Distributed cooperative adaptive tracking control for heterogeneous systems with hybrid nonlinear dynamics


The cooperative leader-following tracking for a group of heterogeneous mechanical systems with nonlinear hybrid order dynamics is studied. The controlled systems are considered to be composed of followers (agents) with hybrid first- and second-order time-varying dynamics. The leader is an unknown nonautonomous nonlinear system and can only give the state information of position and velocity to its neighboring followers. The followers are linked by the directed graph with fixed communication topology. And, not all of them have the information path to the leader directly. The directed information topology graph is required to have at least one spanning tree for position and velocity, respectively. Distributed cooperative adaptive control protocols are developed for all followers with first- or second-order dynamics to achieve the ultimate synchronization to the leader. The control protocols are designed based on the neural networks and the adaptive estimation algorithm for unknown time-varying functions and control coefficients. The convergence and boundedness of the synchronization error is proved by the Lyapunov theory. The simulation example verifies the correctness of the developed distributed control protocols.

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Appendix: Proof of Theorem 1

Appendix: Proof of Theorem 1

Define the Lyapunov function as

$$\begin{aligned} V=V_r +V_W +V_g +V_e \end{aligned}$$

where \(V_r =\frac{1}{2}r^\mathrm{T}Pr\), \(V_W =\frac{1}{2}{\tilde{W}}^\mathrm{T}F^{-1}{\tilde{W}}\), \(V_g =\frac{1}{2}{\tilde{g}}^\mathrm{T}\eta ^{-1}{\tilde{g}}\), \(V_e =\frac{1}{2}\left( {e_p^{l_2 } } \right) ^\mathrm{T}e_p^{l_2 } \).

According to (26), by differentiating \(V_r \), we have

$$\begin{aligned} {\dot{V}}_r= & {} r^\mathrm{T}P{\dot{r}}=r_{n1}^T P^{l_1 }{\dot{r}}_{n1} +r_{n2}^T P^{l_2 }{\dot{r}}_{n2} \nonumber \\= & {} -r_{n1}^T P^{l_1 }(L+B)\left( \left( {{\tilde{W}}^{l_1 }} \right) ^\mathrm{T}\phi ^{_{l_1 } }+\varepsilon ^{l_1 }+{\tilde{g}}^{l_1 }u^{l_1 }\right. \nonumber \\&\left. +\,cr_{n1} +w^{l_1 }-{\underline{1}}^{l_1 }x_{v,0} \right) \nonumber \\&-\,r_{n2}^T P^{l_2 }(L+B)\left( \left( {{\tilde{W}}^{l_2 }} \right) ^\mathrm{T}\phi ^{_{l_2 } }+\varepsilon ^{l_2 }+cr_{n2} \right. \nonumber \\&\left. +\,{\tilde{g}}^{l_2 }u^{l_2 }+w^{l_2 }-{\underline{1}}^{l_2 }f_0 \right) \nonumber \\&+\,r_{n2}^T P^{l_2 }A\left( {D+B} \right) ^{-1}\Lambda e_v \nonumber \\= & {} -r^\mathrm{T}P(L+B)\left[ {{\tilde{W}}^\mathrm{T}\phi +cr+{\tilde{g}}u} \right] \nonumber \\&-\,r_{n1}^T P^{l_1 }(L+B)\left[ {w^{l_1 }+\varepsilon ^{l_1 }-{\underline{1}}^{l_1 }x_{v,0} } \right] \nonumber \\&-\,r_{n2}^T P^{l_2 }(L+B)\left[ {w^{l_2 }+\varepsilon ^{l_2 }-{\underline{1}}^{l_2 }f_0 } \right] \nonumber \\&+\,r_{n2}^T P^{l_2 }A\left( {D+B} \right) ^{-1}\Lambda e_v \nonumber \\= & {} -r^\mathrm{T}P(D+B){\tilde{W}}^\mathrm{T}\phi +r^\mathrm{T}PA{\tilde{W}}^\mathrm{T}\phi \nonumber \\&-\,r^\mathrm{T}P(L+B)cr-r^\mathrm{T}P(D+B){\tilde{g}}u \nonumber \\&+r^\mathrm{T}PA{\tilde{g}}u \nonumber \\&-\,r_{n1}^T P^{l_1 }(L+B)\left[ {w^{l_1 }+\varepsilon ^{l_1 }-{\underline{1}}^{l_1 }x_{v,0} } \right] \nonumber \\&-\,r_{n2}^T P^{l_2 }(L+B)\left[ {w^{l_2 }+\varepsilon ^{l_2 }-{\underline{1}}^{l_2 }f_0 } \right] \nonumber \\&+\,r_{n2}^T P^{l_2 }A\left( {D+B} \right) ^{-1}\Lambda e_v \end{aligned}$$

Considering the fact that \(x^\mathrm{T}y=\mathrm{tr}\left\{ {yx^\mathrm{T}} \right\} \), we have

$$\begin{aligned} {\dot{V}}_r= & {} -\mathrm{tr}\left\{ {{\tilde{W}}^\mathrm{T}\phi r^\mathrm{T}P(D+B)} \right\} \nonumber \\&+\,r^\mathrm{T}PA{\tilde{W}}^\mathrm{T}\phi -r^\mathrm{T}P(L+B)cr \nonumber \\&-\,\mathrm{tr}\left\{ {{\tilde{g}}^\mathrm{T}ur^{T}P(D+B)} \right\} +r^\mathrm{T}PA{\tilde{g}}u\nonumber \\&-\,r_{n1}^T P^{l_1 }(L+B)\left[ {w^{l_1 }+\varepsilon ^{l_1 }-{\underline{1}}^{l_1 }x_{v,0} } \right] \nonumber \\&-\,r_{n2}^T P^{l_2 }(L+B)\left[ {w^{l_2 }+\varepsilon ^{l_2 }-{\underline{1}}^{l_2 }f_0 } \right] \nonumber \\&+\,r_{n2}^T P^{l_2 }A\left( {D+B} \right) ^{-1}\Lambda e_v \end{aligned}$$

Differentiating \(V_W \), \(V_g \) and \(V_e \), we have

$$\begin{aligned} {\dot{V}}_W +{\dot{V}}_g +{\dot{V}}_e= & {} \mathrm{tr}\left\{ {{\tilde{W}}^\mathrm{T}F^{-1}\dot{{\tilde{W}}}} \right\} \nonumber \\&+\,\mathrm{tr}\left\{ {{\tilde{g}}^\mathrm{T}\eta ^{-1}\dot{{\tilde{g}}}} \right\} +\left( {e_p^{l_2 } } \right) ^\mathrm{T}e_v \nonumber \\= & {} -\,\mathrm{tr}\left\{ {{\tilde{W}}^\mathrm{T}F^{-1}\dot{\hat{{W}}}} \right\} -tr\left\{ {{\tilde{g}}^\mathrm{T}\eta ^{-1}\dot{\hat{{g}}}} \right\} \nonumber \\&+\,\left( {e_p^{l_2 } } \right) ^\mathrm{T}e_v \end{aligned}$$

Combining (29) and (30), we have

$$\begin{aligned} {\dot{V}}= & {} {\dot{V}}_r +{\dot{V}}_W +{\dot{V}}_g +{\dot{V}}_e \nonumber \\= & {} -\mathrm{tr}\left\{ {{\tilde{W}}^\mathrm{T}\phi r^\mathrm{T}P(D+B)} \right\} \nonumber \\&+\,r^\mathrm{T}PA{\tilde{W}}^\mathrm{T}\phi -r^\mathrm{T}P(L+B)cr \nonumber \\&-\,\mathrm{tr}\left\{ {{\tilde{g}}^\mathrm{T}ur^{T}P(D+B)} \right\} +r^\mathrm{T}PA{\tilde{g}}u\nonumber \\&-\,r_{n1}^T P^{l_1 }(L+B)\left[ {w^{l_1 }+\varepsilon ^{l_1 }-{\underline{1}}^{l_1 }x_{v,0} } \right] \nonumber \\&-\,r_{n2}^T P^{l_2 }(L+B)\left[ {w^{l_2 }+\varepsilon ^{l_2 }-{\underline{1}}^{l_2 }f_0 } \right] \nonumber \\&+\,r_{n2}^T P^{l_2 }A\left( {D+B} \right) ^{-1}\Lambda e_v \nonumber \\&-\,\mathrm{tr}\left\{ {{\tilde{W}}^\mathrm{T}F^{-1}\dot{\hat{{W}}}} \right\} -tr\left\{ {{\tilde{g}}^\mathrm{T}\eta ^{-1}\dot{\hat{{g}}}} \right\} +\left( {e_p^{l_2 } } \right) ^\mathrm{T}e_v \nonumber \\= & {} -\mathrm{tr}\left\{ {{\tilde{W}}^\mathrm{T}\left[ {F^{-1}\dot{\hat{{W}}}+\phi r^\mathrm{T}P(D+B)} \right] } \right\} \nonumber \\&+\,r^\mathrm{T}PA{\tilde{W}}^\mathrm{T}\phi -r^\mathrm{T}P(L+B)cr \nonumber \\&-\,\mathrm{tr}\left\{ {{\tilde{g}}^\mathrm{T}\left[ {\eta ^{-1}\dot{\hat{{g}}}+ur^\mathrm{T}P(D+B)} \right] } \right\} \nonumber \\&+\,r^\mathrm{T}PA{\tilde{g}}u-r_{n1}^T P^{l_1 }(L+B)\left[ {w^{l_1 }+\varepsilon ^{l_1 }-{\underline{1}}^{l_1 }x_{v,0} } \right] \nonumber \\&-\,r_{n2}^T P^{l_2 }(L+B)\left[ {w^{l_2 }+\varepsilon ^{l_2 }-{\underline{1}}^{l_2 }f_0 } \right] \nonumber \\&+\,r_{n2}^T P^{l_2 }A\left( {D+B} \right) ^{-1}\Lambda e_v +\left( {e_p^{l_2 } } \right) ^\mathrm{T}e_v \end{aligned}$$

The error \({\tilde{g}}_i =g_i -{\underline{g}}_i \ge 0\) if \(\hat{{g}}_i ={\underline{g}}_i \). Then, substituting (24) and (25) into (31), we have

$$\begin{aligned} {\dot{V}}= & {} \mathrm{tr}\left\{ {{\tilde{W}}^\mathrm{T}\kappa \hat{{W}}} \right\} +r^\mathrm{T}PA{\tilde{W}}^\mathrm{T}\phi -r^\mathrm{T}P(L+B)cr \nonumber \\&+\,r^\mathrm{T}PA{\tilde{g}}u-r_{n1}^T P^{l_1 }(L+B)\left[ {w^{l_1 }+\varepsilon ^{l_1 }-{\underline{1}}^{l_1 }x_{v,0} } \right] \nonumber \\&-\,r_{n2}^T P^{l_2 }(L+B)\left[ {w^{l_2 }+\varepsilon ^{l_2 }-{\underline{1}}^{l_2 }f_0 } \right] \nonumber \\&+\,r_{n2}^T P^{l_2 }A\left( {D+B} \right) ^{-1}\Lambda e_v +\left( {e_p^{l_2 } } \right) ^\mathrm{T}e_v \nonumber \\&+\,\mathrm{tr}\left\{ {{\tilde{g}}^\mathrm{T}\theta \hat{{g}}} \right\} \nonumber \\&-\left\{ {\begin{array}{ll} 0,&{}\quad \hbox {if }\hat{{g}}_i >{\underline{g}}_i \\ 0,&{}\quad \hbox {if }\hat{{g}}_i ={\underline{g}}_i \hbox { and }u_i r_i p_i (d_i +b_i )<0 \\ \mathrm{tr}\left\{ {\hbox {diag}\left\{ {{\tilde{g}}_i u_i r_i p_i (d_i +b_i )} \right\} } \right\} ,&{}\hbox { if }\hat{{g}}_i ={\underline{g}}_i \hbox { and }u_i r_i p_i (d_i +b_i )\ge 0 \\ \end{array}} \right. \nonumber \\\le & {} \mathrm{tr}\left\{ {{\tilde{W}}^\mathrm{T}\kappa \hat{{W}}} \right\} \nonumber \\&+\,r^\mathrm{T}PA{\tilde{W}}^\mathrm{T}\phi -r^\mathrm{T}P(L+B)cr \nonumber \\&+\,\mathrm{tr}\left\{ {{\tilde{g}}^\mathrm{T}\theta \hat{{g}}} \right\} +r^\mathrm{T}PA{\tilde{g}}u\nonumber \\&-\,r_{n1}^T P^{l_1 }(L+B)\left[ {w^{l_1 }+\varepsilon ^{l_1 }-{\underline{1}}^{l_1 }x_{v,0} } \right] \nonumber \\&-\,r_{n2}^T P^{l_2 }(L+B)\left[ {w^{l_2 }+\varepsilon ^{l_2 }-{\underline{1}}^{l_2 }f_0 } \right] \nonumber \\&+\,r_{n2}^T P^{l_2 }A\left( {D+B} \right) ^{-1}\Lambda e_v +\left( {e_p^{l_2 } } \right) ^\mathrm{T}e_v \ \end{aligned}$$

Then, according to Lemma 2, we have

$$\begin{aligned} {\dot{V}}\le & {} -\frac{1}{2}cr^\mathrm{T}Qr+\mathrm{tr}\left\{ {{\tilde{W}}^\mathrm{T}\kappa (W-{\tilde{W}})} \right\} \nonumber \\&+\,r^\mathrm{T}PA{\tilde{W}}^\mathrm{T}\phi +\mathrm{tr}\left\{ {{\tilde{g}}^\mathrm{T}\theta (g-{\tilde{g}})} \right\} +r^\mathrm{T}PA{\tilde{g}}u \nonumber \\&-\,r_{n1}^T P^{l_1 }(L+B)\left[ {w^{l_1 }+\varepsilon ^{l_1 }-{\underline{1}}^{l_1 }x_{v,0} } \right] \nonumber \\&-\,r_{n2}^T P^{l_2 }(L+B)\left[ {w^{l_2 }+\varepsilon ^{l_2 }-{\underline{1}}^{l_2 }f_0 } \right] \nonumber \\&+\,\left[ {\left( {e_p^{l_2 } } \right) ^\mathrm{T}+r_{n2}^T P^{l_2 }A\left( {D+B} \right) ^{-1}\Lambda } \right] \nonumber \\&\left( {r_{n2} -\Lambda e_p^{l_2 } } \right) \end{aligned}$$

Taking norm on (33), we have

$$\begin{aligned} {\dot{V}}\le & {} -\frac{1}{2}c{\underline{\sigma } }\left( Q \right) \left\| r \right\| ^{2}+\kappa W_M \left\| {{\tilde{W}}} \right\| _F -\kappa \left\| {{\tilde{W}}} \right\| _F^2 \nonumber \\&+\,\phi _M \bar{{\sigma }}\left( P \right) \bar{{\sigma }}\left( A \right) \left\| r \right\| \left\| {{\tilde{W}}} \right\| _F \nonumber \\&+\,\theta g_M \left\| {{\tilde{g}}} \right\| _F -\theta \left\| {{\tilde{g}}} \right\| _F^2 +u_M \bar{{\sigma }}\left( P \right) \bar{{\sigma }}\left( A \right) \left\| r \right\| \left\| {{\tilde{g}}} \right\| _F \nonumber \\&+\,\bar{{\sigma }}\left( P \right) \bar{{\sigma }}(L+B)B_M \left\| r \right\| -{\underline{\sigma } }\left( \Lambda \right) \left\| {e_p^{l_2 } } \right\| ^{2} \nonumber \\&+\,\frac{\bar{{\sigma }}\left( \Lambda \right) \bar{{\sigma }}\left( {P^{l_2 }} \right) \bar{{\sigma }}\left( A \right) }{{\underline{\sigma } }\left( {D+B} \right) }\left\| r \right\| ^{2}\nonumber \\&+\,\left[ {1+\frac{\bar{{\sigma }}\left( {\Lambda ^{2}} \right) \bar{{\sigma }}\left( {P^{l_2 }} \right) \bar{{\sigma }}\left( A \right) }{{\underline{\sigma } }\left( {D+B} \right) }} \right] \left\| {e_p^{l_2 } } \right\| \left\| r \right\| \end{aligned}$$

where \(B_M =\varepsilon _M +w_M +f_M +x_{0M} \).

Write (34) as

$$\begin{aligned} {\dot{V}}\le -z^\mathrm{T}Sz+K^\mathrm{T}z=-V_z (z) \end{aligned}$$

where \(z=\left[ {{\begin{array}{llll} {\left\| r \right\| }&{}\quad {\left\| {{\tilde{W}}} \right\| _F }&{}\quad {\left\| {e_p^{l_2 } } \right\| }&{}\quad {\left\| {{\tilde{g}}} \right\| _F } \\ \end{array} }} \right] ^\mathrm{T}\), \(K=\left[ {{\begin{array}{llll} {\bar{{\sigma }}\left( P \right) \bar{{\sigma }}(L+B)B_M }&{}\quad {\kappa W_M }&{}\quad 0&{}\quad {\theta g_M } \\ \end{array} }} \right] ^\mathrm{T}\),

$$\begin{aligned} S=\left[ {{\begin{array}{llll} {\frac{1}{2}c{\underline{\sigma } }\left( Q \right) -\psi }&{} \varsigma &{} \vartheta &{} \chi \\ \varsigma &{} \kappa &{} 0&{} 0 \\ \vartheta &{} 0&{} {{\underline{\sigma } }\left( \Lambda \right) }&{} 0 \\ \chi &{} 0&{} 0&{} \theta \\ \end{array} }} \right] . \end{aligned}$$

Then, \(V_z (z)\) is positive definite if \(S>0\) and \(\left\| z \right\| >\frac{\left\| K \right\| }{{\underline{\sigma } }(S)}\).

With Sylvester’s criterion, \(S>0\) if

$$\begin{aligned}&\frac{1}{2}c{\underline{\sigma } }\left( Q \right) -\psi>0 \nonumber \\&\quad \kappa \left( {\frac{1}{2}c{\sigma }\left( Q \right) -\psi } \right) -\varsigma ^{2}>0 \nonumber \\&\quad {\underline{\sigma } }\left( \Lambda \right) \left[ {\kappa \left( {\frac{1}{2}c{\underline{\sigma } }\left( Q \right) -\psi } \right) -\varsigma ^{2}} \right] -\kappa \vartheta ^{2}>0 \nonumber \\&\quad \theta \left\{ {{\underline{\sigma } }\left( \Lambda \right) \left[ {\kappa \left( {\frac{1}{2}c{\underline{\sigma } }\left( Q \right) -\psi } \right) -\varsigma ^{2}} \right] -\kappa \vartheta ^{2}} \right\} \nonumber \\&\qquad -\kappa \chi ^{2}{\underline{\sigma } }\left( \Lambda \right) >0 \end{aligned}$$

Solve the above inequalities, we can get the condition (22).

Define \(B_d \) as

$$\begin{aligned} B_d =\frac{\left\| K \right\| _1 }{{\underline{\sigma } }(S)}=\frac{B_M \bar{{\sigma }}(P)\bar{{\sigma }}(L+B)+\kappa W_M +\theta g_M }{{\underline{\sigma } }(S)} \nonumber \\ \end{aligned}$$

Then, if \(z\ge B_d \), \(\left\| z \right\|>\frac{\left\| K \right\| _1 }{{\underline{\sigma }}(S)}>\frac{\left\| K \right\| }{{\underline{\sigma }}(S)}\) holds. Under condition (22), we have \({\dot{V}}\le -V_z (z)\) with \(V_z (z)\) being positive definite.

By [3], we have

$$\begin{aligned} {\underline{\sigma }}(\Gamma )\left\| z \right\| ^{2}\le V\le \bar{{\sigma }}(\varphi )\left\| z \right\| ^{2} \end{aligned}$$

where \(\Gamma =\hbox {diag}\left( {\frac{{\underline{\sigma } }(P)}{2},\hbox { }\frac{1}{2\bar{{\sigma }}(F)},\frac{1}{2}\hbox { },\frac{1}{2\bar{{\sigma }}(\eta )}} \right) \) and \(\varphi =\hbox {diag}\left( {\frac{\bar{{\sigma }}(P)}{2},\hbox { }\frac{1}{2{\sigma }(F)},\frac{1}{2},\hbox { }\frac{1}{2{\sigma }(\eta )}} \right) \).

According to Theorem 4.18 in [30], we can conclude that for any \(z(t_0 )\) there exists a \(T_0 \) such that

$$\begin{aligned} \left\| {z(t)} \right\| \le \sqrt{\frac{\bar{{\sigma }}(\varphi )}{{\underline{\sigma } }(\Gamma )}}B_d ,\forall t\ge t_0 +T_0 \end{aligned}$$

Define \(d=\min _{\left\| z \right\| \ge B_d } V_z (z)\), then according to [30]

$$\begin{aligned} T_0 =\frac{V(t_0 )-\bar{{\underline{\sigma } }}(\phi )B_d^2 }{d} \end{aligned}$$

With z, (39) implies that the synchronization error r is ultimately bounded. Then, according to Lemma 1, \(\delta _p \) and \(\delta _v \) are CUUB and all nodes in \({{\mathcal {G}}}\) achieve the synchronization to the leader.

For part (2) of Theorem 1, the state \(x_{p,i} ,i\in N\) and\(_{ }x_{v,i} ,i\in l_2 \) are bounded \(\forall t\ge t_0 \). From (35), we have

$$\begin{aligned} {\dot{V}}\le -{\sigma }(S)\left\| z \right\| ^{2}+\left\| K \right\| \left\| z \right\| \end{aligned}$$

According to (38) and (41), we have

$$\begin{aligned} \frac{d}{dt}(\sqrt{V})\le -\frac{{\underline{\sigma } }(S)}{2\bar{{\sigma }}(\varphi )}\sqrt{V}+\frac{\left\| K \right\| }{2\sqrt{{\underline{\sigma } }(\Gamma )}} \end{aligned}$$

Then, we can conclude that V(t) is bounded for \(t\ge t_0 \) under Corollary 1.1 in [31]. Since (27) implies \(\left\| r \right\| ^{2}\le \frac{2V(t)}{{\underline{\sigma } }(P)}\), r(t) is bounded. And, \(x_0 \) is bounded by \(x_{0M} \) in Assumption 1. \(x_{p,i} \hbox { },i\in N\) and\(_{ }x_{v,i} ,i\in l_2 \) are bounded \(\forall t\ge t_0 \). The proof is done. \(\square \)

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Li, X., Shi, P. & Wang, Y. Distributed cooperative adaptive tracking control for heterogeneous systems with hybrid nonlinear dynamics. Nonlinear Dyn 95, 2131–2141 (2019). https://doi.org/10.1007/s11071-018-4681-4

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  • Control synchronization
  • Consensus tracking
  • Neural networks
  • Heterogeneous multi-agent systems
  • Unknown nonlinear dynamics