ON EXTERNAL SPECTRAL FACTORS AND GEOMETRIC CONTROL

P. A. Fuhrmann^{1} Department of Mathematics Ben-Gurion University of the Negev Beer Sheva, Israel e-mail: paf@ivory.bgu.ac.il and A. Gombani LADSEB Corso Stati Uniti 4 Padova, Italy e-mail gombani@ladseb.pd.cnr.it

Abstract

We investigate here how the geometric control theory of Basile, Marro and Wonham can be obtained in a Hilbert space context, as the byproduct of the factorization of a spectral density with no zeros on the imaginary axis. We show how stabilizable and controllability (A, B) − invariant subspaces can be obtained as images of orthogonal projections of coinvariant subspaces onto a seminvariant (markovian) subspace of the Hardy space of square integrable functions analytic in the right half plane.

1 Introduction

The problem we study in this paper has a long history, motivated both by spectral factorization and control problems (they both end up with a Riccati equation). The approach we take is nevertheless new: we exploit the Hilbert space structure (as is done in classical stochastic realization theory), but instead of analyzing the geometry in the stochastic domain, we carry this structure entirely into the frequency setting. The main advantage is a computational one. In fact, we reduce the problem to the study of a set of stable all-pass functions. In particular, in this paper we investigate the relations between minimal spectral in this context factors and the geometric control theory developed by Basile and Marro [1] and Wonham [12].

It is well known (see e.g [10]) that if Φ is a full rank p × p rational spectral density without zeros on the imaginary axis then W is a minimal stable p × m spectral factor if and only if there exist inner functions (see below) Q′ and Q′′ (determined up to a unitary constant matrix) such that W = [W_{−}, 0]Q′ = [W_{+}, 0]Q^{′′∗}, where W_{−} and W_{+} are the minimum- and maximum-phase spectral fac-

tors (see below) and that Q′Q′′ = ( \begin{matrix}Q_{+} & 0 \\ 0 & R \end{matrix} ) where

Q_{+} is an invariant depending only on Φ. It can be shown (see [6]) that this result holds also when the matrix Φ does not have full rank and that the matrix R is equivalent to a divisor of Q_{+}. Let now

( \begin{matrix}A & B \\ C & D \end{matrix} ) be a fixed realization of W: we show

here how the functions Q′ and Q′′ associated to W determine a maximal antistabilizable subspace V and a maximal stabilizable subspace \overline{V} of CI ^{n} which are (A, B) − invariant and output output nulling. Moreover, the supremal output nulling controllability subspace R is the intersection of V and \overline{V}. The step from Q′ to the supremal antistabilizable subspace is quite simple. Let \overline{W} K be the Douglas-Shapiro-Shields factorization (see below) of W; then V is (isomorphic to) the projection of the coinvariant subspace H(Q′) onto H(K). The coinvariant subspace H(K) is quite important in stochastic realization (see e.g. [?] or [11]) because it represents the state space for the realization of W.

One advantage of this approach is computational. The factorizations and the projections can all be computed using state space formulas and solving Lyapunov and Riccati equations. This should not only allow for a simple algorithm for determining the (A, B) − invariant subspaces, but also, through a suitable choice of basis which exploits SVD techniques on the solutions to the Lyapunov equations for the projections, to a solution for approximate problems. Our work has been largely inspired by [10]: we believe that the work exposed is are a natural extension of the results of that paper in the context of the geometric stochastic realization theory of Lindquist and Picci.

2 Preliminaries and notation

We work in the Hilbert space setting of the plane; we define [8] L^{2}( II ) to be the set of the vector or matrix valued (the proper dimension will be clear from the context) square integrable functions on the imaginary axis, and H_{+}^{2} to be the subspace of L^{2} of functions analytic in the right half-plane and such that

sup_{x>0} \frac{1}{2π} ∫_{−∞}^{∞} tr(F^{∗}(x + iy)F (x + iy)) dy < ∞

where ^{∗} denotes transposed conjugate. If F and G are column vectors, the inner product in H_{+}^{2} is

⟨F, G⟩ = \frac{1}{2π} ∫_{−∞}^{∞} G^{∗}(iω) F (iω) dω

Analogously, H_{+}^{∞} is the subspace of L^{2} of functions analytic in the right half-plane and such that

sup_{Re s>0} ∥F (s)∥ < ∞

where ∥F (x + iy)∥ denotes the usual matrix norm. (H_{−}^{2} and H_{−}^{∞} are defined similarly on the left halfplane).

Let F be function of L^{2}: we denote by P^{+}F (P^{−}F) the orthogonal projection of L^{2} onto H_{+}^{2}(H_{−}^{2}).

1Partially supported by GIF under Grant No. I 184 and by CNR

A function Q ∈ H_{+}^{∞} is inner if it is square and Q^{∗}Q = QQ^{∗} = I. It is well known that a column vector space M in H_{+}^{2} is invariant under multiplication by e^{−iωt} for t ≥ 0 if and only if it is of the form M = QH_{+}^{2} (Beurling’s Theorem). Similarly, a row space N is invariant if and only if it is of the form N = H_{+}^{2}Q. We set H_{c}(Q) := (QH_{+}^{2})^{⊥} and H_{r}(Q) := (H_{+}^{2}Q)^{⊥}. Clearly it is P^{+}e^{iωt}H_{c}(Q) ⊂ H_{c}(Q) (similarly for H_{r}(Q)).

Remark: since we will mostly use row vectors in H_{+}^{2}, we will write H(K) for H_{r}(K) when there is no ambiguity. A full column-rank p × m_{0} rational matrix function G in H_{+}^{∞} is said to be minimumphase or outer (on the right) if rank G(s) = m_{0} for Res > 0. It’s well know that a rational function F in H_{+}^{∞} admits an outer inner factorization

F = F_{0}Q

This factorization is unique up to a unitary constant matrix. The factorization W = \overline{W} K of W in H_{+}^{2} is a Douglas-Shapiro-Shields (DSS) factorization of W if \overline{W} ∈ H_{−}^{2}, K is inner and the degree of K is as small as possible. Also this factorization is unique up to a unitary constant matrix. A matrix function W is said to be maximum-phase if its DSS factor \overline{W} is (conjugate) outer in H_{−}^{2}. Consequently W_{+} is the maximum-phase spectral factor of a spectral density Φ if the DSS factor \overline{W}_{+} of W_{+} is minimum-phase in H_{−}^{2}. Given two inner functions P, Q we say that \overline{Q}\overline{P} is a skew-prime factorization (see [4]) if P Q = \overline{Q}\overline{P}, P and \overline{Q} are left coprime and Q and \overline{P} are right coprime.

¿From now on we assume all the functions to be ra-

tional. The notation W = ( \begin{matrix}A & B \\ C & D \end{matrix} ) indicates that

the quadruple (A, B, C, D) is a minimal realization of W. By A^{#} we denote the Moore-Penrose inverse of a matrix A.

The following is a well known characterization of inner functions, see [2], [3] or Fuhrmann and Ober [7].

Proposition 2.1 1. The square proper stable rational matrix function R̂ is inner if and only if it has a minimal realization

R̂ = ( \begin{matrix}A & B \\ B^{∗}P^{−1} & D \end{matrix} )  (1)

where DD^{∗} = I and P solves the Lyapunov equation

AP + P A^{∗} + BB^{∗} = 0.  (2)

2. Similarly R̂ is inner if and only if it has a minimal realization

R̂ = ( \begin{matrix}A & Q^{−1}C^{∗} \\ C & D \end{matrix} )  (3)

where D^{∗}D = I and Q solves the Lyapunov equation

A^{∗}Q + QA + C^{∗}C = 0.  (4)

We define the multiplicity of an m × m rational

inner function K = ( \begin{matrix}A & B \\ − B^{∗}P^{−1} & I \end{matrix} ) function as

the rank of the matrix B.

Corollary 2.1 Let K = ( \begin{matrix}A & B \\ − B^{∗}P^{−1} & I \end{matrix} ) be an

m×m inner matrix of multiplicity m_{0}; then there ex-

ists a unitary matrix V such that KV = ( \begin{matrix}K_{0} & 0 \\ 0 & I \end{matrix} )

and K_{0} is m_{0} × m_{o}

Lemma 2.1 Let

S_{i} = ( \begin{matrix}A_{i} & B_{i} \\ C_{i} & I \end{matrix} ) i = 1, 2

be minimal realizations of two inner functions. Let

A_{e} := ( \begin{matrix}A_{1} & 0 \\ 0 & A_{2} \end{matrix} ) , B_{e} := ( \begin{matrix}B_{1} \\ B_{2} \end{matrix} ) .

Suppose S_{1} and S_{2} are right coprime and define S_{L} = S_{1} ∨_{L} S_{2} to be the least common left multiple of S_{1} and S_{2}, and let \overline{S}_{2} := S_{L}S_{1}^{∗}. Then

P_{H_{r}(\overline{S}_{2})S_{1}}(sI − A_{2})^{−1}B_{2} = (sI − A_{2})^{−1}B̂_{2}S_{1}(s) (5)

where B̂_{2} := −P_{12}^{∗}P_{11}^{−1}B_{1} + B_{2} and let P_{e} =

( \begin{matrix}P_{11} & P_{12} \\ P_{12}^{∗} & P_{22} \end{matrix} ) is the solution to the Lyapunov equa-

tion

A_{e}P_{e} + P_{e}A_{e}^{∗} + B_{e}B_{e}^{∗} = 0  (6)

3 Geometric control

We start with a theorem which characterizes all minimal stable spectral factors of a given dimension of a rational p × p spectral density Φ of rank m_{0} (see [10] and [6]). Let W_{0} and_{+} be the p×m_{0} minimum-phase and maximum-phase spectral factors of W.

Theorem 3.1 Let W be a p × m spectral factor for the spectral density Φ with rank m_{0} ≤ m and no rank drop at infinity. Then there exist inner functions Q′ and Q′′, essentially unique up to a constant unitary matrix factor, such that

W = [W_{−}, 0]Q′ = [W_{+}, 0]Q^{′′∗}  (7)

We will need a simple Lemma about DSS factorizations (see e.g. [7])

Lemma 3.1 Let \overline{W} = W K^{∗} be the DDS factoriza-

tion of W over H_{+}^{∞} and let W = ( \begin{matrix}A & B \\ C & D \end{matrix} ), \overline{W} =

( \begin{matrix}\overline{A} & \overline{B} \\ \overline{C} & \overline{D} \end{matrix} ) and K = ( \begin{matrix}A & B \\ − B^{∗}P^{−1} & I \end{matrix} ) be mini-

mal realizations,with P satisfying AP +P A^{∗}+BB^{∗} = 0. Then

\overline{W} = ( \begin{matrix}−A^{∗} & P^{−1}B \\ DB^{∗} + CP & D \end{matrix} )

and

W = ( \begin{matrix}−\overline{A}^{∗} & P \overline{B} \\ − \overline{D}\overline{B}^{∗} + \overline{C}P^{−1} & \overline{D} \end{matrix} )

Lemma 3.2 Let S_{i} = ( \begin{matrix}A_{i} & B_{i} \\ − B_{i}^{∗}P_{i}^{−1} & I \end{matrix} ) , i =

1, 2, be minimal realizations of inner functions of degree n_{1} and n_{2} respectively.

1. Let P_{12} be the unique solution to the equation

A_{1}P_{12} + P_{12}A_{2}^{∗} + B_{1}B_{2}^{∗} = 0

Then

P_{H_{r}(S_{2})}(sI − A_{1})^{−1}B_{1} = P_{12}P_{2}^{−1}(sI − A_{2})^{−1}B_{2} (8)

2. Similarly, let Q_{21} be the solution to the equation

A_{2}^{∗}Q_{21} + Q_{21}A_{1} + P_{2}^{−1}B_{2}B_{1}^{∗}P_{1}^{−1} = 0

Then

P_{H_{c}(S_{2})}B_{1}P_{1}^{−1}(sI − A_{1})^{−1}  (9)
= B_{2}P_{2}^{−1}(sI − A_{2})^{−1}P_{2}Q_{21}  (10)

Proof: We set

A_{e} := ( \begin{matrix}A_{1} & 0 \\ 0 & A_{2} \end{matrix} ) , B_{e} := ( \begin{matrix}B_{1} \\ B_{2} \end{matrix} )

It is then immediate to check that P_{e} =

( \begin{matrix}P_{1} & P_{21}^{∗} \\ P_{21} & P_{2} \end{matrix} ) is the solution to the Lyapunov equa-

tion

A_{e}P_{e} + P_{e}A_{e}^{∗} + B_{e}B_{e}^{∗} = 0  (11)

So, set u_{i} = e_{i}(sI − A_{1})^{−1}B_{1} for i = 1, …, n_{1} and v_{i} = e_{j}(sI − A_{2})^{−1}B_{2} for j = 1, …, n_{2}, where e_{i} and e_{j} are the standard row unit vectors of dimensions n_{1} and n_{2}. It’s quite well known (see e.g. [2]) that

⟨u_{i}, u_{j}⟩ = [P_{1}]_{ij} ⟨v_{i}, v_{j}⟩ = [P_{2}]_{ij}
⟨u_{i}, v_{j}⟩ = [P_{12}]_{ij}

We decompose now the vector u_{i} = e_{i}(sI − A_{1})^{−1}B_{1} into its components in H_{r}(S_{2}) and its orthogonal complement. Thus we write

u_{i} = ∑_{k=1}^{n} c_{ik}v_{k} + (u_{i} − ∑_{k=1}^{n} c_{ik}v_{k}),

with the coefficients determined by the orthogonality relations (u_{i} − ∑_{k=1}^{n} c_{ik}v_{k}, v_{j}) = 0. Thus we obtain (u_{i}, v_{j}) = ∑_{k=1}^{n} c_{ik}(v_{k}, v_{j}), which is equivalent to P_{12} = CP_{2}. Thus C = P_{12}P_{2}^{−1}. This proves (8). The dual equation is derived similarly. □

We want now to provide a Hilbert space characterization of (A, B) − invariant subspaces. Let (A, B) be a controllable pair. A subspace V of CI ^{n} is said to be (A, B) − invariant if there exists a feedback matrix F such that

(A + BF)V ⊂ V

Given an (A, B) − invariant subspace in R^{n}, define in H_{+}^{2} the subspace X_{V} := {ξ^{∗}P^{−1}(sI − A)^{−1}B; ξ ∈ V} We say that the (A, B) − invariant subspace V is antistabilizable if there exists a feedback F such that (A + BF)_{|V} is antistable. Similarly we say that \overline{V} is stabilizable if there exists a feedback F such that (A + BF)_{|\overline{V}} is stable. Let X ⊂ CI ^{n}: by the notation ⟨A|X ⟩ we indicate the subspace span{A^{k}X; 0 ≤ k ≤ n}. A subspace R ⊂ CI ^{n} is a controllability subspace if there exist a feedback matrix F and a matrix G such that

R = ⟨A + BF |Im BG⟩

A subspace R ⊂ V is a supremal controllability subspace in V if it is not properly contained in any other controllability subspace of V. It is well known (see [12]) that this space is unique. Let therefore R_{V} be the supremal controllability subspace of V, and set V^{0} := V/R_{V}. It is well known (see [12]) that V is antistabilizable if and only if (A + BF)_{|V^{0}} is antistable.

Lemma 3.3 Let (A, B) be a controllable pair, and V an (A, B) − invariant subspaces of CI ^{n}; set K =

( \begin{matrix}A & B \\ − B^{∗}P^{−1} & I \end{matrix} ) to be the inner matrix associated

to (A, B).

1. The (A, B) − invariant subspace V is antistabilizable if and only if there exists an inner function Q′ such that

X_{V} := P_{H(K)}H(Q′)

2. The (A, B) − invariant subspace \overline{V} is stabilizable if and only if there exists an inner function \overline{Q}′′ left coprime with K such that

X\overline{V} := P_{H(K)Q′′}H(\overline{Q}′′)

where Q′′ and K_{+} are uniquely determined by the relation

KQ′′ = \overline{Q}′′K_{+}

Proof:

1. We first show that if X_{V} is as above, then V is antistabilizable (A, B) − invariant). Let Q′ =

( \begin{matrix}A & B \\ − B^{∗}P^{−1} & I \end{matrix} ) be a minimal realizations of

the m × m inner functions Q′. Let

A_{e} = ( \begin{matrix}A & 0 \\ 0 & A \end{matrix} ) , B_{e} = ( \begin{matrix}B \\ B \end{matrix} )

and let P_{e} = ( \begin{matrix}P & P_{12} \\ P_{21} & P \end{matrix} ) be the solution to

the Lyapunov equation

A_{e}P_{e} + P_{e}A_{e}^{∗} + B_{e}B_{e}^{∗} = 0  (12)

we divide the proof into steps.

(a) We claim that the rows of

P_{H(K)}(sI − A)^{−1}B = P_{12}P^{−1}(sI − A)^{−1}B (the equality is true after Lemma 3.2) span an invariant subspace for Â_{1} := P_{12}^{#}AP_{12}. In fact, we can write

P_{12}^{#}ÂP_{12}P^{−1}(s − A)^{−1}B ⊂ P_{H(K)}H(Q′)

since clearly

span{P_{12}P^{−1}(s − A)^{−1}B}

= span{P_{12}^{#}P_{12}P^{−1}(s − A)^{−1}B}

(b) Let now V := Im P_{21}. Then it is true that

V = {ξ : ξ^{∗}P^{−1}(s − A)^{−1}B ∈ P_{H(K)}H(Q′)}

In fact, suppose ξ is such that ξ^{∗}P^{−1}(s − A)^{−1}B ⊂ P_{H(K)}H(Q′). Since a basis for P_{H(K)}H(Q′) is given by the rows of P_{21}P^{−1}(s − A)^{−1}B it is necessarily ξ ∈ Im P_{21}. The converse is obvious.

(c) We have

Â_{1}^{∗}_{|V} = −(A + BB^{∗}P_{21}^{#})_{|V}

The element in the lower right corner of (12) yields

AP_{21} + P_{21}A^{∗} + BB^{∗} = 0

or

AP_{21}P_{21}^{#} + P_{21}A^{∗}P_{21}^{#} + BB^{∗}P_{21}^{#} = 0

Thus we can write

P_{21}A^{∗}P_{21}^{#}_{|V} = −(A + BB^{∗}P_{21}^{#})_{|V}

that is, V is (A, B)-invariant

Suppose  now  V  is  antistabi-

lizable (A, B) − invariant ; then there exists a feedback matrix F such that (A + BF)_{|V} is antistable, and (A + BF)V ⊂ V. Let P_{21} to be any full column-rank matrix having image V. Since P_{21}P_{21}^{#} is a projection onto V,

(A + BF)_{|V} = P_{21}P_{21}^{#}(A + BF)P_{21}P_{21}^{#}

therefore we can define

A^{∗} := −P_{21}^{#}(A + BF)P_{21}

and

B = P_{21}^{∗}F^{∗}. Define Q′ = ( \begin{matrix}A & B \\ − B^{∗}P^{−1} & I \end{matrix} ).

A simple computation yields

AP_{21} + P_{21}A^{∗} + BB^{∗} =

= AP_{21} − P_{21}P_{21}^{#}(A + BF)P_{21} + BF P_{21}
= AP_{21} − (A + BF)P_{21} + BF P_{21} = 0

which is exactly the lower left block of the Lyapunov equation (11) associated to the projection of H(Q′) onto H(K) (cfr. Lemma 3.2).

2. The proof is obtained by duality considerations.

Lemma 3.4 Let W = ( \begin{matrix}A & B \\ C & D \end{matrix} ) be a minimal re-

alization of the spectral factor W, and let K be the associated inner function.

1. Let V be an (A, B) − invariant subspace, and let

Q′ = ( \begin{matrix}A & B \\ − B^{∗}P^{−1} & I \end{matrix} ) be an inner function

such that X_{V} = P_{H(K)}H(Q′). Then W (Q′)^{∗} is stable if and only if

V ⊂ ker(C + DB^{∗}P_{21}^{#})  (13)

where P_{21} satisfies the equation

AP_{21} + P_{21}A^{∗} + BB^{∗} = 0

2. Let \overline{A} be an (A, B) − invariant subspace and let

\overline{Q}′′ = ( \begin{matrix}A & B \\ − B^{∗}P^{−1} & I \end{matrix} ) be a minimal realiza-

tions of the m × m inner functions \overline{Q}′′ such that X\overline{V} = P_{H(K)Q′′}H(\overline{Q}′′)K_{+} then \overline{W} \overline{Q}′′ ∈ H_{−}^{∞} if and only if

V ⊂ ker(C − D(B^{∗}P\overline{Q}_{21}^{#}P^{−1} − B^{∗}P^{−1})) (14)

where \overline{Q}_{21} satisfies

A^{∗}\overline{Q}_{21} + \overline{Q}_{21}A + P^{−1}BB^{∗}\overline{P}

Proof:

1. Let u = ζ^{∗}(sI − A)^{−1}B; then W u^{∗} is analytic in the positive closed halfplane C^{+} and so

∫_{Γ} W (s)u^{∗}(s)ds = 0

for any closed curve Γ contained in C^{+}; in particular,

∫_{Γ_{1}} Du^{∗}(s)ds + ∫_{Γ_{2}} C(sI − A)^{−1}Bu^{∗}(s)ds = 0

for any pair of closed curves Γ_{1} and Γ_{2} containing the poles of u^{∗}. Now it is

∫_{Γ_{1}} Du^{∗}(s)ds

= ∫_{Γ_{2}} DB^{∗}(−sI − A^{∗})^{−1}ζds = −2πiDB^{∗}ζ

since ∫_{Γ_{1}}(sI + A^{∗})^{−1}ζds = 2πiI. The second integral is easily computed observing that, since the two functions are strictly proper, we can integrate along the imaginary axis (with a change of sign, since we are reversing the pattern of integration); but we obtain, in this manner the inner product matrix of the basis (−sI − A^{∗})^{−1}B and (−sI − A^{∗})^{−1}B. We can therefore substitute to (−sI − A^{∗})^{−1}B its projection onto H(K). Thus we obtain

−\frac{1}{2πi} ∫_{Γ_{2}} C(sI − A)^{−1}Bu^{∗}(s)ds

= \frac{1}{2πi} ∫_{I} C(sI − A)^{−1}BB^{∗}(−sI − A)^{−1}ζds

= ∫_{I} C(sI − A)^{−1}BB^{∗}(−sI − A)^{−1}P^{−1}P_{21}ζds

= 2πiCP_{21}ζ

Putting things together we obtain (DB^{∗} + CP_{21})ζ = 0 and remembering that ξ ∈ V if and only if there is a ζ for which ξ = P_{21}ζ we can eventually write

(DB^{∗}P_{21}^{#} + C)ξ = 0  ξ ∈ V (15)

Conversely, suppose V is an (A, B) − invariant

subspace, and that Q′ = ( \begin{matrix}A & B \\ − B^{∗}P^{−1} & I \end{matrix} ) is

an inner function associated to V such that (13) is satisfied; then, by backtracking the above argument, we obtain ∫_{Γ} W (s)u^{∗}(s)ds = 0 for each u ∈ H_{r}(Q′). Since H_{r}(Q′) is coinvariant, this implies that W ∈ H_{+}^{2}Q′, as wanted.

2. The proof is derived by duality considerations and is omitted.

Theorem 3.2 Let W = ( \begin{matrix}A & B \\ C & D \end{matrix} ) and let Q′ and

Q′′ be as (7). Then the maximal antistabilizable (A, B) − invariant output nulling subspace is

V := {ξ; ξ^{∗}P^{−1}(sI − A)^{−1}B ⊂ P_{H(K)}H(Q′)}

and  the  maximal  sta-

bilizable output nulling (A, B) − invariant subspace is

V := \{ξ; ξ^{∗}P^{−1}(sI−A)^{−1}BQ′′ ⊂ P_{H(K)Q′′}H(\overline{Q}′′)K_{+}\}

Proof: In view of Lemma 3.3, V is an antistabilizable (A, B) − invariant subspace, and in view of Lemma 3.4, it is an output nulling subspace. Suppose W = W_{1}Q_{1} and denote by V_{1} the (A, B) − invariant output nulling subspace associated to Q_{1}; since Q′ is the inner factor of W, Q_{1}|_{R}Q′, and therefore it’s not difficult to see that V_{1} ⊂ V. The proof of the other statement is similar.

Corollary 3.1 Let W = ( \begin{matrix}A & B \\ C & D \end{matrix} ), with D full

column rank. Then

(A + BB^{∗}P_{21}^{#})_{|V} = (A − BD^{−L}C)_{|V}

and

(A + B(B^{∗} − \overline{B}^{∗}\overline{P}^{−1}\overline{Q}_{21}^{#})P^{−1})_{|\overline{V}} = (A − BD^{−L}C)_{|V}

Proof: Since D has full column rank, it has a left inverse D^{−L}. This also means that DG = 0 implies G = 0, and therefore the supremal controllability subspace

R^{∗} = ⟨A + BB^{∗}P_{21}^{#}|Im(BG)⟩ = 0

Therefore from (15) we get

D^{−L}C_{|V} = −B^{∗}P_{21}^{#}_{|V}

and eventually

(A + BB^{∗}P_{21}^{#})_{|V} = (A − BD^{−L}C)_{|V}

Similarly for \overline{V}

4 Controllability subspaces

Nota bene: throughout this section, when we write ”P_{Y} X is injective”, we mean that the restricted projection operator (P_{Y})_{|X} is injective.

Definition 4.1 Let K_{+} and Q_{+} be right coprime and

Q′Q′′ = ( \begin{matrix}Q_{+} & 0 \\ 0 & R \end{matrix} ) with R equivalent to a fac-

tor of Q_{+} (i.e. there exists a factorization Q_{+} =

Q′_{1}Q_{2}Q′′_{1} where Q_{2} = ( \begin{matrix}Q_{2}^{i} & 0 \\ 0 & R \end{matrix} ) and R is equiv-

alent to Q_{2}^{i}). Let \overline{Q}′′ satisfy the equation KQ′′ = \overline{Q}′′K_{+}^{e} with K and \overline{Q}′′ left coprime, and define \overline{Q}′ :=

( \begin{matrix}Q_{−} & 0 \\ 0 & R \end{matrix} ) (\overline{Q}′′)^{∗}. We say that the factorization

Q′Q′′ is balanced with respect to K_{+} (and that the factorization \overline{Q}′\overline{Q}′′ is balanced with respect to K_{−}) if

[Q′′, R^{e}]_{R} = I

and

[\overline{Q}′, R^{e}]_{L} = I

We also say that the seminvariant subspace H(K)Q′′ is the state space associated to the factorization.

Theorem 4.1 Let Q′Q′′ be a balanced factorization

w.r. to K_{+}, and let ( \begin{matrix}A & B \\ − B^{∗}P^{−1} & I \end{matrix} ) be a realiza-

tion of K; set X_{R} := P_{H(K)Q′′}H(R^{e}) and R := {ξ : ξ^{∗}P^{−1}(sI − A)^{−1}B ∈ X_{R}. Then

X_{R} = P_{H(K)Q′′}H(Q′)Q′′∩P_{H(K)Q′′}H(\overline{Q}′′)K_{+}^{e} (16)

and R is a controllability subspace for A, B.

Sketch  of  the

proof: we set X_{V} := P_{H(K)Q′′}H(Q′)Q′′ and X\overline{V} := P_{H(K)Q′′}H(\overline{Q}′′)K_{+}^{e}. We know from Lemma 3.3 that V is an antistabilizable (A, B) − invariant subspace and that \overline{V} is a stabilizable (A, B) − invariant subspace. Therefore the intersection is both stabilizable and antistabilizable; thus, if we show that V ∩ \overline{V} is (A, B) − invariant we conclude that it is a controllability subspace. To do this, we show that X_{R} ⊂ X_{V} ∩ X\overline{V}, and that the projection P_{H(K)Q′′}H(R^{e}) is injective; we then show thatdim R ≥ dim(V ∩ \overline{V}). To show that R is (A, B) − invariant , we show that P_{H(K)Q′′}H(R^{e}) = P_{H(K)Q′′}H(Q′_{2})Q′′ for a suitable Q′_{2}; the conclusion then follows from Lemma 3.3. We refer to [6] for a detailed proof.

Theorem 4.2 Let K = ( \begin{matrix}A & B \\ B^{∗}P^{−1} & I \end{matrix} ) an m × m

inner matrix and R be an (A, B) − invariant controllability subspace, R = Im P_{21} , F, G and D such that R = ⟨A + BF |BG⟩, σ(A + BF) ⊂ CI _{+} A, B and Q′ defined by

A^{∗} := −P_{21}^{#}(A + BF)P_{21} B := P_{21}^{∗}F^{∗}

Q′ = ( \begin{matrix}A & B \\ − BP^{−1} & I \end{matrix} )

Let K_{−}^{e} = ( \begin{matrix}A & B_{−}^{e} \\ − (B_{−}^{e})^{∗}P_{−}^{−1} & I \end{matrix} ) and \overline{Q}′ be deter-

mined from K, Q′ by the skew-prime relation

\overline{Q}′K = K_{−}^{e}Q′

with multiplicity of K_{−}^{e} denoted by m_{0} ≤ m. Assume finally that

Im G ⊂ ker B_{−}^{e}  (17)

Then there exist inner functions Q′′ of dimension m and Q_{+} and R of dimension m_{0} and m − m_{0} respec-

tively such that Q′Q′′ = ( \begin{matrix}Q_{+} & 0 \\ 0 & R \end{matrix} ) is a balanced

factorization w.r. to K_{+} and

X_{R} = P_{H(K)}H(Q′) = P_{H(K)Q′′}H(\overline{Q}′)K_{+}^{e}(18)
= P_{H(K)Q′′}H(R^{e})  (19)

Again, we refer to [6] for the proof.

Corollary 4.1 Let R ∈ CI ^{n} , and W =

( \begin{matrix}A & B \\ C & D \end{matrix} ). Then R is an (A, B) − invariant

output-nulling controllability subspace if and only if there exist Q′ and Q′′ inner such that

1. X_{V} = P_{H(K)}H(Q′) is antistabilizable output nulling.

2. X\overline{V} = P_{H(K)Q′′}H(Q′′)K_{+}^{e} is stabilizable output nulling.

3. X_{R} = V ∩ \overline{V}

Proof: The only thing to check is that condition (17) is satisfied, so that we can apply the above theorem. But this follows from the fact that W_{−} =

( \begin{matrix}A & B_{−} \\ C & D_{−} \end{matrix} ) is full column-rank at infinity, and

the output nulling condition [D_{−}, 0]G = 0 implies [B_{−}, 0]G = 0.

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