<Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> control of singular systems via delta operator method

This paper studies the problem of state feedback H _∞ control for singular systems through delta operator approach. A necessary and sufficient condition is presented such that a singular delta operator system is admissible with a prescribed H _∞ performance, which can provide a unified framework of the existing H _∞ performance analysis results for both continuous case and discrete case. The existence condition and explicit expression of a desirable H _∞ controller are also obtained for singular delta operator systems. The proposed design method can be used for both singular continuous systems and singular discrete systems directly. The corresponding design procedures, which simplify the classical approaches, are discussed and presented. All obtained conditions in this paper are in the form of strict linear matrix inequalities whose feasible solutions can be found by standard linear programming method. Numerical examples are provided to illustrate the effectiveness of the theoretical results obtained in this paper.     

Local and global analysis of parametric solid sweeps

In this work, we propose a structured computational framework for modelling the envelope of the swept volume, that is the boundary of the volume obtained by sweeping an input solid along a trajectory of rigid motions. Our framework is adapted to the well-established industry-standard brep format to enable its implementation in modern CAD systems. This is achieved via a “local analysis”, which covers parametrizations and singularities, as well as a “global theory” which tackles face-boundaries, self-intersections and trim curves. Central to the local analysis is the “funnel” which serves as a natural parameter space for the basic surfaces constituting the sweep. The trimming problem is reduced to the problem of surface–surface intersections of these basic surfaces. Based on the complexity of these intersections, we introduce a novel classification of sweeps as decomposable and non-decomposable. Further, we construct an invariant function θ on the funnel which efficiently separates decomposable and non-decomposable sweeps. Through a geometric theorem we also show intimate connections between θ, local curvatures and the inverse trajectory used in earlier works as an approach towards trimming. In contrast to the inverse trajectory approach of testing points, θ is a computationally robust global function. It is the key to a complete structural understanding, and an efficient computation of both, the singular locus and the trim curves, which are central to a stable implementation. Several illustrative outputs of a pilot implementation are included.     

Risk-sensitive filtering, prediction and smoothing for discrete-time singular systems

This paper is concerned with steady-state risk-sensitive filtering, prediction and smoothing problems for discrete-time singular systems. It is shown that a risk-sensitive estimator can be obtained by ensuring the minimum of an indefinite quadratic form to be maximum (minimum) when the risk-sensitivity parameter θ is negative (positive). An auxiliary state-space signal model and an innovation sequence in Krein space are introduced to simplify the derivation of the estimator. The estimator is calculated based on one J-spectral factorization for risk-seeking (θ<0) or one H₂ spectral factorization for risk-averse (θ>0). A numerical example is given to demonstrate the applicability of the result.     

Finite-time H∞ control for discrete-time switched singular time-delay systems subject to actuator saturation via static output feedback

This study employs the multiple Lyapunov-like function method and the average dwell-time concept of switching signal to investigate the finite-time H_∞ static output-feedback (SOF) control problem for a class of discrete-time switched singular time-delay systems subject to actuator saturation. First, sufficient conditions are presented to guarantee the discrete-time switched singular time-delay system regular, causal and finite-time boundedness. Meanwhile, sufficient conditions are presented to ensure the H_∞ disturbance attenuation level, and the design method of H_∞ SOF controller is developed by solving matrix inequalities optimisation problem without any decompositions of system matrices and equivalent transformation. Finally, the effectiveness and merit of the theoretical results are shown through some numerical examples and several vivid illustrations.     

Structural decomposition of linear singular systems: the single-input and single-output case

In this paper, we present a structural decomposition for general single-input and single-output linear singular systems. Such a decomposition has a distinct feature of capturing and displaying all the structural properties, such as the finite and infinite zero structures and redundant dynamics, of the given system. It is expected to be a powerful tool in solving control problems for singular systems, such as H₂ and H_∞ control, model reduction and disturbance decoupling problems, to name a few.     

Delay‐dependent robust H∞ controller synthesis for discrete singular delay systems

In this paper, the problems of delay‐dependent robust stability analysis, robust stabilization and robust H_∞ control are investigated for uncertain discrete‐time singular systems with state delay. First, by making use of the delay partitioning technique, a new delay‐dependent criterion is given to ensure the nominal system to be regular, causal and stable. This new criterion is further extended to singular systems with both delay and parameter uncertainties. Then, without the assumption that the considered systems being regular and causal, robust controllers are designed for discrete‐time singular time‐delay systems such that the closed‐loop systems have the characteristics of regularity, causality and asymptotic stability. Moreover, the problem of robust H_∞ control is solved following a similar line. The obtained results are dependent not only on the delay, but also on the partitioning size and the conservatism is non‐increasing with reducing partitioning size. These results are shown, via extensive numerical examples, to be much less conservative than the existing results in the literature. Copyright © 2010 John Wiley & Sons, Ltd.     

Robust H∞ Sliding Mode Control for a Class of Singular Stochastic Nonlinear Systems

This paper concerns the problem of robust H_∞ sliding mode control for a class of singular stochastic nonlinear systems. Integral sliding mode control is developed to deal with this problem. Based on the integral sliding surface of the design and linear matrix inequality, a sufficient condition which guarantees the sliding mode dynamics is asymptotically mean square admissible and has a prescribed H_∞ performance for a class of singular stochastic nonlinear systems is proposed. Furthermore, a sliding mode control law is synthesized such that the singular stochastic nonlinear system can be driven to the sliding surface in finite time. Finally, a numerical example is proposed to illustrate the effectiveness of the given theoretical results.     

An LPV approach to the robust control of a class of quasi-linear propagation processes

In this paper the trajectory tracking control problem for a certain class of propagation processes modeled as quasi-linear parameter varying systems is considered. The propagation physical models are generally described by means of partial differential equations (PDEs). However in real world control problems the PDE models are usually converted into ordinary differential equations (ODEs) models adopting numerical and/or physical approximations. In many practical problems it happens that the propagation dynamics are linear, while the boundary conditions are described by nonlinear algebraic equations. A trajectory following control scheme is proposed for this class of systems together with a robust performance analysis based on the concept of quadratic stability with an H_∞ norm bound. An LMI based observer synthesis procedure is also proposed to increase the closed loop system performance.     

Solution of the H∞-optimization problems on the basis of the principle of separation

A new approach to the design of the central output H_∞-controller was proposed. It is based on defining the controller’s structure in compliance with the principle of division which is applicable equally to the standard nondegenerate problem and the degenerate problems such as the singular filtration problem (no measurement noise) and singular control problem (no component containing control in the controlled output). Within the framework of the given approach, the general problem of designing the output controller was shown to fall down into two special design problems—under full information and full control. For the degenerate problem, the dimension of the central controller at that drops, and the solution itself is coordinated with the problem of H₂-optimization. An example illustrating solution of the singular filtration problem was presented, and the results obtained were analyzed.     

Adaptive PSO-LS-wavelet H<Subscript>∞</Subscript> control for two-wheeled self-balancing scooter

The current study is concerned with adaptive Particle Swarm Optimization Least Square Wavelet H_∞ for a two-wheel self-balancing scooter that provides a platform in order to balance itself and transport the driver in accordance to its natural lean. In order to keep the rider close to the upright position over smooth and non-smooth surfaces, providing a stable control system is the main challenge for the aforementioned vehicle. For this purpose, H_∞ is combined with adaptive algorithm, Least Square Support Vector Machine (LS-SVM) and Particle Swarm Optimization (PSO) to construct the adaptive control. The most important feature of the proposed control strategy is its inherent robustness and ability to handle the nonlinear behavior of the system. Simulations results indicated that the introduced motion control architecture is capable of providing appropriate control actions to achieve both position control and trajectory tracking satisfactorily.     

基于不完全转移率的连续Markov 跳变奇异系统的H∞ 控制

研究一类不完全转移率信息的Markov跳变奇异系统的H∞控制问题,提出连续Markov跳变奇异系统的新型有界实引理,并将其推广到不完全转移率条件.进一步设计H∞状态反馈控制器,使得闭环系统在转移率部分未知的条件下随机可容许,且满足H∞性能H.所得结论涵盖了奇异矩阵模态依赖情形,且表示为严格线性矩阵不等式形式,利于工程实现.最后,通过仿真算例表明了所提出方法的有效性和优越性.     

Reduced-order filtering for singular systems

This paper solves the problem of reduced-order H_∞ filtering for singular systems. The purpose is to design linear filters with a specified order lower than the given system such that the filtering error dynamic system is regular, impulse-free (or causal), stable, and satisfies a prescribed H_∞ performance level. One major contribution of the present work is that necessary and sufficient conditions for the solvability of this problem are obtained for both continuous and discrete singular systems. These conditions are characterized in terms of linear matrix inequalities (LMIs) and a coupling non-convex rank constraint. Moreover, an explicit parametrization of all desired reduced-order filters is presented when these inequalities are feasible. In particular, when a static or zeroth-order H_∞ filter is desired, it is shown that the H_∞ filtering problem reduces to a convex LMI problem. All these results are expressed in terms of the original system matrices without decomposition, which makes the design procedure simple and directly. Last but not least, the results have generalized previous works on H_∞ filtering for state-space systems. An illustrative example is given to demonstrate the effectiveness of the proposed approach.     

Delay-dependent <Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> control for a class of uncertain time-delay singular Markovian jump systems via hybrid impulsive control

This paper deals with the problem of robust normalization and delay-dependent H _∞ control for a class of singular Markovian jump systems with norm-bounded parameter uncertainties and time delay. A new impulsive and proportional-derivative control strategy with memory is presented, which results in a novel class of hybrid impulsive systems. Sufficient conditions are developed to guarantee that the resultant closed-loop system is not only robust normal and stochastically stable, but also satisfies a prescribed H _∞ performance level for all delays no larger than a given upper bound. In addition, the explicit expression of the desired impulsive control gains is also given together with the design approach. Finally, two numerical examples are provided to illustrate the effectiveness of the proposed methods.     

Robust stability and H∞ control for uncertain discrete Markovian jump singular systems with mode‐dependent time‐delay

The robust stochastic stability, stabilization and H_∞ control for mode‐dependent time‐delay discrete Markovian jump singular systems with parameter uncertainties are discussed. Based on the restricted system equivalent (r.s.e.) transformation and by introducing new state vectors, the singular system is transformed into a standard linear system, and delay‐dependent linear matrix inequalities (LMIs) conditions for the mode‐dependent time‐delay discrete Markovian jump singular systems to be regular, causal and stochastically stable, and stochastically stable with γ‐disturbance attenuation are obtained, respectively. With these conditions, robust stabilization problem and robust H_∞ control problem are solved, and the LMIs sufficient conditions are obtained. A numerical example illustrates the effectiveness of the method given in the paper. Copyright © 2008 John Wiley & Sons, Ltd.     

Finite‐Time H∞ Filtering for Nonlinear Continuous‐Time Singular Semi‐Markov Jump Systems

The stochastic finite‐time H_∞ filtering issue for a class of nonlinear continuous‐time singular semi‐Markov jump systems is discussed in this paper. Firstly, sufficient conditions on singular stochastic H_∞ finite‐time boundedness for the filtering error system are established. The existence of a unique solution for the corresponding system is also ensured. Secondly, based on the bounds of the time‐varying transition rate, without imposing constraints on slack variables, a novel approach to finite‐time H_∞ filter design is proposed in the forms of strict LMIs, which guarantees the filtering error system is singular stochastic H_∞ finite‐time bounded and of a unique solution. Compared with the existing ones, the presented results reveal less conservativeness. Finally, one numerical example is exploited to testify the advantage of the proposed design technique.     

SLIDING-MODE ADAPTIVE CONTROL FOR FLEXIBLE-LINK MANIPULATORS USING A COMPOSITE DESIGN

ABSTRACT

A robust adaptive control scheme for flexible link robotic manipulators is presented. The design is based on considering the flexible mechanical structure as a singular perturbed system, which allows us to assume the existence of slow (rigid) and fast (flexible) modes that separately can be controlled. The rigid dynamics are controlled by means of a robust sliding adaptive approach with well-established stability properties. The flexible dynamics can be controlled using H _∞ or optimal designs, which successfully handle the actual interaction between the slow and fast subsystems. This composite approach achieves good closed-loop tracking properties, both in simulation and experimental results on a laboratory-flexible arm.     

New reduced-order <Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> filter design method for discrete-time singular systems with lossy measurements by strict LMI approach

In this paper, the new design problems of full and reduced-order H _∞ filters including static filters are investigated for discrete-time singular systems with lossy measurements. The lossy measurement is described by a stochastic variable satisfying Bernoulli random binary distribution to model the measured output. Being different from the previous results, the filter existence condition by a new Lyapunov function for singular systems with lossy measurements is given using a strict linear matrix inequality (LMI) in terms of all variables. The proposed full and reduced-order H _∞ filter guarantees regularity, causality, and mean-square stochastic stability with a H _∞ performance bound of filtering error singular system. Moreover, it is shown that the reduced-order H _∞ filter design method is general for both singular systems and non-singular systems with lossy measurements. Numerical examples are worked out to illustrate the effectiveness and applicability of the proposed method.     

Singular nonlinear H∞ optimal control problem

The theory of nonlinear H_∞ of optimal control for affine nonlinear systems is extended to the more general context of singular H_∞ optimal control of nonlinear systems using ideas from the linear H_∞ theory. Our approach yields under certain assumptions a necessary and sufficient condition for solvability of the state feedback singular H_∞ control problem. The resulting state feedback is then used to construct a dynamic compensator solving the nonlinear output feedback H_∞ control problem by applying the certainty equivalence principle.