Study On Cooperative Control Of Lagrangian Systems

The mathematical model of many mechanical systems, such as manipulators ofrobots, autonomous vehicles and satellites, can be represented by Lagrangianequation. It is more reliable that multiple vehicles accomplish complex taskscooperatively than the single one. The multi-agent system that is composed ofmultiple simple mechanical objects is called Lagrangian system. The cooperativecontrol of Lagrangian system is the research front and complex research field ofmulti-agent systems. Lagrangian systems can be classified into cases of no leaders,one leader and multiple leaders. One of our focuses is the research of consensustracking in finite time of one leader’ case to realize states of followers in accordancewith the leader’ states. The other one is the containment control of Lagrangiannetworks in finite time under multiple leaders’ condition, realizing that followersconverge into the convex hull of leaders.The work firstly introduces preliminaries and notations briefly, including graphtheory, the theory of Lyapunov stability and finite time stability theorem. Next, adetail explanation of the modeling process of Lagrangian systems is displayed. Afterthe modeling, the model of simulation is derived.For the case of one leader, designing cooperative algorithms is the goal of theLagrangian system. The algorithms make followers to track state of the leader infinite time, completing the consensus tracking of the whole system. Firstly, theterminal sliding-mode cooperative control law is proposed based on classicalnonlinear theory of sliding-mode control. The effectiveness and robustness ofproposed cooperative law are proved by using finite-time stability theory. Secondly,a fast terminal sliding-mode cooperative control law is proposed to improve thespeed of convergence, which is one of the disadvantages of terminal sliding-modecooperative control. A new non-singular terminal sliding-mode cooperative controllaw is proposed to solve the problem of singularity, which is another shortcoming ofterminal sliding-mode cooperative control. The effectiveness of proposed laws canbe proved by using finite time stability theory. Lastly, the effectiveness andcorrectness of the new three proposed algorithms are demonstrated by mathematicalsimulation results by the software of Matlab.For multiple leaders’ case, the aim of containment control is to actuatefollowers to converge into the convex hull spanned by leaders in finite time. Theuncertainties of parameters and dynamic leaders are taken into consideration in thedesign. Firstly, two sliding-mode estimators of speed and acceleration are designedwhich can evaluate the speed and acceleration of followers exactly in finite time. Secondly, the proposed distributed adaptive control laws can realize containmentcontrol. The convergence of the proposed law is proved by the Barbalat lemma.Thirdly, simulation results of Matlab are given to verify the effectiveness ofproposed algorithm.