In this study, the cluster synchronisation problem of Lur’e networks is focused with non-linear coupling. There are both identical and non-identical nodes in the dynamical system and the coupling matrix is asymmetrical. By using the linear and non-linear negative feedback control schemes, the Lyapunov stability theorem and linear matrix inequality, sufficient conditions are obtained that guarantee the realisation of the cluster synchronisation pattern for all initial values. Numerical simulation results are also given to support the validity of the main results.

A computed force control system using functional link radial basis function network with asymmetric membership function (FLRBFN-AMF) for three-dimension motion control of a piezo-flexural nanopositioning stage (PFNS) is proposed in this study. First, the dynamics of the PFNS mechanism with the introduction of a lumped uncertainty including the equivalent hysteresis friction force are derived. Then, a computed force control system with an auxiliary control is proposed for the tracking of the reference contours with improved steady-state response. Since the dynamic characteristics of the PFNS are non-linear and time varying, a computed force control system using FLRBFN-AMF is designed to improve the control performance for the tracking of various reference trajectories, where the FLRBFN-AMF is employed to estimate a non-linear function including the lumped uncertainty of the PFNS. Moreover, by using the asymmetric membership function, the learning capability of the networks can be upgraded and the number of fuzzy rules can be optimised for the functional link radial basis function network. Furthermore, the adaptive learning algorithms for the training of the parameters of the FLRBFN-AMF online are derived using the Lyapunov stability theorem. Finally, some experimental results for the tracking of various reference contours of the PFNS are given to demonstrate the validity of the proposed control system.

In this study, we have considered the cooperative output synchronisation problem of non-linear multi-agent system. The dynamic of each agent in the network is a class of genuinely non-linear system, named as high-order power integrator that may be uncontrollable after linearisation around the origin. Disturbance is considered in the input channel of each agent as well. Combined Graph theory with the method of adding a power integrator, a distributed controller is designed recursively for each agent to achieve the control objective. The distributed controller of each agent has three parts: state feedback of itself, output information of its neighbours and an adaptive disturbance compensator. It is proved that when the undirected graph is connected, all agents’ outputs in the network can be synchronised, that is, cooperative output synchronisation is achieved. Simulation result is presented to verify its effectiveness.

This study presents a new discrete-time sliding mode control (DSMC) scheme with applications to precise motion control of piezoelectric actuators. Different from existing DSMC algorithms whose implementations rely on the construction of state observers for providing the state feedback, a simple yet effective DSMC strategy is developed based on a discrete-time model without using the state observer. Hence, one distinctive feature of the proposed DSMC lies in that it is very easy to implement. Only a second-order plant model is needed whereas the modelling of piezoelectric non-linearities is not required, which further simplifies the practical implementation process. The local stability of the closed-loop system is proved in theory and the effectiveness of the DSMC is demonstrated by several experimental studies. Results show that the DSMC strategy is superior to proportional-integral-derivative control in terms of transient response speed, positioning accuracy and robustness against external disturbances. The reported method can be extended for precise motion control of other second-order systems as well.

This study presents a new set of formulae for the design of discrete proportional–integral–derivative (PID) controllers under requirements on steady-state performance and robustness specifications, such as phase and the gain margins, as well as the gain crossover frequency. The proposed technique has the advantage of avoiding trial-and-error procedures or approximations connected to an a posteriori discretisation. This method can also be implemented as a graphical design procedure in the Nyquist plane. The plot of suitable regions can be used to check a priori if the problem leads to feasible values of the PID parameters.

Stabilisation design methods for one-sided Lipschitz non-linear differential inclusions are considered in this study. Sufficient conditions of exponential stabilisation for the closed-loop system are given by solving linear matrix inequalities. Two kinds of continuous control laws are designed such that the closed-loop systems are exponentially stable at the origin, respectively. Finally, two numerical examples are given to illustrate the effectiveness of the proposed design technique.

The purpose of this study is to investigate the stability for a class of discrete-time delay Markovian jump systems with stochastic non-linearity and impulses. By using stochastic Lyapunov functionals, some new results are given. Impulse effects on the stability of the systems are revealed. Some examples together with their simulations are also presented to illustrate the effectiveness of the proposed results.

The second-order sliding mode control generates important properties for closed-loop systems, such as robustness, disturbance rejection and finite-time convergence. In this study, it is shown that the adding a power technique plus the nested saturation method will bring in a new second-order sliding mode control scheme for non-linear systems with relative degree two. Based on this, a second-order sliding mode controller is constructed by imposing a natural assumption on the sliding mode dynamics, that is, the uncertainty of the sliding mode dynamics can be bounded by a known function instead of a constant. Under the proposed sliding mode controller, it is proved that the closed-loop system is not only globally convergent, but also locally finite-time stable, which implies the global finite-time stability. Finally, the effectiveness of the proposed method is verified by a numerical example.