# Switched Differential Algebraic Equations

## Abstract

In this chapter, an electrical circuit with switches is modelled as a *switched differential algebraic equation* (switched DAE), i.e. each mode is described by a DAE of the form *Ex*′=*Ax*+*Bu* where *E* is, in general, a singular matrix and *u* is the input. The resulting time-variance follows from the action of the switches present in the circuit, but can also be induced by faults occurring in the circuit. In general, switches or component faults induce jumps in certain state-variables, and it is common to define additional jump-maps based on physical arguments. However, it turns out that the formulation as a switched DAE already implicitly defines these jumps, no additional jump-map must be given. In fact, an easy way to calculate these jumps will be presented in terms of the *consistency projectors*. It turns out that general switched DAEs can have not only jumps in the solutions but also Dirac impulses and/or their derivatives. In order to capture this impulsive behaviour, the space of *piecewise-smooth distributions* is used as an underlying solution space. With this underlying solution space it is possible to show existence and uniqueness of solutions of switched DAEs (including the uniqueness of the jumps induced by the switches). With the help of the consistency projector a condition is formulated whether a switch (or fault) can induce jumps or even Dirac impulses in the solutions. Furthermore, stability of the switched DAE is studied; again the consistency projectors play an important role.

## 6.1 Introduction

*switched differential algebraic equation*(switched DAE; DAEs are also known as descriptor systems or singular systems):

*σ*:ℝ→{1,…,

*p*} is the switching signal,

*E*

_{p},

*A*

_{p}∈ℝ

^{n×n},

*B*

_{p}∈ℝ

^{n×m}for \(p\in\{1,\ldots,{\bar {p}}\}\), \({\bar {p}}\in {\mathbb {N}}\) is the number of different subsystems,

*x*:ℝ→ℝ

^{n}is the state variable and

*u*:ℝ→ℝ

^{m}is the input. In other words, the system is modelled as a time-varying linear differential-algebraic equation whose coefficient matrices are piecewise-constant. The time-variance follows from the action of the switches present in the circuit, but can also be induced by faults occurring in the circuit. Hence the proposed framework can be used to study the behaviour of the circuit for nominal switching as well as in the case of sudden component faults.

For a motivation and an illustration of the notation, consider the following example.

### Example 6.1.1

(ODE vs. DAE description)

*B*=0), where

*x*=

*i*

_{L}or

*x*=(

*i*

_{L},

*v*

_{C})

^{⊤}.

*v*

_{C}does not exist in the first mode’s description at all. Furthermore, it is not clear from the ODE description alone how

*v*

_{C}is initialised after a switch. The underlying problem is that standard circuit analysis eliminates algebraic constraints (like the Kirchhoff’s law) to obtain an ODE description. However, in the presence of switches this elimination might be different for the different modes, so that the resulting ODEs are not compatible anymore. If, on the other hand, the algebraic constraints are not eliminated then this problem disappears. This approach leads to a DAE description of the form \(E\dot{x}=Ax+Bu\) for each mode; for the two modes of the switched circuit from Fig. 6.2, these are given by, with

*x*=(

*i*

_{L},

*v*

_{L},

*i*

_{C},

*v*

_{C})

^{⊤}, and Now the behaviour of the circuit can directly be modelled as a switched DAE of the form (6.1). In addition, if the switch does not change the modes instantaneously but spends some time between the two contacts then this can easily be modelled with a third mode given by

In general, switches or component faults induce jumps in certain state variables, and it is common to define additional jump-maps based on physical arguments [9]. However, it turns out that the appropriate formulation as a switched DAE already implicitly defines these jumps, no additional jump-map must be given. In fact, an easy way to calculate these jumps will be presented in terms of the *consistency projectors*.

In order to allow for jumps in the solution, the problem is embedded into a distributional solution framework. It turns out that general switched DAEs can have not only jumps in the solutions but also Dirac impulses and/or their derivatives. Actually, this is in agreement with the observation of a small experiment (see also the last part of Example 6.1.1): When connecting a coil via a switch to a constant voltage source, one can observe a spark when opening the switch, which can be explained by a voltage peak induced by the rapid drop of the current in the coil. For ideal elements this peak is, in fact, the Dirac impulse because it is the (distributional) derivative of a jump. Unfortunately, it is not possible to just take the classical distribution space as formally introduced by Schwartz in the 1950s [31] as a solution space for switched DAEs. Roughly speaking, the reason is that this space is too large. For example, it is not possible to define a restriction of a general distribution to some interval or, equivalently, to multiply a distribution with a piecewise-constant function, but these operations are needed to formulate the problem as a switched DAEs. To overcome this problem, the smaller space of *piecewise-smooth distributions* [33, 34] is considered as an underlying solution space.

With the right underlying solution space it is possible to study existence and uniqueness of solutions of switched DAEs. The latter is strongly related to the so-called *regularity* of the coefficient matrices and it will turn out that regularity of the matrix pairs is necessary and sufficient for the existence of unique solutions. A remarkable consequence of this general existence and uniqueness result of solutions of switched DAEs is the above mentioned property that the jumps induced by switches (or faults) are already determined uniquely by the switched DAE description.

With the help of the aforementioned consistency projectors it is easy to formulate conditions whether a switch (or fault) can induce jumps or even Dirac impulses in the solutions. This has important application for designing, e.g. fault tolerant systems, because if a component fault can induce Dirac impulses in the solution, this peak of voltages or currents of the circuit might destroy other components, possibly leading to a cascading total destruction of the circuit.

In the context of control theory, asymptotic stability of a switched DAE is an important property. It is well known for switched systems (not in DAE form) that switching between stable subsystems can yield an unstable overall system. This is not the case when one is able to find a common Lyapunov function. It will be shown that this result can be generalised to switched DAEs. The consistency projectors play a prominent role again.

The proposed framework also has limitations. The major drawback is that the distributional solution framework cannot be used in a general nonlinear context. In particular, state dependent switching is not covered by the presented theory. Since diodes yield state dependent switching and play an important role for power converters, extensions of the presented theory to encompass at least diodes is a topic of ongoing research.

## 6.2 Mathematical Preliminaries: Distribution Theory

Before introducing piecewise-smooth distributions in the second part of this section, classical distributions as formalised by Schwartz [31] are summarised and important properties are highlighted. A (real-valued) distribution is a continuous linear map from the space of test functions Open image in new window into the real numbers ℝ, where Open image in new window is the set of all functions *φ*:ℝ→ℝ which are smooth (i.e. arbitrarily often differentiable) and are zero outside some compact set. Continuity of a distribution is defined in terms of a certain topology on the space of test functions Open image in new window, however, this topology is rarely used, instead one works with the following characterisation of continuity.

### Lemma 6.2.1

*A linear map*Open image in new window

*is continuous if and only if*lim

_{n→∞}

*D*(

*φ*

_{n})=0

*for all sequences*Open image in new window

*with the following properties*:

- (C
_{1}) ∃

*compact**K*⊆ℝ ∀*n*∈ℕ ∀*t*∉*K*:*φ*_{n}(*t*)=0*and*- (C
_{2}) \(\forall i\in {\mathbb {N}}:\ \lim_{n\to\infty}\|\varphi^{(i)}_{n}\|_{\infty}=0\),

*where*∥⋅∥

_{∞}

*denotes the supremum norm of a*(

*bounded*)

*function*.

*f*:ℝ→ℝ, (i.e. ∫

_{K}

*f*<∞ for any compact

*K*⊆ℝ) is a “subspace” of \({\mathbb {D}}\).

### Lemma 6.2.2

*Each locally integrable*

*f*:ℝ→ℝ

*induces a distribution*\(f_{{\mathbb {D}}}\in {\mathbb {D}}\)

*given by*

*and the correspondence is one*-

*to*-

*one in the following sense*:

*Distributions induced by locally integrable functions are called*regular distributions.

A very important and useful property of distributions is that all distributions have a derivative within \({\mathbb {D}}\).

### Lemma 6.2.3

*The distributional derivative of*\(D\in {\mathbb {D}}\)

*is given by*

*and is a distribution again*.

*Furthermore*,

*for differentiable*

*f*:ℝ→ℝ

*it holds that*\((f')_{{\mathbb {D}}}= (f_{{\mathbb {D}}})'\).

*Dirac impulse*(or Dirac-Delta) given by In general, the Dirac impulse at

*t*∈ℝ is given by

*δ*

_{t}(

*φ*):=

*φ*(

*t*). The Dirac impulse is the (distributional) derivative of the Heaviside step function Open image in new window, i.e. Open image in new window.

*x*into the switched DAE (6.1) because it is not clear how the product of the piecewise-constant coefficient matrices

*E*

_{σ}and

*A*

_{σ}with \(\dot{x}\) or

*x*should be defined. If one rewrites (6.1) with the help of restrictions to intervals as

*σ*(

*t*)=

*p*

_{i}for

*t*∈[

*t*

_{i},

*t*

_{i+1}) and

*i*∈ℤ then the following remark shows that there is no suitable definition for the terms in (6.1) in a general distributional framework.

### Remark 6.2.4

(Restriction impossible, [34, Rem. 9], see also [33, Thm. 2.2.2])

*φ*(0)≠0 the infinite sum does not converge, hence the restriction is not well defined.

*δ*to the closed interval [0,∞) should be the Dirac impulse itself again, while the restriction to the open interval (0,∞) should be the zero distribution. In order to be able to define a restriction for distributions, it therefore seems reasonable to consider first the space of

*piecewise-regular distributions*

*t*and its derivatives. For a piecewise-regular distribution \(D=f_{{\mathbb {D}}}+\sum_{t\in T}D_{t}\), the restriction to some interval

*M*⊆ℝ is simply defined by

*f*

_{M}:ℝ→ℝ is the restriction of

*f*given by

*f*

_{M}(

*t*)=

*f*(

*t*) if

*t*∈

*M*and

*f*

_{M}(

*t*)=0 otherwise. Although this space is suitable for defining a restriction, it is still too big as a solution space for the switched DAE (6.1) because of the following two reasons:

- 1.
\({\mathbb {D}}_{\mathrm {pw}\mathrm {reg}}\) is not closed under differentiation, i.e. \(x\in {\mathbb {D}}_{\mathrm {pw}\mathrm {reg}}\) does not imply \(\dot{x}\in\nobreak {\mathbb {D}}_{\mathrm {pw}\mathrm {reg}}\), in particular the restriction of \(\dot{x}\) is still not well defined, and

- 2.
It is not possible to specify initial values since neither

*x*(*t*),*x*(*t*−) nor*x*(*t*+) are well defined quantities for \(x\in {\mathbb {D}}_{\mathrm {pw}\mathrm {reg}}\).

*piecewise-continuous distributions*, i.e. he considered piecewise-regular distributions \(D=f_{{\mathbb {D}}}+\sum_{t\in T}D_{t}\) with piecewise-continuous

*f*and defined

*D*(

*t*−):=

*f*(

*t*−) and

*D*(

*t*+):=

*f*(

*t*+). However, Cobb seemed to have overlooked the first problem. The following definition of the space of

*piecewise-smooth distributions*resolves this problem.

### Definition 6.2.5

(Piecewise-smooth distributions, [33, 34])

*piecewise-smooth functions*as

*piecewise-smooth distributions*is defined as

*Fuchssteiner multiplication*[10, 11, 33] which defines an associative (but not commutative) multiplication between two arbitrary piecewise-smooth distributions and which fulfils the Leibniz’s product rule. Here this multiplication will not be used in its full generality because for studying (6.1) only the product of a piecewise-smooth function with a piecewise-smooth distribution is needed. The latter is given by where \(\alpha=\sum_{i\in {\mathbb {Z}}}(\alpha_{i})_{[t_{i},t_{i+1})}\) with Open image in new window. The matrix–vector product in (6.1) is defined in the classical way, i.e. for Open image in new window and Open image in new window the product Open image in new window is a vector of sums of scalar products.

^{1}Note that the Fuchssteiner multiplication makes the standard formulation (6.1) of a switched DAE equivalent to the restriction based formulation (6.2).

### Definition 6.2.6

(Point-wise evaluation)

*t*∈ℝ and \(D=f_{{\mathbb {D}}}+\sum_{\tau\in T} D_{\tau}\) then the left/right evaluation of

*D*at

*t*is given by

*D*at

*t*is

## 6.3 Regularity of Matrix Pairs

It is assumed that each mode of the switched DAE (6.1) is described by a *regular* matrix pair (*E*,*A*). In this section, some properties and consequences of regularity are collected.

### Definition 6.3.1

(Regularity)

A matrix pair (*E*,*A*)∈ℝ^{m×n}×ℝ^{m×n}, *n*,*m*∈ℕ, is called regular if and only if *m*=*n* and the polynomial det(*sE*−*A*) is not the zero polynomial.

In the following theorem, characterisations of regularity of (*E*,*A*) are given which highlight the importance of regularity of (*E*,*A*) with respect to solvability of the corresponding DAE \(E\dot{x}=Ax+f\). Note that here “solution” refers to a classical solution, i.e. a differentiable function *x* fulfilling the DAE.

### Theorem 6.3.2

(Characterisations of regularity)

*The following statements are*equivalent

*for matrix pairs*(

*E*,

*A*)

*with square matrices*

*E*,

*A*∈ℝ

^{n×n}:

- 1.
*The matrix pair*(*E*,*A*)*is regular*. - 2.
*There exist invertible matrices**S*,*T*∈ℝ^{n×n}*such that*(*E*,*A*)*is transformed into a*quasi-Weierstrass form(6.3)*where**J**is some matrix and**N**is a nilpotent matrix*. - 3.
*For all smooth**f*:ℝ→ℝ^{n}*there exists a solution**x**of*\(E\dot{x}=Ax+f\)*and**x**is uniquely given by the value**x*(*t*_{0})*for any fixed**t*_{0}∈ℝ. - 4.
*The only solution**x**of*\(E\dot{x}=Ax\)*with**x*(0)=0*is the trivial solution*.

### Proof

1 ⇔ 2 This is a classical result going back to Weierstrass [36], for a proof see e.g. the textbook [19]. The prefix “quasi” in “quasi-Weierstrass form” is used because it is not assumed here that *J* and *N* are in Jordan canonical form [2].

*f*

_{1},

*f*

_{2}and the solutions are uniquely determined by the values

*v*(

*t*

_{0}) and

*w*(

*t*

_{0}). Classical ODE theory provides this already for

*v*. Consider the operator Open image in new window, which is invertible with inverse given by

*ν*∈ℕ is the nilpotency index of

*N*, i.e.

*ν*is the minimal value such that

*N*

^{ν}=0. Hence the unique solution of \(N\dot{w}=w+f_{2}\) is given by

3 ⇒ 4 Choosing *f*=0 in 3 implies 4.

4 ⇒ 1 This is shown in [19, Thm. 2.14]. The basic idea is to choose *n*+1 different pairs (*λ*_{i},*v*_{i})∈ℝ×ℝ^{n}∖{0}, *i*=1,…,*n*+1 such that (*Eλ*_{i}−*A*)*v*_{i}=0 and a vanishing nontrivial linear combination \(\sum_{i=1}^{n+1}\alpha_{i} v_{i} =0\) for *α*_{i}∈ℝ not all zero. Then \(x(t)=\sum_{i=1}^{n+1} \alpha_{i} v_{i} e^{\lambda_{i} t}\) is not identically zero but solves \(E\dot{x}=Ax\), *x*(0)=0. □

Since the quasi-Weierstrass form (6.3) will play an important role in the following, a convenient method to obtain the transformation matrices *S*,*T* is presented with the following theorem which utilises the *Wong sequences* (named after [37]).

### Theorem 6.3.3

(Wong sequences and the quasi-Weierstrass form [1, 2])

*Let*(

*E*,

*A*)

*be a regular matrix pair*.

*Define the*Wong sequences

*of subspaces by*,

*for*

*i*∈ℕ,

*where*Open image in new window

*denotes the pre*-

*image of the set*Open image in new window

*under the matrix*

*M*∈ℝ

^{n×n}

*and*Open image in new window

*denotes the image of*Open image in new window

*under*

*M*.

*These sequences get stationary after finitely many steps*;

*denote the limits with*Open image in new window

*and*Open image in new window.

*Choose full rank matrices*

*V*,

*W*

*such that*Open image in new window

*and*Open image in new window

*then*

*are invertible and put*(

*E*,

*A*)

*into a quasi*-

*Weierstrass form*(6.3).

## 6.4 Explicit Solution Formula for Non-switched DAE

*Drazin inverses*of

*E*and

*A*. The Drazin inverse approach only works when

*E*and

*A*commute, i.e.

*EA*=

*AE*must hold. This is in general not true, but by multiplying both matrices with (

*Eλ*

_{0}−

*A*)

^{−1}for some

*λ*

_{0}with det(

*Eλ*

_{0}−

*A*)≠0 this assumption always holds [3].

### Definition 6.4.1

(Consistency, differential and impulse projector, [32])

*E*,

*A*) and its quasi-Weierstrass form (6.3) obtained by corresponding transformation matrices

*S*,

*T*∈ℝ

^{n×n}. Let the block sizes in the quasi-Weierstrass form be

*n*

_{1}×

*n*

_{1}and

*n*

_{2}×

*n*

_{2}. The

*consistency projector*is given by the

*differential projector*is given by and the

*impulse projector*is given by where

*I*is the identity matrix of size

*n*

_{1}×

*n*

_{1}for the consistency and differential projector and of size

*n*

_{2}×

*n*

_{2}for the impulse projector.

Note that, in contrast to the consistency projector, the differential and impulse projectors are not projectors in the usual sense because they are in general not idempotent matrices. Furthermore, it is not difficult to see that the definition of the projectors does not depend on the specific choices of the transformations *S* and *T*.

### Definition 6.4.2

(Drazin inverse)

*M*∈ℝ

^{n×n}. Any matrix

*M*

^{D}∈ℝ

^{n×n}is called

*Drazin inverse*of

*M*if

- 1.
*MM*^{D}=*M*^{D}*M*, - 2.
*M*^{D}*MM*^{D}=*M*^{D}, and - 3.
∃

*ν*∈ℕ:*M*^{D}*M*^{ν+1}=*M*^{ν}.

### Lemma 6.4.3

*For all*

*M*∈ℝ

^{n×n}

*there exists a unique Drazin inverse*

*M*

^{D}.

*Furthermore*,

*if*Open image in new window,

*where*

*T*∈ℝ

^{n×n}

*is invertible*, \(C\in\nobreak {\mathbb {R}}^{n_{1}\times n_{1}}\),

*n*

_{1}∈ℕ,

*is an invertible matrix and*\(N\in {\mathbb {R}}^{n_{2}\times n_{2}}\),

*n*

_{2}=

*n*−

*n*

_{1},

*is nilpotent*,

*then*

The solution of the DAE (6.4) based on the above defined projectors as well as on the Drazin inverse is now given in the following theorem.

### Theorem 6.4.4

(Explicit solution formula)

*Let*(

*E*,

*A*)

*be a regular matrix pair and let*

*Π*

_{(E,A)}, \(\varPi ^{\mathrm {diff}}_{(E,A)}\)

*and*\(\varPi ^{\mathrm {imp}}_{(E,A)}\)

*be the consistency*,

*differential and impulse projector*,

*resp*.,

*as in Definition*6.4.1.

*Furthermore*,

*let*

*Then all solutions of*(6.4)

*are given by*,

*for*

*c*∈ℝ

^{n},

*If*,

*in addition*,

*EA*=

*AE*

*then also*

### Proof

*S*,

*T*∈ℝ

^{n×n}yields that

*x*solves \(E\dot{x}=Ax+f\) if and only if Open image in new window solves \(\dot{v}=Jv+[I\ 0]Sf\) and \(N\dot{w}=w+[0\ I]Sf\), hence, together with the step 2 ⇒ 3 in the proof of Theorem 6.3.2, Together with and the first proposed solution formula is obtained. The second solution formula is standard and can be found in [3, 19]. □

### Remark 6.4.5

(Remarks on the solution formulas)

- 1.
If

*EA*=*AE*then both solution formulas are almost identically because then it can be shown, see [2], that \(E^{D}=\varPi _{(E,A)}^{\mathrm {diff}}\) and*E*^{D}*E*=*Π*_{(E,A)}. However, it is in general*not true*that \(A^{D}=\varPi _{(E,A)}^{\mathrm {imp}}\) or*EA*^{D}=*E*^{imp}, therefore the second formula needs the “correction term”*I*−*E*^{D}*E*. - 2.From the solution formula (6.5) it follows thatin particular, the initial value problem (6.4),$$x(0) = \varPi _{(E,A)} c - \sum_{i=0}^{n-1} \bigl(E^\mathrm {imp}\bigr)^i\varPi _{(E,A)}^\mathrm {imp}f^{(i)}(0), $$
*x*(0)=*x*_{0}∈ℝ^{n}has a solution if and only if which characterises consistency of the initial value. - 3.
If

*x*is allowed to have jumps or even Dirac impulses, i.e. Open image in new window, then Theorem 6.4.4 is still true, i.e. considering distributional solutions doesn’t add any new solutions to the problem (for which the DAE \(E\dot{x}=Ax+f\) should hold globally). In particular, also all distributional solutions have consistent initial values. Furthermore, if*f*contains jumps or Dirac impulses, i.e. Open image in new window then the solution formula also holds; however, it is necessary to define the notion of the antiderivative*H*=∫_{0}*D*of the distribution*D*which fulfils*H*′=*D*and*H*(0−)=0, see [34, Prop. 3]. - 4.
The proof of Theorem 6.4.4 reveals that, instead of the consistency projector

*Π*_{(E,A)}in formula (6.5), any matrix*M*with Open image in new window could be used. However, the special choice of*Π*_{(E,A)}yields that this formula holds also true when an inconsistent initial value is given, see the next section.

## 6.5 Inconsistent Initial Values

In the presence of switches, the initial conditions need not to be consistent so that no solution (classical or distributional) with this initial value exists. It is therefore necessary to make precise what a “solution” to an inconsistent initial value problem should be. The viewpoint here is the following.

*t*

_{0}=0). This gives rise to the following

*initial trajectory problem*(

*ITP*), where

*x*

^{0}:(−∞,0)→ℝ

^{n}is some initial trajectory:

If *x*^{0}(0) is not consistent for \(E\dot{x}=Ax+f\) then no classical solution exists; however, it will be shown in the following that a distributional solution exists. Therefore, (6.6) is considered as an equation of piecewise-smooth distributions, in particular, the inhomogeneity *f* and the initial trajectory are also piecewise-smooth distributions.

### Theorem 6.5.1

(Solvability of the ITP)

*Let*(

*E*,

*A*)

*be a regular matrix pair*.

*Then for every initial trajectory*Open image in new window

*and any inhomogeneity*Open image in new window

*the ITP*(6.6)

*has a unique solution*Open image in new window.

*In particular*,

*the jump from*

*x*

^{0}(0−)

*to*

*x*(0+)

*and the impulsive part*

*x*[0]

*is uniquely determined*.

*In fact*,

*if*

*f*[0]=0

*then*

*where*

*Π*

*is the consistency projector and*

*E*

^{imp}=

*Π*

^{imp}

*E*

*with impulse projector*

*Π*

^{imp}

*as in Definition*6.4.1.

*In particular*,

*if*

*f*=0

*then*

*and*,

*on the open interval*(0,∞),

*Furthermore*,

*if*

*f*

*is smooth then the solution*

*x*

*restricted to the open interval*(0,∞)

*is induced by the smooth function given by*(6.5)

*where*

*c*=

*x*

^{0}(0−).

### Proof

*S*,

*T*∈ℝ

^{n×n}invertible such that (

*E*,

*A*) is put into a quasi-Weierstrass form (6.3) with block sizes

*n*

_{1}×

*n*

_{1}and

*n*

_{2}×

*n*

_{2}, resp., and let Open image in new window. In the new coordinates, the ITP (6.6) then decouples into

*v*just yields the same solutions on [0,∞) as the classical solution of the ODE initial value problem \(\dot{v}=Jv+f\),

*v*(0)=

*v*

^{0}(0−). In particular,

*v*(0+)=

*v*(0−) and

*v*[0]=0. As shown in [33, Thm. 3.1.7], the DAE for

*w*can equivalently be written as a special “switched” DAE without initial trajectory

*N*

_{itp}:=

*N*

_{[0,∞)}and \(f_{\mathrm {itp}}=-{w^{0}}_{(-\infty,0)}+ \begin{bmatrix}0 & I \end{bmatrix} Sf_{[0,\infty)}\). Since the operator Open image in new window is still nilpotent, the solution formula derived in the proof of Theorem 6.3.2 still holds, i.e.

*w*is uniquely given by

*x*of the ITP (6.6) and Open image in new window. Some calculations within the piecewise-smooth distributional framework yield

*x*(0+) and

*x*[0]. □

Viewing now the switched DAE (6.1) as a repeated ITP (where the switching times are the initial times) one obtains the following result. Note that it has to be assumed that the switching times do not accumulate, otherwise this approach does not work.

### Corollary 6.5.2

(Existence and uniqueness of solutions of a switched DAE)

*Consider the switched DAE*(6.1)

*with regular matrix pairs*(

*E*

_{p},

*A*

_{p}), \(p\in\{1,\ldots,{\bar {p}}\}\)

*and assume for the switching signal*

*σ*

*Then there exists a globally defined solution*Open image in new window

*which is uniquely given by*

*x*(0−).

### Remark 6.5.3

(On the assumption that *σ*_{(−∞,0)} is constant)

*σ*in (6.1) is constant on (−∞,0) is just a technicality to ensure uniqueness of the solution also backward in time. If one is interested only in the behaviour of the solution on [0,∞) then this assumption is not necessary to obtain the same result. More formally, one could also just study the corresponding ITP with some initial trajectory Open image in new window then it doesn’t matter how

*σ*is defined on (−∞,0).

### Example 6.5.4

(Example 6.1.1 continued)

*E*

_{1},

*A*

_{1},

*B*

_{1}), (

*E*

_{2},

*A*

_{2},

*B*

_{2}), (

*E*

_{3},

*A*

_{3},

*B*

_{3}) as given at the end of Example 6.1.1, i.e. (

*E*

_{1},

*A*

_{1},

*B*

_{1}) correspond to the switch in the left position, (

*E*

_{2},

*A*

_{2},

*B*

_{2}) correspond to the switch in the right position and (

*E*

_{3},

*A*

_{3},

*B*

_{3}) describes the system when the switch is in between. To check whether the switched DAE is uniquely solvable for all input signals

*u*, regularity of the matrix pairs (

*E*

_{1},

*A*

_{1}), (

*E*

_{2},

*A*

_{2}), (

*E*

_{3},

*A*

_{3}) must be checked:

*σ*∈

*Σ*

_{0}and for every input Open image in new window and this solution is uniquely determined by

*x*(0−). To characterise the jumps at the switching instances the Wong sequences are calculated first. The corresponding limit spaces are given by and the consistency, differential and impulse projectors are given by With the help of these projectors the re-initialisation of

*x*(0+) for a given (possibly inconsistent) initial value \(x(0-)=:(i_{L}^{0},u_{L}^{0},i_{C}^{0},u_{C}^{0})\) and the corresponding impulse in the solution can be calculated for each mode by the formula given in Theorem 6.5.1

## 6.6 Stability

*u*can be rewritten as the state equation \(\dot{u}=0\) and a sinusoidal input

*u*can be rewritten as \(\dot{u}=-\omega\overline{u}\), \(\dot{\overline{u}}=\omega u\), for some

*ω*∈ℝ. Hence the switched DAE (6.1) with constant or sinusoidal inputs can be written as the DAE (6.7). Throughout this section, it is assumed that the switching signal

*σ*is such that the switching times do not accumulate.

### Definition 6.6.1

(Asymptotic stability)

- (S)
∀

*ε*>0 ∃*δ*>0: ∥*x*(0−)∥<*δ*⇒∀*t*>0: ∥*x*(*t*±)∥<*ε*, - (A)
*x*(*t*±)→0 as*t*→∞, - (I)
∀

*t*≥0:*x*[*t*]=0.

The assumptions (S) (stability) and (A) (attractivity) are standard for the definition of asymptotic stability. The assumption (I) (impulse-freeness) is motivated by the following observation: Assume (6.7) has a solution *x* with *x*[*t*]≠0 for some *t*. By linearity of (6.7), the scaling of *x* by some *ε*>0 will yield also a solution, in particular, the initial value *εx*(0−) can be made arbitrarily small, but the corresponding impulse *εx*[*t*] does not vanish. Since an impulse can be interpreted as an infinite peak (or the limit of an unbounded sequence of functions) this shows that the classical stability assumption (S) cannot be fulfilled in the sense that small initial values yield a bounded and small solution. The problem whether the switched system (6.7) has impulses in the solution or not, i.e. whether (I) holds or not, is also an interesting question on its own. Consider, for example, the situation where the switched DAE (6.7) models a nominal system with additional sudden faults (like a shortcut in an circuit element). If the corresponding switch is able to produce a Dirac impulse in some of the variables (e.g. voltages and currents) then this impulse might destroy other components of the system, possibly leading to a cascading destruction of the system. Therefore, the topic of impulse-freeness is studied first.

### 6.6.1 Impulse Detection

### Theorem 6.6.2

(Impulse-freeness)

*Consider the switched DAE*(6.7)

*with regular matrix pairs*(

*E*

_{p},

*A*

_{p}), \(p\in\{1,\ldots,{\bar {p}}\}\)

*and a switching from mode*\(p\in\{1,\ldots,{\bar {p}}\}\)

*to mode*\(q\in\{1,\ldots,{\bar {p}}\}\)

*at some switching time*

*t*

_{s}∈ℝ,

*i*.

*e*.

*σ*(

*t*

_{s}−)=

*p*

*and*

*σ*(

*t*

_{s}+)=

*q*.

*Then this switch cannot produce an impulse*,

*i*.

*e*.

*x*[

*t*

_{s}]=0

*for all solutions*Open image in new window

*of*(6.7)

*if the following impulse*-

*freeness*-

*condition holds*:

*where*\(\varPi _{p}:=\varPi _{(E_{p},A_{p})}\), \(\varPi _{q}:=\varPi _{(E_{q},A_{q})}\)

*are the consistency projectors as in Definition*6.4.1.

*In particular*,

*if*(6.8)

*holds for all*\(p,q\in\{1,\ldots,{\bar {p}}\}\)

*then all solutions of*(6.7)

*are impulse*-

*free independently of the switching signal*.

### Proof

*σ*such that

*σ*(

*t*

_{s}−)=

*p*and

*σ*(

*t*

_{s}+)=

*p*. Since Open image in new window, where Open image in new window is the limit of the first Wong sequence corresponding to (

*E*

_{p},

*A*

_{p}), it follows that

*x*(

*t*

_{s}−)=

*Π*

_{p}

*x*(

*t*

_{s}−). The impulse formula from Theorem 6.5.1 now yields □

### Example 6.6.3

(Example 6.1.1 continued)

*u*is constant. As mentioned above, the input is reinterpreted as a state variable given by \(\dot{u}=0\), hence one obtains the following three DAEs with

*x*=(

*u*,

*i*

_{L},

*v*

_{L},

*i*

_{C},

*v*

_{C})

^{⊤}describing the three different modes: It is easily seen that the corresponding consistency projectors

*Π*

_{p},

*p*=1,2,3, are given by where \(\overline{\varPi }_{p}\) is the consistency projector from Example 6.5.4. Hence, the condition (6.8) can easily be checked for each mode change: This is in correspondence to the results obtained in Example 6.5.4 where it was apparent that only a switch to mode three can produce impulses in the solution. However, the method based on Theorem 6.6.2 is simpler (for example, the transformation matrices

*S*

_{p}, \(p\in\{1,\ldots,{\bar {p}}\}\) are not needed) and it also takes into account which mode was active before the switch. For the considered circuit this doesn’t make a difference, but in general the mode before the switch might prohibit certain initial values so that the switch does not produce impulses although it might produce switches for a general initial value. Actually, this is the case when (6.8) holds but

*E*

_{q}(

*I*−

*Π*

_{q})≠0. For a more complex example, see also [7].

### 6.6.2 Lyapunov Functions for Non-switched DAEs

*E*,

*A*)∈ℝ

^{n×n}×ℝ

^{n×n}are studied first. From the quasi-Weierstrass form (6.3) of the pair (

*E*,

*A*) it is immediately clear that (6.9) is asymptotically stable if and only if the matrix

*J*in (6.3) is Hurwitz. A direct method (i.e. without calculating the full quasi-Weierstrass form first) is given by the following result.

### Theorem 6.6.4

(Generalised Lyapunov equation, [27])^{2}

*The DAE*(6.9)

*is asymptotically stable if and only if there exists a positive definite symmetric*

*P*=

*P*

^{⊤}∈ℝ

^{n×n}

*and a symmetric*

*Q*=

*Q*

^{⊤}∈ℝ

^{n×n}

*which is positive definite on*Open image in new window (

*the limit of the first Wong sequence*)

*such that*

*Lyapunov function*as

### 6.6.3 Asymptotic Stability of Switched DAEs

*σ*(

*t*)=

*p*for all

*t*∈ℝ). Hence for studying asymptotic stability for arbitrary switching signals or at least for slow switching it has to be assumed that each mode is asymptotically stable and hence permits a Lyapunov function (6.11). From the theory of switched ODEs (see e.g. [20]), it is known that switching between stable modes can lead to an unstable overall system, hence one might expect a similar behaviour for switched DAEs. This unstable behaviour cannot happen in the switched ODE case when there exists a common Lyapunov function. Surprisingly, this condition is not sufficient any more for the switched DAE case because the possible jumps must be incorporated in the right way, too. The following example shows that the induced jumps are important for the stability (for more examples, see [21]).

### Example 6.6.5

(Unstable switched DAE with the same Lyapunov function for each subsystem)

In general, the jumps must be “compatible” with the Lyapunov function. Furthermore, impulse-freeness must be ensured additionally. Altogether, the following result holds.

### Theorem 6.6.6

(Asymptotic stability under arbitrary switching)

*Consider the switched DAE*(6.7)

*with regular matrix pairs*(

*E*

_{p},

*A*

_{p}), \(p\in\{1,\ldots,{\bar {p}}\}\).

*Let*\(\varPi _{p}:=\varPi _{(E_{p},A_{p})}\)

*be the consistency projector as given by Definition*6.4.1.

*If*,

*for all*\(p\in\{1,\ldots,{\bar {p}}\}\),

- (V
_{p}) \(E_{p}\dot{x}=A_{p} x\)

*is asymptotically stable with Lyapunov function**V*_{p},- (IC)
*E*_{q}(*I*−*Π*_{q})*Π*_{p}=0*for all*\(q\in\{1,\ldots,{\bar {p}}\}\)*and*- (JC)
*V*_{q}(*Π*_{q}*x*)≤*V*_{p}(*x*)*for all*Open image in new window

*then the switched DAE*(6.7)

*is asymptotically stable for all switching signals*.

### Proof

*Π*

_{p}

*x*=

*x*=

*Π*

_{q}

*x*, and therefore

*V*

_{p}(

*x*)=

*V*

_{q}(

*x*) for all Open image in new window. It then follows that

*V*is decreasing along solutions and

*V*(

*x*(

*t*))→0 as

*t*→∞. Finally, positive definiteness of

*V*on each

*V*

_{p}implies

*x*(

*t*)→0 as

*t*→∞. □

As highlighted in the proof, the jump condition (JC) is a generalisation of the common Lyapunov condition for switched ODEs, in fact, if this condition is applied to a switched ODE it reads as *V*_{p}(*x*)≤*V*_{q}(*x*) for all *p*,*q* hence *V*_{p}=*V*_{q}, i.e. it is equivalent to the existence of a common Lyapunov function.

*N*

_{σ}(

*t*

_{1},

*t*

_{2}) the number of switches of

*σ*within the interval [

*t*

_{1},

*t*

_{2}). The class of switching signals with average dwell time

*τ*

_{a}>0 is then given by

*N*

_{0}>0 in the definition of \(\varSigma _{\tau_{a}}\) is the so-called

*chatter bound*and is an upper bound for the number of switches within an interval of length smaller than

*τ*

_{a}. For each switching signal \(\sigma\in \varSigma _{\tau_{a}}\), this chatter bound is finite but it is not bounded for the whole class \(\varSigma _{\tau_{a}}\). The class of switching signals \(\sigma\in \varSigma _{\tau_{a}}\) with chatter bound

*N*

_{0}=1 is exactly the class of switching signals with dwell time

*τ*

_{d}=

*τ*

_{a}.

### Theorem 6.6.7

(Stability under slow switching, [22])

*Consider the switched DAE*(6.7)

*satisfying conditions*(V

_{p})

*and the impulse condition*(IC).

*Then there exists*

*τ*

_{a}>0

*such that*(6.7)

*is asymptotically stable if*\(\sigma\in \varSigma _{\tau_{a}}\).

*In fact*,

*let*

*P*

_{p},

*Q*

_{p}

*be the solutions of the Lyapunov equation*(6.10)

*corresponding to*(

*E*

_{p},

*A*

_{p}),

*let*

*O*

_{p}

*be an orthogonal basis matrix of*Open image in new window

*and let*

*where*

*λ*

_{min}(⋅)

*and*

*λ*

_{max}(⋅)

*denote the minimal and maximal eigenvalue of a symmetric matrix*,

*respectively*.

*Then an average dwell time of*

*guarantees asymptotic stability of*(6.7).

### Example 6.6.8

(Example 6.6.5 continued)

_{p}) holds. Furthermore, it is not difficult to check that the impulse-freeness condition (IC) holds. Since the jump condition (JC) does not hold, Theorem 6.6.6 is not applicable, but Theorem 6.6.7 yields that for switching signals with sufficiently large average dwell time asymptotic stability is guaranteed. In order to calculate a sufficient average dwell time, the Lyapunov functions must be rewritten as

*V*(

*x*)=(

*Ex*)

^{⊤}

*PEx*, where

*P*solves the generalised Lyapunov equation (6.10). This is achieved by choosing and The associated Lyapunov functions are \(V_{1}(x)=2x_{2}^{2}\) and

*V*

_{2}(

*x*)=(

*x*

_{1}+

*x*

_{2})

^{2}which coincide with \(V(x)=x_{1}^{2}+x_{2}^{2}\) on the corresponding consistency spaces. As orthonormal basis of Open image in new window and Open image in new window choose

*μ*:=max

_{p,q}

*μ*

_{p,q}=2 and

*λ*:=min

_{p}

*λ*

_{p}=2. Therefore, the corresponding switched DAE is asymptotically stable for all switching signals

*σ*∈

*Σ*

_{τ}with

*τ*

_{a}>ln2/2. For this case, the bound is actually sharp because it is easily seen that for a periodic switching signal with periodicity ln2/2 the corresponding solution is also periodic (cf. [21]).

### 6.6.4 Commutativity and Asymptotic Stability

For switched ODEs with asymptotically stable subsystems, it is well known [26] that commuting *A*-matrices imply asymptotic stability for arbitrary switching signal and also guarantee a common quadratic Lyapunov function. The aim of this section is to generalise this result to switched DAEs (6.1). Example 6.6.5 shows that commutativity of the *A*-matrix is not the right condition to guarantee asymptotic stability for arbitrary switching. In particular, for switched DAEs instability can be induced by the jumps, hence one would expect that the consistency projectors play a prominent role again. Surprisingly, this is not the case as the following result shows.

### Theorem 6.6.9

(Commutativity of *A*^{diff}-matrices implies stability, [23])

*Consider a switched DAE*(6.7)

*with corresponding matrices*\(A^{\mathrm {diff}}_{p}\)

*as defined in Theorem*6.4.4.

*Assume asymptotic stability of each subsystem*,

*i*.

*e*. (V

_{p}),

*and impulse*-

*freeness*,

*i*.

*e*. (IC),

*then*

*implies asymptotic stability of the switched DAE*(6.7)

*for arbitrary switching signals*.

### Proof

## 6.7 Bibliographical Notes

The theory of DAEs gained popularity in the 1980s, see e.g. the textbooks [3, 4, 6, 16], and was already discussed by Gantmacher in the 1950s [12]. That DAEs are especially suitable to model electrical circuits was highlighted by Verghese et al. [35]. There impulsive behaviour was also discussed and motivated; however, the underlying distributional solution space was not formalised which is also true for the above mentioned textbooks. Later on, the focus of research was on numerics of DAEs and (smoothly) time-varying DAEs, see e.g. the textbook [19] and the references therein.

The existence of impulsive solutions was a reoccurring topic in the literature, see e.g. [5, 30]; for a more detailed overview, see the introduction of [34]. Apart from the author’s own work (presented in this chapter), there is not much work on switched DAEs available [14, 15, 24, 25, 38] and none of these works resolve or even discuss the problem of multiplying a piecewise-constant function with a distribution which naturally occurs when studying switched differential algebraic equations of the form (6.1). Another approach to model electrical circuits with switches uses the complementarity framework and is discussed in the next chapter. Combining both approaches seems fruitful and is a topic of ongoing research.

## 6.8 Summary

In this chapter, it was motivated that switched DAEs are a suitable modelling framework for electrical circuits with switches. The space of piecewise-smooth distributions was introduced as an underlying solution space. Explicit solution formulas were given in particular for the impulsive parts induced by the inconsistent initial values. A simple matrix condition was given which makes it possible to exclude impulsive solutions for arbitrary switching. Finally, stability of switched DAEs was studied. With the help of Lyapunov functions, a sufficient condition for asymptotic stability under arbitrary and slow switching was presented. The theoretical results were illustrated with an examples stemming from a simple electrical circuit.

## Footnotes

- 1.
This is actually a consequence from the standard identification of Open image in new window with the corresponding functional Open image in new window given by

*x*(*φ*)=(*x*_{1}(*φ*),…,*x*_{n}(*φ*))^{⊤}. Some authors [19, 28] identify Open image in new window with Open image in new window and \(x(\varphi):=\sum_{i=1}^{n} x_{i}(\varphi_{i})\). This different viewpoint of the “vector”*x*makes it necessary to define the matrix–vector product differently; however, these authors do not give a motivation for this different viewpoint. - 2.
In [27], only the complex-valued case is studied. However, in the real-valued case it is easily seen that the real part of the complex solutions

*P*and*Q*are also solutions of the Lyapunov equation.

## Notes

### Acknowledgements

This work was supported by the DFG grant Wi1458/10-1. Many thanks to Roman Geiselhart for giving valuable comments on the manuscript of this book chapter.

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