**6. Conclusions**

Due to the complexity of AGV systems, they cause many decision problems, which are difficult to solve. The paper has presented an example of the Flexible Manufacturing System solution with the AGV transport system and discusses some issues related to the design and simulation of such systems. The stage of initial system design optimization is very important, and computer simulation enables the relatively easy elaboration and testing of various variants of manufacturing and logistics systems. On the other hand, excessive simplifications may be applied at the modelling stage, which will make the simulation not reflect the production system properly. It should be noted that detailed modelling is very labour intensive and requires the involvement of experienced specialists. Therefore, choosing what parameters are used in the modelling process and which metric is used to evaluate the

model is very important. In order to make the simulation more accurate and to evaluate the system's productivity, the use of Overall Equipment Effectiveness (OEE) metrics was proposed.

The results obtained from the presented simulations show that the OEE metrics can be used for the modelling and productivity evaluation of manufacturing and logistics systems, with the generalization of Overall Factory Effectiveness (OFE) and Overall Transport Effectiveness (OTE). The use of OEE factors also allows to compare the results obtained from different manufacturing systems. In the real world, most of manufacturing companies have OEE scores closer to 60%, but there are many of them with OEE scores lower than 45%, and a small number of world-class companies that have the OEE value higher than 85% [50]. According to that, the results of simulation can be also used to analyse the costs involved in the implementation of a given project and at the stage of in-depth design of the production system.

**Supplementary Materials:** The following are available online at http://www.mdpi.com/2227-9717/8/12/1648/s1.

**Author Contributions:** Conceptualization, A.K. and G.G.; methodology, A.K.; validation, K.F., G.G.; investigation, K.F.; data curation, G.G.; writing—original draft preparation, A.K.; visualization, G.G. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflict of interest.


**7. Conclusion**

The suggestion for further development of Figure 4 opens up an area for introducing complex mechanisms for reporting on production flows, where the dependency of external inquiries on actually confirmed production offers has to be research, in search for reasons

Analyzing the obtained results, it can be concluded that the total cost of materials on small quantities of one thousand pieces has a smaller share in total price of the product (38.14%), and increasing the amount of two million pieces of material share price increases (51.53%). It's determined that the share prices of materials in the total costs of production has stabilized on quantity of 100000 pieces at 51.23%, for 500000 pieces on 51.48%, and for one million on 51.51%. It also imposes the conclusion that in production of large quantities at one moment comes to stabilizing the ratio of materials cost and total production costs. The moment in which will be stabilized this ratio also depends on the necessary replacement components required in production, or about possible terms of delivery; which condition the additional production capacity. That change of terms of delivery determines the activation of additional production resources, either from its own structure, or from external sources. Compliance the defined terms of planning on the individual resources and departments inside the production structure is one of the main elements of stability in

Further research should be based on the use of new materials and currently available information - production technology. It is also possible that some new materials with good quality characteristics do not satisfied requirements of the market in high production quantities, but in a small zone of production be acceptable. It is also one of the main directions of development of digital - information productions, making a complex and unique products in the minimum quantity (even 1 product) whose realization is fully

The suggestion for further development of Figure 4 opens up an area for introducing complex mechanisms for reporting on production flows, where the dependency of external inquiries on actually confirmed production offers has to be research, in search for reasons

automated. This includes an acceptable competitive price of finished product.

Fig. 8. Change of ratio of total production and cost of material

production process.

**7. Conclusion**

unacceptable on the side of the client. It is also suggested that internal tools be introduced which will construct the inquiry and based on them certain conclusions can be made for new strategic decisions. This opens up a new area for improving production, introducing the system of real simulation of workers according to the effort and productivity in the organization. The system of analysis, comparison and monitoring has to be based on set and realistic values in both information systems, and the results obtained are expected to open new areas for investments.

Such a mechanism demands immediate interaction between users through a web interface with desired time periods and loads in needed resources and workers, comparative prices of production calculations with the post-calculation part, total inquiry calculation, accepted offers, visual representation and numerical values. The results influence daily business decision-making as well as strategic directions in improving the business system. The implementation of such a tool on the existing information structure is possible at the level of the basic model shown in Figure 1, but since such a system does not have qualitative quantitative information from the production of a product itself, adequate conclusions on the production flow is not possible. A concrete application of the suggested mechanism in its full use would be through the model described in Figure 2 where independent inquiries can be placed either to the financial or production system, and can be most outright with targeted inquiries through the integration of both system of the production structure.


**Author details**

Gesessew Kibr Faculty of Agriculture, Department of Food and Nutritional Sciences, Wollega University, Shambu, Ethiopia

\*Address all correspondence to<EMAIL_ADDRESS>
© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*The Health Benefits of Vegetables; Preventive Implications for Chronic Non-Communicable… DOI: http://dx.doi.org/10.5772/intechopen.101303*


**4. Discussion**

The available information on the occurrence of PNSB in colored microbial mats and blooms in the environment has so far been only scattered and fragmentary [8,19–21]. As reported herein, our polyphasic approach to address this issue by culture-independent techniques as well as by conventional cultivation methods has improved information on the distribution of PNSB in colored blooms/mats in terms of quantity and quality. We need to carefully consider that there might be cultivation and PCR biases in the used approach and that the numbers of the *pufM* gene clones sequenced and the isolates phylogenetically identified in this study are not sufficient to describe the phototrophic community structures at the studied sites. However, the results of direct phase-contrast microscopy, quinone profiling, and the clone library analysis match and complement each other relatively well, thereby increasing the reliability of our data.

One of the most important findings in the present study is that there were significant differences in the contribution of PNSB to blooming phenomena and their biodiversity between the coastal environments and wastewater ditches. In the coastal red-pink blooms, PNSB constituted significant proportions of the phototrophic bacterial populations, but usually occurred in smaller CFU numbers than PSB. Actually, direct microscopic observations and quinone profiling data have shown that the

overwhelming majority as the biomass in the coastal red-pink blooms was represented by elemental sulfur globe- and/or gas vacuole-containing PSB, whose main quinones are Q-8 and MK-8. In addition to these findings, the *pufM* gene amplicon analysis has clearly shown that the major phylotypes detected were those assigned to the genera *Thiolamprovum* and *Thiocapsa* in the pink mudflat and to *Marichromatium* and *Thiocapsa* in the red tide pools. Similar PSB members accompanied by smaller numbers of PNSB and aerobic anoxygenic phototrophic bacteria have been found in massive blooms in a brackish lagoon [8]. To our knowledge, therefore, the formation of red-pink blooms in coastal environments is attributable to massive development of PSB, and the contribution of PNSB as the colorants to the bloom formation is less significant.

In contrast, the red-pink blooms/mats in the swine wastewater and sewage ditches exclusively yielded PNSB, as demonstrated by a combination of phase-contrast microscopy, quinone profiling, *pufM* gene clone sequencing, and cultivation-based phylogenetic analysis. Our data have shown that PNSB assigned to the genera *Rhodobacter* ("*Luteovulum*" and "*Phaeovulum*" [61]), *Rhodopseudomonas*, and *Pararhodospirillum* predominated in the ditches, but the PNSB community structure differed from sample to sample. These results fully support the previous observation that visible massive development by PNSB themselves takes place in the environment under specific conditions [19]. Obviously, while coastal blooms commonly harbor *Rhodovulum* species as the major PNSB populations, wastewater blooms/mats may contain different major taxa of PNSB depending on the environmental conditions.

One of the most important environmental factors affecting the proliferation of PNSB is the concentration of sulfide. Since most of the PNSB species are unable to tolerate high concentrations of sulfide, there are few chances for them to grow massively in such sulfide-rich environments as coastal environments. However, marine species belonging to the genus *Rhodovulum* can use relatively high concentrations of sulfide as the electron donor for photolithotrophic growth [64,65], and *Rdv. sulfidophilum* has the genes of the Sox pathway involved in the complete eight-electron oxidation of sulfide to sulfate [66,67]. Our genomic studies have confirmed that *Rhodovulum* sp. strain MB263 as well as *Rdv. sulfidophilum* strains DSM 1374T and DSM 2351 have the complete gene set of the *sox* operon. Also, the previous study has shown that *Rhodovulum* sp. MB263 is capable of photolithotrophic growth with 2 mM sulfide as the electron donor [21]. These facts provide a plausible reason why the group of *Rhodovulum* sp. MB263 and *Rdv. sulfidophilum* occurred in significant phototrophic bacterial populations in the coastal blooms with high concentrations of sulfide. Nevertheless, the biomass density of the *Rhodovulum* members would not become so high as exceeding those of PSB and GSB, both of which have growth advantages, with higher affinities to sulfide as the electron donor for photolithotrophy.

The second important factor controlling PNSB populations is ORP, which is also related to sulfide concentrations. The mudflat and tide pools we studied exhibited an Eh level of −320 to −170 mV with high concentrations of sulfide. Such low Eh levels as well as high sulfide concentrations are apparently more favorable for growth and survival of PSB or GSB than PNSB. On the other hand, a limited higher range of Eh (−93 to 23 mV), as seen in the ditch blooms/mats, may be effective for stimulating the growth of PNSB while suppressing that of PSB and GSB.

The concentration of organic matter, which PNSB can use as energy and carbon sources for photoorganotrophy as their best life mode, should be noted as the third important factor. The coastal blooming mudflat and tide pools studied had low concentrations of organic matter, expressed as COD, whereas the sewage and wastewater ditches were at much higher COD levels. This provides an additional explanation for why PNSB could overgrow PSB in the wastewater ditches but not in the coastal environments. Taken together, it may be logical to conclude that light-exposed, sulfide-deficient water bodies with high-strength simple organic matter and in the limited range of ORP provide the best conditions for the massive growth of PNSB. This also explains the basis for wastewater treatment systems using PNSB for purifying highly concentrated organic matter, where they can compete with co-existing PSB and chemoorganotrophic bacteria under limited aerobic conditions [72,73].

As mentioned above, members of the genus *Rhodovulum*, especially the *Rhodovulum* sp. MB263 genospecies and *Rdv. sulfidophilum*, occurred as the major PNSB populations in coastal blooms, as revealed by *pufM* clone library analysis and cultivation-based phylogenetic studies. Apparently, the ability of the *Rhodovulum* species to utilize sulfide as the electron donor for photolithotrophic growth is an advantage in such sulfide-rich environments as blooming seawater pools and mudflats. In addition, it is noteworthy that *Rhodovulum* sp. MB263 and *Rdv. sulfidophilum* are capable of floc formation depending on the culture conditions [74]. This capacity might be another advantage for the *Rhodovulum* members to protect themselves against environmental stress. Further study with the genomic information obtained in this study should give a more comprehensive understanding of how *Rhodovulum* species respond to various environmental factors in the ecosystem.

*Rhodovulum* sp. strain MB263 was isolated previously from a pink pool in a mudflat [21] in the same area where we found mud blooms Y1 and Y3 in this study. Genomic DNA–DNA hybridization assays in previous work showed that strain MB263 had a similarity level of 57% to *Rdv. sulfidophilum* DSM 1374T as its closest relative, suggesting that the strains are closely related to each other but di ffer at the species level [21]. In the present study, this suggestion is fully supported by genomic and phylogenomic information. An ANI value of 91.21% between the genomes of strains MB263 and DSM 1374T is lower than the recommended threshold value (95–96%) for bacterial genospecies circumscription [52,68]. Genome-wide comparisons of *Rhodovulum* sp. MB263 with other closely related *Rhodovulum* species, e.g., *Rdv. algae*, should provide more definitive information on whether strain MB263 represents a novel species of the genus *Rhodovulum*.


**2.** *In vitro* **assays**

72 Contact Dermatitis

from direct exposure to harmful sensitizing or irritant chemicals (Elias, 2005; Elias, 2007). The barrier function is provided by the uppermost layer of the epidermis, the stratum corneum. The stratum corneum consists of dead, terminally differentiated keratinocytes (corneocytes) embedded in extracellular lipid. The corneocytes and the lipid component of the stratum corneum can be considered as bricks and mortar and form the barrier to the environment and potentially harmful substances (Bouwstra and Ponec, 2006; Elias, 1983; Elias, 2004). In order for a potential sensitizer to cause an allergic reaction it must first penetrate or damage the stratum corneum in order to exert its effect on the viable epidermal and dermal layers below. Once a chemical has penetrated the stratum corneum, it is metabolized by binding to homologous skin proteins. As a result the new antigenic moieties may exert cytotoxic effects on the keratinocytes, and trigger keratinocytes to release alarm signals in the form of cytokines and chemokines. In addition, these hapten-protein complexes may become antigenic for cells of the immune system, such as DC. DC are professional antigen presenting cells, which can efficiently stimulate T cell responses and are therefore important for the initiation and regulation of antigen- or hapten-specific immune responses (Banchereau et al., 2000; Guermonprez et al., 2002; Mellman and Steinman, 2001). In human skin, both epidermal DC (i.e. the Langerhans cells (LC)) as well as dermal DC (DDC) are involved in the initiation of allergic contact dermatitis (Aiba, 2007; Bennett et al., 2005; Kaplan et al., 2005). Following encounter with an allergen, LCs become activated and undergo maturation and differentiate from antigen-capture and processing cells into potent immunostimulatory DCs, able to present antigen effectively to effector T-cells. In order to activate antigen specific acquired immunity leading to the development of effector T-cells, LC migrate to the paracortical area of the regional lymph nodes, where they display the allergenic epitope to naïve T-cells (Aiba et al., 1993; Lanzavecchia and Sallusto, 2001; Nosbaum et al., 2009; Reid et al., 2000; Saint-Mezard et al., 2004). This results in expansion and differentiation of allergen reactive T cells, thereby forming specific effector and memory T cells, which migrate via the efferent lymphatics into the bloodstream and recirculate

There are a number of considerations which should be taken into account when developing an *in vitro* assay for assessment of the sensitizing potential of a compound. In all cases, for an *in vitro* model to replace an animal model it should be able to distinguish a sensitizer from a non-sensitizer to the same degree as the current animal models. Currently it is thought that no single assay will meet these requirements and therefore a battery of assays should be developed which will be used in a tiered manner. This chapter describes the progress on the development of human *in vitro* assays for assessment of the sensitizing potential of a compound, based on the five crucial *in vivo* events in skin

1. The ability of the chemical to penetrate through the stratum corneum: bioavailability 2. The potential of the chemical to metabolize into stable conjugates to create an

3. The ability of a chemical to trigger alarm signals from keratinocytes 4. The ability of a chemical to induce maturation and migration of DCs

5. The ability of a chemical to provoke T-cells responses.

through the body (Sallusto et al., 1999).

sensitization (Fig. 1):

immunogenic complex


5. Conclusions

This chapter dealt with mathematical modeling, system identification, and control of electromagnetic actuators. Actuators are often used in industrial applications such as magnetic levitation, electromagnetic bearings, as well as in fuel injectors in the automotive industry.

After a detailed mathematical model was presented, two different parameter identification techniques were described. The first one is based on the classical current decay test, and the second one is a clustering-based system identification approach. Since the actuator is openloop unstable, the main steps of system identification of the actuators under closed-loop control were presented.

Finally, very simple and easy-to-apply control strategies were discussed, when the armature is controlled around a fixed equilibrium position (PID controller) as well as when the armature is controlled between two extreme positions, armature open and armature closed (gainscheduled PID controller).


**1. Introduction**

It is very important to consider simultaneous estimation of both system states and inaccessi‐ ble modes for hybrid systems with unknown modes [4,7,24,25]. This estimation is called hy‐ brid estimation. By the hybrid estimation we often want to know both a current mode and system state at each time through information of observation. However there exist cases that we want to know distributions of modes on long run time interval rather than each estimate of the modes themselves at each time to grasp global performance over long time intervals, for example, distributions of active modes in solar systems [5,21], distributions of active agents on formation or consensus via hybrid systems representation and so on.

Much work has been done for smoothing theory for both of continuous- and discrete-time systems ([1,2,3,6,8,9,10,12,13,14,15,18,19,20,22,23] and so on). Various researchers have stud‐ ied the smoothing problems by various approach, for example, maximum likelihood ap‐ proach [9,13,19], projection approach [14] and so on. It is well known that smoothers (noncausal estimators) more effectively estimates the states than filters (causal estimators) because of more information of observation.

It is well known that utilization of accumulated information of observation improves esti‐ mation performance. Nevertheless, on research of estimation for hybrid systems, little work has been done from the point of view of the noncausal information of observation, i.e., smoothing. In [9] Helmick et al. have presented a fixed-interval smoothing algorithm for discrete-time Markovian jump systems by maximum likelihood (ML) approach. However they have considered only the case with fully accessible modes and their approach is based on approximate approach to probability density functions (PDFs). Therefore they have pre‐

© 2012 Nakura; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Nakura; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

sented only a nearly optimal smoothing algorithm. While it is significant that optimality is guaranteed for estimation algorithms, in [4] and [7] Costa et al. have presented LMMSE (lin‐ ear minimum mean square estimate) filters to estimate both system states and inaccessible modes for continuous- and discrete-time Markovian jump systems affected by wide sense white noises, but in these LMMSE filters theory the optimality of estimation isn''t always guaranteed in the meaning that these filters aren''t always MMSE (minimum mean square estimate). To the best of the author''s knowledge the optimal smoothing problems in the cas‐ es with inaccessible modes have not yet fully investigated.

*x*(*k* + 1)= *Ad* (*k*, *θ*(*k*))*x*(*k*) + *wd* (*k*, *θ*(*k*)), *x*(0)= *x*0, *θ*(0)=*i*

*y*(*k*)=*Hd* (*k*, *θ*(*k*))*x*(*k*) + *vd* (*k*, *θ*(*k*))

We assume that all these matrices are of compatible dimensions.

date distributions. Let *r* ∈N0 ={1, 2, ⋯, *n*0}, and let P={*ϕ* (1)

lowing performance indices for *r* ∈N0 and *k* ∈ 0, *N* :

*l*=0 *k*−1 ∑ *i*=1 *m ϕ*(*r*) (*l*)(*wd* '

*l*=0 *N* −1 ∑ *i*=1 *m ϕ*(*r*) (*l*)(*wd* '

0 is an initial estimate of *x0* and *x*

noises *wd* ( ⋅ , ⋅ ) and *vd* ( ⋅ , ⋅ ) with the estimates *x*

such candidate distributions on M, i.e., for *r* ∈N0 and *k* ∈ 0, *N* , *ϕ*(*r*)

restricted to be Gaussian.

(*k*)≥0 and ∑

(*x*0, *wd* , *vd* ): =∑

(*x*0, *wd* , *vd* ): = ∑

*i*=1 *m ϕi* (*r*) (*k*)=1.

with *ϕi* (*r*)

> *J*0*k* (*r*)

*J*0*N* (*r*)

where *x* ^ 0

An Approach to Hybrid Smoothing for Linear Discrete-Time Systems with Non-Gaussian Noises

where *x* ∈**Rn** is the state, *wd* ∈**Rn** is the exogenous random noise, *vd* ∈**Rk** is the measure‐ ment noise, and *y* ∈**Rk** is the measured output. *x0* is an unknown initial state and it is as‐ sumed that a distribution of initial modes *i0* is given. The noises *wd* ( ⋅ , ⋅ ) and *vd* ( ⋅ , ⋅ ) aren''t

Let M={1, 2, ⋯, *m*} denote the state space of *θ(k)*. In this chapter it is assumed that the prob‐ ability distribution of *θ(⋅)* is unknown or inaccessible. But it is also assumed that a finite number of candidate distributions and the true probability distribution is among the candi‐

The fixed-interval optimal hybrid estimation problems we address in this chapter for the system (1) are to find the MPT (most probable trajectory) estimate of *x(k)*, *k* ∈ 0, *N* , over the finite horizon *[0,N]*, using the information available on the known part of the observa‐ tion *y*( ⋅ ) for the given distributions of initial mode *i0* and initial state *x0*. We define the fol‐

(*l*, *i*)*Md* (*l*, *i*)*wd* (*l*, *i*) + *vd*

(*l*, *i*)*Md* (*l*, *i*)*wd* (*l*, *i*) + *vd*

*D*0(*x*0 − *x*

^ *N* )′

+ (*x*0 − *x* ^ 0)′

+ (*x*0 − *x* ^ 0)′

+ (*x*(*N* )− *x*

*Nd* (*l*, *i*)≥*O*, *D*0 >*O* and *DN* >*O* are symmetric matrices which reflect the uncertainties on the

mean the energies of noises, initial and terminal estimates under some uncertainties aver‐ aged by the mode distributions for each *r* ∈N0. We consider the optimization problems to

^ 0 and *x* ^

^

'

'

*DN* (*x*(*N* )− *x*

^ *N* )

N is a terminal estimate of *x(N)*. *Md* (*l*, *i*)>*O*,

N. Thus these performance indices

^(0))

*D*0(*x*0 − *x*

^(0))

( ⋅ ), ⋯, *ϕ*(*n*0)

(*k*)=(*ϕ*1 (*r*)

(*l*, *i*)*Nd* (*l*, *i*)*vd* (*l*, *i*))

(*l*, *i*)*Nd* (*l*, *i*)*vd* (*l*, *i*))

( ⋅ )} denote the set of

http://dx.doi.org/10.5772/51385

(*k*), ⋯, *ϕm*

(*r*) (*k*))

(2)

(3)

(1)

165

In this chapter we study hybrid estimation for linear discrete-time systems with non-Gaussi‐ an noises. The concerned systems are general hybrid systems given below which aren''t re‐ stricted to Markovian jump systems [4,5,7] and where added noises aren''t restricted to be Gaussian. It is assumed that modes of the systems are not directly accessible throughout this paper. We consider optimal estimation problems to find both estimated states of the systems and an optimal candidate of the distributions of the modes over the finite time interval. We adopt most probable trajectory (MPT) approach to guarantee the optimality of estimation methods. On this approach, given information of observation, we consider optimal control problems where we seek optimal control by which averaged noises energies are minimized for averaged systems throughout the mode distributions. In [24,25] Zhang has presented hy‐ brid filtering algorithm by MPT approach for the continuous- and discrete-time hybrid sys‐ tems. We consider both filtering and smoothing problems for discrete-time hybrid systems in this chapter. We can expect better estimation performance by taking into consideration noncausal information of observations. The hybrid smoother is realized by two filters ap‐ proach [2,8,12,20,22,23]. Finally we give numerical examples and verify that we can obtain better estimation performance by smoothing than filtering.

The organization of this chapter is as follows. In section 2 we describe the systems and prob‐ lem formulation. In section 3 we present the hybrid estimation algorithms by the MPT ap‐ proach over the finite time interval. In subsection 3.1 we review the hybrid filtering algorithm and in subsection 3.2 we design the backward filters and present the hybrid smoothing algorithm by the two filters approach. In section 4 we consider numerical exam‐ ples and verify the effectiveness of the estimation algorithms presented in this chapter. In the Appendix we present the principles of hybrid optimality, which give the basis of validi‐ ty for the hybrid estimation algorithms presented in this chapter

#### **2. Systems and Problems Formulation**

Let *(Ω, F, P)* be a probability space and, on this space, we consider the following system with mode transitions and noises which aren''t restricted to be Gaussian.

$$\mathbf{x}(k+1) = A\_d(k, \ \theta(k))\mathbf{x}(k) + w\_d(k, \ \theta(k)),$$

$$\mathbf{x}(0) = \mathbf{x}\_{0'} \ \theta(0) = i\_0 \tag{1}$$

$$\mathbf{y}(k) = H\_d(k, \ \theta(k))\mathbf{x}(k) + v\_d(k, \ \theta(k))$$

where *x* ∈**Rn** is the state, *wd* ∈**Rn** is the exogenous random noise, *vd* ∈**Rk** is the measure‐ ment noise, and *y* ∈**Rk** is the measured output. *x0* is an unknown initial state and it is as‐ sumed that a distribution of initial modes *i0* is given. The noises *wd* ( ⋅ , ⋅ ) and *vd* ( ⋅ , ⋅ ) aren''t restricted to be Gaussian.

We assume that all these matrices are of compatible dimensions.

sented only a nearly optimal smoothing algorithm. While it is significant that optimality is guaranteed for estimation algorithms, in [4] and [7] Costa et al. have presented LMMSE (lin‐ ear minimum mean square estimate) filters to estimate both system states and inaccessible modes for continuous- and discrete-time Markovian jump systems affected by wide sense white noises, but in these LMMSE filters theory the optimality of estimation isn''t always guaranteed in the meaning that these filters aren''t always MMSE (minimum mean square estimate). To the best of the author''s knowledge the optimal smoothing problems in the cas‐

In this chapter we study hybrid estimation for linear discrete-time systems with non-Gaussi‐ an noises. The concerned systems are general hybrid systems given below which aren''t re‐ stricted to Markovian jump systems [4,5,7] and where added noises aren''t restricted to be Gaussian. It is assumed that modes of the systems are not directly accessible throughout this paper. We consider optimal estimation problems to find both estimated states of the systems and an optimal candidate of the distributions of the modes over the finite time interval. We adopt most probable trajectory (MPT) approach to guarantee the optimality of estimation methods. On this approach, given information of observation, we consider optimal control problems where we seek optimal control by which averaged noises energies are minimized for averaged systems throughout the mode distributions. In [24,25] Zhang has presented hy‐ brid filtering algorithm by MPT approach for the continuous- and discrete-time hybrid sys‐ tems. We consider both filtering and smoothing problems for discrete-time hybrid systems in this chapter. We can expect better estimation performance by taking into consideration noncausal information of observations. The hybrid smoother is realized by two filters ap‐ proach [2,8,12,20,22,23]. Finally we give numerical examples and verify that we can obtain

The organization of this chapter is as follows. In section 2 we describe the systems and prob‐ lem formulation. In section 3 we present the hybrid estimation algorithms by the MPT ap‐ proach over the finite time interval. In subsection 3.1 we review the hybrid filtering algorithm and in subsection 3.2 we design the backward filters and present the hybrid smoothing algorithm by the two filters approach. In section 4 we consider numerical exam‐ ples and verify the effectiveness of the estimation algorithms presented in this chapter. In the Appendix we present the principles of hybrid optimality, which give the basis of validi‐

Let *(Ω, F, P)* be a probability space and, on this space, we consider the following system

es with inaccessible modes have not yet fully investigated.

164 Advances in Discrete Time Systems

better estimation performance by smoothing than filtering.

ty for the hybrid estimation algorithms presented in this chapter

with mode transitions and noises which aren''t restricted to be Gaussian.

**2. Systems and Problems Formulation**

Let M={1, 2, ⋯, *m*} denote the state space of *θ(k)*. In this chapter it is assumed that the prob‐ ability distribution of *θ(⋅)* is unknown or inaccessible. But it is also assumed that a finite number of candidate distributions and the true probability distribution is among the candi‐ date distributions. Let *r* ∈N0 ={1, 2, ⋯, *n*0}, and let P={*ϕ* (1) ( ⋅ ), ⋯, *ϕ*(*n*0) ( ⋅ )} denote the set of such candidate distributions on M, i.e., for *r* ∈N0 and *k* ∈ 0, *N* , *ϕ*(*r*) (*k*)=(*ϕ*1 (*r*) (*k*), ⋯, *ϕm* (*r*) (*k*)) with *ϕi* (*r*) (*k*)≥0 and ∑ *i*=1 *m ϕi* (*r*) (*k*)=1.

The fixed-interval optimal hybrid estimation problems we address in this chapter for the system (1) are to find the MPT (most probable trajectory) estimate of *x(k)*, *k* ∈ 0, *N* , over the finite horizon *[0,N]*, using the information available on the known part of the observa‐ tion *y*( ⋅ ) for the given distributions of initial mode *i0* and initial state *x0*. We define the fol‐ lowing performance indices for *r* ∈N0 and *k* ∈ 0, *N* :

$$\begin{aligned} \{\mathbf{J}\_{0k}^{(r)}(\mathbf{x}\_0, \mathbf{w}\_{d\prime}, \mathbf{w}\_d) \} &= \sum\_{l=0}^{k-1} \sum\_{i=1}^m \phi^{(r)}(l) \{\mathbf{w}\_d^{'}(l, i)\mathbf{M}\_d(l, i)\mathbf{w}\_d(l, i) + \mathbf{v}\_d^{'}(l, i)\mathbf{N}\_d(l, i)\mathbf{v}\_d(l, i)\} \\ &+ \{\mathbf{x}\_0 - \hat{\mathbf{x}}\_0\}' \mathbf{D}\_0(\mathbf{x}\_0 - \hat{\mathbf{x}}(0)) \end{aligned} \tag{2}$$

$$\begin{aligned} \{\boldsymbol{J}\_{0}^{(r)}(\mathbf{x}\_{0},\mathbf{w}\_{d},\boldsymbol{\upsilon}\_{d})\} & \coloneqq \sum\_{l=0}^{N-1} \sum\_{i=1}^{m} \phi^{(r)}(l) \{\boldsymbol{w}\_{d}^{'}(l,\mathbf{i})\boldsymbol{M}\_{d}(l,\mathbf{i})\boldsymbol{w}\_{d}(l,\mathbf{i}) + \boldsymbol{\upsilon}\_{d}^{'}(l,\mathbf{i})\boldsymbol{N}\_{d}(l,\mathbf{i})\boldsymbol{\upsilon}\_{d}(l,\mathbf{i})\} \\ & + \{\boldsymbol{\upsilon}\_{0} - \hat{\boldsymbol{\mathsf{x}}}\_{0}\}^{\prime} \boldsymbol{D}\_{0}(\mathbf{x}\_{0} - \hat{\boldsymbol{\mathsf{x}}}(\mathbf{0})) \\ & + \{\boldsymbol{\mathsf{x}}(\mathbf{N}) - \hat{\boldsymbol{\mathsf{x}}}\_{N}\}^{\prime} \boldsymbol{D}\_{N}(\mathbf{x}(\mathbf{N}) - \hat{\boldsymbol{\mathsf{x}}}\_{N}) \end{aligned} \tag{3}$$

where *x* ^ 0 is an initial estimate of *x0* and *x* ^ N is a terminal estimate of *x(N)*. *Md* (*l*, *i*)>*O*, *Nd* (*l*, *i*)≥*O*, *D*0 >*O* and *DN* >*O* are symmetric matrices which reflect the uncertainties on the noises *wd* ( ⋅ , ⋅ ) and *vd* ( ⋅ , ⋅ ) with the estimates *x* ^ 0 and *x* ^ N. Thus these performance indices mean the energies of noises, initial and terminal estimates under some uncertainties aver‐ aged by the mode distributions for each *r* ∈N0. We consider the optimization problems to decide *wd* ( ⋅ , *i*), *vd* ( ⋅ , *i*) and *r* ∈N0 minimizing *J*0*k* (*r*) and *J*0*N* (*r*) utilizing the known parts of the observed information Y*N* ={*y*(*l*)|0≤*l* ≤ *N* }.

We consider the optimal control problems to minimize *Jf*

(*r*)

(*r*)

(*k*, *x*, *wd* )

(*k*, *x*, *wd* )

(*r*) (*k*, *x*)

(*r*)

(*r*)

^ *f* (*k* ))

(*r*) (*k*, *x* ^ *f* (*r*) (*k*))

(*k*, *x*, *wd* , *f* (*r*)\* (*k*)) ≤ *Jf* (*r*)

(*r*)

(*r*) (*k*, *x* ^ *s* (*r*) (*k*))

(*r*)

^ *f* (*k*)) ≤*Vf*

(*r*)

(*k*, *x*): *x* ∈*R n*},

(*k*):*r* ∈N0}.

^ *s*(*k* ))

(*r*) (*k*, *x* ^ *f* (*r*) (*k*))

(*k*, *x*, *wd* , *f* (*r*)\* (*k*)) ≤ *Jf* (*r*)

( ⋅ ). Let *x* ^ *s*(*k*)= *x* ^ *s* (*r* ^ *s*(*k* ))

Now we define the following optimal estimators in the sense of most probable trajectory (MPT).

(*k*, *x*, *wd* (*k*)):*w* ∈*R n*}

(*k*, *x*, *wd* (*k*)):*w* ∈*R n*}

(*k*, *x*, *wd* (*k*)):*w* ∈*R n*}

(*k*):*r* ∈N0}.

( ⋅ ). Let *x*

^ *f* (*k*)= *x* ^ *f* (*r* ^ *f* (*k* ))

(*k*, *x*, *wd* (*k*)).

(*k*, *x*, *wd* (*k*)).

(*k*) and we have

(*k*) and we have

(*k*, *x*): *x* ∈*R n*},

(*k*, *x*) + *Vb*

(*r*)

(*r*)

(*r*)

(*k*): =arg min {*Vf*

(*r*) (*k*, *x* ^ *f* (*r*) (*k*))

(*k*): =*Vf*

*f* (*k*): =*arg*min{*Vf*

^ *f* (*k*)) ≤*Vf*

(*r*)

(*k*, *x*)= *Jf*

(*k*): =*arg*min{*Vs*

(*k*): =*Vs*

*Vs* (*r*)

*s*(*k*): =*arg*min{*Vs*

(*k*, *x*)= *Jf*

(*k*, *x*) and *Vb*

(*k*, *x*): = inf *wd* (⋅) *Jf* (*r*)

(*k*, *x*): = inf *wd* (⋅) *Jb* (*r*)

(*k*, *x*): =*Vf*

*x* ^ *f* (*r*)

*Vf* (*r*)

^

Then the most probable distribution is *ϕ* (*r*

(*r* ^ *f* (*k* )) (*k*, *x*

≤*Vf* (*r*)

*x* ^ *s* (*r*)

^

Then the most probable distribution is *ϕ* (*r*

(*r* ^ *f* (*k* )) (*k*, *x*

≤*Vf* (*r*)

(*k*): =arg min{*Jf*

(*k*): =arg min{*Jb*

(*k*): =arg min{*Js*

(*r*)

*Vf* (*r*)

*Vb* (*r*)

*Vs* (*r*)

*wd* , *f* (*r*)\*

*wd* ,*b* (*r*)\*

*wd* ,*s* (*r*)\*

parts of Y*N* . Let *Vf*

follows:

Then define

*r*

*Vf*

*r*

*Vf*

Also define

and

and

(*r*) and *Js* (*r*) = *Jf* (*r*) + *Jb* (*r*)

An Approach to Hybrid Smoothing for Linear Discrete-Time Systems with Non-Gaussian Noises

(*k*, *x*) be the value functions of these control problems as

for the given

167

http://dx.doi.org/10.5772/51385

Since the mode *θ(k)* at each time is inaccessible, we cannot directly design estimators for the system (1) including the unknown modes. Also, even if the modes are accessible, the com‐ putational complexity can exponentially increase with *k* if we directly design the estimators for the system (1) including *θ(k)* explicitly. Hence we introduce the system averaged through the mode distributions for each *r* ∈N0.

For notational simplicity, we adopt the following notation.

$$\overline{F}^{(r)}(k) = \sum\_{i=1}^{m} \phi\_i^{(r)}(k) F(k, i)$$

$$\text{for a matrix function } F(k, i) \text{ and } r \in \mathbb{N}\_0. \text{ Similarly } \overline{F\_1 F\_2}^{(r)}$$

for a matrix function *F(k,i)* and *r* ∈N0. Similarly *F*1*F*2 (*k*)= ∑ *i*=1 *m ϕi* (*r*) (*k*)*F*1(*k*, *i*)*F*2(*k*, *i*) for matrix functions *F1(k,i)* and *F2(k,i)* and so on. Using these notations, we can shift the drift term in the system (1) to *Ad* ¯(*r*) (*k*) as follows:

\*\*System\*\* (1)  $\mathcal{O}\,A\_d$  ( $n$ ) as 10000w.

$$\propto (k+1) = \overline{A\_d}^{(r)}(k)\,\text{x}(k) + w\_d(k)$$

where

Where
$$w\_d(k) = w\_d^{(r)}(k) = (A\_d(k, \theta(k)) - \overline{A\_d}^{(r)}(k))x(k) + w\_d(k, \theta(k)).$$

Bv replacing the system noise  $w\_d(k, i)$  by  $(\overline{A\_d}^{(r)}(k) - A\_d(k, i))x(k) + w\_d(k)$  and

By replacing the system noise *wd* (*k*, *i*) by (*Ad* (*k*)− *Ad* (*k*, *i*))*x*(*k*) + *wd* (*k*) and the observation noise *vd* (*k*, *i*) by *y*(*k*)−*Hd* (*k*, *i*)*x*(*k*) in the performance indices (2) and (3), we define

$$\begin{split} \mathcal{L} & \stackrel{(r)}{=} \langle k, \,\,\,\mathbf{x}, \,\,w\_{d'} \,\, \mathbf{y} \rangle \\ \coloneqq & \sum\_{i=1}^{m} \phi\_{i}^{(r)}(k) \underline{\mathbf{f}}(\overline{A\_{d}}^{(r)}(k) - A\_{d}(k, \,\,i)) \mathbf{x} + \boldsymbol{w}\_{d} \mathbf{1}^{'} \\ & \quad \times \boldsymbol{M}\_{d}(k, \,\,i) \underline{\mathbf{f}}(\overline{A\_{d}}^{(r)}(k) - A\_{d}(k, \,i)) \mathbf{x} + \boldsymbol{w}\_{d} \mathbf{1}^{'} \\ & \quad + (y - H\_{d}(k, \,i)\mathbf{x})^{'} \boldsymbol{N}\_{d}(k, \,i)(y - H\_{d}(k, \,i)\mathbf{x})). \end{split}$$

Then we can define the following performance indices:

$$\{J\_f^{(r)}(\mathbf{k}, \ x, \; w\_d(\cdot \cdot)) \coloneqq \sum\_{l=0}^{k-1} L^{(r)}(l, \ x(l), \; w\_d(l), \; y(l)) + \Phi\_0(\mathbf{x}(0))\tag{4}$$

$$\{f\_b^{(r)}(\mathbf{k}, \ x, w\_d(\cdot))\} \coloneqq \sum\_{l=k}^{N-1} L^{(r)}(l, \ge(l), \ w\_d(l), \ y(l)) + \Phi\_N(\ge(N))\tag{5}$$

$$J\_s^{(r)}(\mathbf{k}, \mathbf{x}, w\_d(\cdot \cdot)) \coloneqq J\_f^{(r)}(\mathbf{k}, \mathbf{x}, w\_d(\cdot \cdot)) + J\_b^{(r)}(\mathbf{k}, \mathbf{x}, w\_d(\cdot \cdot))$$

$$\text{where } \Phi\_0(\mathbf{x}(\cdot \cdot)) = (\mathbf{x}(\cdot) - \stackrel{\wedge}{\mathbf{x}\_0})^\prime D\_0(\mathbf{x}(\cdot \cdot) - \stackrel{\wedge}{\mathbf{x}\_0}) \text{ and } \Phi\_N(\mathbf{x}(\cdot \cdot)) = (\mathbf{x}(\cdot) - \stackrel{\wedge}{\mathbf{x}\_N})^\prime D\_N(\mathbf{x}(\cdot \cdot) - \stackrel{\wedge}{\mathbf{x}\_N}).$$

We consider the optimal control problems to minimize *Jf* (*r*) and *Js* (*r*) = *Jf* (*r*) + *Jb* (*r*) for the given parts of Y*N* . Let *Vf* (*r*) (*k*, *x*) and *Vb* (*r*) (*k*, *x*) be the value functions of these control problems as follows:

$$\begin{split} &V\_{f}^{(r)}(k,\ \mathbf{x}) := \inf\_{w\_{d}(\cdot)} \limits\_{w\_{d}(\cdot)} \langle k,\ \mathbf{x},w\_{d}\rangle \\ &V\_{b}^{(r)}(k,\ \mathbf{x}) := \inf\_{w\_{d}(\cdot)} I\_{b}^{(r)}(k,\ \mathbf{x},w\_{d}) \\ &V\_{s}^{(r)}(k,\ \mathbf{x}) := V\_{f}^{(r)}(k,\ \mathbf{x}) + V\_{b}^{(r)}(k,\ \mathbf{x}) \\ &w\_{d,f}^{(r)\*}(k) := \arg\min \Big[ \big| J\_{f}^{(r)}(k,\ \mathbf{x},w\_{d}(k)) : w \in \mathbb{R}^{n} \Big] \\ &w\_{d,b}^{(r)\*}(k) := \arg\min \Big[ \big| J\_{b}^{(r)}(k,\ \mathbf{x},w\_{d}(k)) : w \in \mathbb{R}^{n} \Big] \\ &w\_{d,s}^{(r)\*}(k) := \arg\min \Big[ \big| J\_{s}^{(r)}(k,\ \mathbf{x},w\_{d}(k)) : w \in \mathbb{R}^{n} \Big] \end{split}$$

Then define

$$\begin{aligned} \widehat{\boldsymbol{x}}\_{f}^{(r)}(k) &:= \arg\min \left\{ \boldsymbol{V}\_{f}^{(r)}(k,\,\,\mathbf{x}) : \boldsymbol{x} \in \mathbb{R}^{n} \right\}, \\ \boldsymbol{V}\_{f}^{(r)}(k) &:= \boldsymbol{V}\_{f}^{(r)}(k,\,\,\widehat{\boldsymbol{x}}\_{f}^{(r)}(k)) \end{aligned}$$

and

decide *wd* ( ⋅ , *i*), *vd* ( ⋅ , *i*) and *r* ∈N0 minimizing *J*0*k*

through the mode distributions for each *r* ∈N0.

*F*

¯(*r*)

*wd* (*k*)=*wd*

*x*(*k* + 1)= *Ad*

By replacing the system noise *wd* (*k*, *i*) by (*Ad*

(*k*, *x*, *wd* , *y*)

(*k*)( (*Ad* ¯(*r*)

Then we can define the following performance indices:

(*k*, *x*, *wd* ( ⋅ )):=∑

(*k*, *x*, *wd* ( ⋅ )):= ∑

(*k*, *x*, *wd* ( ⋅ )):= *Jf*

^ 0)′

*L* (*r*)

:=∑ *i*=1 *m ϕi* (*r*)

*Jf* (*r*)

*Jb* (*r*)

(*r*)

where Φ0(*x*( ⋅ ))=(*x*( ⋅ )− *x*

*Js*

system (1) to *Ad*

where

For notational simplicity, we adopt the following notation.

for a matrix function *F(k,i)* and *r* ∈N0. Similarly *F*1*F*2

(*k*) as follows:

¯(*r*)

(*r*)

(*k*)=∑ *i*=1 *m ϕi* (*r*)

¯(*r*)

(*k*)=(*Ad* (*k*, *θ*(*k*))− *Ad*

(*k*)− *Ad* (*k*, *i*))*x* + *wd*

′

(*k*, *x*, *wd* ( ⋅ )) + *Jb*

(*r*)

0) and Φ*N* (*x*( ⋅ ))=(*x*( ⋅ )− *x*

(*k*, *x*, *wd* ( ⋅ ))

^ *N* )′

¯(*r*)

×*Md* (*k*, *i*) (*Ad*

+ (*y* −*Hd* (*k*, *i*)*x*)

*l*=0 *k*−1 *L* (*r*)

*l*=*k N* −1 *L* (*r*)

(*r*)

^

*D*0(*x*( ⋅ )− *x*

noise *vd* (*k*, *i*) by *y*(*k*)−*Hd* (*k*, *i*)*x*(*k*) in the performance indices (2) and (3), we define

¯(*r*)

observed information Y*N* ={*y*(*l*)|0≤*l* ≤ *N* }.

166 Advances in Discrete Time Systems

(*r*)

(*k*)*F* (*k*, *i*)

functions *F1(k,i)* and *F2(k,i)* and so on. Using these notations, we can shift the drift term in the

(*k*)*x*(*k*) + *wd* (*k*)

¯(*r*)

′

(*k*)− *Ad* (*k*, *i*))*x* + *wd*

*Nd* (*k*, *i*)(*y* −*Hd* (*k*, *i*)*x*)).

(*k*))*x*(*k*) + *wd* (*k*, *θ*(*k*)).

(*l*, *x*(*l*), *wd* (*l*), *y*(*l*)) + Φ0(*x*(0)) (4)

(*l*, *x*(*l*), *wd* (*l*), *y*(*l*)) + Φ*N* (*x*(*N* )) (5)

*DN* (*x*( ⋅ )− *x*

^ *N* ).

(*k*)− *Ad* (*k*, *i*))*x*(*k*) + *wd* (*k*) and the observation

¯(*r*)

(*k*)= ∑ *i*=1 *m ϕi* (*r*)

Since the mode *θ(k)* at each time is inaccessible, we cannot directly design estimators for the system (1) including the unknown modes. Also, even if the modes are accessible, the com‐ putational complexity can exponentially increase with *k* if we directly design the estimators for the system (1) including *θ(k)* explicitly. Hence we introduce the system averaged

 and *J*0*N* (*r*)

utilizing the known parts of the

(*k*)*F*1(*k*, *i*)*F*2(*k*, *i*) for matrix

$$\stackrel{\wedge}{r}\_f(k) \coloneqq \arg\min \{ V\_f^{(r)}(k) \colon r \in \mathbb{N}\_0 \}.$$

Then the most probable distribution is *ϕ* (*r* ^ *f* (*k* )) ( ⋅ ). Let *x* ^ *f* (*k*)= *x* ^ *f* (*r* ^ *f* (*k* )) (*k*) and we have

$$\begin{aligned} &V\_{f}^{(r)}{}^{(r)}{}^{(k)}\{\mathbf{k},\ \stackrel{\frown}{\mathbf{x}}\_{f}(\mathbf{k})\} \leq V\_{f}^{(r)}{}^{(r)}\{\mathbf{k},\ \stackrel{\frown}{\mathbf{x}}\_{f}^{(r)}(\mathbf{k})\} \\ \leq &V\_{f}^{(r)}{}^{(r)}\{\mathbf{k},\ \mathbf{x}\} = \mathbf{J}\_{f}^{(r)}{}^{(r)}\{\mathbf{k},\ \mathbf{x},\ \stackrel{(r)\*}{w}\_{d,f}^{(r)}(\mathbf{k})\} \leq \mathbf{J}\_{f}^{(r)}{}^{(r)}\{\mathbf{k},\ \mathbf{x},\ \stackrel{\frown}{w}\_{d}(\mathbf{k})\}. \end{aligned}$$

Also define

$$\begin{aligned} \widehat{\boldsymbol{\mathfrak{X}}}\_{s}^{(r)}(k) &:= \arg\min \{ \boldsymbol{V}\_{s}^{(r)}(k, \boldsymbol{\mathfrak{x}}) : \boldsymbol{x} \in \mathbf{R}^{n} \}, \\ \boldsymbol{V}\_{s}^{(r)}(k) &:= \boldsymbol{V}\_{s}^{(r)}(k, \overset{\wedge}{\boldsymbol{\mathfrak{X}}}\_{s}^{(r)}(k)) \end{aligned}$$

and

$$\widehat{r}\_s(k) \coloneqq \arg\min \{ V\_s^{(r)}(k) \colon r \in \mathbb{N}\_0 \}.$$

Then the most probable distribution is *ϕ* (*r* ^ *s*(*k* )) ( ⋅ ). Let *x* ^ *s*(*k*)= *x* ^ *s* (*r* ^ *s*(*k* )) (*k*) and we have

$$\begin{aligned} &V\_{\boldsymbol{f}}^{(r)}(\boldsymbol{k},\mathop{\mathbf{x}}\_{\boldsymbol{f}}^{\star}(\boldsymbol{k})) \leq V\_{\boldsymbol{f}}^{(r)}(\mathop{\mathbf{k}},\mathop{\mathbf{x}}\_{\boldsymbol{f}}^{\star(\boldsymbol{r})}(\mathop{\mathbf{k}})) \\ \leq &V\_{\boldsymbol{f}}^{(r)}(\boldsymbol{k},\mathop{\mathbf{x}}) = \sf{J}\_{\boldsymbol{f}}^{(r)}(\mathop{\mathbf{k}},\mathop{\mathbf{x}},\mathop{\mathbf{w}}\_{\boldsymbol{d},\boldsymbol{f}}^{(r)\boldsymbol{\lambda}}(\mathop{\mathbf{k}})) \leq \sf{J}\_{\boldsymbol{f}}^{(r)}(\mathop{\mathbf{k}},\mathop{\mathbf{x}},\mathop{\mathbf{w}}\_{\boldsymbol{d}}(\mathop{\mathbf{k}})). \end{aligned}$$

Now we define the following optimal estimators in the sense of most probable trajectory (MPT).

#### **2.1 Definition**

Given the matrices *Md*, *Nd*, *D0* and *DN*, (*r* ^ *f* (*k*), *x* ^ *f* (*k*)), *k* ≥0, is called an optimal filter (in the MPT sense) if it minimizes *Vf* (*r*) (*k*, *x*). (*r* ^ *s*(*k*), *x* ^ *s*(*k*)), 0≤*k* ≤ *N* is called an optimal smoother (in the MPT sense) if it minimizes *Vs* (*r*) (*k*, *x*).

*enough.*

minimize *Jf*

*Vf* (*r*)

Let

where

**3.1 Optimal Hybrid Filtering**

(*k* + 1, *x*)=min

*wd*

*Vf* (*r*)

(*r*)

scalar equations with initial conditions:

(*k* + 1)=*Md*

(*k* + 1)= − *Hd*

(*k* + 1)= −*M*¯

(*r*) and *qf*

{*L* (*r*)

(*k*, *x*)= *x* ′

(*r*)∗(*k*, *x*)= *x* − *Ad*

(*k*)= *Ad* ′ *Md Ad*

¯(*r*)

*d Ad* (*r*) (*k*)*Sd*

¯(*r*)

*Nd*

'

(*r*)

for some functions *Kf*

minimizing *wd* ( ⋅ ).

*wd* , *f*

*Sd*

*Kf* (*r*)

*pf* (*r*)

*qf* (*r*) *Ad* ¯(*r*)

*If we assume that Ac(t, i) is uniformly bounded, then Ad*

(*k*)=∑ *i*=1 *m ϕi* (*r*)

= *I* + *h* ∑ *i*=1 *m ϕi* (*r*)

=∑ *i*=1 *m ϕi* (*r*)

with regard to *wd* ( ⋅ ) are given as follows:

¯(*r*),−1

(*k*, *Ad*

*Kf* (*r*)

¯(*r*)

¯(*r*)

(*k*)−*Md Ad*

¯(*r*)

(*r*) *Hd* '

(*k*)*y*(*k*)− *pf*

+ *y* '

(*k*)*Sd* (*r*) (*k*)(*Ad* ′ *Md*

(*k*) + *Hd* ′ *NdHd*

> (*k*)*Sd* (*r*) (*k*)*Ad* ′ *Md*

¯(*r*)

*Nd*

(*r*) (*k*) ' *Sd* (*r*) (*k*) *Hd* '

(*k*)*Nd* ¯(*r*)

(*k*)*x* + 2(*pf*

(*k*)*Ad* (*k*, *i*)

An Approach to Hybrid Smoothing for Linear Discrete-Time Systems with Non-Gaussian Noises

(*k*) *I* + *h Ac*(*kh* , *i*)

The dynamic programming (DP) equations associated with the forward control problem to

(*k*)*Ac*(*kh* , *i*).

*¯(r)*

(*k*)(*x* −*wd* ), *wd* , *y*(*k*)) + *Vf*

*Vf* (*r*)

¯(*r*)

+ *Kf* (*r*) (*k*) −1 .

¯(*r*)

(*k*)*y*(*k*)− *pf*

(*k*)*y*(*k*) + *qf*

(*r*) (*k*)) ′ *x* + *qf* (*r*)

¯

Then we obtain the following matrix difference equations, forward vector equations and

(*r*) (*k*, *x*)}

(0, *x*)=Φ0(*x*), *r* ∈N0

(*r*) with appropriate dimensions. Then we obtain the following

′ *Nd*

¯(*r*)

(*k*)*x* + *Hd*

(*k*), *Kf* (*r*)

(*r*) (*k*) , *pf* (*r*)

*Nd*

(*r*) (*k*), *qf* (*r*) (0)= *x* ^ 0*D*0*x* ^ 0

¯(*r*)

*(k), k = 0, 1, ⋯ is invertible for h small*

http://dx.doi.org/10.5772/51385

169

(*k*) (6)

(*k*)*y*(*k*)− *pf*

(*r*) (*k*))

(0)=*D*0 (7)

0 (8)

(9)

(0)= −*D*0*x* ^

> (*r*) (*k*)

(*k*)*y*(*k*)− *pf*

Then we formulate the following optimal hybrid estimation problems for the performance indices (4) and (5).

#### **The Optimal Hybrid Filtering Problem for Linear Discrete-Time Systems:**

Find the pair (*r* ^ *f* (*l*), *x* ^ *f* (*r* ^ *f* (*l*)) (*l*)), *l* ∈ 0, *k* minimizing the performance index (4) based on the causal part Y*k* ={*y*(*l*)|0≤*l* ≤*k*} of the observed information Y*N* .

#### **The Optimal Hybrid Smoothing Problem for Linear Discrete-Time Systems:**

Find the pair (*r* ^ *s*(*k*), *x* ^ *s* (*r* ^ *s*(*k* )) (*k*)), *k* ∈ 0, *N* minimizing the performance index (4)+(5) based on the whole observed information Y*N* .

**Remark 2.1.** *In general, if we directly adopt dynamic programming (DP) method for mode-depend‐ ent systems, it can arise that computational complexity increases exponentially with time k ([5,11]). On the other hand in this chapter we consider the averaged systems and averaged performance indices for them with regard to the candidates of the mode distributions. Note that this introduction of the averaged systems and performance indices prevents the computational complexity from increasing ex‐ ponentially by applying the dynamical programming (DP) method as seen in the next section.*