A NOVEL OPTIMAL PROBABILITY DENSITY FUNCTION TRACKING FILTER DESIGN 1

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1 A NOVEL OPTIMAL PROBABILITY DENSITY FUNCTION TRACKING FILTER DESIGN 1 Jinglin Zhou Hong Wang, Donghua Zhou Department of Automation, Tsinghua University, Beijing , P. R. China Control Systems Centre, The University of Manchester, Manchester M60 1QD, United Kingdom Institute of Automation Chinese Academy of Science, Beijing , P. R. China Abstract: A new optimal tracing filtering algorithm for a class of multivariate dynamic stochastic systems is presented. New concepts such as hybrid probability density functions PDFs) and hybrid characteristic functions CFs) are introduced to describe the stochastic nature of the dynamic estimation errors. New relationships between the hybrid characteristic functions HCFs) of the multivariate stochastic input and the outputs, and the properties of the HCF are established. A new performance index of the tracing filter is then constructed based on the form of the HCF of the estimation error. An analytical solution, which guarantees the filter gain matrix to be an optimal one, is then obtained. A simulation case study is included to show the effectiveness of the proposed algorithm and encouraging results have been obtained. Keywords: Dynamic stochastic systems; Characteristic functions; Optimal tracing control; Hybrid random vectors; Non-Gaussian variables; Optimal filtering. 1. INTRODUCTION State estimation or filtering) for dynamic stochastic systems has always been one of the main focuses in signals processing. Many approaches have been developed and widely used successfully in real applications following the development of the Kalman filtering KF) algorithm Anderson and Moore, 1979; Astrom, 1970; Goodwin and Sin, 1984). It has been observed that the standard KF algorithms may lead to poor performance when the system is subjected to uncertainties, nonlinearities or non-gaussian disturbance inputs. Motivated by this problem, many alternative methods, such as the extended Kalman filters EKF) has been developed. The EKFs Einice and White, 1999; Rahim and Dan, 2002) focus on nonlinear systems with Gaussian white noise random disturbances. The EKF applies the KF to nonlinear 1 This wor was mainly supported by NSFC Grant No , , ) systems by simply linearizing all the nonlinear models so that the traditional linear KF equations can be applied again with the performance index being the mean and the variance of the estimation errors. However, due to the presence of the nonlinearity in the system dynamics, the system output and the estimation error can generally be non-gaussian even if the system is subjected to Gaussian inputs. In this context, the mean and the variance of the estimation error are not sufficient to characterize the filtering performance for nonlinear systems. Based upon the above discussions, it can be concluded that current EKF design techniques have used the mean and variance criterion for optimizing their estimation errors and residuals under the assumption that the estimation errors can be treated as Gaussian random processes. However, this assumption is strict for many real systems and therefore limits the use of the filter to non-gaussian and nonlinear systems because the mean and variance cannot capture all features of non-gaussian noise and

2 determine the shape of its PDF. In this paper, we aim to develop a new filtering algorithm by shaping the conditional PDF of the estimation error signals. In the following, except for specially pointing out, matrices are assumed to have appropriate dimensions. For two real vectors v 1 and v 2, the notation v 1 v 2 is used to denote that every element of v 2 is no less than the corresponding one of v 1. v 1 v 2 and v 1 v 2 have the similar meanings. diag{ } is used to denote a diagonal matrix. E ) and Var ) represent mathematical expectation and variance of random variables. df, z D ) denotes df, z D + ). = + + has appropriate dimensions. 2. PRELIMINARIES 2.1 System model and estimation error Consider the following stochastic nonlinear systems, x +1 = A x + G w +1 y = Hx ) + v 1) where x R m is the state, y R l is the output, w R n and v R l are the random disturbance. A and G are two nown time-varying system matrices. w can be an arbitrary bounded independent random vectoror hybrid random vector, see the following definitions) rather than Gaussian input. The following assumptions are required in this paper. Assumption 1. {w } and {v } are bounded, stationary processes. {w }, {v } and x 0 are mutually independent. w has a nown CF denoted by ϕ w x) with Ew ) < +, and Varw ) < +. v has a nown bounded mean value Ev ) < +. Assumption 2. Guo and Wang, 2006) H ) is a nown Borel measurable and smooth vector-value nonlinear function of its arguments. For the dynamic system given by 1), a filter can be adopted as follows, ˆx +1 = A ˆx + U y ŷ ) ŷ = ŷ + Ev ). where U R m l is a gain matrix to be determined. Denote e = x ˆx, then the estimation error equation satisfies e +1 = A e + G w +1 U y ŷ ) 2) where e R m. A desired filter should ensure that a measure of e be minimized. As Guo and Wang Guo and Wang, 2006) pointed out, in 2), e +1 can be repressed by a sum of two independent random vectors A e s and G w +1 q +1, as well as a measurable term U y ŷ ). Thus, the probability of e +1 is a conditional probability related to the probabilities of both e and w +1 for a set of given A, G, y, ŷ and U. For simplicity, γ e ) and ϕ e ) are used to represent the conditional joint PDF and CF of e, respectively. The purpose of filtering is to use available information of the system input and output to estimate x. The criteria that can be used to assess the accuracy of such a filtering algorithm relies on the statistic nature of the estimation error e, which is comprehensively embedded in the PDF or characteristic function) of the estimation error e. Thus, the filter design can be performed by minimizing the following performance function at every sampling time. J = γ e x) gx)) 2 dx + U T R U 3) where gx) is a pre-specified PDF for the estimation error PDF γ e x) to follow. In practice gx) can be selected as a narrowly distributed Gaussian PDF. This means that the filtering design should be such that the error PDF is made as narrow and as Guassian as possible. In this context, the first term in 3) provides a direct metric to measure the difference between γ e x) and gx), and the second term reflects the constraints of the filter gain matrices. This means that the actual error distribution is made as close as possible to its desired distribution gx) whilst the energy of the filter gain matric will be minimized. This performance function has been used by Wang Wang, 2000; Guo and Wang, 2006) and the filter is called to be an optimal tracing filter. However, it is very difficult to directly obtain a compact mathematical form that lins the PDF γ e x) to those of the noise. Therefore, the filter design using directly performance function 3) will be very difficult to be carried out. In order to solve this problem, the CF is employed here. In this context, it is important to formulate the CF of the estimation error using equation 2). This will enable us to establish a new performance function. 2.2 Hybrid characteristic functions In order to study the stochastic behavior of the multivariate mapping, the existing theory needs to be extended. For this purpose, the following definitions on hybrid random vectors, hybrid probabilities and HCFs are introduced to generalize some conventional concepts in probability theory. If a random variable z only has a value Pz=c)=1, then it is called as a degenerated distribution and the variable is referred to as a degenerated variablema and etc., 2001). In fact, the degenerated distribution taes a constant as a discrete random variable. In the sequel, without a special indication, the discrete random variables can contain some degenerated variables. Definition 1. A random vector Z R m is called as a hybrid random vector if it contains both continuoustime and discrete-time random variables. Let z C

3 R m 1 and z D R m 2 be a continuous-time random vector and a discrete-time random vector, respectively, with m = m 1 + m 2. Then their related probability, which is referred to as a hybrid probability, is defined as P z C δ, z D = σ i ) = P z C δ)p z D = σ i ) 4) where δ R m1, σ i R m2, i = 1, 2,, N. Similarly, its hybrid probability distribution function is defined as F δ, z D σ i ) = z D P z C δ)p z D σ i ) 5) where z D = z D σ i. In this context, the corresponding PDF is called as a hybrid probability density function HPDF) Guo and Wang, 2006). Definition 2. The CF of a hybrid random vector Z = [z T C R1 m 1, z T D R1 m 2 ] T is called as an HCF, which is defined by ϕ Z t 1, t 2,, t m ) = E{expjt C z C )}E{expjt D z D )} N = e jt D ) ) Cm 1 σ p exp j t z )df δ, z D ) =1 =C1 6) where j = 1 is the imaginary number unit, t C R 1 m1, t D R 1 m2, σ R m2, P z D = σ ) = p, = 1, 2,, N, C1 and Cm 1 represent the first and the last variable of the continuous time sub-vector. In the third equation, z D + rather than z D σ i as shown in equation 5) because the discrete time subvector can include both finite and infinite components in the HCF 6). Definition 3. If a variable contains a continuous-time part and a discrete-time part then the variable is referred as a mixed random variable denoted by z M. Similarly, the related probability, which is taen as the mixed probability denoted by P z MC δ, z MD = σ i ), is defined as P z MC δ, z MD = σ i ) = P z MC δ)p z MD = σ i ) 7) A hybrid random vector Z which contains some mixed random variables z M R m 3 ) is called as systemoutput-type hybrid SOTH) random vector. The corresponding characteristic function is therefore refereed to as an SOTH Characteristic Function and is defined as ϕ Z t 1, t 2,, t m ) = exp j Cm 1 +m 3 ) =C1 N1 e jt D ) ) σ p =1 t z df δ, [z D, z MD ]) 8) where z C R m1, z D R m2, m 1 + m 2 + m 3 = m, t T D Rm 2+m 3, σ R m 2+m 3 and P [z T D, zt MD ]T = σ ) = p, = 1, 2,, N 1. C1 and Cm 1 + m 3 ) represent the first and the last variable of the contained continuous-time sub-vector. Definition 4. If a random vector Z only contains continuous time variables and degenerated variables then it is called as a strict system-output-type hybrid SSOTH) random vector Guo and Wang, 2006). The corresponding CF is therefore refereed to as an SSOTH CF. Based upon the definition of the HCF, the SSOTH characteristic function is given by ϕ Z t 1, t 2,, t m ) = exp j Cm 1 =C1 t z )df δ, z D )e Dm 2 j i=d1 σ i t i 9) where D1 and Dm 2 represent the first and the last variable of the degenerated variables. In order to simplify the descriptions and technical formulation procedures, in the following expressions only SSOTH random vectors will be considered. An SSOTH random vector can be transformed from other SSOTH random vectors. The ey issue here is how this hybrid characteristic function can be calculated. In the next section, we will present the solutions on the linear transformation and the algebraic sum operation among SSOTH random vectors. 3. ERRORS CHARACTERISTIC FUNCTION Under Assumptions 1 and 2, from 2) it can be seen that one ey tas is to calculate the CF of e +1 using the CF of A e and G w +1. In this regard, the computation of HCFs of the linear transformation and the algebraic sum operation among hybrid random vectors will be described. For this purpose, the following propositions is given. Proposition 5. Let ϕ Z x) = ϕ Z x 1, x 2,, x n ) be an SSOTH CF of SSOTH random vector Z R n, and denote A R m n and b = [b 1, b 2,..., b m ] T as two constant matrices. Then m dimensional random vector Y = AZ + b is still an SSOTH random vector with its CF being given by ϕ Y t 1, t 2,, t m ) = e jtb ϕ Z ta) 10) where ϕ Y t 1, t 2,, t m ) is the HCF of Y. Proposition 6. Let an SSOTH random vector Z be represented by the following algebraic sum Z = Z 1 + Z 2 11) where Z 1 and Z 2 are two independent SSTOH random vectors with the same dimension. Then its SSTOH CF can be represented as follows ϕ Z t) = ϕ Z1 t)ϕ Z2 t) 12) where ϕ Z1 t) and ϕ Z2 t) are the HCFs of Z 1 and Z 2, respectively.

4 Proposition 5 and 6 provide a way to compute the HCF of the algebraic sum of any two SSOTH random vectors. Thus, the following result can be readily obtained, Lemma 7. Under assumptions 1 and 2, the HCF of e +1 can be formulated recursively by ϕ e+1 t) = e ju y ŷ )t ϕ s t)ϕ q+1 t) 13) where ϕ s t) and ϕ q+1 t) are the corresponding hybrid characteristic functions of s and q +1 which can be calculated by using proposition PDF tracing filtering 4. FILTERING DESIGN For the required filtering algorithm design such as the minimum entropy filter Guo and Wang, 2006) and the filter designed using the performance index 3), the PDF of the e +1 need to be obtained as a starting point. If the hybrid characteristic function of e +1 has been formulated using Lemma 7, then the following proposition can be established so as to obtain its corresponding hybrid probability density function. It is shown that the design of the filter based on the PDF shaping needs another transformation which will possibly increase the computation time and thus may not be suited for the real time filtering. On the other hand, the filter design can directly use the CF simply because controlling the shape of conditional hybrid probability density function is equivalent to the shape control of its HCF. Thus, in the rest of the paper, we will only consider the filter design based on the hybrid characteristic function by selecting a filtering gain matrix U. For this purpose, denote ϕ g t) = + + e jtx gx)dx 14) as the transform function of the given target distribution function, gx). As a result, the aim of the filter design is to select U such that ϕ e t) is made as close as possible to ϕ g t). 4.2 Reselection of performance indexes Since the multi-dimensional multiplications are involved in equation 13), the direct use of performance index 3) for the filter design can be too complicated to be used in practice. An alternative performance index should therefore be formed which measures directly the difference between ϕ e t) and ϕ g t). In addition, since the SSOTH CF can still exhibit the same basic properties of normal CFs such as ϕt) 1 where is the complex modulus),and ϕ t) = ϕt). We can use the CFs for SSOTH random variables to define the distance between two CFs. In this context, the following lemma can be established, Lemma 8. Let ϕ a t) and ϕ b t) be any two SSOTH CFs. Denote ψ = ϕ a t) log ϕ at) dt 15) ϕ b t) where t T R m and log z log z + j arg z, arg z π/2 16) in which arg z represents the angler of a complex number z. Then ψ is a real number. Using this lemma, as Wang in Wang and Wang, 2002), the following new performance index { J = Kt)ϕ g t) log ϕ } 2 gt) ϕ e t) dt 17) + U T R U = J 0 + U T R U can be defined, where the first term is similar to the well nown Kullbac-Leibler distance widely used in the information functional measure. The use of such a new performance index allows the transfer of the multiplication operations in equation 13) into a simple algebraic sum by using a logarithm operator. Simultaneously, by minimizing this term, ϕ e t) can be made as close as possible to ϕ g t). The first term is zero for ϕ g t) = ϕ e t) almost surely) and infinite if there is a set of a positive Lebesgue measure on which ϕ e t) 0 Wang and Wang, 2002). The second term in equation 17) is again the constraint on the filter gain matrix with R > 0 being a pre-specified weighting matrix. In equation 17), J 0 = { Kt)ϕ gt) log ϕgt) ϕ e t) dt } 2, and Kt) in equation 17) is a weighting function which should be selected so as to mae J 0 a real number and to guarantee the boundness of J 0. Since ϕt) 1, it can be shown that { J 0 = { { ϕ e t) dt Kt)ϕ g t) log ϕ gt) Kt) log ϕ } gt) 2 dt Kt) } 2 ϕ e t) log 2 ϕ g t) ϕ e t) + π2 4 dt }2 18) To guarantee the uniform boundness of J 0, it is sufficient to select the weighting function Kt) > 0 such that the following inequality Kt) log 2 ϕ g t) ϕ e t) + π2 4 M 1e tm2t T 19) holds, where t T R m, M 1 > 0 and M 2 = M2 T R m m are pre-specified numbers and positive definite matrix, respectively. Using 19), it can be further shown that if the following inequality 0 < Kt) M 1e tm 2t T, M 1 2 π M 1 20)

5 holds, then the boundedness of J 0 can be guaranteed. Moreover, if K t) = Kt), then J 0 is a real number. To summarize, we have the following theorem. Theorem 9. Suppose the weighting function Kt) has been selected so that 0 < Kt) M 1e tm 2t T and K t) = Kt), then J 0 is a real number and J 0 M1 2 π n / n i=1 ξ i, where ξ i, i = 1, 2,, n) are the diagonal elements of M Optimal filter gain matrix Once the performance index J is selected, the filter design can be readily carried out by directly minimizing the selected performance index. In order to provide the filter with a simple structure, the instantaneous performance index J 3 in 17) is considered firstly in the design. For this purpose, U should be calculated from J 3 U = 0, which leads to 2R U + J 0 U = 0 21) To obtain an analytical solution of equation 21), J 0 should be analyzed first. Indeed, it can be shown that where J 0 = a 2 + 2a b U + U T b T b U 22) a = b = Kt)ϕ g t) log ϕ g t)dt Kt)ϕ g t) log ϕ s t)dt Kt)ϕ g t) log ϕ q+1 t)dt; Kt)ϕ g t)jty ŷ )dt. Substituting equation 22) into equation 21) yields a b + b T b + R ) U = 0 23) and the filter gain matrix can be obtained immediately as follows. U = b T b + R ) 1 a b ) 24) From equation 23), the following condition on the second-order derivative of J 3 should also be satisfied at every sampling time in order to guarantee the minimization result, 2 J 3 U 2 = b T b + R > 0 25) It shows that the solution of equation 24) is a global optimal solution. As such, to mae the filter design process clear, the solution route in the filter design can be summarized as the following four steps. 1) Describe the stochastic behavior of e and w +1 in Equation 2) using Definition 1; 2) Compute the hybrid characteristic function of e +1 in terms of Lemma 7; 3) Select a narrowly distributed Gaussian PDF as the target distribution function and a weight function Kt) which satisfies Theorem 9; 4) Compute the filtering gain matrices using Equation 24); 5. SIMULATION EXAMPLE Although the SSOTH random vectors are considered, all the propositions, lemmas, and theorems can be applied to SOTH random vectors. The following simple example on SOTH random vectors will be used to demonstrate the proposed filtering algorithm where the system has the following dynamics case 1). [ ] [ x1, arctan0.04) = x 2, arctan1 + ) 1 ] [ ] ) x1, x 2, [ ] [ w1, ) 1 ) w 2,+1 26a) y = sinx 1, ) + x 2, + x 1, x 2, + 0.2x 2 1, v 26b) where w 1 N0.1, 1) is normal distribution, w 2 P 1) is a Poisson distribution with parameter 1 and their characteristic functions are given respectively by ϕ w1 t) = exp0.1tj t 2 /2) 27) ϕ w2 t) = exp 1 e jt )) 28) Remar 10. In most of the engineering processes, except of some general noise Gaussian or non- Gaussian), the systems may be disturbed by some swashing disturbance. therefore in the simple model 26a), we introduce a random vector w = [w 1,, w 2, ] T to describe this cases. Also, random variable v, = 0, 1, ) is assumed to be subjected to a uniform distribution defined on [ 1, 1]. The target characteristic function is selected as ) ϕ g t) = exp tbtt 29) 2000 where B = diag{1.1, 1.1}. Such a target distribution shape is close to a narrowly distributed Gaussian PDF. In the simulation, the weighting function Kt) is selected as follows, Kt) = 1 55 expjtµ tm 2t T ) 30) where µ = [0.0001, ] T, M 2 = diag{ , }. The weighting matrix and the initial condition are selected as R = diag{ ), )}, x0) 10 = [2, 1] T and ˆx0) = [0, 0] T. ]

6 Performance Index Fig. 1. Sequence of tracing performance function x Fig. 2. State x 1 x Fig. 3. State x 2 Actual PDFTF Actual PDFTF The response of the J 0 and the state vectors are shown in Fig. 1 to Fig. 3. From those figures, it can be seen that a convergent filtering result has been achieved in line with the proposed filtering algorithm. 6. CONCLUSIONS Using the concept of PDF shaping, a new optimal tracing filter design for multivariate stochastic systems subjected to non-gaussian noise is presented in this paper, where the ey idea is to select the filtering gain so that the PDFs of the filtering error can be made to follow a target distribution shape. This has therefore extended the existing minimum variance KF and EKF etc.) based filtering algorithms. Indeed, if the targeted distribution is a narrowly distributed Gaussian PDF, then the proposed filter aims at obtaining a state estimation error whose PDFs is made as close as possible to a Gaussian shape. To effectively characterize the stochastic property of multivariate mappings in the filtering design, new concepts such as hybrid random vectors variables) and hybrid characteristic functions are introduced. The relationship between hybrid CF of the hybrid random vectors and that after multivariate mapping are firstly established. These relationships are then employed to establish the filtering error dynamical systems, where the error CFs are represented by the measured output and the hybrid characteristic function of the random input. To construct an optimal tracing filtering, a new index is constructed by using the Kullbac- Leibler distance and the property of the characteristic function. Under this performance function, an optimal filtering algorithm is obtained, where the filtering gain can be nicely represented by a compact form as shown in 23). Moreover, a simulated example is given to illustrate the effectiveness of the proposed filtering algorithm. 7. REFERENCES Anderson, B. D. O. and J. B. Moore 1979). Optimal Filtering. Prentice Hall, Englewood Cliffs N. J. Astrom, K. J. 1970). Introduction to Stochastic Control Theory. Academin Press, New Yor. Einice, C. A. and L. B. White 1999). Robust extended Kalman filtering. IEEE Trans. Signal Processing 47, Goodwin, G. C. and K. S. Sin 1984). Adaptive Filtering Prediction and Control. Prentice-Hall Englewood Cliffs NJ. New Jersey. Guo, L. and H. Wang 2006). Minimum entropy filtering for multivariate stochastic systems with non-gaussian noises. IEEE Trans. Auto. Contr. 51, Ma, Z. H. and etc. 2001). Handboo of Modern Application Mathmatical: The Volume of Probability Statistics and Stochastic Process. Tsighua University Publishing Company. Beijing. Rahim, J. Z. and N. Dan 2002). Extended Kalman filter Based sensor fusion for operational space control of a robot arm. IEEE Trans. Instrumentation and Measurement 51, Wang, H. 2000). Bounded Dynamic Stochastic Systems: Modelling and Control. Springer Verlag Ltd.. London. Wang, Y. and H. Wang 2002). Output PDF control of linear stochastic systems with arbitrarily bounded random parameters, a new application of the laplace transforms. Proc. of the American Control Conference 5,