Inverse optimal design of input-to-state stabilisation for affine nonlinear systems with input delays

We establish robustness of the predictor feedback control law to perturbations appearing at the system input for affine nonlinear systems with time-varying input delay and additive disturbances. Furthermore, it is shown that it is inverse optimal with respect to a differential game problem. All of the stability and inverse optimality proofs are based on the infinite-dimensional backstepping transformation and an appropriate Lyapunov functional. A single-link manipulator subject to input delays and disturbances is given to illustrate the validity of the proposed method.     

Finite-time stabilisation of a class of stochastic nonlinear systems with dead-zone input

This paper addresses the problem of finite-time stabilisation by state feedback for a class of stochastic nonlinear systems in strict-feedback form. Remarkably, the system under consideration possesses high-order and low-order nonlinear growth rates and the dead-zone input. By delicately combining sign function with adding a power integrator technique, a state feedback controller is designed such that the closed-loop system is finite-time stable in probability. Simulation results of the parallel active suspension system are provided to show the effectiveness of the proposed method.     

Method of stochastic sensitivity synthesis in a stabilisation problem for nonlinear discrete systems with incomplete information

A problem of the synthesis of the assigned probabilistic distribution near the equilibrium of stochastic nonlinear discrete-time system with incomplete state information is considered. We construct a static feedback regulator which provides a required stochastic sensitivity of this equilibrium. It is shown that this problem is reduced to the solution of some quadratic matrix equations. The solvability of these quadratic equations is analysed, and attainability sets are described. In the two-dimensional case, two variants of the control input are discussed and compared. These general results are applied to the suppression of large-amplitude oscillations around the equilibria of the stochastically forced Henon model with noisy observations. We show how suggested control technique, by minimising the stochastic sensitivity, allows us to suppress chaos and provide a structural stabilisation.     

Output feedback stabilisation for uncertain nonlinear time-delay systems subject to input constraints

Robust stabilisation of a class of imperfectly known systems with time-varying time-delays via output feedback is investigated. The systems addressed are composed of a nonlinear nominal system influenced by nonlinear perturbations which may be time-, state-, delayed state- and/or input-dependent. The output of the system is modelled by a nonlinear function, which may depend on the delayed states, and inputs, together with a feed-through term. Using bounding information on the perturbations, in terms of specified growth conditions, classes of unconstrained and constrained output feedback controllers are designed in order to guarantee a prescribed stability property for the closed-loop systems, provided appropriate stability criteria hold. Two stability criteria are given: one in terms of a linear matrix inequality (LMI) and the other is algebraic in nature, obtained using a Ger?gorin Theorem.     

Global adaptive stabilisation for nonlinear systems with unknown control directions and input disturbance

This paper addresses the global adaptive stabilisation via switching and learning strategies for a class of uncertain nonlinear systems. Remarkably, the systems in question simultaneously have unknown control directions, unknown input disturbance and unknown growth rate, which makes the problem in question challenging to solve and essentially different from those in the existing literature. To solve the problem, an adaptive scheme via switching and learning is proposed by skilfully integrating the techniques of backstepping design, adaptive learning and adaptive switching. One key point in the design scheme is the introduction of the learning mechanism, in order to compensate the unknown input disturbance, and the other one is the design of the switching mechanism, through tuning the design parameters online to deal with the unknown control directions, unknown bound and period of input disturbance and unknown growth rate. The designed controller guarantees that all the signals of the resulting closed-loop systems are bounded, and furthermore, the closed-loop system states globally converge to zero.     

Finite-time output feedback stabilisation for a class of feedforward nonlinear systems with input saturation

This paper considers the problem of global finite-time stabilisation by output feedback for a class of feedforward (upper triangular) nonlinear systems with input saturation. Based on the finite-time stability theorem, and by skillfully using the homogeneous domination approach and the nested saturation technique, a saturated output feedback controller is successfully constructed, which renders the origin of the closed-loop system globally finite-time stable. In simulation studies, a numerical example is illustrated to show the effectiveness of the control scheme. Moreover, the design strategy is successfully applied to solve the saturated finite-time control problem for vertical wheel on rotating table.     

Global stabilisation of nonlinear delay systems with a compact absorbing set

Predictor-based stabilisation results are provided for nonlinear systems with input delays and a compact absorbing set. The control scheme consists of an inter-sample predictor, a global observer, an approximate predictor, and a nominal controller for the delay-free case. The control scheme is applicable even to the case where the measurement is sampled and possibly delayed. The input and measurement delays can be arbitrarily large but both of them must be constant and accurately known. The closed-loop system is shown to have the properties of global asymptotic stability and exponential convergence in the disturbance-free case, robustness with respect to perturbations of the sampling schedule, and robustness with respect to measurement errors. In contrast to existing predictor feedback laws, the proposed control scheme utilises an approximate predictor of a dynamic type that is expressed by a system described by integral delay equations. Additional results are provided for systems that can be transformed to systems with a compact absorbing set by means of a preliminary predictor feedback.     

Inverse optimal control for strict‐feedforward nonlinear systems with input delays

We consider inverse optimal control for strict‐feedforward systems with input delays. A basic predictor control is designed for compensation for this class of nonlinear systems. Furthermore, the proposed predictor control is inverse optimal with respect to a meaningful differential game problem. For a class of linearizable strict‐feedforward system, an explicit formula for compensation for input delay, which is also inverse optimal with respect to a meaningful differential game problem, is also acquired. A cart with an inverted pendulum system is given to illustrate the validity of the proposed method.     

Two‐stage stability control for strict‐feedback systems with input delays and input nonlinear uncertainties

In this paper, a two‐stage control procedure is proposed for stabilization of a class of strict‐feedback systems with unknown constant time delays and nonlinear uncertainties in the input. A nominal controller is first designed to compensate input time delays without considering input nonlinear uncertainties. Extended from backstepping algorithm, input delay compensation is realized by means of predicted states that are computed through integration of cascaded system dynamics, making the nominal closed‐loop system asymptotically stable. Based on the nominal controller presented for the input delay system, a multi‐timescale system is subsequently developed to estimate the unknown input nonlinearity and make the estimate approach the nominal control input as fast as possible. It is proved that the proposed control scheme can make states of the strict‐feedback systems converge to zero and all the signals of the closed‐loop systems are guaranteed to be bounded in the presence of input time delays and nonlinear uncertainties. Simulation verification is carried out to illuminate the effectiveness of the proposed control approach.     

Stabilisation for switched linear systems with time-varying delay and input saturation

This article investigates the stabilisation problems for continuous-time and discrete-time switched systems with time-varying delay and saturated control input. Based on dwell time switching signals and multiple Lyapunov functional method, stabilisation conditions are well obtained in the context of linear matrix inequalities. To estimate attractive regions as large as possible, the feasibility problems are translated into optimisation problems. In addition, the corresponding results are presented for linear time-delay systems and switched delay-free systems, which improve and supplement some existing ones in the literature. Finally, numerical examples and simulations are given to illustrate the effectiveness and values of the proposed results.     

Delay compensation for linear systems with both state and distinct input delays

This paper is concerned with the stabilization of linear systems with both state and distinct input delays. Nested predictor feedback controllers are designed to predict the future states such that the distinct input delays that can be arbitrarily large yet bounded are compensated completely. It is shown that the compensated closed‐loop system possesses the same characteristic equation as the closed‐loop system without distinct input delays. Both continuous‐time and discrete‐time time‐delay systems are studied in this paper. Moreover, the safe implementation problem for the continuous‐time nested predictor feedback controller is solved via adding input filters. Three numerical examples show the effectiveness of the proposed approaches.     

Observer‐predictor feedback for consensus of discrete‐time multiagent systems with both state and input delays

This article is concerned with the consensus problem for discrete‐time multiagent systems with both state and input delays. Single observer‐predictor‐based protocols and multiple observer‐predictors feedback protocols are simultaneously established to predict the future state such that the input delay that can be arbitrarily large yet bounded is completely compensated. It is shown that the consensus of the multiagent system can be achieved by the single/multiple observer‐predictors feedback protocol. Moreover, sufficient conditions guaranteeing the consensus of the multiagent system are provided in terms of the stability of some simple observer‐error systems, and the separation principle is discovered. Finally, a numerical example is worked out to illustrate the effectiveness of the proposed approaches.     

Global output feedback stabilisation of stochastic high-order feedforward nonlinear systems with time-delay

This paper considers the problem of output feedback stabilisation for stochastic high-order feedforward nonlinear systems with time-varying delay. By using the homogeneous domination theory and solving several troublesome obstacles in the design and analysis, an output feedback controller is constructed to drive the closed-loop system globally asymptotically stable in probability.     

Linear quadratic regulation for linear time-varying systems with multiple input delays

This paper studies the classic linear quadratic regulation (LQR) problem for both continuous-time and discrete-time systems with multiple input delays. For discrete-time systems, the LQR problem for systems with single input delay has been studied in existing literature, whereas a solution to the multiple input delay case is not known to our knowledge. For continuous-time systems with multiple input delays, the LQR problem has been tackled via an infinite dimensional system theory approach and a frequency/time domain approach. The objective of the present paper is to give an explicit solution to the LQR problem via a simple and intuitive approach. The main contributions of the paper include a fundamental result of duality between the LQR problem for systems with multiple input delays and a smoothing problem for an associated backward stochastic system. The duality allows us to obtain a solution to the LQR problem via standard projection in linear space. The LQR controller is simply constructed by the solution of one backward Riccati difference (for the discrete-time case) or differential (for the continuous-time case) equation of the same order as the plant (ignoring the delays).     

Robust predictor feedback for discrete-time systems with input delays

This work studies the design problem of feedback stabilisers for discrete-time systems with input delays. A backstepping procedure is proposed for disturbance-free discrete-time systems. The feedback law designed by using backstepping coincides with the predictor-based feedback law used in continuous-time systems with input delays. However, simple examples demonstrate that the sensitivity of the closed-loop system with respect to modelling errors increases as the value of the delay increases. The paper proposes a Lyapunov redesign procedure that can minimise the effect of the uncertainty. Specific results are provided for linear single-input discrete-time systems with multiplicative uncertainty. The feedback law that guarantees robust global exponential stability is a nonlinear, homogeneous of degree 1 feedback law.     

State feedback stabilisation for stochastic high-order feedforward nonlinear systems with input time-delay

This paper devotes to solve the problem of state feedback stabilisation for a class of stochastic high-order feedforward nonlinear systems with input time-delay. First, the delay-dependent control input is skillfully handled by using variable transformation technique. Then, by introducing the homogeneous domination approach and the appropriate Lyapunov--Krasovskii functional to deal with the time-delay, we construct a state feedback controller through a backstepping recursive design. It is shown that the proposed controller can guarantee the closed-loop system to be globally asymptotically stable (GAS) in probability. Finally, an example is given to verify the effectiveness of the obtained analytical results.     

Stabilisation and consensus of linear systems with multiple input delays by truncated pseudo-predictor feedback

This paper provides a new approach referred to as pseudo-predictor feedback (PPF) for stabilisation of linear systems with multiple input delays. Differently from the traditional predictor feedback which is from the model reduction appoint of view, the proposed PPF utilises the idea of prediction by generalising the corresponding results for linear systems with a single input delay to the case of multiple input delays. Since the PPF will generally lead to distributed controllers, a truncated pseudo-predictor feedback (TPPF) approach is established instead, which gives finite dimensional controllers. It is shown that the TPPF can compensate arbitrarily large yet bounded delays as long as the open-loop system is only polynomially unstable. The proposed TPPF approach is then used to solve the consensus problems for multi-agent systems characterised by linear systems with multiple input delays. Numerical examples show the effectiveness of the proposed approach.     

Predictor-based control of linear systems with large and variable measurement delays

This paper concerns the problem of the control of linear systems by means of feedback from delayed output, where the delay is known and time-varying. The main advantage of the approach is that it can be applied to systems with any delay bound, i.e. not only small delays. The predictor is based on a combination of finite-dimensional elementary predictors whose number can be suitably chosen to compensate any delay. The single-predictor element is an original proposal, and the class of delays to which the schema can be applied includes, but it is not limited to, continuous delay functions.