Global set‐point tracking control for a class of nonlinear systems with iISS inverse dynamics

In this paper, we consider the problem of global set‐point tracking control for a class of nonlinear systems with dynamic uncertainty. Unlike the existing works, the investigated system is with the integral input‐to‐state stable (iISS) inverse dynamics and more general uncertain nonlinearities. By using a recursive design method, a partial‐state feedback controller is designed. The tuning function technique is applied in this procedure to avoid the overparametrization. It is shown that the developed control procedure could guarantee that the tracking error is driven to the origin and the other signals are bounded. In addition, it can also reduce to a linear or even a classical PI control law under some sufficient conditions. Simulation results are illustrated to show the effectiveness of the proposed algorithm. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

A mixed numerical–analytical stable pseudo‐inversion method aimed at attaining an almost exact tracking

This paper considers the problem of computing the input u(t) of an internally asymptotically stable, possibly non‐minimum phase, linear, continuous time system Σ yielding a very accurate tracking of a pre‐specified desired output trajectory . The main purpose of the new approach proposed here is to alleviate some limitations that inherent the classical methods developed in the framework of the preview‐based stable inversion, which represents an important reference context for this class of control problems. In particular, the new method allows one to deal with arbitrary and possibly uncertain initial conditions and does not require a pre‐actuation. The desired output to be exactly tracked in steady state is here assumed to belong to the set of polynomials, exponential, and sinusoidal time functions. The desired transient response is specified to obtain a fast and smooth transition toward the steady‐state trajectory , without under and/or overshoot in the case of a set point reset. The transient control input u_t(t) is a priori assumed to be given by a piecewise polynomial function. Once has been specified, this allows the computation of the unknown u_t(t) as the approximate least squares solution of the Fredholm's integral equation corresponding to the explicit formula of the output forced response. The steady‐state input u_s(t) is analytically computed exploiting the steady‐state output response expressions for inputs belonging to the same set of . Copyright © 2013 John Wiley & Sons, Ltd.     

Spline Based Pseudo‐Inversion of Sampled Data Non‐Minimum Phase Systems for an Almost Exact Output Tracking

This paper considers the problem of achieving a very accurate tracking of a pre‐specified desired output trajectory , for linear, multiple input multiple output, non‐minimum phase and/or non hyperbolic, sampled data, and closed loop control systems. The proposed approach is situated in the general framework of model stable inversion and introduces significant novelties with the purpose of reducing some theoretical and numerical limitations inherent in the methods usually proposed. In particular, the new method does not require either a preactuation or null initial conditions of the system. The desired and the corresponding sought input are partitioned in a transient component ( and u_t(k), respectively) and steady‐state ( and u_s(k), respectively). The desired transient component is freely assigned without requiring it to be null over an initial time interval. This drastically reduces the total settling time. The structure of u_t(k) is a priori assumed to be given by a sampled smoothing spline function. The spline coefficients are determined as the least‐squares solution of the over‐determined system of linear equations obtained imposing that the sampled spline function assumed as reference input yield the desired output over a properly defined transient interval. The steady‐state input u_s(k) is directly analytically computed exploiting the steady‐state output response expressions for inputs belonging to the same set of .     

Multiple‐model adaptive control using set‐valued observers

A multiple‐model adaptive control methodology is proposed that is able to provide stability and performance guarantees, for uncertain linear parameter‐varying plants. The identification problem is addressed by taking advantage of recent advances in model falsification using set‐valued observers (SVOs). These SVOs provide set‐valued estimates of the state of the system, according to its dynamic model. If such estimate is the empty set, the underlying dynamic model is invalidated, and a different controller is connected to the loop. The behavior of the proposed control algorithm is demonstrated in simulation, by resorting to a mass–spring–dashpot system. As a caveat, the computational burden associated with the SVOs can be prohibitive under some circumstances. Copyright © 2013 John Wiley & Sons, Ltd.     

Robust set‐point constrained regulation via dynamic inversion

In this paper we present a novel methodology, based on dynamic system inversion, for the set‐point constrained regulation of a scalar plant in presence of structured uncertainties. The approach consists in choosing a polynomial as the output function and then designing both the controller and the reference input in order to minimize the worst‐case settling‐time of the set‐point regulation, with constraints on the maximum absolute value of the control variable and on the maximum overshoot of the output. A comparison with a traditional methodology, which is based on the step‐response, shows how the overall design is significantly simplified and how the worst‐case settling‐time can be greatly reduced. Optimization has been performed by means of genetic algorithms. Copyright © 2001 John Wiley & Sons, Ltd.     

Pre‐Actuation and Optimal State Transition Based Precise Tracking for Maximum Phase System

Precise trajectory tracking for the maximum phase system can be achieved by the stable inversion method. However, its side effect is requiring the sufficient extended zero trajectory. To overcome this drawback, firstly, this paper introduces the concept of stable initial state which is of vital importance to achieve precise output tracking. And then several methods have been presented to obtain the stable initial state. Both the standard method and the pre‐actuation method can be seen as the simplified stable inversion method under certain conditions, but the two methods only perform approximate output tracking. To achieve precise trajectory tracking, the optimal state transition method and the combination method consisting of the pre‐actuation and the optimal state transition are introduced. Finally, the optimal combination method is proposed to obtain the best overall tracking effect. In brief, different methods are fit for different situation but the optimal combination method is the finest method in practical situation. The effectiveness of the proposed methods are verified through numerical simulations for the maximum phase system.     

A nonlinear set‐membership filter for on‐line applications

A guaranteed estimator for a general class of nonlinear systems and on‐line usage is developed and analysed. This filter bounds the linearization error, then applies a linear set‐membership filter such that stability guarantees hold for nonlinear systems. A tight bound on the linearization error is found using interval analysis. This filter recursively estimates an ellipsoidal set in which the true state lies. General assumptions include the use of bounded noises and twice continuously differentiable dynamics. When the system is uniformly observable, it is proven that the nonlinear set‐membership filter is stable. In addition, if no noise is present and the initial error is small, the error between the centre of the estimated set and the true value converges to zero. The result is an estimator which is computationally attractive and can be implemented robustly in real‐time. The proposed method is applied to a two‐state example to demonstrate the theoretical results. Copyright © 2003 John Wiley & Sons, Ltd.     

Sensor‐fault tolerance using robust MPC with set‐based state estimation and active fault isolation

In this paper, a sensor fault‐tolerant control scheme using robust model predictive control (MPC) and set‐theoretic fault detection and isolation (FDI) is proposed. The robust MPC controller is used to control the plant in the presence of process disturbances and measurement noises while implementing a mechanism to tolerate faults. In the proposed scheme, fault detection (FD) is passive based on interval observers, while fault isolation (FI) is active by means of MPC and set manipulations. The basic idea is that for a healthy or faulty mode, one can construct the corresponding output set. The size and location of the output set can be manipulated by adjusting the size and center of the set of plant inputs. Furthermore, the inputs can be adjusted on‐line by changing the input‐constraint set of the MPC controller. In this way, one can design an input set able to separate all output sets corresponding to all considered healthy and faulty modes from each other. Consequently, all the considered healthy and faulty modes can be isolated after detecting a mode changing while preserving feasibility of MPC controller. As a case study, an electric circuit is used to illustrate the effectiveness of the proposed scheme. Copyright © 2016 John Wiley & Sons, Ltd.     

A homotopy method for exact output tracking of some non‐minimum phase nonlinear control systems

This paper presents a method for non‐causal exact dynamic inversion for a class of non‐minimum phase nonlinear systems, which seems to be an alternative to those existing in the literature. This method is based on a homotopy procedure that allows to find a ‘small’ periodic solution of a desired equation by a continuous deformation of a known periodic solution of a simpler auxiliary system. This method allows to face the exact output tracking problem for some non‐minimum phase systems that are well known in the literature, such as the inverted pendulum, the motorcycle and the CTOL aircraft. Copyright © 2008 John Wiley & Sons, Ltd.     

Semi‐global finite‐time observers for multi‐output nonlinear systems

Finite‐time stability is investigated for nonlinear systems, which satisfy uniqueness of solution. First, a new sufficient condition for local finite‐time stability is presented. Next, by using the high‐gain observers and carefully selecting the homogeneity powers and weights, the problem of semi‐global and finite‐time stable observers is studied for multi‐output nonlinear systems with uniform observability and a triangular structure. Then, a design procedure is worked out for such observers. Finally, two numerical examples further verify the validity of the proposed approach. Copyright © 2009 John Wiley & Sons, Ltd.     

Active fault detection based on set‐membership approach for uncertain discrete‐time systems

Active fault detection facilitates determination of the fault characteristics by injecting proper auxiliary input signals into the system. This article proposes an observer‐based on‐line active fault detection method for discrete‐time systems with bounded uncertainties. First, the output including disturbances, measurement noise and interval uncertainties at each sample time is enclosed in a zonotope. In order to reduce the conservativeness in the fault detection process, a zonotopic observer is designed to estimate the system states allowing to generate the output zonotopes. Then, a proper auxiliary input signal is designed to separate the output zonotopes of the faulty model from the healthy model that is injected into the system to facilitate the detection of small fault . Since the auxiliary input signal generation leads to a nonconvex optimization problem, it is transformed into a mixed integer quadratic programming problem. Finally, a case study based on a DC motor is used to show the effectiveness of the proposed method.     

A UD factorization‐based nonlinear adaptive set‐membership filter for ellipsoidal estimation

The extended set‐membership filter (ESMF) for nonlinear ellipsoidal estimation suffers from numerical instability, computation complexity as well as the difficulty in filter parameter selection. In this paper, a UD factorization‐based adaptive set‐membership filter is developed and applied to nonlinear joint estimation of both time‐varying states and parameters. As a result of using the proposed UD factorization, combined with a new sequential and selective measurement update strategy, the numerical stability and real‐time applicability of conventional ESMF are substantially improved. Furthermore, an adaptive selection scheme of the filter parameters is derived to reduce the computation complexity and achieve sub‐optimal estimation. Simulation results have shown the efficiency and robustness of the proposed method. Copyright © 2007 John Wiley & Sons, Ltd.     

Optimal fixed‐structure control for linear non‐negative dynamical systems

In this paper, we develop optimal output feedback controllers for set‐point regulation of linear non‐negative dynamical systems. Specifically, using a constrained fixed‐structure control framework we develop optimal output feedback control laws that guarantee that the trajectories of the closed‐loop system remain in the non‐negative orthant of the state space for non‐negative initial conditions. In addition, we characterize domains of attraction predicated on closed and open Lyapunov level surfaces contained in the non‐negative orthant for unconstrained optimal linear‐quadratic output feedback controllers. Output feedback controllers for compartmental systems with non‐negative inputs are also given. Copyright © 2004 John Wiley & Sons, Ltd.     

Design of optimal interval observers using set‐theoretic methods for robust state estimation

This article aims to design an optimal interval observer for discrete linear time‐invariant systems. Particularly, the proposed design method first transforms the interval observer into a zonotopic set‐valued observer by establishing an explicit mathematical relationship between the interval observer and the zonoptopic set‐valued observer. Then, based on the established mathematical relationship, a locally optimal observer gain is designed for the interval observer via the equivalent zonotopic set‐valued observer structure and the Frobenious norm‐based size of zonotopes. Third, considering that the dynamics of the optimal interval observer becomes a discrete linear time‐varying system due to the designed time‐varying optimal gain, an optimization problem to obtain a coordinate transformation matrix and the locally optimal observer gain for the interval observer is formulated and handled. Finally, a theoretic comparison on the conservatism of the interval observer and the zonotopic set‐valued observer is made. At the end of this article, a microbial growth bioprocess is used to illustrate the effectiveness of the proposed method.     

Application of the Hirsch Generic Convergence Statement to the Set‐Point Regulation of Excitable‐Transparent‐Monotone Systems

Monotone systems are dynamical systems whose solutions preserve an ordering, relative to the initial data. This paper focuses on the set‐point regulation problem for excitable‐transparent‐monotone single‐input single‐output systems with well‐defined static characteristics. The suggested technique for designing a static feedback controller is inspired from the Hirsch Theorem for monotone autonomous systems. It is shown that the closed‐loop system is strongly monotone and its generic solution converges to the desired set‐point. A remark is given on the application of the suggested design methodology to delay systems, as well as to systems with reaction‐diffusion equations. A simulation example from the biological world demonstrates the effectiveness of the proposed control strategy in real applications.     

Precise Tracking for Continuous‐Time Non‐Minimum Phase Systems

Stable inversion based precise tracking for continuous‐time square or nonsquare non‐minimum phase systems is studied. However, high precision trajectory tracking of non‐minimum phase systems can be obtained by the stable inversion method but requiring large enough extended time interval. In order to solve this problem of large extended time restriction, a novel approach to precise trajectory tracking of non‐minimum phase systems is proposed, it is called the improved stable inversion (ISI) method, using an optimal integration of the pre‐actuation and the optimal state transition (OST) techniques. The ISI method can obtain precise trajectory tracking with a smaller extended time interval as compared to the stable inversion method. The proposed method achieves better validation through numerical simulations for the non‐minimum phase system.     

Multi‐Objective Cost‐Detectability in Unfalsified Adaptive Control

Vector‐valued controller cost functions that are solely data‐dependent and reflect multiple objectives of a control system are examined within the framework of unfalsified adaptive control. The notion of Pareto optimality of vector‐valued cost functions and the conditions under which they are cost‐detectable are discussed. A sampled data/discrete‐time Level‐Set controller switching algorithm is investigated which allows for the relaxation of the assumption that the controller cost function be monotonically nondecreasing in time. This opens up the possibility of the use of fading memory cost functions which are nonmonotone. When an active controller is falsified at the current threshold cost level, the Level‐Set switching algorithm replaces it by an effectively unique solution of the weighted Tchebycheff method, thus ensuring the selection of an unfalsified Pareto optimal controller. Theoretical results for convergence and stability of the adaptive system are given. Simulation results validate the use of cost‐detectable multi‐objective cost functions. An example of a cost‐detectable cost function which uses fading memory norm of the fictitious tracking error as a performance measure is shown. This allows for computation of performance of nonactive controllers with respect to a reference model.     

Transient performance for discrete‐time singular systems with actuators saturation via composite nonlinear feedback control

This paper is concerned with the transient performance improvement in tracking control problems for linear multivariable discrete‐time singular systems subject to actuators saturation. A composite nonlinear feedback control strategy is considered, and the resulting controller consists of a linear feedback law and a nonlinear feedback law without any switching element. The nonlinear term leads to a varying damping ratio of the closed‐loop system and yields a small overshoot as the output approaches the target reference, whereas the linear component is designed to achieve a quick response of the closed‐loop system. Two composite nonlinear feedback control laws by both state feedback and measurement output feedback are addressed. An illustrative example is included to show the validity of the obtained results. Copyright © 2012 John Wiley & Sons, Ltd.     

Prescribed performance control of strict‐feedback systems under actuation saturation and output constraint via event‐triggered approach

In this work, we study the performance‐guaranteed event‐triggered control for a class of uncertain nonlinear systems in strict‐feedback form subject to input saturation and output constraint. The prescribed performance (ie, convergence rate, tracking error accuracy) and output constraint are firstly taken into account for nonlinear systems with event‐triggered input. By blending a speed transformation into the barrier Lyapunov function and introducing an intermediate variable to the system, two different event‐triggered control schemes are proposed for systems with and without saturation, respectively. Each scheme has two rules to determine triggering time sequences, one for control signal updating and the other for control signal transmission with the latter being a subsequence of the first. Meanwhile, it is proved that the tracking error converges to a preset compact set around zero at the prescribed decay rate and the output is maintained within a given bound at all times. Simulation verification also confirms the effectiveness of the proposed approach.     

Symbolic execution of floating‐point computations

Symbolic execution is a classical program testing technique which evaluates a selected control flow path with symbolic input data. A constraint solver can be used to enforce the satisfiability of the extracted path conditions as well as to derive test data. Whenever path conditions contain floating‐point computations, a common strategy consists of using a constraint solver over the rationals or the reals. Unfortunately, even in a fully IEEE‐754‐compliant environment, this leads not only to approximations but also can compromise correctness: a path can be labelled as infeasible although there exists floating‐point input data that satisfy it. In this paper, the peculiarities of symbolic execution of programs with floating‐point numbers are addressed. Issues in the symbolic execution of this kind of program are carefully examined and a constraint solver is described that supports constraints over floating‐point numbers. Preliminary experimental results demonstrate the value of the approach proposed. Copyright © 2005 John Wiley & Sons, Ltd.