A link between input-output stability and Lyapunov stability

It is shown that incremental boundedness of an input-output operator ensures global asymptotic stability of any motion. A reciprocal statement is set.     

On characterizations of the input-to-state stability property

We show that the well-known Lyapunov sufficient condition for “input-to-state stability” (ISS) is also necessary, settling positively an open question raised by several authors during the past few years. Additional characterizations of the ISS property, including one in terms of nonlinear stability margins, are also provided.     

Improved results on stability analysis and controller synthesis for T-S fuzzy systems

This paper is concerned with the stability analysis and synthesis issues for T-S fuzzy systems. By fully using the properties of fuzzy weighting functions and matrix inequalities, two improved sufficient stability conditions are derived based on the common Lyapunov function and fuzzy Lyapunov function, respectively. It is not necessary to require every fuzzy subsystem to be stable in these conditions. Following the analysis, both parallel and non-parallel distributed compensation controllers are obtained by solving linear matrix inequalities. Finally, four examples are given to show the effectiveness of our proposed approach and the advantages of the approach over existing methods.     

New dilated LMI characterizations for continuous-time multiobjective controller synthesis

This paper provides new dilated linear matrix inequality (LMI) characterizations for continuous-time controller synthesis. The dilated LMIs enable us to parametrize controllers without involving the Lyapunov variables in the parametrizations. Taking advantage of this feature, we can readily design multiobjective controllers with non-common Lyapunov variables, whereas we are forced to employ a common one in the well-known Lyapunov shaping paradigm. In particular, it is shown that the proposed dilated-LMI-based approach to H₂/D-stability synthesis encompasses the corresponding Lyapunov shaping paradigm as a special case. In this sense, the results of this paper can be viewed as partial counterparts of those obtained in the discrete-time setting.     

Backstepping controller design for a class of stochastic nonlinear systems with Markovian switching

A more general class of stochastic nonlinear systems with irreducible homogenous Markovian switching are considered in this paper. As preliminaries, the stability criteria and the existence theorem of strong solutions are first presented by using the inequality of mathematic expectation of a Lyapunov function. The state-feedback controller is designed by regarding Markovian switching as constant such that the closed-loop system has a unique solution, and the equilibrium is asymptotically stable in probability in the large. The output-feedback controller is designed based on a quadratic-plus-quartic-form Lyapunov function such that the closed-loop system has a unique solution with the equilibrium being asymptotically stable in probability in the large in the unbiased case and has a unique bounded-in-probability solution in the biased case.     

速度受限的全方向康复步行训练机器人Backstepping补偿跟踪控制

全方向康复步行训练机器人需要跟踪医生指定的训练轨迹、针对运动过程中速度过快而影响康复者安全性问题,提出一种基于Backstepping补偿方法的速度受限型控制器设计方案,目的是保证康复者的安全训练.通过限制实际运动速度的幅值,并根据所提出轨迹跟踪误差和速度跟踪误差的实时补偿方法,实现了全方向康复步行训练机器人在安全速度下的轨迹跟踪.基于李亚普诺夫稳定性理论和LaSalle不变性原理,证明了补偿误差系统和跟踪误差系统的渐近稳定性.通过仿真结果对比分析,表明了所提控制器设计方法的有效性和可行性.     

Attractors and partial stability of nonlinear dynamical systems

This paper represents a study of the key concepts of attracting sets (submanifolds) and partially stable systems aimed at establishing relations between the local properties and the unification of the methodologies of their analysis. The use of recent results of stability theory and techniques of geometric control enables one to generalize Lyapunov-like sufficient conditions and propose simplified solutions for the problems of set attractivity and partial stability.     

Adaptive control of nonlinear systems with nonlinear parameterization

An adaptive feedback regulation scheme is proposed for a class of single input nonlinear systems, with nonlinear parameterizations. A proof of local regulation is given. The results are validated through a simulation study.     

Backstepping design for global stabilization of switched nonlinear systems in lower triangular form under arbitrary switchings

This paper is concerned with the global stabilization problem for switched nonlinear systems in lower triangular form under arbitrary switchings. Two classes of state feedback controllers and a common Lyapunov function (CLF) are simultaneously constructed by backstepping. The first class uses the common state feedback controller which is independent of switching signals; the other class utilizes individual state feedback controllers for the subsystems. As an extension of the designed method, the global stabilization problem under arbitrary switchings for switched nonlinear systems in nested lower triangular form is also studied. An example is given to show the effectiveness of the proposed method.     

A new discrete-time robust stability condition

A new robust stability condition for uncertain discrete-time systems with convex polytopic uncertainty is given. It enables to check stability using parameter-dependent Lyapunov functions which are derived from LMI conditions. It is shown that this new condition provides better results than the classical quadratic stability. Besides the use of a parameter-dependent Lyapunov function, this condition exhibits a kind of decoupling between the Lyapunov and the system matrices which may be explored for control synthesis purposes. A numerical example illustrates the results.     

The advanced-step NMPC controller: Optimality, stability and robustness

Widespread application of dynamic optimization with fast optimization solvers leads to increased consideration of first-principles models for nonlinear model predictive control (NMPC). However, significant barriers to this optimization-based control strategy are feedback delays and consequent loss of performance and stability due to on-line computation. To overcome these barriers, recently proposed NMPC controllers based on nonlinear programming (NLP) sensitivity have reduced on-line computational costs and can lead to significantly improved performance. In this study, we extend this concept through a simple reformulation of the NMPC problem and propose the advanced-step NMPC controller. The main result of this extension is that the proposed controller enjoys the same nominal stability properties of the conventional NMPC controller without computational delay. In addition, we establish further robustness properties in a straightforward manner through input-to-state stability concepts. A case study example is presented to demonstrate the concepts.     

Diagonal stability of a class of cyclic systems and its connection with the secant criterion

We consider a class of systems with a cyclic interconnection structure that arises, among other examples, in dynamic models for certain biochemical reactions. We first show that a “secant” criterion for local stability, derived earlier in the literature, is in fact a necessary and sufficient condition for diagonal stability of the corresponding class of matrices. We then revisit a recent generalization of this criterion to output strictly passive systems, and recover the same stability condition using our diagonal stability result as a tool for constructing a Lyapunov function. Using this procedure for Lyapunov construction we exhibit classes of cyclic systems with sector nonlinearities and characterize their global stability properties.     

Lyapunov-based exact stability analysis and synthesis for linear single-parameter dependent systems

We propose a class of polynomially parameter-dependent quadratic (PPDQ) Lyapunov functions for assessing the stability of single-parameter-dependent linear, time-invariant, (s-PDLTI) systems in a non-conservative manner. It is shown that the stability of s-PDLTI systems is equivalent to the existence of a PPDQ Lyapunov function. A bound on the degree of the polynomial dependence of the Lyapunov function in the system parameter is given explicitly. The resulting stability conditions are expressed in terms of a set of matrix inequalities whose feasibility over a compact and connected set can be cast as a convex, finite-dimensional optimisation problem. Extensions of the approach to state-feedback controller synthesis are also provided.     

A power system nonlinear adaptive decentralized controller design

In this paper, a novel excitation control is designed for improvement of transient stability of power systems. The control algorithm is based on the adaptive backstepping method in a recursive way without linearizing the system model. Lyapunov function method is applied in designing the controller to ensure the convergence of the power angle, relative speed of the generator and the active electrical power delivered by the generator when a large fault occurs. Compared with the existing nonlinear decentralized control approaches, the proposed controller has no requirement for the bounds of interconnections in the power system. And the new approach does not need the existence of solution of a designed algebraic Riccati equation. Furthermore, the transient stability performance of power systems can also be improved by the designed control approach. The efficacy of the designed controller has been demonstrated in a multimachine power system. Simulation results show transient stability enhancement of a power system in the face of a large sudden fault.     

Output-to-state stability and detectability of nonlinear systems

The notion of input-to-state stability (ISS) has proved to be useful in nonlinear systems analysis. This paper discusses a dual notion, output-to-state stability (OSS). A characterization is provided in terms of a dissipation inequality involving storage (Lyapunov) functions. Combining ISS and OSS there results the notion of input/output-to-state stability (IOSS), which is also studied and related to the notion of detectability, the existence of observers, and output injection.     

Global stabilization of MIMO minimum-phase nonlinear systems without strict triangular conditions

An extension of backstepping to a class of multivariable minimum-phase nonlinear systems is proposed. The systems are assumed to be in a special interlaced form which includes a lower triangular form as a special case. The extension involves the recursive application of backstepping and augmentation.     

Operability analysis of nonlinear processes based on incremental dissipativity

Process operability can be defined as the ability of a process to reject disturbances at a specified operating point and/or to move quickly and smoothly from one operating point to another operating point using a feedback control system. Unlike linear processes, the properties of nonlinear processes (e.g., stability, minimum phase condition, etc.) are different around different equilibria. Most existing operability analysis for nonlinear systems focuses on one particular operating point of interest. This paper addresses the issues of dynamic process operability at various operating points, including the reachability of all equilibrium points or output trajectories in an operating region, regardless of initial conditions. In this work, a nonlinear analysis approach is developed based on the concept of incremental stability. Conditions for incremental stability are derived based on incremental dissipativity. The links between input and output multiplicities and incremental dissipativity are explored. The dynamic control performance achievable in terms of the speed of the response of the closed-loop system and offset minimization is studied. A method for determination of incremental dissipativity using Linear Differential Inclusion (LDI) is also presented, to facilitate the dissipativity based operability analysis.     

板球系统的参数自调整反步控制

为解决系统初始值大范围变化,且控制量受到饱和特性约束的板球系统镇定控制问题,提出一种自动调整待设计参数的反步控制方法.分析了反步法偏差系统的平衡点特性;说明了反步法待设计参数与偏差系统动态间的关系;构造了Mamdani模糊逻辑,以自动调整反步法待设计参数;考虑了控制量受饱和特性限制而引入的约束条件,进而以遗传算法优化了模糊规则.仿真和实验结果均表明,自动调整待设计参数的反步控制方法降低了镇定控制中的位置偏差和位置超调.     

Applications of non-uniform in time robust global asymptotic output stability to robust partial state feedback stabilization

The problem of adding an integrator is considered for time-varying control systems. Sufficient conditions for the solution of this problem are given, which are weaker than the corresponding conditions given in the literature. To this end, the notion of non-uniform in time robust global asymptotic output stability (RGAOS) is used. Applications to problems of partial state feedback global stabilization are considered.