# Duality of switched DAEs

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## Abstract

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.

### Keywords

Duality Switched systems Differential-algebraic equations## 1 Introduction

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.

- 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).

- D2
The classical duality between (some form of) controllability and (some form of) observability holds.

- D3
There is some formal justification of duality in terms of the solution trajectories.

*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

*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:

*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

## 2 Mathematical preliminaries

### 2.1 Regular matrix pairs

*x*,

*u*,

*y*). For existence and uniqueness of solutions, the following notion of regularity of the matrix pair (

*E*,

*A*) 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

- 1.
The matrix pair (

*E*,*A*) is regular. - 2.There exist invertible matrices \(S,T\in {\mathbb {R}}^{n\times n}\) transforming (
*E*,*A*) into quasi-Weierstrass form (QWF), that iswith \(N\in \mathbb {R}^{n_N\times n_N}\) nilpotent, \(J\in \mathbb {R}^{n_J\times n_J}\), \(n_N+n_J=n\), and$$\begin{aligned} \left( SET,SAT\right) = \left( \begin{bmatrix} I&\quad 0 \\ 0&\quad N \end{bmatrix} , \begin{bmatrix} J&\quad 0 \\ 0&\quad I \end{bmatrix} \right) \end{aligned}$$(6)*I*an identity matrix of appropriate size. - 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.
The only solution of (5) with \(u=0\) and \(x(0)=0\) is \(x=0\).

*S*,

*T*, the Wong sequences [28] are useful:

*V*,

*W*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 (

*E*,

*A*) to QWF. They can also be used to construct the following “projectors”:

*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:

*differential*and

*impulsive parts*of

*E*,

*A*,

*B*, and

*C*, respectively.

*consistency space*, we denote the space of all consistent states of the homogeneous system:

*augmented consistency space*:

### 2.2 Distributional solutions

### Definition 4

*piecewise-smooth*iff

\({\alpha }_{\mathbb {D}}\) 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.

*a*are well-defined [28]. The space of piecewise-smooth distributions is denoted by \(\mathbb {D}_{\text { pw}\mathcal {C}^{\infty } }\).

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

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

*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

*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

*r*.

*behavior*is given by

### Theorem 6

*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

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

*switched DAE with impacts*is a system of the form

*x*,

*u*, 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 \((r,\infty )\).

*G*. For (13), a solution formula similar to the one given in Theorem 6 holds:

### Remark 8

*impact*is related to the corresponding switched ODEs. In [31], it was shown that the ODE with jumps

*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

## 4 Adjoint and dual system

### 4.1 Adjointness

*adjoint system*of (18).

*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

### Definition 9

*adjoint system*is

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

*p*, i.e., the largest behavior \(\mathcal {B}\) which is a behavioral adjoint of the above DAE is given by

- 1.
The coefficient matrix \(E_\sigma \) is inside of the derivative operator.

- 2.
The matrix-vector product is reversed in order.

*A*,

*B*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

*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

- 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*.

### 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

*time inversion of D at time T*is defined by

*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

### Proof

See Appendix B. \(\square \)

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

### Lemma 16

### Proof

See Appendix B. \(\square \)

### 4.3 Definition of the dual system

*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 \(\mathcal {B}_\sigma ^{\text {T-dual}}\).

### Remark 18

*T*]. In that case, we have

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

*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

*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 \(\forall \left( u,x,y\right) \in \mathcal {B}_{\sigma }\):$$\begin{aligned} u= 0 \wedge y_{[0,T]} = 0 \; \Rightarrow x_{(T,\infty )} = 0. \end{aligned}$$

### Example 21

*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

*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

*s*and

*t*are

*not switching times*of \(\sigma \), then

### Proof

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

*i*the Kalman matrices (here \([A/B]:=[A^\top ,B^\top ]^\top \))

*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

*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

*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

### Proof

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 \(x(t_i^+)=x_i\).

### Theorem 27

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 \(\{0\} = \widetilde{\mathcal {N}}_1^m\).

### Proof

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 (*E*, *A*) 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 \(O_i^\text { imp}\varPi _i^\text { imp}= O_i^\text { imp}\).

*E*,

*A*) 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

*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

- 1.
a multiplication of the systems Eq. (13) from the left with some \(S_\sigma \), \(S_k\) invertible, and

- 2.
the precise value of the inversion time \(T>t_m\).

*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

*S*,

*T*of the quasi-Weierstrass form (QWF, see Sect. 2.1):

### 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*].

### Proof

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 \(\widehat{H}_{i} = H_{i+1}^\top \).

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

## 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.

### References

- 1.Balla K, März R (2002) A unified approach to linear differential algebraic equations and their adjoints. Z Anal Anwend 21:783–802MathSciNetCrossRefMATHGoogle Scholar
- 2.Barabanov NE (1995) Stability of inclusions of linear type. In: American Control Conference, Proceedings of the 1995, vol 5, pp 3366–3370. doi:10.1109/ACC.1995.532231
- 3.Basile G, Marro G (1992) Controlled and conditioned invariants in linear system theory. Prentice-Hall, Englewood CliffsMATHGoogle Scholar
- 4.Berger T, Trenn S (2012) The quasi-Kronecker form for matrix pencils. SIAM J Matrix Anal Appl 33(2):336–368. doi:10.1137/110826278 MathSciNetCrossRefMATHGoogle Scholar
- 5.Berger T, Trenn S (2014) Kalman controllability decompositions for differential-algebraic systems. Syst Control Lett 71:54–61. doi:10.1016/j.sysconle.2014.06.004 MathSciNetCrossRefMATHGoogle Scholar
- 6.Berger T, Reis T, Trenn S (2016) Observability of linear differential-algebraic systems. In: Ilchmann A, Reis T (eds) Surveys in differential-algebraic equations IV, Differential-algebraic equations forum. Springer-Verlag, Berlin-Heidelberg (to appear)Google Scholar
- 7.Campbell SL (1980) Singular systems of differential equations I. Pitman, New YorkMATHGoogle Scholar
- 8.Campbell SL, Nichols NK, Terrell WJ (1991) Duality, observability, and controllability for linear time-varying descriptor systems. Circ Syst Signal Process 10(4):455–470. doi:10.1007/BF01194883 MathSciNetCrossRefMATHGoogle Scholar
- 9.Cobb JD (1984) Controllability, observability and duality in singular systems. IEEE Trans Autom Control AC 29:1076–1082. doi:10.1109/TAC.1984.1103451
- 10.Frankowska H (1990) On controllability and observability of implicit systems. Syst Control Lett 14:219–225. doi:10.1016/0167-6911(90)90016-N MathSciNetCrossRefMATHGoogle Scholar
- 11.Kalman RE (1961) On the general theory of control systems. In: Proceedings of the first international congress on automatic control, Moscow 1960. Butterworth’s, London, pp 481–493Google Scholar
- 12.Knobloch HW, Kappel F (1974) Gewöhnliche Differentialgleichungen. Teubner, StuttgartCrossRefMATHGoogle Scholar
- 13.Küsters F (2015) On duality of switched DAEs. Master’s thesis, TU KaiserslauternGoogle Scholar
- 14.Küsters F, Trenn S (2015) Duality of switched ODEs with jumps. In: Proc. 54th Conf. on Decision and Control. IEEE, Osaka, Japan, pp 4879–4884. doi:10.1109/CDC.2015.7402981
- 15.Küsters F, Ruppert MGM, Trenn S (2015) Controllability of switched differential-algebraic equations. Syst Control Lett 78:32–39. doi:10.1016/j.sysconle.2015.01.011 MathSciNetCrossRefMATHGoogle Scholar
- 16.Lamour R, März R, Tischendorf C (2013) Differential algebraic equations: a projector based analysis, differential-algebraic equations forum, vol 1. Springer-Verlag, Heidelberg-BerlinCrossRefMATHGoogle Scholar
- 17.Lawrence D (2010) Duality properties of linear impulsive systems. In: Proc. 49th IEEE Conf. Decis. Control, Atlanta, pp 6028–6033. doi:10.1109/CDC.2010.5717765
- 18.Li Z, Soh CB, Xu X (1999) Controllability and observability of impulsive hybrid dynamic systems. IMA J Math Control Inf 16(4):315–334. doi:10.1093/imamci/16.4.315 MathSciNetCrossRefMATHGoogle Scholar
- 19.Linh V, März R (2015) Adjoint pairs of differential-algebraic equations and their lyapunov exponents. J Dyn Differ Equ, pp 1–30. doi:10.1007/s10884-015-9474-6
- 20.Meng B (2006) Observability conditions of switched linear singular systems. In: Proceedings of the 25th Chinese control conference, Harbin, pp 1032–1037Google Scholar
- 21.Petreczky M, Tanwani A, Trenn S (2015) Observability of switched linear systems. In: Djemai M, Defoort M (eds) Hybrid dynamical systems, Lecture notes in control and information sciences, vol 457. Springer-Verlag, Switzerland, pp 205–240. doi:10.1007/978-3-319-10795-0_8
- 22.van der Schaft AJ (1991) Duality for linear systems: external and state space characterization of the adjoint system. In: Bonnard B, Bride B, Gauthier JP, Kupka I (eds) Analysis of controlled dynamical systems, Progress in systems and control theory, vol 8. Birkhäuser, Boston, pp 393–403. doi:10.1007/978-1-4612-3214-8_35
- 23.Schwartz L (1957, 1959) Théorie des Distributions. Hermann, ParisGoogle Scholar
- 24.Sun Z, Ge SS (2005) Switched linear systems. Communications and control engineering. Springer-Verlag, London. doi:10.1007/1-84628-131-8 CrossRefGoogle Scholar
- 25.Tanwani A, Trenn S (2010) On observability of switched differential-algebraic equations. In: Proc. 49th IEEE Conf. Decis. Control, Atlanta, pp 5656–5661. doi:10.1109/CDC.2010.5717685
- 26.Tanwani A, Trenn S (2012) Observability of switched differential-algebraic equations for general switching signals. In: Proc. 51st IEEE Conf. Decis. Control, Maui, USA, pp 2648–2653, doi:10.1109/CDC.2012.6427087
- 27.Tanwani A, Trenn S (2016) Determinability and state estimation for switched differential-algebraic equations, Automatica (under review)Google Scholar
- 28.Trenn S (2009) Distributional differential algebraic equations. PhD thesis, Institut für Mathematik, Technische Universität Ilmenau, Universitätsverlag Ilmenau, Germany. http://www.db-thueringen.de/servlets/DocumentServlet?id=13581
- 29.Trenn S (2009) Regularity of distributional differential algebraic equations. Math Control Signals Syst 21(3):229–264. doi:10.1007/s00498-009-0045-4 MathSciNetCrossRefMATHGoogle Scholar
- 30.Trenn S (2012) Switched differential algebraic equations. In: Vasca F, Iannelli L (eds) Dynamics and control of switched electronic systems—advanced perspectives for modeling, simulation and control of power converters, chap 6. Springer-Verlag, London, pp 189–216. doi:10.1007/978-1-4471-2885-4_6
- 31.Trenn S, Willems J (2012) Switched behaviors with impulses—a unifying framework. In: Proc. 51st IEEE Conf. Decis. Control, Maui, pp 3203–3208. doi:10.1109/CDC.2012.6426883
- 32.Trumpf J (2003) On the geometry and parametrization of almost invariant subspaces and observer theory. PhD thesis, Universität WürzburgGoogle Scholar