RU2453904C2 - Method of controlling value of alpha-anisotropic standard of memoryless linear system - Google Patents

Abstract

FIELD: information technology.

SUBSTANCE: invention relates to testing and monitoring elements of a control system, monitoring parameters of devices performing linear conversion of signals, as well as to generation of test input data. In the method of controlling said standard of a memoryless linear system with m inputs and p outputs, n sets of test signals are transmitted in series to the input of the system, said test signals being such that a matrix X=(x₁…x_n), composed of values of components of said test signals, is the generating matrix for an m-dimensional random Gaussian sequence whose anisotropy does not exceed said value α. The root-mean-square value of the system gain is calculated as the square root of the ratio of the sum of squares of all p components of all output signals, corresponding to all n said input signals, to the sum of squares of all m components of all n said input signals. The obtained root-mean-square value of the system gain on said test signals is taken as the lower-bound estimate for said value of the α-anisotropic standard of the linear system and the permissibility of the value of the α-anisotropic standard of said memoryless linear system is determined based on said estimate value.

EFFECT: high reliability of determining the lower-bound estimate of the value of the α-anisotropic standard of a linear system using a fixed number of input signals.

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Description

The invention relates to methods for controlling the parameters of devices performing linear signal transformations, to methods for generating test input data, and also to methods for testing and monitoring elements of control systems. The invention can be used in the development of functional control systems for elements of control systems.

In modern theory and practice of designing and analyzing control systems, the notion of anisotropy of the disturbance of input signals arriving at the input of a linear system is used. The anisotropy of the n-dimensional random signal characterizes the Kullback-Leibler information deviation of a given signal from a random signal with a Gaussian probability distribution having a scalar covariance matrix. Moreover, a situation often occurs when for a given linear system A without memory [Mesarovich M, Y. Takahara. General theory of systems: mathematical foundations, M .: Mir, 1978, p. 41, definition 3.5], which functions as a part of a larger system B, the upper estimate of the anisotropy of the input n-dimensional random signal arriving at the input of system A is known. In this case, for the specified larger system to function correctly, it is necessary B for linear system A to function correctly at least on the corresponding input sets s signals, characterized specify the level of anisotropy. Thus, the task of testing the performance of system A on random sequences of input signals characterized by a given level of anisotropy is relevant.

In the general case, a linear system A without memory implements a linear operator L _A that translates the input signals x∈R ^m into output signals y = L _A x∈R ^p , and nothing can be known about the internal structure of system A, i.e. System A can be seen as a black box. Physically, system A can be performed both on a digital basis and on the basis of analog elements. Possible malfunctions in system A can lead to the fact that instead of implementing the nominal operator L _A , the system begins to implement another real operator L ' _A = L _A + ΔL _A. The reason for this can be both the failures of discrete elements and the systematic drift of the parameters of the analog elements of system A. If system A does not have its own self-test system, then system B must have the means to supply input inputs of system A that would correspond to working input influences during the operation of system A as part of system B, and means of analyzing the values of the output signals at the outputs of system A with such input influences that allow us to draw conclusions about the degree the health of system A.

As a criterion for the operability of system A, the smallness of the difference between the a- anisotropic norm of the real operator L ' _A and the a- anisotropic norm of the nominal operator L _{A can be used} , where a is a positive number, which is an upper estimate of the anisotropy of the input signals arriving at the input of system A when it functioning as part of system B. According to [Vladimirov IG, Diamond F, Kloeden P. Anisotropic analysis of robust quality of non-stationary discrete systems on a finite time interval // http://www.math.uni-frankfurt.de/~numerik/kloeden / Art icle / ait_paper_final.pdf] the concept of a -anisotropic norm of an operator is defined as follows. Below, E denotes the mathematical expectation of a random variable.

For any λ> 0, we denote by p _{m, λ the} Gaussian density on R ^m with zero mean and the scalar covariance matrix λI _m , where I _m is the identity matrix of order m, i.e.

класс R^m-значных квадратично интегрируемых случайных векторов, распределенных абсолютно непрерывно относительно m-мерной лебеговой меры mes_m. Для любого

с плотностью распределения вероятностей f:R^m→R₊ относительно mes_m его относительная энтропия относительно (1) принимает видDenote by

the class of R ^m- valued quadratically integrable random vectors distributed absolutely continuously with respect to the m-dimensional Lebesgue measure mes _m . For anyone

with probability density f: R ^m → R ₊ with respect to mes _m its relative entropy with respect to (1) takes the form

- дифференциальная энтропия случайного вектора W в смысле лебеговой меры mes_m. Определим анизотропию A(W) случайного вектора

как минимальное информационное уклонение его распределения от гауссовских распределений на R^m с нулевым средним и скалярными ковариационными матрицами. Непосредственное вычисление показывает, что минимум в (2) по всевозможным λ>0 достигается при λ=E|W|²/m и, следовательно,Where

is the differential entropy of the random vector W in the sense of the Lebesgue measure mes _m . We determine the anisotropy A (W) of a random vector

as the minimum information deviation of its distribution from Gaussian distributions on R ^m with zero mean and scalar covariance matrices. A direct calculation shows that the minimum in (2) for all possible λ> 0 is achieved for λ = E | W | ² / m and therefore

Let F∈R ^{p × m} be an arbitrary fixed matrix. The root mean square gain of the operator F with respect to W is defined as

функция Q(F, W) задает норму на R^p×m. В случае произвольного а≥0 значение а-анизотропийной нормы матрицы F определяется какFor any fixed

the function Q (F, W) defines the norm on R ^{p × m} . In the case of arbitrary a ≥ 0, the value of the a -anisotropic norm of the matrix F is defined as

Thus, the problem arises of controlling the value of the a- anisotropic norm of a linear system without memory with m inputs and p outputs, where a is a given positive number, by applying test signals to the system input and determining the rms value of the system gain on the indicated test signals.

с заданным уровнем анизотропии а. Недостатком этого способа является то, что получаемый результат является только статистической оценкой а-анизотропийной нормы, причем для получения приемлемого уровня точности и достоверности число входных воздействий должно быть достаточно большим и является неограниченным при приближении длины доверительного интервала к нулю. Хуже всего то, что получаемое случайное значение оценки может оказаться больше истинного значения a-анизотропийной нормы, т.е. оно не обеспечивает надежного получения даже оценки снизу для а-анизотропийной нормы. Кроме того, в случае, когда реальный оператор, реализуемый системой, неизвестен, для получения достоверного значения а-анизотропийной нормы необходимо производить определение значений Q(F, W) для достаточно большого множества распределений случайных сигналов W с уровнем анизотропии а, что приводит к очень высокой трудоемкости этого способа.A natural way to control the α- anisotropic norm of a linear system that implements the linear operator F is to directly use the definition of (3) the mean-square gain of the operator F with respect to W by estimating the mathematical expectations by the Monte Carlo method using random signals to the inputs of the linear system

with a given level of anisotropy a . The disadvantage of this method is that the result obtained is only a statistical estimate of the α- anisotropic norm, and in order to obtain an acceptable level of accuracy and reliability, the number of input actions must be sufficiently large and is unlimited when the length of the confidence interval approaches zero. Worst of all, the resulting random estimate may turn out to be greater than the true value of the a -anisotropic norm, i.e. it does not provide reliable obtaining even lower bounds for a -anisotropic norms. In addition, in the case when the real operator implemented by the system is unknown, in order to obtain a reliable value of the a -anisotropic norm, it is necessary to determine the values of Q (F, W) for a sufficiently large set of distributions of random signals W with the level of anisotropy a , which leads to a very the high complexity of this method.

In the above work, Vladimirova I.G. et al. A method for controlling the value of the a -anisotropic norm of a linear system without memory with m inputs and p outputs is proposed if the matrix F of the linear system is known. In this case, it is possible to explicitly determine the covariance matrix of a random vector with an anisotropy equal to a , on which the upper bound (4) is reached. The disadvantage of this method is that it is not applicable in the case when the matrix F of the linear system is unknown.

In view of the foregoing, it is the development of a method for monitoring the value of the a- anisotropic norm of a linear system without memory with n inputs and m outputs, where a is a given positive number by applying test signals to the system input and determining the rms value of the system gain on the specified test signals, in which to determine the lower α- anisotropic norm of the specified system, a deterministic sequence of input signals is used, the length of which would be limited. In addition, the urgent task is to find such a determinate sequence of signals in the case when the exact value of the matrix of the specified linear system is unknown.

The proposed method for controlling the value of the a- anisotropic norm of a linear system without memory with m inputs and p outputs, where a is a given positive number, by supplying n sets of test signals to the system input, where n≥m, and determining the rms value of the system gain at the indicated test signals consists in the fact that n sets of test signals x _i = (x _i1 , ..., x _im ) ^T , 1≤i≤n, such that the matrix X = (x ₁ ... x _n ), composed from the values of the components of these test signals, is formative ith matrix for an m-dimensional random Gaussian sequence whose anisotropy does not exceed the indicated value of a . The root mean square value of the system gain on the indicated test signals is determined as the square root of the ratio of the sum of the squares of all p components of all output signals corresponding to all n specified input signals to the sum of the squares of all m components of all n specified input signals. Finally, the obtained rms value of the system gain on the indicated test signals is taken as a lower bound for the indicated value of the a- anisotropic norm of the linear system. At the same time, a technical result is achieved, consisting in a reliable determination of the lower estimate of the value of the a -anisotropic norm of the linear system using a fixed number of input signals.

The implementation of the invention

The value of the a -anisotropic norm of a linear system A without memory with m inputs and p outputs, where a is a given positive number, is controlled by applying test signals to the system input and determining the rms value of the system gain on the indicated test signals. To do this, n sets of test signals x _i = (x _i1 , ..., x _im ) ^T , 1≤i≤n, where n≥m, such that the matrix X = (x ₁ ... x _n ), are sequentially fed to the input of system A, composed of the values of the components of these test signals, is the forming matrix for the m-dimensional random Gaussian sequence, the anisotropy of which does not exceed the specified value of a . This means a random vector

where (ξ ₁ , ..., ξ _n ) is a set of independent normally distributed (Gaussian) random variables, is a random vector whose anisotropy does not exceed a . According to the above work, Vladimirova I.G. et al. the value of the anisotropy of such a random vector is

-

Thus, as the matrix X, we can take any non-degenerate matrix of size m × n, for which the inequality

-

When a test signal x _{i is} applied to the input of system A, an output signal y _i = (y _i1 , ..., y _ip ) ^T = Ax _i arises at the output of system A. For each such output signal, the value of the square of the norm of this signal is determined, equal to the sum of the squares of the components of this signal:

.

.

After that, the root-mean-square value of the gain K of system A is determined on the test signals (x ₁ , ..., x _n ) as the square root of the ratio of the sum of squares of all m components of all output signals corresponding to all n specified input signals to the sum of squares of all m components of n specified input signals

The obtained root-mean-square value K of the system gain on the indicated test signals as a lower estimate for the indicated value of the a -anisotropic norm of the linear system.

We show that the above method correctly calculates a lower bound for the a -anisotropic norm of a linear system, i.e. that the indicated value of K is indeed a lower bound for the a -anisotropic norm of the linear system A being equal to the mean square gain of the operator F realized by system A with respect to W according to (3).

Indeed, let (F _ij ) = (A _ij ) be the matrix of the linear operator F. Then the equalities

Since ξ _i are independent random variables with zero mean and unit variance, E (ξ _i ξ _r ) = δ _ir , where δ _ir = 1 for i = r and δ _ir = 0 for i ≠ r, whence the last five-fold sum is

It is similarly verified that

Therefore, according to (3) and (5)

which proves our claim.

We note that the minimum number n of sets of test signals ensuring the fulfillment of the condition det (XX ^T ) ≠ 0 is m, which corresponds to the case of the classical forming filter for an m-dimensional random vector. In principle, situations are possible where, due to certain design restrictions on the input of a linear system, only the number of sets of test signals that is a multiple of any value that is not a divisor of m can be input. In this case, the described method can be applied with an arbitrary number n of input signal sets greater than m. In the absence of such restrictions, the choice of n = m provides a minimal set of test signals for monitoring the a- anisotropic norm of a linear system without memory.

The described method for monitoring the α- anisotropic norm of a linear system without memory provides a technical result consisting in the reliable determination of a lower estimate of the value of the α- anisotropic norm of a linear system using a fixed number of input signals. At the same time, reliability is ensured due to the fact that the set of input signals is deterministic, which eliminates statistical errors associated with the need to take into account the probability distributions of input signals and take into account confidence intervals associated with the usual statistical estimation method.

From the proof of the correctness of the above method, it follows that the value of the average gain of a linear system without memory on a given set of sets of input vectors depends exclusively on the covariance matrix obtained as a sample covariance matrix of a given set of input vectors. Thus, the set of components of the sample covariance matrix is a set of sufficient statistics to determine the α- anisotropic norm of the linear system. This suggests that the theory of a- anisotropic norm is part of the correlation theory and does not need to involve information-theoretic considerations, as was previously considered in the anisotropic theory of linear systems.

Claims (4)

1. A method of controlling the value of the a-anisotropic norm of a linear system without memory with m inputs and p outputs, where a is a given positive number, by applying n sets of test signals to the system input, where n≥m, and determining the root mean square value of the system gain by specified test signals, in which:
sequentially fed to the input of the system n sets of test signals x _i = (x _i1 , ..., x _im ) ^t , 1≤i≤n, such that the matrix X = (x ₁ ... x _n ), composed of the values of the components of these test signals , is the forming matrix for the m-dimensional random Gaussian sequence W, the anisotropy of which does not exceed the specified value of a;
determining the rms value of the system gain on the indicated test signals as the square root of the ratio of the sum of the squares of all m components of all output signals corresponding to all n specified input signals to the sum of the squares of all m components of all n specified input signals;
taking the obtained rms value of the system gain on the indicated test signals as a lower bound for the indicated value of the a-anisotropic norm of the linear system, and judging by the magnitude of this estimate, the admissibility of the value of the a-anisotropic norm of the indicated linear system without memory is judged. 2. The method according to claim 1, in which the specified matrix X is an arbitrary matrix satisfying the condition -

3. The method according to claim 1, in which n = m. 4. The method according to claim 1, in which n> m.