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On Linear Systems Disturbed by Wide-Band Noise

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On Linear Systems Disturbed by Wide-Band Noise On linear systems disturbed by wide-band noise Agamirza E. Bashirov Department of Mathematics, Eastern Mediterranean University Gazimagusa, Mersin 10, Turkey [email protected] Institute of Cybernetics, Azerbaijan Academy of Sciences F. Agayev St. 9, Baku-370141, Azerbaijan Keywords: Wide-band noise, white noise, stochastic sys- function £ is called an autocovariance function. The wide- tem, filtering. band noise process ³ is said to be stationary if the autoco- × " variance function £ depends only on (see [1]). If is so small that it is normally assumed to be 0, then the wide-band Abstract noise process ³ is transformed into a white noise process. As it was mentioned, in many fields such substitution gives rise An approach, based on a distributed delay, is presented to to tangible distortions. handle wide-band noise processes. This approach is closely There are different techniques of handling and working related with the semigroups of right translation and allows with the wide-band noise processes. For example, in [2] the a reduction of a wide-band noise driven linear system to a approximate approach is used. Another approach, based on white noise driven linear system. This reduction is applied a representation of wide-band noise as a distributed delay of to the Kalman-Bucy filtering in order to modify it to a wide- white noise, is suggested in [3]. band noise driven linear system. In [3, 4] the Kalman-Bucy estimation results are modified to a wide-band noise driven systems. The proofs of respec- tive results are based on the duality principle and, technically, 1 Introduction they are routine being useless if one tries to modify them to nonlinear systems. It becomes important to develop a more The modern stochastic optimal control and filtering theories handle technique of working with the wide-band noise pro- use white noise driven systems. The results such as the sep- cesses represented through a distributed delay. aration principle and the Kalman-Bucy filtering are based on Some attempts in this way are made in [5] where a re- the white noise model. Indeed, white noise, being a math- duction of a wide-band nose driven system to a white noise ematical idealization, gives only an approximate description driven system is presented. The reduction from [5] requires to actual noise. In some fields the parameters of actual noise an unbounded observation system which does not allow to are near to the parameters of white noise and so the math- obtain a complete set of formulae for the respective optimal ematical methods of control and filtering for white noise filter. driven systems can be satisfactorily applied in them. But in In this paper we establish another reduction of a wide-band many fields white noise is a grude approximation to actual nose driven system to a white noise driven system which does noise. Consequently, the theoretical optimal controls and the not require an unbounded observation system. We apply this theoretical optimal estimators for white noise driven systems reduction to obtain a complete set of formulae for the re- become not optimal and, indeed, might be quite far from be- spective optimal filter. This reduction gives hope to push on ing optimal. It becomes important to develop the problems studying nonlinear systems disturbed by wide-band noise. of systems theory for a noise model that describe actual noise more adequately. 2 Wide-band noise The issue is that actual noise is marked with a property which ensures the correlation of its values within a small time Consider the random process interval, i.e. if we denote such noise by ³, then Z Ø ´× ص dÛ ´×µ; Ø ¼; ´Øµ= ³ (2) Ø; ×µ ¼ ×<" £´ if ÑaÜ´¼;Ø "µ Ú[³´Ø · ×µ;³´Øµ] = cÓ (1) ¼ otherwise; " > ¼ ¾ Ä ´ "; ¼µ Û where , ¾ and is a stationary pro- ¼ × ¼ "> ¼ Û ´Øµ=Ø where Ø and are time moments, is a small cess with orthogonal increments satisfying cov and ³ EÛ ´Øµ = ¼ Ø ¼ value and £ is a nonzero function. A random process with for all . One can easily verify that the the property (1) is called a wide-band noise process. The random process ³, as defined by (2), is a wide-band noise ¼ ´ µ process which becomes stationary starting the time moment F is positive-definite, i.e. it is continuous at zero, it sat- ¼ ¼ = " F ´ µ´ Ö µ=F ´ µ´Ö µ Ö Ø with the autocovariance function isfies for all real numbers and Z × Ò X ¼ ´Ö µ´Ö · ×µ dÖ; ¼ × ": ´×µ= (3) ¼ F ´ µ´Ö Ö µ i j i j " i;j =½ Formally, the wide-band noise process (2) can be inter- Ò ´Ö ;:::;Ö µ Ò Ò for all -tuples ½ of real numbers and for all - preted as vibration. At moment Ø a vibration that is formed ´ ;:::; µ Ò tuples ½ of complex numbers, where is the con- " by the action of white noise during the time between Ø = ¾ jugate of . Since ½ for any pair of complex num- ½ ¾ Ø " and Ø affects the system. Values of white noise until ¼ F ´ µ ¾ do not take part in the formation of this vibration at moment bers ½ and , from the positive-definiteness of , one ¼ ½=¾ ´ µ can easily obtain that the function F is also positive- Ø, because their weight is sufficiently small and we can ne- definite. Thus, again by the Bochner’s theorem, the inverse glect them in the model (2). Consequently, the function ¼ ½ ¼ ½=¾ = F ´F ´ µ µ stands for the coefficient of relaxing the initial effect at dif- Fourier transformation is real-valued. ¼ ferent time moments. By this interpretation, the model (2) If is non-zero in some subset of the real line being in out- ;"] corresponds to real cases when the vibration generated by side of [¼ and having nonzero Lebesgue measure, then ¼ [¼;"] white noise stands to affect a system starting with the initial the function does not vanish in outside of . Letting ¼ ´Ö µ= ´ Ö µ " Ö ¼ , , we obtain a desired solution = ¼ time Ø (a reason for that may be the switching on the system from resting state to dynamic state which changes its of the equation (3). ¼ <Ø <" Ä ´¼;"µ sensitivity to random disturbances). Hence, when Thus, given a function in ¾ , there is a variety of the wide-band noise proces (2) is formed by values of white wide-band noise processes in the form (2) which have the [¼;Ø] " noise on . same autocovariance function starting the time moment . In practice wide-band noise is identified by autocovariance This variety is reached by different solutions of the equation function. Therefore, to get the representation (2) one has to (3) and also by different processes Û . solve the equation (3) in under given . "> ¼ ¾ Ä ´¼;"µ Theorem 1. Let and let ¾ . Then there 3 Notation µ exists a ´non-unique in general solution of the equation X Z À À À G Ä ´ "; ¼µ ¾ Let , , , ½ , and be separable Hilbert spaces, (3) in ¾ . ´X; Z µ let Ä be the space of linear bounded operators from ¼ ¼ Proof. Define and by X Z B ´a; b; Ä´X; Z µµ to and let ¾ be the space of equiva- lence classes of strongly Lebesgue measurable and square ´×µ ¼ × " ¼ if ´×µ= a; b] Ä´X; Z µ integrable operator-valued functions from [ to . ¼ otherwise From the discussion given in Section 2, it is reasonable to and define a Hilbert space version of the representation (2) and, ´ Ö µ ¼ Ö " ³ ³ X Z ¼ if ¾ hence, we let ½ and be - and -valued wide-band ´Ö µ= ¼ otherwise. noise processes in the form ¼ ¼ Z In terms of and the equation (3) has the form Ø ¨ ´× ص dÛ ´×µ; Ø ¼; ³ ´Øµ= Z ½ ½ ½ (5) ½ ÑaÜ´¼;Ø "µ ¼ ¼ ¼ ´Ö µ ´Ö ×µ dÖ = ´×µ; ½ <× < ½: (4) Z Ø ½ ³ ´Øµ= ¨ ´× ص dÛ ´×µ; Ø ¼; ¾ ¾ ¾ (6) ¼ ÑaÜ´¼;Ø Æ µ Since the left-hand side of (4) is the convolution of by ¼ ¼ ? itself (we denote this convolution by ), a solution of the " > ¼ Æ > ¼ Û Û À À ¾ ½ ¾ where , , ½ and are - and - equation (4) can be derived by use of the direct and inverse ¨ ¨ ¾ valued Wiener processes, ½ and are strongly differen- ½ F Fourier tramsforms F and as follows: "; ¼] [ Æ; ¼] tiable operator-valued functions from [ and to Ä´À ;Xµ Ä´À ;Zµ ½ ¾ ¼ ¼ ¼ ¼ ¼ ¼ and , respectively, with ? = µ F ´ ? µ=F ´ µ ¼ ¾ ¼ d µ F ´ µ = F ´ µ ¨ ¾ B ´ "; ¼; Ä´À ;Xµµ; ¨ ´ "µ=¼; ½ ¾ ½ ½ ¼ ¼ ½=¾ d µ F ´ µ=F ´ µ ¡ d ¼ ½=¾ ¼ ½ ¨ ¾ B ´ Æ; ¼; Ä´À ;Zµµ; ¨ ´ Æ µ=¼: : F ´ µ µ = F ¾ ¾ ¾ ¾ d« ¼ ½=¾ ´ µ Note that the function F is multi-valued. To be Note that by the interpretation of the wide-band noise pro- certain, we take the principal branch of the square root cess (2) given above, the function is the coefficient of re- ¼ ½=¾ ¼ ´ µ = in F . It remains to show that the function laxing the initial effect at distinct time moments. Therefore, ¡ ½ ¼ ½=¾ F ´ µ F is real-valued and it vanishes in outside of it should be a decreasing (hence, almost everywhere differ- ¼ ;"] F ´ µ ´ "µ=¼ [¼ . Since is a Fourier transformation of the real- entiable) function with . In view of this, the con- ¼ ¨ ¨ ¾ valued function , by the well-known Bochner’s theorem, ditions on ½ and are reasonable. ~ ~ Z Consider the partially observable linear system to X and , respectively. We will use the notation ¾ ¿ dܴص=´Aܴص·³ ´ØµµdØ ·¨dÛ ´Øµ; ½ A ¼ ½ ~ 4 5 ¼ d=d ¼ : A = Ü´¼µ = Ü ; Ø ¼; ¼ (7) ¼ ¼ d=d« dÞ ´Øµ=´Cܴص·³ ´ØµµdØ ·©aÚ ´Øµ; ¾ ´¼µ = ¼; Ø ¼; Þ (8) One can easily verify that ¾ ¿ £ Þ A where Ü is a state process, is an observation process, A ¼ ¼ £ £ ~ 5 is the infinitesimal generator of a strongly continuous semi- 4 A = d=d ¼ ; ½ X group Í of linear bounded operators on , for that we use ¼ ¼ d=d« ¾ Ë ´X µ C ¾ Ä´X; Z µ ¨ ¾ Ä´À; X µ the symbol Í , , , £ © ¾Ä´G; Z µ ³ ³ A A d=d d=d« ¾ , ½ and are wide-band noise processes de- where is the adjoint of , and are differential Ü X fined by (5) and (6), ¼ is an -valued Gaussian random operators from Û Ú À G © variable, and are - and -valued Wiener processes so ¨ ~ ~ D ´d=d µ= f ¾ X :´d=d µf ¾ X; f ´¼µ = ¼ Ü ´Û; Û ;Û µ Ú Û ½ ¾ that ¼ , and are mutually independent, and , Û Û ¾ ½ and are correlated.
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