An underactuated mechanical system (UMS) is a system which has fewer independent control actuators than degrees of freedom to be controlled. Control of UMS is considered as one of the most active fields of research because of its diverse engineering applications. This survey reviews UMS from its history to the state-of-the-art research on modelling, classification, control, and to some extent, provides some unique insights for bottleneck issues and future research directions.

Many practical non-linear systems can be described by non-linear auto-regressive moving average (NARMA) system models, whose stabilisation problem is challenging in the presence of large parametric uncertainties and non-parametric uncertainties. In this work, to address this challenging problem for a wide class of discrete-time NARMA systems, in which there are uncertain periodic parameters as well as uncertain non-linear part with unknown periodic time delays, we develop adaptive predictive control laws using the key ideas of ‘future outputs prediction’ and ‘nearest-neighbour compensation’, among which the former is carried out to overcome the non-causalness problem and the latter novel idea is proposed to completely compensate for the effect of non-linear uncertainties as well as unknown time delays. To achieve the desired asymptotic tracking performance in the presence of semi-parametric uncertainties with time delays, an ‘n-step parameter update law’ is first designed, based on which an ‘one-step update law’ is then elaborately constructed to obtain smoother closed-loop signals. This study in general develops a systematic adaptive control framework for periodic NARMA systems with guaranteed boundedness stability and asymptotic tracking performance, which are established by rigorous theoretic proof and verified by simulation studies.

At beginning, the kinematic model and dynamic model of a differential mobile robot (DMR), and the dynamic model of left- and right-wheel DC motors are combined to be the controlled system. The control inputs of the proposed controlled system are the input voltages for the left- and right-wheel motors. The (indirect) outputs are the two-dimensional (2D) position and orientation of a DMR. Owing to the under-actuated characteristic, the direct reference input (i.e. two desired motor currents) using the first sliding surface is designed, so that the 2D position and orientation of the DMR are simultaneously controlled by two motor currents (i.e. the direct output). On the other hand, the second sliding surface is designed as the linear dynamics of tracking error of motor currents. Under completely (or partially) known frictions and uncertainties, the hierarchical sliding-mode under-actuated control with suitable conditions is designed, such that two motor currents asymptotically track two desired motor currents, respectively. Then the asymptotic tracking for the 2D position and orientation of a DMR is achieved. The simulations for various trajectories, completely and partially known frictions and uncertainties, and control parameters are presented to evaluate the effectiveness and robustness of the proposed method.

For electric bikes, this work develops linear parameter-varying (LPV) game-theoretic synthesis to regulate the trade-off between energy consumption per distance and propulsion capability in transience. Such a regulation plays like the transmission in transient state, compared to the gear transmission in steady state. Here, the propulsion dynamics is identified with LPV parameterisation that perfectly captures the non-linearity of the dynamics. Incorporation of this LPV plant with per-distance energy-motion performance forms a generalised plant with multiple L ₂-gain objectives. This generalised plant is specially constructed such that energy and motion objectives can share the same control and estimation storages without bringing conservatism to numerical solutions. Embedding the transient transmission into a dsPIC microcontroller, the authors simulate and experiment to credit such a LPV parameterisation and game-theoretic control for DC electric propulsion.

Stable-protocol (SP) output consensus analysis and design problems for high-order linear time-invariant swarm systems with time-varying delays are investigated. First, a dynamic output feedback consensus protocol is proposed on the basis of the observable decomposition, and a necessary and sufficient condition for SP output consensus is given, which transforms the SP output consensus problem into asymptotical stability problems of multiple lower-dimensional subsystems. Then, in terms of linear matrix inequalities (LMIs), a sufficient condition for SP output consensualisation is presented, which can guarantee the scalability of swarm systems since it includes only five LMI constrains independent of the number of agents. Furthermore, an explicit expression of the output consensus function, which is independent of the time-varying delay, is shown and the impacts of initial states of agents and consensus protocols on the output consensus function are determined, respectively. Finally, a numerical example is given to demonstrate theoretical results.

Output-feedback control of general underactuated mechanical systems is currently considered a major open problem. This study is focused on the output-feedback stabilisation control problems for a special class of underactuated mechanical systems, which appear in robotics and aerospace applications. For the synthesis of controller, first, the considered underactuated mechanical system is explicitly transformed into two cascade connected subsystems, and then an auxiliary filter-based virtual stabilisation controller is developed to locally asymptotically stabilise the first subsystem. Further, the designed virtual controller is again involved into the second subsystem using backstepping procedure to construct the actual control law, in which a series of auxiliary time-varying first-order low-pass filters are also implemented to avoid using the derivative of the system non-linear functions. Moreover, in the second step, finite-time observer technique is utilised to precisely reconstruct the immeasurable states to achieve the finite-time stabilisation control in the sense of output feedback. Lyapunov analysis shows the local asymptotic stability of the closed-loop system through the cascade system stability criteria. Simulation results are presented by using two benchmark non-linear underactuated mechanical systems to demonstrate the feasibility and the effectiveness of the proposed controller.

In non-linear system identification, results from traditional non-parametric identification techniques contain both linear and non-linear contributions. When Gaussian excitation signals (including random-phased multisines) are used, the non-linear contributions are noise-like and therefore not easy to distinguish from environment noise and measurement noise. In contrast, when excitation signals based on binary maximum-length sequences (m-sequences) are used, a particular property of the sequences results in the non-linear contributions being structured. It is shown in this study that it is possible to take advantage of this structure by using a median-based averaging technique, rather than the more traditional arithmetic mean-based averaging, to obtain better identification performance.

This study is concerned with dynamic output feedback (DOF) H _∞ control for continuous-time networked control systems (NCSs). The packet dropouts and network-induced delays in the sensor-to-controller channel, and network-induced delays in the controller-to-actuator channel are taken into account. A linear estimation-based method is proposed to estimate the measurement output, and the non-uniform distribution characteristic of the measurement output arrival instant is taken into full consideration to establish a new model for the considered NCSs. Based on the newly established model and a new Lyapunov functional, the problem of DOF controller design is studied. A numerical example is given to illustrate the effectiveness of the proposed DOF controller design.

This study is concerned with observability and accessibility of discrete-time non-linear switched impulsive systems, for which the problem is more complicated than that for general discrete systems and has novel features. A new method from the combination of differential geometric theory and Lie group analysis is developed. Specifically, under the geometric framework of the system, infinitesimal invariance for the observability and accessibility is investigated, respectively. Based on the invariance investigation, explicit criteria for local observability and local accessibility are derived. Numerical examples are discussed to show the effectiveness of the proposed methods, and the features of observability and accessibility for discrete switched impulsive systems.

This study presents new sufficient conditions for Hurwitz and Schur stability of interval matrices. Tight bounds for the spectrum of interval matrices are estimated using computationally simple optimisation problems. The conservativeness is reduced further by application of ordinary similarity transformation. A necessary and sufficient vertex based criterion for the stability of a subclass of interval systems in continuous and discrete-time cases is also proposed. This enables the spectra for this class of interval systems to be determined exactly. A selection of various examples adopted from existing literature is used to demonstrate the utility of the proposed criteria.

This study considers the problem of robust H _∞ control using decentralised state feedback controllers for a class of large-scale systems with Markov jump parameters. A sufficient condition is developed to design controllers using local system states and local system operation modes. The sufficient condition is given in terms of rank constrained linear matrix inequalities. An illustrative numerical example is given to demonstrate the developed theory.

This study examines the control of a passive plant utilising strictly passive feedback as motivated by the passivity theorem. A state representation of the plant is assumed with very few conditions imposed. No assumptions are made on the state representation of the feedback control law. However, some mild additional input–output properties of the feedback control law are assumed. Global stability of the closed-loop system and asymptotic convergence of a subset of the states is proven using invariance principles. The theoretical results in the study are applied to a number of application examples, demonstrating that a much broader class of controllers can deliver closed-loop global stability and asymptotic convergence, unlike the previous works in the literature where the examples are taken from. In one of the examples, it is demonstrated that actuator saturation constraints can readily be handled using the theory presented in this study.

This study investigates a distributed adaptive consensus tracking problem for multiple non-linear strict-feedback systems with parametric uncertainties under a directed communication graph topology. It is assumed that the information of a leader is available for only a limited subset of followers. The distributed adaptive dynamic surface design approach is presented to design the local controller of each follower to guarantee that all networked followers with unmatched parametric uncertainties synchronise to the leader regardless of a lack of the shared information on the communication links. From the Lyapunov stability theorem, it is shown that the consensus tracking errors are cooperatively semiglobally uniformly ultimately bounded and converge to an adjustable neighbourhood of the origin.

Finite-time stability and finite-time boundedness for a class of switched linear systems with time-varying delay are studied. Sufficient conditions which guarantee switched linear systems with time-varying delay finite-time stable or finite-time bounded are presented. These conditions are delay-dependent and are given in terms of linear matrix inequalities. Average dwell time of switching signals is also given such that switched linear systems are finite-time stable or finite-time bounded. Moreover, finite-time weighted L ₂-gain of switched linear systems with time-varying delay are also given to measure its disturbance tolerance capability in the fixed time interval. Detail proofs are accomplished by using multiple Lyapunov-like functions. An example is employed to verify the efficiency of the proposed method.