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Summary: Optimal robust state estimation (SE) and fault detection (FD) methods of discrete‐time linear parameter varying systems with measurement error‐affected scheduling variables are proposed under the boundedness assumption of system uncertainties. By using the weighted Frobenius norm of the generator matrix of SE zonotope to characterize the set size, the optimal observer gain can be computed by a zonotopic Kalman filter (ZKF) procedure for the purpose of observation. Meanwhile, by minimizing the influence of system uncertainties while maximizing that of faults on SE to enhance the sensitivity of FD, the optimal FD criterion is characterized based on an on‐line fractional programming problem, which can be equivalently transformed into a parametric quadratic programming (pQP) problem. The pQP problem can be efficiently solved by searching the root of its nonlinear characteristic equation using secant method. In general, as long as sensors with sufficiently high precision are equipped to measure the scheduling variables, the bounds of measurement errors of scheduling variables can be less conservative than those direct bounds of scheduling variables, which can reduce the conservatism of FD or SE in this way. At the end of this article, a case study based on a practical circuit model is used to illustrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

Summary: 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. [ABSTRACT FROM AUTHOR]

This paper proposes an invariant set-based robust fault detection (FD) and optimal fault estimation (FE) method for discrete-time linear parameter varying (LPV) systems with bounded uncertainties. Firstly, a novel invariant-set construction method for discrete-time LPV systems is proposed if and only if the system is poly-quadratically stable, which need not satisfy the condition that there must exist a common quadratic Lyapunov function for all vertex matrices of the system compared to the traditional invariant-set construction methods. Furthermore, by using a shrinking procedure, we provide minimal robust positively invariant (mRPI) set approximations that are always positively invariant at each step of iteration and allow a priori desired precision to obtain a high sensitivity of FD. Owing to the existence of invariant set-based FD phase, the assumption that the initial faults should be bounded by a given set can be avoided for FE. We compute an optimal parametric matrix gain by minimising the Frobenius norm-based size of the corresponding FE set to obtain the optimal FE performance. Theoretically, any trajectory in the FE tube can be chosen as a specific-value estimation for the real fault signals. Finally, a vehicle dynamics system is used to illustrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

This paper focuses on comparing fault detection (FD) conservatism of the set-theoretic unknown input observer (SUIO) based method and the interval observer (IO) based method. The former is implemented by combining both the set theory and the unknown input observer (UIO), where the set theory and the UIO are used to decouple disturbances and noises in passive and active ways, respectively. The latter is based on the set theory and the Luenberger-structure observer, which completely relies on the set-based passive decoupling of disturbances and noises to obtain FD robustness. The objective of this paper is to compare their FD conservatism using systematic mathematical analysis based on zonotopes and a mathematical criterion comparing their FD conservatism is obtained. Moreover, by establishing a general residual framework for conservatism analysis, the proposed method comparing the FD conservatism is also extended to compare the FD conservatism of a wide variety of set-based robust FD methods. At the end of this paper, a two-tank system example is used to illustrate the effectiveness of the obtained results. [ABSTRACT FROM AUTHOR]

This paper proposes a confidence set-based computational method of minimal detectable fault (MDF) based on the confidence set-separation condition between the healthy and faulty residual sets for discrete linear time-invariant systems. The state-estimation-error dynamics for the analysis of MDF under hybrid random and bounded uncertainties is divided into two sub-dynamics. The first sub-dynamics is only affected by the bounded uncertainties. Under the precondition of Schur stability, an outer-approximation of minimal robust positively invariant set with any given precision is obtained. While the second sub-dynamics is only affected by the random uncertainties following the Gaussian distributions. It is proved that the behavior of the second sub-dynamics at steady stage also follows a certain Gaussian distribution, which can be bounded by confidence zonotopes given a proper confidence level. MDF for actuator and sensor faults can be obtained by solving a non-convex optimization problem to minimize the magnitude of fault subjected to the residual set-separation constraints, which is equivalent to compute a distance from the origin to hyperplane along a fixed direction by exploiting the geometric property. At the end of this paper, a two-link manipulator model and a vehicle model are used to verify the effectiveness of our proposed method. [ABSTRACT FROM AUTHOR]

• A method to establish invariant set separation-based guaranteed FDI conditions for single optimal SUIO is proposed, which reduces the FDI conservatism of single observer but simultaneously obtains robust FDI guarantees. • A method to design an optimal observer configuration for a bank of SUIOs is proposed, which aims to minimize the effect of all unknown inputs on guaranteed robust FDI to reduce the conservatism of a bank of observers. This paper proposes an optimal guaranteed robust fault detection and isolation (FDI) method combing the set-theoretic unknown input observer (SUIO) and the robust positively invariant (RPI) sets for linear time-invariant (LTI) systems. The optimality of the proposed FDI method is achieved under a two-layer framework. The first layer allows to design a single optimal FDI-oriented SUIO with RPI set separation-based guaranteed FDI conditions to reduce the FDI conservatism of single observer. The second layer consists in designing an optimal observer configuration for a bank of optimal SUIOs to reduce the FDI conservatism of all the SUIOs together. Finally, the effectiveness of the proposed guaranteed robust FDI method is illustrated by using a four-tank system. [ABSTRACT FROM AUTHOR]

This paper considers the problem of integrating fault estimation (FE) and fault-tolerant control (FTC) for linear parameter varying (LPV) systems with inexact scheduling variables from a new perspective. Unlike the conventional passive FE approaches that maintain the performance of FE by only adjusting parametric matrices, the proposed method guarantees the precision and optimality of FE by actively designing optimal inputs at each time instant under the set-theoretic framework, which is named as active fault estimation (AFE). Meanwhile, by constructing a quadratic Lyapunov function, the stability of augmented estimation-error dynamics can be guaranteed based on H ∞ setting by solving a group of matrix inequalities. In addition, the optimal inputs are also designed to implement the objective of integrated AFE and FTC for LPV systems, which leads to a balance between the goal of AFE and the reference output tracking performance. Although the original optimization problem computing the optimal inputs is not convex, it is proved that this problem can be equivalently transformed to a family of convex quadratic programming problems with linear constraints. At the end, a physical vehicle model is used to verify the effectiveness of our proposed integrated AFE and FTC method. [ABSTRACT FROM AUTHOR]

This paper proposes a novel geometry optimization strategy for the online robust active fault diagnosis (AFD) of discrete-time linear parameter varying (LPV) systems under set-theoretic framework. By establishing a bank of zonotopic set-value observers to match healthy/faulty system modes, the key of the optimization strategy comes down to the design of the optimal system inputs and gain matrices simultaneously. The criterion on the design of optimal inputs is characterized by fully utilizing the geometry property of zonotopes, which formulates a non-convex fractional programming problem able to be solved under 0–1 mixed integer quadratic programming framework based on a series of transformations. While the optimal gain matrices can be obtained analytically based on a so-called zonotopic kalman filtering process. The proposed method can improve the sensitivity of AFD as much as possible while ensuring the stability of the designed observer. At the end, two physical models are used to verify the effectiveness of our proposed method. [ABSTRACT FROM AUTHOR]

Continuum space manipulators (CSMs) have key application prospects in the maintenance of spacecrafts in narrow environments and space debris removal. Similar to conventional articulated space manipulators (ASMs), CSMs encounter a number of kinematic, dynamic, and control problems caused by the dynamic coupling between the continuum manipulator and its microsatellite platform. However, the continuum manipulator has no obvious kinematic joints. In addition, its motion is generated by the continuum deformation of its elastic body; thus, the virtual manipulator (VM) model of ASM in the free-floating state becomes inappropriate for CSMs. In this study, we propose a kinematic equivalent method for a free-floating CSM, which is equivalent to a virtual discrete-jointed manipulator (VDJM) connected to a virtual ground. First, the constraint relationship is used to establish the virtual vector manipulator (VVM) model for the CSM such that its center of mass in the free-floating mode remains unchanged. Furthermore, on the basis of the VVM model and D-H parameters of the CSM, virtual discrete joints are introduced, and a VDJM model for the CSM is proposed. Compared with the VVM model, the developed VDJM model can more intuitively express the motion characteristics of the CSM. Finally, we demonstrate the application potential of the developed VDJM model in the inverse kinematics solution, workspace, and dynamic coupling of CSMs. • A VVM model for CSMs using the basic principle of conservation of mass centers. • Similar to VM model but the length and angle of position vectors change constantly. • Angle and rotation direction can be simultaneously calculated using proposed method. • Simulation proves that VDJM model can accurately describe position of end-effector. • Kinematic properties of free-floating CSMs are readily analyzed using the model. [ABSTRACT FROM AUTHOR]

This paper proposes to design an unknown input observer (UIO) for the linear-parameter-varying (LPV) system on the basis of the set theory, which is named as the set-theoretic UIO (SUIO). The advantage of the SUIO consists in that it combines active and passive approaches to obtain robustness in state estimation (SE) and fault detection (FD). The active approach is based on the use of UIO to decouple unknown inputs, while the passive approach is based on the set theory to bound uncertain factors that cannot be actively decoupled. As a result, the effect of both unknown inputs (process disturbances, modeling errors, etc.) and measurement noises can be appropriately handled in the residual signals compared with the standard UIO-based SE and FD approaches. The design of SUIO can overcome the limitations of the traditional UIO design conditions, which can significantly broaden the application of the UIO-based SE and FD theory. Moreover, this paper proposes a generalized framework that can provide more flexibility in the design of SUIO guaranteeing their stability by means of a group of matrix inequalities. Because the LPV system uses a collection of online obtainable scheduling variables to embed nonlinearities, the design of SUIO for the LPV system can be used to address the SE and FD problems of nonlinear systems. At the end of this paper, two case studies are used to illustrate the effectiveness of the proposed approach. Copyright © 2017 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The fault diagnosis module is a key component integrated in active fault-tolerant control systems. However, the classical methods usually design fault diagnosis and control modules, separately, which ignore the existing interactions between the two key modules in active fault-tolerant control systems. In order to deal with this problem, this paper establishes a framework for integrated design of active fault diagnosis (AFD) and control while ensuring the performance of tracking control. In particular, the input is designed to balance the goal of AFD with the reference output tracking performance. The design of inputs is formulated to minimize the sum of a quadratic function and a non-convex quadratic fractional function at each step, whose global optimality can be obtained under the tight condition of convex relaxation. Furthermore, despite the global optimal condition is not achieved, a locally optimal solution could still be obtained by using our proposed local linearization search algorithm, whose suboptimality can be strictly guaranteed theoretically. At the end, a practical four-tank system example is used to verify the effectiveness of our proposed method. [ABSTRACT FROM AUTHOR]

This paper focuses on designing, via the minimization of a suitable cost function, an input sequence to guarantee active fault diagnosis (AFD) of discrete-time linear parameter-varying systems regardless of how the scheduling variables change in the scheduling set. Convex polyhedrons are used to characterize the sets of system uncertainties such that the designed input sequence leads to outputs which are consistent with a unique fault mode or healthy mode. Considering the smoothness and energy of the input sequence simultaneously, the optimal input sequence is obtained by solving a family of linear constrained quadratic programming (LCQP) problems in a brute-force way, which will lead to inefficiency when the number of facets of the interest region becomes large. In order to handle this problem, by analyzing a geometric property of the objective function, the original optimization problem is shown to be equivalent to the minimization of the radius of a hypersphere containing the origin based on a simple linear mapping, which is much more efficient than the brute force-based resolution. At the end of this paper, a numerical example and a vehicle model are used to illustrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

This paper characterises and analyses the minimum detectable fault (MDF) (named generalised MDF here) for multiple multiplicative actuator faults based on set-separation constraints of healthy and faulty residual sets for discrete-time linear parameter varying (LPV) systems with bounded inputs and uncertainties. First, multiple multiplicative actuator faults are equivalently transformed into the form of multiple additive actuator faults for the sake of obtaining the simplified faulty residual set explicitly. Then, two different generalised MDF for different multiple actuator faults are, respectively, defined, which can be computed via solving simple linear programming (LP) problems. Moreover, we propose three detectability conditions to guarantee the detection of multiple multiplicative actuator faults by using the invariant set-based method. Finally, we provide effective selection guidance for the detection of different multiple multiplicative actuator faults by systematically analysing the conservatism and characteristics of these three guaranteed detectability conditions. At the end of this paper, an electrical circuit example is used to illustrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]