ON A CLASS OF CHARACTERISTIC FUNCTIONS AND THEIR SYSTEM APPLICATIONS

P. A. Fuhrmann^{∗}^{†}

Department of Mathematics Ben-Gurion University of the Negev Beer Sheva, Israel

Recently, see Fuhrmann and Ober [1993b], normalized coprime factorizations over H^{∞}, the normalization being with respect to degenerate, definite and indefinite metrics, have been derived for several classes of functions. This generalizes results in Vidyasagar [1985,1988], Meyer and Franklin [1987]. Using the Youla-Kucera parametrization, in each case this led to a canonical pair of solutions of the Bezout equations exhibiting the coprimeness of the factors. This costruction leads directly to the introduction of several characteristic functions, all of them stable. We refer to Ober and Fuhrmann [1993] for some of the details.

We will desribe several diverse applications of the various characteristics. For the class of antistable functions the S-characteristic relates to the inversion of intertwining maps and hence to the inversion of Hankel operators, considered as restricted to maps from the cokernels to their image. It is a trivial observation that this operation involves singular value inversion.

For the class of minimal systems the norm of the L-characteristic relates to robust stabilization under normalized coprime factor uncertainty. See Glover and McFarlane [1989] and Georgiou and Smith [1990]. We put the study of the L-characteristic in a wider context and develop around it a theory of duality between problems of robust stabilization and those of model reduction.

∗Earl Katz Family Chair in Algebraic System Theory

†Partially supported by the Israeli Academy of Sciences under Grant No. 249/90 and by GIF under Grant No. I 184.

It is well known that some robust stabilization problems are solved by use of Nehari complements. We will describe how the B-characteristic, associated with the class of bounded real functions, can be used to derive a special suboptimal Nehari complement and furthermore to parametrize the whole set.

Finally, for the class of positive real systems, we relate the P -characteristic to the phase function associated with a stationary, Gaussian, stochastic process.

In all cases under study we have a bijective map between the class under study and the set of proper antistable functions. State space formulas are derived for the characteristic map and its inverse. The characteristic map is derived via the use of Riccati equations, whereas for its inverse we need to solve Lyapunov equations. The connection between solutions of Riccati and Lyapunov equations lead to several results on balanced realizations.

1 Introduction

In an ever widening field of research, as control theory evidently is, it is of paramount importance to structure the field in order to enable us to see the forest as well as the trees. Thus the search for an underlying unity that puts some order in an otherwise, seemingly random, collection of facts.

It is in this spirit that the work presented in this paper has been conceived. Since the subject area addressed in this work is quite large, no attempt will be made to present the mass of details. Rather, we look at this as some road map describing, from a given starting point, the various directions in which the interested reader can proceed. Our feeling is that not a single one of these directions has been exhausted. The methods developed and utilized in this work are not overly difficult, yet very powerful. The further application of the same methods is bound to be as fruitful in the future as it has been in the past.

The initial motivation that led to most of this research was the attempt, made by R. Ober and the author, to gain a better understanding of the result, due to Glover and McFarlane [1989], that gave an explicit formula for the maximal robust stability margin for normalized cofactor uncertainty.

Let us review this particular result. Given a rational matrix function G, we consider the left and right normalized coprime factorizations over H_{+}^{∞}, i.e.

G = \overline{M}^{−1}\overline{N} = NM^{−1},  (1)

where M, N, \overline{N}, \overline{M} ∈ H_{+}^{∞}. The normalizations are given by requiring that

\{ \begin{matrix}\overline{M}\overline{M}^{∗} + \overline{N}\overline{N}^{∗} & = & I \\ M^{∗}M + N^{∗}N & = & I \end{matrix}  (2)

Let

( \begin{matrix}\overline{V} & −\overline{U} \\ −\overline{N} & \overline{M} \end{matrix} ) ( \begin{matrix}M & U \\ N & V \end{matrix} ) = ( \begin{matrix}I & 0 \\ 0 & I \end{matrix} )  (3)

be any doubly coprime factorization arising out of the normalized coprime factorizations (1). This relates of course to the Youla-Kucera parametrization of all stabilizing controllers for G. Now the solutions of the two Bezout equations, namely

\{ \begin{matrix}\overline{V} M − \overline{U}N & = & I \\ \overline{M}V − \overline{N}U & = & I \end{matrix}  (4)

are anything but unique. However the nonuniqueness is easily parametrized by an H_{+}^{∞} function Q. In fact, if (V_{0}, U_{0}) solves the second Bezout equation in (4), then the general solution is given by

( \begin{matrix}U \\ V \end{matrix} ) = ( \begin{matrix}U_{0} \\ V_{0} \end{matrix} ) + ( \begin{matrix}M \\ N \end{matrix} ) Q  (5)

and a similar parametrization holds for the other Bezout equation. Using (5) and the normalization condition, we get

M^{∗}U + N^{∗}V = M^{∗}U_{0} + N^{∗}V_{0} + Q.  (6)

Since the Hankel operator with an analytic symbol is trivial, it follows that the Hankel operator H_{M^{∗}U+N^{∗}V} is independent of the particular solution of the Bezout equation, and thus depends only on the normalized coprime factorization which is essentially unique. Hence this Hankel operator is a system invariant. We can use the freedom of the parameter Q in equation (5) in order to achieve the condition R^{∗} = M^{∗}U +N^{∗}V ∈ H_{+}^{∞}, and moreover to fix its value at ∞. This particular R^{∗} is termed the LQG characteristic in Fuhrmann and Ober [1993a]. The derivation presented here is the prototype for similar derivations in other classes of functions.

Now, it can be shown that for a system with normalized coprime factorizations given by (1), the optimally robust controller with respect to normalized coprime factor

uncertainty is given by the controller that minimizes the H_{+}^{∞} norm of ( \begin{matrix}U \\ V \end{matrix} ), or alternatively, of ( \overline{V} −\overline{U} ). The following striking formulas for the maximal robust stability margin were derived by Glover and McFarlane.

ϵ_{max} = (1 + ∥H_{M^{∗}U+N^{∗}V} ∥^{2})^{− \frac{1}{2}} = (1 − ∥H_{( _{M^{∗} N^{∗}} )}∥^{2})^{\frac{1}{2}}

An inkling of how to understand this striking formula can be gained from the commutative diagram given in section 5.

When analyzing other classes of functions, different normalized coprime factorizations are used. However the process of defining the corresponding characteristic functions is as previously described. The characteristic function is a convenient tool in the study of the original transfer function. First, it characterizes the original function in a bijective way. Secondly, and more importantly, being antistable it is associated with a Hankel operator, the study of which sheds light on important properties of the original function.

The paper is organized as follows. In section 2 we study normalized coprime factorizations for several classes of functions. Next, in section 3, we define thecharacteristic function and derive the characteristic functions for all these classes. We also present state space formulas which are dependent on solution of Riccati equations.

The derivation of the characteristic function proceeds always via the solution of two Riccati equations. Conversely, with each characteristic function there are associated two Lyapunov equations. It turns out that there are simple relations between these pairs of equations. In particular the (Lyapunov) balancing of the realization of the characteristic function leads to the derivation of various balanced canonical forms for these classes of functions. These canonical forms were first derived in Ober [1993]. In this connection see also Fuhrmann [1991] and Hoffmann and Fuhrmann [1994].

Next, in sections 4 - 7, we describe specific applications for the various characteristic functions. For the class of antistable functions we relate the characteristic function to the inversion of maps intertwining model operators. This can also be interpreted as the inversion of Hankel operators, when considered as maps from their cokernel to the image. These results are immediately applied in section 5 to the development of duality theory. This puts model reduction and robust control within the same framework.

We proceed, in section 6, to the study of suboptimal Nehari complements. This leads to a particularly short route to the parametrization of all suboptimal Nehari complements. Moreover, it clarifies the connections between several diverse approaches to this problem. Finally, in section 7, we show some connections to the study of spectral factorizations and relate the phase function to the characteristic function.

The results presented in this paper have been derived in collaboration with Raimund Ober, or have their roots in this collaboration. More complete accounts can be found in Fuhrmann and Ober [1993a,b], Ober and Fuhrmann [1993] and Fuhrmann [1994a,b].

2 Coprime Factorizations Over H^{∞}

We denote by H_{+}^{∞} the Banach space of all bounded, matrix valued, analytic functions in the open right half plane. In order not to encumber the notation, we omit the reference to the size of the matrices as this should always be clear from the context. In the case of square matrices H_{+}^{∞} is a Banach algebra. By a theorem of Fatou, these functions have nontangential boundary values a.e. on the imaginary axis. Thus H_{+}^{∞} can be identified with a closed subspace of L^{∞} of the imaginary axis.

Two elements M, N ∈ H_{+}^{∞} are called (strongly) right coprime if there exists an H_{+}^{∞} solution to the Bezout equation

\overline{V} M − \overline{U}N = I.  (7)

Similarly, \overline{M}, \overline{N} ∈ H_{+}^{∞} are called (strongly) left coprime if there exists an H_{+}^{∞} solution to the Bezout equation

\overline{M}V − \overline{N}U = I.  (8)

Note that M, N need not have the same size, only the same number of columns. There is also a weaker notion of coprimeness in H_{+}^{∞}, see Fuhrmann [1981] for the details, but we will not discuss it here as we focus on rational functions and in this case the two notions of coprimeness coincide.

Let G be a proper rational matrix-valued function. Then the factorization G = NM^{−1} is called a right factorization of G, if N, M are stable rational functions and M is invertible with proper inverse. If N, M are right coprime, then the factorization is called a right coprime factorization. Similarly we define a left coprime factorization G = \overline{M}^{−1}\overline{N}. The two proper stable rational block matrices

( \begin{matrix}M & U \\ N & V \end{matrix} ) , ( \begin{matrix}\overline{V} & −\overline{U} \\ −\overline{N} & \overline{M} \end{matrix} ) ,

with M (\overline{M}) having a proper inverse, form a doubly coprime factorization of the proper rational function G, if

( \begin{matrix}\overline{V} & −\overline{U} \\ −\overline{N} & \overline{M} \end{matrix} ) ( \begin{matrix}M & U \\ N & V \end{matrix} ) = ( \begin{matrix}I & 0 \\ 0 & I \end{matrix} )

and

G = NM^{−1} = \overline{M}^{−1}\overline{N}.

Doubly coprime factorizations are related to the Youla-Kucera parametrization of all stabilizing controllers. This accounts for the importance of coprime factorizations over H_{+}^{∞}. We note that given a particular doubly coprime factorization, the general solution of the Bezout equation (7) is given by

( \begin{matrix}U \\ V \end{matrix} ) = ( \begin{matrix}U_{0} \\ V_{0} \end{matrix} ) + ( \begin{matrix}M \\ N \end{matrix} ) Q  (9)

where Q can be chosen arbitrarily in H_{+}^{∞}. Similarly

( \overline{V} −\overline{U} ) = ( \overline{V}_{0} −\overline{U}_{0} ) + Q ( −\overline{N} \overline{M} )  (10)

Now let us specify the classes of functions we shall consider. We start with the class of antistable functions which we shall identify with H_{−}^{∞}. Thus, at least in the rational case these are functions with poles in the open right half plane and which are bounded at ∞. The class of minimal functions we consider the set of all proper rational functions, without restrictions on the pole locations. The bounded real functions are the subset of H_{+}^{∞} functions which are strictly contractive, i.e. satisfy I−G(s)^{∗}G(s) > 0 for Res > 0. Positive real functions are, in our case, rational functions in H_{+}^{∞} satisfying G(s) + G(s)^{∗} > 0 in Res > 0.

For these classes of functions H_{+}^{∞} we will look at normalized coprime factorizations with respect to various metrics. In all cases G = NM^{−1} = \overline{M}^{−1}\overline{N}, however the normalization is with respect to an appropriate metric. We sum up the definitions in the following table.

Class  Metric  Normalization
Antistable  J_{S} = ( \begin{matrix}I & 0 \\ 0 & 0 \end{matrix} )  \begin{aligned}M_{S}^{∗}M_{S} & = & I \\ \overline{M}_{S}\overline{M}_{S}^{∗} & = & I \end{aligned}
Minimal  J_{L} = ( \begin{matrix}I & 0 \\ 0 & I \end{matrix} )  \begin{aligned}M_{L}^{∗}M_{L} + N_{L}^{∗}N_{L} & = & I \\ \overline{M}_{L}\overline{M}_{L}^{∗} + \overline{N}_{L}\overline{N}_{L}^{∗} & = & I \end{aligned}
Bounded real J_{B} = ( \begin{matrix}I & 0 \\ 0 & −I \end{matrix} ) \begin{aligned}M_{B}^{∗}M_{B} − N_{B}^{∗}N_{B} & = & I \\ \overline{M}_{B}\overline{M}_{B}^{∗} − \overline{N}_{B}\overline{N}_{B}^{∗} & = & I \end{aligned}
Positive real J_{P} = ( \begin{matrix}0 & I \\ I & 0 \end{matrix} )  \begin{aligned}M_{P}^{∗} N_{P} + N_{P}^{∗} M_{P} & = & I \\ \overline{M}_{P} \overline{N}_{P}^{∗} + \overline{N}_{P} \overline{M}_{P}^{∗} & = & I \end{aligned}

The existence of all these normalized coprime factorizations is most easily proved via spectral factorization techniques. However, from a computational point of view, it is desirable to have state space formulas available. A general approach to state space formulas for coprime factorizations over H_{+}^{∞} was given in Khargonekar and Sontag [1982]. For normalized coprime factorizations the modified formulas were derived by Meyer and Franklin [1987] and Vidyasagar [1988]. A comprehensive study is given in Fuhrmann and Ober [1993b]. In the next table we list the various Riccati equations that arise naturally, through spectral factorizations, in the derivation of the state space formulas for the appropriate normalized coprime factorizations. Usually one Riccati equation arises for the left factorization and the other for the right one.

Class  Riccati Equations
Antistable  A^{∗}X + XA − XBB^{∗}X = 0
AZ + ZA^{∗} − ZC^{∗}CZ = 0
Minimal  A^{∗}X + XA + C^{∗}C − XBB^{∗}X = 0
AZ + ZA^{∗} + BB^{∗} − ZC^{∗}CZ = 0
Bounded real  A^{∗}X + XA + C^{∗}C + XBB^{∗}X = 0
AZ + ZA^{∗} + BB^{∗} + ZC^{∗}CZ = 0
Positive real (A^{∗} − C^{∗}R_{D}^{−1}B^{∗})X + X(A − BR_{D}^{−1}C) + C^{∗}R_{D}^{−1}C + XBR_{D}^{−1}B^{∗}X = 0

(A − BR_{D}^{−1}C)Z + Z(A^{∗} − C^{∗}R_{D}^{−1}B^{∗}) + BR_{D}^{−1}B^{∗} + ZC^{∗}R_{D}^{−1}CZ = 0

Next we turn to the state space realizations of the normalized coprime factors. We note that for the class of antistable functions the formulas are apparently due to Doyle, while for minimal systems to Meyer and Franklin [1987]. For the bounded real and positive real case the formulas were derived in Fuhrmann and Ober [1993].

In the next table we summarize the state space formulas for the normalized coprime factors and the Riccati equations that need to be solved. In all cases we need to choose the stabilizing solution of the Riccati equation. We begin by summarizing the Riccati equations.

8 Fuhrmann Antistable system

( \begin{matrix}M_{S} \\ N_{S} \end{matrix} ) = ( \begin{matrix}A − BB^{∗}X & B \\ −B^{∗}X & I \\ C & 0 \end{matrix} )

( ^{\overline{N}_{S} \overline{M}_{S}} ) = ( \begin{matrix}A − ZC^{∗}C & −B & ZC^{∗} \\ −C & 0 & I \end{matrix} )

Minimal systems

( \begin{matrix}M_{L} \\ N_{L} \end{matrix} ) = ( \begin{matrix}A − BB^{∗}X & B \\ −B^{∗}X & I \\ C & 0 \end{matrix} )

( ^{−\overline{N}_{L} \overline{M}_{L}} ) = ( \begin{matrix}A − ZC^{∗}C & B & ZC^{∗} \\ −C & 0 & I \end{matrix} )

Bounded real T_{D} = (I − D^{∗}D)^{\frac{1}{2}} , \overline{T}_{D} = (I − DD^{∗})^{\frac{1}{2}}

( \begin{matrix}M_{B} \\ N_{B} \end{matrix} ) = ( \begin{matrix}A + BT_{D}^{−1}(B^{∗}X + D^{∗}C) & BT_{D}^{− \frac{1}{2}} \\ (B^{∗}X + D^{∗}C)T_{D}^{−1} & T_{D}^{− \frac{1}{2}} \\ C + DT_{D}^{−1}(B^{∗}X + D^{∗}C) & DT_{D}^{− \frac{1}{2}} \end{matrix} )

\begin{matrix}( −\overline{N}_{B} \overline{M}_{B} ) = \\ ( & \begin{matrix}A + (ZC^{∗} + BD^{∗})\overline{T} _{D}^{−1}C & B + (ZC^{∗} + BD^{∗})\overline{T} _{D}^{−1}D & (ZC^{∗} + BD^{∗})\overline{T} _{D}^{−1} \\ \overline{T} _{D}^{− \frac{1}{2}} C & \overline{T} _{D}^{− \frac{1}{2}} D & \overline{T} _{D}^{− \frac{1}{2}} \end{matrix} & ) \end{matrix}

Positive real R_{D} = (D + D^{∗})^{\frac{1}{2}}

( \begin{matrix}M_{P} \\ N_{P} \end{matrix} ) = ( \begin{matrix}A − BR_{D}^{−1}(C − B^{∗}X) & BR_{D}^{− \frac{1}{2}} \\ −R_{D}^{−1}(C − B^{∗}X) & R_{D}^{− \frac{1}{2}} \\ D^{∗}R_{D}^{−1}C + DR_{D}^{−1}B^{∗}X & DR_{D}^{− \frac{1}{2}} \end{matrix} )

\begin{matrix}( −\overline{N}_{P} \overline{M}_{P} ) = \\ ( \begin{matrix}A − (B − ZC^{∗})R_{D}^{−1}C & −[BR_{D}^{−1}D^{∗} + ZC^{∗}R_{D}^{−1}D] & (B − ZC^{∗})R_{D}^{−1} \\ R_{D}^{− \frac{1}{2}} C & −R_{D}^{− \frac{1}{2}} D & R_{D}^{− \frac{1}{2}} \end{matrix} ) , \end{matrix}

3 Characteristic Functions

We proceed to derive the various characteristic functions. We do this within one framework. Assume G is rational and G = NM^{−1} = \overline{M}^{−1}\overline{N} are normalized coprime factorizations with respect to the metric J. The metric varies depending on the class of functions. The general solution to the Bezout equation \overline{M}V − \overline{N}U = I is given, in terms of a particular solution by

( \begin{matrix}U \\ V \end{matrix} ) = ( \begin{matrix}U_{0} \\ V_{0} \end{matrix} ) + ( \begin{matrix}M \\ N \end{matrix} ) Q  (11)

Multiplying on the left by ( M^{∗} N^{∗} ) J we get

( M^{∗} N^{∗} ) J ( \begin{matrix}U \\ V \end{matrix} ) = ( M^{∗} N^{∗} ) J ( \begin{matrix}U_{0} \\ V_{0} \end{matrix} ) + Q

Since we are in the rational case, using partial fraction decompositions, we can choose Q in such a way that

R^{∗} = ( M^{∗} N^{∗} ) J ( \begin{matrix}U \\ V \end{matrix} ) ∈ H_{−}^{∞}  (12)

and is moreover strictly proper. This determines R^{∗} uniquely. We call this the char-

acteristic function of G. The controller ( \begin{matrix}U_{c} \\ V_{c} \end{matrix} ) that leads to the characteristic

function will be called the special controller. A similar construction leads to

\overline{R}^{∗} = ( \overline{V}_{c} −\overline{U}_{c} ) J ( \begin{matrix}\overline{M} & ∗ \\ −\overline{N}^{∗} \end{matrix} ) ∈ H_{−}^{∞}.  (13)

It is easy to show that \overline{R} = R. The form of the characteristic function is given in

Class  Normalization  Characteristic
Antistable  \begin{aligned}M_{S}^{∗}M_{S} & = & I \\ \overline{M}_{S}\overline{M}_{S}^{∗} & = & I \end{aligned}  \begin{aligned}χ_{S} & = & M_{S}^{∗}U_{S} \\ & = & \overline{U}_{S}\overline{M}_{S}^{∗} \end{aligned}
Minimal  \begin{aligned}M_{L}^{∗}M_{L} + N_{L}^{∗}N_{L} & = & I \\ \overline{M}_{L}\overline{M}_{L}^{∗} + \overline{N}_{L}\overline{N}_{L}^{∗} & = & I \end{aligned}  \begin{aligned}χ_{L} & = & M_{L}^{∗}U_{L} + N_{L}^{∗}V_{L} \\ & = & \overline{U}_{L}\overline{M}_{L}^{∗} + \overline{V}_{L}\overline{N}_{L}^{∗} \end{aligned}
Bounded real \begin{aligned}M_{B}^{∗}M_{B} − N_{B}^{∗}N_{B} & = & I \\ \overline{M}_{B}\overline{M}_{B}^{∗} − \overline{N}_{B}\overline{N}_{B}^{∗} & = & I \end{aligned} \begin{aligned}χ_{B} & = & M_{B}^{∗}U_{B} − N_{B}^{∗}V_{B} \\ & = & \overline{U}_{B}\overline{M}_{B}^{∗} − \overline{V}_{B}\overline{N}_{B}^{∗} \end{aligned}
Positive real \begin{aligned}M_{P}^{∗} N_{P} + N_{P}^{∗} M_{P} & = & I \\ \overline{M}_{P} \overline{N}_{P}^{∗} + \overline{N}_{P} \overline{M}_{P}^{∗} & = & I \end{aligned}  \begin{aligned}χ_{B} & = & M_{P}^{∗} V_{P} + N_{P}^{∗} U_{P} \\ & = & \overline{V}_{P} \overline{M}_{P}^{∗} + \overline{U}_{P} \overline{N}_{P}^{∗} \end{aligned}

The doubly coprime factorization equations, taken together with the definition of the characteristic function leads to interesting equalities. We sum these up in the following table.

Class  Equation
Antistable  \begin{aligned}( \begin{matrix}U_{S} \\ V_{S} \end{matrix} ) & = & ( \begin{matrix}M_{S} \\ N_{S} \end{matrix} ) R_{S}^{∗} + ( \begin{matrix}0 \\ −\overline{M}^{∗} \end{matrix} ) \\ ( \overline{V}_{S} −\overline{U}_{S} ) & = & R_{S}^{∗} ( \overline{N}_{S} −\overline{M}_{S} ) + ( M_{S}^{∗} 0 ) \end{aligned}

Minimal  \begin{aligned}( \begin{matrix}U_{L} \\ V_{L} \end{matrix} ) & = & ( \begin{matrix}M_{L} \\ N_{L} \end{matrix} ) R_{L}^{∗} + ( \begin{matrix}−\overline{N}_{L}^{∗} \\ \overline{M} & {∗ \atop L} \end{matrix} ) \\ ( \overline{V}_{L} −\overline{U}_{L} ) & = & R_{L}^{∗} ( \overline{N}_{L} −\overline{M}_{L} ) + ( M_{L}^{∗} N_{L}^{∗} ) \end{aligned}

Bounded real \begin{aligned}( \begin{matrix}U_{B} \\ V_{B} \end{matrix} ) & = & ( \begin{matrix}M_{B} \\ N_{B} \end{matrix} ) R_{B}^{∗} + ( \begin{matrix}\overline{N} & {∗ \atop B} \\ \overline{M} & {∗ \atop B} \end{matrix} ) \\ ( \overline{V}_{B} −\overline{U}_{B} ) & = & R_{B}^{∗} ( \overline{N}_{B} −\overline{M}_{B} ) + ( N_{B}^{∗} M_{B}^{∗} ) \end{aligned}

Positive real \begin{aligned}( \begin{matrix}U_{P} \\ V_{P} \end{matrix} ) & = & ( \begin{matrix}M_{P} \\ N_{P} \end{matrix} ) R_{P}^{∗} + ( \begin{matrix}−\overline{M}_{P}^{∗} \\ \overline{N} & {∗ \atop P} \end{matrix} ) \\ ( \overline{V}_{P} −\overline{U}_{P} ) & = & R_{P}^{∗} ( \overline{N}_{P} −\overline{M}_{P} ) + ( N_{P}^{∗} M_{P}^{∗} ) \end{aligned}

For each class, these equalities yield important information. However, before getting into that, we note that all these equalities have one of the dual forms

\{ \begin{matrix}G_{1} & = & GK^{∗} + L^{∗} \\ G_{1} & = & K^{∗}G + L^{∗} \end{matrix}  (14)

with K, L ∈ H_{+}^{∞}. These equations can be interpreted in terms of Hankel operators.

Let L^{∞} denote the space of measurable, essentially bounded functions on the imaginary axis. H_{+}^{2} will denote both the Hardy space of vector valued, analytic functions in the open right half plane which satisfy

∥f∥^{2} = sup_{x>0} ∫_{−∞}^{∞} ∥f(x + iy)∥^{2}dy < ∞,

and the subspace of L^{2} of their boundary values. Similarly we dfine H_{−}^{2}. As a cosequence of the Fourier-Plancherel and the Paley-Wiener theorems we have the orthogonal direct sum decomposition

L^{2} = H_{+}^{2} ⊕ H_{−}^{2}.

We denote by P_{+} and P_{−} the orthogonal projections of L^{2} onto H_{+}^{2} and H_{−}^{2} respectively. Given a function G ∈ L^{∞} the Hankel operator H_{G} : H^{2} ⟶ H_{−}^{2} is defined by

H_{G}f = P_{−}Gf f ∈ H_{+}^{2}.

We will denote by Ĥ_{Φ} : H_{−}^{2} ⟶ H_{+}^{2} the involuted Hankel operator defined by

Ĥ_{Φ}f = P_{+}(Φ^{∗}f), for f ∈ H_{−}^{2}.  (15)

Clearly we have, for Φ ∈ L^{∞}(iR),

(H_{Φ})^{∗} = Ĥ_{Φ^{∗}}  (16)

In fact it has been proved, in Fuhrmann [1981], that

\begin{matrix}ImH_{G_{1}} ⊂ ImH_{G} ⇔ G_{1} = GK + L, K, L ∈ H_{+}^{∞} \\ KerH_{G_{1}} ⊃ KerH_{G} ⇔ G_{1} = KG + L, K, L ∈ H_{+}^{∞} \end{matrix}  (17)

Thus the equalities in the previous table imply

\begin{matrix}ImH_{( \begin{matrix}U_{c} \\ V_{c} \end{matrix} )} ⊂ ImH_{( \begin{matrix}M_{c} \\ N_{c} \end{matrix} )} \\ KerH_{( _{\overline{V}_{c} −\overline{U}_{c}} )} ⊃ KerH_{( _{−\overline{N}_{c} \overline{M}_{c}} )}. \end{matrix}  (18)

Using the realization theory based on model operators, see Fuhrmann [1981] for details, we can derive state space formulas for the special controllers. In all cases the matrices X, Z are the stabilizing solutions of the corresponding Riccati equations. This is summed up in 12 Fuhrmann

Special Controllers: State Space Equations

\begin{aligned}( \begin{matrix}U_{S} \\ V_{S} \end{matrix} ) & = & ( \begin{matrix}A − BB^{∗}X & ZC^{∗} \\ −B^{∗}X & 0 \\ C & I \end{matrix} ) \\ ( \overline{V}_{S} −\overline{U}_{S} ) & = & ( \begin{matrix}A − ZC^{∗}C & B & ZC^{∗} \\ B^{∗}X & I & 0 \end{matrix} ) \end{aligned}

\begin{aligned}( \begin{matrix}U_{L} \\ V_{L} \end{matrix} ) & = & ( \begin{matrix}A − BB^{∗}X & ZC^{∗} \\ −B^{∗}X & 0 \\ C & I \end{matrix} ) \\ ( \overline{V_{L}} −\overline{U_{L}} ) & = & ( \begin{matrix}A − ZC^{∗}C & B & ZC^{∗} \\ B^{∗}X & I & 0 \end{matrix} ) \end{aligned}

\begin{aligned}( \begin{matrix}U_{B} \\ V_{B} \end{matrix} ) & = & ( \begin{matrix}A + BB^{∗}X & −ZC^{∗} \\ B^{∗}X & 0 \\ C & I \end{matrix} ) \\ ( \overline{V}_{B} −\overline{U}_{B} ) & = & ( \begin{matrix}A + ZC^{∗}C & −B & ZC^{∗} \\ B^{∗}X & I & 0 \end{matrix} ) \end{aligned}

\begin{matrix}( \begin{matrix}U_{P} \\ V_{P} \end{matrix} ) = ( \begin{matrix}A − BR_{D}^{−1}(C − B^{∗}X) & −ZC^{∗}R_{D}^{− \frac{1}{2}} \\ −R_{D}^{−1}(C − B^{∗}X) & −R_{D}^{− \frac{1}{2}} \\ D^{∗}R_{D}^{−1}C + DR_{D}^{−1}B^{∗}X & D^{∗}R_{D}^{− \frac{1}{2}} \end{matrix} ) \\ ( & \overline{V}_{P} −\overline{U}_{P} & ) \\ = ( \begin{matrix}A − (B − ZC^{∗})R_{D}^{−1}C & −[BR_{D}^{−1}D^{∗} + ZC^{∗}R_{D}^{−1}D] & (B − ZC^{∗})R_{D}^{−1} \\ R_{D}^{− \frac{1}{2}} B^{∗}X & R_{D}^{− \frac{1}{2}} D^{∗} & R_{D}^{− \frac{1}{2}} \end{matrix} ) \end{matrix}

The next step is the derivation of state space formulas for the characteristic function. We organize these in the following table. In all cases we assume G = ( \begin{matrix}A & B \\ C & D \end{matrix} ) and that X, Z are the stabilizing solutions of the respective Riccati equations.

Class  Realization

Antistable  \begin{aligned}R_{S} & = & ( \begin{matrix}A − BB^{∗}X & B \\ CZX & 0 \end{matrix} ) \\ & = & ( \begin{matrix}A − ZC^{∗}C & ZXB \\ C & 0 \end{matrix} ) \end{aligned}

Minimal  \begin{aligned}R_{S} & = & ( \begin{matrix}A − BB^{∗}X & B \\ CZX & 0 \end{matrix} ) \\ & = & ( \begin{matrix}A − ZC^{∗}C & ZXB \\ C & 0 \end{matrix} ) \end{aligned}

Bounded real  \begin{aligned}R_{B} & = & ( \begin{matrix}A + BB^{∗}X & B \\ −C(I − ZX) & 0 \end{matrix} ) \\ & = & ( \begin{matrix}A + ZC^{∗}C & (I − ZX)B \\ −C & 0 \end{matrix} ) \end{aligned}

Positive real \begin{aligned}R_{P} & = & ( \begin{matrix}A − BR_{D}^{−1}(C − BX) & BR_{D}^{− \frac{1}{2}} \\ −R_{D}^{− \frac{1}{2}} C(I − ZX) & 0 \end{matrix} ) \\ & = & ( \begin{matrix}A − (B − ZC)R_{D}^{−1}C & (I − ZX)BR_{D}^{− \frac{1}{2}} \\ −R_{D}^{− \frac{1}{2}} C & 0 \end{matrix} ) \end{aligned}

With these realizations are associated Lyapunov equations. In fact if R_{c} = ( \begin{matrix}A & B \\ C & 0 \end{matrix} )

is the stable form of the characteristic function then we consider the Lyapunov equations

\begin{matrix}A^{∗}P + P A = −C^{∗}C \\ QA^{∗} + AQ = −BB^{∗} \end{matrix}  (19)

One might expect that there is a connection between the solutions of these Lyapunov equations and the Riccati equations that led to the characteristic functions. This is indeed the case and these connections can be summarized as follows.

Realization  Equation

Antistable

( \begin{matrix}A − BB^{∗}X & B \\ CZX & 0 \end{matrix} )  Q = X^{−1}
P = XZX
( \begin{matrix}A − ZC^{∗}C & ZXB \\ C & 0 \end{matrix} )  Q = ZXZ
P = Z^{−1}

Minimal

( \begin{matrix}A − BB^{∗}X & B \\ CZX & 0 \end{matrix} )  Q = (I + ZX)^{−1}Z = Z(I + XZ)^{−1}
P = X(I + ZX) = (I + XZ)X
( \begin{matrix}A − ZC^{∗}C & ZXB \\ C & 0 \end{matrix} ) Q = (I + ZX)^{−1}Z = Z(I + XZ)^{−1}
P = X(I + ZX) = (I + XZ)X

Bounded real

( \begin{matrix}A + BB^{∗}X & B \\ −C(I − ZX) & 0 \end{matrix} ) P = (I − ZX)^{−1}Z = Z(I − XZ)^{−1}
Q = X(I − ZX) = (I − XZ)X
( \begin{matrix}A − ZC^{∗}C & ZXB \\ C & 0 \end{matrix} )  P = (I − ZX)Z = Z(I − XZ)
Q = X(I − ZX)^{−1} = (I − XZ)^{−1}X

These equalities have several implications. First, it is easy to invert the characteristic map χ, i.e the map from G to R. This can be done in two different ways. One via state space formulas, the other using J-spectral factorizations. For the details see Ober and Fuhrmann [1993]. These maps provide diffeomorphic maps between each class of functions and H_{−}^{∞}. The second implication is in regard to balancing. In fact if the realization of the characteristic function is Lyapunov balanced then P = Q are bot diagonal which implies that X, Z are both diagonal, though not necessarily equal. An elementary, diagonal, state space isomorphism leads now to balanced realizations in the original framework. Thus we get LQG , bounded real and positive real balancing. See Ober [1993] and Ober and Fuhrmann [1993] for details. Hoffmann and Fuhrmann [1994] is also relevant.

4 Antistable Functions

Given a rational function G ∈ H_{−}^{∞} we consider the J_{S}-normalized coprime factorizations

G = N_{S}M_{S}^{−1} = \overline{M}_{S}^{−1}\overline{N}_{S}.

The factorizations reduce in this case to what is now referred to as DSS factorizations, see Douglas, Shapiro and Shields [1970] and Fuhrmann [1975,1981]. Here, due to the specific normalization, the denominator matrices M_{S}, \overline{M}_{S} are inner functions in H_{+}^{∞}. Inner functions provide of course, via the Beurling-Lax-Halmos theorem, a parametrization of all invariant subspaces of H_{+}^{2}. In fact the inner functions relate to the kernel and image of the Hankel operator H_{G}. Indeed we have

\{ \begin{matrix}KerH_{G} & = & M_{S}H_{+}^{2} \\ ImH_{G} & = & H_{−}^{2} ⊖ \overline{M}_{S}^{∗}H_{−}^{2} \end{matrix}

The equality N_{S}M_{S}^{−1} = \overline{M}_{S}^{−1}\overline{N}_{S} is equivalent to

\overline{N}_{S}M_{S} = \overline{M}_{S}N_{S}  (20)

which relates to the commutant lifting theorem, see Sz.-Nagy and Foias [1970], and is the key to the analysis of intertwining maps for model operators. Such an intertwining map Z : \{M_{S}H_{+}^{2}\}^{⊥} ⟶ \{\overline{M}_{S}H_{+}^{2}\}^{⊥} is defined by

Zf = P_{\{\overline{M}_{S}H_{+}^{2}\}}_{⊥} \overline{N}_{S}f.

If we rewrite the doubly coprime factorization in the form

( \begin{matrix}M_{S} & U_{S} \\ N_{S} & V_{S} \end{matrix} ) ( \begin{matrix}\overline{V}_{S} & −\overline{U}_{S} \\ −\overline{N}_{S} & \overline{M}_{S} \end{matrix} ) = ( \begin{matrix}I & 0 \\ 0 & I \end{matrix} )

we obtain a new pair of Bezout equations, describing the left coprimeness of M_{S}, U_{S} and the right coprimeness of \overline{U}_{S}, \overline{M}_{S}. Moreover we get the intertwining relation

U_{S}\overline{M}_{S} = M_{S}\overline{U}_{S}.

In fact this leads to the inversion of Z, the inverse map being given by

Z^{−1}g = −P_{{M_{S}H_{+}^{2}}}_{⊥}U_{S}g.

For more on this see Fuhrmann [1981] or Fuhrmann [1968a,b] where the connection between invertibility of intertwining maps, coprimeness and the Carleson [1962] corona theorem have been originally established.

Given a function G ∈ H_{−}^{∞} admitting the DSS factorizations (20), it is easy to relate the intertwining map to the Hankel operator H_{G}. The connection is given by Z = \overline{M}_{S}H_{G}|_{\{M_{S}H_{+}^{2}\}}_{⊥}. Considering the rational case, then Z has singular values μ_{i}. It

follows that Z^{−1} has singular values μ_{i}^{−1}. However, Z and H_{G} have the same singular values, so also Z^{−1} and H_{−M_{S}^{−1}U_{S}} = H_{−\overline{U}_{S} \overline{M}_{S}^{−1}} have the same singular values. Thus

in the passage from G to R_{S}^{∗} = \overline{U}_{S}\overline{M}_{S}^{−1} = M_{S}^{−1}U_{S} the Hankel singular values are inverted. This has direct implications towards the development of duality theory.

5 Minimal Functions

The idea of studying linear systems via their coprime factorizations over H_{+}^{∞} apparently originated with Youla and Kucera who used this, and the related Bezout equations, to parametrize all stabilizing controllers of a given system. In this connection see Vidyasagar [1985]. The advantage in this approach is that by passing to, in particular a normalized, coprime factorization we have objects that behave well as far as stability is concerned. This allows us to apply well established operator theoretic techniques. In particular we can apply the body of knowledge about Hankel operators to the study of arbitrary systems. In the introduction we already described how these methods lead to the establishment of the robust stability margin with respect to normalized coprime factor uncertainty.

We outline now a framework, put forward in Fuhrmann and Ober [1993a], for the study of various problems using thhese techniques. To this end, let G be an arbitrary, proper rational function having the J_{L}-normalized coprime factorizations G = N_{L}M_{L}^{−1} = \overline{M}_{L}^{−1}\overline{N}_{L}. We consider the two Hankel operators H( \begin{matrix}−\overline{N}_{L}^{∗} \\ \overline{M} & {∗ \atop L} \end{matrix} ) and

H_{( _{M_{L}^{∗} N_{L}^{∗}} )}. Let R_{L}^{∗} = M_{L}^{∗}U_{L}+N_{L}^{∗}V_{L} be the L-characteristic function and let R_{L}^{∗} =

Φ_{K}S_{K}^{−1} = S_{I}^{−1}Φ_{I} be its coprime DSS factorizations. Obviously KerH_{R^{∗}} = {S_{K}H_{+}^{2}}^{⊥} and ImH_{R^{∗}} = {S_{I}^{∗}H_{−}^{2}}^{⊥}. However, more is true. We have also ImH_{( _{M_{L}^{∗} M_{L}^{∗}} )} =

ImH_{R^{∗}} and KerH( \begin{matrix}−\overline{N}_{L}^{∗} \\ \overline{M} & {∗ \atop L} \end{matrix} ) = KerH_{R^{∗}}. Setting KerH_{( _{M_{L}^{∗} N_{L}^{∗}} )} = Ω_{K}H_{+}^{2} and

ImH( \begin{matrix}−\overline{N}_{L}^{∗} \\ \overline{M} & {∗ \atop L} \end{matrix} ) = \{Ω_{I}^{∗}H_{−}^{2}\}^{⊥}, we get the following commutative diagram

[figure: \{S_{K}H_{+}^{2}\}^{⊥}; \{Ω_{K}H_{+}^{2}\}^{⊥}; \{Ω_{I}^{∗}H_{−}^{2}\}^{⊥}; \{S_{I}^{∗}H_{−}^{2}\}^{⊥}; Z_{K}; Y_{I}^{∗}; H_{R^{∗}}; H( \begin{matrix}−\overline{N}_{L}^{∗} \\ \overline{M} & {∗ \atop L} \end{matrix} )_{( _{−\overline{N}_{L} \overline{M}_{L}} )}; H_{(M_{L}^{∗} N_{L}^{∗})}; H( \begin{matrix}−\overline{N}_{L}^{∗} \\ \overline{M} & {∗ \atop L} \end{matrix} )]

Here Z_{K} : {S_{K}H_{+}^{2}}^{⊥} ⟶ {Ω_{K}H_{+}^{2}}^{⊥} and Y_{I} : {Ω_{I}^{∗}H_{−}^{2}}^{⊥} → {S_{I}^{∗}H_{−}^{2}}^{⊥} are the inter-

twining maps defined by Z_{K}f = P_{\{Ω_{K}H_{+}^{2}\}^{⊥}} ( \begin{matrix}U_{L} \\ V_{L} \end{matrix} ) f and Y_{I} ( \begin{matrix}g_{1} \\ g_{2} \end{matrix} ) = −P_{\{S_{I}^{∗}H_{−}^{2}\}^{⊥}}(M^{∗} N^{∗}) ( \begin{matrix}g_{1} \\ g_{2} \end{matrix} ).

The key diagram is highly structured and contains a lot of information. In particular, the singular values of H_{R^{∗}} determine the singular values of all other maps in a very simple way. Moreover, the singular vectors of all the operators are shared in a natural way. For more on this see Fuhrmann and Ober [1993a].

Due to the limited scope of this paper we will not go into the details of duality theory. We will indicate however one key result.

Let G be a proper rational function. Define G^{∗}(s) = G(−\overline{s})^{∗}. It is of interest to compare the L-characteristics of G and G^{∗}. In fact it can be proved, see Fuhrmann

[1994a] that R_{L}(G^{∗}), the L-characteristic of G^{∗}, is equal to the S-characteristic of R_{L}(G)^{∗}.

This simple result can be used to obtain the doubly coprime factorizations of G^{∗} as well as the key diagram associated with G^{∗}. This leads to the clarification of the connections between robust control and model reduction. For a comprehensive exposition, see Fuhrmann [1994a].

6 Bounded Real Functions

Nehari’s theorem plays a central role both in operator theory as well as control theory. In operator theory, in its multivariable version, it is equivalent to the commutant lifting theorem and plays a key role in the deep study of Hankel operators initialized by Adamyan, Arov and Krein [1968a,b,1971,1978]. In control theory it appears in the model matching problem, in robust stabilization and in model reduction.

To pose the problem briefly, let us consider a function F ∈ L^{∞}. We look for a function Q ∈ H_{+}^{∞} for which the norm ∥F + Q∥_{∞} is minimal. Since H_{F} = H_{F +Q}, it follows that ∥H_{F} ∥ ≤ inf_{Q∈H_{+}^{∞}}∥F + Q∥_{∞} . An optimal Nehari complement is a Q ∈ H_{+}^{∞} for which ∥H_{F} ∥ = ∥F + Q∥_{∞}. A ρ-suboptimal complement would be, for ∥H_{F} ∥ < ρ, any complement for which ∥F + Q∥_{∞} ≤ ρ. It is convenient to normalize the roblem by letting ρ = 1 and assume ∥H_{F} ∥ < 1, i.e. all the singular values of H_{F} are less than 1.

Let us make now a slight detour. Assume G ∈ H_{+}^{∞} is a rational function that is strictly contractive. Let G = N_{B}M_{B}^{−1} = \overline{M}_{B}^{−1}\overline{N}_{B} be its J_{B}-normalized coprime factorizations, and let R_{B}^{∗} = M_{B}^{∗}U_{B} − N_{B}^{∗}V_{B} be its B-characteristic.

Now, from the equations, obtained in section 3, it follows that

\{ \begin{matrix}U_{B} & = & M_{B}R_{B}^{∗} + \overline{N}_{B}^{∗} \\ \overline{U}_{B} & = & R_{B}^{∗}\overline{M}_{B} + N_{B}^{∗} \end{matrix}

or

\{ \begin{matrix}R_{B}^{∗} − M_{B}^{−1}U_{B} & = & −M_{B}^{−1}\overline{N}_{B}^{∗} \\ R_{B}^{∗} − \overline{U}_{B}\overline{M}_{B}^{−1} & = & −N_{B}^{∗}\overline{M}_{B}^{−1} \end{matrix}  (21)

It follows from the reversed form of the doubly coprime factorization of G that −M_{B}^{−1}\overline{N}_{B}^{∗} = −N_{B}^{∗}\overline{M}_{B}^{−1} is a strict contraction in L^{∞}. Hence all the singular values of H_{R_{B}^{∗}} are less than 1. But equations (21) say more. They actually point out a particular Nehari complement, namely M_{B}^{−1}U_{B}. This opens up the following way to Nehari complementation. Suppose, given a rational R^{∗} ∈ H_{−}^{∞} which we further assume to be strictly proper, that we can identify it as the B-characteristic of an H_{+}^{∞} contraction G, then we automatically have at least one Nehari complement. But this is possible, for it has been shown in Ober and Fuhrmann [1993] that the map from the set of bounded real functions in H_{+}^{∞} to the set of H_{−}^{∞} functions whose Hankel singular values are < 1 is bijective. There are several ways that this map can be inverted.

The first method involves state space computations. In fact, with ∥H_{R^{∗}}∥ < 1, let

R = ( \begin{matrix}A & B \\ C & 0 \end{matrix} ) be a minimal realization. Let P, Q be the positive definite solutions

of the Lyapunov equations

\{ \begin{matrix}AP + P A^{∗} & = & −BB^{∗} \\ A^{∗}Q + QA & = & −C^{∗}C \end{matrix}

then

G == ( \begin{matrix}A − BB^{∗}Q(I − P Q)^{−1} & B \\ −C(I − P Q)^{−1} & 0 \end{matrix} ) = ( \begin{matrix}A − P (I − QP)^{−1}C^{∗}C & −(I − P Q)^{−1}B \\ C & 0 \end{matrix} )

are minimal realizations of a bounded real contraction whose B-characteristic is R^{∗}.

As R determines G, it also determines the J_{B}-normalized coprime factorizations of G. All we need is to solve the bounded real Riccati equations. However, the solution to these equations are derivable from the solution to the previous Lyapunov equations. This immediately leads to the state space formulas

\begin{matrix}( \begin{matrix}M_{B} \\ N_{B} \end{matrix} ) = ( \begin{matrix}A & B \\ B^{∗}Q(I − P Q)^{−1} & I \\ −C(I − P Q)^{−1} & 0 \end{matrix} ) . \\ ( −\overline{N}_{B} \overline{M}_{B} ) = ( \begin{matrix}A & (I − P Q)^{−1}B & (I − P Q)^{−1}P C^{∗} \\ C & 0 & I \end{matrix} ) \end{matrix}

Another method uses J_{B}-spectral factorizations. Under the previous assumptions

on R, and setting Γ = ( \begin{matrix}I & −R^{∗} \\ 0 & I \end{matrix} ), we solve the spectral factorization problem

\{ \begin{matrix}Γ_{+}J_{B}Γ_{+}^{∗} = ΓJ_{B}Γ^{∗} \\ Γ_{+}(∞) = ( \begin{matrix}I & 0 \\ 0 & I \end{matrix} ) \end{matrix}

Denoting the solution by

\begin{matrix}Γ_{+} = ( \begin{matrix}\overline{V}_{B} & −\overline{U}_{B} \\ −\overline{N}_{B} & \overline{M}_{B} \end{matrix} ) ∈ H_{+}^{∞} \\ Γ_{+}^{−1} = ( \begin{matrix}M_{B} & U_{B} \\ N_{B} & V_{B} \end{matrix} ) \end{matrix}

we get G by letting G = N_{B}M_{B}^{−1} = \overline{M}_{B}^{−1}\overline{N}_{B}.

Finally, there is a method put forward by Krein and Melik-Adamyan [1984a,b]. This is given by solving the systems of equations

\begin{matrix}( \begin{matrix}I & H_{R^{∗}}^{∗} \\ H_{R^{∗}} & I \end{matrix} ) ( \begin{matrix}Φ_{−}^{∗} \\ Φ_{+}^{∗} \end{matrix} ) = ( \begin{matrix}−R \\ 0 \end{matrix} ) \\ ( \begin{matrix}I & H_{R^{∗}}^{∗} \\ H_{R^{∗}} & I \end{matrix} ) ( \begin{matrix}\overline{Φ} & {∗ \atop −} \\ \overline{Φ} & {∗ \atop +} \end{matrix} ) = ( \begin{matrix}0 \\ −R^{∗} \end{matrix} ) \end{matrix}

and setting

\{ \begin{matrix}M_{B} & = & I + Φ_{+}, & N_{B} & = & \overline{Φ}_{+} \\ \overline{M}_{B} & = & I + \overline{Φ}_{−}^{∗}, & \overline{N}_{B} & = & Φ_{−}^{∗} \end{matrix}

In terms of any of these methods, it is fairly straightforward to show that the parametrization of all suboptimal Nehari complements is given by

\{R^{∗} + E|E ∈ H_{+}^{∞}, ∥R^{∗} + E∥_{∞} < 1\} = \{−(N_{B}^{∗} + M_{B}^{∗}Q)(\overline{M}_{B} + \overline{N}_{B}Q)^{−1}|Q ∈ BH_{+}^{∞}\}

For the full details that have been omitted in this brief presentation we refer to Fuhrmann [1993].

7 Positive Real Functions

Let Z be a proper rational, strictly positive real function. Let the spectral function be defined by

Φ = Z + Z^{∗}.

Then Φ is a function symmetric with respect to the imaginary axis. We assume that it is coercive, i.e. Φ(it) ≥ c > 0 for some constant c. Let Z = N_{P} M_{P}^{−1} = \overline{M}_{P}^{−1}\overline{N}_{P} be a J_{P} -normalized coprime factorization. Then we can compute

Φ = Z + Z^{∗} = N_{P} M_{P}^{−1} + M_{P}^{−∗}N_{P}^{∗}
= M_{P}^{−∗}(N_{P}^{∗} M_{P} + M_{P}^{∗} N_{P})M_{P}^{−1}
= M_{P}^{−∗}M_{P}^{−1}

Similarly, Φ = \overline{M}_{P}^{−1}\overline{M}_{P}^{−∗}. Now, for a positive real function Z, the denominator matrices satisfy M_{P}^{±1}, \overline{M}_{P}^{±1} ∈ H_{+}^{∞}, so \overline{W}_{+} = M_{P}^{−∗} is the antistable, maximum phase spectral factor whereas W_{−} = \overline{M}_{P}^{−1} is the stable, minimum phase spectral factor.

The phase function T_{0} is defined by

T_{0} = \overline{W} _{+}^{−1}W_{−} = M_{P}^{∗} \overline{M}_{P}^{−1} = M_{P}^{−1}\overline{M}_{P}^{∗} .

Now we can utilize the equation, derived in section (3), that is

( \begin{matrix}U_{P} \\ V_{P} \end{matrix} ) = ( \begin{matrix}M_{P} \\ N_{P} \end{matrix} ) R_{P}^{∗} + ( \begin{matrix}−\overline{M}_{P}^{∗} \\ \overline{N} & {∗ \atop P} \end{matrix} )

to obtain \overline{M}_{P}^{∗} = M_{P} R_{P}^{∗} − U_{P}, or equivalently

T_{0} = M_{P}^{−1}\overline{M}_{P}^{∗} = R_{P}^{∗} − M_{P}^{−1}U_{P} .

Since M_{P}^{−1}U_{P} ∈ H_{+}^{∞},we see that R_{P}^{∗} is the antistable part in the partial fraction decomposition of the phase function.

This simple result facilitates many state space computations and is relevant to stochastic balancing. It also indicates the importance of the phase function. For the positive real function Z and hence also the spectral function Φ are reconstructible from the phase function. This can be done both by state space methods as well as via J_{P} -spectral factorization.

For more information on the spectral factorization problem in relation to the factorizations of the phase function, see Fuhrmann [1994].

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