**Appendix: Principle of Hybrid Optimality**

With regard to the optimal control problems considered in this chapter, it is obvious that principle of optimality does not hold for the optimal trajectory *x* <sup>∗</sup>( ⋅ ) with optimal control input *wd* (*r*)∗( ⋅ ) and each performance index for each mode distribution candidate *r* ∈N0. However, for the pair (*r* ^( <sup>⋅</sup> ), *<sup>x</sup>*( <sup>⋅</sup> )) of the optimal mode distribution candidate and optimal trajectory with the optimal control inputs *wd* (*r*)∗( ⋅ ), the following principles of hybrid opti‐ mality hold. These principles give a basis of validity for the hybrid estimation algorithms presented in this chapter.

Consider the following system

**Figure 7.** The square errors between the state and filtered

**Figure 9.** The filtered mode distributions **Figure 10.** The smoothed mode distributions

In this chapter we have studied the state and mode estimation problems for linear discretetime hybrid systems over the fixed time interval. The systems aren''t restricted to the Marko‐ vian jump systems and the added noises aren''t restricted to be Gaussian. With regard to concrete examples of the systems considered in this chapter, refer to [24,25]. Those examples show that the systems and estimation algorithms presented in this chapter cover extreme broad classes of dynamical systems affected by the noises not to be restricted to be Gaussian. We have adopted the MPT approach. The state and mode estimation approach adopted in this paper guarantees the optimality of estimation performance in the meaning of MPT dif‐

In this chapter we have considered the problems that both system state and modes are esti‐ mated. However we have considered the problems that the distributions of the modes over the fixed time interval not the modes themselves are estimated to grasp the global behavior of the hybrid systems over the long time intervals. In order to estimate both the system state and distributions of the modes we have introduced the averaged performance indices with respect to the candidates of the mode distributions for the averaged systems. This introduc‐

**Figure 8.** The square errors between the state and

smoothed values: 2nd components

values: 2nd components

176 Advances in Discrete Time Systems

**5. Concluding Remarks**

ferent from the previous work ([4,7]).

$$
\mathbf{x}(k+1) = \overline{A}\_d^{(r)}(k)\mathbf{x}(k) + w\_d(k) \tag{20}
$$

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

and the performance indices

$$\mathbf{U}\_f^{(r)}(\mathbf{k},\ \mathbf{x},\ \mathbf{w}\_d(\cdot \ \cdot)) = \sum\_{l=0}^{k-1} L^{(r)}(l,\ \mathbf{x}(l),\ \mathbf{w}\_d(l),\ \mathbf{y}(l)) + \Phi\_0(\mathbf{x}(0)) \tag{21}$$

*for the system (20) with the initial state x(0) on the partial time interval [0,τ] and then let the pair of*

**Proposition B (Principle of Hybrid Optimality (B))** *Consider the optimal control problem (B) on the time interval [k,N]. Also consider the optimal control problem minimizing the performance index*

*for the system (20) with the initial state x(τ) on the partial time interval τ, N and then let the pair*

**Theorem C (Principle of Hybrid Optimality (C))** *Consider the optimal control problem (C) on the fixed time interval [0,N]. Split the performance index (23) into the two parts (21) and (22) and also consider the optimal control problems (A) on [0,k] and (B) on [k,N] for the system (20) with initial state*

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

*(l)), l ∈ k + 1, N hold.*

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

(*l*)) *l* ∈ 0, *τ*

) *on* 0, *τ*

) and so the pair of control inputs consisting of (*r*

^

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

(*l*)) *l* ∈ *τ* + 1, *k*

In this appendix we give only a proof of Proposition 7.1. The others can be shown by the

^

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

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

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

) gives less value of the performance index (21) than (*r*

) , *l* ∈ 0, *τ* holds. (Q.E.D.).

*l ∈ k, N are optimal input minimizing the values of the (21) and (22) to be used in order to compose the solution of the fixed-interval optimal control problem (C), i.e., any time k ∈ 0, N*

*(l)) , l ∈ τ, N holds.*

*^*

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

*(l).), l ∈ 0, τ . Then (r*

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

(*l*, *x*(*l*), *wd* (*l*), *y*(*l*)) + Φ*<sup>N</sup>* (*x*(*N* )), *k* <*τ* < *N* −1 (25)

*(l)) , l ∈ 0, k and (r*

*(l)), l ∈ 0, k and*

*^*

*(l)) , l ∈ τ, N . Then*

*^*

(*l*)), *l* ∈ 0, *k* of the optimal control

) giving less value of the performance

^

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

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

(*l*)), *l* ∈ 0, *k* . Therefore

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

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

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

179

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

*optimal control inputs be (r*

*(l)) , l ∈ 0, τ holds.*

*l*=*τ N* −1 *L* (*r*)

*of optimal control inputs be (r*

*x(0) and x(k) respectively. Then at each time k(r*

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

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

**(Proof of Proposition A)** We split the pair (*r*

(*r* ^

(*r* ^

^

Then there exists the pair of control inputs (*r*

*<sup>f</sup>* ,1, *wd* , *<sup>f</sup>* (*r* ^ f,1,)\*

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

^

*<sup>f</sup>* ,1, *wd* , *<sup>f</sup>* ,1 (*r* ^ *<sup>f</sup>* ,1)\* )=(*r* ^

*<sup>f</sup>* ,2, *wd* , *<sup>f</sup>* ,2 (*r* ^ *<sup>f</sup>* ,2)\* )=(*r* ^

*<sup>f</sup>* ,1, *wd* , *<sup>f</sup>* ,1 (*r* ^ *<sup>f</sup>* ,1)\* )≠(*r* ^ ^ *<sup>f</sup>* , *wd* , *<sup>f</sup>* (*r* ^^ *<sup>f</sup>* )\*\*

*l* ∈ 0, *k* . This contradicts with the optimality of (*r*

*(l)) = (r ^*

*(l)) = (r ^*

inputs into the following two parts:

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

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

*(r ^ ^*

*(r ^*

*(r ^*

*(r ^*

(*r* ^

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

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

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

similar arguments.

Now we assume

index (24) than (*r*

*<sup>f</sup>* ,2, *wd* , *<sup>f</sup>* ,2 (*r* ^ *<sup>f</sup>* ,2)\*

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

(*r*

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

> *Jb* (*r*)

$$\left(\Box f\_{b}^{(r)}(\mathbf{k},\ \mathbf{x},\ \mathbf{w}\_{d}(\cdot \ \cdot))\right) = \sum\_{l=k}^{N-1} L^{\ast(r)}(l,\ \mathbf{x}(l),\ \mathbf{w}\_{d}(l),\ \mathbf{y}(l)) + \Phi\_{N}(\mathbf{x}(N))\tag{22}$$

$$J\_s^{(r)}(\mathbf{k}, \; \mathbf{x}, \; w\_d(\cdot \cdot)) = J\_f^{(r)}(\mathbf{k}, \; \mathbf{x}, \; w\_d(\cdot \cdot)) + J\_b^{(r)}(\mathbf{k}, \; \mathbf{x}, \; w\_d(\cdot \cdot)) \tag{23}$$

where

$$\begin{split} \mathbf{L} \stackrel{(r)}{=} & (\mathbf{k}, \mathbf{\color[rgb]{0,0,0}{0}} \mathbf{x}\_{d'}, \mathbf{\color[rgb]{0,0,0}{0}} \mathbf{y}) \\ = & \sum\_{i=1}^{\overline{\phantom{0}{\phantom{0}{\phantom{0}}}}} \phi\_{i}^{(r)}(\mathbf{k}) \mathbf{(}\mathbf{i}(\overline{\phantom{0}{\phantom{0}}}^{\phantom{0}{\phantom{0}}}(\mathbf{k}) - A\_{d}(\mathbf{k}, \mathbf{i})) \mathbf{x} + \mathbf{w}\_{d} \mathbf{1} ) \\ & \times M\_{d}(\mathbf{k}, \mathbf{i}) \mathbf{f}(\overline{A\_{d}}^{\phantom{0}{\phantom{0}}}(\mathbf{k}) - A\_{d}(\mathbf{k}, \mathbf{i})) \mathbf{x} + \mathbf{w}\_{d} \mathbf{1} ) \\ & \quad + (\mathbf{y} - H\_{d}(\mathbf{k}, \mathbf{i}) \mathbf{x}) \mathbf{<} N\_{d}(\mathbf{k}, \mathbf{i}) (\mathbf{y} - H\_{d}(\mathbf{k}, \mathbf{i}) \mathbf{x})). \end{split}$$

We consider the following three optimal control problems for the system (20) and the per‐ formance indices (21)-(23):

#### **Problem (A): Forward Optimal Control Problem**

Consider the system (20) with the initial state *x*(0). Find the pair (*r* ^ *<sup>f</sup>* (*l*), *wd* , *<sup>f</sup>* (*r* ^ *<sup>f</sup>* (*l*))\* (*l*)) , *l* ∈ 0, *k* minimizing the value of the performance index (21) based on the causal part Y*<sup>k</sup>* ={*y*(*l*)|0≤*l* ≤*k*} of the observed information Y*N* .

#### **Problem (B): Backward Optimal Control Problem**

Consider the system (20) with the initial state *x*(*k*). Find the pair (*r* ^ *<sup>b</sup>*(*l*), *wd* ,*<sup>b</sup>* (*r* ^ *<sup>b</sup>*(*l*))\* (*l*)), *l* ∈ *k*, *N* minimizing the value of the performance index (22) based on the anti-causal part *Y*¯ *<sup>k</sup>* ={*y*(*l*)|*k* ≤*l* ≤ *N* } of the observed information Y*N* .

#### **Problem (C): Fixed-Interval Optimal Control Problem**

Consider the system (20) with the initial state *x*(0). Find the pair (*r* ^ *<sup>s</sup>*(*k*), *wd* ,*<sup>s</sup>* (*r* ^ *<sup>s</sup>*(*k* ))\* (*k*)) , *k* ∈ 0, *N* minimizing the value of the performance index (23) based on the whole observed information Y*N* .

**Proposition A (Principle of Hybrid Optimality (A))** *Consider the optimal control problem (A) on the time interval [0,k]. Also consider the optimal control problem minimizing the performance index*

$$\{f\_f^{(r)}(\tau,\infty,w\_d(\cdot)) = \sum\_{l=0}^{\tau-1} L^{(r)}(l,\ge(l), w\_d(l),\ y(l)) + \Phi\_0(\ge(0)), \ 1 \le \tau \le \mathbf{k} \tag{24}$$

*for the system (20) with the initial state x(0) on the partial time interval [0,τ] and then let the pair of optimal control inputs be (r ^ ^ <sup>f</sup> (l) wd , <sup>f</sup> (r ^^ <sup>f</sup> (l))\*\* (l).), l ∈ 0, τ . Then (r ^ <sup>f</sup> (l), wd , <sup>f</sup> (r ^ <sup>f</sup> (l))\* (l))= (r ^ ^ <sup>f</sup> (l), wd , <sup>f</sup> (r ^^)\*\* (l)) , l ∈ 0, τ holds.*

**Proposition B (Principle of Hybrid Optimality (B))** *Consider the optimal control problem (B) on the time interval [k,N]. Also consider the optimal control problem minimizing the performance index*

$$\mathbb{E}\left[J\_b^{(r)}(\tau,\infty,\varpi\_d(\cdot,\cdot))=\sum\_{l=\tau}^{N-1}L^{(r)}(l,\ge(l),\ w\_d(l),\ w\_d(l))+\varPhi\_N(\ge(N)),\ k\le\tau\le N-1\tag{25}$$

*for the system (20) with the initial state x(τ) on the partial time interval τ, N and then let the pair of optimal control inputs be (r ^ ^ <sup>b</sup>(l)wd ,b (r ^^ <sup>b</sup>(l))\*\* (l)) , l ∈ τ, N . Then (r ^ <sup>b</sup>(l), wd ,b (r ^ <sup>b</sup>(l))\* (l)) = (r ^ ^ <sup>b</sup>(l)wd ,b (r ^^ <sup>b</sup>(l))\*\* (l)) , l ∈ τ, N holds.*

**Theorem C (Principle of Hybrid Optimality (C))** *Consider the optimal control problem (C) on the fixed time interval [0,N]. Split the performance index (23) into the two parts (21) and (22) and also consider the optimal control problems (A) on [0,k] and (B) on [k,N] for the system (20) with initial state x(0) and x(k) respectively. Then at each time k(r ^ <sup>f</sup> (l) wd , <sup>f</sup> (r ^ <sup>f</sup> (l))\* (l)) , l ∈ 0, k and (r ^ <sup>b</sup>(l), wd ,b (r ^ <sup>b</sup>(l))\* (l)), l ∈ k, N are optimal input minimizing the values of the (21) and (22) to be used in order to compose the solution of the fixed-interval optimal control problem (C), i.e., any time k ∈ 0, N (r ^ <sup>s</sup>(l), wd ,s (r ^ <sup>f</sup> (l))\* (l)) = (r ^ <sup>f</sup> (l), wd , <sup>f</sup> (r ^ <sup>f</sup> (l))\* (l)), l ∈ 0, k and (r ^ <sup>s</sup>(l), wd ,s (r ^ <sup>f</sup> (l))\* (l)) = (r ^ <sup>b</sup>(l), wd , <sup>f</sup> (r ^ <sup>b</sup>(l))\* (l)), l ∈ k + 1, N hold.*

In this appendix we give only a proof of Proposition 7.1. The others can be shown by the similar arguments.

**(Proof of Proposition A)** We split the pair (*r* ^ *<sup>f</sup>* (*l*)*wd* , *<sup>f</sup>* (*r* ^ *<sup>f</sup>* (*l*))\* (*l*)), *l* ∈ 0, *k* of the optimal control inputs into the following two parts:

$$\begin{array}{ll} \mbox{(\hat{\boldsymbol{r}}\_{\boldsymbol{f},1\boldsymbol{\omega}}\mbox{\ \boldsymbol{w}}\_{d,\boldsymbol{f},\boldsymbol{\jmath}^{\boldsymbol{\omega}}\mbox{\ \ 1}}^{\mbox{(\hat{\boldsymbol{r}}\_{\boldsymbol{f},1\boldsymbol{\upbeta}}\mbox{\ \ 1})^{\star}}\mbox{)} = (\mbox{\stackrel{\scriptstyle\mbox{\}}{}}\mbox{\ \ \{\}}\mbox{\ \ \{\}}\mbox{\ \ \{\}}\mbox{\ \ \{\}}\mbox{\ \ \{\}}\mbox{)}\mbox{\ \ \{\}}\mbox{\ \ \{\}}\mbox{\ \ \{\}}\mbox{\ \ \{\}}\mbox{\ \ \{\}}\mbox{)}\mbox{\ \ \{\}}\mbox{\ \ \{\}}\mbox{\ \ \{\}}\mbox{\ \ \{\}}\mbox{\ \ \{\}}\mbox{\ \ \{\}}\mbox{\ \ \{\}}\mbox{\ \ \{\}}\mbox{\ \ \{\}}\mbox{\ \ \{\}}\mbox{\ \ \{\}}\mbox{\ \ \{\}}\mbox{\ \end{array}$$

Now we assume

*Jf* (*r*)

178 Advances in Discrete Time Systems

*Jb* (*r*)

formance indices (21)-(23):

where

*Y*¯

information Y*N* .

*Jf* (*r*) *Js* (*r*)

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

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

*L* (*r*)

**Problem (A): Forward Optimal Control Problem**

Y*<sup>k</sup>* ={*y*(*l*)|0≤*l* ≤*k*} of the observed information Y*N* .

**Problem (B): Backward Optimal Control Problem**

*<sup>k</sup>* ={*y*(*l*)|*k* ≤*l* ≤ *N* } of the observed information Y*N* .

**Problem (C): Fixed-Interval Optimal Control Problem**

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

*l*=0 *τ*−1 *L* (*r*)

=∑ *i*=1 *m ϕi* (*r*)

(*k*, *x*, *wd* ( ⋅ ))= *Jf*

*l*=0 *k*−1 *L* (*r*)

*l*=*k N* −1 *L* (*r*)

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

(*k*)( (*Ad* ¯(*r*)

Consider the system (20) with the initial state *x*(0). Find the pair (*r*

Consider the system (20) with the initial state *x*(*k*). Find the pair (*r*

Consider the system (20) with the initial state *x*(0). Find the pair (*r*

(*r*)

(*k*, *x*, *wd* ( ⋅ )) + *Jb*

(*k*)− *Ad* (*k*, *i*))*x* + *wd*

′

We consider the following three optimal control problems for the system (20) and the per‐

minimizing the value of the performance index (21) based on the causal part

minimizing the value of the performance index (22) based on the anti-causal part

*k* ∈ 0, *N* minimizing the value of the performance index (23) based on the whole observed

**Proposition A (Principle of Hybrid Optimality (A))** *Consider the optimal control problem (A) on the time interval [0,k]. Also consider the optimal control problem minimizing the performance index*

¯(*r*)

×*Md* (*k*, *i*) (*Ad*

+ (*y* −*Hd* (*k*, *i*)*x*)

(*r*)

′

(*k*)− *Ad* (*k*, *i*))*x* + *wd*

^

^

(*l*, *x*(*l*), *wd* (*l*), *y*(*l*)) + Φ0(*x*(0)), 1<*τ* <k (24)

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

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

^

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

(*l*)) , *l* ∈ 0, *k*

(*l*)), *l* ∈ *k*, *N*

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

(*l*, *x*(*l*), *wd* (*l*), *y*(*l*)) + Φ0(*x*(0)) (21)

(*l*, *x*(*l*), *wd* (*l*), *y*(*l*)) + Φ*<sup>N</sup>* (*x*(*N* )) (22)

(*k*, *x*, *wd* ( ⋅ )) (23)

$$(\stackrel{\diamond}{r}\_{f,1^{\prime}} \; w\_{d,f,1^{\prime}}^{(\hat{r}\_{f,1^{\prime}})^{\ast}}) \star (\stackrel{\diamond}{r}\_{f^{\prime}} \; w\_{d,f}^{(\hat{r}\_{f,1^{\prime}})^{\ast \ast}}) \; on \; \mathsf{f}\,\mathsf{0},\ \mathsf{r}\,\mathsf{J}$$

Then there exists the pair of control inputs (*r* ^ ^ *<sup>f</sup>* , *wd* , *<sup>f</sup>* (*r* ^ *<sup>f</sup>* )\*\* ) giving less value of the performance index (24) than (*r* ^ *<sup>f</sup>* ,1, *wd* , *<sup>f</sup>* (*r* ^ f,1,)\* ) and so the pair of control inputs consisting of (*r* ^ ^ *f* , *wd* , *<sup>f</sup>* (*r* ^^ *<sup>f</sup>* ,)\*\* ) and (*r* ^ *<sup>f</sup>* ,2, *wd* , *<sup>f</sup>* ,2 (*r* ^ *<sup>f</sup>* ,2)\* ) gives less value of the performance index (21) than (*r* ^ *<sup>f</sup>* (*l*), *wd* , *<sup>f</sup>* (*r* ^ *<sup>f</sup>* ,(*l*))\* (*l*)) , *l* ∈ 0, *k* . This contradicts with the optimality of (*r* ^ *<sup>f</sup>* (*l*), *wd* , *<sup>f</sup>* (*r* ^ *<sup>f</sup>* ,(*l*))\* (*l*)), *l* ∈ 0, *k* . Therefore (*r* ^ *<sup>f</sup>* (*l*), *wd* , *<sup>f</sup>* (*r* ^ *<sup>f</sup>* )\* )=(*r* ^ ^ *<sup>f</sup>* (*l*), *wd* , *<sup>f</sup>* (*r* ^^ *<sup>f</sup>* (*l*))\*\* ) , *l* ∈ 0, *τ* holds. (Q.E.D.).
