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This note presents new results on the design of the process noise covariance matrix in Kalman filtering. The proposed design belongs to the realm of worst case methods: the optimal covariance is sought such as to maximize an upper bound on the estimation error matrix under a total budget constraint. Optimality conditions are derived along with a numerical algorithm for iterative approximation of the optimal solution. The optimal solution features a few time points and the associated covariances. Broadly speaking, the process noise is only “active” at the time points that enhance its controllability. The importance of the method is that it bounds from above the performance of any Kalman filter with a different design. Monte Carlo simulations on a double harmonic system numerical example illustrate the good convergence, performance, and low sensitivity to initial guesses of the proposed method. Moreover, in terms of estimation accuracy, the Kalman filter with the optimal design outperforms a robust $H_2$ filter designed to cope with an equivalent uncertainty. [ABSTRACT FROM AUTHOR]

Inspired by the interesting findings regarding the hierarchy and coordination of pigeons in Nagy et al.  [Nature, 2010], we revisit the discrete-time leader–follower consensus problem with switching topologies. The purpose of this paper is to investigate the impact of hierarchical topologies and followers’ self-loops on the convergence performance of leader–follower consensus, including the convergence rate and robustness to switching topologies. We first study the fixed topology case, and show that the followers converge to the leader's state in finite time if and only if each follower has no self-loops and the topology is hierarchical. However, we show via counterexample, that leader–follower consensus may not be achieved when some followers have no self-loops and the topology is switching; even if each interaction graph has a spanning tree rooted at the leader. With the aid of binary relation theory, we further develop a new approach to present a novel sufficient condition for leader–follower consensus with switching topologies and with some followers having no self-loops. We prove that when no followers have self-loops and the same hierarchical organization is kept under the switching topologies, then the fastest rate of convergence in leader–follower consensus can be achieved; even in the presence of complex dynamic topologies. This is consistent with the natural phenomena found in pigeons by Nagy et al. [ABSTRACT FROM AUTHOR]

A fault-tolerant observer design methodology is proposed. The aim is to guarantee a minimum level of closed-loop performance under all possible sensor fault combinations while optimizing performance under the nominal, fault-free condition. A novel approach is proposed to tackle the combinatorial nature of the problem, which is computationally intractable even for a moderate number of sensors, by recasting the problem as a robust performance problem, where the uncertainty set is composed of all combinations of a set of binary variables. A procedure based on an elimination lemma and an extension of a semidefinite relaxation procedure for binary variables is then used to derive sufficient conditions (necessary and sufficient in the case of one binary variable) for the solution of the problem which significantly reduces the number of matrix inequalities needed to solve the problem. The procedure is illustrated by considering a fault-tolerant observer switching scheme in which the observer outputs track the actual sensor fault condition. A numerical example from an electric power application is presented to illustrate the effectiveness of the design. [ABSTRACT FROM PUBLISHER]

The integer order differentiation by integration method based on the Jacobi orthogonal polynomials for noisy signals was originally introduced by Mboup, Join and Fliess. We propose to extend this method from the integer order to the fractional order to estimate the fractional order derivatives of noisy signals. Firstly, two fractional order differentiators are deduced from the Jacobi orthogonal polynomial filter, using the Riemann–Liouville and the Caputo fractional order derivative definitions respectively. Exact and simple formulae for these differentiators are given by integral expressions. Hence, they can be used for both continuous-time and discrete-time models in on-line or off-line applications. Secondly, some error bounds are provided for the corresponding estimation errors. These bounds allow to study the design parameters' influence. The noise error contribution due to a large class of stochastic processes is studied in discrete case. The latter shows that the differentiator based on the Caputo fractional order derivative can cope with a class of noises, whose mean value and variance functions are polynomial time-varying. Thanks to the design parameters analysis, the proposed fractional order differentiators are significantly improved by admitting a time-delay. Thirdly, in order to reduce the calculation time for on-line applications, a recursive algorithm is proposed. Finally, the proposed differentiator based on the Riemann–Liouville fractional order derivative is used to estimate the state of a fractional order system and numerical simulations illustrate the accuracy and the robustness with respect to corrupting noises. [ABSTRACT FROM AUTHOR]

This paper presents a new approach to event-triggered control for nonlinear uncertain systems by using the notion of input-to-state stability (ISS) and the nonlinear small-gain theorem. The contribution of this paper is threefold. First, it is proved that infinitely fast sampling can be avoided if the system is input-to-state stabilizable with the sampling error as the external input and the corresponding ISS gain is locally Lipschitz. No assumption on the existence of known ISS-Lyapunov functions is made in the discussions. Moreover, the forward completeness problem with event-triggered control is studied systematically by using ISS small-gain arguments. Second, the proposed approach gives rise to a new self-triggered sampling strategy for a class of nonlinear systems subject to external disturbances. If an upper bound of the external disturbance is known, then the closed-loop system can be designed to be robust to the external disturbance, and moreover, the system state globally asymptotically converges to the origin if the external disturbance decays to zero. Third, a new design method is developed for event-triggered control of nonlinear uncertain systems in the strict-feedback form. It is particularly shown that the ISS gain with the sampling error as the input can be designed to satisfy the proposed condition for event-triggered control and self-triggered control. [ABSTRACT FROM PUBLISHER]