Finite-time consensus protocols for multi-dimensional multi-agent systems

Abstract

A finite-time consensus protocol is proposed for multi-dimensional multi-agent systems, using direction-preserving signum controls. Filippov solutions and nonsmooth analysis techniques are adopted to handle discontinuities. Sufficient and necessary conditions are provided to guarantee infinite-time convergence and boundedness of the solutions. It turns out that the number of agents which have continuous control law plays an essential role in finite-time convergence. In addition, it is shown that the unit balls introduced by $\ell _p$ norms, where $p\in [1,\infty ]$, are invariant for the closed loop.

Appendix

Proof of Lemma 2

We divide the proof into two parts, discussing the cases $p\in [1,\infty )$ and $p=\infty$ separately.

• 1) Let $p\in [1,\infty )$. We introduce a Lyapunov function candidate

  $$\begin{aligned} V(x) = \max _{i\in {\mathcal {I}}} \dfrac{1}{p}\Vert x_i\Vert _p^p \end{aligned}$$

  (A1)

  and note that $V(x)\leqslant \dfrac{1}{p}C^p$ implies that $x\in {\mathcal {S}}_p(C)$. Since the function $(\cdot )^p$ is convex on $\mathbb {R}_{\geqslant 0}$, it can be observed that V is convex and, hence, regular. In the remainder of the proof, we will show that V(x(t)) is nonincreasing along all Filippov solutions of (14), implying strong invariance of the set ${\mathcal {S}}_p(C)$ for any $C>0$.

Let $\alpha (x)$ denote the set of indices that achieve the maximum in (45) as

$$\begin{aligned} \alpha (x) = \{ i\in {\mathcal {I}}\,|\,\Vert x_i\Vert _p^p = pV(x) \}. \end{aligned}$$

(A2)

Then, the generalized gradient of V in (45) is given by

$$\begin{aligned} \partial V(x)=\mathrm {co} \{ e_i\otimes \psi (x_i)\,|\,i\in \alpha (x) \} \end{aligned}$$

(A3)

where

$$\begin{aligned} \psi (x_i) = \left[ \begin{array}{c} |x_{i,1}|^{p-1}{\mathcal {F}}[{{\,\mathrm{sgn}\,}}](x_{i,1}) \\ \vdots \\ |x_{i,k}|^{p-1}{\mathcal {F}}[{{\,\mathrm{sgn}\,}}](x_{i,k})\end{array}\right] \end{aligned}$$

(A4)

and $x_i = [x_{i,1} ~~\cdots ~~ x_{i,k}]^{\mathrm{T}}\in \mathbb {R}^k$.

Next, let $\varPsi$ be defined as

$$\begin{aligned} \varPsi = \Big\{ t\geqslant 0\,\big|\,{\dot{x}}(t) \text { and } \dfrac{\mathrm{d}}{{\mathrm{d}}t}V(x(t)) \text { exist} \Big\} . \end{aligned}$$

(A5)

Since x is absolutely continuous (by definition of Filippov solutions) and V is locally Lipschitz, by Lemma 1 in [29] it follows that $\varPsi =\mathbb {R}_{\geqslant 0}{\setminus }{\bar{\varPsi }}$ for a set ${\bar{\varPsi }}$ of measure zero and

$$\begin{aligned} \dfrac{\mathrm{d}}{{\mathrm{d}}t}V(x(t))\in {\mathcal {L}}_{{\mathcal {F}}[h]}V(x(t)) \end{aligned}$$

(A6)

for all $t\in \varPsi$, such that the set ${\mathcal {L}}_{{\mathcal {F}}[h]}V(x(t))$ is nonempty for all $t\in \varPsi$. For $t\in {\bar{\varPsi }}$, we have that ${\mathcal {L}}_{{\mathcal {F}}[h]}V(x(t))$ is empty, and hence $\max {\mathcal {L}}_{{\mathcal {F}}[h]}V(x(t)) = -\infty < 0$ by definition. Therefore, we only consider $t\in \varPsi$ in the rest of the proof.

Next, we will consider the cases $x\in {\mathcal {C}}$ and $x\notin {\mathcal {C}}$ separately.

First, for $x\in {\mathcal {C}}$, it can be observed that $\alpha (x) = {\mathcal {I}}$. Then, the following two cases can be distinguished.

• i) $|{\mathcal {I}}| \geqslant 2$ and $|{\mathcal {I}}_c|\geqslant 1$. As there is at least one agent with continuous vector field, there exists $i\in {\mathcal {I}}$ such that $f_i$ is locally Lipschitz and direction preserving. Then, by definition of the Filippov set-valued map, it follows that $\nu _i = 0$ for all $\nu \in {\mathcal {F}}[h](x)$ (recall $x\in {\mathcal {C}}$). As ${\mathcal {L}}_{{\mathcal {F}}[h]}V(x(t))$ is nonempty (by considering $t\in \varPsi$), there exists $a\in {\mathcal {L}}_{{\mathcal {F}}[h]}V(x(t))$ such that $a = \zeta ^{\mathrm{T}}\nu$ for all $\zeta \in \partial V(x(t))$, see definition (6). Choosing $\zeta = e_i\otimes \psi (x_i(t))$, it follows that $a = (e_i\otimes \psi (x_i(t)))^{\mathrm{T}}\nu = 0$, which implies that $\max {\mathcal {L}}_{{\mathcal {F}}[h]}V(x(t)) = 0 \leqslant 0$, i.e., V(x) is nonincreasing for any $x\in {\mathcal {C}}$.

• ii) $|{\mathcal {I}}| = 2$ and $|{\mathcal {I}}_c| = 0$. In this case, system (13) can be written as

  $$\begin{aligned} \left\{ \begin{aligned} {\dot{x}}_1&= \dfrac{x_2-x_1}{\Vert x_2-x_1\Vert _p}, \\ {\dot{x}}_2&= \dfrac{x_1-x_2}{\Vert x_1-x_2\Vert _p}. \end{aligned}\right. \end{aligned}$$

  (A7)

  Then, by using definition (3), it can be shown that, for $x_1 = x_2$ (i.e., $x\in {\mathcal {C}}$), any element $\nu$ in the Filippov set-valued map of (51) satisfies $\nu _1 = -\nu _2$. Stated differently, the following implication holds with $\nu = [\nu _1^{\mathrm{T}}~~\nu _2^{\mathrm{T}}]^{\mathrm{T}}$:

  $$\begin{aligned} \nu \in {\mathcal {F}}[h](x),\; x\in {\mathcal {C}} \;\Rightarrow \; \nu _1 = -\nu _2. \end{aligned}$$

  (A8)

  Next, by recalling that $\alpha (x) = {\mathcal {I}}$ (see (46)), it follows from (47) that

  $$\begin{aligned} \partial V(x) = \mathrm {co}\{e_1\otimes \psi (x_1(t)), e_2\otimes \psi (x_2(t))\} \end{aligned}$$

  (A9)

  with $x_1 = x_2$. Now, following a similar reasoning as in item (i) on the basis of the definition of the set-valued Lie derivative in (6), it can be concluded that $a = \zeta ^{\mathrm{T}}\nu$ is necessarily 0, such that $\max {\mathcal {L}}_{{\mathcal {F}}[h]}V(x(t)) = 0 \leqslant 0$ for all $x\in {\mathcal {C}}$.

Second, the case $x\notin {\mathcal {C}}$ is considered. For this case, Theorem 1 in [30] is applied to obtain

$$\begin{aligned} {\mathcal {F}}[h](x) \subset \times _{j=1}^N {\mathcal {F}}[f_i](-{\bar{L}}_ix) =: \bar{{\mathcal {F}}}(x), \end{aligned}$$

(A10)

after which it follows from the definition of the set-valued Lie derivative (6) that

$$\begin{aligned} {\mathcal {L}}_{{\mathcal {F}}}V(x) \subset {\mathcal {L}}_{\bar{{\mathcal {F}}}}V(x). \end{aligned}$$

(A11)

Therefore, in the remainder of the proof for the case $x\notin {\mathcal {C}}$, we will show that $\max {\mathcal {L}}_{\bar{{\mathcal {F}}}[h]}V(x(t)) \leqslant 0$, which implies the desired result by (55). As before, it is sufficient to consider the set $\varPsi$ such that ${\mathcal {L}}_{\bar{{\mathcal {F}}}[h]}V(x(t))$ is nonempty for all $t\in \varPsi$.

Now, take an index $i\in \alpha (x)$ such that ${\bar{L}}_ix \ne {\mathbf {0}}$. Note that such i indeed exists. Namely, assume in order to establish a contradiction that ${\bar{L}}_ix = {\mathbf {0}}$ for all $i\in \alpha (x)$. If $\alpha (x)={\mathcal {I}}$, then there exists $\ell \in \{1,\ldots ,k\}$ such that

$$\begin{aligned} \beta (\ell ):=\arg \max _{j\in {\mathcal {I}}}x_{j,\ell } \subsetneq {\mathcal {I}}, \end{aligned}$$

(A12)

i.e., there exists a state component $\ell$ that does not have the same value for all agents. Otherwise, $x\in {\mathcal {C}}$, which is a contradiction. Then, for any $i\in \beta (\ell )$ with $j\in N_i{\setminus } \beta (\ell )$, we have ${\bar{L}}_ix \ne {\mathbf {0}}$, where $N_i$ is the set of neighbors of agent i. If $\alpha (x)\subsetneq {\mathcal {I}}$ and ${\bar{L}}_ix = {\mathbf {0}}$ for all $i\in \alpha (x)$, then for any $i\in \alpha (x)$ with $j\in N_i{\setminus }\alpha (x)$, we have

$$\begin{aligned} 0= \psi^{\mathrm{T}} (x_i){\bar{L}}_ix \end{aligned}$$

(A13)

$$\begin{aligned}&\quad= \textstyle \sum \limits _{j\in N_i} (\Vert x_i\Vert _p^p - \textstyle \sum \limits _{\ell =1}^{k}|x_{i,\ell }|^{p-1}{\mathcal {F}}[{{\,\mathrm{sgn}\,}}](x_{i,\ell })x_{j,\ell }) \nonumber \\& \quad\geqslant \textstyle \sum \limits _{j\in N_i} (\Vert x_i\Vert _p^p - \textstyle \sum \limits _{\ell =1}^{k}|x_{i,\ell }|^{p-1} |x_{j,\ell }|) \nonumber \\&\quad \geqslant \textstyle \sum \limits _{j\in N_i} (\Vert x_i\Vert _p^p - \Vert x_j\Vert _p \Vert x_i\Vert _p^{p-1} )\nonumber \\&\quad > 0 ,\end{aligned}$$

(A14)

where inequality (58) is based on Hölder’s inequality, and the last inequality is implied by $\Vert x_i\Vert >\Vert x_j\Vert$ for any $j\in N_i{\setminus }\alpha (x)$. This is a contradiction.

For the index $i\in \alpha (x)$ satisfying ${\bar{L}}_ix\ne {\mathbf {0}}$, it follows from Assumption 1 that there exists $\gamma >0$ such that

$$\begin{aligned} {\mathcal {F}}[f_i](-{\bar{L}}_ix) = \{-\gamma {\bar{L}}_ix \}, \end{aligned}$$

(A15)

i.e., for any $\nu \in \bar{{\mathcal {F}}}(x)$ it holds that $\nu _i = -\gamma {\bar{L}}_ix$. Note that this is a result of the direction-preserving property of either the direction-preserving signum (for a nonzero argument, then $\gamma = \dfrac{1}{\Vert {\bar{L}}_ix\Vert _p})$ or the Lipschitz continuous function (by Assumption 1). Then, choosing $\zeta \in \partial V(x)$ as $\zeta = e_i\otimes \psi (x_i)$ (recall that $i\in \alpha (x)$), it follows from (6) that

$$\begin{aligned} {\mathcal {L}}_{\bar{{\mathcal {F}}}[h]}V(x) = \{-\gamma \psi ^{\mathrm{T}}(x_i){\bar{L}}_ix\}. \end{aligned}$$

(A16)

Next, by observing (58), we have

$$\begin{aligned} \psi ^{\mathrm{T}}(x_i){\bar{L}}_ix \geqslant 0, \end{aligned}$$

(A17)

which implies ${\mathcal {L}}_{\bar{{\mathcal {F}}}[h]}V(x)\subset \mathbb {R}_{\leqslant 0}$.

Summarizing the results of the two cases leads to the condition

$$\begin{aligned} \max {\mathcal {L}}_{{\mathcal {F}}}V(x)\leqslant 0 \end{aligned}$$

(A18)

for all $x\in \mathbb {R}^{kn}$, which proves strong invariance of ${\mathcal {S}}_p(C)$ for all $C>0$.

• 2) Let $p=\infty$. Consider

  $$\begin{aligned} V(x) = \max _{i\in {\mathcal {I}}} \Vert x_i\Vert _\infty \end{aligned}$$

  (A19)

  as a Lyapunov function candidate. Since the proof shares the same structure and reasoning as the case $p\in [1,\infty )$, we only provide a sketch of the proof.

In this case, the set $\alpha (x)$ in (46) is

$$\begin{aligned} \alpha (x) = \{ i\in {\mathcal {I}}\,|\,\Vert x_i\Vert _\infty = V(x) \}, \end{aligned}$$

(A20)

whereas the generalized gradient of V reads

$$\begin{aligned} \partial V(x)=\mathrm {co} \{&e_i\otimes ({\mathcal {F}}[{{\,\mathrm{sgn}\,}}](x_{i,\ell })e_\ell )\,|\,e_i\in \mathbb {R}^n, \nonumber \\&e_\ell \in \mathbb {R}^k, |x_{i,\ell }|=V(x) \}. \end{aligned}$$

(A21)

For the case $x\in {\mathcal {C}}$, we have $\max {\mathcal {L}}_{{\mathcal {F}}[h]}V(x(t)) = 0 \leqslant 0$ by using the same argument as the case $p\in [1,\infty )$. Hence, we omit the details.

For the case $x\notin {\mathcal {C}}$, we first show that there exists an index $i\in \alpha (x)$ such that ${\bar{L}}_ix \ne {\mathbf {0}}$ by using contradiction. If $\alpha (x)={\mathcal {I}}$, the conclusion follows as the case (1). If $\alpha (x)\subsetneq {\mathcal {I}}$ and ${\bar{L}}_ix = {\mathbf {0}}$ for all $i\in \alpha (x)$, then for any $i\in \alpha (x)$ with $j\in N_i{\setminus }\alpha (x)$ we have

$$\begin{aligned} 0= ({\mathcal {F}}[{{\,\mathrm{sgn}\,}}](x_{i,\ell })e_\ell )^{\mathrm{T}}{\bar{L}}_ix \end{aligned}$$

(A22)

$$\begin{aligned}&\quad= \textstyle \sum \limits _{j\in N_i} ( |x_{i,\ell }| - {\mathcal {F}}[{{\,\mathrm{sgn}\,}}](x_{i,\ell })x_{j,\ell } ) \\& \quad\geqslant \textstyle \sum \limits _{j\in N_i} (\Vert x_i\Vert _\infty - \Vert x_{j}\Vert _\infty ) \\&\quad> 0, \end{aligned}$$

(A23)

where $e_i\otimes ({\mathcal {F}}[{{\,\mathrm{sgn}\,}}](x_{i,\ell })e_\ell )\in \partial V(x)$ and the last inequality is implied by $\Vert x_i\Vert >\Vert x_j\Vert$ for any $j\in N_i{\setminus }\alpha (x)$. This is a contradiction.

For the index $i\in \alpha (x)$ satisfying ${\bar{L}}_ix\ne {\mathbf {0}}$, it follows from Assumption 1 that there exists $\gamma >0$ such that for any $\nu \in \bar{{\mathcal {F}}}(x)$, it holds that $\nu _i = -\gamma {\bar{L}}_ix$. Using the same reasoning as in case (1), by choosing $\zeta \in \partial V(x)$ as $\zeta = e_i\otimes ({\mathcal {F}}[{{\,\mathrm{sgn}\,}}](x_{i,\ell })e_\ell )$ (recall that $i\in \alpha (x)$), it follows from (6) that

$$\begin{aligned} {\mathcal {L}}_{\bar{{\mathcal {F}}}[h]}V(x) = \{-\gamma ({\mathcal {F}}[{{\,\mathrm{sgn}\,}}](x_{i,\ell })e_\ell )^{\mathrm{T}}{\bar{L}}_ix\}. \end{aligned}$$

(A24)

Next, by observing (67), we have $({\mathcal {F}}[{{\,\mathrm{sgn}\,}}](x_{i,\ell })e_\ell )^{\mathrm{T}}{\bar{L}}_ix \geqslant 0,$ which implies ${\mathcal {L}}_{\bar{{\mathcal {F}}}[h]}V(x)\subset \mathbb {R}_{\leqslant 0}$.

In summary, we have

$$\begin{aligned} \max {\mathcal {L}}_{{\mathcal {F}}}V(x)\leqslant 0 \end{aligned}$$

(A25)

for all $x\in \mathbb {R}^{kn}$, which proves strong invariance of ${\mathcal {S}}_p(C)$ for all $C>0$.