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In this article, the consensus problem of linear systems is revisited from a novel geometric perspective. The interaction network of these systems is assumed to be piecewise fixed. Moreover, it is allowed to be disconnected at any time but holds a quite mild joint connectivity property. The system matrix is marginally stable and the input matrix is not of full-row rank. By directly examining the subspace determined by the network, we first establish convergence by resorting to an observability condition. Then, according to joint connectivity, we are able to extend this convergence uniformly to the entire orthogonal complement of the consensus manifold. In this way, we work out the necessary and sufficient condition for exponential consensus. It turns out that, with a suitably designed feedback matrix, exponential consensus can be realized globally and uniformly if and only if a jointly $(\delta,T)$ -connected condition and an observability condition relying only on the system and input matrices are satisfied. We also characterize the lower bound of the convergence rate. Simple yet effective examples are presented to illustrate the findings. [ABSTRACT FROM AUTHOR]

Differential game is an effective technique to describe the negotiation between the humans and robots, which is widely used to realize the trajectory tracking tasks in the human-robot interaction (HRI). However, most existing works consider the control-affine HRI systems and assume the desired trajectory is available to both the human and the robot, which limit the scope of applications. To overcome these difficulties, this work focuses on the nonaffine HRI system and supposes that the desired trajectory is not available to the robot. A novel differential game framework encoding the desired trajectory estimator is proposed, where the desired trajectory is estimated via the Gaussian process regression (GPR) technique. To address the challenge arising from the nonlinearity of the HRI system, we equivalently transform the original problem into the one in a differentially flat space, and seek the equilibrium strategies for the transformed problem substitutionally. We further prove that the trajectory tracking error satisfies a probabilistic bound, whose confidence interval tightens as the decrease of noise variance during the interaction. Comparative simulation results show that our method outperforms the learning-based method in terms of robustness, parameters setting, and time consumption. Experiment results further show that the tracking error under the proposed human-robot cooperative algorithm is reduced by 55% compared to the human direct control.