数理解析研究所講究録
第 822 巻 1993 年 14-26

14Dual Connections towards Information Geometry of
Stable State Feedback Systems

Atsumi Ohara (小原敦美)

Department of Systems Engineering, Osaka University
1-1, Machikane-yama, Toyonaka, Osaka 560, JAPAN
Abstract- This paper gives new approach to investigate differential geometric structuresof stable and stable state feedback systems. For this purpose pairs of dual connections are
introduced. Some of these connections are found to characterize the geometric structuresof stable state feedback systems well.
1. Introduction
There have been many differential geometric approaches to investigate structures of linear(dynamical) systems (e.g. $[1]-[3]$ ). However, it seems that differential geometric structuresof feedback systems, which are very important in the engineering sense, have not been
deeply studied yet.
As is widely known, the parametrization of stabilizing controllers by Youla et al [4] andKucera [5] has given great advantage to the control theory and its applications. Since
designers of controllers must optimize various performance indices on the parametrized setof stabilizing controllers, studying its geometric structures gives useful insights.

Recently, it has been shown that the set of stable matrices and the set of stabilizing
state feedback gains are diffeomorphic to some kind of vector bundles, and the set of state
feedback system matrices are parametrically imbedded as a submanifold of this vector
bundle [6].
In this paper, we define metrics and connections on these vector bundles to analyze thedifferential geometric structures of stable state feedback systems. The main tool used hereis the theory of dual connections, what is called Information Geometry, extensively studied
by Amari [8] in statistics. Using this theory, we can reveal the simple structures of the setsof stable systems and stabilizing state feedback gains, which can not be elucidated by usual
Riemannian geometry.
In this paper, $PD(n),$ $Skew(n)$ and $S(n)$ denote, respectively, the set of positive definite
real matrices, skew symmetric real matrices $(M=-M^{T})$ and stable real matrices (all the
eigenvalues are located in the open left half complex plane), the size of which are all by
.
$n$

$n$

2. Parametrization of Stabilizing State Feedback Gains and Stable matrices

This section will give parametrizations of stabilizing state feedback gains and stable
matrices using Lyapunov equations. Complete derivations of the results in this section canbe found in [6].

15Consider an n-dimensional linear system with $m$ inputs represented by a state spaceequation:(2.1)
dr $=Ax+Bu$ ,where $(A, B)$ is controllable and is of column full rank.
Letdenote the set of stabilizing state feedback gains $F$ , i.e., $\mathcal{F}_{S}(A, B):=\{F|A+$$B$$\mathcal{F}_{S}(A, B)$$BF\in S(n)\}$.

Definition
be in $PD(n)$ . The set of positive definite matrices
1. Letequation:$Q$$P$

that satisfy the following(2.2)$(I-BB^{\uparrow})(AP+PA^{T}+Q)(I-BB^{\uparrow})=0$

is denoted by $PD(n;A, B, Q)$ .
2. The set of skew symmetric matrices$S$$BB^{\uparrow}SBB^{\uparrow}=S$

that satisfy the following equation:, (or equivalently $BB^{\uparrow}S=S$ )(2.3)

is denoted by Skew $(n;B)$ .
Here,
represents a pseudo inverse of matrix.
$\bullet\dagger$$B)$ )
Proposition 1: (Parametrization ofi) Let be in $PD(n)$ . All stabilizing state feedback gains
by$\mathcal{F}_{S}(A,$$Q$$F\in \mathcal{F}_{S}(A, B)$

are parametrized$F=-B^{\uparrow}(AP+PA^{T}+Q)(I- \frac{1}{2}BB^{\uparrow})P^{-1}-B^{\uparrow}SP^{-1}$(2.4)

using $P\in PD(n;A, B, Q)$ and $S\in Skew(n;B)$ .
ii) The mapping: $PD(n;A, B, Q)\cross Skew(n;B)arrow \mathcal{F}_{S}(A, B)$ defined by (2.4) is diffeomorphic.
The feedback gain represented as (2.4) satisfies the following Lyapunov equation:$\psi_{Q}$$F$$\ddot{u}i)$$(A+BF)P+P(A+BF)^{T}+Q=0$
for $P\in PD(n;A, B, Q)$ that is just parametrizing$F$in (2.4).

Proposition 2:i) The set $PD(n;A, B, Q)$ is an $m(2n-m+1)/2(=:N_{P})$ -dimensional submanifold of $PD(n)$ .
ii) The set Skew $(n;B)$ is an $m(m-1)/2(=:N_{s})$ -dimensional vector subspace of Skew $(n)$ .
Proposition 3: (Parametrization of $S(n)$ )i) Letbe in $PD(n)$ . All stable matrices $A_{s}\in S(n)$ are parametrized by$Q$$A_{s}=- \frac{1}{2}QP^{-1}+SP^{-1}$using $P\in PD(n)$ and $S\in Skew(n)$ .
ii) The mapping: $PD(n)\cross Skew(n)arrow S(n)$ defined by (2.5) is diffeomorphic.
$\phi_{Q}$

(2.5)

16
iii) The stable matrix$A_{s}$

represented as (2.5) satisfies the following Lyapunov equation:$A_{s}P+PA_{s}^{T}+Q=0$ .

Remark 1: Proposition 3 ii) shows that the set $S(n)$ can be treated as a vector bundle
which consists of $PD(n)$ as a base manifold and Skew $(n)$ as a fibre of vector space. We use
a notation $Skew_{P}(n)$ to represent each fibre attached to the element $P$ in the base manifold$PD(n)$ . Each fibre $Skew_{P}(n)$ has its own metric depending on $P$ (Theorem 3.2). This is
one of the main reason why we will treat $PD(n)\cross Skew(n)$ as a vector bundle rather thana mere product set. The interpretation of decomposition (2.5) from point of view of system
dynamics is discussed in [6], [9].
Define the set of state feedback. system matrices as$S_{f}(A, B)$$:=\{A+BF|F\in \mathcal{F}_{S}(A, B)\}\subset S(n)$and a linear mapping$\chi$: $\mathcal{F}_{S}(A, B)\ni F\mapsto A+BF\in S_{f}(A, B)$ .

, we can characterize structures of $S_{f}(A, B)$ in $PD(n)\cross Skew(n)$ .
Using
From (2.4) we can get$\phi_{Q}^{-1}$$A+BF=- \frac{1}{2}QP^{-1}+(S_{0}(P)-S)P^{-1}$ ,

(2.6)

where
$S_{0}(P)$ $:=AP-BB^{\uparrow}(AP+PA^{T}+Q)(I- \frac{1}{2}BB^{\uparrow})+\frac{1}{2}Q$$P\in PD(n,\cdot A, B, Q)$

,

,

(2.7)

$S\in Skew(n;B)$ .

Since $P\in PD(n;A, B, Q),$ $S_{0}(P)$ is proved to be skew symmetric using (2.2) and so is$S_{0}(P)-S$ .

Let $Skew_{P}(n;B)$ denote the set of all $S\in Skew_{P}(n)$ that satisfy (2.3). Comparing (2.5)
and (2.6), we now know how $S_{f}(A, B)$ is imbedded in $PD(n)\cross Skew(n)$ by:$\phi_{Q}^{-1}$

Proposition 4: The set of stable state feedback system matrices $S_{f}(A, B)$ is imbedded in
the vector bundle $PD(n)\cross Skew(n)$ as follows:i) in the base manifold $PD(n),$ $S_{f}(A, B)$ is restricted to the submanifold $PD(n;A, B, Q)$ ,ii) in each fibre $Skew_{P}(n)$ s.t. $P\in PD(n;A, B, Q),$ $\phi_{Q}^{-1}(S_{f}(A, B))$ is restricted to $S_{0}(P)+$$Skew_{P}(n;B)$.

In other words, $S_{f}(A, B)$ is diffeomorphic to a submanifold$\phi_{Q}^{-1}(S_{f}(A, B))=\bigcup_{P\in PD(n;A,B,Q)}\{S_{0}(P)+Skew_{P}(n;B)\}$

contained in $PD(n)\cross Skew(n)$ . (See Figure 1.)

,

(2.8)

17

Fig. 1 Geometric Structures of $\phi_{Q}^{-1}(S_{f}(A, B))$ in Vector Bundle $PD(n)\cross Skew(n)$3. Transformation Invariant Metrics
To explore metric structures of $PD(n)\cross Skew(n)$ , we introduce metrics to two vector
bundles. One is a tangent bundle $TPD(n)= \bigcup_{P\in PD(n)}T_{P}PD(n)$ , where $T_{P}PD(n)$ denotes the tangent vector space at a point $P\in PD(n)$ . The other is $PD(n)\cross Skew(n)=$$\bigcup_{P\in PD(n)}Skew_{P}(n)$ itself. A metric $g(P)$ of $TPD(n)$ is called a Riemannian metric on$PD(n)$ and we refer to a metric $f(P)$ of $PD(n)\cross Skew(n)$ as a fibre metric.
For the basis transformations of state space, $\sim x(t)=Tx(t)$ , where $T\in GL(n;R)$ , thematrices and in the parametrizations are transformed congruently as
$P$

$S$

$(P, S)arrow(\overline{P},\overline{S})=$( $TPT^{T},$ TST ).

(3.1)

For consistency with linear systenrs theory, we should define metrics for $PD(n)\cross Skew(n)$invariant against the above congruent transformations.

3.1 Transformation Invariant Riemannian Metric on $PD(n)$be the matrix with one as the $(p, q)$ -th element and zero otherwise. Now, weLetdefine , the basis matrices of vector space $Sym(n)$ , by
$E_{pq}$$E_{i}$$E_{i}$$:=E_{\sigma(p,q)}=\{\begin{array}{l}E_{pp}E_{pq}+E_{qp}\end{array}$$p=qp<q$,18
is an appropriate rule to assign integers to the pairs $(p, q)$ , i.e., $\sigma(p, q)=i$ , where$1\leq p\leq q\leq n$ and $1\leq i\leq N$ $:=n(n+1)/2$ .
. uniquely, where
Using $E;$ , we can represent any $P\in PD(n)$ as
that satisfies the positive definiteness. Hence, we
belongs to some open subset ofas a global coordinate system of N-dimensional manifold $PD(n)$ . Then,considercan be identified with , i.e.,natural basis of tangent vector fields

Here,$\sigma$$P= \sum_{i=1}^{N}\eta^{i}E_{1}$$(\eta^{i})$$R^{N}$$(\eta^{i})$$\partial_{i};=\partial/\partial\eta^{i}$$E_{i}$$\mathcal{X}(PD(n))\ni\partial_{i}\sim E;\in Sym(n)$$1\leq i\leq N$

,

(3.2)

where $\mathcal{X}(PD(n))$ denotes the set of tangent vector fields on $PD(n)$ . Using (3.2), we shallhereafter identify $\mathcal{X}(PD(n))$ with the set of $Sym(n)$ -valued differentiable function $X(P)$on $PD(n)$ :$\mathcal{X}(PD(n))\ni\sum_{i=1}^{N}a^{i}(P)\frac{\partial}{\partial\eta^{i}}\sim X(P)$$:= \sum_{i=1}^{N}a^{i}(P)E_{i}$,$a^{i}(P)\in C(PD(n))$,

where $C(PD(n))$ denotes the set of differentiable functions on $PD(n)$ , and tangent vector
space at denoted by $T_{P}PD(n)$ with $Sym(n)$ :$P$$X_{P}$$:= \sum_{i=1}^{N}a^{i}(\frac{\partial}{\partial\eta^{1}})_{P}\in T_{P}PD(n)\sim X$$:= \sum_{i=1}^{N}a^{i}E;\in Sym(n)$,$a^{i}\in R$

.

(3.3)

First we shall make clear what invariance is required for a Riemannian metric on $PD(n)$ .
( , ) on each tangent
A Riemannian metric $g(P)=[gij(P)]$ defines an inner product,space $T_{P}PD(n)$ . To represent
as an inner product of $Sym(n)$ , we use the same
notation. Then these inner products are defined by$g_{P}$$g_{P}(\bullet$$\bullet$$g_{P}(\partial_{i}, \partial_{j})=g_{P}(E_{i}, E_{j})$$\bullet$$\bullet$$)$$:=gij(P)$ ,$1\leq i\leq N,$ $1\leq j\leq N$

.

(3.4)

Denote the congruent transformation (3.1) by$\tau_{T}$: $P TPT^{T}$ ,$T\in GL(n;R)$ ,(3.5)

which is induced by the basis transformation of the state space (1.11). Then using theidentification (3.3), the differential: $T_{P}PD(n)\mapsto T_{r_{T}}{}_{(P)}PD(n)$ is represented as atransformation in $Sym(n)$ by$X TXT^{T}$ ,(3.6)$\tau_{T^{*}}$$\tau_{\tau*}:$, respectively.
where $X$ and $TXT^{T}$ indicate $X_{P}\in T_{P}PD(n)$ and
The invariance we require here of a Riemannian metric $g(P)$ is that the following equation:$\tau_{T^{*}}(X_{P})\in T_{\tau_{T}}{}_{(P)}PD(n)$$g_{P}(X_{P}, Y_{P})=g_{\tau_{T}(P)}(\tau_{T^{*}}(X_{P}), \tau_{\tau*}(Y_{P}))$should be satisfied for any $P\in PD(n),$equivalent to$X_{P},$$Y_{P}\in T_{P}PD(n)$(3.7)

and $T\in GL(n;R)$ . This is$g_{P}(X, Y)=g_{TPT^{T}}(TXT^{T}, TYT^{T})$for any $P\in PD(n),$ $X,$ $Y\in Sym(n)$ and $T\in GL(n;R)$ .
Now the invariant Riemannian metric on $PD(n)$ is obtained.

(3.8)

19
Theorem 3.1: Define $gij(P)$ by$gij(P):= \frac{1}{2}tr(P^{-1}E;P^{-1}E_{j})$

(3.9)

,

then $g(P)=[g;j(P)]$ is a Riemannian metric on $PD(n)$ invariant under the basis transformation of the state space.
Proof) Because of (3.4) and (3.9), the inner product on $Sym(n)$ is represented by$g_{P}(X, Y)= \frac{1}{2}tr(P^{-1}XP^{-1}Y)$.

(3.10)

Hence the invariance condition (3.8) can be easily confirmed as$g_{TPT^{T}}$( $TXT^{T},$ TYT ) $= \frac{1}{2}tr\{T^{-T}(P^{-1}XP^{-1}Y)T^{T}\}$$= \frac{1}{2}tr\{(P^{-1}XP^{-1}Y)T^{T}T^{-T}\}=g_{P}(X, Y)$.

(3.11)To show the positive definiteness of $g;;(P)$ , consider the basis transformation $T\in$$GL(n;R)$ satisfying$TPT^{T}=I$ .
(3.12)
Let $X$$:= \sum_{i=1}^{N}a^{i}E_{i}$, then from the invariance we can get$g_{P}(X, X)= \sum_{i,j=1}^{N}gti(P)a^{i}a^{j}=\frac{1}{2}tr\{(P^{-1}X)^{2}\}=\frac{1}{2}tr(X^{/2})$,$X’$ $:=TXT^{T}$

is identical to the square of the Euclid norm
Since
is also a symmetric matrix,$X’$$X’=0$. This means $g;;(P)$ is positive definite.
of , which is always positive except that.
Moreover, the differentiability of $g\{j(P)$ follows from that ofThus, $g(P)$ is proved to be an invariant Riemannian metric on $PD(n)$ .
$X^{/}$$tr(X^{\prime 2})$

$P^{-1}$$\blacksquare$

3.2 Transformation Invariant Fibre Metric of $PD(n)\cross Skew(n)$On the other hand, define the basis matrices
of $n(n-1)/2$ -dimensional vector space
Skew $(n)$ by,$p<q$ ,$\tilde{E}_{\mu}$$\tilde{E}_{\mu}$$:=\tilde{E}_{\overline{\sigma}(p,q)}=E_{pq}-E_{qp}$is an appropriate rule to as sign integers to the pair $(p, q)$ , i.e.,$1\leq p<q\leq n$ and $1\leq\mu\leq\tilde{N};=n(n-1)/2$ . Then any skew symmetric matrix
in each fibre space $Skew_{P}(n)$ is represented bywhere$\tilde{\sigma}(p, q)=\mu$$\tilde{\sigma}$$S= \sum_{\mu=1}^{\overline{N}}\tilde{\eta}^{\mu}\tilde{E}_{\mu}$,, where$S=(\tilde{\eta}^{\mu})$$\tilde{\eta}^{\mu}\in R$

Furthermore, a Skew $(n)$ -valued differentiable function $S(P)$ on $PD(n)$ is just a crosssection of $PD(n)\cross Skew(n)$ :$S(P)= \sum\tilde{\eta}^{\mu}(P)\tilde{E}_{\mu}\tilde{N}\in\Gamma(PD(n)\cross Skew(n)))$$\tilde{\eta}^{\mu}(P)\in C(PD(n))$$\mu=1$,

20
where $\Gamma(PD(n)\cross Skew(n))$ denotes the set of cross sections of $PD(n)\cross Skew(n)$ . We
as a (constant) basis cross section.
regard( , e) on each fibre $Skew_{P}(n)$A fibre metric $f(P)=[f_{\mu\lambda}(P)]$ defines an inner product$\tilde{E}_{\mu}$$f_{P}$$f_{P}(\tilde{E}_{\mu},\tilde{E}_{\lambda}):=f_{\mu\lambda}(P)$,$\bullet$$\tilde{E}_{\mu},\tilde{E}_{\lambda}\in Skew_{P}(n)$Note that the basis transformation of the state space causes the congruence transformation
which maps $S\in Skew_{P}(n)$ to $TST^{T}\in Skew_{TPT^{T}}(n)$ , or equivalently,$\tau_{T}$Then we find a fibre metric: $(P, S)$ ( $TPT^{T},$ TST ).

$f_{\mu\lambda}(P)$is required to satisfy the invariance such that$f_{P}(S, R)=f_{TPT^{T}}$($TST^{\prime r},$TRT )for any $P\in PD(n),$ $S,$ $R\in Skew_{P}(n)$ and $T\in GL(n;R)$ .
An invariant fibre metric $f(P)=[f_{\mu\lambda}(P)]$ can be derived in the similar manner to
Theorem 3.1.
Theorem 3.2: Define$f_{\mu\lambda}(P)$by$f_{\mu\lambda}(P):=- \frac{1}{2}tr(P^{-1}\tilde{E}_{\mu}P^{-1}\tilde{E}_{\lambda})$,then $f(P)=[f_{\mu\lambda}(P)]$ is a fibre metric of $PD(n)\cross Skew(n)$ invariant under the basis
transformation of the state space.
Proof) The invariance and the differentiability ofare proved in the same way ofTheorem 2. In contrast to Theorem 2, skew symmetry guarantees the positive definiteness.
Consider the same basis transformation (3.12), then$f_{\mu\lambda}(P)$.

$f_{P}(S, S)= \sum_{\mu,\lambda=1}^{\overline{N}}f_{\mu\lambda}(P)\tilde{\eta}^{\mu}\tilde{\eta}^{\lambda}=-\frac{1}{2}tr\{(P^{-1}S)^{2}\}=-\frac{1}{2}tr(S^{;2})=\frac{1}{2}tr(S^{;^{T}}S’)=\frac{1}{2}||S’||^{2}$

Hence, $f(P)=[f_{\mu\lambda}(P)]$ is positive definite.

$\blacksquare$

4. Dual Connections on $PD(n)\cross Skew(n)$Let be in an interval $[0, t_{1}]\subset R$ and : $trightarrow c(t)=P(t)$ be a smooth curve in $PD(n)$to. For all , consider a linear isomorphism
from: $T_{P_{0}}PD(n)arrow T_{P(t)}PD(n)$called a parallel displacement along the curve . Let $X(P)$ be a tangent vector field on$PD(n)$ , then covariant derivative atfor the direction $\dot{c}(0)=\dot{P}(0)$ is obtained from theparallel displacementas$t$$P_{0}$$c$$P_{1}$$II_{C}(t)$

$t$

$c$

$P_{0}$$\square _{c}(t)$$\nabla_{\dot{P}(0)}X=\lim_{tarrow 0}\frac{1}{t}\{\Pi_{C}(t)^{-1}X(P(t))-X(P_{0})\}$.

Using the parallel displacement along all the smooth curve on $PD(n)$ , we can define thecovariant derivative vector field on $PD(n)$ .

21In the same way, a parallel displacement along the curve can be introduced to thevector bundle $PD(n)\cross Skew(n)$ . It is characterized as a linear isomorphism fi $c(t)$ :$Skew_{P_{0}}(n)arrow Skew_{P(\ell)}(n)$ , and the corresponding covariant derivative for a cross section$S(P)\in\Gamma(PD(n)\cross Skew(n))$ is expressed by$c$$\tilde{\nabla}_{\dot{P}(0)}S=\lim_{tarrow 0}\frac{1}{t}\{\Pi_{C}(t)^{-1}S(P(t))-S(P_{0})\}\sim$.

The above shows that if parallel displacements
in $TPD(n)$ and 11 in $PD(n)\cross Skew(n)$are defined for any piecewise smooth curve in $PD(n)$ , we can derive affine connectionsof $TPD(n)$ and fibre connections V of $PD(n)\cross Skew(n))$ respectively.
$II_{c}$$c$$\nabla$$c$Consider two parallel displacements$II_{C}(t)X=X$ ,$\Pi_{C}$and II: of $TPD(n)$ defined by$\Pi_{c}^{*}(t)X=P(t)P_{0}^{-1}XP_{0}^{-1}P(t)$

,

(4.1)

for any curve and $X\in Sym(n)$ using the identification (3.3). Let
denoteandthe corresponding affine connections. It is easily proved from (3.9) that these two parallel
displacements satisfy$\nabla$$c$$g_{P_{0}}(X, Y)=g_{P(t)}(II_{c}(t)X, \Pi_{C}^{*}(t)Y)^{\forall}X,$ $Y\in

Sym(n)$$\nabla^{*}$.

Such a pair of parallel displacements ( , II:) and a pair of the derived connections
are said to be mutually dual [8]. The obtained results are as follows.
$\Pi_{C}$$(\nabla, \nabla^{*})$

Theorem 4.1: The covariant derivatives with respect to the parallel displacements
satisfy$II_{C}$

and$\Pi_{C}^{*}$$\nabla_{E_{i}}E_{j}=0$,$\nabla_{E_{i}}^{*}E_{j}=-E_{i}P^{-1}E_{j}-E_{j}P^{-1}E;$

respectively. Here, we are identifying the vector field$\partial/\partial\eta^{i}$and$E_{i}$

,

(4.2)

.

Proof) Since the parallel displacement
does not change basis tangent vectors as $II_{c}E;=$, the equation
is obvious.
To obtain the, we consider the curve $\gamma:P(t)$ defined by$II_{C}$$\nabla_{E_{1}}E_{j}=0$$E_{i}$$\nabla_{E_{i}}^{*}E_{j}$$P(t)$ $:=\tau_{P^{1/2}}\exp(Xt)=P^{\frac{1}{2}}\exp(Xt)P^{1}2$

The curve$\gamma$.

(4.3): $P(t)$ is found to satisfy$P(0)=P$

and$\dot{P}(0)=\tau_{P^{1/2^{*}}}X=P^{\frac{1}{2}}XP^{\frac{1}{2}}$

,

where
is the differential of. Hence, to calculate the covariant derivative
using the curve $\gamma:P(t)$ , we shall set$\tau_{P^{1/2^{*}}}$$\tau_{P^{1/2}}$$X$$:=P^{-\frac{1}{2}}E_{i}P^{-\frac{1}{2}}$$\nabla_{E_{j}}^{*}E_{j}$.

Then from the definition of covariant derivative,$\nabla_{E_{i}}^{*}E_{j}=\lim_{tarrow 0}\frac{1}{t}${$\Pi_{\gamma}^{*}(t)^{-1}$

Ej–Ej}.

(4.4)

22
Since

$\Pi_{\gamma}^{*}(t)^{-1}E_{j}$is represented via (4.1) and (4.3) by,$II_{\gamma(t)}^{*-1}E_{j}=PP(t)^{-1}E_{j}P(t)^{-1}P=P^{\frac{1}{2}}\exp(-Xt)P^{-\frac{1}{2}}E_{j}P^{-\frac{1}{2}}\exp(-Xt)P^{\frac{1}{2}}$we substitute this expression in (4.4) to get$\nabla_{E_{j}}^{*}E_{j}=\frac{d}{dt}\{P2\exp(-xt)P^{-}2E_{j}P^{-}2\iota\iota\iota_{\exp(-Xt)P^{\frac{1}{2}}\}1_{t=0}}$$=-P^{\frac{1}{2}}XP^{-L}2E_{j}-E_{j}P^{-\frac{1}{2}}XP^{\frac{1}{2}}$$=-E;P^{-1}E_{j}-E_{j}P^{-1}E;$.

$\blacksquare$

Theorem 4.2:i) The manifold $PD(n)$ is torsion free and flat with respect to the affine connection . (We)
shall call the latter propertiesii) The manifold $PD(n)$ is torsion free and flat with respect to the affine connection)(We shall call the latter properties$\nabla$$\nabla- flat,$$\nabla^{*}$$\nabla^{*}- flat.$

Proof)i) From (4.2), the coefficients of affine connection , which is defined by, vanish. This means the statement i).
. We have, be the torsion tensor of the affine connectionii) First, let$\nabla$$\Gamma;jk$$:=$$g_{P}(\nabla_{E}E_{j}, E_{k})$$T^{*}(\bullet$$\bullet$$\nabla^{*}$$)$$T_{ijk}^{*}(P):=g_{P}(T^{*}(E;, E_{j}),$ $E_{k}$Here, the coefficients of affine connection$\nabla^{*}$)$=g_{P}(\nabla_{E_{i}}^{*}E_{j}-\nabla_{E_{j}}^{*}E;, E_{k})$denoted by$\Gamma_{ijk}^{*}$.

is obtained by$\Gamma_{ijk}^{*}(P):=g_{P}(\nabla_{E}^{*}E_{j}, E_{k})=\frac{1}{2}tr\{P^{-1}(-E;P^{-1}E_{j}-E_{j}P^{-1}E;)P^{-1}E_{k}\}$$=-tr(P^{-1}E;P^{-1}E_{j}P^{-1}E_{k})$using the symmetry of $P,$

$E_{i},$$E_{j}$

and$E_{k}$. Hence, we get$T_{ijk}^{*}(P)=\Gamma_{ijk}^{*}(P)-\Gamma_{jik}^{*}(P)=0$, then we find it depends not on the curve it passes
Secondly, recall the definition ofalong but on the points where it starts and finishes. Furthermore, when the curve is closed,
II: is proved to map a tangent vector $X\in T_{P_{0}}PD(n)$ to itself. This means ii) is true
because the curvature tensors are geometrically interpreted as changes of tangent vectors
by parallel displacements along infinitesimally small closed curves. Thus the statement ii)
follows.
$II_{c}^{*}$$\blacksquare$

Similarly a pair of parallel displacements$PD(n)\cross Skew(n)$ :$\tilde{\Pi}_{C}(t)S=S$

And we shall also call$(\tilde{\Pi}_{C}, II_{c}^{*})\sim$,$(II_{c}, II_{c}^{*})\sim\sim$for any curve can be defined on$\tilde{\Pi}_{c}^{*}(t)S=P(t)P_{0}^{-1}SP_{0}^{-1}P(t)$.

, or a pair of the corresponding fibre connections$(\tilde{\nabla},\tilde{\nabla}^{*})$,

23
mutually dual because they satisfy.

$f_{P_{0}}(S, R)=f_{P(\ell)}(I^{\sim}I_{C}(t)S, II_{c}^{*}(t)R)\sim,$ $\forall_{S,R}\in Skew_{P_{0}}(n)$We can show the followings similarly to Theorem 4.1 and 4.2.

Theorem 4.3: The covariant derivatives with respect to the paraUel displacements fi andsatisfy,,$c$$\tilde{\Pi}|c*$$\tilde{\nabla}_{E_{i}}\tilde{E}_{\mu}=0$$\tilde{\nabla}_{E_{i}}^{*}\tilde{E}_{\mu}=-E;P^{-1}\tilde{E}_{\mu}-\tilde{E}_{\mu}P^{-1}E;$

respectively. Here, we are identifying$\tilde{E}_{\mu}$as the basis cross section.

Theorem 4.4: The vector bundle $PD(n)\cross Skew(n)$ is flat with respect to both fibre
connections
and V’. We shall call these properties, respectively.
and$\tilde{\nabla}$$\tilde{\nabla}- flat$$\tilde{\nabla}^{*}- flat$

Remark 2: Bothandare non-metric affine connections. However, we see they
qualify $PD(n)$ as a torsion free, flat manifold in Theorem. Similarly, non-metric fibre
connections V andendow the vector bundle $PD(n)\cross Skew(n)$ with flatness.
The coordinate system
is called affine with respect tobecause its basis vector$\nabla_{E}E_{j}=0$. The coordinate systemfields satisfies
is also natural with respect to V
in the sense that its basis cross sections satisfies. On the other hand, we canalso introduce affine and “natural” coordinate systems with respect toand. Thesecoordinate systems are called dual coordinate systems[8]. Such a pairs of primal and dual
coordinate systems play important roles in the theory of dual connections, e.g., defining a
pseudo-distance called divergences[8].
Another remark is that we can define the family of connections using)
and (V,in the same way to [8]. Define the connections depending one parameterby$\nabla$$\nabla^{*}$$\tilde{\nabla}^{*}$$(\eta^{i})$$\nabla$$(\tilde{\eta}^{\mu})$$\tilde{\nabla}_{E}\tilde{E}_{\mu}:=0$$\nabla^{*}$$\tilde{\nabla}^{*}$$\tilde{\nabla}^{*}$$(\nabla, \nabla^{*})$$\alpha\in R$$\nabla:=\alpha\frac{1-\alpha}{2}\nabla+\frac{1+\alpha}{2}\nabla^{*}$,$\tilde{\nabla}:=\alpha\frac{1-\alpha}{2}\tilde{\nabla}+\frac{1+\alpha}{2}\tilde{\nabla}^{*}$$\alpha$We call$(\nabla,\nabla)\alpha-\alpha$$\nabla\alpha$

and V

and

$\alpha$$(\tilde{\nabla},\tilde{\nabla})\alpha-\alpha$-affine and$\alpha- fibre$

connections, respectively. The pairs of connections

are mutually dual, respectively. Particularly, affine connection

fibre connection are metric. It is well known thatconnection of $PD(n)$ .
$\tilde{\nabla}0$$\nabla 0$$\nabla 0$andis called a Riemannian (Levi-Civita)

5. Differential Geometry of Parameter Space forand $S_{f}(A, B)$
In the previous sections, we have defined metrics and connections, the fundamental quantities for the differential geometric structures of $PD(n)\cross Skew(n)$ . Now using thesequantities, we shall exploit the geometric structures of vector bundle $PD(n;A, B, Q)\cross$$Skew(n;B)=\psi_{Q}^{-1}(\mathcal{F}_{S}(A, B))$ , which parametrizes
and imbedded submanifold$\phi_{Q}^{-1}(S_{f}(A, B))$ in $PD(n)\cross Skew(n)$ , which parametrizes $S_{f}(A, B)$ .
$\{a, b, \ldots\},$In this section, indicesandattached to quantities means that the quantities are components with respect to $PD(n),$ $PD(n;A, B, Q)$ ,$\mathcal{F}_{S}(A, B)$$\mathcal{F}_{S}(A, B)$$\{i, j, \ldots\},$$\{\mu, \lambda, \ldots\}$$\{\alpha, \beta, \ldots\}$

24
Skew $(n)$ and Skew $(n;B)$ , respectively.
Geometry of $PD(n;A, B, Q)\cross Skew(n;B)$ induced from $PD(n)\cross Skew(n)$Using the equation (2.2) which specifies the submanifold $PD(n;A, B, Q)$ in $PD(n)$ , wefor $PD(n;A, B, Q)$ and then define induced
first construct the coordinate system
Riemannian metric and connections.
Since (2.2) is linear equations with respect to the components of , i.e., $\eta=(\eta^{i})\in R^{N}$ ,then it can be rewritten as$K\eta=w$ ,(5.1)$(\gamma^{a})$$P$

where $K\in R^{(N-N_{P})\cross N},$ $w\in R^{N-N_{P}}$ are some constant matrix and vector determinedfrom (2.2)[6].
Let be in $GL(N;R)$ which satisfies $KT=[I 0]$ and define $:=T^{-1}\eta$ . Then (5.1) istransformed and solved as$T$$\eta’$[I$0$] $\eta’=w$ ,.

$\eta’=\{\begin{array}{l}w\gamma\end{array}\}$The free parameter $\gamma=(\gamma^{a})\in R^{N_{P}}$ is just the coordinate system of the submanifold$PD(n;A, B, Q)$ . The relation between
andis$(\eta^{i})$$(\gamma^{a})$$\eta(\gamma)=T\eta’=[T_{1} T_{2}]\{\begin{array}{l}w\gamma\end{array}\}$.

Hence, the Jacobian matrix
is a constant matrix. Then, the relation
between basis tangent vector fields
of $PD(n)$ andof $PD(n;A, B, Q)$ isfound to be $E_{a}=J_{a^{i}}E;$ .
Using this relation, geometrical quantities on submanifold $PD(n;A, B, Q)$ are naturally
induced from those of $PD(n)$ . The induced metric
on $PD(n;A, B, Q)$ is given by$J$$:=(\partial\eta^{i}/\partial\gamma^{a})=T_{2}$$E_{a}\sim\partial/\partial\gamma^{a}$$E_{i}$$g_{a}b(\gamma)$$g_{ab(\gamma)=J_{a}^{1}J_{b}^{j}g_{ij}(\eta)}$

Generally, the coefficients$\Gamma_{abc}(\gamma)$

.

(5.2)

of induced affine connections are obtained by(5.3)$\Gamma_{ab_{C}}(\gamma)=g_{P(\gamma)}(\nabla_{E_{a}}E_{b}, E_{c})=J_{a}^{1}J_{b}^{j}J_{C}^{k}\Gamma_{ijk}(\eta)+(\partial_{a}J_{b}^{i})J_{C}^{j}gij$

whereis the coefficients of any affine connection on $PD(n)$ .
By means of (5.2), (5.3), we can obtain the following results:$\Gamma_{ijk}(\eta)$

Theorem 5.1: $PD(n;A, B, Q)$ isand
manifold in itself, namely its RiemannChristofell curvature tensor $R_{abcd}$ andvanish.
$\nabla- flat$$\nabla^{*}- flat$$R_{abcd}^{*}$

Proof) Since Jacobian matrix in (5.3) is constant, its partial derivatives byvanish,. From this and $\Gamma_{ijk}=0$ (Theorem 4.1), (5.3) means $\Gamma_{abc}=0$ , i.e., the coefficientsof affine connection for submanifold $PD(n;A, B, Q)$ vanish. Since Riemann-Christoffel$J$$\gamma^{a}$$\partial_{a}J_{b^{i}}=0$

curvature tensor of$\nabla$-connection$R_{abcd}$$R_{abcd}=g_{P}(R(E_{a}, E_{b})E_{c},$ $E_{d}$)is defined by$=(\partial_{a}\Gamma_{bc}^{e}-\partial_{b}\Gamma_{ac}^{e})g_{e}d+(\Gamma_{aed}\Gamma_{bc}^{e}-\Gamma_{bed}\Gamma_{ac}^{e})$,25
it also vanishes. This shows $PD(n;A, B, Q)$ isfrom $\nabla- flatness[8]$ .

$\nabla- flat$. The

follows automatically$\nabla^{*}- flatness$$\blacksquare$In the same way, using the equation (2.3) which specifying Skew $(n;B)$ , we can constructof Skew $(n;B)$ . In also this fibre case, constantthe coordinate system.
Jacobian matrix denoted by is obtained:
We can induce fibre metrics and connections on $PD(n;A, B, Q)\cross Skew(n;B)$ . Since
is constant,$\tilde{\gamma}=(\tilde{\gamma}^{\alpha})\in R^{N_{S}}$$\tilde{J}$$\tilde{\eta}=\tilde{J}\tilde{\gamma}$$\tilde{J}$$f_{\alpha\beta}(\gamma)=\tilde{J}_{\alpha}^{\mu}\tilde{J}_{\beta}^{\lambda}f_{\mu\lambda}(\eta)$.

$\tilde{\Gamma}_{a\alpha\beta}(\gamma)=f_{P}(\tilde{\nabla}_{Ea}\tilde{E}_{\alpha},\tilde{E}_{\beta})=J_{a}^{i}\tilde{J}_{\alpha}^{\mu}\tilde{J}_{\beta}^{\lambda}\Gamma_{i\mu\lambda}+(\partial_{a}\tilde{J}_{\alpha}^{\mu})\tilde{J}_{\beta}^{\lambda}f_{\alpha\beta}$$=J_{a}^{i}\tilde{J}_{\alpha}^{\mu}\tilde{J}_{\beta}^{\lambda}\tilde{\Gamma}_{i\mu\lambda}(\eta)$

Then we can show the following similarly to Theorem 5.1.
Theorem 5.2: The vector bundle $PD(n;A, B, Q)\cross Skew(n;B)$ isbundle in itself, i.e., its curvatures vanish.

ノ$\sim-$

andハゴ*-flat vector

5.2 Geometry of $\phi_{Q}^{-1}(S_{f}(A, B))$ imbedded in $PD(n)\cross Skew(n)$In section 2, we have seen $S_{f}(A, B)$ is imbedded in $PD(n)\cross Skew(n)$ byas (2.8).
We can get some results using this imbedding.
To see the structures of $PD(n;A, B, Q)$ as a canonically imbedded submanifold in $PD(n)$ ,it is enough to calculate Euler-Schouten (imbedding) curvature tensor. Let$N$ ) be the basis of orthogonally complement subspace of $T_{P}PD(n;A, B, Q)$ in$N_{P}+1,$$T_{P}PD(n)$ , then Euler-Schouten curvature tensor
is defined by$\phi_{Q}$$\{E_{\overline{k}}\}(\overline{k}=$$\ldots,$$H_{ab\overline{k}}$$H_{ab\overline{k}}=g_{P}(\nabla_{E_{a}}E_{b}, E_{\overline{k}}),$(5.4)$H_{ab\overline{k}}^{*}=g_{P}(\nabla_{E_{\sigma}}^{*}E_{b}, E_{\overline{k}})$This quantities show how curved $PD(n;A, B, Q)$ is in $PD(n)$ .
Theorem 5.3: The submanifold $PD(n;A, B, Q)$ is an autoparallel submanifold in $PD(n)$with respect to , namely, its Euler-Schouten (imbedding) curvature is identicaJly vanishes.
Hence, $PD(n;A, B, Q)$ is totally geodesic submanifold with respect to , which means that
$PD(n;A, B, Q)$ consists of $al1$ the -geodesics whose tangent vectors belong to the tangent
space $T_{P}PD(n;A, B, Q)$ .
$\nabla$$\nabla$$\nabla$Proof) Calculating (5.4), Euler-Schouten curvature is$H_{ab\overline{k}}=J_{a}^{i}J_{b}^{j}J \frac{k}{k}\Gamma_{jik}+(\partial_{a}J_{b}^{j})J\frac{k}{k}g_{jk}$(5.5)

is constant,is found to be zero as Theorem
From the facts that $\Gamma_{ijk}=0$ and5.1. Since autoparallel submanifold is always totally geodesic [10], the statement follows.
$(J_{b}^{k})$$H_{ab\overline{k}}$$\blacksquare$be the basis of orthogonally complement subspace of
Let
$(n)$
Skew $(n;B)$ in Skew . On a canonically imbedded submanifold $PD(n;A, B, Q)\cross Skew(n$ ;$B)-PD(n)\cross Skew(n)$ , which has a subvector bundle structure, we can define curvature$\{\tilde{E}_{\overline{\kappa}}\},\overline{\kappa}=N_{s}+1,$$\ldots,\tilde{N}$26tensor$\tilde{H}_{a\mu\overline{\kappa}}$and$\tilde{H}_{a\mu\overline{\kappa}}^{*}$, in the same manner to Euler-Schouten curvature tensor,,$\tilde{H}_{a\mu\overline{\kappa}}=f_{P}(\nabla_{E_{a}}\tilde{E}_{\mu},\tilde{E}_{\overline{\kappa}})$$\tilde{H}_{a\mu\overline{\kappa}}^{*}=f_{P}(\nabla_{E_{\sigma}}^{*}\tilde{E}_{\mu},\tilde{E}_{\overline{\kappa}})$.

vanishes and the subvector bundle $PD(n;A, B, Q)\cross$
It is also easy to observe only
$Skew(n;B)$ consists of V-geodesics. However, when it is imbedded into $PD(n)\cross Skew(n)$, which just emas the submanifold $\bigcup_{P\in PD(n;A,B,Q)}\{S_{0}(P)+Skew_{P}(n;B)\}$ bybodies the structures of $S_{f}(A, B)$ in $S(n)$ , this manifold (not subvector bundle) is generally
curved even in V-flat sense because of the origin-shifted term $S_{0}(P)$ .
$\tilde{H}_{a\mu\overline{\kappa}}$$\phi_{Q}^{-1}0\chi 0\psi_{Q}$

6. Conclusions
In this paper, we have dually introduced affine and fibre connections on $PD(n)\cross Skew(n)$ ,which is diffeomorphic to $S(n)$ . Then, using this connections, geometric structures ofand V are proved tostable state feedback systems have been discussed. Connections
characterize them well. Analysis by divergences and applications of obtained results will
be found in another place.
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“$\Pi$