Sampled-data output regulation of unstable well-posed infinite-dimensional systems with constant reference and disturbance signals

Abstract

We study the sample-data control problem of output tracking and disturbance rejection for unstable well-posed linear infinite-dimensional systems with constant reference and disturbance signals. We obtain a sufficient condition for the existence of finite-dimensional sampled-data controllers that are solutions of this control problem. To this end, we study the problem of output tracking and disturbance rejection for infinite-dimensional discrete-time systems and propose a design method of finite-dimensional controllers by using a solution of the Nevanlinna–Pick interpolation problem with both interior and boundary conditions. We apply our results to systems with state and output delays.

Appendices

A Nevanlinna–Pick interpolation problem

In this section, we obtain a necessary and sufficient condition for the solvability of the interpolation problem to which we reduce the design problem of regulating controllers. In the process, we also show how to construct a solution of the interpolation problem. Although we consider $H^{\infty }({\mathbb {E}}_1,{\mathbb {C}}^{p \times q})$ in Sect. 2, the standard theory of the Nevanlinna–Pick interpolation problem uses $H^{\infty }({\mathbb {D}},{\mathbb {C}}^{p \times q})$. Hence, it is convenient to map ${\mathbb {E}}_1$ to ${\mathbb {D}}$ via the bilinear transformation $\varphi : {\mathbb {E}}_1 \rightarrow {\mathbb {D}}:z \mapsto 1/z$.

In Sect. A.1, we recall basic facts on the Nevanlinna–Pick interpolation problem only with conditions on the interior ${\mathbb {D}}$. Section A.2 is devoted to solving the Nevanlinna–Pick interpolation problem with conditions on both the interior ${\mathbb {D}}$ and the boundary ${\mathbb {T}}$. As in [30, 48], we transform this problem into the Nevanlinna–Pick interpolation problem only with conditions on the boundary ${\mathbb {T}}$, which is always solvable.

1.1 A.1 Interpolation problem only with interior conditions

First we consider interpolation problems only with interior interpolation conditions.

Problem A.1

(Chapter 18 in [4], Section II in [21]) Suppose that $\alpha _1,\dots ,\alpha _n \in {\mathbb {D}}$ are distinct. Let vector pairs $(\xi _\ell , \eta _\ell )\in {\mathbb {C}}^p \times {\mathbb {C}}^q$ satisfy

$$\begin{aligned} \Vert \xi _\ell \Vert _{{\mathbb {C}}^p} > \Vert \eta _\ell \Vert _{{\mathbb {C}}^q} \qquad \forall \ell \in \{1,\dots , n\}. \end{aligned}$$

(A.1)

Find $\varPhi \in H^{\infty }({\mathbb {D}}, {\mathbb {C}}^{p\times q})$ such that $\Vert \varPhi \Vert _{H^{\infty }({\mathbb {D}})} < 1$ and

$$\begin{aligned} \xi _\ell ^* \varPhi (\alpha _\ell )&= \eta _\ell ^*\qquad \forall \ell \in \{1,\dots , n\}. \end{aligned}$$

We call this problem the Nevanlinna–Pick interpolation problem with n interpolation data $(\alpha _\ell , \xi _\ell ,\eta _\ell )_{\ell =1}^n$. The solvability of Problem A.1can be characterized by the so-called Pick matrix.

Theorem A.2

(Theorem 18.2.3 in [4], Theorem 2 in [21]) Consider Problem A.1. Define the Pick matrix P by

$$\begin{aligned} P := \begin{bmatrix} P_{1,1}&\cdots&P_{1,n} \\ \vdots&\vdots \\ P_{n,1}&\cdots&P_{n,n} \end{bmatrix}, \quad \text {where } P_{j,\ell } := \frac{\xi _j^*\xi _\ell - \eta _j^*\eta _\ell }{1-\alpha _j{\bar{\alpha }}_\ell }\quad \forall j,\ell \in \{1,\dots ,n\}. \end{aligned}$$

Problem A.1 is solvable if and only if P is positive definite.

Let us next introduce an algorithm to construct a solution of Problem A.1. To this end, define

$$\begin{aligned} {\mathcal {B}} := \{E \in {\mathbb {C}}^{p \times q} : \Vert E\Vert _{{\mathbb {C}}^{p\times q}} < 1 \}. \end{aligned}$$

Let $I_p$ and $I_q$ be the identity matrix with dimension p and q, respectively. For a matrix $E \in {\mathcal {B}}$, define

$$\begin{aligned} A(E)&:= (I_p-EE^*)^{-1/2},\quad B(E) := -(I_p-EE^*)^{-1/2}E \end{aligned}$$

(A.2a)

$$\begin{aligned} C(E)&:= -(I_q-E^*E)^{-1/2}E^*,\quad D(E) := (I_q-E^*E)^{-1/2}, \end{aligned}$$

(A.2b)

where $M^{-1/2}$ denotes the inverse of the Hermitian square root of a positive definite matrix M. Define the maps $U_E$ and $V_E$ by

$$\begin{aligned} U_E&:{\mathbb {C}}^{p} \times {\mathbb {C}}^{q} \rightarrow {\mathbb {C}}^{p}: (\xi ,\eta ) \mapsto A(E)\xi + B(E)\eta \\ V_E&:{\mathbb {C}}^{p} \times {\mathbb {C}}^{q} \rightarrow {\mathbb {C}}^{q}: (\xi ,\eta ) \mapsto C(E)\xi + D(E)\eta . \end{aligned}$$

The mapping $T_E$ in the lemma below is useful for solving Problem A.1.

Lemma A.3

(Lemma 6.5.10 in [47]) For a matrix $E \in {\mathcal {B}}$, define the matrices A(E), B(E), C(E), and D(E) by (A.2). The mapping

$$\begin{aligned} T_E:{\mathcal {B}} \rightarrow {\mathcal {B}}:X \mapsto \big (A(E)X+B(E)\big ) \big (C(E)X+D(E)\big )^{-1} \end{aligned}$$

(A.3)

is well defined and bijective.

A routine calculation shows that the inverse of $T_E$ is given by

$$\begin{aligned} T_E^{-1}(Y)&= \big (A(E)-YC(E)\big )^{-1} \big (YD(E)-B(E)\big ) \nonumber \\&= \big (A(E)Y-B(E)\big ) \big (\!-C(E)Y+D(E)\big )^{-1}. \end{aligned}$$

(A.4)

Lemma A.4

(Lemma 1 in [21]) Consider Problem A.1 with n interpolation data $(\alpha _\ell , \xi _\ell ,\eta _\ell )_{\ell =1}^n$. Set $E := \xi _1 \eta _1^*/\Vert \xi _1\Vert ^2_{{\mathbb {C}}^p}$ and define A(E), B(E), C(E), and D(E) as in (A.2). Define also $\nu := U_E(\xi _1,\eta _1)$ and

$$\begin{aligned} \kappa (z)&:= {\left\{ \begin{array}{ll} \frac{|\alpha _1|}{\alpha _1}\frac{z- \alpha _1}{1-{\bar{\alpha }}_1 z} &{} \text {if } \alpha _1 \not = 0 \\ z &{} \text {if } \alpha _1 = 0 \end{array}\right. },\quad X := I_p + (\kappa -1)\frac{\nu \nu ^*}{\Vert \nu \Vert ^2_{{\mathbb {C}}^p} }. \end{aligned}$$

(A.5)

Problem A.1 with n interpolation data $(\alpha _\ell , \xi _\ell ,\eta _\ell )_{\ell =1}^n$ is solvable if and only if Problem A.1 with $n-1$ interpolation data

$$\begin{aligned} \big (\alpha _\ell , X(\alpha _\ell )^*U_E(\xi _\ell ,\eta _\ell ),V_E(\xi _\ell ,\eta _\ell )\big )_{\ell =2}^n \end{aligned}$$

(A.6)

is solvable. Moreover, if $\varPhi _{n-1}$ is a solution of the problem with $n-1$ interpolation data given in (A.6), then

$$\begin{aligned} \varPhi _n := T_{-E}\left( X \varPhi _{n-1}\right) = \big (A(E)X \varPhi _{n-1}-B(E)\big ) \big (\!-C(E)X \varPhi _{n-1}+D(E)\big )^{-1} \end{aligned}$$

(A.7)

is a solution $\varPhi _n$ of the original problem with n interpolation data $(\alpha _\ell , \xi _\ell ,\eta _\ell )_{\ell =1}^n$.

The iterative algorithm derived from Lemma A.4 is called the Schur–Nevanlinna algorithm. Lemma A.4 also shows that if the problem is solvable, then there exist always solutions whose elements are rational functions.

Note that $\nu$ given in Lemma A.4 is nonzero. In fact, since $\Vert \xi _1\Vert _{{\mathbb {C}}^p} > \Vert \eta _1\Vert _{{\mathbb {C}}^q} ,$ it follows that

$$\begin{aligned} A(E)^{-1}\nu = \xi _1 - E \eta _1 = \xi _1 - \frac{\Vert \eta _1\Vert _{{\mathbb {C}}^q}^2}{\Vert \xi _1\Vert _{{\mathbb {C}}^p}^2} \xi _1 \not = 0, \end{aligned}$$

and hence $\nu \not = 0$. Furthermore, the matrix X defined by (A.5) satisfies $X(\lambda )^{-1} = X(\lambda )^*$ for all $\lambda \in {\mathbb {T}}$ and $\Vert X(z)\Vert _{{\mathbb {C}}^{p\times p}} < 1$ for all $z \in {\mathbb {D}}$.

1.2 A.2 Interpolation problem with both interior and boundary conditions

We next study interpolation problems with both interior and boundary conditions.

Problem A.5

Suppose that $\alpha _1,\dots ,\alpha _n \in {\mathbb {D}}$ and $\lambda _1,\dots ,\lambda _m \in {\mathbb {T}}$ are distinct. Consider vector pairs $(\xi _\ell , \eta _\ell )\in {\mathbb {C}}^p \times {\mathbb {C}}^q$ for $\ell \in \{1,\dots , n\}$ and matrices $F_j,G_j \in {\mathbb {C}}^{p\times q}$ for $j\in \{1,\dots , m\}$, and suppose that

$$\begin{aligned} \Vert \xi _\ell \Vert _{{\mathbb {C}}^p}&> \Vert \eta _\ell \Vert _{{\mathbb {C}}^q}\qquad \forall \ell \in \{1,\dots , n\} \end{aligned}$$

(A.8a)

$$\begin{aligned} \Vert F_j\Vert _{{\mathbb {C}}^{p\times q}}&< 1 \qquad \forall j\in \{1,\dots , m\}. \end{aligned}$$

(A.8b)

Find a rational function $\varPhi \in H^{\infty }({\mathbb {D}}, {\mathbb {C}}^{p\times q})$ such that $\Vert \varPhi \Vert _{H^{\infty }({\mathbb {D}})} < 1$ and

$$\begin{aligned} \xi _\ell ^* \varPhi (\alpha _\ell )&= \eta _\ell ^*\qquad \forall \ell \in \{1,\dots , n\} \end{aligned}$$

(A.9a)

$$\begin{aligned} \varPhi (\lambda _j)&= F_j,\quad \varPhi ^{\prime }(\lambda _j) = G_j \qquad \forall j \in \{1,\dots , m\}. \end{aligned}$$

(A.9b)

Problem A.5 is called the Nevanlinna–Pick interpolation problem with interior interpolation data $(\alpha _\ell , \xi _\ell ,\eta _\ell )_{\ell =1}^n$ and boundary interpolation data $(\lambda _j,F_j,G_j)_{j=1}^m$. The scalar-valued case $p=q=1$ with more general interpolation conditions has been studied in [30].

The following theorem implies that the solvability of Problem A.5 depends only on its interior interpolation data.

Theorem A.6

Problem A.5 with interior interpolation data $(\alpha _\ell , \xi _\ell ,\eta _\ell )_{\ell =1}^n$ and boundary interpolation data $(\lambda _j,F_j,G_j)_{j=1}^m$ is solvable if and only if Problem A.1 with interpolation data $(\alpha _\ell ,\xi _\ell ,\eta _\ell )_{\ell =1}^n$ is solvable.

To solve Problem A.5, we transform it to the following problem with boundary conditions only:

Problem A.7

Suppose that $\lambda _1 ,\dots ,\lambda _m \in {\mathbb {T}}$ are distinct. Consider matrices $F_j,G_j \in {\mathbb {C}}^{p\times q}$ for $j\in \{1,\dots , m\}$, and suppose that

$$\begin{aligned} \Vert F_j\Vert _{{\mathbb {C}}^{p\times q}} < 1 \qquad \forall j\in \{1,\dots , m\}. \end{aligned}$$

(A.10)

Find a rational function $\varPhi \in H^{\infty }({\mathbb {D}}, {\mathbb {C}}^{p\times q})$ such that $\Vert \varPhi \Vert _{H^{\infty }({\mathbb {D}})} < 1$ and

$$\begin{aligned} \varPhi (\lambda _j)&= F_j,\quad \varPhi ^{\prime }(\lambda _j) = G_j \qquad \forall j \in \{1,\dots , m\}. \end{aligned}$$

This problem is referred to as the boundary Nevanlinna–Pick interpolation problem with interpolation data $(\lambda _j,F_j,G_j)_{j=1}^m$. The condition (A.10) is necessary for the solvability for Problem A.7, and the lemma below shows that the condition (A.10) is also sufficient. We can prove the sufficiency by extending the Schur–Nevanlinna algorithm in Lemma A.4.

Lemma A.8

Problem A.7 is always solvable.

Proof

Consider Problem A.7 with interpolation data $(\lambda _j,F_j,G_j)_{j=1}^m$. We first find $m-1$ interpolation data such that if Problem A.7 with these $m-1$ data is solvable, then the original problem with m interpolation data $(\lambda _j,F_j,G_j)_{j=1}^m$ is also solvable. To that purpose, we extend the technique developed in [30] for the scalar-valued case.

Define $A:= A(F_1)$, $B:= B(F_1)$, $C:= C(F_1)$, and $D:= D(F_1)$ as in (A.2). For $\epsilon >0$, set

$$\begin{aligned} \kappa _{\epsilon }(z)&:= \frac{1}{\lambda _1}\frac{z-\lambda _1}{(1+\epsilon )-\bar{\lambda }_1 z} \\ {\widehat{F}}_1&:= \epsilon \lambda _1(I_p-F_1F_1^*)^{-1/2} G_1 (I_q-F_1^*F_1)^{-1/2} \end{aligned}$$

and

$$\begin{aligned} {\widehat{F}}_j&:= \frac{1}{\kappa _{\epsilon }(\lambda _j)}T_{F_1}(F_j) \\ {\widehat{G}}_j&:= \frac{1}{\kappa _{\epsilon }(\lambda _j)} (A-\kappa _\epsilon (\lambda _j){\widehat{F}}_j C) G_j (CF_j +D)^{-1} - \frac{\kappa ^{\prime }_{\epsilon }(\lambda _j)}{\kappa _{\epsilon }(\lambda _j)} {\widehat{F}}_j \end{aligned}$$

for $j \in \{2,\dots ,m\}$. Let us show that there exists $\epsilon >0$ such that

$$\begin{aligned} \Vert {\widehat{F}}_j\Vert _{{\mathbb {C}}^{p\times q}}<1\qquad \forall j \in \{1,\dots ,m \}. \end{aligned}$$

(A.11)

By definition,

$$\begin{aligned} \Vert {\widehat{F}}_1\Vert _{{\mathbb {C}}^{p\times q}} \le \epsilon \Vert G_1\Vert _{{\mathbb {C}}^{p\times q}} \cdot \big \Vert (I_p-F_1F_1^*)^{-1/2} \big \Vert _{{\mathbb {C}}^{p\times p}} \cdot \big \Vert (I_q-F_1^*F_1)^{-1/2} \big \Vert _{{\mathbb {C}}^{q\times q}}, \end{aligned}$$

and hence if

$$\begin{aligned} \epsilon < \frac{1}{\Vert G_1\Vert _{{\mathbb {C}}^{p\times q}} \cdot \big \Vert (I_p-F_1F_1^*)^{-1/2}\big \Vert _{{\mathbb {C}}^{p\times p}} \cdot \big \Vert (I_q-F_1^*F_1)^{-1/2} \big \Vert _{{\mathbb {C}}^{q\times q}}}, \end{aligned}$$

(A.12)

then $\Vert {\widehat{F}}_1\Vert _{{\mathbb {C}}^{p\times q}} < 1$. Let $j \in \{2,\dots ,m \}$ be given. We obtain

$$\begin{aligned} \Vert {\widehat{F}}_j \Vert _{{\mathbb {C}}^{p\times q}} \le \left( 1+ \frac{\epsilon }{| \lambda _j - \lambda _1|} \right) \Vert T_{F_1}(F_j)\Vert _{{\mathbb {C}}^{p\times q}}. \end{aligned}$$

(A.13)

Since $F_j \in {\mathcal {B}}$, it follows that $\Vert T_{F_1}(F_j)\Vert _{{\mathbb {C}}^{p\times q}} < 1$ by Lemma A.3. If we choose $\epsilon > 0$ so that

$$\begin{aligned} \epsilon < \min _{j=2,\dots ,m} \left( |\lambda _j - \lambda _1| \left( \frac{1}{\Vert T_{F_1}(F_j)\Vert _{{\mathbb {C}}^{p\times q}} } - 1 \right) \right) , \end{aligned}$$

(A.14)

then $\Vert {\widehat{F}}_j\Vert _{{\mathbb {C}}^{p\times q}} < 1$ for every $j \in \{2,\dots ,m \}$. Thus, we obtain the desired inequality (A.11) for $\epsilon >0$ satisfying (A.12) and (A.14).

Assume that there exists a rational solution $\varPsi _{m-1} \in H^{\infty }({\mathbb {D}}, {\mathbb {C}}^{p\times q})$ such that

$$\begin{aligned}&\Vert \varPhi _{m-1}\Vert _{H^{\infty }({\mathbb {D}})}<1 \end{aligned}$$

(A.15a)

$$\begin{aligned}&\varPsi _{m-1} (\lambda _j) = {\widehat{F}}_j\qquad \forall j \in \{1,\dots ,m \} \end{aligned}$$

(A.15b)

$$\begin{aligned}&\varPsi _{m-1} ^{\prime }(\lambda _j) = {\widehat{G}}_j\qquad \forall j \in \{2,\dots ,m\} \end{aligned}$$

(A.15c)

We shall show that $\varPsi _{m} := T_{F_1}^{-1}(\kappa _{\epsilon }\varPsi _{m-1})$ is a solution of the original problem with m interpolation data $(\lambda _j, F_j,G_j)_{j=1}^m$. By definition, $\varPsi _m$ is rational. Since $\Vert \kappa _{\epsilon }\Vert _{H^{\infty }({\mathbb {D}})} < 1$ and $\Vert \varPsi _{m-1}\Vert _{H^{\infty }({\mathbb {D}})} < 1$, it follows that

$$\begin{aligned} \kappa _{\epsilon }(z)\varPsi _{m-1}(z) \in {\mathcal {B}}\qquad \forall z \in \mathrm{cl}({\mathbb {D}}). \end{aligned}$$

Together with this, Lemma A.3 yields $\varPsi _m \in H^{\infty }({\mathbb {D}}, {\mathbb {C}}^{p\times q})$ and $\Vert \varPsi _m\Vert _{H^{\infty }({\mathbb {D}})}<1$.

We now prove that $\varPsi _m$ satisfies the interpolation conditions $\varPsi _m(\lambda _j) = F_j$ and $\varPsi _m'(\lambda _j) = G_j$ for every $j \in \{ 1,\dots ,m\}$. For the case $j=1$, $\kappa _{\epsilon }(\lambda _1) = 0$ yields

$$\begin{aligned} \varPsi _m (\lambda _1) = T_{F_1}^{-1}\big (\kappa _{\epsilon }(\lambda _1)\varPsi _{m-1}(\lambda _1)\big ) =F_1. \end{aligned}$$

By (A.4), we obtain

$$\begin{aligned} (A- \kappa _{\epsilon }\varPsi _{m-1}C ) \varPsi _m = \kappa _{\epsilon } \varPsi _{m-1}D-B, \end{aligned}$$

which implies

$$\begin{aligned} (\kappa _{\epsilon } \varPsi _{m-1}^{\prime }+ \kappa _{\epsilon }^{\prime }\varPsi _{m-1}) (C\varPsi _m+D)= (A-\kappa _{\epsilon } \varPsi _{m-1} C)\varPsi _m^{\prime }. \end{aligned}$$

(A.16)

Therefore,

$$\begin{aligned} \varPsi _m^{\prime }(\lambda _1) = \kappa _{\epsilon }^{\prime }(\lambda _1) A^{-1} {\widehat{F}}_1(CF_1+D). \end{aligned}$$

Since

$$\begin{aligned} \kappa _{\epsilon }^{\prime }(z) = \frac{1}{\lambda _1} \frac{\epsilon }{\big ((1+\epsilon ) - \bar{\lambda }_1 z \big )^2}, \end{aligned}$$

it follows that $\kappa _{\epsilon }^{\prime }(\lambda _1) = 1/(\epsilon \lambda _1 )$. Using

$$\begin{aligned} A^{-1} = (I_p-F_1F_1^*)^{1/2},\quad CF_1+D = (I_q-F_1^*F_1)^{1/2}, \end{aligned}$$

we derive $\varPsi ^{\prime }_m(\lambda _1) = G_1.$

For $j\in \{2,\dots ,m\}$, we have by the definition of ${\widehat{F}}_j$ that

$$\begin{aligned} \varPsi _m(\lambda _j) = T_{F_1}^{-1}(\kappa _{\epsilon }(\lambda _j){\widehat{F}}_j) = T_{F_1}^{-1}\big (T_{F_1}(F_j)\big ) = F_j. \end{aligned}$$

Using (A.16) again, we obtain

$$\begin{aligned} \kappa _{\epsilon }(\lambda _j) {\widehat{G}}_j+ \kappa _{\epsilon }^{\prime }(\lambda _j) {\widehat{F}}_j = (A - \kappa _{\epsilon }(\lambda _j) {\widehat{F}}_j C) \varPsi _m^{\prime }(\lambda _j) (CF_j +D)^{-1}. \end{aligned}$$

By the definition of ${\widehat{G}}_j$, we find that

$$\begin{aligned} \varPsi _m^{\prime }(\lambda _j) = G_j\qquad \forall j \in \{2,\dots ,m \}. \end{aligned}$$

Thus $\varPhi _m$ is a solution of the original problem with m interpolation conditions.

If we apply this procedure again to the resulting interpolation problem, i.e., the problem of finding a rational solution $\varPsi _{m-1} \in H^{\infty }({\mathbb {D}}, {\mathbb {C}}^{p\times q})$ such that the conditions given in (A.15) hold, then the interpolation condition at $z = \lambda _1$ is removed. Therefore, Problem A.7 with m interpolation data can be reduced to Problem A.7 with $m-1$ interpolation data. Continuing in this way, we finally obtain Problem A.7 with no interpolation conditions, which always admits a solution. Thus Problem A.7 is always solvable. $\square$

By Lemmas A.4 and A.8, we obtain a proof of Theorem A.6.

Proof of Theorem A.6

The necessity is straightforward. We prove the sufficiency. To this end, it is enough to show that the following problem always has a solution:

Problem A.9

Assume that Problem A.1 with n interior interpolation data $(\alpha _\ell ,\xi _\ell ,\eta _\ell )_{\ell =1}^n$ is solvable and that $\Vert F_j\Vert _{{\mathbb {C}}^{p \times q}} < 1$ for every $j \in \{1,\dots ,m \}$. Find a solution of Problem A.5 with n interior interpolation data $(\alpha _\ell ,\xi _\ell ,\eta _\ell )_{\ell =1}^n$ and m boundary interpolation data $(\lambda _j, F_j, G_j)_{j=1}^m$.

Suppose that Problem A.1 with n interior interpolation data $(\alpha _\ell ,\xi _\ell ,\eta _\ell )_{\ell =1}^n$ is solvable. Define the matrix E and the function X as in Lemma A.4. Then this lemma shows that Problem A.1 with $n-1$ interior interpolation data

$$\begin{aligned} \big (\alpha _\ell ,X(\alpha _\ell )^*U_E(\xi _\ell ,\eta _\ell ),V_E(\xi _\ell ,\eta _\ell )\big )_{\ell =2}^n \end{aligned}$$

(A.17)

is solvable. Set $A:= A(E)$, $B:= B(E)$, $C:= C(E)$, and $D:= D(E)$ as in (A.2). For $j \in \{1,\dots ,m\}$, define also

$$\begin{aligned} {\widehat{F}}_j&:= X(\lambda _j)^{-1} T_{-E}^{-1}(F_j) \\ {\widehat{G}}_j&:= X(\lambda _j)^{-1} (A+F_jC)^{-1}G_j(-CX(\lambda _j){\widehat{F}}_j + D) - X(\lambda _j)^{-1} X^{\prime }(\lambda _j){\widehat{F}}_j. \end{aligned}$$

Since $X(\lambda _j)^{-1} = X(\lambda _j)^*$ for every $j \in \{1,\dots ,m \}$, we obtain $\Vert X(\lambda _j)^{-1}\Vert _{{\mathbb {C}}^{p\times p}} = 1$ and hence $\Vert {\widehat{F}}_j\Vert _{{\mathbb {C}}^{p\times p}} < 1$ for every $j \in \{1,\dots ,m \}$. Suppose that $\varPhi _{n-1}$ is a solution of Problem A.5 with $n-1$ interior interpolation data given in (A.17) and m boundary interpolation data $(\lambda _j, {\widehat{F}}_j, {\widehat{G}}_j)_{j=1}^m.$ Then $\varPhi _n := T_{-E}(X \varPhi _{n-1})$ is a solution of Problem A.5 with n interior interpolation data $(\alpha _\ell ,\xi _\ell ,\eta _\ell )_{\ell =1}^n$ and m boundary interpolation data $(\lambda _j, F_j, G_j)_{j=1}^m$. In fact, Lemma A.4 shows that $\varPhi _n$ satisfies $\Vert \varPhi _n\Vert _{H^{\infty }({\mathbb {D}})} < 1$ and $\xi _{\ell }^* \varPhi _n(\alpha _\ell ) = \eta _\ell ^*$ for every $\ell \in \{1,\dots ,n\}$. It remains to show that the boundary conditions hold. We obtain

$$\begin{aligned} \varPhi _n(\lambda _j) = T_{-E}\big (X(\lambda _j) {\widehat{F}}_j \big ) = T_{-E}\big (T_{-E}^{-1}(F_j) \big ) =F_j\qquad \forall j \in \{1,\dots ,m \}. \end{aligned}$$

By the definition of $T_{-E}$, we obtain

$$\begin{aligned} \varPhi _n(-CX\varPhi _{n-1}+D) = (AX\varPhi _{n-1}-B), \end{aligned}$$

and hence

$$\begin{aligned} \varPhi _n^{\prime } (-CX \varPhi _{n-1}+D) = (A+\varPhi _nC)(X\varPhi _{n-1})^{\prime }. \end{aligned}$$

This yields

$$\begin{aligned} \varPhi _n^{\prime }(\lambda _j) = (A+F_jC)(X(\lambda _j) {\widehat{G}}_j + X^{\prime }(\lambda _j) {\widehat{F}}_j) (-CX(\lambda _j) {\widehat{F}}_j +D)^{-1} = G_j. \end{aligned}$$

Thus, we can reduce Problem A.9 with n interior data to that with $n-1$ interior data. Continuing in this way, we reduce Problem A.5 to Problem A.7, which is always solvable by Lemma A.8. This completes the proof. $\square$

In the construction of regulating controllers in Sect. 2, a rational function ${\mathbf {Y}}_+\in H^{\infty }({\mathbb {E}}_1,{\mathbb {C}}^{p \times p})$ needs to satisfy the interpolation condition ${\mathbf {Y}}_+(\infty ) = 0$. Its counterpart in $H^{\infty }({\mathbb {D}},{\mathbb {C}}^{p \times p})$ under the transformation $\varphi : {\mathbb {E}}_1 \rightarrow {\mathbb {D}}:z \mapsto 1/z$ is given by the interpolation condition $({\mathbf {Y}}_+\circ \varphi ^{-1} )(0) = 0$. Such a condition is excluded in Problem A.5, but we can easily incorporate it into the problem.

Corollary A.10

Suppose that $\alpha _1,\dots ,\alpha _n \in {\mathbb {D}}{\setminus } \{0\}$ and $\lambda _1,\dots ,\lambda _m \in {\mathbb {T}}$ are distinct. Consider vector pairs $(\xi _\ell , \eta _\ell )\in {\mathbb {C}}^p \times {\mathbb {C}}^q$ for $\ell \in \{1,\dots , n\}$ and matrices $F_j,G_j \in {\mathbb {C}}^{p\times q}$ for $j\in \{1,\dots , m\}$, and suppose that the norm conditions (A.8) are satisfied. Then the following three statements are equivalent:

• (a) There exists a rational function $\varPhi \in H^{\infty }({\mathbb {D}}, {\mathbb {C}}^{p\times q})$ such that $\Vert \varPhi \Vert _{H^{\infty }({\mathbb {D}})} < 1$, $\varPhi (0) = 0$, and the interpolation conditions (A.9a) and (A.9b) hold.

• (b) There exists a rational function $\varPhi \in H^{\infty }({\mathbb {D}}, {\mathbb {C}}^{p\times q})$ such that $\Vert \varPhi \Vert _{H^{\infty }({\mathbb {D}})} < 1$, $\varPhi (0) = 0$, and the interpolation conditions (A.9a) hold.

• (c) The Pick matrix P defined by

  $$\begin{aligned} P := \begin{bmatrix} P_{1,1}&\cdots&P_{1,n} \\ \vdots&\vdots \\ P_{n,1}&\cdots&P_{n,n} \end{bmatrix}, ~~ \text {where } P_{j,k} := \frac{\alpha _j \bar{\alpha }_\ell \xi _j^*\xi _\ell - \eta _j^*\eta _\ell }{1- \alpha _j\bar{\alpha }_\ell } ~~ \forall j,\ell \in \{1,\dots ,n\} \end{aligned}$$

  is positive definite.

Proof

By a straightforward calculation, we have the following fact: A rational function $\varPhi \in H^{\infty }({\mathbb {D}}, {\mathbb {C}}^{p\times q})$ satisfies the conditions of (a) if and only if ${\widehat{\varPhi }}(z) := \varPhi (z)/z$ is a solution of Problem A.5 with interior interpolation data $(\alpha _\ell , {\bar{\alpha }}_\ell \xi _\ell , \eta _\ell )_{\ell =1}^n$ and boundary interpolation data $(\lambda _j, F_j/\lambda _j, G_j/\lambda _j - F_j/\lambda _j^2)_{j=1}^m$. This fact together with Theorem A.6 shows that (a) is true if and only if Problem A.1 with interpolation data $(\alpha _\ell , \bar{\alpha }_\ell \xi _\ell , \eta _\ell )_{\ell =1}^n$ is solvable. Hence, we obtain (a) $\Leftrightarrow$ (c) by Theorem A.2. Using the fact mentioned above again, we obtain (a) $\Leftrightarrow$ (b). This completes the proof. $\square$

Remark A.11

Suppose that the interpolation data have conjugate symmetry in Problem A.5. In other words, suppose that both $(\alpha , \xi ,\eta )$ and $({\bar{\alpha }}, {\bar{\xi }}, {\bar{\eta }})$ are in its interior interpolation data and that $(\lambda ,F,G)$ and $({\bar{\lambda }},{\bar{F}},{\bar{G}})$ are in its boundary interpolation data. If the interpolation problem is solvable, then there exists a solution that is a rational function with real coefficients. In fact, for every rational function $\varPhi$, there uniquely exist rational functions $\varPhi _R$ and $\varPhi _I$ with real coefficients such that $\varPhi = \varPhi _R + i \varPhi _I$. If a rational function $\varPhi$ is a solution of the interpolation problem, then one can easily prove that its real part $\varPhi _R$ is also a solution.

Remark A.12

Let $\lambda \in {\mathbb {T}}$. For a vector pair $(\xi , \eta ) \in {\mathbb {C}}^p \times {\mathbb {C}}^q$, define a matrix $F:=\xi \eta ^*/\Vert \xi \Vert _{{\mathbb {C}}^{p}}^2$. If $\Vert \xi \Vert _{{\mathbb {C}}^{p}} > \Vert \eta \Vert _{{\mathbb {C}}^{q}}$, then $\Vert F\Vert _{{\mathbb {C}}^{p \times q}} < 1$. Further, if a rational function $\varPsi \in H^{\infty }({\mathbb {D}}, {\mathbb {C}}^{p\times q})$ satisfies $\varPsi (\lambda ) = F$, then $\xi ^* \varPsi (\lambda ) = \eta ^*$. In this way, we can transform the tangential interpolation condition $\xi ^* \varPsi (\lambda ) = \eta ^*$ to the matrix-valued interpolation condition $\varPsi (\lambda ) = F$. This transformation is used in the design procedure of regulating controllers in Sect. 2 if unstable eigenvalues of A lie on the boundary ${\mathbb {T}}$. Moreover, the above observation and Theorem A.6 indicate that for $\lambda \in {\mathbb {T}}$ and $(\xi , \eta ) \in {\mathbb {C}}^p \times {\mathbb {C}}^q$ with $\Vert \xi \Vert _{{\mathbb {C}}^{p}} > \Vert \eta \Vert _{{\mathbb {C}}^{q}}$, boundary interpolation conditions of the form $\xi ^* \varPsi (\lambda ) = \eta ^*$ can be also ignored when we determine the solvability of the Nevanlinna–Pick interpolation problem.

B $\varLambda$-extension of output operator of delay systems

Consider the delay system (4.1), and define x as in (4.2). The objective of this section is to show for a.e. $t \ge 0$,

$$\begin{aligned} \sum _{\ell =1}^{\widehat{q}} c_\ell z(t-{\widehat{h}}_\ell ) = C_{\varLambda }x(t). \end{aligned}$$

(B.1)

Since $x(t) \in X_1$ for every $t \ge h_q$ and since $C_{\varLambda } \zeta = C \zeta$ for every $\zeta \in X_1$, it suffices to show (B.1) a.e. on $[0,h_q)$. For simplicity of notation, we consider the case ${\widehat{q}} = 1$ and define ${\widehat{h}} := {\widehat{h}}_1$ and $c := c_{1}$.

By Lemma 2.4.5 of [10], there exists $s_0 >0$ such that

$$\begin{aligned} (sI-A)^{-1} x(t) =\begin{bmatrix} g_1(t) \\ g_2(t) \end{bmatrix} \qquad \forall s > s_0,~\forall t \in [0,h_q), \end{aligned}$$

where

$$\begin{aligned} g_1(t)&:= \varDelta (s)^{-1} \left( z(t) + \sum _{j=1}^{q} \int ^0_{-h_j} e^{-s(\theta +h_j) }A_jz(t+\theta ) \text {d}\theta \right) \\ \big (g_2(t)\big )(\theta )&:= e^{s\theta } g_1(t) - \int ^\theta _0 e^{s(\theta -\nu )} z(t+\nu )\text {d}\nu \qquad \forall \theta \in [-h_q,0]. \end{aligned}$$

Hence for every $s>s_0$ and every $t \in [0,h_q)$, we obtain

$$\begin{aligned} Cs(sI-A)^{-1}x(t)&= s c \big ( g_2(t) \big )(-{\widehat{h}}) \\&= sc \left( e^{-s {\widehat{h}}} g_1(t) + \int ^{\widehat{h}}_0 e^{-s({\widehat{h}}-\nu )} z(t-\nu )\text {d}\nu \right) . \end{aligned}$$

Since

$$\begin{aligned} \lim _{s\rightarrow \infty ,~\!\! s \in {\mathbb {R}}} s\varDelta (s)^{-1} = I \quad \text {and} \quad z \in L^1 ([-h_q,h_q], \mathbb {C}^n), \end{aligned}$$

Lebesgue’s dominated convergence theorem implies that in the case ${\widehat{h}} = 0$,

$$\begin{aligned}&\lim _{s\rightarrow \infty ,~\!\!s \in {\mathbb {R}}} sc\left( e^{-s {\widehat{h}}} g_1(t) + \int ^{\widehat{h}}_0 e^{-s({\widehat{h}}-\nu )} z(t-\nu )\text {d}\nu \right) \\&\quad = \lim _{s\rightarrow \infty ,~\!\!s \in {\mathbb {R}}} s c\varDelta (s)^{-1} \left( z(t) + \sum _{j=1}^{q} \int ^0_{-h_j} e^{-s(\theta +h_j) }A_jz(t+\theta ) \text {d}\theta \right) \\&\quad = cz(t) \qquad \forall t \in [0,h_q). \end{aligned}$$

Thus, we obtain $cz(t- {\widehat{h}} ) = C_{\varLambda }x(t)$ for every $t \in [0,h_q)$ if ${\widehat{h}} = 0$.

In the case ${\widehat{h}} \in (0,h_q)$, we obtain

$$\begin{aligned} \lim _{s \rightarrow \infty ,~\!\! s\in {\mathbb {R}}} s e^{-s \widehat{h}} g_1(t) = 0\qquad \forall t \in [0,h_q). \end{aligned}$$

Since $B \in {\mathcal {L}}(\mathbb {C},X)$, it follows that $x(t) \in \mathrm{dom}(C_{\varLambda })$ for a.e. $t\ge 0$ and

$$\begin{aligned} C_{\varLambda }x(t)&= \lim _{s \rightarrow \infty ,~\!\!s\in {\mathbb {R}}}Cs(sI-A)^{-1}x(t) \nonumber \\&= \lim _{s \rightarrow \infty ,~\!\!s\in {\mathbb {R}}} s \int ^{{\widehat{h}}}_0 e^{-s({\widehat{h}}-\nu )} \zeta (t-\nu )\text {d}\nu \qquad \text {a.e. } t\ge 0, \end{aligned}$$

(B.2)

where $\zeta := cz$. For each $n \in {\mathbb {N}}$, define

$$\begin{aligned} f_n(t) := n\int ^{\widehat{h}}_0 e^{-n({\widehat{h}} - \nu )} \zeta (t-\nu )\text {d}\nu \qquad \forall t \in [0,h_q). \end{aligned}$$

We will show that there exists a subsequence $\{f_{n_{\ell }}:\ell \in {\mathbb {N}}\}$ such that $\lim _{\ell \rightarrow \infty }f_{n_\ell }(t)= \zeta (t - {\widehat{h}})$ for a.e. $t \in [0,h_q)$. Together with (B.2), this yields $\zeta (t-{\widehat{h}}) = C_{\varLambda }x(t)$ for a.e. $t \in [0,h_q)$ in the case ${\widehat{h}} \in (0,h_q)$.

Let $s > s_0$. Define

$$\begin{aligned} \varphi (s) := s\int ^{\widehat{h}}_0 e^{-s({\widehat{h}} - \nu )} \text {d}\nu = 1 - e^{-{\widehat{h}}s}. \end{aligned}$$

Since $\zeta \in L^1(-h_q,h_q)$, it follows from Fubini’s theorem that

$$\begin{aligned}&\int ^{h_q}_0 \left| \zeta (t-{\widehat{h}}) - s\int ^{\widehat{h}}_0 e^{-s({\widehat{h}} - \nu )} \zeta (t-\nu )\text {d}\nu \right| \text {d}t\\&\quad \le \int ^{h_q}_0 \left| \big (1 - \varphi (s)\big )\zeta (t-{\widehat{h}})\right| \text {d}t + s \int ^{h_q}_0 \int ^{\widehat{h}}_0 e^{-s({\widehat{h}} - \nu )} \big |\zeta (t- {\widehat{h}}) - \zeta (t-\nu ) \big | \text {d}\nu \text {d}t \\&\quad \le e^{-\widehat{h} s} \Vert \zeta \Vert _{L^1(-h_q,h_q)} + s \int ^{\widehat{h}}_0 e^{-s({\widehat{h}} - \nu )} \int ^{h_q}_0 \big |\zeta (t-{\widehat{h}}) - \zeta (t-\nu ) \big | \text {d}t \text {d}\nu . \end{aligned}$$

Choose $\varepsilon >0$ arbitrarily. By the strong continuity of the left translation semigroup on $L^1(-h_q,h_q)$ (see, e.g., Example I.5.4 in [11]), there exists $\delta _0 \in (0, {\widehat{h}})$ such that

$$\begin{aligned} \int ^{h_q}_0 | \zeta (t-{\widehat{h}}) - \zeta (t-{\widehat{h}} +\delta )|\text {d}t < \varepsilon \qquad \forall \delta \in [0,\delta _0). \end{aligned}$$

Therefore,

$$\begin{aligned} s \int ^{\widehat{h}}_{{\widehat{h}} - \delta _0} e^{-s({\widehat{h}} - \nu )} \int ^{h_q}_0 \big |\zeta (t-{\widehat{h}}) - \zeta (t-\nu ) \big | \text {d}t \text {d}\nu< \varepsilon (1-e^{-\delta _0 s}) < \varepsilon . \end{aligned}$$

Since

$$\begin{aligned}&s \int ^{{\widehat{h}} - \delta _0}_{0} e^{-s({\widehat{h}} - \nu )} \int ^{h_q}_0 \big |\zeta (t-{\widehat{h}}) - \zeta (t-\nu ) \big | \text {d}t \text {d}\nu \\&\quad \le 2\Vert \zeta \Vert _{L^1(-h_q,h_q)} (e^{-\delta _0 s} - e^{-{\widehat{h}} s}), \end{aligned}$$

it follows that there exists $s_1 > s_0$ such that for every $s > s_1$,

$$\begin{aligned} e^{-\widehat{h} s} \Vert \zeta \Vert _{L^1(-h_q,h_q)}< \varepsilon ,\quad s \int ^{\widehat{h} - \delta _0}_{0} e^{-s({\widehat{h}} - \nu )} \int ^{h_q}_0 \big |\zeta (t-{\widehat{h}}) - \zeta (t-\nu ) \big | \text {d}t \text {d}\nu < \varepsilon . \end{aligned}$$

Hence we obtain

$$\begin{aligned}&\int ^{h_q}_0 \left| \zeta (t-{\widehat{h}}) - s\int ^{\widehat{h}}_0 e^{-s({\widehat{h}} - \nu )} \zeta (t-\nu )\text {d}\nu \right| \text {d}t < 3\varepsilon . \end{aligned}$$

Since $\varepsilon >0$ was arbitrary, we have that $\lim _{n \rightarrow \infty }\Vert \zeta (\cdot - {\widehat{h}}) - f_n\Vert _{L^1(0,h_q)} = 0$. Then there exists a subsequence $\{f_{n_{\ell }}:\ell \in {\mathbb {N}}\}$ such that $\lim _{\ell \rightarrow \infty }f_{n_\ell }(t)= \zeta (t - {\widehat{h}})$ for a.e. $t \in [0,h_q)$; see, e.g., Theorem 3.12 in [41]. This completes the proof. $\square$