Composite Finite-Time Containment Control for Disturbed Second-Order Multi-agent Systems

Abstract

In this chapter, the distributed finite-time containment control problem is investigated for second-order multi-agent systems with external disturbances. By combining finite-time control and finite-time disturbance observer techniques together, a kind of feedforward-feedback composite distributed controllers are proposed. Under these distributed controllers, the effects of the disturbances on the system states are removed in a finite time and the followers globally converge to the convex hull spanned by the leaders in a finite time as well. Simulations illustrate the effectiveness of the proposed control algorithms.

Appendix

Proof of Proposition 8.2: For brevity, denote $d_{F}=[d^{T}_{1},\ldots ,d^{T}_{n}]^{T},\hat{d}_{F}=[\hat{d}^{T}_{1},\ldots ,\hat{d}^{T}_{n}]^{T},\tilde{d}_{F}=[\tilde{d}^{T}_{1},\ldots ,\tilde{d}^{T}_{n}]^{T}$. By Lemma 8.5, the DO (8.12) is finite-time convergent. So there is a constant h such that $\Vert \tilde{d}_i(t)\Vert _2\le h,\forall i\in F,t\in [0,+\infty )$. Define $W(x_{F},v_{F})=\frac{1}{2}x^{T}_{F}x_{F}+\frac{1}{2}v^{T}_{F}v_{F}$. $\dot{W}$ along system (8.3) satisfies

$$\begin{aligned} \dot{ W}&=x^{T}_{F}v_{F}+v^{T}_{F}(u_{F}+d_{F})\le W+\sum ^{n}_{i=1}\Vert v_i\Vert _2(\Vert u^*_i\Vert _2+h). \end{aligned}$$

(8.34)

where $u^*_i=u_i+\hat{d}_i$. From (8.11), it can be obtained that

$$\begin{aligned} \Vert u^*_{i}\Vert _2\le&~\Vert \dot{\hat{v}}^d_i\Vert _2+k_{2}\Vert \zeta _i^{2q-1}\Vert _2,~i\in F. \end{aligned}$$

(8.35)

where $\zeta _i=(v_{i}-\hat{v}^d_i)^{1/q}+k_{1}^{1/q}\sum _{j\in F\cup L}a_{ij}(x_{i}-x_{j})$. According to Lemma 8.2, $\forall y=[y_{1},\ldots ,y_{p}]^{T}\in \mathbb {R}^{p},a\ge 0$, the following holds

$$\begin{aligned} \Vert y^{a}\Vert _2=\left[ \sum ^{p}_{k=1}(y^{a}_{k})^{2}\right] ^{1/2}\le \sum ^{p}_{k=1}|y_{k}|^{a}\le p\left( \sum ^{p}_{k=1}y^{2}_{k}\right) ^{a/2}=p\Vert y\Vert _2^{a}. \end{aligned}$$

(8.36)

Clearly, (8.36) holds by letting $y=(v_{i}-\hat{v}^d_i)^{1/q} +k_{1}^{1/q}\sum _{j\in F\cup L}a_{ij}(x_{i}-x_{j})\in \mathbb {R}^{p},a=2q-1$ or $y=v_{i}-\hat{v}^d_i\in \mathbb {R}^{p},a=1/q$. Based on the fact $0<2q-1<1$ and Lemma 8.2, it follows that $p\Vert (v_{i}-\hat{v}^d_i)^{1/q} +k_{1}^{1/q}\sum _{j\in F\cup L}a_{ij}(x_{i}-x_{j})\Vert _2^{2q-1} \le p\Vert (v_{i}-\hat{v}^d_i)^{1/q}\Vert _2^{2q-1}+pk_{1}^{2-1/q}\Vert \sum _{j\in F\cup L}a_{ij}(x_{i}-x_{j})\Vert _2^{2q-1}$. Note that $0<2-1/q<1$. Based on (8.36) and Lemma 8.2, it can be verified that $\Vert (v_{i}-\hat{v}^d_i)^{1/q}\Vert _2^{2q-1}\le p^{2q-1}(\Vert v_{i}\Vert _2+\Vert \hat{v}^d_i\Vert _2)^{2-1/q}\le p^{2q-1}(\Vert v_{i}\Vert _2^{2-1/q}+\Vert \hat{v}^d_i\Vert _2^{2-1/q})$. In addition, $\Vert \sum _{j\in F\cup L}a_{ij}(x_{i}-x_{j})\Vert _2\le \beta \sum ^{n+m}_{j=1}(\Vert x_{i}\Vert _2+\Vert x_{j}\Vert _2)=\beta (n+m)\Vert x_{i}\Vert _2+\beta \sum ^{n}_{j=1}\Vert x_{j}\Vert _2+\beta \sum ^{n+m}_{j=n+1}\Vert x_{j}\Vert _2$, where $\beta =\max _{\forall i\in F}\left\{ \sum _{j\in F\cup L}a_{ij}\right\}$. Then it can be obtained from (8.36) that

$$\begin{aligned} p\Vert \zeta _i\Vert _2^{2q-1} \le&~ p^{2q}\left( \Vert v_{i}\Vert _2^{2-1/q}+\Vert \hat{v}^d_i\Vert _2^{2-1/q}\right) +pk_{1}^{2-1/q}\beta ^{2q-1}\nonumber \\&\times \left[ (n+m)^{2q-1}\Vert x_{i}\Vert _2^{2q-1}+\sum ^{n}_{j=1}\Vert x_{j}\Vert _2^{2q-1}+\sum ^{n+m}_{j=n+1}\Vert x_{j}\Vert _2^{2q-1}\right] ,~i\in F. \end{aligned}$$

(8.37)

Then, putting (8.35)–(8.37) together yields

$$\begin{aligned} \Vert u^*_{i}\Vert _2\le&~\Vert \dot{\hat{v}}^d_i\Vert _2+k_{2}p^{2q}\left( \Vert v_{i}\Vert _2^{2-1/q}+\Vert \hat{v}^d_i\Vert _2^{2-1/q}\right) +k_{2}pk_{1}^{2-1/q}\beta ^{2q-1}\nonumber \\&\times \left[ (n+m)^{2q-1}\Vert x_{i}\Vert _2^{2q-1}+\sum ^{n}_{j=1}\Vert x_{j}\Vert _2^{2q-1}+\sum ^{n+m}_{j=n+1}\Vert x_{j}\Vert _2^{2q-1}\right] \nonumber \\ \le&~ \delta _{1}+k_{2}p^{2q}\Vert v_{i}\Vert _2^{2-1/q}\nonumber \\&+k_{2}pk_{1}^{2-1/q}\beta ^{2q-1}\left[ (n+m)^{2q-1}\Vert x_{i}\Vert _2^{2q-1}+\sum ^{n}_{j=1}\Vert x_{j}\Vert _2^{2q-1}\right] ,~i\in F, \end{aligned}$$

(8.38)

where $\delta _{1}=\Vert \dot{\hat{v}}^d_i\Vert _2+k_{2}p^{2q}\Vert \hat{v}^d_i\Vert _2^{2-1/q}+k_{2}pk_{1}^{2-1/q}\beta ^{2q-1}\sum ^{n+m}_{j=n+1}\Vert x_{j}\Vert _2^{2q-1}.$ Note that $\dot{\hat{v}}^d_i=-\rho _1\mathrm{sig}^{\alpha }\left( \sum _{j\in F\cup L}a_{ij}(\hat{v}^d_i-\hat{v}^d_j)\right) -\rho _2\mathrm{sgn}\left( \sum _{j\in F\cup L}a_{ij}(\hat{v}^d_i-\hat{v}^d_j)\right)$. Due to global convergence of observer (8.4) and Assumption 6, the existence of $\delta _{1}$ is guaranteed. Then it follows from (8.34), (8.38) and Lemma 8.3 that

$$\begin{aligned} \dot{ W}\le&~ W+(\delta _{1}+h)\sum ^{n}_{i=1}\Vert v_{i}\Vert _2+k_{2}p^{2q}\sum ^{n}_{i=1}\Vert v_{i}\Vert _2^{3-1/q}\nonumber \\&+\frac{k_{2}pk_{1}^{2-1/q}\beta ^{2q-1}}{2q}\sum ^{n}_{i=1}[(n+m)^{2q-1}+n]\Vert v_{i}\Vert _2^{2q}\nonumber \\&+\frac{k_{2}pk_{1}^{2-1/q}\beta ^{2q-1}(2q-1)}{2q}\sum ^{n}_{i=1} \left[ (n+m)^{2q-1}\Vert x_{i}\Vert _2^{2q}+\sum ^{n}_{j=1}\Vert x_{j}\Vert _2^{2q}\right] \nonumber \\ \le&~ W+(\delta _{1}+h)\sum ^{n}_{i=1}\Vert v_{i}\Vert _2+k_{2}p^{2q}\sum ^{n}_{i=1}\Vert v_{i}\Vert _2^{3-1/q}\nonumber \\&+\frac{k_{2}pk_{1}^{2-1/q}\beta ^{2q-1}[(n+m)^{2q-1}+n]}{2q}\sum ^{n}_{i=1}(\Vert x_{i}\Vert _2^{2q}+\Vert v_{i}\Vert _2^{2q}). \end{aligned}$$

(8.39)

In addition, it holds that $\max \{\Vert x_{i}\Vert _2^{a},\Vert v_{i}\Vert _2^{a}\}\le (\Vert x_{i}\Vert _2^{2}+\Vert v_{i}\Vert _2^{2})^{a/2},\forall \ a\ge 0$. Then based on the fact $0<q,(3-1/q)/2<1$ and (8.39), it follows that

$$\begin{aligned} \dot{ W}\le&~ W+(\delta _{1}+h)\sum ^{n}_{i=1}(\Vert x_{i}\Vert _2^{2}+\Vert v_{i}\Vert _2^{2})^{1/2}+k_{2}p^{2q}\sum ^{n}_{i=1}(\Vert x_{i}\Vert _2^{2}+\Vert v_{i}\Vert _2^{2})^{(3-1/q)/2}\nonumber \\&+\frac{k_{2}pk_{1}^{2-1/q}\beta ^{2q-1}[(n+m)^{2q-1}+n]}{q}\sum ^{n}_{i=1}(\Vert x_{i}\Vert _2^{2}+\Vert v_{i}\Vert _2^{2})^{q}. \end{aligned}$$

(8.40)

Note that $\Vert x\Vert _{p}=\left( \sum ^{m}_{l=1}|x_{l}|^{p}\right) ^{1/p},\forall x=[x_{1},\ldots ,x_{m}]^{T}$ with $p\ge 1,m\in N^{+}$ denotes p-norm in $\mathbb {R}^{m}$. Based on the equivalence between any two different norms in $\mathbb {R}^{p}$ and Lemma 8.2, there is $\delta _{2}>0$ such that $\sum ^{n}_{i=1}(\Vert x_{i}\Vert _2^{2}+\Vert v_{i}\Vert _2^{2})^{1/2}\le \sum ^{n}_{i=1}(\Vert x_{i}\Vert _2+\Vert v_{i}\Vert _2)\le \delta _{2} W^{1/2}$. Similarly, both $\sum ^{n}_{i=1}(\Vert x_{i}\Vert _2^{2}+\Vert v_{i}\Vert _2^{2})^{(3-1/q)/2}\le \delta _{3} W^{(3-1/q)/2}$ and $\sum ^{n}_{i=1}(\Vert x_{i}\Vert _2^{2}+\Vert v_{i}\Vert _2^{2})^{q}\le \delta _{4} W^{q}$ hold with appropriate $\delta _{3}>0,\delta _{4}>0$. Then it follows from (8.40) that

$$\begin{aligned} \dot{ W}\le&~ W+(\delta _{1}+h)\delta _{2} W^{1/2}+k_{2}p^{2p}\delta _{3} W^{(3-1/q)/2}\nonumber \\&+\frac{k_{2}pk_{1}^{2-1/q}\beta ^{2q-1}[(n+m)^{2q-1}+n]\delta _{4}}{q} W^{q}. \end{aligned}$$

(8.41)

By Lemma 8.3, it holds that $W^{b}= W^{b}\cdot 1^{1-b}\le b W+1-b,\forall \ 0<b\le 1$. Then it can be obtained from (8.41) that

$$\begin{aligned} \dot{ W}\le \delta _{5} W+\delta _{6}, \end{aligned}$$

(8.42)

where $\delta _{5}=1+\frac{(\delta _{1}+h)\delta _{2}}{2}+\frac{3q-1}{2q}k_{2}p^{2q}\delta _{3}+k_{2}pk_{1}^{2-1/q}\beta ^{2q-1}[(n+m)^{2q-1}+n]\delta _{4}$ and $\delta _{6}=\frac{(\delta _{1}+h)\delta _{2}}{2}+\frac{1-q}{2q}k_{2}p^{2q}\delta _{3}+\frac{k_{2}pk_{1}^{2-1/q}\beta ^{2q-1}[(n+m)^{2q-1}+n]\delta _{4}(1-q)}{q}$. By noting that $\delta _{5},\delta _{6}\in (0,+\infty )$, it follows from (8.42) that W is bounded, which implies that $x_{i}(t),v_{i}(t),i\in F$ are bounded $\forall \ t\in [0,+\infty )$. This completes the proof.    $\square$