**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 *<sup>r</sup>* <sup>∈</sup>N<sup>0</sup> ={1, 2, <sup>⋯</sup>, *<sup>n</sup>*0}, and let P={*<sup>ϕ</sup>* (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* ): <sup>=</sup>∑

(*x*0, *wd* , *vd* ): <sup>=</sup> ∑

*i*=1 *m ϕi* (*r*) (*k*)=1.

with *ϕ<sup>i</sup>* (*r*)

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

*J*0*<sup>N</sup>* (*r*)

where *x* ^ 0

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

where *<sup>x</sup>* <sup>∈</sup>**Rn** is the state, *wd* <sup>∈</sup>**Rn** is the exogenous random noise, *vd* <sup>∈</sup>**R<sup>k</sup>** 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*<sup>0</sup> − *x*

^ *N* )′

+ (*x*<sup>0</sup> − *x* ^ 0)′

+ (*x*<sup>0</sup> − *x* ^ 0)′

+ (*x*(*N* )− *x*

*Nd* (*l*, *i*)≥*O*, *D*<sup>0</sup> >*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

^ <sup>0</sup> 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*<sup>0</sup> − *x*

^(0))

( ⋅ ), ⋯, *ϕ*(*n*0)

(*k*)=(*ϕ*<sup>1</sup> (*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*), ⋯, *ϕ<sup>m</sup>*

(*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 *<sup>x</sup>* <sup>∈</sup>**Rn** is the state, *wd* <sup>∈</sup>**Rn** is the exogenous random noise, *vd* <sup>∈</sup>**R<sup>k</sup>** 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 *<sup>r</sup>* <sup>∈</sup>N<sup>0</sup> ={1, 2, <sup>⋯</sup>, *<sup>n</sup>*0}, and let P={*<sup>ϕ</sup>* (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*<sup>1</sup> (*r*) (*k*), ⋯, *ϕ<sup>m</sup>* (*r*) (*k*)) with *ϕ<sup>i</sup>* (*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*<sup>0</sup> >*O* and *DN* >*O* are symmetric matrices which reflect the uncertainties on the noises *wd* ( ⋅ , ⋅ ) and *vd* ( ⋅ , ⋅ ) with the estimates *x* ^ <sup>0</sup> 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* ∈N<sup>0</sup> minimizing *J*0*<sup>k</sup>* (*r*) and *J*0*<sup>N</sup>* (*r*) utilizing the known parts of the observed information Y*<sup>N</sup>* ={*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*)

^ *<sup>f</sup>* (*k* ))

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

(*k*, *x*, *wd* , *<sup>f</sup>* (*r*)\* (*k*)) ≤ *Jf* (*r*)

(*r*)

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

(*r*)

^ *<sup>f</sup>* (*k*)) <sup>≤</sup>*Vf*

(*r*)

(*k*, *x*): *x* ∈*R <sup>n</sup>*},

(*k*):*r* ∈N0}.

^ *<sup>s</sup>*(*k* ))

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

(*k*, *x*, *wd* , *<sup>f</sup>* (*r*)\* (*k*)) ≤ *Jf* (*r*)

( ⋅ ). Let *x* ^ *<sup>s</sup>*(*k*)= *x* ^ *s* (*r* ^ *<sup>s</sup>*(*k* ))

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

(*k*, *<sup>x</sup>*, *wd* (*k*)):*<sup>w</sup>* <sup>∈</sup>*<sup>R</sup> <sup>n</sup>*}

(*k*, *<sup>x</sup>*, *wd* (*k*)):*<sup>w</sup>* <sup>∈</sup>*<sup>R</sup> <sup>n</sup>*}

(*k*, *<sup>x</sup>*, *wd* (*k*)):*<sup>w</sup>* <sup>∈</sup>*<sup>R</sup> <sup>n</sup>*}

(*k*):*r* ∈N0}.

( ⋅ ). Let *x*

^ *<sup>f</sup>* (*k*)= *<sup>x</sup>* ^ *f* (*r* ^ *<sup>f</sup>* (*k* ))

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

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

(*k*) and we have

(*k*) and we have

(*k*, *x*): *x* ∈*R <sup>n</sup>*},

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

(*r*)

(*r*)

(*r*)

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

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

(*k*): =*Vf*

*<sup>f</sup>* (*k*): =*arg*min{*Vf*

^ *<sup>f</sup>* (*k*)) <sup>≤</sup>*Vf*

(*r*)

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

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

(*k*): =*Vs*

*Vs* (*r*)

*<sup>s</sup>*(*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* (*<sup>r</sup>*

(*r* ^ *<sup>f</sup>* (*k* )) (*k*, *x*

≤*Vf* (*r*)

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

^

Then the most probable distribution is *ϕ* (*<sup>r</sup>*

(*r* ^ *<sup>f</sup>* (*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* , *<sup>f</sup>* (*r*)\*

*wd* ,*<sup>b</sup>* (*r*)\*

*wd* ,*<sup>s</sup>* (*r*)\*

parts of Y*<sup>N</sup>* . Let *Vf*

follows:

Then define

*r*

*Vf*

*r*

*Vf*

Also define

and

and

(*r*) and *Js* (*r*) = *Jf* (*r*) <sup>+</sup> *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*<sup>2</sup> (*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*) <sup>+</sup> *Jb* (*r*) for the given parts of Y*<sup>N</sup>* . 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* ∈N<sup>0</sup> minimizing *J*0*<sup>k</sup>*

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*, *<sup>x</sup>*, *wd* ( <sup>⋅</sup> )):=∑

(*k*, *<sup>x</sup>*, *wd* ( <sup>⋅</sup> )):= ∑

(*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*<sup>2</sup>

(*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*<sup>N</sup>* (*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*<sup>N</sup>* ={*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*<sup>N</sup>* (*x*(*N* )) (5)

*DN* (*x*( ⋅ )− *x*

^ *<sup>N</sup>* ).

(*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*<sup>N</sup>* (*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* (*<sup>r</sup>* ^ *<sup>f</sup>* (*k* )) ( ⋅ ). Let *x* ^ *<sup>f</sup>* (*k*)= *<sup>x</sup>* ^ *f* (*r* ^ *<sup>f</sup>* (*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* (*<sup>r</sup>* ^ *<sup>s</sup>*(*k* )) ( ⋅ ). Let *x* ^ *<sup>s</sup>*(*k*)= *x* ^ *s* (*r* ^ *<sup>s</sup>*(*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* ^ *<sup>f</sup>* (*k*), *x* ^ *<sup>f</sup>* (*k*)), *<sup>k</sup>* <sup>≥</sup>0, is called an optimal filter (in the MPT sense) if it minimizes *Vf* (*r*) (*k*, *x*). (*r* ^ *<sup>s</sup>*(*k*), *x* ^ *<sup>s</sup>*(*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*, *<sup>x</sup>*)= *<sup>x</sup>* <sup>−</sup> *Ad*

(*k*)= *Ad* ′ *Md Ad*

¯(*r*)

*<sup>d</sup> Ad* (*r*) (*k*)*Sd*

¯(*r*)

*Nd*

'

(*r*)

for some functions *Kf*

minimizing *wd* ( ⋅ ).

*wd* , *<sup>f</sup>*

*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*) <sup>−</sup><sup>1</sup> .

¯(*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* ∈N<sup>0</sup>

(*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* ^ <sup>0</sup>*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*<sup>0</sup> (7)

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

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

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

**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.*
