Duality of switched DAEs

Original Article


We present and discuss the definition of the adjoint and dual of a switched differential-algebraic equation (DAE). For a proper duality definition, it is necessary to extend the class of switched DAEs to allow for additional impact terms. For this switched DAE with impacts, we derive controllability/reachability/determinability/observability characterizations for a given switching signal. Based on this characterizations, we prove duality between controllability/reachability and determinability/observability for switched DAEs.


Duality Switched systems Differential-algebraic equations 

1 Introduction

We study duality of switched differential-algebraic equations (DAEs) of the form
$$\begin{aligned} \begin{aligned} E_\sigma \dot{x}&= A_\sigma x + B_\sigma u,\\ y&= C_\sigma x, \end{aligned} \end{aligned}$$
for a given switching signal \(\sigma : \mathbb {R}\rightarrow \mathbb {N}\). It will be necessary to generalize this system class later on to allow for dualization.

Duality is a classical research subject in linear system theory, and apart from being of theoretical interest, it has applications in optimal control. First introduced by Kalman [11], it was later generalized to other system classes, in particular unswitched DAEs [9, 10], linear differential inclusions [2], switched linear ODEs (with switching signal as input) [24], linear (continuously) time-varying DAEs [8], non-switched impulsive systems [17], and hybrid systems (including jumps) with periodic switching signal [18]. A concept closely related to duality is adjointness, and we will discuss, in detail, the connection between both in the context of switched DAE. Adjointness for homogeneous (continuously) time-varying DAEs is still an active research field, see the recent article [19] and the references therein. Although switched DAEs with given switching signal are also time-varying DAEs, the discontinuities due to switching pose significant challenges in the theoretical analysis; nevertheless, our approach is inspired by the results on duality/adjointness of linear time-varying DAEs, and it may even be possible to unify these results, but this is outside the scope of our paper.

For constant-coefficient DAEs, the recent survey [6] gives duality results for different notions of controllability and observability. Cobb [9] and Frankowska [10] use notions that do not coincide with ours and whose generalization to switched DAEs does not lead to duality. Using more appropriate notations for observability and controllability, our result differs from [9] and [10] even in the unswitched case. In addition to the non-canonical controllability/observability definitions for DAEs, there is also some choice in how to treat the switching signals in the definitions of controllability/observability, see the survey [21] on different observability concepts for switched systems. Some duality result for switched DAEs is claimed in [20]; however, therein, a rigorous solution theory is missing, and furthermore, the observability definition requires to choose the switching signal depending on the initial value.

A priori it is not clear which generalizations of controllability and observability are most natural for switched DAEs; however, with our chosen notions (see Definition 17 and 19), we are able to show the very satisfying duality statement (Theorem 30)There are certain pitfalls towards obtaining this duality result. A first challenge is to find an appropriate definition of the dual system (see Sect. 4). A “correct” definition of duality should have the following properties:
  1. D1

    The dual of the dual is the original system; in particular, the dual is an element of the same system class (otherwise, the original duality definition cannot be applied to the dual system).

  2. D2

    The classical duality between (some form of) controllability and (some form of) observability holds.

  3. D3

    There is some formal justification of duality in terms of the solution trajectories.

The above-mentioned works do not elaborate on the derivation of the dual system, but merely state the dual system and show that D1 and D2 hold. Following this approach, a naive definition (motivated by the definition of the dual of a non-switched DAE and, indeed, proposed in [20]) of the dual system of (1) is
$$\begin{aligned} \begin{aligned} E_\sigma ^\top \dot{z}&= A_\sigma ^\top z + C_\sigma ^\top u_d,\\ y_d&= B_\sigma ^\top z. \end{aligned} \end{aligned}$$
Considering the time-varying nature of switched DAEs, another possible definition for the dual of (1) would additionally reverse the time, resulting in
$$\begin{aligned} \begin{aligned} E_{\overline{\sigma }}^\top \dot{z}&= A_{\overline{\sigma }}^\top z + C_{\overline{\sigma }}^\top u_d,\\ y_d&= B_{\overline{\sigma }}^\top z \end{aligned}\end{aligned}$$
with \(\overline{\sigma }(t):=\sigma (T-t)\), where [0, T], \(T>0\), is the compact time interval of interest. The following example shows that both approaches do not yield a satisfying duality definition as property D2 is not satisfied.

Example 1

Consider the switched DAE (1) with switching signal
$$\begin{aligned} \sigma (t) = {\left\{ \begin{array}{ll} 0,&{}\quad t\in (-\infty ,1),\\ 1,&{}\quad t\in [1,2),\\ 2,&{}\quad t\in [2,\infty ), \end{array}\right. } \end{aligned}$$
and modes given by
$$\begin{aligned} \begin{aligned} (E_0,A_0,B_0,C_0)&= \left( \begin{bmatrix}1&\quad 0 \\ 0&\quad 0 \end{bmatrix}, \begin{bmatrix} 0&\quad 0 \\ 0&\quad 1 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 0&\quad 0 \end{bmatrix}\right) ,\\ (E_1,A_1,B_1,C_1)&= \left( \begin{bmatrix}1&\quad 0 \\ 0&\quad 0 \end{bmatrix}, \begin{bmatrix} 0&\quad 0 \\ 1&\quad 1 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 0&\quad 0 \end{bmatrix}\right) ,\\ (E_2,A_2,B_2,C_2)&= \left( \begin{bmatrix}1&\quad 0 \\ 0&\quad 1 \end{bmatrix}, \begin{bmatrix} 0&\quad 0 \\ 0&\quad 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 0&\quad 1 \end{bmatrix}\right) . \end{aligned} \end{aligned}$$
In all three modes, it holds \(\dot{x}_1 = 0\), and hence, the solution satisfies \(x_1(t) = x_1^0\) for all \(t\in \mathbb {R}\) and some \(x_1^0\in \mathbb {R}\). The variable \(x_2\) is zero on \((-\infty ,1)\), on [1, 2) it holds that \(x_2=-x_1\), and afterwards, \(x_2\) remains constant, because \(\dot{x}_2=0\). Altogether, we have
$$\begin{aligned} x(t) = {\left\{ \begin{array}{ll} \begin{pmatrix} x_0^1 \\ 0 \end{pmatrix},&{}\quad t\in (-\infty ,1),\\ \begin{pmatrix} x_0^1 \\ -x_0^1 \end{pmatrix},&{}\quad t\in [1,\infty ).\\ \end{array}\right. } \end{aligned}$$
The output is then
$$\begin{aligned} y(t) = {\left\{ \begin{array}{ll} 0,&{}\quad t\in (-\infty ,2),\\ -x_0^1,&{}\quad t\in [2,\infty ). \end{array}\right. } \end{aligned}$$
Hence, we can uniquely deduce from the output (e.g., observed on the interval [0, 3]) the value \(x_0^{1}\), and therefore, we can reconstruct the whole state trajectory, i.e., this particular switched DAE is observable.
The naive dual (3) of the switched DAEs satisfies
$$\begin{aligned} \begin{aligned} \text {on }&(-\infty ,1)&\qquad \text {on }&[1,2)&\qquad \text {on }&[2,\infty )\\ \dot{z}_1&= 0,&\dot{z}_1&= z_2,&\dot{z}_1&= 0, \\ 0&= z_2,&0&= z_2,&\dot{z}_2&= u_d. \end{aligned} \end{aligned}$$
Clearly, \(\dot{z}_1 = 0\) holds for all times, i.e., \(z_1(t) = z_1^0\) for all \(t\in \mathbb {R}\) and some \(z_0^1\in \mathbb {R}\). Hence, this switched DAE is neither controllable nor reachable. Reversing the switching signal with respect to the interval [0, 3] results in
$$\begin{aligned} \begin{aligned} \text {on }&(-\infty ,1)&\qquad \text {on }&[1,2)&\qquad \text {on }&[2,\infty )\\ \dot{z}_1&= 0,&\dot{z}_1&= z_2,&\dot{z}_1&= 0, \\ \dot{z}_2&= u_d,&0&= z_2,&0&= z_2. \end{aligned} \end{aligned}$$
Again, \(z_1(t) = z_1^0\) for all \(t\in \mathbb {R}\) and some \(z_0^1\in \mathbb {R}\), and also, this switched DAE is neither controllable nor reachable.
It will turn out that (in contrast to non-switched system) it is helpful to clearly distinguish between the notion of an adjoint and a dual system. In view of the desired duality property D3, the definition of the adjoint system is derived based on an adjointness condition in terms of the solution trajectories (Definition 9). This results in the following adjoint system of (1):
$$\begin{aligned} \tfrac{\mathrm{d}}{\mathrm{d}t} \left( p^\top E_\sigma \right) = - p^\top A_\sigma + u_a^\top C_\sigma , \quad y_a = p^\top B_\sigma . \end{aligned}$$
The resulting system is not causal, i.e., the solution p is not uniquely defined on \([t,\infty )\) by its past \(p_{(-\infty ,t)}\) and the input u. Reversing the time with respect to an interval [0, T], \(T>0\), we arrive (also paying special attention to the distributional multiplications involved) at a causal system:
$$\begin{aligned} \tfrac{\mathrm{d}}{\mathrm{d}t} \left( E_{\overline{\sigma }}^{\top } z \right) = A_{\overline{\sigma }}^{\top } z + C_{\overline{\sigma }}^{\top } u_d,\quad y_d = B_{\overline{\sigma }}^{\top } z \end{aligned}$$
which is then called the T-dual system (Definition 17). This dual system is not a DAE of the form (1), because using the product rule, the term \((E_{\overline{\sigma }}^\top )' z\) occurs. Since \(E_{\overline{\sigma }}\) has jumps, its derivative contains Dirac impulses. Fortunately, this occurrence of Dirac impulses in the coefficient matrices is covered by the distributional solution framework in [29], and in view of [31], we call the enlarged system class switched DAEs with impacts, given by
$$\begin{aligned} E_{\sigma } \dot{x} = A_{\sigma } x + B_{\sigma } u+ G[\cdot ] x, \quad y = C_{\sigma } x, \end{aligned}$$
where \(G[\cdot ]\) is a sum of Dirac impulses; details are discussed in Sect. 3. For this extended system class, the above derivation of adjointness and duality have to be repeated (see Sect. 4), and furthermore, controllability and observability notions have to be generalized to switched DAEs with impacts (see Sect. 5). Finally, we are able to precisely state and prove our duality result (2) in Sect. 6.

2 Mathematical preliminaries

2.1 Regular matrix pairs

We first recall properties of the (unswitched) DAE
$$\begin{aligned} \begin{aligned} E\dot{x}&= Ax + Bu,\\ y&= Cx \end{aligned} \end{aligned}$$
with matrices \(E, A \in \mathbb {R}^{n \times n}, B \in \mathbb {R}^{n \times q}, C \in \mathbb {R}^{r \times n}\), and classical (smooth) solutions (xuy). For existence and uniqueness of solutions, the following notion of regularity of the matrix pair (EA) is crucial:

Definition 2

(Regularity) Let \(E,A\in {\mathbb {R}}^{n\times n}\). The matrix pair \(\left( E,A\right) \) is called regular iff \(\det \left( sE-A\right) \in {\mathbb {R}}[s]\) is not the zero polynomial. The DAE (5) is called regular iff the corresponding matrix pair \(\left( E,A\right) \) is regular.

Lemma 3

(Regularity characterizations, [30, Thm. 6.3.2]) For a DAE (5), the following is equivalent:
  1. 1.

    The matrix pair (EA) is regular.

  2. 2.
    There exist invertible matrices \(S,T\in {\mathbb {R}}^{n\times n}\) transforming (EA) into quasi-Weierstrass form (QWF), that is
    $$\begin{aligned} \left( SET,SAT\right) = \left( \begin{bmatrix} I&\quad 0 \\ 0&\quad N \end{bmatrix} , \begin{bmatrix} J&\quad 0 \\ 0&\quad I \end{bmatrix} \right) \end{aligned}$$
    with \(N\in \mathbb {R}^{n_N\times n_N}\) nilpotent, \(J\in \mathbb {R}^{n_J\times n_J}\), \(n_N+n_J=n\), and I an identity matrix of appropriate size.
  3. 3.

    For all smooth \(u:\mathbb {R}\rightarrow \mathbb {R}^q\) there exists a solution x of (5) and x is uniquely determined by \(x(t_0)\) for any \(t_0\in \mathbb {R}\).

  4. 4.

    The only solution of (5) with \(u=0\) and \(x(0)=0\) is \(x=0\).

In the following, we will assume the DAE (5) to be regular. To obtain the transformation matrices ST, the Wong sequences [28] are useful:
$$\begin{aligned} \mathcal {V}_0&:= \mathbb {R}^n,&\; \mathcal {V}_{i+1}&:=A^{-1}(E\mathcal {V}_i),&\; i&=0,1,2,\ldots ,\\ \mathcal {W}_0&:= \{0\},&\; \mathcal {W}_{i+1}&:= E^{-1}(A\mathcal {W}_i),&\; i&=0,1,2,\ldots . \end{aligned}$$
These sequences converge after finitely many steps. Their limits are denoted by
$$\begin{aligned} \mathcal {V}^*_{} :=\bigcap _{i\in \mathbb {N}} \mathcal {V}_i \; \text { and } \; \mathcal {W}^*_{} := \bigcup _{i\in \mathbb {N}} \mathcal {W}_i. \end{aligned}$$
By choosing full rank matrices VW with \({{\mathrm{im}}}V = \mathcal {V}^*_{} \) and \({{\mathrm{im}}}W = \mathcal {W}^*_{}\), we can define \(T:=[V,W]\), \(S=[EV,AW]^{-1}\). These matrices transform (EA) to QWF. They can also be used to construct the following “projectors”:
$$\begin{aligned}&\text {the {consistency projector}}&\varPi&:= T \begin{bmatrix} I&\quad 0 \\ 0&\quad 0 \end{bmatrix} T^{-1}, \\&\text {the {differential projector} }&\varPi ^\text { diff}&:= T \begin{bmatrix} I&\quad 0 \\ 0&\quad 0 \end{bmatrix} S, \\&\text {the {impulsive projector}}&{\varPi }^\text { imp}&:= T \begin{bmatrix} 0&\quad 0 \\ 0&\quad I \end{bmatrix} S, \end{aligned}$$
where the block structure of all three objects corresponds to the quasi-Weierstrass form. Only the consistency projector is a projector, as the other two are not idempotent. The definitions above do not depend on the specific choice of S and T (see [28, Section 4.2.2] for \(\varPi \); the proof for \({\varPi }^\text { diff}\) and \({\varPi }^\text { imp}\) is analogous). Using these projectors, the following matrices can be defined:
$$\begin{aligned} {E}^\text { diff}&:= {\varPi }^\text { diff}E,&{A}^\text { diff}&:= {\varPi }^\text { diff}A,&{B}^\text { diff}&:= {\varPi }^\text { diff}B,\\ E^\text { imp}&:= {\varPi }^\text { imp}E,&A^\text { imp}&:= {\varPi }^\text { imp}A,&B^\text { imp}&:= {\varPi }^\text { imp}B, \end{aligned}$$
$$\begin{aligned} {C}^\text { diff}:=C \varPi , \qquad C^\text { imp}:= C (I-\varPi ). \end{aligned}$$
We call these matrices the differential and impulsive parts of EAB,  and C, respectively.
As consistency space, we denote the space of all consistent states of the homogeneous system:
$$\begin{aligned} \left\{ x_0\in \mathbb {R}^n\ \left| \ \phantom {x_0\in \mathbb {R}^n} \exists \text { smooth solution { x} of } E\dot{x}=Ax \text { with } x(0)=x_0\right. \right\} . \end{aligned}$$
The consistency space of the inhomogeneous system is called augmented consistency space:
$$\begin{aligned} \left\{ x_0\in \mathbb {R}^n\ \left| \ \phantom {x_0\in \mathbb {R}^n} \exists \text { smooth } (x,u) \text { solving } (5)~\text { with } x(0)=x_0\right. \right\} . \end{aligned}$$
For the DAE (5), the consistency space (7) is given by \( \mathcal {V}^*_{} ={{\mathrm{im}}}\varPi \), and the augmented consistency space (8) is (see [5, Corollary 4.5])
$$\begin{aligned} \overline{ \mathcal {V}^*_{} } := \mathcal {V}^*_{} \oplus \langle E^\text { imp},B^\text { imp}\rangle . \end{aligned}$$
All solutions of (5) have the form
$$\begin{aligned} x(t) = \mathrm {e}^{{A}^\text { diff}t}\varPi c + \int _{0}^{t} \mathrm {e}^{{A}^\text { diff}(t-s)}{B}^\text { diff}u(s) \mathrm {d}s -\sum _{i=0}^{n-1} \left( E^\text { imp}\right) ^i B^\text { imp}u^{(i)}(t) \end{aligned}$$
for some \(c\in \mathbb {R}^n\) ([30, Theorem 6.4.4]).

2.2 Distributional solutions

A switched DAE (1) usually does not have a classical solution, as the consistency spaces of different modes do not need to coincide. Thus, a switch might result in an inconsistent initial condition, which—even for homogeneous systems—may produce jumps or even Dirac impulses in the state variables [30]. A distributional solution framework is, therefore, necessary. Recall that the space of distributions (or generalized functions) is defined as (following Schwartz [23]):
$$\begin{aligned} \mathbb {D}:=\left\{ D:\mathcal {C}^\infty _0\rightarrow \mathbb {R}\ \left| \ \phantom {D:\mathcal {C}^\infty _0\rightarrow \mathbb {R}} D\text { is linear and continuous}\right. \right\} , \end{aligned}$$
where \(\mathcal {C}^\infty _0\) is the space of smooth functions with compact support (so called test functions) and is equipped with a suitable locally convex topology. Every distribution \(D\in \mathbb {D}\) has a derivative in \(\mathbb {D}\) given by \(D'(\varphi ):=-D(\varphi ')\), \(\varphi \in \mathcal {C}^\infty _0\). However, it turns out that the whole space of distributions is not an appropriate solution space for (1), because it is “too large” [28]. To overcome this problem, we follow [28] and introduce the space of piecewise-smooth distributions. The latter can be seen as the “differential closure” of the space of piecewise-smooth functions defined as follows:
$$\begin{aligned} \mathcal {C}_{\text { pw}}^{\infty }:= \left\{ \alpha = \sum _{i\in {\mathbb {Z}}} \left( \alpha _i\right) _{[t_i,t_{i+1})}\ \left| \ \phantom {\alpha = \sum _{i\in {\mathbb {Z}}} \left( \alpha _i\right) _{[t_i,t_{i+1})}} \begin{aligned}&\left\{ t_i\ \left| \ \phantom {t_i} i\in {\mathbb {Z}}\right. \right\} \text { locally finite with }\\&t_i<t_{i+1},\ \alpha _i\in \mathcal {C}^\infty ,\ i\in {\mathbb {Z}} \end{aligned}\right. \right\} , \end{aligned}$$
where \(\mathcal {C}^\infty \) denotes the space of smooth functions \(\alpha :\mathbb {R}\rightarrow \mathbb {R}\) and \(\alpha _I\) denotes the restriction of the function \(\alpha \) to the interval \(I\subseteq \mathbb {R}\) given by \(\alpha _I(t)=\alpha (t)\) for \(t\in I\) and \(\alpha _I(t)=0\) otherwise. The precise definition of piecewise-smooth distributions is then:

Definition 4

A distribution \(a\in \mathbb {D}\) is called piecewise-smooth iff
$$\begin{aligned} a = {\alpha }_{\mathbb {D}} + a[\cdot ] := {\alpha }_{\mathbb {D}} + \sum _{t\in \varGamma }a_t, \end{aligned}$$
  • \({\alpha }_{\mathbb {D}}\) is the regular distribution induced by a piecewise-smooth function \(\alpha \in \mathcal {C}_{\text { pw}}^{\infty }\), i.e. \({\alpha }_{\mathbb {D}}: \mathcal {C}_0^\infty \ni \varphi \mapsto \int _{\mathbb {R}} \alpha (t) \varphi (t) {\mathrm {d}}t\),

  • \(a_t\in \text {span}\left\{ \delta _t,\delta _t',\delta _t'',\ldots \right\} \) where \(\delta _t:\mathcal {C}_0^\infty \ni \varphi \mapsto \varphi (t)\) is the Dirac impulse with support \(\{t\}\),

  • \(\varGamma \subseteq \mathbb {R}\) is locally finite.

We denote \(a(t^-)=\lim _{s\nearrow t} \alpha (s)\), \(a(t^+)=\lim _{s\searrow t}\alpha (s)\) and \(a[t]=a_t\) if \(t\in \varGamma \) and \(a[t]=0\) otherwise. These “evaluations” of a are well-defined [28]. The space of piecewise-smooth distributions is denoted by \(\mathbb {D}_{\text { pw}\mathcal {C}^{\infty } }\).
As mentioned above, \(\mathbb {D}_{\text { pw}\mathcal {C}^{\infty } }\) is a subspace of \(\mathbb {D}\) which is closed under differentiation and, in addition, for which restrictions to intervals are well-defined [28]. Furthermore, there exist exactly two (non-commutative) multiplications \(*\) on \(\mathbb {D}_{\text { pw}\mathcal {C}^{\infty } }\) that satisfy
  1. 1.

    \(\forall \alpha ,\beta \in \mathcal {C}_{\text { pw}}^{\infty }: {\alpha }_{\mathbb {D}} *{\beta }_{\mathbb {D}} = {\left( \alpha \beta \right) }_{\mathbb {D}}\) (generalization of multiplication on \(\mathcal {C}_{\text { pw}}^{\infty }\)),

  2. 2.

    \(\forall a,b,c\in \mathbb {D}_{\text { pw}\mathcal {C}^{\infty } }: \left( a*b\right) *c=a*\left( b*c\right) \) (associativity),

  3. 3.

    \(\forall a,b\in \mathbb {D}_{\text { pw}\mathcal {C}^{\infty } }: \left( a*b\right) ' = a'*b+a*b'\) (differentiation rule of multiplication),

  4. 4.

    \(\forall t\in \mathbb {R}\ \forall \varphi \in \mathcal {C}^\infty _0: \left( {{\mathbbm {1}}_{[0,\infty )}}_\mathbb {D}*\delta _0\right) (\varphi ) = \left( {{\mathbbm {1}}_{[t,\infty )}}_\mathbb {D}*\delta _t\right) \big (\varphi (\cdot -t)\big )\) (condition for shift-invariance),

see [28, Section 2.4.1]. Here, \({\mathbbm {1}}_{[t,\infty )}\) denotes the characteristic function of the interval \([t,\infty )\), i.e., \({\mathbbm {1}}_{[t,\infty )}(\tau ) = 0\) if \(\tau <t\) and \({\mathbbm {1}}_{[t,\infty )}(\tau ) = 1\) otherwise. We denote these two multiplications by \(*_c,*_{ac}\). They are uniquely characterized by
$$\begin{aligned} \begin{aligned} {{\mathbbm {1}}_{[0,\infty )}}_\mathbb {D}*_c \delta _0&= \delta _0,\\ {{\mathbbm {1}}_{[0,\infty )}}_\mathbb {D}*_{ac} \delta _0&= 0,\\ \end{aligned} \end{aligned}$$
and are called causal and anticausal Fuchssteiner multiplication, respectively. If not stated otherwise, the causal Fuchssteiner multiplication will be used in the following. We will shortly write ab instead of \(a *_c b\) and \(\alpha a\) instead of \(\alpha _\mathbb {D}*_c a\).

The solution formula (10) for DAEs still holds true when allowing \(\mathbb {D}_{\text { pw}\mathcal {C}^{\infty } }\)-solutions if one introduces the notion of antiderivative for piecewise-smooth distributions [30, Remark 6.4.5 (3)].

3 Switched DAEs with impacts

Switched DAEs are now considered within the space of piecewise-smooth distributions \(\mathbb {D}_{\text { pw}\mathcal {C}^{\infty } }\). This makes it necessary to slightly restrict the set of switching signals:

Definition 5

(Switching Signal) \(\sigma : \mathbb {R}\rightarrow \mathbb {N}, \, t \mapsto \sigma (t)\) is called a suitable switching signal iff it is right-continuous, piecewise constant with locally only finitely many discontinuities (jumps), and constant on \((-\infty ,0)\). Without restriction (e.g., by appropriate relabeling of the matrices), we can assume that
$$\begin{aligned} \begin{aligned} \sigma _{(-\infty ,t_1)}&= 0, \\ \sigma _{[t_i,t_{i+1})}&= i \; \text { for } i=1,2,\ldots , \end{aligned}\end{aligned}$$
where \(0\le t_1< t_2 < \cdots \) are the switching times of \(\sigma \) with dwell times \(\tau _i:=t_{i+1}-t_i\) (with the convention \(t_0:=0\) and \(t_{m+1}:=T\) when the discontinuities are only allowed in the open interval (0, T) for some \(T>0\)). Note that this notation does not exclude an artificial introduction of switching times, because \((E_i,A_i,B_i,C_i)=(E_j,A_j,B_j,C_j)\) for \(i\ne j\) is allowed. Finally, for some \(r\ge 0\), we define the restriction \(\sigma _{>r}\) of a switching signal \(\sigma \) by
$$\begin{aligned} \sigma _{>r}(t) = {\left\{ \begin{array}{ll} \sigma (t) , &{}\quad t> r,\\ \sigma (r^+), &{}\quad t\le r. \end{array}\right. } \end{aligned}$$
In particular, the restriction \(\sigma _{>r}\) does not have a jump at r.
For a switched DAE (1), the corresponding solution behavior is given by
$$\begin{aligned} \mathcal {B}_{\sigma }= \left\{ (u,x,y) \ \left| \ \phantom { (u,x,y) } (u,x,y) \in \left( \mathbb {D}_{\text { pw}\mathcal {C}^{\infty } }\right) ^{q+ n + r} \text { solves}(1)\right. \right\} . \end{aligned}$$

Theorem 6

([30, Theorem 6.5.1 and Corollary 6.5.2]) The switched DAE (1) with suitable switching signal \(\sigma \) has a solution x for any input \(u\in \mathbb {D}_{\text { pw}\mathcal {C}^{\infty } }^q\), which is uniquely determined by \(x(0^-)\in \overline{ \mathcal {V}^*_{\sigma (0^-)} } \). For any consistent initial state \(x_0\in \overline{ \mathcal {V}^*_{\sigma (0^-)} } \), there exists \((u,x,y)\in \mathcal {B}_{\sigma }\) with \(x(0^-)=x_0\). If \(u_{[t_i,t_i+\varepsilon )}=0\) for some \(\varepsilon >0\), it holds
$$\begin{aligned} x(t_i^+)= & {} \varPi _i x(t_i^-), \\ x[t_i]= & {} - \sum _{j=0}^{n-1} \left( E_i^\text { imp}\right) ^{j+1}\left( I-\varPi _i\right) x(t_i^-) \delta _{t_i}^{(j)}. \end{aligned}$$

As motivated in the introduction, switched DAEs of the form (1) are not general enough to define a dual within the same system class. Therefore, we introduce the following larger system class:

Definition 7

A switched DAE with impacts is a system of the form
$$\begin{aligned} \begin{aligned} E_{\sigma } \dot{x}&= A_{\sigma } x + B_{\sigma } u+\sum _{i\ge 1} G_{t_i}\delta _{t_i} x, \\ y&= C_{\sigma } x \end{aligned} \end{aligned}$$
where \(E_i, A_i \in \mathbb {R}^{n \times n}, B_i \in \mathbb {R}^{n \times q}, C_i \in \mathbb {R}^{r \times n}\) for all \(i\in \mathbb {N}\) and some \(n,q,r \in \mathbb {N}\); \(\sigma : \mathbb {R}\rightarrow \mathbb {N}\) is a suitable switching signal according to Definition 5 with switching times \(t_i\), \(i\in \mathbb {N}\) and \(G_{t_i}\in \mathbb {R}^{n\times n}\). Furthermore, we assume that \(\left( E_i,A_i\right) \) is regular for each \(i\in \mathbb {N}\), and that xu,  and y are vectors of piecewise-smooth distributions, i.e., \(x\in \mathbb {D}_{\text { pw}\mathcal {C}^{\infty } }^n\), \(u\in \mathbb {D}_{\text { pw}\mathcal {C}^{\infty } }^q\) and \(y\in \mathbb {D}_{\text { pw}\mathcal {C}^{\infty } }^r\).

The behavior for a switched DAE with impact is defined in the same way as for switched DAEs. Note that for a restricted switching signal \(\sigma _{>r}\) also the impacts \(G_{t_i}\) are restricted to the interval \((r,\infty )\).

$$\begin{aligned} G = G[\cdot ] := \sum _{i\ge 1} G_{t_i}\delta _{t_i}\in \mathbb {D}_{\text { pw}\mathcal {C}^{\infty } }^{n\times n}, \end{aligned}$$
we can rewrite (13) as a distributional DAE [29]
$$\begin{aligned} E_\sigma \dot{x} = \mathcal {A}x + B_\sigma u, \end{aligned}$$
where \(\mathcal {A}:={A_\sigma }_\mathbb {D}+G[\cdot ]\in \mathbb {D}_{\text { pw}\mathcal {C}^{\infty } }^{n\times n}\). Hence, by [29, Thm. 21] existence and uniqueness of solutions, as stated in Theorem 6, follow for switched DAEs with impacts, independently of the choice of G. For (13), a solution formula similar to the one given in Theorem 6 holds:
$$\begin{aligned} \displaystyle x(t_i^+)= & {} \varPi _i^\text { diff}\left( E_i + G_{t_i} \right) x(t_i^-), \end{aligned}$$
$$\begin{aligned} \displaystyle x[t_i]= & {} - \sum _{j=0}^{n-1} \left( E_i^\text { imp}\right) ^j \varPi _i^\text { imp}\left( E_i + G_{t_i} \right) x(t_i^-) \delta _{t_i}^{(j)}. \end{aligned}$$
This can be seen as follows: We can restrict our attention to the initial trajectory problem with \(u=0\) and \(G[\cdot ]=G_0\delta _0\)
$$\begin{aligned} \begin{aligned} x_{(-\infty ,0)}&= x^0_{(-\infty ,0)}, \\ \left( E \dot{x} \right) _{[0,\infty )}&= \left( A x + G_0 \delta _0 x\right) _{[0,\infty )}. \end{aligned} \end{aligned}$$
Note that it holds \(G_0 \delta _0 x=G_0 \delta _0 x^0(0^-)\). By linearity, one can write \(x=\tilde{x} + \hat{x}\), where \(\tilde{x}\) solves (15) without the term \(G_0\delta _0 x\) and \(\hat{x}\) solves (15) with \(\hat{x}_{(-\infty ,0)}=0\). The solution formula for \(\hat{x}\) follows by [30, Theorem 6.4.4 and Remark 6.4.5(3)]:
$$\begin{aligned} \hat{x}(0^+)&= {\varPi }^\text { diff}G_0 x^0(0^-),&\hat{x}[0]&= - \sum _{i=0}^{n-1} \left( E^\text { imp}\right) ^i {\varPi }^\text { imp}G_0 x^0(0^-)\delta _0^{(i)}. \end{aligned}$$
Rewriting the solution of \(\tilde{x}\) as given in Theorem 6 gives
$$\begin{aligned} \tilde{x}(0^+)&= {\varPi }^\text { diff}E x^0(0^-),&\tilde{x}[0]&= - \sum _{i=0}^{n-1} \left( E^\text { imp}\right) ^i {\varPi }^\text { imp}E x^0(0^-)\delta _0^{(i)}. \end{aligned}$$

Remark 8

The expression impact is related to the corresponding switched ODEs. In [31], it was shown that the ODE with jumps
$$\begin{aligned} \dot{x} = A x + Bu, \; x(t_i^+)=J_i x(t_i^-) \; \text { for } i \in {\mathbb {Z}} \end{aligned}$$
and the distributional ODE
$$\begin{aligned} \dot{x} = A x + B u + \left( \sum _{i\in {\mathbb {Z}}} \left( J_i-I\right) \delta _{t_i}\right) x \end{aligned}$$
have the same solutions. Solutions of (16) are assumed to be piecewise-smooth functions. [31] showed that (17) has the same solutions (in \(\mathbb {D}_{\text { pw}\mathcal {C}^{\infty } }\)) as (16) if we assume the input to be piecewise smooth. Duality for switched ODEs with jumps has been dealt with in [14].
The feasible space at time \(t^\pm \) is the set of all values, the system can obtain at time \(t^\pm \) [15, Remark 2.10], that is
$$\begin{aligned} \left\{ x_t\in \mathbb {R}^n\ \left| \ \phantom {x_t\in \mathbb {R}^n} \exists (u,x,y)\in \mathcal {B}_{\sigma }\text { with } x(t^\pm )=x_t\right. \right\} . \end{aligned}$$
A consequence of Theorem 6 and the subsequent considerations is that for \(t=0^-\), the feasible space of (1) and (13) is given by \( \overline{ \mathcal {V}^*_{\sigma (0^-)} } \). This does not hold true, in general, for \(t>0\) for either of the systems.

4 Adjoint and dual system

The dual of a non-switched DAE is given in [9, 10] as
$$\begin{aligned} E^\top \dot{x} = - A^\top x + C^\top u, \quad y = B^\top x \end{aligned}$$
or, in the (continuously) time-varying case [8, 19], as
$$\begin{aligned} \tfrac{\mathrm{d}}{\mathrm{d}t} \left( E^\top x\right) = - A^\top x + C^\top u, \quad y = B^\top x. \end{aligned}$$
The references lack a motivation that would suffice to generalize the dual to switched systems. As we have seen in the introduction, a naive dualization does not work. In contrast to the unswitched case, time inversion is crucial. To point out this, we distinct between the adjoint and the dual system in the subsequent derivation.

4.1 Adjointness

To derive an adjointness condition, we will first recall adjointness for linear systems
$$\begin{aligned} \begin{aligned} \dot{x}&=Ax+Bu,\\ y&= Cx. \end{aligned} \end{aligned}$$
In [22], the following adjointness condition for (18) was derived:
$$\begin{aligned} \tfrac{\mathrm{d}}{\mathrm{d}t} \left( p^{\top } x \right) -y_a^{\top } u + u_a^{\top } y = 0 \; \; \forall \left( u,x,y\right) \in \mathcal {B}. \end{aligned}$$
The set of all \(\left( u_a,p,y_a\right) \) solving (19) is precisely the behavior of the system
$$\begin{aligned} \begin{aligned} \dot{p}&=-A^\top p - C^\top u_a,\\ y_a&= B^\top p. \end{aligned} \end{aligned}$$
Thus, we call (20) the adjoint system of (18).
Adjointness for homogeneous time-varying DAEs (with continuous coefficients) was considered in [1, 16]. The condition given there was (adapted to our notion)
$$\begin{aligned} p^\top E_\sigma x = \mathrm{const}. \end{aligned}$$
Together with (19), this leads us to the following adjointness condition for switched DAEs with impacts (13):
$$\begin{aligned} \boxed { \tfrac{\mathrm{d}}{\mathrm{d}t} \left( p^{\top } E_\sigma x \right) -y_a^{\top } u + u_a^{\top } y = 0 \; \; \forall \left( u,x,y\right) \in \mathcal {B}_\sigma .} \end{aligned}$$
We call any linear subspace \(\mathcal {B}\subseteq \mathbb {D}^{r+n+q}\) a behavioral adjoint of (13) if for any \((u_a,p,y_a)\in \mathcal {B}\), the adjointness condition (21) holds. Invoking the differentiation rule of the Fuchssteiner multiplication and inserting (13), we obtain the equivalent adjointness condition
$$\begin{aligned} \begin{aligned} \left( \tfrac{\mathrm{d}}{\mathrm{d}t} \left( p^{\top } E_\sigma \right) + p^\top A_\sigma + u_a^\top C_\sigma + p^\top G[\cdot ] \right) x + \left( p^\top B_\sigma - y_a^{\top }\right) u&=0 \\ \forall \left( u,x,y\right)&\in \mathcal {B}_\sigma . \end{aligned} \end{aligned}$$
This motivated the following definition of the adjoint system of (13):

Definition 9

For the switched DAE with impacts (13) and suitable switching signal \(\sigma \), the adjoint system is
$$\begin{aligned} \boxed {\begin{aligned} \tfrac{\mathrm{d}}{\mathrm{d}t} \left( p^{\top } E_{\sigma }\right)&= -p^{\top } A_{\sigma } -u_a^{\top } C_{\sigma } - p^\top G[\cdot ], \\ y_a^{\top }&= p^{\top } B_{\sigma }. \end{aligned}} \end{aligned}$$
The corresponding behavior is denoted by \(\mathcal {B}_\sigma ^{\text {adj}}\).

Obviously, \(\mathcal {B}_\sigma ^{\text {adj}}\) is a behavioral adjoint of (13). However, in contrast to ODEs, the condition (22) does not uniquely yield the adjoint system (23), and this is already a problem in the unswitched case:

Example 10

Consider the DAE \(0=x+0\,u\), \(y=0\) which has only the zero solution, i.e. \(\mathcal {B}_\sigma =\left\{ (u,0,0)\ \left| \ \phantom {(u,0,0)} u\in \mathbb {D}_{\text { pw}\mathcal {C}^{\infty } }^q\right. \right\} \). Hence, condition (21) reduces to \(y_a=0\), and there are no constraints on \(u_a\) and p, i.e., the largest behavior \(\mathcal {B}\) which is a behavioral adjoint of the above DAE is given by
$$\begin{aligned} \left\{ (u_a,p,0)\ \left| \ \phantom {(u_a,p,0)} u_a\in \mathbb {D}_{\text { pw}\mathcal {C}^{\infty } }^r,p\in \mathbb {D}_{\text { pw}\mathcal {C}^{\infty } }^n\right. \right\} . \end{aligned}$$
This is not the solution behavior of any regular DAE.
Another problem of the adjoint system (23) is that it is not in the form of (13) for two reasons:
  1. 1.

    The coefficient matrix \(E_\sigma \) is inside of the derivative operator.

  2. 2.

    The matrix-vector product is reversed in order.

The first problem can be resolved easily, as \(E_\sigma ^\top \) is piecewise constant and \(p^\top \tfrac{\mathrm{d}}{\mathrm{d}t} E_\sigma ^\top \) fits to the impact term in (23) (which has been introduced precisely for this reason). The second problem is more severe, because for the Fuchssteiner multiplication, it is not true, in general, that \(\left( A B\right) ^{\top } = B^{\top } A^{\top }\) for AB matrices over \(\mathbb {D}_{\text { pw}\mathcal {C}^{\infty } }\). In fact, the reversed order leads to an acausal behavior, as the following two examples illustrate.

Example 11

$$\begin{aligned} \tfrac{\mathrm{d}}{\mathrm{d}t} \left( p {\mathbbm {1}}_{(-\infty ,0)}\right) = p {\mathbbm {1}}_{[0,\infty )}. \end{aligned}$$
or, equivalently, \(\dot{p}{\mathbbm {1}}_{(-\infty ,0)} = p{\mathbbm {1}}_{[0,\infty )} + p \delta _0\). Invoking the calculus of piecewise-smooth distribution, we can conclude
$$\begin{aligned} \begin{aligned} \dot{p}&= 0&\quad \text {on }&(-\infty ,0),\\ 0&= p&\quad \text {on }&(0,\infty ),\\ \dot{p}[0]&= p \delta _0. \end{aligned} \end{aligned}$$
Since \(p\delta _0 = p(0^+)\delta _0 = 0\), it follows that \(\dot{p}[0]=0\), which implies that p cannot have a jump at \(t=0\) and \(p=0\) is the only solution. Hence, the past (\(p_{(-\infty ,0)}\)) is restricted by the future (\(p_{[0,\infty )}=0\)), i.e., the system is not causal.

Example 12

Consider now
$$\begin{aligned} \tfrac{\mathrm{d}}{\mathrm{d}t} \left( p {\mathbbm {1}}_{[0,\infty )}\right) = p{\mathbbm {1}}_{(-\infty ,0)}, \end{aligned}$$
or, equivalently, \(\dot{p} {\mathbbm {1}}_{[0,\infty )} = p{\mathbbm {1}}_{(-\infty ,0)} - p \delta _0\). This gives
$$\begin{aligned} \begin{aligned} 0&= p&\quad \text {on }&(-\infty ,0),\\ \dot{p}&= 0&\quad \text {on }&(0,\infty ),\\ p[0]&= p \delta _0. \end{aligned} \end{aligned}$$
Hence, \(p[0]=p(0^+)\delta _0\), and there is no constraint on the jump at \(t=0\), i.e., from a unique past \(p_{(-\infty ,0)}\), the future \(p_{[0,\infty )}\) is not determined uniquely.
The system in Example 11 does not have a solution for every initial trajectory \(p_{(-\infty ,0)}\), while the system in Example 12 has multiple solutions for the same initial trajectory \(p_{(-\infty ,0)}\). Therefore, we have to add a third problem concerning the adjoint system (23) to our list:
  1. 3.

    In general, the adjoint system is not causal, i.e., a solution p is not uniquely defined on \([t,\infty )\) by its past \(p_{(-\infty ,t)}\) and the input u.

We will resolve the third problem by reverting time in the next section. As a by-product, this will also resolve problem 2 mentioned above.

Remark 13

Another approach to derive the adjoint of the ODE (18) was given in [12, 32]. There, the system is identified with three mappings (input-to-state, initial-state-to-final-state, and state-to-output) and the adjoint is then defined by the adjoint operators of these mappings. While the approach works well for switched ODEs with jumps [14], it seems to fail for the more general class of switched DAEs [13].

The dual system given in [6, 8, 9, 10] fits to our notion of adjoint system (23)—despite the order of multiplication and possibly some signs, on which the references also do not agree. A main difference to these references is that the adjoint system which we have derived so far is not causal. This is a consequence of the considered solution space. For the non-causal system (23), it does not make sense to consider system properties, such as controllability and observability. Therefore, we introduce a time inversion of the adjoint system to arrive at a causal system of the form (13), which we then call the dual system.

4.2 Time inversion

For linear systems (18), the adjoint can be considered as a system going backwards in time [12]. We, therefore, define a time inversion for distributions:

Definition 14

Let \(D\in \mathbb {D}\) a distribution and \(T\in \mathbb {R}\). The time inversion of D at time T is defined by
$$\begin{aligned} \mathscr {T}_T\left\{ D\right\} (\varphi ) := D(\varphi (T-\cdot )) \text { for all } \varphi \in {\mathcal {C}}_0^\infty . \end{aligned}$$
\({\mathscr {T}}_T\) is a linear operation on the space of distributions. For the differentiation, it holds
$$\begin{aligned} \mathscr {T}_T\left\{ D'\right\} = -\left( \mathscr {T}_T\left\{ D\right\} \right) ' \text { for } D\in {\mathbb {D}}. \end{aligned}$$
The space of piecewise-smooth distributions is closed under time inversion, as it holds \(\mathscr {T}_T\left\{ \alpha _{\mathbb {D}}\right\} =\alpha (T-\cdot )_{\mathbb {D}}\) for a regular distribution \(\alpha _{\mathbb {D}}\) and \(\mathscr {T}_T\left\{ \delta ^{(k)}_t\right\} = (-1)^k \delta ^{(k)}_{T-t}\) for the Dirac impulse and its derivatives. In particular, it holds
$$\begin{aligned} \mathscr {T}_T\left\{ a\right\} ((T-t)^\pm )=a(t^\mp ) \text { for } t\in \mathbb {R}. \end{aligned}$$
and \(\mathscr {T}_T\left\{ a\right\} [T-t]=a[t]\) if a does not contain derivatives of Dirac impulses. Applying the time inversion to a product of piecewise-smooth distributions yields an anticausal multiplication:

Lemma 15

Let \(a,b\in \mathbb {D}_{\text { pw}\mathcal {C}^{\infty } }\) and \(T\in \mathbb {R}\). Then, it holds
$$\begin{aligned} \mathscr {T}_T\left\{ a *_{c} b\right\} = \mathscr {T}_T\left\{ a\right\} *_{ac} \mathscr {T}_T\left\{ b\right\} \text { and } \mathscr {T}_T\left\{ a *_{ac} b\right\} = \mathscr {T}_T\left\{ a\right\} *_{c} \mathscr {T}_T\left\{ b\right\} . \end{aligned}$$


See Appendix B. \(\square \)

The following lemma will be helpful for rewriting the time inversion of the adjoint system.

Lemma 16

Let \(A \in \left( \mathbb {D}_{\text { pw}\mathcal {C}^{\infty } }\right) ^{n_1 \times n_2}, B \in \left( \mathbb {D}_{\text { pw}\mathcal {C}^{\infty } }\right) ^{n_2\times n_3}\) matrices over \(\mathbb {D}_{\text { pw}\mathcal {C}^{\infty } }\) for some \(n_1,n_2,n_3\in \mathbb {N}\), then it holds
$$\begin{aligned} \left( A *_c B \right) ^{\top } = B^{\top } *_{ac} A^{\top }. \end{aligned}$$


See Appendix B. \(\square \)

4.3 Definition of the dual system

Applying a time inversion at time \(T>0\) to the adjoint system (23), and using Lemmas 15 and 16 gives:
$$\begin{aligned} \tfrac{\mathrm{d}}{\mathrm{d}t} \left( \mathscr {T}_T\left\{ E_\sigma ^{\top }\right\} *_{c} \mathscr {T}_T\left\{ p\right\} \right)= & {} \mathscr {T}_T\left\{ A_\sigma ^{\top }\right\} *_{c} \mathscr {T}_T\left\{ p\right\} + \mathscr {T}_T\left\{ C_\sigma ^{\top }\right\} *_{c} \mathscr {T}_T\left\{ u_a\right\} \\&+ \mathscr {T}_T\left\{ G[\cdot ]^\top \right\} *_{c} \mathscr {T}_T\left\{ p\right\} ,\\ \mathscr {T}_T\left\{ y_a\right\}= & {} \mathscr {T}_T\left\{ B_\sigma ^{\top }\right\} *_{c} \mathscr {T}_T\left\{ p\right\} . \end{aligned}$$
As \(E_\sigma \) is piecewise constant, it holds \(\mathscr {T}_T\left\{ E_\sigma \right\} = E_{\overline{\sigma }}\) for the time-inverted switching signal \(\overline{\sigma }\) with
$$\begin{aligned} \boxed {\overline{\sigma }(t):=\sigma (T-t)}\quad \forall t\in \mathbb {R}. \end{aligned}$$
The same holds true for \(A_\sigma \), \(B_\sigma \), and \(C_\sigma \). Hence, the time inversion at \(T>0\) of (23) is given by
$$\begin{aligned} \begin{aligned} \tfrac{\mathrm{d}}{\mathrm{d}t} \left( E_{\overline{\sigma }}^{\top } z \right)&= A_{\overline{\sigma }}^{\top } z + C_{\overline{\sigma }}^{\top } u_d + G[T-\cdot ]^\top z, \\ y_d&= B_{\overline{\sigma }}^{\top } z \end{aligned} \end{aligned}$$
with \(\left( u_d,z,y_d\right) =\left( \mathscr {T}_T\left\{ u_a\right\} ,\mathscr {T}_T\left\{ p\right\} ,\mathscr {T}_T\left\{ y_a\right\} \right) \). Here, we used that G does not contain derivatives of Dirac impulses.

Definition 17

For a switched DAE with impacts (13) with suitable switching signal \(\sigma \), the dual system with inversion time\(T>0\) (or, short, T-dual) is defined by (27) with inverted switching signal \(\overline{\sigma }\) given by (26). Its behavior is denoted by \(\mathcal {B}_\sigma ^{\text {T-dual}}\).

Note that the inverted switching signal does not have the form (11). In particular, the system switches to mode \(i-1\) at time \(s_i:=T-t_i\) (cf. forthcoming Fig. 1).
Fig. 1

Jump matrices \(H_{1}\), \(\widehat{H}_{0}\) for the switched DAE with impacts and its dual

The derivation of the dual system shows
$$\begin{aligned} \left( u_a,p,y_a\right) \in \mathcal {B}_\sigma ^{\text {adj}} \Leftrightarrow \left( u_d,z,y_d\right) =\left( \mathscr {T}_T\left\{ u_a\right\} ,\mathscr {T}_T\left\{ p\right\} ,\mathscr {T}_T\left\{ y_a\right\} \right) \in \mathcal {B}_\sigma ^{\text {{ T}-dual}}. \end{aligned}$$

Remark 18

The T-dual (27) can be written in the form of a switched DAE with impulses:
$$\begin{aligned} \boxed {\begin{aligned} E_{\overline{\sigma }}^{\top } \tfrac{\mathrm{d}}{\mathrm{d}s} z&= A_{\overline{\sigma }}^{\top } z + C_{\overline{\sigma }}^{\top } u_d + \left( G[T-\cdot ]^\top - \tfrac{\mathrm{d}}{\mathrm{d}s} E_{\overline{\sigma }}^\top \right) z, \\ y_d&= B_{\overline{\sigma }}^{\top } z. \end{aligned}} \end{aligned}$$
As \(E_{\overline{\sigma }}\) is piecewise constant, it holds \( \tfrac{\mathrm{d}}{\mathrm{d}s} E_{\overline{\sigma }}^\top = \sum _{i} \left( E_{i-1} - E_i \right) ^\top \delta _{t_i}\). The reversed switching signal is (by definition) a suitable switching signal only if it is constant in the past. This is only the case if the original switching signal \(\sigma \) is constant on \((T,\infty )\), i.e., all jumps of \(\sigma \) (and hence, \(\overline{\sigma }\)) are contained in the interval [0, T]. In that case, we have
$$\begin{aligned} \boxed {\left( \mathcal {B}_\sigma ^{\text {{ T}-dual}}\right) ^{\text {{ T}-dual}} = \mathcal {B}_\sigma ,} \end{aligned}$$
i.e., the desired duality property D1 (see Sect. 1) holds. Note that, in general, the dual of a switched DAE (without impacts) is a switched DAE with impacts. Hence, the enlargement of the considered system class was, in fact, necessary.

The T-dual (28) depends on the time \(T>0\) chosen for time inversion. In the sequel, we will assume that the switching signal \(\sigma \) is constant on \([T,\infty )\). Since our forthcoming definitions of controllability/reachability/observability/determinability are with respect to a finite interval [0, T] anyway, this is not a restriction of generality. Furthermore, it guarantees that the T-dual is again a switched DAE with impacts in the sense of Definition 7. We assume, additionally, that 0 and T are not switching times. This will be necessary for the duality result, see Remark 31. In particular, we only need to consider finitely many switching times \(t_1,\ldots ,t_m\) within the interval (0, T).

5 System theoretic properties

In this section, we will recall the system theoretic properties controllability, observability, and determinability, as given in [15, 21, 25, 26, 27] for switched DAEs and introduce a notion of reachability. We define and characterize these concepts for switched DAEs with impacts. Compared to the references, some changes in the notation were necessary—due to the effect of the impacts on jumps and impulses and also for a more convenient derivation of the duality.

5.1 Definitions

Definition 19

A switched DAE with impacts (13) is called
  • controllable on [0, T], \(T>0\), iff it holds
    $$\begin{aligned} \forall \omega ,\hat{\omega }\in \mathcal {B}_{\sigma }\; \exists \tilde{\omega }\in \mathcal {B}_{\sigma }: \omega _{(-\infty ,0)}=\tilde{\omega }_{(-\infty ,0)}, \;\hat{\omega }_{(T,\infty )} = \tilde{\omega }_{(T,\infty )}; \end{aligned}$$
  • reachable on [0, T], \(T>0\), iff it holds
    $$\begin{aligned} \forall \omega \in \mathcal {B}_{\sigma },\; \hat{\omega }\in \mathcal {B}_{\sigma (T^+)} \; \exists \tilde{\omega }\in \mathcal {B}_{\sigma }: \omega _{(-\infty ,0)}=\tilde{\omega }_{(-\infty ,0)} \text { and } \hat{\omega }_{(T,\infty )}=\tilde{\omega }_{(T,\infty )}; \end{aligned}$$
  • observable on [0, T], \(T>0\), iff it holds
    $$\begin{aligned} \forall \left( u,x,y\right) ,\left( \hat{u},\hat{x},\hat{y}\right) \in \mathcal {B}_{\sigma }: u = \hat{u} \wedge y_{[0,T]}= \hat{y}_{[0,T]} \quad \Rightarrow \quad x = \hat{x}; \end{aligned}$$
  • determinable on [0, T], \(T>0\), iff it holds
    $$\begin{aligned} \forall \left( u,x,y\right) ,\left( \hat{u},\hat{x},\hat{y}\right) \in \mathcal {B}_{\sigma }:\; u = \hat{u} \wedge y_{[0,T]}= \hat{y}_{[0,T]} \quad \Rightarrow \quad x_{(T,\infty )} = \hat{x}_{(T,\infty )}. \end{aligned}$$

The difference of controllability and reachability is that for the latter all feasible solutions \(\hat{\omega }\) of the last mode are considered, while the first only considers feasible solutions of the switched system.

To see the difference between observability and determinability, note that the solution might contain singular jumps. Hence, it might be possible to reconstruct the state after a certain time from input and output, but not the whole state trajectory.

The system theoretic properties can be simplified to zero-controllability, zero-reachability, etc:

Lemma 20

A switched DAE with impacts (13) is
  • controllable on [0, T], iff \(\forall \omega \in \mathcal {B}_{\sigma }\)\(\exists \tilde{\omega }\in \mathcal {B}_{\sigma }\):
    $$\begin{aligned} \omega _{\left( -\infty ,0\right) } = \tilde{\omega }_{\left( -\infty ,0\right) }, \; 0_{\left( T,\infty \right) } = \tilde{\omega }_{\left( T,\infty \right) }. \end{aligned}$$
  • reachable on [0, T] iff \(\forall \hat{\omega }\in \mathcal {B}_{\sigma (T^+)}\)\(\exists \tilde{\omega }\in \mathcal {B}_{\sigma }\):
    $$\begin{aligned} 0_{\left( -\infty ,0\right) } = \tilde{\omega }_{\left( -\infty ,0\right) }, \; \hat{\omega }_{\left( T,\infty \right) } = \tilde{\omega }_{\left( T,\infty \right) }. \end{aligned}$$
  • observable on [0, T] iff it holds \(\forall \left( u,x,y\right) \in \mathcal {B}_{\sigma }\):
    $$\begin{aligned} u= 0 \wedge y_{[0,T]} = 0 \; \Rightarrow x = 0. \end{aligned}$$
  • determinable on [0, T] iff it holds \(\forall \left( u,x,y\right) \in \mathcal {B}_{\sigma }\):
    $$\begin{aligned} u= 0 \wedge y_{[0,T]} = 0 \; \Rightarrow x_{(T,\infty )} = 0. \end{aligned}$$
Lemma 20 implies
$$\begin{aligned} \begin{aligned} (13)\,\text {is observable} \;&\Rightarrow \; (13)\,\text {is determinable},\\ (13)\,\text {is reachable} \;&\Rightarrow \; (13)\,\text {is controllable}. \end{aligned} \end{aligned}$$
The reverse does not hold true, as the following example shows.

Example 21

All solutions of the switched DAE
$$\begin{aligned} \left( {\mathbbm {1}}_{(-\infty ,t_1)}+{\mathbbm {1}}_{[t_2,\infty )}\right) \dot{x} = {\mathbbm {1}}_{[t_1,t_2)}x+0 u, \; y = 0x \end{aligned}$$
have the form \(x = {\mathbbm {1}}_{(-\infty ,t_1)}c\) for some \(c\in \mathbb {R}\). In particular, x is zero for \(t\ge t_1\). Hence, the system is (trivially) determinable and controllable on [0, T], \(T>t_1\). However, it is not observable as the output y is always zero. It is not reachable on [0, T], \(T>t_1\), as \( \overline{ \mathcal {V}^*_{2} } \ne \{0\}\) but \(x_{[t_2,\infty )}=0\) for any solution.

For duality, one usually considers only controllability and observability. However, they are not dual for switched DAEs, as the Example 22 shows. This can be interpreted as a problem with the time inversion of the dual system, which does not have any effect for unswitched ODE systems.

Example 22

The system \(\dot{x} = - \delta _{t_1}x + 0 u, \; y = 0x\) has solutions of the form \(x = c {\mathbbm {1}}_{(-\infty ,t_1)}\) for \(c\in \mathbb {R}\). The system is controllable on [0, T] (\(T>t_1\)), as each solution x is zero on \([t_1,\infty )\). Its dual \(\dot{z}=-\delta _{s_1}z + 0 u_d, \; y_d = 0 z\) is not observable as the output is zero and there are non-zero solutions \(z=c{\mathbbm {1}}_{(-\infty ,s_1)}\), \(c\in \mathbb {R}\).

We can characterize the system theoretic properties with the following spaces:

Definition 23

Let \(0\le s < t\) and define
$$\begin{aligned} \begin{aligned} \mathcal {C}_\sigma ^{[s,t]}&:= \left\{ x_s \in \mathbb {R}^n\ \left| \ \phantom { x_s \in \mathbb {R}^n} \exists (u,x,y)\in \mathcal {B}_{\sigma }: x(s^-)=x_s \ \wedge \ x(t^+)=0\right. \right\} ,\\ \mathcal {C}_\sigma ^{(s,t)}&:= \left\{ x_s \in \mathbb {R}^n\ \left| \ \phantom { x_s \in \mathbb {R}^n} \exists (u,x,y)\in \mathcal {B}_{\sigma }: x(s^+)=x_s \ \wedge \ x(t^-)=0 \right. \right\} , \\ \mathcal {R}_\sigma ^{[s,t]}&:= \left\{ x_t \in \mathbb {R}^n\ \left| \ \phantom { x_t \in \mathbb {R}^n} \exists (u,x,y)\in \mathcal {B}_{\sigma }: x(s^-)=0 \ \wedge \ x(t^+)=x_t \right. \right\} ,\\ \mathcal {R}_\sigma ^{(s,t)}&:= \left\{ x_t \in \mathbb {R}^n\ \left| \ \phantom { x_t \in \mathbb {R}^n} \exists (u,x,y)\in \mathcal {B}_{\sigma }: x(s^+)=0 \ \wedge \ x(t^-)=x_t \right. \right\} , \\ \mathcal {UO}_\sigma ^{[s,t]}&:= \left\{ x_s \in \mathbb {R}^n\ \left| \ \phantom { x_s \in \mathbb {R}^n} \exists (0,x,y)\in \mathcal {B}_{\sigma }: x(s^-)=x_s \ \wedge \ y_{[s,t]} = 0 \right. \right\} ,\\ \mathcal {UO}_\sigma ^{(s,t)}&:= \left\{ x_s \in \mathbb {R}^n\ \left| \ \phantom { x_s \in \mathbb {R}^n} \exists (0,x,y)\in \mathcal {B}_{\sigma }: x(s^+)=x_s \ \wedge \ y_{(s,t)} = 0 \right. \right\} , \\ \mathcal {UD}_\sigma ^{[s,t]}&:= \left\{ x_t \in \mathbb {R}^n\ \left| \ \phantom { x_t \in \mathbb {R}^n} \exists (0,x,y)\in \mathcal {B}_{\sigma }: x(t^+)=x_t \ \wedge \ y_{[s,t]}=0 \right. \right\} ,\\ \mathcal {UD}_\sigma ^{(s,t)}&:= \left\{ x_t \in \mathbb {R}^n\ \left| \ \phantom { x_t \in \mathbb {R}^n} \exists (0,x,y)\in \mathcal {B}_{\sigma }: x(t^+)=x_t \ \wedge \ y_{(s,t)}=0 \right. \right\} . \end{aligned} \end{aligned}$$
These spaces are called controllable, reachable, unobservable, and undeterminable space, respectively.

A switched DAE with impacts is controllable on [0, T] iff \(\mathcal {C}_\sigma ^{[0,T]} = \overline{ \mathcal {V}^*_{\sigma (0^-)} } \), reachable on [0, T] iff \(\mathcal {R}_\sigma ^{[0,T]}= \overline{ \mathcal {V}^*_{\sigma (T^+)} } \), observable on [0, T] iff \(\mathcal {UO}_\sigma ^{[0,T]}=\{0\}\), and determinable on [0, T] iff \(\mathcal {UD}_\sigma ^{[0,T]}=\{0\}\).

Controllable, reachable, unobservable, and undeterminable spaces were not only defined for the interval [0, T], but also for the open interval (0, T). The first notion fits to the definition of the system theoretic properties, while the second helps to interpret the derivation of the system theoretic properties of switched DAEs. The spaces are related as follows:

Lemma 24

Consider the switched DAE with impacts (13) with switching signal \(\sigma \). Let \(\mathcal {A}\in \{\mathcal {C},\mathcal {R},\mathcal {UO},\mathcal {UD}\}\). If s and t are not switching times of \(\sigma \), then
$$\begin{aligned} \mathcal {A}_\sigma ^{[s,t]}&= \mathcal {A}_\sigma ^{(s,t)}. \end{aligned}$$


It is sufficient to consider smooth control functions [15, Remark 2.12] if s and t are not switching times. Hence, x is also smooth at s and t. \(\square \)

Example 25

(Time inversion) Controllability and observability can be interpreted as properties of the states at time \(t=0\), reachability and determinability as properties of the states at time \(t=T\). This does, however, not mean that there is a relation between these properties if the switching signal is inverted. In fact, the system
$$\begin{aligned} \begin{aligned} \left( E_0,A_0,B_0,C_0\right)&= \left( \begin{bmatrix} 1&\quad 0 \\ 0&\quad 1 \end{bmatrix} , \begin{bmatrix} 0&\quad 0 \\ 0&\quad 0 \end{bmatrix} ,\begin{bmatrix}0\\1\end{bmatrix},\begin{bmatrix}1&0\end{bmatrix} \right) \\ \text { and } \left( E_1,A_1,B_1,C_1\right)&= \left( \begin{bmatrix}0&\quad 1\\0&\quad 0\end{bmatrix},\begin{bmatrix}0&\quad 0\\1&\quad 1\end{bmatrix},\begin{bmatrix}0\\0\end{bmatrix},\begin{bmatrix}0&\quad 0\end{bmatrix}\right) . \end{aligned} \end{aligned}$$
with \(\sigma _1:={\mathbbm {1}}_{[t_1,\infty )}\) is reachable, controllable, not observable, and not determinable. The same system with inverted switching signal \(\overline{\sigma }_1={\mathbbm {1}}_{(-\infty ,T-t_1)}\) is not reachable, not controllable, observable and determinable.
For a switched DAE with impacts (13) define for each mode i the Kalman matrices (here \([A/B]:=[A^\top ,B^\top ]^\top \))
$$\begin{aligned} K_i^{\text { diff}}&:= \left[ B_i^{\text { diff}}, A_i^{\text { diff}} B_i^{\text { diff}}, \ldots , \left( A_i^{\text { diff}}\right) ^{n-1} B_i^{\text { diff}} \right] ,\\ K_i^{\text { imp}}&:= \left[ B_i^{\text { imp}}, E_i^{\text { imp}} B_i^{\text { imp}}, \ldots , \left( E_i^{\text { imp}}\right) ^{n-1} B_i^{\text { imp}} \right] , \\ O_i^{\text { diff}}&:= \left[ C_i^{\text { diff}} / C_i^{\text { diff}} A_i^{\text { diff}} / \ldots / C_i^{\text { diff}} \left( A_i^{\text { diff}}\right) ^{n-1} \right] ,\\ O_i^{\text { imp}}&:= \left[ C_i^{\text { imp}} / C_i^{\text { imp}} E_i^{\text { imp}} / \ldots / C_i^{\text { imp}} \left( E_i^{\text { imp}}\right) ^{n-1} \right] , \end{aligned}$$
for the differential and the impulsive parts, respectively. The jump matrix is defined as \( H_{i} := E_i + G_{t(i)}\), where t(i) is the time mode i is entered (i.e., \(t(i)=t_i\) for systems with switching signals of the form (11) and \(t(i)=T-t_{i+1}\) for the corresponding reversed switching signal). Finally, define
$$\begin{aligned} \mathcal {C}_i&:= {{\mathrm{im}}}K_i^\text { diff}\oplus {{\mathrm{im}}}K_i^\text { imp}, \\ \mathcal {U}_i&:= \ker O_i^\text { diff}\cap \ker O_i^\text { imp}, \\ \mathcal {U}^{\text { H}}_{i}&:= \ker \left( O_i^\text { diff}\varPi _i^\text { diff}H_{i} \right) \cap \ker \left( O_i^\text { imp}\varPi _i^\text { imp}H_{i} \right) . \end{aligned}$$
Defining \(H_i\) via t(i) seems unusual, but it is necessary when dealing with the dual system, as this system does not have a switching signal of the form (11). Using Remark 18, it holds \(\widehat{H}_{i} = \widehat{G}_{s_{i+1}} + E_i^\top = G_{t_{i+1}}^\top - E_i^\top + E_{i+1}^\top + E_i^\top = H_{i+1}^\top \) (see Fig. 1). The circumflex ( \(\hat{}\) ) refers to the dual system.

5.2 System theoretic properties and duality of unswitched DAEs

For a switched DAE with impacts (13) with constant \(\sigma =0\), the notions of controllability and reachability as well as the notions of observability and determinability coincide. It holds (c.f. [15, 25])
$$\begin{aligned} \begin{aligned} \mathcal {C}_\sigma ^{[0,T]}&= \mathcal {R}_\sigma ^{[0,T]}= {{\mathrm{im}}}K_0^\text { diff}+ {{\mathrm{im}}}K_0^\text { imp}, \\ \mathcal {UO}_\sigma ^{[0,T]}&= \mathcal {UD}_\sigma ^{[0,T]}= {{\mathrm{im}}}\varPi _0 \cap \ \ker O_0^\text { diff}, \end{aligned} \end{aligned}$$
hence, controllability/reachability and observability/determinability are characterized by the conditions:
$$\begin{aligned} {{\mathrm{im}}}K_0^\text { diff}+ {{\mathrm{im}}}K_0^\text { imp}= \overline{ \mathcal {V}^*_{0} } \quad \text { and } \ker O_0^\text { diff}\cap {{\mathrm{im}}}\varPi _0 = \{0\}, \end{aligned}$$
respectively. These characterizations do not directly appear to be dual. However, it is easily seen that only the differential part of the DAE is relevant: An equivalent condition for controllability/reachability is given by
$$\begin{aligned} {{\mathrm{im}}}K_0^\text { diff}+ \ker \varPi _0 = \mathbb {R}^n. \end{aligned}$$
Now, duality for the unswitched case is apparent, because for the dual, we have \(\widehat{K}_0^\text { diff}= {O^\text { diff}_0}^\top \), and in general, we have \(({{\mathrm{im}}}M)^\bot = \ker M^\top \) for some matrix M.

The duality result for unswitched systems differs from those given in [9, 10], as these papers use different definitions for controllability (and in case of [9] also a different definition of observability). The duality in [10] requires the technical assumption \(\ker E \cap \ker C \subseteq \ker A\), which is not motivated there. This assumption is equivalent to \({{\mathrm{im}}}\hat{E} + {{\mathrm{im}}}\hat{B} \supseteq {{\mathrm{im}}}\hat{A}\) for the dual system, for which controllability is considered. This condition, however, implies that the augmented consistency space of the dual is the whole space; under this condition, the controllability notions of [10] and our notion coincide.

5.3 Characterizations

In this section, the system properties controllability, reachability, observability and determinability are characterized by their corresponding spaces. We start with the single switch case.

Lemma 26

(Single switch) Consider the switched DAE with impacts (13) with switching signal \(\sigma _1:={\mathbbm {1}}_{[t_1,\infty )}\), \(t_1>0\). For \(T>t_1\), the controllable/reachable/unobservable/undeterminable spaces are given by
$$\begin{aligned} \mathcal {C}_{\sigma _1}^{(0,T)}&= \left( \mathcal {C}_0 + \mathrm {e}^{-A^\text { diff}_0 \tau _0} \left( \varPi _1^\text { diff}H_{1} \right) ^{-1} \mathcal {C}_1\right) \cap \overline{ \mathcal {V}^*_{0} } , \end{aligned}$$
$$\begin{aligned} \mathcal {R}_{\sigma _1}^{(0,T)}&= \mathcal {C}_1 + \mathrm {e}^{A^\text { diff}_1 \tau _1} \Pi ^\text { diff}_{1} H_{1} \mathcal {C}_0, \end{aligned}$$
$$\begin{aligned} \mathcal {UO}_{\sigma _1}^{(0,T)}&= {{\mathrm{im}}}\varPi _0 \cap \ker O_0^\text { diff}\cap \mathrm {e}^{-A_0^\text { diff}\tau _0} \mathcal {U}^{\text { H}}_{1}, \end{aligned}$$
$$\begin{aligned} \mathcal {UD}_{\sigma _1}^{(0,T)}&= \mathrm {e}^{A_1^\text { diff}\tau _1} \Pi ^\text { diff}_{1} H_{1} \left( {{\mathrm{im}}}\varPi _0 \cap \ker O_0^\text { diff}\cap \mathcal {U}^{\text { H}}_{1} \right) , \end{aligned}$$


See Appendix C. \(\square \)

The single switch result is now used to derive a recursive formula for the multi-switch case. For controllability, we have to go backwards in time, i.e., start with the last switch. To make use of the single switch result, we have to consider a switching signal whose switches are restricted to the interval \((t_{m-1},T)\), as otherwise, we would have to care about feasibility of (consistent) states. Using the restricted switching signal \(\sigma _{>t_i}\) guarantees that any \(x_i\in \overline{ \mathcal {V}^*_{i} } \) is a feasible state at time \(t_i^+\), i.e., there exists \((u,x,y)\in \mathcal {B}_{\sigma _{>t_i}}\) with \(x(t_i^+)=x_i\).

For a switched DAE with impacts (13) and switchting signal (11) with switching times \(0<t_1<\cdots<t_m<T\), this leads to the recursion
$$\begin{aligned} \begin{aligned} \mathcal {P}_m^m&:= \mathcal {C}_m, \\ \mathcal {P}_{i}^m&:= \mathcal {C}_{i} + \mathrm {e}^{-A^\text { diff}_{i}\tau _{i}}\left( \varPi _{i+1}^\text { diff}H_{i+1}\right) ^{-1} \mathcal {P}_{i+1}^m \;\text { for } i=m-1,\ldots ,0. \end{aligned} \end{aligned}$$
For reachability, the recursion goes forward in time. Hence, a restriction of (the switches of) the switching signal is not required:
$$\begin{aligned} \begin{aligned} \mathcal {Q}_0^0&:= \mathcal {C}_0, \\ \mathcal {Q}_0^{i}&:= \mathcal {C}_{i} + \mathrm {e}^{A^\text { diff}_{i}\tau _{i}} \Pi ^\text { diff}_{i} H_{i} \mathcal {Q}_0^{i-1} \; \text { for } i=1,\ldots ,m. \end{aligned} \end{aligned}$$
The single switch result on observability motivates the definition of the local unobservable space at \(t_i^-\) for (13) and switching signal (11):
$$\begin{aligned} \widetilde{\mathcal {M}}_i = {{\mathrm{im}}}\varPi _{i-1} \cap \ker O_{i-1}^\text { diff}\cap \mathcal {U}^{\text { H}}_{i}. \end{aligned}$$
With this, a recursion for the unobservable space can be given as
$$\begin{aligned} \begin{aligned} \widetilde{\mathcal {M}}_m^m&:= \mathrm {e}^{-A^\text { diff}_{m-1}\tau _{m-1}}\widetilde{\mathcal {M}}_m, \\ \widetilde{\mathcal {M}}_i^m&:= \mathrm {e}^{-A^\text { diff}_{i-1}\tau _{i-1}}\left( \widetilde{\mathcal {M}}_i \cap \left( \varPi ^\text { diff}_{i} H_{i}\right) ^{-1} \widetilde{\mathcal {M}}_{i+1}^m \right) \; \text { for } i=m-1,\ldots ,1. \end{aligned} \end{aligned}$$
A recursion for the undeterminable space is given by
$$\begin{aligned} \begin{aligned} \widetilde{\mathcal {N}}_1^1&:= \mathrm {e}^{A^\text { diff}_1\tau _1} \Pi ^\text { diff}_{1} H_{1} \widetilde{\mathcal {M}}_1, \\ \widetilde{\mathcal {N}}_1^i&:= \mathrm {e}^{A^\text { diff}_i\tau _i} \Pi ^\text { diff}_{i} H_{i} \left( \widetilde{\mathcal {M}}_i \cap \widetilde{\mathcal {N}}_1^{i-1}\right) \; \text { for } i=2,\ldots ,m. \end{aligned} \end{aligned}$$

Theorem 27

(General switching signal) Consider the switched DAE with impacts (13) with switching signal (11) and switching times \(0<t_1<\cdots<t_m<T\). Then, it holds
$$\begin{aligned} \begin{aligned} \mathcal {C}_{\sigma _{>t_i}}^{(t_i,T)}&= \mathcal {P}^m_i \cap \overline{ \mathcal {V}^*_{i} }&\text { for } i=0,\ldots ,m, \\ \mathcal {R}_\sigma ^{(0,t_{i+1})}&= \mathcal {Q}_0^{i}&\text { for } i=0,\ldots ,m, \\ \mathcal {UO}_{\sigma _{>t_{i-1}}}^{(t_{i-1},T)}&= \widetilde{\mathcal {M}}^m_i&\text { for } i=1,\ldots ,m, \\ \mathcal {UD}^{(0,t_{i+1})}_\sigma&= \widetilde{\mathcal {N}}_1^i&\text { for } i=1,\ldots ,m. \end{aligned} \end{aligned}$$
In particular, the system is
  • controllable on [0, T] iff \( \overline{ \mathcal {V}^*_{0} } \subseteq \mathcal {P}_0^m\),

  • reachable on [0, T] iff \( \overline{ \mathcal {V}^*_{m} } = \mathcal {Q}_0^m\),

  • observable on [0, T] iff \(\{0\} = \widetilde{\mathcal {M}}^m_1 \),

  • determinable on [0, T] iff \(\{0\} = \widetilde{\mathcal {N}}_1^m\).


See Appendix C. \(\square \)

5.4 Normalization and explicit formulas

To give rather simple explicit formulas for the system theoretic properties we introduce the notion of a normalized system. This will also be helpful for proving duality.

Definition 28

A DAE (5) is called normalized iff \(E=E^\text { diff}+E^\text { imp}\) and \(A=A^\text { diff}+ A^\text { imp}\). The switched system (13) is called normalized iff each mode is normalized.

Normalizedness is equivalent to \(\left( {\varPi }^\text { diff}+{\varPi }^\text { imp}\right) \left( \lambda E + A\right) = \lambda E + A\) for all \(\lambda \in \mathbb {R}\). As (EA) is regular, this gives \({\varPi }^\text { diff}+ {\varPi }^\text { imp}=I\) or, in terms of S and T from the QWF, \(TS=I\). Hence, for normalized DAEs, it holds \({\varPi }^\text { diff}=\varPi \), \({\varPi }^\text { imp}=I-\varPi \), \(\mathcal {U}^{\text { H}}_{i} = H_{i}^{-1}\mathcal {U}_i\), \(O_i^\text { diff}\varPi _i^\text { diff}= O_i^\text { diff}\), and \(O_i^\text { imp}\varPi _i^\text { imp}= O_i^\text { imp}\).

It is easily seen that any DAE can be transformed into a normalized DAE via
$$\begin{aligned} \begin{aligned} \left( T_{}S_{}\right) ^{}E \dot{x}&= \left( T_{}S_{}\right) ^{} A x + \left( T_{}S_{}\right) ^{} B u, \\ y&= C x \end{aligned} \end{aligned}$$
where \(\left( S,T\right) \) transforms the matrix pair (EA) to QWF. Note that for normalized DAEs, the matrices E and A commute (the converse is not true in general), and the premultiplication to obtain (38) has some similarity with the often used trick from [7, Lem. 3.1.1] to obtain commutativity of E and A.

Theorem 29

(Explicit formulas) A normalized system (13) with switching signal (11) and switching times \(0<t_1<\cdots<t_m<T\) is
  • controllable on [0, T] iff
    $$\begin{aligned} \varPi _0^{-1} \left( \mathcal {C}_0 \!+ \!\mathrm {e}^{-A^\text { diff}_0\tau _0} H_{1}^{-1}\varPi _1^{-1} \left( \ldots \left( \mathcal {C}_{m-1} \!+\! \mathrm {e}^{-A^\text { diff}_{m-1}\tau _{m-1}} H_{m}^{-1}\varPi _m^{-1}\mathcal {C}_m \right) \ldots \right) \right) =\! \mathbb {R}^n; \end{aligned}$$
  • reachable on [0, T] iff
    $$\begin{aligned} \ker \varPi _m + \mathcal {C}_m+ \mathrm {e}^{A_m^\text { diff}\tau _m} H_{m} \sum _{j=0}^{m-1} \left( \prod _{k=1}^{m-1-j} \mathrm {e}^{A^\text { diff}_{m-k}\tau _{m-k}} \varPi _{m-k}H_{m-k}\right) \mathcal {C}_j = \mathbb {R}^n; \end{aligned}$$
  • observable on [0, T] iff
    $$\begin{aligned} {{\mathrm{im}}}\varPi _0 \cap \mathcal {U}_0 \cap \mathrm {e}^{-A_0^\text { diff}\tau _0} H_{1}^{-1} \left( \bigcap \limits _{j=1}^m \left( \prod \limits _{k=1}^{j-1} \mathrm {e}^{-A_k^\text { diff}\tau _k} \varPi _k^{-1} H_{k+1}^{-1} \right) \mathcal {U}_j \right) = \{0\}; \end{aligned}$$
  • determinable on [0, T] iff
    $$\begin{aligned} \varPi _m \left( \mathcal {U}_m \cap \mathrm {e}^{A_m^\text { diff}\tau _m} H_{m} \varPi _{m-1} \left( \ldots \left( \mathcal {U}_1 \cap \mathrm {e}^{A_1^\text { diff}\tau _1} H_{1} \varPi _0 \mathcal {U}_0 \right) \ldots \right) \right) = \{0\}. \end{aligned}$$

6 Duality statement

When defining the dual system, there were two sources for non-uniqueness:
  1. 1.

    a multiplication of the systems Eq. (13) from the left with some \(S_\sigma \), \(S_k\) invertible, and

  2. 2.

    the precise value of the inversion time \(T>t_m\).

Neither of those has an influence on the system theoretic properties and, thus, on the duality result. Multiplying (13) from the left with some \(S_\sigma \), \(S_k\) invertible, yields a state transformation (\(\tilde{z} = S_{\overline{\sigma }}^\top z\)) of the dual system. In particular, this does not influence the system theoretic properties of the dual. Hence, we can assume the system and, thus, also its dual to be normalized. To see that the system theoretic properties do not depend on the precise value of \(T>t_m\), observe that the statements in Theorem 29 do not depend on \(\tau _0\) and \(\tau _m\). The terms \({\mathrm {e}}^{A_0^\text { diff}\tau _0}\) and \({\mathrm {e}}^{A_m^\text { diff}\tau _m}\) can be removed from the equations using the \({\mathrm {e}}^{A_i^\text { diff}\tau _i}\)-invariance of \(\mathcal {C}_i\) and \(\mathcal {U}_i\). Thus, the system theoretic properties on [0, T] and on \([0,T']\) are the same for \(T,T'>t_m\). In addition, the properties of the T-dual and the properties of the \(T'\)-dual are the same.

6.1 Normalization and the dual system

Using the above considerations, we can assume the system to be normalized for the duality result. This simplifies the subsequent calculations, as the following indicates: The matrices \(\widehat{\varPi },\widehat{\varPi }^\text { diff},\widehat{E}^\text { imp},\ldots \) of the dual system and the corresponding matrices of the original system are related via the transformation matrices ST of the quasi-Weierstrass form (QWF, see Sect. 2.1):
$$\begin{aligned} \widehat{\varPi } = \left( T_{}S_{}\right) ^{\top }{\varPi }^\top \left( T_{}S_{}\right) ^{-\top }, \; \widehat{\varPi }^\text { diff}= \left( {\varPi }^\text { diff}\right) ^{\top }, \; \widehat{E}^\text { diff}= \left( T_{}S_{}\right) ^{\top } \left( E^\text { diff}\right) ^{\top } \left( T_{}S_{}\right) ^{-\top }. \end{aligned}$$
This can be seen by straightforward calculations, as \((T^\top ,S^\top )\) transform the dual to QWF. For normalized systems, it holds \(S=T^{-1}\). Thus, the matrices and spaces of the dual system simplify to \(\widehat{\varPi }={\varPi }^\top \) and
$$\begin{aligned} \widehat{\mathcal {A}}^\text { part}= \left( {\mathcal {A}}^\text { part}\right) ^\top \text { for } {\mathcal {A}}\in \left\{ E,A,B,C\right\} \text { and } \text { part}\in \left\{ \text { diff},\text { imp}\right\} \end{aligned}$$
where we used \(\widehat{B} = C^{\top }\) and \(\widehat{C} = B^{\top }\). Consequently, it holds for normalized systems
$$\begin{aligned} \widehat{\mathcal {C}} = \mathcal {U}^\perp \; \text { and }\; \widehat{\mathcal {U}} = \mathcal {C}^\perp . \end{aligned}$$

6.2 Duality statement

In contrast to switched ODEs with jumps [14] one cannot work directly with the recursions from Sect. 5. One reason is that the conditions for reachability (\(\mathcal {R}_\sigma ^{[0,T]}= \overline{ \mathcal {V}^*_{\sigma (T^+)} } \)) and observability (\(\mathcal {UO}_\sigma ^{[0,T]}=\{0\}\)) are not complementary as \( \overline{ \mathcal {V}^*_{\sigma (T^+)} } \ne \mathbb {R}^n\), in general.

Theorem 30

(Duality of switched DAEs with impacts) For a switched DAE with impacts (13) with switching signal (11) whose switching times are contained in (0, T) and its T-dual (28), the duality statement (2) holds, i.e., (13) is observable (determinable) on [0, T] if and only its T-dual (28) is reachable (controllable) on [0, T].


We assume the system to be normalized, as this does not have any influence on the system theoretic properties of the system or its dual. Note that for the dual system, it holds \(\widehat{H}_{i} = H_{i+1}^\top \).

By Theorem 29, a normalized switched DAE with impacts is reachable iff
$$\begin{aligned} \mathbb {R}^n= \ker \varPi _m + \mathcal {C}_m + \mathrm {e}^{A_m^\text { diff}\tau _m} H_{m} \sum _{j=0}^{m-1} \left( \prod _{k=1}^{m-1-j} \mathrm {e}^{A^\text { diff}_{m-k}\tau _{m-k}} \varPi _{m-k}H_{m-k}\right) \mathcal {C}_j. \end{aligned}$$
Hence, a dual system is reachable iff
$$\begin{aligned} \mathbb {R}^n&= \ker \widehat{\varPi }_0 + \widehat{\mathcal {C}}_0 + \mathrm {e}^{\widehat{A}_0^\text { diff}\tau _0} \widehat{H}_0 \sum _{j=1}^{m} \left( \prod _{k=1}^{j-1} \mathrm {e}^{\widehat{A}^\text { diff}_{k}\tau _{k}} \widehat{\varPi }_{k}\widehat{H}_{k}\right) \widehat{\mathcal {C}}_j\\&= \ker \varPi _0^\top + \mathcal {U}_0^\perp + \mathrm {e}^{\left( A_0^\text { diff}\right) ^\top \tau _0} H_{1}^\top \sum _{j=1}^{m} \left( \prod _{k=1}^{j-1} \mathrm {e}^{\left( A^\text { diff}_k\right) ^\top \tau _k} \varPi _k^\top H_{k+1}^\top \right) \mathcal {U}_j^\perp \\&= \left( {{\mathrm{im}}}\varPi _0 \cap \mathcal {U}_0 \cap \mathrm {e}^{-A_0^\text { diff}\tau _0} H_{1}^{-1} \bigcap \limits _{j=1}^{m} \left( \prod _{k=1}^{j-1} \mathrm {e}^{-A_k^\text { diff}\tau _k} \varPi _k^{-1} H_{k+1}^{-1} \right) \mathcal {U}_j \right) ^\perp . \end{aligned}$$
This is the observability condition of the original system (Theorem 29). Hence, a switched DAE with impacts is observable iff its dual is reachable.
The same theorem gives that a normalized switched DAE with impacts is controllable iff
$$\begin{aligned} \mathbb {R}^n= & {} \varPi _0^{-1} \left( \mathcal {C}_0 + \mathrm {e}^{-A^\text { diff}_0\tau _0} H_{1}^{-1}\varPi _1^{-1} \left( \ldots \left( \mathcal {C}_{m-1} + \mathrm {e}^{-A^\text { diff}_{m-1}\tau _{m-1}} H_{m}^{-1}\varPi _m^{-1}\mathcal {C}_m \right) \ldots \right. \right) . \end{aligned}$$
Hence, a dual system is controllable iff
$$\begin{aligned} \mathbb {R}^n&= \widehat{\varPi }_m^{-1} \left( \widehat{\mathcal {C}}_m + \mathrm {e}^{-\widehat{A}^\text { diff}_m\tau _m} \widehat{H}_{m-1}^{-1} \widehat{\varPi }_{m-1}^{-1} \left( \ldots \left( \widehat{\mathcal {C}}_1 + \mathrm {e}^{-\widehat{A}^\text { diff}_1 \tau _1} \widehat{H}_0^{-1} \widehat{\varPi }_0^{-1} \widehat{\mathcal {C}}_0 \right) \ldots \right. \right) \\&= \varPi _m^{-\top } \left( \mathcal {U}_m^\perp + \mathrm {e}^{-\left( A_m^\text { diff}\right) ^\top \tau _m} H_{m}^{-\top } \varPi _{m-1}^{-\top } \left( \ldots \left( \mathcal {U}_1^\perp \right. \right. \right. \\&\quad +\left. \left. \left. \mathrm {e}^{-\left( A_1^\text { diff}\right) ^\top \tau _1} H_{1}^{-\top } \varPi _0^{-\top } \mathcal {U}_0^\perp \right) \ldots \right. \right) \\&= \left( \varPi _m \left( \mathcal {U}_m \cap \mathrm {e}^{A_m^\text { diff}\tau _m} H_{m} \varPi _{m-1} \left( \ldots \left( \mathcal {U}_1 \cap \mathrm {e}^{A_1^\text { diff}\tau _1} H_{1} \varPi _0 \mathcal {U}_0 \right) \ldots \right. \right. \right) ^\perp . \end{aligned}$$
This is the determinability condition of the original system (again Theorem 29). Hence, a switched DAE with impacts is determinable iff its dual is controllable.

Applying these results to the dual of a switched DAE with impacts and the dual’s dual, which is again the original system (Remark 18), gives that a switched DAE with impacts is controllable iff its dual is determinable and reachable iff its dual is observable. \(\square \)

Remark 31

(Assumptions on the switching signal) We assumed the switching signal to have only finitely many switches and not to have a switch at \(t=0\). Neither of these assumptions is used in [15, 21, 25], but both are crucial here. The switching signal is assumed to be constant on \((-\infty ,0)\) to ensure that the feasibility set at \(t=0^-\) is \( \overline{ \mathcal {V}^*_{\sigma (0^-)} } \). A restriction on the switchings in \((-\infty ,0)\) is necessary to ensure that non-trivial solutions exist. To achieve the same for the dual system with the time-inverted switching signal \(\overline{\sigma }\), the switching signal \(\sigma \) has to be constant after some time \(T>0\). Hence, only finitely many switches are allowed.

The switching time \(t=0\) is excluded in this work, as it does not give rise to the duality statement: The system \(\dot{x}=0, y={\mathbbm {1}}_{(-\infty ,0)}x\) is not determinable, but its dual is controllable (and reachable) via instantaneous control (see [15, Lemma 2.11]). One can also find a switched DAE that is not reachable, but whose dual is observable (see [13, Example 7.3.1]).

Remark 32

(Other notions of the system theoretic properties for switched systems) So far, we fixed a switching signal and considered the system theoretic properties of the system for the given switching signal \(\sigma \). Other notions for the system theoretic properties of a switched system are that the relevant property hold for all switching signals or that there exists a switching signal such that the property holds. For observability, this is called strong observability and controlled observability [21], respectively. Strong observability (strong controllability, etc.) is equivalent to observability (controllability, etc.) of each mode. Hence, strong observability and strong determinability as well as strong reachability and strong controllability coincide. Furthermore, the duality follows directly form the duality for unswitched DAEs [6]. Duality for the first notion can be derived from Theorem 30. Assume that the system (13) is contr./reach./obsv./det. for a given switching signal \(\sigma \). Then, there exists a switching signal that has only switches in (0, T) for some \(T>0\) and that yields the same property. By Theorem 30, the dual system is then det./obsv./reach./contr. for the switching signal \(\overline{\sigma }\).

Remark 33

Due to the results for switched DAEs, we conjecture that a suitable adjointness condition for (smoothly) time-varying DAEs should be
$$\begin{aligned} \tfrac{\mathrm{d}}{\mathrm{d}t} \left( p^\top E x \right) - y_a^\top u + u_a^\top y =0, \end{aligned}$$
which the authors have not found in the literature so far.

7 Conclusion

We have established a duality result between controllability/reachability and determinability/observability for switched DAEs. It turned out that the problem of duality cannot be solved within the class of switched DAEs, as the dual of a switched DAE does not belong to this system class. Thus, we considered the more general class of switched DAEs with impacts, for which characterizations of controllability, reachability, observability, and determinability are derived. These are then the basis for our duality result.

It is well known that the dual of a system plays an fundamental role in optimal control, and it is, therefore, a canonical topic for future research whether the duality presented here is a fruitful basis for studying optimal control for switched DAEs.

Another line of future research is an relaxation on the assumption that the switching signal is given. The observability/determinability notions would then include the ability to detect the current mode and the controllability/reachability notions may allow for the switching signal to be an additional input. Our approach will presumably not carry over directly, because in our framework, we could still exploit the linearity of the (time-varying) dynamical system, which will not be possible when the switching signal is not known.


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Copyright information

© Springer-Verlag London 2016

Authors and Affiliations

  1. 1.Fraunhofer Institute for Industrial MathematicsKaiserslauternGermany
  2. 2.Technomathematics GroupUniversity of KaiserslauternKaiserslauternGermany

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