Impulse-controllability of system classes of switched differential algebraic equations

Abstract

In this paper, impulse-controllability of system classes containing switched DAEs is studied. We introduce several notions of impulse-controllability of system classes and provide a characterization of strong impulse-controllability of system classes generated by arbitrary switching signals. For a system class generated by switching signals with a fixed-mode sequence, it is shown that either almost all systems are impulse-controllable, or almost all systems are impulse-uncontrollable. Sufficient conditions for all systems in this system class to be impulse-controllable or impulse-uncontrollable are presented. Furthermore, it is observed that even if all systems in a system class are impulse-controllable, knowledge of all the switching times is generally necessary to construct an input that ensures impulse-free solutions. Consequently, it is impossible to design a controller in real time if the switching times in the future are unknown. This phenomenon can be regarded as a causality issue. Therefore, the concept of (quasi-) causal impulse-controllability is introduced, and system classes which are (quasi-) causal are characterized. Finally, necessary and sufficient conditions for a system class to be causal given some dwell-time are stated.

1 Introduction

We consider switched differential algebraic equations (switched DAEs) of the form

$$\begin{aligned} E_\sigma =A_\sigma x+B_\sigma u, \qquad x(0^-)=x_0, \end{aligned}$$

(1)

where $\sigma :\mathbb {R}\rightarrow \mathbb {N}$ is the switching signal and $E_p,A_p\in \mathbb {R}^{n\times n}$, $B_p \in \mathbb {R}^{n\times m}$, for $p,n,m\in \mathbb {N}$. In general, trajectories of switched DAEs exhibit jumps (or even Dirac impulses), which may exclude classical solutions from existence. Therefore, we adopt the piecewise-smooth distributional solution framework introduced in [1]. For many applications, it is of crucial importance that Dirac impulses in the state are avoided as they can cause damage to the system in the form of, e.g., hydraulic shocks in water distribution systems or sparks in electronic circuits. To that extent, the ability to choose an input in such a way that no Dirac impulses are induced by the switches, i.e., impulse-controllability, is highly relevant. In this contribution, we will extend our recently established results [2] for the case of fixed switching signals to the case where the switching times are not known.

Differential algebraic equations (DAEs) arise naturally when modeling physical systems with certain algebraic constraints on the state variables; examples of applications of DAEs in electrical circuits (with distributional solutions) can be found, e.g., in [3]. In the case of abrupt structural changes in a system, several DAEs are necessary to describe each mode of the system and a switched DAE is obtained. For non-switched systems, the algebraic constraints are often eliminated such that the system is described by ordinary differential equations (ODEs). However, in the case of switched systems, the elimination process of the constraints is in general different for each individual mode, and therefore, there does not exist a description as a switched ODE with a common state variable for every mode in general. This problem can be overcome by studying switched DAEs directly.

Several structural properties of switched DAEs have been studied recently, such as controllability [4], stability/stabilizability [5,6,7,8] and observability/detectability in [9,10,11]. Impulse-controllability has been studied in the non-switched case [12,13,14,15] and in the switched case in [2] for fixed switching signals. Controllability of switched linear systems has also been studied in [16, 17], but therein the switching signal is considered as an (additional) input signal. Our approach differs from related studies on (impulse-)controllability of switched systems by the viewpoint that the switching is externally given and not directly under our control. In particular, we consider (1) as a (piecewise-constant) time-varying linear system and systems of the form (1) for which the switching signals are not identical, are considered to be different systems. Consequently, this viewpoint allows for studying to what extent properties such as (impulse-) controllability depend on the switching signal, which is of particular interest for applications in which the switching signal is not known.

In the case of, for example, component failure or cyber-physical attacks, the instances at which structural changes in the system occur, c.q. switching times, are often unknown and they are often unpredictable. This poses a problem when Dirac impulses in the state are to be avoided, since impulse-controllability of switched DAEs is in general dependent on the switching times [2]. However, in some cases the existence of impulse-free solutions for all initial values does not depend on the switching signal. Consider, for example, any system of the form (1) with two modes for which the system matrices are given by:

$$\begin{aligned} \begin{aligned} E_0&={\left[ \begin{array}{ccc} 1&{}0\\ 0&{}0\end{array}\right] } \;&A_0&= {\left[ \begin{array}{ccc} 1&{}0\\ 0&{}1\end{array}\right] },&B_0&={\left[ \begin{array}{ccc} 0\\ 1\end{array}\right] }, \\ E_1&={\left[ \begin{array}{ccc} 1&{}1\\ 0&{}0\end{array}\right] }, \;&A_1&={\left[ \begin{array}{ccc} 1&{}0\\ 1&{}1\end{array}\right] }, \;&B_1&={\left[ \begin{array}{ccc} 0\\ 0\end{array}\right] }. \end{aligned} \end{aligned}$$

i.e., each mode is given by

(2)

For every switching time $t_1\in (0,\infty )$ and any order in which the modes appear the corresponding switched DAE is impulse-controllable, since the input $u=x_1$ on $[t_0,t_1)$ and $u=0$ on $[t_1,\infty )$ ensures impulse-free solutions. Hence, regardless of the switching signal $\sigma$, the system will be impulse-controllable. Alternatively, after collecting all systems of the form (1) with the modes given by (2) in a system class, the system class can be called impulse-controllable as all systems contained in it are impulse-controllable. Moreover, under the assumption that a switch can be observed as soon as it occurs, only knowledge of the current mode is required to design a control input that guarantees impulse-free solutions. That is, it is not necessary to know what the switching time $t_1$ is to ensure impulse-freeness of solutions, as long as we know at $t_1$ that $t_1$ is a switching time.

Motivated by this example, the aim of this paper is to characterize the system classes for which any system contained in it is impulse-controllable. We will present necessary and sufficient conditions under which there exist impulse-free solutions of any switched system, with modes governed by the matrices $E_p,A_p$ and $B_p$ and $p\in \{0,1,...,\texttt{n}\}$, regardless of the mode durations and the sequence in which the modes appear. Furthermore, we will investigate system classes containing switched systems for which the order in which the modes appears is fixed, i.e., for a particular class of switching signals. For those system classes, we will show that either all systems, almost all, none or almost none of the systems are impulse-controllable. Then, it is shown that although every system in such a system class is impulse-controllable, an input that guarantees impulse-free solution might depend on the switching times in the future. That is, the switching time $t_i$ needs to be known at $t<t_i$ to guarantee impulse-freeness. This causes a causality issue in the sense that knowledge of future events is necessary to decide on the current action required. From a control design perspective, such a causality issue is problematic if the future switching times are unknown. Therefore, we introduce the concepts of (quasi-) causal impulse-controllability of system classes and provide characterizations. Finally, necessary and sufficient conditions for system classes to be causally impulse-controllable given some dwell-time are presented.

The remainder of the paper is structured as follows: The mathematical preliminaries are given in Sect. 2. The result regarding impulse-controllability of system classes is contained by Sect. 3, and (quasi-) causal impulse-controllability is considered in Sect. 4. Conclusions and direction for further research are given in Sect. 5.

2 Mathematical preliminaries

In this section, we recall some notation and properties related to the non-switched DAE:

$$\begin{aligned} E\dot{x}=Ax+Bu. \end{aligned}$$

(3)

2.1 Properties and definitions for regular matrix pairs

In the following, we call a matrix pair (E, A) and the associated DAE (3) regular iff the polynomial $\det (sE-A)$ is not the zero polynomial. Recall the following result on the quasi-Weierstrass form [18].

Proposition 1

A matrix pair $(E,A)\in \mathbb {R}^{n\times n} \times \mathbb {R}^{n\times n}$ is regular if, and only if, there exists invertible matrices $S,T\in \mathbb {R}^{n\times n}$ such that

$$\begin{aligned} (SET,SAT)=\left( \begin{bmatrix} I&{}0\\ 0&{}N\end{bmatrix}, \begin{bmatrix} J&{}0\\ 0&{}I\end{bmatrix} \right) , \end{aligned}$$

(4)

where $J\in \mathbb {R}^{n_1\times n_1}$, $0\le n_1\le n$, is some matrix and $N\in \mathbb {R}^{n_2\times n_2}$, $n_2:=n-n_1$, is a nil-potent matrix.

The matrices S and T can be calculated by using the so-called Wong sequences [18, 19]:

$$\begin{aligned} \begin{aligned} \mathcal {V}_0&:=\mathbb {R}^n,&\mathcal {V}_{i+1}&:=A^{-1}(E\mathcal {V}_i),&i&=0,1,...\\ \mathcal {W}_0&:=\{0\},&\mathcal {W}_{i+1}&:=E^{-1}(A\mathcal {W}_i),&i&=0,1,... \end{aligned} \end{aligned}$$

The Wong sequences are nested and get stationary after finitely many iterations. The limiting subspaces are defined as follows:

$$\begin{aligned} \mathcal {V}^*:=\bigcap _i \mathcal {V}_i, \qquad \mathcal {W}^*:=\bigcup \mathcal {W}_i. \end{aligned}$$

For any full rank matrices V, W with ${{\,\textrm{im}\,}}V=\mathcal {V}^*$ and ${{\,\textrm{im}\,}}W=\mathcal {W}^*$, the matrices $T:=[V,W]$ and $S:=[EV,AW]^{-1}$ are invertible and (4) holds.

Based on the Wong sequences, we define the following projector and selectors.

Definition 2

Consider the regular matrix pair (E, A) with corresponding quasi-Weierstrass form (4). The consistency projector of (E, A) is given by:

$$\begin{aligned} \Pi _{(E,A)}:=T\begin{bmatrix} I&{}0\\ 0&{}0\end{bmatrix} T^{-1}, \end{aligned}$$

(5)

the differential and impulse selector are given by

$$\begin{aligned} \Pi ^{\textrm{diff}}_{(E,A)}:=T\begin{bmatrix} I&{}0\\ 0&{}0\end{bmatrix} S,\; \Pi ^{\textrm{imp}}_{(E,A)}:=T\begin{bmatrix} 0&{}0\\ 0&{}I\end{bmatrix} S. \end{aligned}$$

(6)

In all three cases, the block structure corresponds to the block structure of the quasi-Weierstrass form. Furthermore, we define

$$\begin{aligned} \begin{aligned} A^{\textrm{diff}}&:=\Pi ^{\textrm{diff}}_{(E,A)}A,\quad&E^{\textrm{imp}}&:=\Pi ^{\textrm{imp}}_{(E,A)}E,\\ B^{\textrm{diff}}&:=\Pi ^{\textrm{diff}}_{(E,A)}B,\quad&B^{\textrm{imp}}&:=\Pi ^{\textrm{imp}}_{(E,A)}B. \end{aligned} \end{aligned}$$

Note that all the above-defined matrices do not depend on the specifically chosen transformation matrices S and T; they are uniquely determined by the original regular matrix pair (E, A).

Remark 3

It follows directly from (4), (5) and (6) that

$$\begin{aligned} E^{\textrm{imp}}&= E^{\textrm{imp}}(I-\Pi _{(E,A)})= T^{-1}\begin{bmatrix} 0&{}0\\ 0&{}N\end{bmatrix}T,\\ A^{\textrm{diff}}&= A^{\textrm{diff}}\Pi _{(E,A)}= T^{-1} \begin{bmatrix} J&{}0\\ 0&{}0 \end{bmatrix} T. \end{aligned}$$

Therefore, we can also immediately conclude

$$\begin{aligned} {{\,\textrm{im}\,}}A^{\textrm{diff}}\subseteq {{\,\textrm{im}\,}}\Pi _{(E,A)}, \text { and } \ker E\subseteq \ker \Pi _{(E,A)}. \end{aligned}$$

An important feature for DAEs is the so-called consistency space, defined as follows:

Definition 4

Consider the DAE (3); then, the consistency space is defined as:

$$\begin{aligned} \mathcal {V}_{(E,A)}:=\left\{ x_0\in \mathbb {R}^n\,\left| \,\begin{aligned}&\exists \; \text {smooth solution} \;x\; \text { of}\;\\ {}&E\dot{x}=Ax,\; \text {with}\; x(0)=x_0 \end{aligned}\,\right. \!\!\right\} , \end{aligned}$$

and the augmented consistency space is defined as

$$\begin{aligned} \mathcal {V}_{(E,A,B)}:=\left\{ x_0\in \mathbb {R}^n\,\left| \,\begin{aligned} \exists \text { smooth solutions } (x,u)\; \text {of } \\ E\dot{x}=Ax+Bu \text { and }x(0)=x_0 \end{aligned}\,\right. \!\!\right\} . \end{aligned}$$

In order to express (augmented) consistency spaces in terms of the Wong limits, we need the following notation for matrices A,B of suitable sizes:

$$\begin{aligned} \langle A\mid B\rangle := {{\,\textrm{im}\,}}\begin{bmatrix} B&AB&\ldots&A^{n-1}B\end{bmatrix}. \end{aligned}$$

Proposition 5

([20]) Consider the regular DAE (3), then $\mathcal {V}_{(E,A)} = \mathcal {V}^* = {{\,\textrm{im}\,}}\Pi _{(E,A)} = {{\,\textrm{im}\,}}\Pi ^{\textrm{diff}}_{(E,A)}$ and $\mathcal {V}_{(E,A,B)} = \mathcal {V}^*\oplus \langle E^{\textrm{imp}}\mid B^{\textrm{imp}}\rangle$.

2.2 Distributional solutions of switched DAEs

For studying impulsive solutions, we consider the space of piecewise-smooth distributions from [1] as a solution space, defined as

$$\begin{aligned} \mathbb {D}_{\text {pw}\mathcal {C}^\infty }:= \left\{ D=f_\mathbb {D}+\sum _{t\in T} D_t \left| \begin{array}{l} f\in \mathcal {C}^\infty _{\textrm{pw}}, \; T\subseteq \mathbb {R}\text { locally finite, }\\ \forall t\in T:D_t\in \textrm{span} \{\delta _t, \delta _t^\prime , \delta _t^{\prime \prime },...,\} \end{array}\right. \right\} . \end{aligned}$$

where $\mathcal {C}^\infty _{\textrm{pw}}$ denotes the space of piecewise-smooth functions, $f_{\mathbb {D}}$ denotes the regular distribution induced by f, $\delta _t$ denotes the Dirac impulse with support $\{t\}$ and $\delta _t^\prime$ denotes distributional derivatives of $\delta _t$.

For a piecewise-smooth distribution $D\in \mathbb {D}_{\text {pw}\mathcal {C}^\infty }$, three types of “evaluation at time t" are defined as follows.

Definition 6

Let $t\in \mathbb {R}$ and $D=f_\mathbb {D}+\sum _{\tau \in T} D_\tau$, then the left/right evaluation of D at t is given by

$$\begin{aligned} D(t^-):=f(t^-)=\lim _{\varepsilon \searrow 0} f(t-\varepsilon ), \qquad D(t^+):=f(t^+)= f(t) \end{aligned}$$

and the impulsive part of D at t is

$$\begin{aligned} D[t]={\left\{ \begin{array}{ll} D_t &{}t\in T,\\ 0&{} t\not \in T.\end{array}\right. } \end{aligned}$$

Given the evaluation at t of a distribution, we can define the concept of impulse-free distributions.

Definition 7

A distribution $D\in \mathbb {D}_{\text {pw}\mathcal {C}^\infty }$ is said to be impulse-free at t if $D[t]=0$. Furthermore, if $D[t]=0$ for all $t\in \mathbb {R}$, then D is said to be an impulse-free distribution.

Solving the DAE (3) with an inconsistent initial value is reinterpreted as the problem of finding a solution $(x,u)\in (\mathbb {D}_{\text {pw}\mathcal {C}^\infty })^{n+m}$ to the following initial-trajectory problem (ITP):

$$\begin{aligned} x_{(-\infty ,0)}&=x^0_{(-\infty ,0)}, \end{aligned}$$

(7a)

$$\begin{aligned} (E\dot{x})_{[0,\infty )}&=(Ax+Bu)_{[0,\infty )}, \end{aligned}$$

(7b)

where $x^0\in (\mathbb {D}_{\text {pw}\mathcal {C}^\infty })^n$ is some initial trajectory, and $f_\mathcal {I}$ denotes the restriction of a piecewise-smooth distribution f to an interval $\mathcal {I}$. In [1], it is shown that the ITP (7) has a unique solution for any initial trajectory if, and only if, the matrix pair (E, A) is regular. In fact, the following explicit solution expression can be obtained.

Theorem 8

([21] Theorem 5.1) Let (E, A) be a regular matrix pair. Then, for any initial trajectory $x^0\in (\mathbb {D}_{\text {pw}\mathcal {C}^\infty })^n$ and any input $u\in (\mathbb {D}_{\text {pw}\mathcal {C}^\infty })^m$ the ITP (7) has a unique solution $x\in (\mathbb {D}_{\text {pw}\mathcal {C}^\infty })^n$. In particular, the jump from $x^0(t_0^-)$ to $x(t_0^+)$ and the impulsive part $x[t_0]$ is uniquely determined. In the case that the input u is impulse-free, it follows that

$$\begin{aligned} x(t_0^+)&=\Pi x^0(t_0^-)-\sum _{i=0}^{\nu -1} (E^{\textrm{imp}})^i B^{\textrm{imp}}u^{(i)}(t_0^+),\\ x[t_0]&=-\sum _{i=0}^{\nu -1}(E^{\textrm{imp}})^{i+1}\left( x^0(t_0^-)\delta ^{(i)}+\sum _{j=0}^i B^{\textrm{imp}}u^{(i-j)}(t_0^+) \delta ^{(j)}\right) \\&=-\sum _{i=0}^{\nu -1}\left( E^{\textrm{imp}}\right) ^{i+1} \left( x^0(t_0^-)-x(t_0^+)\right) \delta ^{(i)} \end{aligned}$$

and for $t\in (t_0,\infty )$

$$\begin{aligned} x(t^-)=e^{A^{\textrm{diff}}t}\Pi x^0(t_0^-)+\int _{t_0}^t e^{A^{\textrm{diff}}(t-\tau )} B^{\textrm{diff}}u(\tau )\; \textrm{d}\tau -\sum _{i=0}^{\nu -1} (E^{\textrm{imp}})^i B^{\textrm{imp}}u^{(i)}(t^-), \end{aligned}$$

where $\Pi$ is the consistency projector as in (5) and $E^{\textrm{imp}}=\Pi ^{\textrm{imp}}E$ with the impulse selector $\Pi ^{\textrm{imp}}$ as in  (6).

As a direct consequence, the switched DAE (1) with regular matrix pairs is also uniquely solvable for any switching signal with locally finitely many switches, where we adopt the following solution concept for a switched DAE.

Definition 9

A distribution $(x,u)\in \mathbb {D}_{\text {pw}\mathcal {C}^\infty }^{(n+m)}$ is called a solution to the switched DAE (1) with regular matrix pairs $(E_p,A_p)\in \mathbb {R}^{n\times n}\times \mathbb {R}^{n\times n}$ and a switching signal $\sigma$ with switching times $t_0,t_1,...,t_p$ if (x, u) is a local (distributional) solution to (3) on each interval $[t_p,t_{p+1})$ with $E=E_p$, $A=A_p$ and $B=B_p$.

Note that in the above definition $\dot{x}[t_p]$ depends on $x(t_p^-)$; in particular, the above solution definition can equivalently be formulated as a repeated ITP of the form (7).

2.3 Properties of DAEs

Recall the following definitions and characterization of (impulse-) controllability [20].

Proposition 10

The reachable space of the regular DAE (3) defined as

$$\begin{aligned} \mathcal {R}:= \left\{ x_T\in \mathbb {R}^n\,\left| \,\begin{array}{l}\exists T>0\ \text {and } \exists \text { a smooth solution } (x,u) \text { of (}3\text {)}\\ \text {with } x(0) = 0 \text { and } x(T) = x_T\end{array}\,\right. \!\!\right\} \end{aligned}$$

satisfies $\mathcal {R}=\langle A^{\textrm{diff}}\mid B^{\textrm{diff}}\rangle \oplus \langle E^{\textrm{imp}}\mid B^{\textrm{imp}}\rangle$.

It is easily seen that the reachable space for (3) coincides with the controllable space, i.e.,

$$\begin{aligned} \mathcal {R}= \left\{ x_0\in \mathbb {R}^n\,\left| \,\begin{array}{l}\exists T>0\ \text {and } \exists \text { a smooth solution } (x,u) \text { of (}3\text {)}\\ \text {with } x(0) = x_0 \text { and } x(T) = 0\end{array}\,\right. \!\!\right\} . \end{aligned}$$

Corollary 11

The augmented consistency space of (3) satisfies $\mathcal {V}_{(E,A,B)}=\mathcal {V}_{(E,A)}+\mathcal {R}= \mathcal {V}_{(E,A)} \oplus \langle E^{\textrm{imp}},B^{\textrm{imp}}\rangle$.

Definition 12

The DAE (3) is impulse-controllable if for all initial conditions $x_0\in \mathbb {R}^n$ there exists a solution (x, u) of the ITP (7) such that $x(0^-)=x_0$ and $(x,u)[0] = 0$, i.e., (x, u) is impulse-free at $t=0$. The space of impulse-controllable states of the DAE (3) is given by

$$\begin{aligned} \mathcal {C}^{\textrm{imp}}_{(E,A,B)}:=\left\{ x_0\in \mathbb {R}^n\,\left| \, \begin{array}{l} \exists \text { solution } (x,u)\in \mathbb {D}_{\text {pw}\mathcal {C}^\infty }\text { of (}7\text {)}\\ \text {s.t. } x(0^-)=x_0 \text { and }(x,u)[0]= 0. \end{array}\,\right. \!\!\right\} . \end{aligned}$$

In particular, the DAE (3) is impulse-controllable if and only if $\mathcal {C}^{\textrm{imp}}_{(E,A,B)} =\mathbb {R}^n$.

Impulse-controllability can be characterized geometrically as follows (cf. [15, 22]).

Lemma 13

The regular DAE (3) is impulse-controllable if and only if

$$\begin{aligned} {{\,\textrm{im}\,}}E+A\ker E+{{\,\textrm{im}\,}}B=\mathbb {R}^n. \end{aligned}$$

Furthermore,

$$\begin{aligned} \mathcal {C}^{\textrm{imp}}_{(E,A,B)}&=\mathcal {V}_{(E,A,B)}+\ker E=\mathcal {V}_{(E,A)} + \mathcal {R}+ \ker E\\&=\mathcal {V}_{(E,A)} \oplus \mathcal {D}^\textrm{imp}= {{\,\textrm{im}\,}}\Pi _{(E,A)} \oplus \mathcal {D}^\textrm{imp}. \end{aligned}$$

where $\mathcal {D}^\textrm{imp}:=\langle E^{\textrm{imp}}\mid B^{\textrm{imp}}\rangle + \ker E$.

In the case of a family of matrix triplets $\{(E_p,A_p,B_p)\}_{p=0}^\texttt{n}$ for some $\texttt{n}\in \mathbb {N}$, we will adopt the shorthand notation $\Pi _p:=\Pi _{(E_p,A_p)}$, $\Pi ^{\textrm{diff}}_p:=\Pi ^{\textrm{diff}}_{(E_p,A_p)}$ and $\Pi ^{\textrm{imp}}_p:=\Pi ^{\textrm{imp}}_{(E_p,A_p)}$ for the consistency projector and the consistency selectors. The matrices $A^{\textrm{diff}}_p$, $B^{\textrm{diff}}_p$, $E^{\textrm{imp}}_p$, $B^{\textrm{imp}}_p$ are defined accordingly. The impulse-controllable space for mode p is denoted by $\mathcal {C}^{\textrm{imp}}_p:=\mathcal {C}^{\textrm{imp}}_{(E_p,A_p,B_p)}$.

3 Impulse-controllability of system classes

3.1 System classes

The concepts introduced above will be used in the following to study impulse-controllability of system classes containing switched DAEs. We will focus our attention on finite time intervals with finitely many mode changes within this interval. Since we do not want to fix the length of the interval of interest a priori, we simply assume that the last mode remains active until $t=\infty$. In other words, we restrict our attention to classes of switching signals which are defined on the interval $[t_0,\infty )$ and have finitely many mode changes. The corresponding class of switching signals with at most $\texttt{n}\in \mathbb {N}$ mode changes is formally defined as follows.

Definition 14

(Arbitrary switching signals) The class of (arbitrary) switching signals $\overline{\mathcal {S}}_\texttt{n}$ is defined as the set of all $\sigma : \mathbb {R}\rightarrow \{0,1,...,\texttt{n}\}$ of the form

$$\begin{aligned} \sigma (t)=q_p \quad t\in [t_p,t_{p+1}) \end{aligned}$$

(8)

where $\textbf{q}:= (q_0,q_1,\ldots ,q_\texttt{n})\in \{0,1,\ldots ,\texttt{n}\}^{\texttt{n}+1}$ is the mode sequence of $\sigma$ and $t_1<t_2<...<t_\texttt{n}$ are the $\texttt{n}\in \mathbb {N}$ switching times in $(0,\infty )$ with $t_0:=0$ and $t_{\texttt{n}+1}:=\infty$ for notational convenience. Furthermore, for a given sequence of switching times, let $\tau _i:=t_{i+1}-t_{i}$, $i=0,1,\ldots ,\texttt{n}-1$ and

$$\begin{aligned} \varvec{\tau }:=(\tau _0, \tau _1, \ldots , \tau _{\texttt{n}-1}) \in \mathbb {R}^{\texttt{n}}_{>0}, \end{aligned}$$

(9)

the sequence of (finite) mode-durations.

Note that in the above definition, we do not exclude the situation that $q_p=q_{p+1}$ for some p, effectively leading to a switching signal with less than $\texttt{n}$ switches. Consequently, for such a switching signal the mode duration $\varvec{\tau }$ is not uniquely defined, as the switching time $t_{p+1}$ can be altered without changing the actual switching signal. Nevertheless, this does not lead to any technical problems in the following and we will use $\sigma \in \overline{\mathcal {S}}_\texttt{n}$ and the corresponding pair $(\textbf{q},\varvec{\tau })\in \mathbb {N}^{\texttt{n}+1}\times \mathbb {R}^\texttt{n}_{>0}$ interchangeably.

Definition 15

(Fixed mode sequence switching signals) The class of switching signals with fixed mode sequence $\textbf{q}\in \mathbb {N}^{\texttt{n}+1}$ is denoted by $\mathcal {S}_{\textbf{q}}$, i.e. $\mathcal {S}_{\textbf{q}}$ contains all switching signals associated with $(\textbf{q},\varvec{\tau })$ for some $\varvec{\tau }\in \mathbb {R}^{\texttt{n}}_{>0}$. For the canonical mode sequence $\textbf{q}=(0,1,2,\ldots ,\texttt{n})$, we simply write $\mathcal {S}_{\texttt{n}}:=\mathcal {S}_{(0,\ldots ,\texttt{n})}$.

Definition 16

(System classes) For a family of matrix triplets $\{(E_p,A_p,B_p)\}_{p=0}^\texttt{n}$ with regular pairs $(E_p,A_p)$, the system class $\overline{\Sigma }_\texttt{n}$ of associated switched (regular) DAEs (1) under arbitrary switching is given by:

$$\begin{aligned} \overline{\Sigma }_\texttt{n}:= \left\{ (E_\sigma , A_\sigma ,B_\sigma )\,\left| \,\sigma \in \overline{\mathcal {S}}_\texttt{n}\,\right. \!\!\right\} , \end{aligned}$$

where $(E_\sigma ,A_\sigma ,B_\sigma )$ is understood as a triplet of (piecewise-constant) time-varying matrices for each specific switching signal $\sigma :(t_0,\infty )\rightarrow \{0,1,\ldots ,\texttt{n}\}$.

The corresponding system class $\Sigma _{\texttt{n}}$ of switched DAEs with fixed mode sequence ${\textbf {q}}=(0,1,\ldots ,\texttt{n})$ is given by:

$$\begin{aligned} \Sigma _\texttt{n}:= \left\{ (E_\sigma , A_\sigma ,B_\sigma )\,\left| \,\sigma \in \mathcal {S}_\texttt{n}\,\right. \!\!\right\} . \end{aligned}$$

3.2 Strong impulse-controllability of $\overline{\Sigma }_\texttt{n}$

For an individual switched DAE (1) given by the (time-varying) matrix triplet $(E_\sigma ,A_\sigma ,B_\sigma )$, impulse-controllability is defined as the property that Dirac impulses can be avoided regardless of the initial condition. This is formalized as follows.

Definition 17

(Impulse-controllability) The switched DAE $(E_\sigma ,A_\sigma ,B_\sigma )$ for a fixed switching signal $\sigma \in \overline{\mathcal {S}}_n$ is called impulse-controllable iff for all $x_0\in \mathcal {V}_{( E_{q_0}, A_{q_0}, B_{q_0} )}$ there exists a solution $(x,u)\in \mathbb {D}_{\text {pw}\mathcal {C}^\infty }^{n+m}$ with $x(t_0^+)=x_0$ which is impulse-free.

In the case all systems $(E_\sigma ,A_\sigma ,B_\sigma )\in \overline{\Sigma }_\texttt{n}$ are impulse-controllable, the system class itself is called impulse-controllable.

Definition 18

(Strong impulse-controllability) The whole system class $\overline{\Sigma }_\texttt{n}$ associated with the family $\{(E_p,A_p,B_p)\}_{p=0}^\texttt{n}$ is called strongly impulse-controllable, if $(E_\sigma ,A_\sigma ,B_\sigma )$ is impulse-controllable for all $\sigma \in \overline{\mathcal {S}}_\texttt{n}$.

Remark 19

The definition of impulse-controllability of an individual switched DAEs is very similar to the definition in [2], which is restricted to a bounded interval and is in fact equivalent when considering the finite interval $[t_0,t_f)$ for some $t_f>t_\texttt{n}$. Furthermore, note that an individual switched system with constant switching signal is by definition always impulse-controllable, because only consistent initial values are considered (cf. the discussion after [2, Def. 9]).

Some system classes are trivially strongly impulse-controllable (e.g., when each individual mode is impulse-controllable or the switched DAEs is in fact non-switching because $(E_p,A_p,B_p)=(E_q,A_q,B_q)$ for all p, q, cf. the discussion after [2, Def 9]).

However, the following example shows that there exist non-trivial examples of strongly impulse-controllable system classes.

Example 20

Consider a switched DAE (1) with mode triplets

$$\begin{aligned} \begin{aligned} (E_0,A_0,B_0)&=\left( \begin{bmatrix} 1&{}0\\ 0&{}1\end{bmatrix}, \begin{bmatrix} 0&{}0\\ 0&{}0\end{bmatrix}, \begin{bmatrix} 0\\ 1\end{bmatrix} \right) \\ (E_1,A_1,B_1)&=\left( \begin{bmatrix} 0&{}1\\ 0&{}0\end{bmatrix}, \begin{bmatrix} 1&{}0\\ 0&{}1\end{bmatrix}, \begin{bmatrix} 0\\ 0\end{bmatrix}\right) . \end{aligned} \end{aligned}$$

(10)

It is easily seen that the corresponding augmented consistency and impulse-controllable spaces satisfy $\mathcal {V}_0 = \mathcal {C}^{\textrm{imp}}_0 = \mathbb {R}^2$ and $\mathcal {V}_1 = \mathcal {C}^{\textrm{imp}}_1= {{\,\textrm{im}\,}}{\left[ \begin{array}{ccc} 1\\ 0\end{array}\right] }$.

The corresponding system class $\overline{\Sigma }_1$ is strongly impulse-controllable, which can be seen by considering all possible cases for the switching signals: switching signals with $\textbf{q}=(0,0)$ or $\textbf{q}=(1,1)$ result in non-switched DAEs which are impulse-controllable by definition; for switched DAEs with a mode sequence given by $\textbf{q} = (0,1)$ it is possible to choose a smooth input on $(t_0,t_1)$ such that $x_2(t_1^-)=0$ and hence no impulse occurs at the switching time $t_1$; for switched DAEs with a mode sequence given by (1, 0) the input $u(t)=0$ will result in an impulse-free solution for all initial values in $\mathcal {V}_1={{\,\textrm{im}\,}}{\left[ \begin{array}{ccc} 1\\ 0\end{array}\right] }$ $\diamond$

In the case of switched DAEs with a single switch, the following characterization of impulse-controllability is a simple consequence from the results in [2].

Lemma 21

(cf. [2, Thm. 14 & Lem. 17]) A switched DAE $(E_\sigma ,A_\sigma ,B_\sigma ) \in \Sigma _1$ with mode sequence $\textbf{q}=(0,1)$ is impulse-controllable if, and only if,

$$\begin{aligned} {{\,\textrm{im}\,}}\Pi _0\subseteq \mathcal {C}^{\textrm{imp}}_1+\mathcal {R}_0. \end{aligned}$$

(11)

The single-switch result can directly be used to arrive at a characterization of strong impulse-controllability as follows.

Theorem 22

Consider the system class $\overline{\Sigma }_\texttt{n}$ associated with $\{E_p,A_p,B_p\}_{p=0}^\texttt{n}$ with corresponding (individual) consistency projectors $\Pi _p$, impulse-controllable spaces $\mathcal {C}^{\textrm{imp}}_p$ and reachability spaces $\mathcal {R}_p$. Then $\overline{\Sigma }_\texttt{n}$ is strongly impulse-controllable if, and only if,

$$\begin{aligned} {{\,\textrm{im}\,}}\Pi _i \subseteq \mathcal {C}^{\textrm{imp}}_j+\mathcal {R}_i \end{aligned}$$

(12)

for all $i,j\in \{0,1,...,\texttt{n}\}$.

Proof

Necessity of (12) is clear by considering switching signals with mode sequences of the form $\textbf{q} = (i,j,q_2,\ldots ,q_\texttt{n})$ together with Lemma 21 and the obvious fact that an impulse-free solution needs to be impulse-free on the initial interval $[t_0,t_2)$ as well.

Sufficiency of (12) is also clear by regarding each switched system $(E_\sigma ,A_\sigma ,B_\sigma )$ as a concatenation of switched DAEs with a single switch and considering the ability to choose the input independently around the switching times to ensure impulse-freeness at each individual switch (as a consequence of Lemma 21). $\square$

Remark 23

The characterization of strong impulse-controllability of $\overline{\Sigma }_\texttt{n}$ via (12) is much simpler than the characterization of impulse-controllability of an individual switched system as given in [2, Thm. 21], which is based on a rather complicated recursive subspace sequence (discussed in detail in the next subsection, see (14)) and depends on the specific mode durations $\varvec{\tau }$. The underlying reason is that strong impulse-controllability is by definition independent from the mode durations and, furthermore, can be reduced to the single switch case (as utilized in the proof of Theorem 22).

3.3 Impulse-controllability of $\Sigma _n$

As can be seen from Theorem 22, verifying whether a system class $\overline{\Sigma }_\texttt{n}$ is strongly impulse-controllable can be done by verifying impulse-controllability of all possible single switch switched DAEs. However, if a mode sequence is fixed, these conditions are only sufficient and not necessary in general. In fact, defining strong impulse-controllability for $\Sigma _n$ analogously as in Definition 17 (see also the forthcoming Definition 26), we have the following consequence from Lemma 21.

Corollary 24

The system class $\Sigma _n$ of switched systems with fixed mode sequence $\textbf{q}=(0,1,2,\ldots ,\texttt{n})$ is strongly impulse-controllable if

$$\begin{aligned} {{\,\textrm{im}\,}}\Pi _{k} \subseteq \mathcal {C}^{\textrm{imp}}_{k+1}+\mathcal {R}_k\quad \forall k\in \{0,1,\ldots ,\texttt{n}-1\}. \end{aligned}$$

(13)

The following examples show that (13) is indeed only sufficient and not necessary in general.

Example 25

Consider the system class $\Sigma _\texttt{n}$ with $\texttt{n}=2$ and modes $(E_0,A_0,B_0) = (I, 0, {\left[ \begin{array}{ccc} 1 \\ 0 \end{array}\right] })$ $(E_1,A_1,B_1) = (I, 0, 0)$ $(E_2,A_2,B_2) = ({\left[ \begin{array}{ccc} 0 &{} 0 \\ 1 &{} 0 \end{array}\right] },I,0)$. It is easily seen that $\Sigma _\texttt{n}$ is strongly impulse-controllable; in fact, for any switching time $t_1$ and any initial value it is possible to choose the input u on $[0,t_1)$ such $x_1(t_1^-) = 0$, in the second mode the state then remains constant and hence $x_1(t_2^-)=x_1(t_1^-)=0$, which then implies that at the last switch $x_1$ does not jump and hence no Dirac impulse is induced. However, condition (13) is not satisfied for the mode pair (1, 2); indeed, ${{\,\textrm{im}\,}}\Pi _1 = \mathbb {R}^2$ is not contained in $\mathcal {C}^{\textrm{imp}}_2+\mathcal {R}_1 = {{\,\textrm{im}\,}}{\left[ \begin{array}{ccc} 0 \\ 1 \end{array}\right] } + \{0\}$.

The above example shows that characterization of impulse-controllability of $\Sigma _\texttt{n}$ cannot simply be reduced to the single switch case anymore. In particular, it will turn out that it is possible that a switched system with fixed mode sequence has some isolated mode duration for which impulse-controllability is lost, but for all remaining mode duration it is impulse-controllable. Furthermore, for arbitrary switching signals it is not possible that none of the systems in $\overline{\Sigma }_\texttt{n}$ are impulse-uncontrollable (see Remark 19); however, for a fixed mode sequence it is indeed possible, that all of the systems in $\Sigma _\texttt{n}$ are not impulse-controllable. Finally, it is also possible that for some specific mode durations a system in $\Sigma _\texttt{n}$ is impulse-controllable, while for all remaining mode durations the systems are not impulse-controllable. This motivates us to introduce the following different notions of impulse-controllability for the system class $\Sigma _\texttt{n}$.

Definition 26

(Strong and essential impulse (un-)controllability for $\Sigma _\texttt{n}$) Consider the class $\Sigma _\texttt{n}$ of switched systems (1) with fixed mode sequence $\textbf{q}=(0,1,2,\ldots ,\texttt{n})$ and arbitrary mode durations $\varvec{\tau }=(\tau _0,\tau _1,\ldots ,\tau _{\texttt{n}-1}) \in \mathbb {R}^{\texttt{n}}_{>0}$.

• $\Sigma _\texttt{n}$ is called strongly impulse-controllable if all $(E_\sigma ,A_\sigma ,B_\sigma )\in \Sigma _\texttt{n}$ are impulse-controllable.

• $\Sigma _\texttt{n}$ is called essentially impulse-controllable if the set of all mode durations $\varvec{\tau }\in \mathbb {R}^{\texttt{n}}_{>0}$ of $(E_\sigma ,A_\sigma ,B_\sigma )\in \Sigma _\texttt{n}$ which are not impulse-controllable has measure zero in $\mathbb {R}^{\texttt{n}}_{>0}$.

• $\Sigma _\texttt{n}$ is called strongly impulse-uncontrollable if all $(E_\sigma ,A_\sigma ,B_\sigma )\in \Sigma _\texttt{n}$ are not impulse-controllable.

• $\Sigma _\texttt{n}$ is called essentially impulse-uncontrollable if the set of all mode durations $\varvec{\tau }\in \mathbb {R}^{\texttt{n}}_{>0}$ of $(E_\sigma ,A_\sigma ,B_\sigma )\in \Sigma _\texttt{n}$ which are impulse-controllable has measure zero in $\mathbb {R}^{\texttt{n}}_{>0}$.

First note that clearly every strongly impulse (un-)controllable system class is also essentially impulse (un-)controllable.

Secondly, note that if the switching signal is considered to be an input, a switched system can be said to be impulse-controllable if for all consistent initial values there exists a switching signal for which Dirac impulses are avoidable. Consequently, a switched system for which the switching signal is an input is impulse-controllable if there exists an essentially impulse-(un)controllable system class $\Sigma _\texttt{n}\subseteq \overline{\Sigma }_\texttt{n}$.

Example 25 already provides a non-trivial example for a strongly impulse-controllable $\Sigma _\texttt{n}$, and every $\Sigma _\texttt{n}$ with two modes, which do not satisfy the single-switch impulse-controllability condition (11), is an example for a strongly impulse-uncontrollable $\Sigma _\texttt{n}$. In order to justify the introduction of the notion of essential impulse (un-)controllability, we will provide in the following examples which are essentially impulse (un-)controllable but not strongly impulse (un-)controllable.

Example 27

(Essentially, but not strongly, impulse-controllable class) Consider the switched system class $\Sigma _2$ with modes

$$\begin{aligned} (E_0,A_0,B_0)&=(I,0,{\left[ \begin{array}{ccc} 1\\ 0 \end{array}\right] }),\\ (E_1,A_1,B_1)&=(I,{\left[ \begin{array}{ccc} 0&{}1\\ -1 &{}0 \end{array}\right] },0),\\ (E_2,A_2,B_2)&=({\left[ \begin{array}{ccc} 0&{}1\\ 0&{}0\end{array}\right] },I,0). \end{aligned}$$

For any mode duration $\varvec{\tau }=(\tau _0,\tau _1)$, we see that the solution of the corresponding switched DAE (3) with (arbitrary) initial value $x(0^+) = {\left( \begin{array}{c} x_{01}\\ x_{02}\end{array}\right) }$ is given by

$$\begin{aligned}\begin{aligned} x(t)&= \begin{pmatrix} x_{01} + \int _{0}^t u \\ x_{02}\end{pmatrix}, \quad t\in (0,t_1),\\ x[t_1]&=0,\\ x(t)&= {\left[ \begin{array}{ccc} \cos (t-t_1) &{} \sin (t-t_1) \\ -\sin (t-t_1) &{} \cos (t-t_1)\end{array}\right] } x(t_1^-),\quad t\in (t_1,t_2),\\ x[t_2]&= -{\left[ \begin{array}{ccc} 0 &{} 1 \\ 0 &{} 0 \end{array}\right] } x(t_2^-) \delta _{t_2},\\ x(t)&= 0,\quad t>t_2. \end{aligned} \end{aligned}$$

For the specific mode duration $\tau _2=2\pi$, we see that $x(t_2^-) = x(t_1^-)$; hence, the second component of $x(t_2^-)$ is $x_{02}$, independently of the choice of the input u. However, for $x_{02}\ne 0$ this leads to an unavoidable Dirac impulse at $t=t_2$, i.e., $\Sigma _{\texttt{n}}$ is not strongly impulse-controllable. On the other hand, for all $\tau _2\ne k\pi$, it is easily seen that there exists an input u on $(0,t_1)$ resulting in a suitable first entry of $x(t_1^-)$ such that the rotation in mode 1 leads to $x_2(t_2^-)$ having a zero second component and hence resulting in an impulse-free switch at $t=t_2$. This shows that $\Sigma _{\texttt{n}}$ is indeed essentially impulse-controllable.

Example 28

(Essentially, but not strongly, impulse-uncontrollable class) Consider the switched system class $\Sigma _{2}$ with modes

$$\begin{aligned} (E_0,A_0,B_0)&=({\left[ \begin{array}{ccc} 1&{}0\\ 0&{}0\end{array}\right] },{\left[ \begin{array}{ccc} 0&{}0\\ 0&{}1\end{array}\right] },0),\\ (E_1,A_1,B_1)&=(I,{\left[ \begin{array}{ccc} 0&{}1\\ -1 &{}0\end{array}\right] },0),\\ (E_2,A_2,B_2)&=({\left[ \begin{array}{ccc} 0&{}1\\ 0&{}0\end{array}\right] },I,0). \end{aligned}$$

Note that for this example the input is not effecting the dynamics at all, so impulse-controllability reduces to impulse-freeness. Clearly, the solution in the initial mode is given by $x(t) = {\left[ \begin{array}{ccc} x_{01} \\ 0 \end{array}\right] }$ and afterwards the solutions are given as in Example 27 (because modes 1 and 2 are identical to the ones there). Consequently, for $\tau _1=2\pi$ we have $x(t_2^-)=x(t_1^-) = {\left[ \begin{array}{ccc} x_{01} \\ 0 \end{array}\right] }$, which results in an impulse-free solution of the switched DAE, i.e., $\Sigma _2$ is not strongly impulse-uncontrollable. Nevertheless, for any $\tau _1\ne k\pi$ we see that the second component of $x(t_2^-)$ is nonzero (if $x_{01}\ne 0$), and hence, a Dirac impulse occurs at $t=t_2$. This means that $\Sigma _{2}$ is essentially impulse-uncontrollable.

We are now ready to formulate our first main result concerning impulse-controllability of the class of switched DAEs with fixed mode sequence.

Theorem 29

Consider a class $\Sigma _\texttt{n}$ of switched systems (1) with fixed mode sequence $\textbf{q}=(0,1,2,\ldots ,\texttt{n})$. Then, $\Sigma _\texttt{n}$ is either essentially impulse-controllable or essentially impulse-uncontrollable.

Before presenting the proof of Theorem 29, we highlight some consequences and introduce a certain sequence of subspaces.

Remark 30

Theorem 29 states that the classes of switched DAEs with fixed mode sequences fall into four disjoint categories: 1) strongly impulse-controllable, 2) essentially (but not strongly) impulse-controllable, 3) essentially (but not strongly) impulse-uncontrollable, 4) strongly impulse-uncontrollable. Interestingly, there are only three categories for the notions of observability and controllability for switched systems with a fixed mode sequences (cf. [23] for observability, which by the duality arguments of [24] also carry over to controllability, see also [16]). The underlying reason is that the characterization of impulse-controllability is expressed in terms of sums and intersections of certain subspaces (see the forthcoming discussion), which can result in a singular dimension drop as well as a singular dimension increase in the involved duration-dependent subspaces; this in contrast to the observability (reachability) subspaces, which only involve intersections (sums).

The proof of Theorem 29 utilizes the following sequence of subspaces, which are inspired by the backward approach from [2]. For each switched DAE $(E_\sigma ,A_\sigma ,B_\sigma )\in \Sigma _\texttt{n}$ with corresponding mode durations $\varvec{\tau }=(\tau _0,\tau _1,\ldots ,\tau _{\texttt{n}-1})\in \mathbb {R}^\texttt{n}_{>0}$ define

$$\begin{aligned} \begin{aligned} \mathcal {K}^{\varvec{\tau }}_\texttt{n}&:= \mathcal {C}^{\textrm{imp}}_\texttt{n},\\ \mathcal {K}^{\varvec{\tau }}_{i-1} \!\!&:= \!\! \left( {{\,\textrm{im}\,}}\Pi _{i-1}\!\cap \!(e^{-A^{\textrm{diff}}_{i-1}\tau _{i-1}} \mathcal {K}^{\varvec{\tau }}_{i} + \mathcal {R}_{i-1} )\right) \oplus \mathcal {D}^\textrm{imp}_{i-1},\\&\quad i=\texttt{n},\texttt{n}-1,\ldots ,1. \end{aligned} \end{aligned}$$

(14)

In view of invertibility of each exponential term $e^{-A^{\textrm{diff}}_{i-1}\tau _{i-1}}$ in (14) and $A^{\textrm{diff}}_{i-1}$-invariance of the subspaces ${{\,\textrm{im}\,}}\Pi _{i-1}$ and $\mathcal {R}_{i-1}$, it follows that the recursive definition (14) can equivalently be written as:

$$\begin{aligned} \mathcal {K}^{\varvec{\tau }}_{i-1} = e^{-A^{\textrm{diff}}_{i-1}\tau _{i-1}}\left( {{\,\textrm{im}\,}}\Pi _{i-1}\cap ( \mathcal {K}^{\varvec{\tau }}_{i} + \mathcal {R}_{i-1} )\right) \oplus \mathcal {D}^\textrm{imp}_{i-1}. \end{aligned}$$

We are now ready to present the proof of Theorem 29.

Proof of Theorem 29

The proof utilizes properties of analytic matrices, which are recalled in Appendix.

Case 1: All systems in $\Sigma _{\texttt{n}}$ are impulse-controllable.

By definition, $\Sigma _{\texttt{n}}$ is then strongly impulse-controllable and in particular essentially impulse-controllable.

Case 2: There exists at least one impulse-uncontrollable system in $\Sigma _{\texttt{n}}$.

In view of Lemma 52 in Appendix, we can choose an analytic matrix $N_0:\mathbb {R}^{\texttt{n}}\rightarrow \mathbb {R}^{n\times k_0}$ with generically full rank such that ${{\,\textrm{im}\,}}N_0(\varvec{\tau })=\mathcal {K}_0^{\varvec{\tau }}$ for a.a. $\varvec{\tau }\in \mathbb {R}^\texttt{n}$.

Case 2a: For all impulse-uncontrollable mode durations $\overline{\varvec{\tau }}\in \mathbb {R}^{\texttt{n}}_{>0}$, we have that ${{\,\textrm{im}\,}}N_0(\overline{\varvec{\tau }})\ne \mathcal {K}_0^{\overline{\varvec{\tau }}}$ or $N_0(\overline{\varvec{\tau }})$ does not have full rank.

In this case, the set of impulse-uncontrollable mode durations is contained in a set of measure zero; hence, $\Sigma _{\texttt{n}}$ is essentially impulse-controllable.

Case 2b: There exists an impulse-uncontrollable mode duration $\overline{\varvec{\tau }}\in \mathbb {R}^{\texttt{n}}_{>0}$ such that ${{\,\textrm{im}\,}}N_0(\overline{\varvec{\tau }})=\mathcal {K}_0^{\overline{\varvec{\tau }}}$ and $N_0(\overline{\varvec{\tau }})$ has full rank.

Since impulse-controllability for a specific switching signal is equivalent to (15), we have

$$\begin{aligned} \mathcal {V}_{( E_0, A_0, B_0)}\not \subseteq \mathcal {K}_0^{\overline{\varvec{\tau }}}={{\,\textrm{im}\,}}N_0(\overline{\varvec{\tau }}). \end{aligned}$$

Hence, there exists a vector $v\in \mathcal {V}_{( E_0, A_0, B_0)}$ such that $M(\varvec{\tau }):={{\,\textrm{rank}\,}}{}[N(\varvec{\tau }), v ]$ has full rank for $\varvec{\tau }=\overline{\varvec{\tau }}$. In particular, M is an analytic matrix for which $\varvec{\tau }\mapsto \det M(\varvec{\tau })^\top M(\varvec{\tau })$ is not identically zero, i.e. M is generically full rank. Consequently, $v\not \in {{\,\textrm{im}\,}}N(\varvec{\tau })$ for a.a. $\varvec{\tau }\in \mathbb {R}^\texttt{n}_{>0}$, and hence,

$$\begin{aligned} \mathcal {V}_{( E_0, A_0, B_0)}\not \subseteq {{\,\textrm{im}\,}}N_0(\varvec{\tau })=\mathcal {K}_0^{\varvec{\tau }}\quad \text {for a.a. }\varvec{\tau }\in \mathbb {R}^{\texttt{n}}_{>0}. \end{aligned}$$

This implies that almost all systems in $\Sigma _{\texttt{n}}$ are impulse-uncontrollable, i.e. $\Sigma _{\texttt{n}}$ is essentially impulse-uncontrollable. This concludes the proof as no other cases are possible. $\square$

In the investigation of the different notions of impulse-controllability for the system class $\Sigma _\texttt{n}$, the sequences (14) will be exploited further. The relevance of the subspaces $\mathcal {K}_{i-1}^{\varvec{\tau }}$ is highlighted by the following result.

Lemma 31

(Cf. [2, Lem. 19]) Consider a switched DAE $(E_\sigma ,A_\sigma ,B_\sigma )\in \Sigma _\texttt{n}$ with mode duration $\varvec{\tau }=(\tau _0,\tau _1,\ldots ,\tau _{\texttt{n}-1})\in \mathbb {R}^\texttt{n}_{>0}$ and $\mathcal {K}_i^{\varvec{\tau }}$ given by (14). Then,

$$\begin{aligned} \mathcal {K}_i^{\varvec{\tau }} = \left\{ x_i\in \mathbb {R}^n\,\left| \,\begin{aligned}&\exists \text { impulse-free sol. }(x,u)\text { of (}1\text {)}\\&\text {on }[t_i,t_f)\text { with }x(t_i^-)=x_i\end{aligned}\,\right. \!\!\right\} . \end{aligned}$$

Proof

The proof follows inductively with the same arguments as used in the proof of [2, Lem. 19] and is therefore omitted.

Corollary 32

([2, Thm. 21]) The switched DAE $(E_\sigma ,A_\sigma ,B_\sigma )\in \Sigma _\texttt{n}$ with fixed mode sequence and with mode duration $\varvec{\tau }\in \mathbb {R}^{\texttt{n}}_{>0}$ is impulse-controllable if, and only if

$$\begin{aligned} \mathcal {V}_{(E_0,A_0,B_0)} \subseteq \mathcal {K}^{\varvec{\tau }}_0. \end{aligned}$$

(15)

An obvious characterization of strong impulse (un-)controllability of the system class $\Sigma _\texttt{n}$ is therefore the condition that (15) does (not) hold for all $\varvec{\tau }\in \mathbb {R}^{\texttt{n}}_{>0}$. However, this characterization is not very insightful and impracticable because uncountably many subspace sequence need to be calculated. We can obtain more practicable (sufficient) conditions for strong impulse (un-)controllability, by using the fact that for any subspace $\mathcal {S}$, any matrix A and any $t\in \mathbb {R}$ we have

$$\begin{aligned} \langle \mathcal {S}\mid A \rangle \subseteq e^{At} \mathcal {S}\subseteq \langle A \mid \mathcal {S}\rangle , \end{aligned}$$

(16)

where $\langle \mathcal {S}\mid A \rangle$ denotes the largest A-invariant subspace contained in $\mathcal {S}$ and $\langle A \mid \mathcal {S}\rangle$ denotes the smallest A-invariant subspace containing $\mathcal {S}$. In fact, we can construct an over- and underestimation of $\mathcal {K}^{\varvec{\tau }}_i$ as follows:

$$\begin{aligned}{} & {} \overline{\mathcal {K}}_{i-1}:= \left\langle A^{\textrm{diff}}_{i-1} \mid {{\,\textrm{im}\,}}\Pi _{i-1}\cap (\overline{\mathcal {K}}_{i} + \mathcal {R}_{i-1} )\right\rangle \oplus \mathcal {D}^\textrm{imp}_{i-1}, \end{aligned}$$

(17)

$$\begin{aligned}{} & {} \underline{\mathcal {K}}_{i-1}:= \left\langle {{\,\textrm{im}\,}}\Pi _{i-1}\cap ( \underline{\mathcal {K}}_{i} + \mathcal {R}_{i-1} )\mid A^{\textrm{diff}}_{i-1}\right\rangle \oplus \mathcal {D}^\textrm{imp}_{i-1}, \end{aligned}$$

(18)

each for $i=\texttt{n},\texttt{n}-1,\ldots ,1$ and with $\overline{\mathcal {K}}_\texttt{n}= \underline{\mathcal {K}}_\texttt{n}= \mathcal {C}^{\textrm{imp}}_\texttt{n}$. By construction, we have $\underline{\mathcal {K}}_{i}\subseteq \mathcal {K}^{\varvec{\tau }}_{i}\subseteq \overline{\mathcal {K}}_{i}$, which immediately leads to the following sufficient condition for strong impulse (un-)controllability.

Corollary 33

The system class $\Sigma _\texttt{n}$ is strongly impulse-controllable if

$$\begin{aligned} \mathcal {V}_{(E_0,A_0,B_0)} \subseteq \underline{\mathcal {K}}_0 \end{aligned}$$

and it is strongly impulse-uncontrollable if

$$\begin{aligned} \mathcal {V}_{(E_0,A_0,B_0)} \not \subseteq \overline{\mathcal {K}}_0. \end{aligned}$$

Remark 34

It is also possible to obtain under- and overestimation of $\mathcal {K}^{\varvec{\tau }}_i$ by using (16) directly in (14); however, it turns out that this leads to smaller underestimations and bigger overestimations and hence leads to more conservative sufficient conditions.

Remark 35

(Sufficient condition for essential impulse (un-) controllability) It seems that there is not a simple, weaker sufficient condition compared to the ones provided in Corollary 33 to guarantee essential impulse (un-) controllability. However, if condition (15) is satisfied for some random set of duration times, then $\Sigma _\texttt{n}$ is essentially impulse-controllable with probability one and if (15) is not satisfied, then $\Sigma _\texttt{n}$ is essentially impulse-uncontrollable with probability one. In practice, this seems to be a reliable way to check for essential impulse (un-)controllability.

4 (Quasi)-causal impulse-controllability

So far we have presented several sufficient conditions for strong impulse-controllability, which is concerned with the existence of an input (depending on the initial value) resulting in an impulse-free solution. Clearly, this “impulse-avoiding” input generally depends on the switching signal. In particular, for the system class $\Sigma _\texttt{n}$ with known-mode sequence it is not clear whether an impulse-avoiding input can be constructed independently of the (unknown) mode durations. The following example shows that indeed the impulse-avoiding input may depend on future mode durations.

Example 36

(Non-causal impulse-controllability) Consider the class $\Sigma _2$ of switched systems with fixed mode sequence $\textbf{q}=(0,1,2)$ and with modes given by

$$\begin{aligned} (E_0,A_0,B_0)&=(I,0,{\left[ \begin{array}{ccc} 0\\ 1 \end{array}\right] }),\\ (E_1,A_1,B_1)&=(I,{\left[ \begin{array}{ccc} 0&{}0\\ 0 &{}1\end{array}\right] },0),\\ (E_2,A_2,B_2)&=({\left[ \begin{array}{ccc} 0&{}0\\ 1&{}1\end{array}\right] },{\left[ \begin{array}{ccc} 1&{}1\\ 0&{}1\end{array}\right] },0). \end{aligned}$$

For a given switching signal with mode durations $\varvec{\tau }=(\tau _0,\tau _1) \in \mathbb {R}^2_{>0}$, the sequence (14) is given by:

$$\begin{aligned} \mathcal {K}_2^{\varvec{\tau }}&= \mathcal {C}^{\textrm{imp}}_2={{\,\textrm{im}\,}}{\left[ \begin{array}{ccc} 1 \\ -1\end{array}\right] },\\ \mathcal {K}_1^{\varvec{\tau }}&= {{\,\textrm{span}\,}}\left\{ e^{A_1\tau _1}{\left[ \begin{array}{ccc} 1\\ -1\end{array}\right] } \right\} ={{\,\textrm{span}\,}}\left\{ {\left[ \begin{array}{ccc} 1 \\ e^{-\tau _1}\end{array}\right] }\right\} ,\\ \mathcal {K}_0^{\varvec{\tau }}&=\mathcal {K}_1^{\varvec{\tau }} +\mathcal {R}_0={{\,\textrm{span}\,}}\left\{ {\left[ \begin{array}{ccc} 1 \\ e^{-\tau _1}\end{array}\right] },{\left[ \begin{array}{ccc} 0\\ 1\end{array}\right] }\right\} = \mathbb {R}^2. \end{aligned}$$

Hence, the system class is strongly impulse-controllable. However, for two mode durations $\varvec{\tau }=(\tau _0,\tau _1)$ and $\overline{\varvec{\tau }}=(\overline{\tau }_0,\overline{\tau }_1)$ with $\tau _1\ne \overline{\tau }_1$ we have that

$$\begin{aligned} \mathcal {K}_1^{\varvec{\tau }}\cap \mathcal {K}_1^{\overline{\varvec{\tau }}}=\{0\}. \end{aligned}$$

Since the first mode is not null-controllable, this means that the value of the state $x(t_1^-)$ explicitly depends on the future mode-duration in order to guarantee impulse-freeness. For example, for the (consistent) initial condition $x(0^+)={\left[ \begin{array}{ccc} 1\\ 0\end{array}\right] }$, it follows for the first state component that $x_1(t_2^-)=1$ as $\dot{x}_1=0$ in the zeroth and first mode. Hence, in order to ensure an impulse-free solution it is required that the second state component satisfies $x_2(t_2^-)=-1$. This is achieved if and only if $x_2(t_1^-)=e^{-\tau _1}$. Consequently, the control on the interval $(0,t_1)$ needs to ensure that $x_2(t_1^-)=e^{-\tau _1}$ and therefore necessarily depends on the future mode duration $\tau _1$. $\diamond$

4.1 Quasi-causality of $\Sigma _\texttt{n}$

In some application, it may be the case that the current mode duration is known once the mode is activated, but the mode durations of the future modes are not known yet; for example, if a switch is induced by shutting down or decoupling components for scheduled maintenance whose duration is known upfront. In this case, causality of the input means that it should be independent from the future mode durations, but it can utilize the knowledge when the next switch happens. This somewhat weaker notion of causal impulse-controllability is called quasi-causal impulse-controllability and is defined in terms of the existence of a family of input-defining maps

$$\begin{aligned} \mathcal {U}_t:(\sigma _{(t_0,t)},x_0)\mapsto u_{(t_0,t)}, \end{aligned}$$

such that for all $\sigma \in \mathcal {S}_{\texttt{n}}$ and all initial values $x_0\in \mathcal {V}_{(E_0, A_0,B_0)}$ the corresponding solution $(x,u)_{(t_0,t)}$ of $(E_\sigma ,A_\sigma ,B_\sigma )$ on $(t_0,t)$ satisfying $x(t_0^+)=x_0$ is impulse-free. Additionally, we have to require that the map $\mathcal {U}_t$ is itself quasi-causal, i.e., for all switching times $t_i$ and $s>t_i$ the following holds

$$\begin{aligned} \mathcal {U}_{t_i}(\sigma _{(t_0,t_i)},x_0) = \mathcal {U}_{s}(\sigma _{(t_0,s)},x_0)_{(t_0,t_i)}. \end{aligned}$$

(19)

Observe that for two switching signals $\sigma ,\bar{\sigma } \in \overline{\mathcal {S}}_\texttt{n}$ satisfying $\sigma _{(t_0,s)}=\bar{\sigma }_{(t_0,s)}$ for some $s\in (t_i,t_{i+1})$ it may occur that $\mathcal {U}_s(\sigma _{(t_0,s)},x_0)\ne \mathcal {U}_s(\bar{\sigma }_{(t_0,s)},x_0)$.

Before presenting conditions for quasi impulse-controllability, we will present the following lemma, which is required in the proofs to come.

Lemma 37

For all $p\in \{0,1,...,,\texttt{n}-1\}$ and $\underline{\mathcal {K}}_{p}$ as in (18), we have

$$\begin{aligned} {\underline{\mathcal {K}}_{p}=\left\{ x_p\in \mathbb {R}^n\left| \begin{array}{l} \forall \tau >0\;\exists \; \text {impulse-free solution} \;(x,u)\\ \text {on }[t_p,t_p+\tau )\text { of } E_{p}\dot{x}=A_{p}x+B_{p}u, \\ \text {with}\; x(t_p^-)=x_p \text { and } x((t_p+\tau )^-)\in \underline{\mathcal {K}}_{p+1} \end{array}\right. \right\} }, \end{aligned}$$

i.e. the subspace $\underline{\mathcal {K}}_p$ consists of all initial states for mode p which can be controlled impulse-freely into the subspace $\underline{\mathcal {K}}_{p+1}$ within a given time duration $\tau >0$.

Before providing the proof, we want to highlight that in the statement above the impulse avoiding input in general depends on $\tau$, i.e., on the mode duration of the current mode, whereas the subspaces given by (18) are independent from the mode duration (but depend on the mode sequence).

Proof

Let $x_p\in \underline{\mathcal {K}}_{p}$. Then, $x_p=w+v$ for some $w\in \langle {{\,\textrm{im}\,}}\Pi _{p}\cap \big (\underline{\mathcal {K}}_{p+1} +\mathcal {R}_{p}\big )\mid A^{\textrm{diff}}_p\rangle$ and $v\in \mathcal {D}_{p}^\textrm{imp}$. Recall that any $v\in \mathcal {D}_{p}^\textrm{imp}$ can be impulse-freely controlled to zero with a smooth input for any given time duration $\tau >0$. Let $u_{v,\tau }$ be such an input.

By linearity of solutions, it follows that for the input $u=u_{v,\tau }+u_{w,\tau }$ where $u_{w,\tau }$ is some impulse-free input the solution $(x,u)=(x_v,u_{v,\tau })+(x_w,u_{w,\tau })$ with $x_v(t_p^-)=v$ and $x_w(t_p^-)=w$. Furthermore, it satisfies

$$\begin{aligned} x((t_p+\tau )^-)=x_v((t_p+\tau )^-)+x_w((t_p+\tau )^-)=x_w(t_p+\tau ^-). \end{aligned}$$

Hence, $x(t_p^-)=x_p$ can be steered to $\underline{\mathcal {K}}_{p+1}$ if w can be steered to $\underline{\mathcal {K}}_{p+1}$. Consequently, it suffices to consider the case $x_p \in \langle {{\,\textrm{im}\,}}\Pi _{p}\cap \big (\underline{\mathcal {K}}_{p+1} +\mathcal {R}_{p}\big )\mid A^{\textrm{diff}}_p\rangle$. Note that as $x_p\in {{\,\textrm{im}\,}}\Pi _p$ it follows that $x_p=\Pi _px_p$, since $\Pi _p$ is a projector. Furthermore, it follows then from $A^{\textrm{diff}}_p$-invariance that for $\tau \in \mathbb {R}$

$$\begin{aligned} e^{A^{\textrm{diff}}_p\tau } \Pi _p x_p=k_{p+1}^\tau +\eta ^\tau . \end{aligned}$$

for some $k_{p+1}^\tau \in \underline{\mathcal {K}}_{p+1}$ and $\eta ^\tau \in \mathcal {R}_p$. As $\eta ^\tau$ is contained in the reachable space, there exists a smooth input u defined on $[t_p,t_p+\tau )$ which steers the origin smoothly to $-\eta ^\tau$. Applying this input for the initial value $x(0^-)=x_p$ results in solution (x, u) satisfying

$$\begin{aligned} x(\tau ^-)&=e^{A^{\textrm{diff}}_p\tau }\Pi _p x_p-\eta ^\tau , \\&=k^\tau _{p+1}+\eta ^\tau -\eta ^\tau ,\\&=k^\tau _{p+1}, \end{aligned}$$

as desired.

Conversely, let $x_p$ be such that for all $\tau$ there exists an impulse-free solution (x, u) of $E_{p}\dot{x} = A_{p}x+B_{p} u$ with $x(t_p^-)=x_p$ and $x((t_p+\tau )^-)\in \underline{\mathcal {K}}_{p+1}$. Using the same inductive arguments as in Lemma 31 and utilizing $A^\textrm{diff}_p$ invariance of ${{\,\textrm{im}\,}}\Pi _p$, $\mathcal {R}_p$, $\mathcal {D}^\textrm{imp}_p$, it then follows for all $\tau \in \mathbb {R}$ that

$$\begin{aligned} x_p&\in {{\,\textrm{im}\,}}\Pi _p\cap \big (e^{-A^{\textrm{diff}}_{p}\tau }\underline{\mathcal {K}}_{p+1}+\mathcal {R}_{p}\big ) \oplus \mathcal {D}_p^\textrm{imp}\\&=e^{-A^{\textrm{diff}}_{p}\tau }\left( {{\,\textrm{im}\,}}\Pi _p\cap \big (\underline{\mathcal {K}}_{p+1}+\mathcal {R}_{p}\big ) \oplus \mathcal {D}_p^\textrm{imp}\right) \end{aligned}$$

As this holds for all $\tau >0$, we obtain

$$\begin{aligned} x_p&\in \bigcap _{\tau >0} e^{-A^{\textrm{diff}}_{p}\tau }\left( {{\,\textrm{im}\,}}\Pi _p\cap \big (\underline{\mathcal {K}}_{p+1}+\mathcal {R}_{p}\big ) \oplus \mathcal {D}_p^\textrm{imp}\right) \\&= \langle {{\,\textrm{im}\,}}\Pi _{p}\cap (\underline{\mathcal {K}}_{p+1} +\mathcal {R}_{p}) \oplus \mathcal {D}_{p}^\textrm{imp} \mid A^{\textrm{diff}}_p\rangle = \underline{\mathcal {K}}_p, \end{aligned}$$

which follows from the general facts that $\bigcap _{\tau >0} e^{-A\tau }\mathcal {V}= \left\langle \mathcal {V}\mid A\right\rangle$ and $\langle \mathcal {V}+ \mathcal {W}\mid A \rangle = \langle \mathcal {V}\mid A \rangle + \mathcal {W}$ for any matrix $A\in \mathbb {R}^{n\times n}$ and any subspaces $\mathcal {V},\mathcal {W}\subseteq \mathbb {R}^n$ of which $\mathcal {W}$ is A-invariant. This concludes the proof. $\square$

Given this result, we can present the following simple characterization of quasi-causally impulse-controllable system classes.

Theorem 38

The system class $\Sigma _\texttt{n}$ is quasi-causally impulse-controllable if and only if

$$\begin{aligned} \mathcal {V}_{(E_0, A_0, B_0)}\subseteq \underline{\mathcal {K}}_0 \end{aligned}$$

Proof

$(\Rightarrow )$ Suppose the system class is quasi-causally impulse-controllable. Consider the solution (x, u) of (1) with $x(t_0^+)=x_0$ and $u_{(t_0,t_f)}$ given by $\mathcal {U}_{t_f}(\sigma _{(t_0,t_f)},x_0)$. Then by definition, the solution (x, u) is impulse-free on $(t_0,t_f)$, in particular, $x(t_\texttt{n}^-)\in \mathcal {C}^{\textrm{imp}}_\texttt{n}=\underline{\mathcal {K}}_\texttt{n}$ for all possible switching signals.

In the following, we want to show by induction that $x(t_i^-)\in \underline{\mathcal {K}}_i$ for $i\in \{\texttt{n}-1,\ldots ,1,0\}$. Hence, inductively, we may assume that if (x, u) satisfies $x(t_0^+)=x_0$ and u is defined by $\mathcal {U}_{t_i}(\sigma _{(t_0,t_i)},x_0)$, then $x(t_i^-)\in \underline{\mathcal {K}}_i$ for all switching signals. We want to show that $x(t_{i-1}^-)\in \underline{\mathcal {K}}_{i-1}$ for any solution (x, u) of (1) with $x(t_0^+)=x_0$ and u given by $\mathcal {U}_{t_{i-1}}(\sigma _{(t_0,t_{i-1})},x_0)$. For any $\tau >0$, consider the switching signal $\bar{\sigma }$ with $\bar{\sigma }_{(t_0,t_{i-1})}=\sigma _{(t_0,t_{i-1})}$ and $\bar{t}_i = \bar{t}_{i-1} + \tau = t_{i-1} + \tau$. Let $\bar{u}$ be given by $\mathcal {U}_{\bar{t}_i}(\bar{\sigma }_{(t_0,\bar{t}_i)},x_0)$, then the corresponding solution $(\bar{x},\bar{u})$ is impulse-free and by induction assumption satisfies $\bar{x}(\bar{t}_i)\in \underline{\mathcal {K}}_i$. Since $\tau >0$ was arbitrary, Lemma 37 yields that $\bar{x}(t_{i-1}^-)\in \underline{\mathcal {K}}_{i-1}$ By causality, $u_{(t_0,t_{i-1})} = \bar{u}_{(t_0,t_{i-1})}$ and hence $x(t_{i-1}^-)=\bar{x}(t_{i-1}^-)$ which concludes the inductive proof. Since for all $x_0\in \mathcal {V}_{( E_0, A_0, B_0)}$ there exists an impulse-free solution (x, u) satisfying $x(t_0^+) = x(t_0^-)=x_0$ we can conclude that $x_0\in \underline{\mathcal {K}}_0$ and hence

$$\begin{aligned} \mathcal {V}_{( E_0, A_0, B_0)}\subseteq \underline{\mathcal {K}}_0. \end{aligned}$$

$(\Leftarrow )$ Let $\sigma \in \mathcal {S}_\texttt{n}$. Recall that by definition for all $\sigma \in \mathcal {S}_\texttt{n}$, for each mode $p\in \{0,1,...,\texttt{n}-1\}$ and each $x_p\in \underline{\mathcal {K}}_p$ there exists an input $u^p(\cdot ,x_p)$ on $[t_p,t_{p+1})$ such that the solution x of mode p satisfies $x(t_p^-)=x_p$ and $x(t_{p+1}^-)\in \underline{\mathcal {K}}_{p+1}$. Now, concatenate these inputs inductively as follows: $u(t):= u^0(t,x_0)$ for $t\in [t_0,t_1)$ and $u(t):= u^p(t,x(t_p^-))$ for $t\in [t_p,t_{p+1})$ where $x(t_p^-)$ is the value of the solution x corresponding to the already defined input u on $[t_0,t_p)$. Finally, by assumption $x(t_\texttt{n}^-)\in \mathcal {C}^{\textrm{imp}}_\texttt{n}$, hence the input u can be extended on $[t_\texttt{n},\infty )$ in such a way that the solution remains impulse-free. Altogether we can define $\mathcal {U}_{t_i}(\sigma _{(t_0,t_i)},x_0):=u_{(t_0,t_i)}$ which satisfies the quasi-causality properties for all switching signals and all $x_0$. Hence, the system class is quasi-causally impulse-controllable. $\square$

4.2 Causal impulse-controllability of $\Sigma _\texttt{n}$

Knowledge of the current mode duration cannot always be assumed; hence, we want to provide in this subsection a characterization of a more strict causality notion. In particular, we make the above definition of quasi-causal impulse-controllability stronger by requiring the causality property (19) of $\mathcal {U}_t$ to hold for all $t\in (t_0,\infty )$ and not only for the switching times $t=t_i$ of the corresponding switching signal. That is, we require for all t and all $s>0$

$$\begin{aligned} \mathcal {U}_{t_i}(\sigma _{(t_0,t)},x_0) = \mathcal {U}_{s}(\sigma _{(t_0,s)},x_0)_{(t_0,t)}. \end{aligned}$$

A key idea to characterize this stronger notion of causality are so called controlled invariant subspaces which are subspaces associated with a DAE $E\dot{x}=Ax+Bu$ which have the property that for any initial value in such a subspace there exists an input u such that the trajectory x remains in that subspace, cf. [25, 26]. It is well known that any controlled invariant subspace $\mathcal {V}\subseteq \mathcal {V}_{E,A,B}$ is (A, E, B)-invariant, i.e. $A\mathcal {V}\subseteq E\mathcal {V}+ {{\,\textrm{im}\,}}B$; in particular, the augmented consistency space $\mathcal {V}_{E,A,B}$ is the largest controlled invariant subspace. For the class $\Sigma _\texttt{n}$ of switched DAEs with a known-mode sequence to be causally impulse controllable, it is now intuitively clear that at the switch the state trajectory has to jump immediately into a controlled invariant subspace, which is contained in a suitable subspace for the following mode. This intuition is formalized by the following sequence of subspaces

$$\begin{aligned} \underline{\mathcal {C}}_{i-1}:= \langle \underline{\mathcal {C}}_i \mid A_{i-1},E_{i-1},B_{i-1}\rangle + \ker E_{i-1}, \end{aligned}$$

for $i\in \{\texttt{n},\texttt{n}-1,...,1\}$ and with $\underline{\mathcal {C}}_\texttt{n}:=\mathcal {C}^{\textrm{imp}}_\texttt{n}$; furthermore, $\langle \underline{\mathcal {C}_i}\mid A_{i-1},E_{i-1},B_{i-1}\rangle$ denotes the largest $( A_{i-1},E_{i-1}, B_{i-1} )$ invariant subspace contained in $\overline{\mathcal {C}}_i$. Note that such a subspace can be calculated with a subspace sequence similar to the Wong sequences, see [14, Thm. 10].

Theorem 39

The system class $\Sigma _\texttt{n}$ is causally impulse-controllable if, and only if,

$$\begin{aligned} \mathcal {V}_{(E_0,A_0,B_0)}\subseteq \underline{\mathcal {C}}_0. \end{aligned}$$

(20)

Proof

$(\Rightarrow )$ Suppose the system class $\Sigma _{\texttt{n}}$ is causally impulse-controllable. Then for any given switching signal $\sigma \in \mathcal {S}_\texttt{n}$ there exists an impulse-free solution (x, u) where $u_{[t_0,t)}=\mathcal {U}_t(\sigma _{[t_0,t)},x_0)$.

We will prove by induction that $x(t_{i}^-)\in \underline{\mathcal {C}}_{i}$ for all $i\in \{\texttt{n}, \texttt{n}-1,...,1\}$. Since (x, u) is impulse-free, it follows that $x(t_\texttt{n}^-)\in \mathcal {C}^{\textrm{imp}}_\texttt{n}=\underline{\mathcal {C}}_\texttt{n}$. Hence, we assume that the statement holds for i and continue to proof the statement for $i-1$. Consider now another switching signal $\tilde{\sigma }\in \mathcal {S}_{\texttt{n}}$ such that $\sigma _{(t_0,t_i)} = \tilde{\sigma }_{(t_0,t_i)}$ (in particular, $\tilde{t}_i \ge t_i$) and with corresponding impulse-free solution $(\tilde{x},\tilde{u})$, where $\tilde{u}_{[t_0,t)}=\mathcal {U}_t(\tilde{\sigma }_{[t_0,t)},x_0)$. By the inductive assumption, we have $\tilde{x}(\tilde{t}_i^-)\in \underline{\mathcal {C}}_i$. Consequently, we can always find an input $\tilde{u}$ on $[t_i,\tilde{t}_i)$ which ensures that the trajectory $\tilde{x}$ which starts at $x(t_i^-)\in \underline{\mathcal {C}}_i$ stays in the same subspace for arbitrary $\tilde{t}_i>t_i$ under the dynamics of $E_{i-1}\dot{x} = A_{i-1} x + B_{i-1} u$. Consequently, $x(t_i^-)$ must be contained in the largest controlled invariant subspace within $\overline{\mathcal {C}}_i$, i.e. $x(t_i^-)\in \langle \underline{\mathcal {C}}_i\mid A_{i-1},E_{i-1},B_{i-1}\rangle$. Since this is true for any mode duration $t_i-t_{i-1}$, it follows that $x(t_{i-1}^+)\in \langle \underline{\mathcal {C}}_i\mid A_{i-1},E_{i-1},B_{i-1}\rangle$. Since x is impulse-free, it follows that $x(t_{i-1}^-) - x(t_{i-1}^+)\in \ker E_{i-1}$ (otherwise the Dirac impulse occurring in $\dot{x}$ at $t_{i-1}$ must also occur on the right-hand side of the DAE, which is not possible because x and u are impulse-free), this shows that $x(t_{i-1}^-)\in \underline{\mathcal {C}}_{i-1}$. Now we can conclude that $x_0\in \underline{\mathcal {C}}_0$ and since this holds for all $x_0\in \mathcal {V}_{(E_0, A_0, B_0)}$ we have shown the necessity part of the statement. $(\Leftarrow )$ Let $x_i\in \underline{\mathcal {C}}_i$, then there exists $x_i^+\in \langle \underline{\mathcal {C}}_{i+1}\mid A_i,E_i,B_i \rangle \subseteq \mathcal {V}_{(E_i,A_i,B_i)}$ and $\xi _i\in \ker E_i$ such that $x_i = x_i^+ +\xi _i$. Choose an input u on $[t_i,\infty )$ such that the solution x of $E_i\dot{x}=A x_i + B_i u$ with consistent initial condition $x(t_i^+)=x_i^+$ satisfies $x(t^+)\in \langle \underline{\mathcal {C}}_{i+1}\mid A_i,E_i,B_i \rangle \subseteq \underline{\mathcal {C}}_{i+1}$ for all $t\in [t_i,\infty )$. Furthermore, observe that the zero distribution on $[t_i,\infty )$ is an (impulse-free) solution of the (inconsistent) initial value problem $E_i\dot{x}=A_i x$, $x(t_i^-)=\xi _i$. Consequently, the previously chosen (x, u) is also a solution of $E_i\dot{x}=A x_i + B_i u$ with inconsistent initial value $x(t_i^-)=x_i$. Hence, for a given switching signal and (consistent) initial condition $x_0\in \underline{\mathcal {C}}_0$, we can successively construct an input (independent of the mode durations), such that the resulting solution x is impulse-free and satisfies $x(t_i^-)\in \underline{\mathcal {C}}_{i-1}$. In particular, $x(t_\texttt{n}^-)\in \mathcal {C}^{\textrm{imp}}_\texttt{n}$ which implies that u can be defined on $[t_\texttt{n},\infty )$ such that the resulting solution remains impulse-free, which concludes the proof. $\square$

The above condition on causal impulse-controllability is in most situation too restrictive because the controller must be designed in such a way that at a switch the correct input must be chosen to avoid a Dirac impulse and at the same time the state right after the switch must be an element of a controlled invariant subspace contained in the impulse-controllable subspace of the next mode. This is required because if some non-instantaneous control action is needed to drive the state into a suitable subspace, then this control input (which needs a duration $d>0$ to arrive at that subspace) would not work for a switching duration smaller than this d (and hence causality would be violated). However, in most practical situation, a dwell time for the switching signal can be assumed, i.e. there exists $d>0$ such that $t_{i+1}-t_i\ge d$ for all switching times. Under such a dwell-time condition, we are able to prove a less restrictive characterization of causal impulse-controllability. Toward this goal, we define an enlarged version of the subspace sequence (20) for the system class $\Sigma _{\texttt{n}}$ as follows:

$$\begin{aligned} \overline{\mathcal {C}}_{i-1} := \left\langle \overline{\mathcal {C}}_{i} \mid A_{i-1},E_{i-1},B_{i-1}\right\rangle + \mathcal {R}_{i-1} +\ker E_{i-1} \end{aligned}$$

(21)

for $i\in \{\texttt{n},\texttt{n}-1,...,1\}$ and with $\overline{\mathcal {C}}_\texttt{n}= \mathcal {C}^{\textrm{imp}}_\texttt{n}$.

Theorem 40

The system class $\Sigma _\texttt{n}$ with some dwell time $d>0$ is causally impulse-controllable if and only if

$$\begin{aligned} \mathcal {V}_{(E_0,A_0,B_0)} \subseteq \overline{\mathcal {C}}_0. \end{aligned}$$

Note that the above characterization of causal impulse-controllability is independent of the dwell time $d>0$; however, the input map $\mathcal {U}_t$ will depend on it. The proof of the theorem utilizes the following property of (A, E, B)-invariant subspaces.

Lemma 41

Let (E, A) be a regular matrix pair with corresponding consistency projector $\Pi$ and flow matrix $A^{\textrm{diff}}$. Then for any (A, E, B) invariant subspace $\mathcal {V}$ we have

• a) $\Pi \mathcal {V}\subseteq \langle \mathcal {V}+\mathcal {R}\mid A^{\textrm{diff}}\rangle \subseteq \mathcal {V}+ \mathcal {R}$,

• b) $A^{\textrm{diff}}\mathcal {V}\subseteq \mathcal {V}+ \mathcal {R}$.

Proof

• (a) Let $x\in \mathcal {V}$. Then, there exists an input such that $x(t)\in \mathcal {V}$ for all $t\ge 0$. Consequently,

  $$\begin{aligned} e^{A^{\textrm{diff}}t}\Pi x_0\in \mathcal {V}+\mathcal {R}\end{aligned}$$

  for all $t\ge 0$, i.e. $\Pi x_0 \in \bigcap _{t>0} e^{-A^{\textrm{diff}}t}(\mathcal {V}+\mathcal {R})$. Hence, $\Pi x_0 \in \langle \mathcal {V}+\mathcal {R}\mid A^{\textrm{diff}}\rangle$.

• (b) Recall Remark 3, which notices $A^{\textrm{diff}}\Pi = A^{\textrm{diff}}$. Hence, it follows from a) that for each $x\in \mathcal {V}$,

  $$\begin{aligned}\begin{aligned} A^{\textrm{diff}}x = A^{\textrm{diff}}\Pi x&\in A^{\textrm{diff}}\langle \mathcal {V}+\mathcal {R}\mid A^{\textrm{diff}}\rangle \subseteq \langle \mathcal {V}+\mathcal {R}\mid A^{\textrm{diff}}\rangle \subseteq \mathcal {V}+\mathcal {R}. \end{aligned} \end{aligned}$$

$\square$

Proof of Theorem 40

$(\Rightarrow )$ Suppose the system class $\Sigma _{\texttt{n}}$ with dwell time $d>0$ is causally impulse-controllable. Then for any given switching signal $\sigma \in \mathcal {S}_\texttt{n}$ (with dwell time $d>0$) there exists an impulse-free solution (x, u) where $u_{[t_0,t)}=\mathcal {U}_t(\sigma _{[t_0,t)},x_0)$.

We will proof by induction that $x(t_{i}^-)\in \underline{\mathcal {C}}_{i}$ for all $i\in \{\texttt{n}, \texttt{n}-1,...,1\}$. Since (x, u) is impulse-free, it follows that $x(t_\texttt{n}^-)\in \mathcal {C}^{\textrm{imp}}_\texttt{n}=\underline{\mathcal {C}}_\texttt{n}$. Hence, we assume that the statement holds for i and continue to proof the statement for $i-1$. Using the same arguments as in the proof of Theorem 39, we can show that $x(t_i^-)\in \langle \overline{\mathcal {C}}_i\mid A_{i-1},E_{i-1},B_{i-1}\rangle$.

It follows from the solution formula for differential algebraic equations that

$$\begin{aligned} e^{A^{\textrm{diff}}_{i-1} \tau _{i-1}}\Pi _{i-1} x(t_{i-1}^-)\in \langle \overline{\mathcal {C}}_i\mid A_{i-1},E_{i-1},B_{i-1}\rangle +\mathcal {R}_{i-1} \end{aligned}$$

and hence

$$\begin{aligned} \begin{aligned} \Pi _{i-1} x(t_{i-1}^-)&\in e^{-A^{\textrm{diff}}_{i-1} \tau _{i-1}}\langle \overline{\mathcal {C}}_i\mid A_{i-1},E_{i-1},B_{i-1}\rangle +e^{-A^{\textrm{diff}}_{i-1} \tau _{i-1}}\mathcal {R}_{i-1}\\&\subseteq \langle \overline{\mathcal {C}}_i\mid A_{i-1},E_{i-1},B_{i-1}\rangle +\mathcal {R}_{i-1}, \end{aligned} \end{aligned}$$

where we utilized that $\mathcal {R}_{i-1}$ is $A^{\textrm{diff}}_{i-1}$-invariant together with Lemma 41.b).

Since (x, u) is impulse-free, it follows that $x(t_{i-1}^-)\in \mathcal {C}^{\textrm{imp}}_{i-1}$. It follows from Lemma 13 that $\mathcal {C}^{\textrm{imp}}_{i-1} = {{\,\textrm{im}\,}}\Pi _{i-1}\oplus \mathcal {D}^\textrm{imp}_{i-1}$. Hence, we can conclude that

$$\begin{aligned} (I-\Pi _{i-1})x(t_{i-1}^-)\in \mathcal {D}^\textrm{imp}_{i-1}\subseteq \mathcal {R}_{i-1}+\ker E_{i-1}. \end{aligned}$$

Altogether, we conclude the inductive proof by observing that

$$\begin{aligned} x(t_{i-1}^-)&=\Pi _{i-1} x(t_{i-1}^-)+(I-\Pi _{i-1})x(t_{i-1}^-)\\&\in \langle \overline{\mathcal {C}}_i\mid A_{i-1},E_{i-1},B_{i-1}\rangle +\mathcal {R}_{i-1}+\ker E_{i-1}=\overline{\mathcal {C}}_{i-1}. \end{aligned}$$

Now we can conclude that $x_0\in \overline{\mathcal {C}}_0$, and since this holds for all $x_0\in \mathcal {V}_{( E_0, A_0, B_0)}$, we have shown the necessity part of the statement.

$(\Leftarrow )$ Let $x_i\in \overline{\mathcal {C}}_i\subseteq \mathcal {V}_{(E_i,A_i,B_i)}+\ker E_i=\mathcal {C}^{\textrm{imp}}_i$, hence there exists an input $\widehat{u}$ on $[t_i,t_{i+1}+d)$ such that the corresponding solution $\widehat{x}$ of mode i with (inconsistent) initial condition $\widehat{x}(t_i^-)=x_i$ is impulse-free. Furthermore, $\widehat{x}((t_i+d)^-) = e^{A_i^\textrm{diff}d}\Pi _i x_i + \widehat{\eta }_i$ for some $\widehat{\eta }_i\in \mathcal {R}_i$. Combining the observation in Remark 3 that $\ker E_i\subseteq \ker \Pi _i$ and the result of Lemma 41 leads to

$$\begin{aligned}\begin{aligned} e^{A^{\textrm{diff}}_{i}d} \Pi _i x_i&\in e^{A^{\textrm{diff}}_{i}d} \Pi _i \left( \langle \overline{\mathcal {C}}_{i+1} \mid A_i,E_i,B_i\rangle + \mathcal {R}_i + \ker E_i \right) \\&\subseteq \langle \overline{\mathcal {C}}_{i+1} \mid A_i,E_i,B_i\rangle + \mathcal {R}_i. \end{aligned} \end{aligned}$$

Consequently, $e^{A^{\textrm{diff}}_{i}d} \Pi _i x_i = c_{i+1} + \eta _i$ for some $c_{i+1}\in \langle \overline{\mathcal {C}}_{i+1} \mid A_i,E_i,B_i\rangle$ and $\eta _i\in \mathcal {R}_i$.

Now choose a smooth input $\widetilde{u}$ on $[t_i,t_i+d)$ such that corresponding solution $\widetilde{x}$ of mode i with initial condition $\widetilde{x}(t_i^-)=0$, satisfies $\widetilde{x}((t_i+d)^-)=-\eta _i-\widehat{\eta }_i$. Now let $u:=\widehat{u}+\widetilde{u}$ then, by linearity, the corresponding solution x of mode i with (inconsistent) initial condition $x(t_i^-)=x_i$ is impulse-free and satisfies

$$\begin{aligned} \begin{aligned} x((t_i+d)^-)&= \widehat{x}((t_i+d)^-) + \widetilde{x}((t_i+d)^-)\\&= e^{A_i^\textrm{diff}d}\Pi _i x_i + \widehat{\eta }_i - \eta _i-\widehat{\eta }_i = c_{i+1}. \end{aligned} \end{aligned}$$

Due to the controlled invariance of $\langle \overline{\mathcal {C}}_{i+1} \mid A_i,E_i,B_i\rangle$, it is possible to extend u onto $[t_i,t_{i+1})$ such that the corresponding solution satisfies $x(t^-)\in \overline{\mathcal {C}}^\textrm{imp}_{i+1}$ for all $t\in [t_i+d,t_{i+1})$. Now, concatenate these inputs inductively with the corresponding initial conditions $x(t_i^-)$ obtained from the previous input it follows that the overall input is causal (in particular, independent of the mode duration) and achieves and impulse-free solution on $[t_0,t_i)$ with $x(t_i^-)\in \overline{\mathcal {C}}_i$, $i=1,2,\ldots \texttt{n}$. Finally, by assumption $x(t_{\texttt{n}}^-)\in \mathcal {C}^{\textrm{imp}}_\texttt{n}$, hence the input u can by extended also on $[t_\texttt{n},\infty )$ in such a way that the solution remains impulse-free. Altogether, we can define $\mathcal {U}(\sigma _{[t_0,t)},x_0):= u_{[t_0,t)}$ which satisfies the causality properties with a dwell time for all switching signals and all $x_0$. $\square$

Remark 42

Since $\langle {{\,\textrm{im}\,}}\Pi _0\cap (\underline{\mathcal {K}}_1+\mathcal {R}_0)\mid A^{\textrm{diff}}_0\rangle \subseteq {{\,\textrm{im}\,}}\Pi _0\subseteq \mathcal {V}_{(E_0,A_0,B_0)}$, $\langle \overline{\mathcal {C}}_1\mid A_0,E_0,B_0\rangle \subseteq \mathcal {V}_{(E_0,A_0,B_0)}$ and $\langle \underline{\mathcal {C}}_1\mid A_0,E_0,B_0\rangle \subseteq \mathcal {V}_{(E_0,A_0,B_0)}$ and, by Lemma 13,

$$\begin{aligned} \mathcal {C}^{\textrm{imp}}_0={{\,\textrm{im}\,}}\Pi _0+\mathcal {R}_0+\ker E_0=\mathcal {V}_{(E_0,A_0,B_0)}+\ker E_0, \end{aligned}$$

it follows that

$$\begin{aligned} \ker E_0 \subseteq \underline{\mathcal {C}}_0 \subseteq \overline{\mathcal {C}}_0\subseteq \underline{K}_0 \subseteq \mathcal {C}^{\textrm{imp}}_0. \end{aligned}$$

Consequently, we have the following equivalent characterizations for quasi-causal impulse-controllability, causal impulse-controllability and causal impulse-controllability with a dwell-time of $\Sigma _\texttt{n}$, respectively:

$$\begin{aligned} \mathcal {C}^{\textrm{imp}}_0&= \underline{\mathcal {K}}_0, \end{aligned}$$

(22)

$$\begin{aligned} \mathcal {C}^{\textrm{imp}}_0&=\underline{\mathcal {C}}_0, \end{aligned}$$

(23)

$$\begin{aligned} \mathcal {C}^{\textrm{imp}}_0&=\overline{\mathcal {C}}_0. \end{aligned}$$

(24)

4.3 Causal impulse-controllability for $\overline{\Sigma }_\texttt{n}$

We conclude this section by considering causality also for the case of unknown mode sequence, i.e. for the system class $\overline{\Sigma }_\texttt{n}$. The definition of (quasi)-causality given above carries over to the system class $\overline{\Sigma }_\texttt{n}$ without change (apart from considering switching signals in $\overline{\mathcal {S}}_\texttt{n}$ instead of $\mathcal {S}_\texttt{n}$). Since $\overline{\Sigma }_\texttt{n}$ contains all switched systems with a single switch, we can immediately state necessary conditions for (quasi-) causal impulse-controllability (with dwell time). In fact, similar as in Theorem 22 these necessary conditions turn out to be sufficient as well.

Corollary 43

Consider the system class of switched systems $\overline{\Sigma }_\texttt{n}$ of switched DAEs with arbitrary mode sequence and arbitrary mode durations.

• a) $\overline{\Sigma }_\texttt{n}$ is quasi-causally impulse-controllable if, and only if, for all $i,j\in \{0,1,\ldots ,\texttt{n}\}$

  $$\begin{aligned} \mathcal {C}^{\textrm{imp}}_i = \langle {{\,\textrm{im}\,}}\Pi _i \cap (\mathcal {C}^{\textrm{imp}}_j + \mathcal {R}_i)\mid A^{\textrm{diff}}_i\rangle \oplus \mathcal {D}^\textrm{imp}_{i}. \end{aligned}$$

  (25)

• b) $\overline{\Sigma }_\texttt{n}$ is causally impulse-controllable if, and only if, for all $i,j\in \{0,1,\ldots ,\texttt{n}\}$

  $$\begin{aligned} \mathcal {C}^{\textrm{imp}}_i = \langle \mathcal {C}^{\textrm{imp}}_j\mid E_i,A_i,B_i\rangle +\ker E_i. \end{aligned}$$

• c) $\overline{\Sigma }_\texttt{n}$ with dwell time $d>0$ is causally impulse-controllable if, and only if, for all $i,j\in \{0,1,\ldots ,\texttt{n}\}$

  $$\begin{aligned} \mathcal {C}^{\textrm{imp}}_i = \langle \mathcal {C}^{\textrm{imp}}_j\mid E_i,A_i,B_i\rangle + \mathcal {R}_i+\ker E_i. \end{aligned}$$

Proof

In the following, we will prove the statement only for quasi-causality; the other two cases follow analogously.

In view of Remark 42, it suffices to show that (22) is satisfied for all possible mode sequences if, and only if, (25) holds. Sufficiency follows by considering the definition (18) recursively, taking (25) into account; necessity follows from considering the specific mode sequence $\varvec{q}=(i,j,j,...,j)$ and observing that

$$\begin{aligned}{} & {} \langle {{\,\textrm{im}\,}}\Pi _j\cap (\mathcal {C}^{\textrm{imp}}_j+\mathcal {R}_j)\mid A^{\textrm{diff}}_j \rangle \oplus \mathcal {D}^\textrm{imp}_j=\langle {{\,\textrm{im}\,}}\Pi _j\cap \mathcal {C}^{\textrm{imp}}_j\mid A^{\textrm{diff}}_j \rangle \oplus \mathcal {D}^\textrm{imp}_j\\{} & {} \quad ={{\,\textrm{im}\,}}\Pi _j \oplus \mathcal {D}^\textrm{imp}_j =\mathcal {C}^{\textrm{imp}}_j. \end{aligned}$$

From (18), it follows then recursively that $\underline{\mathcal {K}}_0=\langle {{\,\textrm{im}\,}}\Pi _i \cap (\mathcal {C}^{\textrm{imp}}_j + \mathcal {R}_i)\mid A^{\textrm{diff}}_i\rangle \oplus \mathcal {D}^\textrm{imp}_{i}$, hence (25) implies (22). $\square$

5 Conclusion

In this paper, impulse-controllability of system classes of switched DAEs has been considered. It was shown that strong impulse-controllability of system classes generated by arbitrary switching signals is equivalent to impulse-controllability of every switched system with a single switch. In the case, the system class contains systems with a fixed mode sequence, either all or almost all systems are impulse-(un)controllable and sufficient conditions for strong impulse-(un)controllability are given. Finally, we considered the notions of (quasi-) causal impulse-controllability and controllability and characterized system classes with these properties.

A natural direction of research is to design controllers that achieve impulse-free solutions. In the case of causal impulse-controllable systems, it seems that there should exist a switched feedback controller that guarantees impulse-free solutions. However, for systems in a class that is causally impulse-controllable given some dwell-time or quasi-causally impulse-controllable, the controller design seems not so straight forward. Furthermore, it remains an open question whether simple necessary conditions for essential impulse-(un)controllability of system classes can be stated.

A Proofs

1.1 A.1 Proof of Theorem 29

The proof of Theorem 29 relies on utilizing properties of analytic functions, which are recalled first.

Definition 44

A function $f:\mathbb {R}^p\rightarrow \mathbb {R}$ is called analytic if for each $x\in \mathbb {R}^p$ the function f may be presented by a convergent power series in some neighborhood of x.

A useful property of analytic functions is the following well-known result.

Lemma 45

(Cf. [27, Cor I.A.10]) The zero-set of a non-trivial analytic function $f:\mathbb {R}^p\rightarrow \mathbb {R}$ has (Lebesgue) measure zero.

The notion of analyticity can be extended to matrix-valued function as follows.

Definition 46

The matrix-valued function $M:\mathbb {R}^p \rightarrow \mathbb {R}^{m\times n}$ is called an analytic matrix if each entry $m_{ij}:\mathbb {R}^p\rightarrow \mathbb {R}$ of M is an analytic function.

Definition 47

A analytic matrix $M:\mathbb {R}^p \rightarrow \mathbb {R}^{m\times n}$ is called generically full rank if either $\det \left( M(\varvec{\tau })^\top M(\varvec{\tau })\right) \ne 0$ for almost all¹$\varvec{\tau }\in \mathbb {R}^p$ or $\det \left( M(\varvec{\tau }) M(\varvec{\tau })^\top \right) \ne 0$ for a.a. $\varvec{\tau }\in \mathbb {R}^p$.

Lemma 48

Let $A\in \mathbb {R}^{n\times n}$, $W:\mathbb {R}^p \rightarrow \mathbb {R}^{n\times k}$ a generically full-rank analytic matrix and $\mathcal {R}\subseteq \mathbb {R}^n$ some subspace. Then, there exists an analytic matrix $N:\mathbb {R}^{p+1}\rightarrow \mathbb {R}^{n\times q}$ which is generically full rank and such that for a.a. $(\tau _0,\varvec{\tau }) \in \mathbb {R}^{p+1}$

$$\begin{aligned} {{\,\textrm{im}\,}}N(\tau _0,\varvec{\tau })=e^{A \tau _0}{{\,\textrm{im}\,}}W(\varvec{\tau })+\mathcal {R}. \end{aligned}$$

(26)

Proof

We use $\mathcal {N}_{\tau _0,\varvec{\tau }}\subseteq \mathbb {R}^n$ as short-hand notation for the right-hand side of (26) in the following. Pick any $(\overline{\tau }_0,\overline{\varvec{\tau }})\in \mathbb {R}^{p+1}$ such that $\dim \mathcal {N}_{\overline{\tau }_0,\overline{\varvec{\tau }}} = \max _{(\tau _0,\varvec{\tau })} \dim \mathcal {N}_{\tau _0,\varvec{\tau }}=:q$ and let $r_1,...,r_l\in \mathbb {R}^n$ be a basis of $\mathcal {R}$. Choose $B_{W}\in \mathbb {R}^{k\times (q-l)}$ such that $[\overline{w}_1,\ldots ,\overline{w}_{q-l}]=W(\overline{\varvec{\tau }}) B_W$ yields a basis

$$\begin{aligned} r_1, \ldots , r_{l}, e^{A \overline{\tau }_0} \overline{w}_1, \ldots , e^{A \overline{\tau }_0} \overline{w}_{q-l} \end{aligned}$$

of $\mathcal {N}_{\overline{\tau }_0,\overline{\varvec{\tau }}}$. Consider now the matrix-valued function $N:\mathbb {R}^{p+1} \rightarrow \mathbb {R}^{n\times q}$ defined by

$$\begin{aligned} N(\tau _0,\varvec{\tau }):=\left[ r_1, \ldots , r_{l}, e^{A \tau _0} W(\varvec{\tau }) B_W \right] . \end{aligned}$$

This matrix is analytic because the matrix exponential is analytic and the product of two analytic matrices is again analytic. By construction,

$$\begin{aligned} \det \left( N(\overline{\tau }_0,\overline{\tau })^\top N(\overline{\tau }_0,\overline{\tau })\right) \ne 0, \end{aligned}$$

and hence the analytic function $(\tau _0,\varvec{\tau })\mapsto \det \left( N(\tau _0,\varvec{\tau })^\top N(\tau _0,\varvec{\tau })\right)$ is not identically zero. In view of Lemma 45, it therefore follows that N is generically full rank.

It remains to be shown that (26) holds. By construction, ${{\,\textrm{im}\,}}N(\tau _0,\varvec{\tau })\subseteq \mathcal {N}_{\tau _0,\varvec{\tau }}$ for all $(\tau _0,\varvec{\tau }) \in \mathbb {R}^{p+1}$. Furthermore, since $\dim \mathcal {N}_{\tau _0,\varvec{\tau }}\le q$ and $\dim {{\,\textrm{im}\,}}N(\tau _0,\varvec{\tau }) = q$ for a.a. $(\tau _0,\varvec{\tau })\in \mathbb {R}^{p+1}$ the claim follows. $\square$

Remark 49

It is indeed possible that for some specific $(\tau _0,\varvec{\tau })$ we have ${{\,\textrm{im}\,}}N(\tau _0,\varvec{\tau })\subsetneq N_{\tau _0,\varvec{\tau }}$. As an example, consider for $\alpha >0$

$$\begin{aligned}&W(\tau _1) = {{\,\textrm{span}\,}}\left\{ {\left[ \begin{array}{ccc} e^\tau _1 \\ e^\tau _1-e^\alpha \end{array}\right] }, {\left[ \begin{array}{ccc} 0 \\ e^\tau _1 \end{array}\right] } \right\} :={{\,\textrm{span}\,}}\{w_1(\tau _1),w_2(\tau _2)\},\\&\mathcal {R}={{\,\textrm{span}\,}}\left\{ {\left[ \begin{array}{ccc} 1\\ 0\end{array}\right] }\right\} :={{\,\textrm{span}\,}}\{r_1\}, \quad A=0. \end{aligned}$$

Then clearly, $e^{A\tau _0}W(\tau _1)+\mathcal {R}= \mathbb {R}^2$ for all $(\tau _0,\tau _1)\in \mathbb {R}^2$. However, while the choice $N(\tau _0,\tau _1):=[r_1, w_1(\tau _1)]$ satisfies

$$\begin{aligned} {{\,\textrm{im}\,}}N(\tau _0,\tau _1) = e^{A\tau _0}W(\tau _1)+\mathcal {R}= \mathbb {R}^2\quad \text { for a.a. }(\tau _0,\tau _1), \end{aligned}$$

for $\tau _1=\alpha$ we have

$$\begin{aligned} {{\,\textrm{im}\,}}N(\tau _0,\alpha ) = {{\,\textrm{span}\,}}\{r_1\} \ne \mathbb {R}^2. \end{aligned}$$

Lemma 50

Let $W:\mathbb {R}^p\rightarrow \mathbb {R}^{q\times n}$, $n>q$, be an analytic matrix with generically full rank. Then, there exists an analytic matrix $N:\mathbb {R}^p\rightarrow \mathbb {R}^{n\times (n-q)}$ with generically full rank such that ${{\,\textrm{im}\,}}N(\varvec{\tau })= \ker W(\varvec{\tau })$ for a.a. $\varvec{\tau }\in \mathbb {R}^p$.

Proof

By considering the field of meromorphic functions (i.e. fractions of scalar-valued analytic functions), we can apply Gauss–Jordan eliminations on $W(\varvec{\tau })$ to obtain a reduced row echelon form (RREF), which contains meromorphic entries and whose kernel for a.a. $\varvec{\tau }\in \mathbb {R}^p$ equals $\ker W(\varvec{\tau })$. Identically as for constant matrices, a full rank matrix $\overline{N}(\varvec{\tau })\in \mathbb {R}^{n\times (n-q)}$, can be easily constructed from the (meromorphic) entries of the obtained RREF such that $W(\varvec{\tau })\overline{N}(\varvec{\tau }) = 0$ for all $\varvec{\tau }$ for which $\overline{N}(\varvec{\tau })$ is well-defined. As a final step, let $N(\varvec{\tau }) = \overline{N}(\varvec{\tau }) {\left[ \begin{array}{ccc}\alpha _1(\varvec{\tau })\\ &{}\ddots \\ &{}&{}\alpha _{n-q}(\varvec{\tau })\end{array}\right] }$, where $\alpha _i(\varvec{\tau })$ is the product of all denominators of the entries in the i-th column of $\overline{N}(\varvec{\tau })$. Then $M(\varvec{\tau })N(\varvec{\tau }) = 0$ for a.a. $\varvec{\tau }\in \mathbb {R}^p$ and $\varvec{\tau }\mapsto N(\varvec{\tau })$ is an analytic matrix and has generically the same rank as $\overline{N}$, i.e. N is generically full rank. $\square$

Lemma 51

Let $W:\mathbb {R}^p\rightarrow \mathbb {R}^{n\times k}$, $k\le n$, be an analytic matrix with generically full rank. Then for any $\Pi \in \mathbb {R}^{n\times n}$ there exists an analytic matrix $N: \mathbb {R}^p\rightarrow \mathbb {R}^{n\times m}$ with generically full rank such that ${{\,\textrm{im}\,}}N(\varvec{\tau })= {{\,\textrm{im}\,}}\Pi \cap {{\,\textrm{im}\,}}W(\varvec{\tau })$ for a.a. $\varvec{\tau }\in \mathbb {R}^p$.

Proof

By Lemma 50, there exists an analytic matrix $\overline{N}: \mathbb {R}^p\rightarrow \mathbb {R}^{n\times q}$ with generically full rank and ${{\,\textrm{im}\,}}\overline{N}(\varvec{\tau })=\ker W(\varvec{\tau })^\top$ for a.a. $\varvec{\tau }\in \mathbb {R}^p$. Consequently,

$$\begin{aligned} \left( {{\,\textrm{im}\,}}\Pi \cap {{\,\textrm{im}\,}}W(\varvec{\tau })\right) ^\bot&=\ker \Pi ^\top +\ker W(\varvec{\tau })^\top ,=\ker \Pi ^\top + {{\,\textrm{im}\,}}\overline{N}(\varvec{\tau }). \end{aligned}$$

Applying Lemma 48 for $\mathcal {R}=\ker \Pi ^\top$ and $A=0$, we find an analytic matrix $\widetilde{N}:\mathbb {R}^p\rightarrow \mathbb {R}^{n\times \widetilde{q}}$ with generically full rank such that ${{\,\textrm{im}\,}}\widetilde{N}(\varvec{\tau })=\ker \Pi ^\top +{{\,\textrm{im}\,}}\overline{M}(\varvec{\tau })$ for a.a. $\varvec{\tau }$. Finally, using Lemma 50 again we can find an analytic matrix $N:\mathbb {R}^p\rightarrow \mathbb {R}^{n\times q}$, $q=n-\widetilde{q}$ with generically full rank such that ${{\,\textrm{im}\,}}N(\varvec{\tau })=\ker \widetilde{N}(\varvec{\tau })^\top$ for a.a. $\varvec{\tau }$. Altogether, we have for a.a. $\varvec{\tau }$

$$\begin{aligned} {{\,\textrm{im}\,}}\Pi \cap {{\,\textrm{im}\,}}W(\varvec{\tau })=\left( {{\,\textrm{im}\,}}\widetilde{N}(\varvec{\tau })\right) ^\bot =\ker \widetilde{N}(\varvec{\tau })^\top ={{\,\textrm{im}\,}}N(\varvec{\tau }). \end{aligned}$$

$\square$

Lemma 52

Consider the sequence (14). Then for all $i \in \{0,1,...,\texttt{n}\}$ there exists an analytic matrix $N_i:\mathbb {R}^{\texttt{n}-i}\rightarrow \mathbb {R}^{n\times k_i}$ with generically full rank such that ${{\,\textrm{im}\,}}N_i(\varvec{\tau })= \mathcal {K}_i^{\varvec{\tau }}$ for a.a. $\varvec{\tau }\in \mathbb {R}^{\texttt{n}-i}$.

Proof

For $i=\texttt{n}$, we use the convention that a constant full-rank matrix is interpreted as an analytic matrix depending on an empty tuple $\varvec{\tau }=()\in \mathbb {R}^0$, then the claim is correct by simply choosing the columns of $N_{\texttt{n}}(\varvec{\tau })$ as a (constant) basis of $\mathcal {C}^{\textrm{imp}}_\texttt{n}$. We now proceed inductively and assume the claim is correct for some $i\in \{1,2,\ldots ,\texttt{n}\}$. Let $\mathcal {N}_{\tau _{i-1},\varvec{\tau }}:=e^{-A^{\textrm{diff}}_{i-1}\tau _{i-1}}{{\,\textrm{im}\,}}N_i(\varvec{\tau })+\mathcal {R}_{i-1}$ and $\mathcal {R}^\textrm{imp}_{i-1}:=\langle E^{\textrm{imp}}_{i-1}\mid B^{\textrm{imp}}_{i-1}\rangle + \ker E_{i-1}$, then

$$\begin{aligned} \mathcal {K}^{(\tau _{i-1},\varvec{\tau })}_{i-1} = \left( {{\,\textrm{im}\,}}\Pi _{i-1}\cap \mathcal {N}_{\tau _{i-1},\varvec{\tau }} \right) + \mathcal {R}^\textrm{imp}_{i-1} \end{aligned}$$

for a.a. $\varvec{\tau }\in \mathbb {R}^{\texttt{n}-i}$ and all $\tau _{i-1}\in \mathbb {R}$. Utilizing Lemmas 48 and 51, we find analytic and generically full-rank matrices $\widetilde{N}_{i-1}:\mathbb {R}^{\texttt{n}-(i-1)}\rightarrow \mathbb {R}^{n\times \widetilde{k}_i}$, $\overline{N}_{i-1}:\mathbb {R}^{\texttt{n}-i+1}\rightarrow \mathbb {R}^{n\times \overline{k}_i}$, $N_{i-1}:\mathbb {R}^{\texttt{n}-(i-1)}\rightarrow \mathbb {R}^{n\times k_i}$ such that a.a. $(\tau _{i-1},\varvec{\tau })\in \mathbb {R}^{\texttt{n}-(i-1)}$

$$\begin{aligned} \begin{aligned} {{\,\textrm{im}\,}}\widetilde{N}_{i-1}(\tau _{i-1},\varvec{\tau })&= \mathcal {N}_{\tau _{i-1},\varvec{\tau }},\\ {{\,\textrm{im}\,}}\overline{N}_{i-1}(\tau _{i-1},\varvec{\tau })&= {{\,\textrm{im}\,}}\Pi _{i-1} \cap {{\,\textrm{im}\,}}\widetilde{N}_{i-1}(\tau _{i-1},\varvec{\tau }),\\ {{\,\textrm{im}\,}}N_{i-1}(\tau _{i-1},\varvec{\tau })&= {{\,\textrm{im}\,}}\overline{N}_{i-1}(\tau _{i-1},\varvec{\tau }) + \mathcal {R}^\textrm{imp}_{i-1}, \end{aligned} \end{aligned}$$

i.e. $\mathcal {K}^{(\tau _{i-1},\varvec{\tau })}_{i-1} = {{\,\textrm{im}\,}}N_{i-1}(\tau _{i-1},\varvec{\tau })$ as desired. $\square$

With these preliminary results related to analytic matrices, we are now in the position to proof Theorem 29.