , from step to step changes slowly enough) the
problem of parametric identification is solved for a given structure of the mathematical process
description (i.e. system (1)). The resulting model in the form of an algorithm Aa is used for subsequent
control. Otherwise, as control information becomes available, the model used for control (algorithm
Aa) is specified (or – adapted). Further, we will use methods of parametric identification of
inertiafree objects [
Using results of calculations on the basic algorithm Ab obtained to the (i+1)-th step of integration, we can determine parameters of the control algorithm Aa on the (i+1)-th step of the system of algebraic equations
CQ = B (10)
For the ectrapolation control method which uses evaluation of the derivative at the point ti-1+α, the values included in expression (10) are determined on the (i+1)-th step of integration as follows Q = (a−1 , a1 , ..., a p1 , ..., b0 , b1 , ..., bs1 )T ,
B = ( yi , yi−1 , ..., yi−k +1 )T , yi−1 ... yi− p1−1 y i−2+ y i−2 ... y i−s1−1 C = yi−2 ... yi− p1−2 y i−3+ y i−3 ... y i−s1−2 ... ... . .. ... ... ... ... ... yi−k ... yi− p1−k y i−k −1+ y i−k −1 ... y i−s1−k For the interpolation method of control: control method
Q = (a−1 , a1 ,..., a p1 ,..., b−1 , b0 , b1 ,..., bs−11 )T ,
B = ( yi−1 , yi−2 , ..., yi−k )T ,
If the matrix C is not degenerate and the number of its rows is equal to the dimension of the parameter vector (for the extrapolation method Π ers = (a−1, a1, a2 ,..., a p , b0 , b1,..., bs ) and interpolation method Π int = ( a1 , a2 ,..., a p , b1 ,..., bs ), respectively) defined by the value of k, then the evaluation Qˆ of the parameter matrix Q (for both extrapolation and interpolation methods) is the solution of the system of algebraic equations
Qˆ = C −1B and the results obtained by the main and control algorithms on the sequence yi , yi−1 , ..., yi−k +1 coincide.
If the matrix C is rectangular, we can use the method of least squares to determine the estimate of the parameter vector.
Example. As an example, consider the Runge-Kutta algorithm of second order precision [
y = f (Y,U , t ) The solution of problem (11) boils down to the following: 1. Do the calculations by Runge-Kutta algorithm
k1 = h f (yi , ui , ti ), k2 = h f yi + k1 , u(ti + h), ti + h,
i yi+1 = yi + r k r ,
r=1 where , , – parameters of the algorithm. 2. (11) (12) 2. Determine the control value of the function by the first-order extrapolation formula of the form (6)
yi+1 = ayi − hby i−1+ (13) where y i−1+ = f yi + k1 , u(ti + h), ti + h– is the value of estimation of derivative, obtained by algorithm Ab on the i-th step of integration. At the first step of integration coefficients a, b are determined from the condition of exact result according to the control algorithm Ak for the polynomial of the first degree of accuracy. 3.
3. Determine the value of
= yi+1 − yi+1 4. Check condition 5. Form the matrix sup , (14) where sup – is a permissible value of . If condition (14) is not satisfied, then correct the solution (for example, by replacing with yi+1 на yi+1). If t tk , where tk – integration time, then proceed to item 5, otherwise carry out the end of the calculation process. integration. The value defines the number of matrix rows.
B = (yi , ..., yi−k −1 )T 7. Determine the coefficients of the control algorithm (13) from the system of algebraic equations
CQ = B where Q = (a, b)T .
As can be seen from the solution procedure above, additional calculations are required to determine the current value of coefficients a, b. It is possible to simplify the control procedure by determining not the whole vector Q, but its part, assuming that the remaining vector components have equal values obtained from the exact result of the control formula for a polynomial of a possibly high degree. Thus, in the considered procedure, it was possible to determine the first coefficient a by assuming the other b=1. This corresponds to the transformation of the model (truncation of the model by parameters).
The main purpose of the considered algorithms is to detect failures during the numerical solution of the equations of dynamics in real-time simulation and control systems.
Since both the basic Ab and the control Ak algorithms are executed by different programs, the proposed control methods also allow detection of hardware failures during computations.
Spending time on control depend on the method of control, the order of accuracy of the control formula, the accuracy measure of calculations and are determined by the number of multiplication and addition operations on the control formula, the time of determination of the accuracy measure of calculations and the operation of comparing the obtained value of the accuracy measure of calculations with tolerances.
The main advantages of the proposed methods of control should be the simplicity and low cost of time for control at a relatively high control accuracy, which can be regulated by both changing the method of control and changing the order of accuracy of control formulas.
The disadvantages of the proposed methods include "inherent noise" (i.e., mismatch of results at the integration step obtained by the basic Ab and control Ak algorithms), which reduces the control accuracy. The characteristics of "intrinsic noise" of the control algorithms Ak are the initial data for choosing the control method and defining the control quality indices.
The fact, that the model of the controlled process (i.e. equation of dynamics of the form (1)) is known, is essentially used in suggested methods of control. This allows to refuse from application of finite differences for determination of derivatives and to organize iterative control. When finite differences are used, it is difficult to organize iterative control because the controlled variable and the control variable can be used together to determine the derivative estimate.
The proposed control methods may be used to control dynamic processes, the mathematical model of which is known. For example, with their help it is possible to organize control of sensors of the state of a dynamic object.