On an invariance principle for differential-algebraic equations with jumps and its application to switched differential-algebraic equations

Pablo Ñañez; Ricardo G. Sanfelice; Nicanor Quijano

doi:10.1007/s00498-016-0185-2

Mathematics of Control, Signals, and Systems

March 2017, 29:5 | Cite as

On an invariance principle for differential-algebraic equations with jumps and its application to switched differential-algebraic equations

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Authors and affiliations

Pablo ÑañezEmail author
Ricardo G. Sanfelice
Nicanor Quijano

Original Article

First Online: 20 January 2017

Received: 18 September 2015

Accepted: 19 December 2016

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1 Citations

Abstract

We investigate the invariance properties of a class of switched systems where the value of a switching signal determines the current mode of operation (among a finite number of them) and, for each fixed mode, its dynamics are described by a Differential-Algebraic Equation (DAE). Motivated by the lack of invariance principles of switched DAE systems, we develop such principles for switched DAE systems under arbitrary and dwell-time switching. By obtaining a hybrid system model that describes the switched DAE system, we build from invariance results, for hybrid systems, the invariance principles for such switched systems. Examples are included to illustrate the results.

Keywords

Differential-algebraic equations Switched systems Hybrid systems Invariance principle Descriptor systems Singular systems

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Appendix

A.1 Supplementary definitions

Definition 9

(Wong sequences ⁸) Consider a regular matrix pair $(E_{{\sigma }},A_{{\sigma }})$ where ${\sigma }\in \varSigma $. The associated Wong sequences are defined by

$$\begin{aligned} \begin{array}{ll} \nu _0 := {\mathbb {R}}^n,\quad \nu _{i+1}:= A_{{\sigma }}^{-1}(E_{{\sigma }}\nu _i),\quad \nu ^*_{{\sigma }}:= \bigcap _{i\in {\mathbb {N}}}\nu _i,\ \forall i\in {\mathbb {N}}\\ \omega _0 := \{0\},\quad \omega _{i+1}:= E_{{\sigma }}^{-1}(A_{{\sigma }}\omega _i),\quad \omega ^*_{{\sigma }}:= \bigcup _{i\in {\mathbb {N}}}\omega _i,\ \forall i\in {\mathbb {N}} \end{array} \end{aligned}$$

which are nested subspaces and get stationary after exactly $\varsigma _{{\sigma }}$ steps, namely, $\nu _{i+1}\subseteq \nu _i,\ \omega _{i+1}\supseteq \omega _i$ for each $i\in {\mathbb {N}}$ and $\nu _{i+1}=\nu _{i}=\nu ^*_{{\sigma }},\ \omega _{i+1}=\omega _{i}=\omega ^*_{{\sigma }}$ for all $i\ge \varsigma _{{\sigma }}$, where the integer $\varsigma _{{\sigma }}$ is called the index of the ${\sigma }$-th DAE.

Theorem 4

(The quasi-Weierstrass form⁹) Suppose that $(E_{{\sigma }},A_{{\sigma }})$ is regular, where ${\sigma }\in \varSigma $. Then, there exist matrices $S_{{\sigma }}$ and $T_{{\sigma }}$ that transform $(E_{{\sigma }},A_{{\sigma }})$ into the quasi-Weierstrass form

$$\begin{aligned} (S_{{\sigma }}E_{{\sigma }}T_{{\sigma }},S_{{\sigma }}A_{{\sigma }}T_{{\sigma }})&= \left( \begin{bmatrix}I_{n_1^{{\sigma }}}&\quad 0\\ 0&\quad N_{{\sigma }} \end{bmatrix}, \begin{bmatrix}J_{{\sigma }}&\quad 0\\ 0&\quad I_{n_2^{{\sigma }}} \end{bmatrix}\right) \end{aligned}$$

for some $J_{{\sigma }} \in {\mathbb {R}}^{n_1^{{\sigma }}\times n_1^{{\sigma }}}$ and a nilpotent matrix $N_{{\sigma }} \in {\mathbb {R}}^{n_2^{{\sigma }}\times n_2^{{\sigma }}}$, where $(N_{{\sigma }})^{n_2^{{\sigma }}}=0$, $n_1^{{\sigma }}+n_2^{{\sigma }}=n$, $\varsigma _{{\sigma }}\in {\mathbb {N}}$ is the smallest number, such that $(N_{{\sigma }})^{\varsigma _{{\sigma }}} = 0$. The identity matrices $I_{n_1^{{\sigma }}}$, $I_{n_2^{{\sigma }}}$, and zero matrices have the proper dimensions.

A.2 Details of the proof of Lemma 1

Proof

Given the two representations $\hat{\mathcal {H}}_{\mathrm{DAE}}$ and $\mathcal {H}_{\mathrm{DAE}}$, $\mathcal {S}_{\hat{\mathcal {H}}_{\mathrm{DAE}}}$ and $\mathcal {S}_{\mathcal {H}_{\mathrm{DAE}}}$ are the set of maximal solutions (according to [8, Definition 2.7]) to $\hat{\mathcal {H}}_{\mathrm{DAE}}$ and $\mathcal {H}_{\mathrm{DAE}}$, respectively. For all $\phi \ \in \ \mathcal {S}_{\mathcal {H}_{\mathrm{DAE}}}$ (for all consistent and inconsistent initial conditions) we have:

(i)
Consistent initial value: following Definition 5, given a consistent initial condition $\xi _0\in {\mathfrak {O}}_{\sigma _0}$ the solution $\phi _\xi $ to the DAE $E_{\sigma _0} \dot{\phi }_\xi (t) = A_{\sigma _0}\phi _\xi (t)$ is such that $t\mapsto \phi _\xi (t)\in {\mathfrak {O}}_{\sigma _0}$. In addition, consider the following cases of consistent initial conditions to (8) and (30):
1. (a)
  
  $x^* = (\xi _{t^*},\chi _{t^*},\sigma _{t^*})\in ({\mathfrak {O}}_{\sigma _{t^*}}\times {\mathbb {R}}^m\times \{\sigma _{t^*}\})\cap C_{\sigma _{t^*}}$: Notice that given the construction of F in (8) and (30), for the time interval $[t^*,t^*+\tau )\times \{j^*\}$ and $\tau >0$, the solutions $\phi : [t^*,t^*+\tau )\times \{j^*\}\rightarrow {\mathbb {R}}^{n+m+1}$ and $\hat{\phi }: [t^*,t^*+\tau )\times \{j^*\}\rightarrow {\mathbb {R}}^{n+m+1}$ are such that $\phi (t,j)=\hat{\phi }(t,j)\in ({\mathfrak {O}}_{\phi _\sigma (t,j)}\times {\mathbb {R}}^m\times \{\phi _\sigma (t,j)\})$ for all $(t,j)\in [t^*,t^*+\tau )\times \{j^*\}$.
2. (b)
  
  $x^* = (\xi _{t^*},\chi _{t^*},\sigma _{t^*})\in ({\mathfrak {O}}_{\sigma _{t^*}}\times {\mathbb {R}}^m\times \{\sigma _{t^*}\})\cap D_{\sigma _{t^*}}$: Given the construction of G and $\hat{G}$ in (8) and (30), respectively,
  $$\begin{aligned} G(x^*)=\hat{G}(x^*)\subset \bigcup _{\tilde{\sigma }\in \varphi (\xi _{t^*},\chi _{t^*},\sigma _{t^*})}({\mathfrak {O}}_{\tilde{\sigma }}\times {\mathbb {R}}^m\times \{\tilde{\sigma }\}). \end{aligned}$$
  Notice that conditions (a) and/or (b) are fulfilled for all (t, j) in $\mathrm{dom}\,{\phi }$; therefore, from (a) and (b) we conclude that given an initial condition $\phi (0,0)=\hat{\phi }(0,0)=(\xi _0,\chi _0,\sigma _0)\in ({\mathfrak {O}}_{\sigma _0}\times {\mathbb {R}}^m\times \{\sigma _0\})\cap \left( C_{\sigma _0}\cup D_{\sigma _0}\right) $, we have that $\phi (t,j),\hat{\phi }(t,j)\in {{\mathrm{\cup }}}_{\tilde{\sigma }\in \varSigma }\big (({\mathfrak {O}}_{\tilde{\sigma }}\times {\mathbb {R}}^m\times \{\tilde{\sigma }\})\cap (C_{\tilde{\sigma }}\cup D_{\tilde{\sigma }})\big )$ and $\phi (t,j)\equiv \hat{\phi }(t,j)$ for all $(t,j)\in \mathrm{dom}\,{\phi }=\mathrm{dom}\,{\hat{\phi }}$, namely, at flows and jumps the solutions $\phi $ and $\hat{\phi }$ match and $\hat{\phi }\ \in \ \mathcal {S}_{\hat{\mathcal {H}}_{\mathrm{DAE}}}$.
(ii)

Inconsistent initial value: for all nontrivial solutions $\phi $ to $\mathcal {H}_{\mathrm{DAE}}$, if $\phi (0,0)=(\xi _0,\chi _0,\sigma _0)\in ({\mathbb {R}}^n{\setminus }{\mathfrak {O}}_{\sigma _0})\times {\mathbb {R}}^m\times \{\sigma _0\}$ and using the definition of G in (8), we have
$$\begin{aligned} \phi (0,1) \in G(\xi _0,\chi _0,\sigma _0)&= \bigcup _{\tilde{\sigma }\in \varphi (\xi _0,\chi _0,\sigma _0)} \begin{bmatrix}{g}(\xi _0,\chi _0,\sigma _0,\tilde{\sigma })\\ \chi _0\\ \tilde{\sigma } \end{bmatrix}\\&= \bigcup _{\tilde{\sigma }\in \varphi (\xi _0,\chi _0,\sigma _0)} \begin{bmatrix}\varPi _{\tilde{\sigma }}\xi _0\\ \chi _0\\ \tilde{\sigma } \end{bmatrix}. \end{aligned}$$
If
$$\begin{aligned} \phi (0,1) \in \bigcup _{\tilde{\sigma }\in \varphi (\xi _0,\chi _0,\sigma _0)}\left( ({\mathfrak {O}}_{\tilde{\sigma }}\times {\mathbb {R}}^m\times \{\tilde{\sigma }\})\cap \left( C_{\tilde{\sigma }}\cup D_{\tilde{\sigma }}\right) \right) \end{aligned}$$
and considering $\phi (0,1)$ as the consistent initial condition required in (i), then, according to (i), there exists a solution $\hat{\phi }\ \in \ \mathcal {S}_{\hat{\mathcal {H}}_{\mathrm{DAE}}}$ from $\hat{\phi }(0,0) =\phi (0,1) \in G(\xi _0,\chi _0,\sigma _0)$, such that $ \mathrm{dom}\,{\hat{\phi }}\subset \mathrm{dom}\,{\phi } $ and $ \hat{\phi }(t,j-1) = \phi (t,j) $ for all $(t,j)\ \in \ \mathrm{dom}\,{\phi }{\setminus }\{(0,0)\}$.

$\square $

A.3 Details of the proof of Lemma 2.

Since, for each $\sigma \in \varSigma $, the matrix pairs $(E_\sigma ,A_\sigma )$ are regular (see Definition 1), there exist matrices $T_\sigma $ and $S_\sigma $ that put $(S_\sigma E_\sigma T_\sigma ,S_\sigma A_\sigma T_\sigma )$ into the quasi-Weierstrass form (see Lemma 5).

Notice that given the construction of $\hat{\mathcal {H}}_{\mathrm{DAE}}$, G maps to ${\mathfrak {O}}_{ p }$ after each jump, where p is the value to which $\sigma $ is updated to. This is due to ${\mathfrak {C}}_{ p } =\text{ im }(\varPi _{ p })$ and ${\mathfrak {O}}_{ p } := \{\xi \ |\ \xi \in \text{ span }({\mathfrak {C}}_{ p }),\ \sigma \equiv p \}$, then at a jump at $(\xi ,\sigma ) \in D_\sigma \cap \mathfrak {O}_\sigma $, thus $(\xi ^+, p )\in {\mathfrak {O}}_{ p }$.

Furthermore (according to Definitions 2, 3, and [13, Definition 2.1 and references therein]), given a consistent initial condition, the “classical” solution to $E_{ p }\dot{\xi }(t)={A_{ p }\xi (t)}$ belongs to the consistency space ${\mathfrak {C}}_{ p } $ for the open interval $(0,\infty )$. Then, given a solution $\phi $ to $\hat{\mathcal {H}}_{\mathrm{DAE}}$, we have that $(\xi (t,j),\sigma (t,j))\in {\mathfrak {O}}_{\sigma (t,j)}\ \forall (t,j)\in \mathrm{dom}\,{\phi }$. For each j, such that $I^j\times \{j\}\subset \mathrm{dom}\,{\phi }$, and $I^j$ has a positive Lebesgue measure, the solution component $\xi $ satisfies

$$\begin{aligned} {E_{\sigma (t,j)}\dot{\xi }(t,j) = A_{\sigma (t,j)}\xi (t,j)} \ \forall (t,j)\in I^j\times \{j\}, \end{aligned}$$

(48)

where $I^j\times \{j\}:=\{t\in {\mathbb {R}}_{\ge 0}\ :\ (t,j)\in \mathrm{dom}\,{\phi }\}\times \{j\}$. In fact, over the interval $I^j\times \{j\}$, $\xi $ is given by the “classical” solution to the DAE $E_{\sigma (t,j)}\dot{\xi }={A_{\sigma (t,j)}\xi }$ (see Appendix A.4). Applying Lemma 5 to (48) for all open intervals $I^j\times \{j\}\in \mathrm{dom}\,{\phi }$, for each interval $I^j\times \{j\}$, $\xi (t,j)$ satisfies

$$\begin{aligned} {\dot{\xi }(t,j) = T_{\sigma (t,j)}\begin{bmatrix}I&\quad 0\\0&\quad 0\end{bmatrix}S_{\sigma (t,j)} A_{\sigma (t,j)} \xi (t,j)}. \end{aligned}$$

(49)

To establish that $\phi $ is also a solution to $\widetilde{\mathcal {H}}$, notice that given the construction of G (in $\hat{\mathcal {H}}_{\mathrm{DAE}}$ and $\widetilde{\mathcal {H}}$), the initial value of $\xi $ after each jump belongs to the consistency set ${\mathfrak {O}}_\sigma $, in other words $(\xi (t,j+1),\sigma (t,j+1))\in {\mathfrak {O}}_{\sigma (t,j+1)}$ holds for each $(t,j)\in \mathrm{dom}\,{\phi }$, such that $(t,j+1)\in \mathrm{dom}\,{\phi }$.

In addition, a function $t\mapsto \xi (t)$ that is a solution to $\dot{\xi } = \varPi ^{\tiny \text{ diff }}_\sigma A_\sigma \xi $ if and only if is a solution to $E_\sigma \dot{\xi } = A_\sigma \xi $; therefore, following the same steps above, a solution to $\widetilde{\mathcal {H}}$ is also a solution to $\hat{\mathcal {H}}_{\mathrm{DAE}}$.

A.4 Classical solutions for invariant regular-DAEs

Lemma 5

(Explicit solution formula for switched DAE systems [22, Theorem 6.4.4]) Let $I\subset [\underline{t},\overline{t}] \subset {\mathbb {R}}_{\ge 0}$ have nonempty interior and the regular matrix pair $(E_{\sigma ^*},A_{\sigma ^*})$. In addition, let the projectors defined in Definition 4. The smooth functions $\xi ,h : I\rightarrow {\mathbb {R}}^n$ satisfy

$$\begin{aligned} E_{\sigma ^*}\dot{\xi }(t)=A_{\sigma ^*}\xi (t)+h(t) \end{aligned}$$

if and only if they satisfy,

$$\begin{aligned} \xi (t)&= e^{\varPi _{\sigma ^*}^{\tiny \text{ diff }}A_{\sigma ^*} (t-\underline{t})}\varPi _{\sigma ^*} c + \int _{\underline{t}}^t e^{\varPi _{\sigma ^*}^{\tiny \text{ diff }}A_{\sigma ^*}(t-\tau )}\varPi _{\sigma ^*}^{\tiny \text{ diff }}h(\tau )\mathrm{d}\tau -\sum _{k=0}^{\varsigma }\left( \varPi _{\sigma ^*}^{\tiny \text{ imp }}E_{\sigma ^*}\right) ^k\varPi _{\sigma ^*}^{\tiny \text{ imp }}h^{(k)}(t), \end{aligned}$$

(50)

for $c\in {\mathbb {R}}^n$ and an initial condition $\xi (\underline{t}) = \xi _0$, such that

$$\begin{aligned} \xi _0+\sum _{k=0}^{\varsigma }\left( \varPi _{\sigma ^*}^{\tiny \text{ imp }}E_{\sigma ^*}\right) ^k\varPi _{\sigma ^*}^{\tiny \text{ imp }}h^{(k)}(\underline{t}) \in {\mathfrak {O}}_{\sigma ^*}. \end{aligned}$$

(51)

Furthermore, if $h\equiv 0$ then (50) can be replaced by (52).

$$\begin{aligned} \dot{\xi }(t)&= T_{\sigma ^*}\begin{bmatrix}I&\quad 0\\0&\quad 0\end{bmatrix} S_{\sigma ^*} A_{\sigma ^*}\xi (t). \end{aligned}$$

(52)

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Pablo Ñañez
- 1
Email authorView author's OrcID profile
Ricardo G. Sanfelice
- 2
Nicanor Quijano
- 3

1.Universidad de los AndesBogotáColombia
2.Department of Computer EngineeringUniversity of CaliforniaSanta CruzUSA
3.Universidad de los AndesBogotáColombia

On an invariance principle for differential-algebraic equations with jumps and its application to switched differential-algebraic equations

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