LMI optimization approach to robust H∞ observer design and static output feedback stabilization for discrete‐time nonlinear uncertain systems

A new approach for the design of robust H_∞ observers for a class of Lipschitz nonlinear systems with time‐varying uncertainties is proposed based on linear matrix inequalities (LMIs). The admissible Lipschitz constant of the system and the disturbance attenuation level are maximized simultaneously through convex multiobjective optimization. The resulting H_∞ observer guarantees asymptotic stability of the estimation error dynamics and is robust against nonlinear additive uncertainty and time‐varying parametric uncertainties. Explicit norm‐wise and element‐wise bounds on the tolerable nonlinear uncertainty are derived. Also, a new method for the robust output feedback stabilization with H_∞ performance for a class of uncertain nonlinear systems is proposed. Our solution is based on a noniterative LMI optimization and is less restrictive than the existing solutions. The bounds on the nonlinear uncertainty and multiobjective optimization obtained for the observer are also applicable to the proposed static output feedback stabilizing controller. Copyright © 2008 John Wiley & Sons, Ltd.     

Gain‐scheduled H∞ filtering of parameter‐varying systems

This paper deals with the gain‐scheduled H_∞ filtering problem for a class of parameter‐varying systems. A sufficient condition for the existence of a gain‐scheduled filter, which guarantees the asymptotic stability with an H_∞ noise attenuation level bound for the filtering error system, is given in terms of a finite number of linear matrix inequalities (LMIs). The filter is designed to be parameter‐varying and have a nonlinear fractional transformation structure. A numerical example is presented to demonstrate the application of the proposed method. Copyright © 2006 John Wiley & Sons, Ltd.     

Distributed Observer‐based Stabilization of Nonlinear Multi‐agent Systems with Sampled‐data Control

In this paper, the distributed observer‐based stabilization problem of multi‐agent systems under a directed graph is investigated. Distributed observer‐based control protocol with sampled‐data information is proposed. The dynamics of each agent contain a nonlinear part, which is supposed to be general Lipschitz. In order to stabilize the states of the whole network, all the nodes utilize the relative output estimation error at sampling instants and only a small fraction of nodes use the absolute output estimation error additionally. By virtue of the input‐to‐state stability (ISS) property and the Lyapunov stability theory, an algorithm to design the control gain matrix, observer gain matrix, coupling strength as well as the allowable sampling period are derived. The conditions are in the form of LMIs and algebraic inequality, which are simple in form and easy to verify. Some further discussions about the solvability of obtained linear matrix inequalities (LMIs) are also given. Lastly, an example is simulated to further validate the obtained results.     

Separation principle for quasi‐one‐sided Lipschitz nonlinear systems with time delay

In this article, we address the problem of output stabilization for a class of nonlinear time‐delay systems. First, an observer is designed for estimating the state of nonlinear time‐delay systems by means of quasi‐one‐sided Lipschitz condition, which is less conservative than the one‐sided Lipschitz condition. Then, a state feedback controller is designed to stabilize the nonlinear systems in terms of weak quasi‐one‐sided Lipschitz condition. Furthermore, it is shown that the separation principle holds for stabilization of the systems based on the observer‐based controller. Under the quasi‐one‐sided Lipschitz condition, state observer and feedback controller can be designed separately even though the parameter (A,C) of nonlinear time‐delay systems is not detectable and parameter (A,B) is not stabilizable. Finally, a numerical example is provided to verify the efficiency of the main results.     

An H∞non‐fragile observer‐based adaptive sliding mode controller design for uncertain fractional‐order nonlinear systems with time delay and input nonlinearity

In this paper the problem of non‐fragile adaptive sliding mode observer design is addressed for a class of nonlinear fractional‐order time‐delay systems with uncertainties, external disturbance, exogenous noise, and input nonlinearity. An H_∞ observer‐based adaptive sliding mode control considering the non‐fragility of the observer is proposed for this system. The sufficient asymptotic stability conditions are derived in the form of linear matrix inequalities. It is proven that the sliding surface is reachable in finite time. An illustrative example is provided which corroborates the effectiveness of the theoretical results.     

Full‐Order and Reduced‐order Observer Design for a Class of Fractional‐order Nonlinear Systems

The paper is concerned with problem of the full‐order and reduced‐order observer design for a class of fractional‐order one‐sided Lipschitz nonlinear systems. By introducing a continuous frequency distributed equivalent model and using indirect Lyapunov approach, the sufficient condition for asymptotic stability of the full‐order observer error dynamic system is presented. Furthermore, the proposed design method was extended to reduced‐order observer design for fractional‐order nonlinear systems. All the stability conditions are obtained in terms of LMI, which are less conservative than some existing ones. Finally, a numerical example demonstrates the validity of this approach.     

Interconnected jumping time‐delay systems: Mode‐dependent decentralized stability and stabilization

In this paper, we deal with the problems of mode‐dependent decentralized stability and stabilization with ??_∞ performance for a class of continuous‐time interconnected jumping time‐delay systems. The jumping parameters are governed by a finite state Markov process and the delays are unknown time‐varying and mode‐dependent within interval. The interactions among subsystems satisfy quadratic bounding constraints. To characterize mode‐dependent local stability behavior, we employ an improved Lyapunov–Krasovskii functional at the subsystem level and express the stability conditions in terms of linear matrix inequalities (LMIs). A class of local decentralized state‐feedback controllers is developed to render the closed‐loop interconnected jumping system stochastically stable. Then, we extend the feedback strategy to dynamic observer‐based control and establish the stochastic stabilization via LMIs. It has been established that the developed results encompass several existing results as special cases which are illustrated by simulation of examples. Copyright © 2011 John Wiley & Sons, Ltd.     

A dual‐observer approach for global output feedback stabilization of planar nonlinear systems with output‐dependent growth rates

In this paper, we investigate the problem of global output feedback stabilization for a class of planar nonlinear systems under a more general growth condition, which encompasses both lower‐order and higher‐order state growths with output‐dependent rates. For more accurate estimation, two new observers with nonlinear gains are constructed to estimate the states on the lower‐order and higher‐order scales, respectively. The estimates produced from the dual‐observer are used delicately in the output feedback control law with both lower‐order and higher‐order modes. The overall stability of the system is guaranteed by rigorously choosing these nonlinear gains in the control law and the dual‐observer.     

Passivity‐based sliding mode controller/observer for second‐order nonlinear systems

A passivity‐based sliding mode control for a class of second‐order nonlinear systems with matched disturbances is proposed in this paper. Firstly, a nonlinear sliding surface is designed using feedback passification, in which the passivity is employed to guarantee the closed‐loop system's stability. The passivity‐based controller comprising a discontinuous term guarantees globally asymptotical convergence to the sliding surface. A sliding mode‐based control law that satisfies the reaching and sliding condition is also developed. Moreover, the passivity‐based sliding mode observer is also developed to effectively estimate the system states. Compared with conventional sliding mode control, the proposed control scheme has a shorter reaching time; and hence, the system performance is less affected by disturbances, thus eliminating the need to increase the control input gain. Finally, simulation results demonstrate the validity of the proposed method.     

Peaking free output‐feedback exact tracking of uncertain nonlinear systems via dwell‐time and norm observers

The control of uncertain nonlinear systems by high‐gain observer based output feedback is addressed. Two tracking sliding mode controllers are designed for a broad class of uncertain nonlinear systems with arbitrary relative degree and unmatched polynomial nonlinearities in the unmeasured states. The proposed strategies are based either on dwell‐time for control activation or on simple norm state observers to remove the peaking phenomenon related with high‐gain observers, depending on the nonlinearity growth conditions. In contrast with previous works, exact tracking is also achieved by means of a switching strategy based on locally exact differentiators. Global or semi‐global stability is proved by using Lyapunov theory and on small‐gain analysis. Simulations show that the proposed methodologies provide better and uniform transient behavior, larger regions of attraction, performance recovery with significantly smaller observer gains and good robustness properties with respect to exogenous disturbances and measurement noise. Copyright © 2011 John Wiley & Sons, Ltd.     

Composite disturbance‐observer‐based control and H ∞ control for nonlinear time‐delay systems

A novel type of control scheme combined the distance‐observer‐based control (DOBC) with H_∞ control is proposed for a class of nonlinear time‐delay systems subject to disturbances. The disturbances are supposed to include two parts. One in the input channel is generated by an exogenous system with uncertainty, which can represent the harmonic signals with modeling perturbations. The other is supposed to have the bounded H₂ norm. The disturbance observers based on regional pole placement and D‐stability theory are presented, which can be designed separately from the controller design. By integrating disturbance‐observer‐based control with H_∞ control laws, the disturbances can be rejected and attenuated, simultaneously, the desired dynamic performances can be guaranteed for nonlinear time‐delay systems with unknown nonlinear dynamics. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Resilient decentralized stabilization of interconnected time‐delay systems with polytopic uncertainties

Decentralized delay‐dependent local stability and resilient feedback stabilization methods are developed for a class of linear interconnected continuous‐time systems. The subsystems are time‐delay plants which are subjected to convex‐bounded parametric uncertainties and additive feedback gain perturbations while allowing time‐varying delays to occur within the local subsystems and across the interconnections. The delay‐dependent local stability conditions are established at the subsystem level through the construction of appropriate Lyapunov–Krasovskii functional. We characterize decentralized linear matrix inequalities (LMIs)‐based delay‐dependent stability conditions by deploying an injection procedure such that every local subsystem is delay‐dependent robustly asymptotically stable with an γ‐level ??₂‐gain. Resilient decentralized state‐feedback stabilization schemes are designed, which takes into account additive gain perturbations such that the family of closed‐loop feedback subsystems enjoys the delay‐dependent asymptotic stability with a prescribed γ‐level ??₂‐gain for each subsystem. The decentralized feedback gains are determined by convex optimization over LMIs. All the developed results are tested on representative examples. Copyright © 2010 John Wiley & Sons, Ltd.     

Global output feedback sampled‐data stabilization for upper‐triangular nonlinear systems with improved maximum allowable transmission delay

This paper addresses the problem of output feedback sampled‐data stabilization for upper‐triangular nonlinear systems with improved maximum allowable transmission delay. A class of hybrid systems are firstly introduced. The transmission delay may be larger than the sampling period. Then, sufficient conditions are proposed to guarantee global exponential stability of the hybrid systems. Based on these sufficient conditions and a linear continuous‐discrete observer, an output feedback control law is presented to globally exponentially stabilize the feedforward nonlinear system. The improved maximum allowable transmission delay is also given. The results are also extended to output feedback sampled‐data stabilization for lower‐triangular nonlinear systems. Finally, illustrative examples are used to verify the effectiveness of the proposed design methods. Copyright © 2016 John Wiley & Sons, Ltd.     

Global output‐feedback stabilization for a class of stochastic nonlinear systems via sampled‐data control

In this paper, the global sampled‐data output‐feedback stabilization problem is considered for a class of stochastic nonlinear systems. First, based on output‐feedback domination technique and emulation approach, a systematic design procedure for sampled‐data output‐feedback controller is proposed for a class of stochastic lower‐triangular nonlinear systems. It is proved that the proposed sampled‐data output‐feedback controller will stabilize the given stochastic nonlinear system in the sense of mean square exponential stability. Because of the domination nature of the proposed control approach, it is shown that the proposed control approach can also be used to handle the global sampled‐data output‐feedback stabilization problems for a more general class of stochastic non‐triangular nonlinear systems. Finally, simulation examples are given to demonstrate the effectiveness of the proposed method. Copyright © 2017 John Wiley & Sons, Ltd.     

Output‐feedback tracking in causal nonminimum‐phase nonlinear systems using higher‐order sliding modes

Asymptotic output‐feedback tracking in a class of causal nonminimum phase uncertain nonlinear systems is addressed via sliding mode techniques. Sliding mode control is proposed for robust stabilization of the output tracking error in the presence of a bounded disturbance. The output reference profile and the unknown input/disturbance are supposed to be described by unknown linear exogenous systems of a given order. Local asymptotic stability of the output tracking error dynamics along with the boundedness of the internal states are proven. The unstable internal states are estimated asymptotically via the proposed multistage observer that is based on the method of extended system center. A higher‐order sliding mode observer/differentiator is used for the exact estimation of the input–output states in a finite time. The bounded disturbance is reconstructed asymptotically. A numerical example illustrates the efficiency of the proposed output‐feedback tracking approach developed for causal nonminimum phase nonlinear systems. Copyright © 2009 John Wiley & Sons, Ltd.     

Output feedback finite‐time stabilization of a class of nonlinear time‐delay systems in the p‐normal form

This article investigates the finite‐time output feedback stabilization problem for a class of nonlinear time‐varying delay systems in the p‐normal form. First, a reduced‐order state observer is designed to estimate the unmeasurable state. Then, an output feedback controller is constructed, with the help of the finite‐time Lyapunov stability theorem, it is proved that the state of the resulting closed‐loop system converges to the origin in finite time. Two simulation examples are given to verify the effectiveness of the proposed scheme.     

Global adaptive finite‐time stabilization for a class of p‐normal nonlinear systems via an event‐triggered strategy

This article addresses the problem of global adaptive finite‐time control for a class of p‐normal nonlinear systems via an event‐triggered strategy. A state feedback controller is first designed for the nominal system by adding a power integrator method. Then, by the skillful design of adaptive dynamic gain mechanism, a novel event‐triggered controller is constructed for uncertain nonlinear system without homogeneous growth condition. It is proved that the global finite‐time stabilization of p‐normal nonlinear systems is guaranteed and the Zeno phenomenon is excluded. Finally, two examples are presented to indicate the effectiveness of the proposed control scheme.     

Nonlinear H∞ observer design for one‐sided Lipschitz discrete‐time singular systems with time‐varying delay

This paper investigates the H_∞ observer design problem for a class of nonlinear discrete‐time singular systems with time‐varying delays and disturbance inputs. The nonlinear systems can be rectangular and the nonlinearities satisfy the one‐sided Lipschitz condition and quadratically inner‐bounded condition, which are more general than the traditional Lipschitz condition. By appropriately dealing with these two conditions and applying several important inequalities, a linear matrix inequality–based approach for the nonlinear observer design is proposed. The resulting nonlinear H_∞ observer guarantees asymptotic stability of the estimation error dynamics with a prescribed performance γ. The synthesis condition of H_∞ observer design for nonlinear discrete‐time singular systems without time delays is also presented. The design is first addressed for one‐sided Lipschitz discrete‐time singular systems. Finally, two numerical examples are given to show the effectiveness of the present approach.     

Synchronization of A Class of Uncertain Chaotic Systems with Lipschitz Nonlinearities Using State‐Feedback Control Design: A Matrix Inequality Approach

This paper proposes a new state‐feedback stabilization control technique for a class of uncertain chaotic systems with Lipschitz nonlinearity conditions. Based on Lyapunov stabilization theory and the linear matrix inequality (LMI) scheme, a new sufficient condition formulated in the form of LMIs is created for the chaos synchronization of chaotic systems with parametric uncertainties and external disturbances on the slave system. Using Barbalat's lemma, the suggested approach guarantees that the slave system synchronizes to the master system at an asymptotical convergence rate. Meanwhile, a criterion to find the proper feedback gain vector F is also provided. A new continuous‐bounded nonlinear function is introduced to cope with the disturbances and uncertainties and obtain a desired control performance, i.e. small steady‐state error and fast settling time. Several criteria are derived to guarantee the asymptotic and robust stability of the uncertain master–slave systems. Furthermore, the proposed controller is independent of the order of the system's model. Numerical simulation results are displayed with an expected satisfactory performance compared to the available methods.     

Event‐triggered practical finite‐time output feedback stabilization of a class of uncertain nonlinear systems

This paper studies the event‐triggered practical finite‐time output feedback stabilization problem for a class of uncertain nonlinear systems with unknown control gains. First, a reduced‐dimensional observer is employed to implement the reconstruction of the unavailable states. Furthermore, a novel event‐triggered output feedback control strategy is proposed based on the idea of backstepping design and sign function techniques. It is shown that the practical finite‐time stability of the closed‐loop systems is ensured by Lyapunov analysis and related stability criterion. Compared with the existing methods, the main advantage of this strategy is that the observer errors and event‐trigger errors can be processed simultaneously to achieve the practical finite‐time stability. Finally, an example is adopted to demonstrate the validity of the proposed scheme.