On the existence of a continuous storage function for dissipative systems

Conditions for nonlocal existence of a continuous storage function for nonlinear dissipative system are presented. More precisely, it is shown that under the local w-uniform reachability assumption at one point x, the required supply function is continuous on the set of points reachable from x. Conditions for the local w-uniform reachability based on the local controllability properties of the system are provided.     

-like control for nonlinear stochastic systems

In this paper we develop a H_∞-type theory, from the dissipation point of view, for a large class of time-continuous stochastic nonlinear systems. In particular, we introduce the notion of stochastic dissipative systems analogously to the familiar notion of dissipation associated with deterministic systems and utilize it as a basis for the development of our theory. Having discussed certain properties of stochastic dissipative systems, we consider time-varying nonlinear systems for which we establish a connection between what is called the L₂-gain property and the solution to a certain Hamilton–Jacobi inequality (HJI), that may be viewed as a bounded real lemma for stochastic nonlinear systems. The time-invariant case with infinite horizon is also considered, where for this case we synthesize a worst case-based stabilizing controller. Stability in this case is taken to be in the mean-square sense. In the stationary case, the problem of robust state feedback control is considered in the case of norm-bounded uncertainties. A solution is then derived in terms of linear matrix inequalities.     

Determining Dissipativity of Switched Nonlinear Systems Using Linearization

Local strict QSR‐dissipativity of a switched nonlinear system is studied using the linearization technique in this paper. We obtain local strict QSR‐dissipativity of a switched system even if each subsystem is not locally strictly QSR dissipative by designing a switching law. The derived dissipative sufficient condition is characterized by a modified Lyapunov‐Metzler inequality that can be simplified as an LMI by assuming specific forms. Two special forms of local strict QSR‐dissipativity, local input state strict passivity and local L₂‐gain, are considered. When the approximate errors of a switched affine system satisfy certain conditions, local strict passivity can be drawn from its linearization. Finally, a numerical example is given to illustrate how to apply the proposed method to achieve passivity of switched nonlinear systems.     

A parametrization for dissipative behaviors

The behavioral equivalent of single input single output (SISO) systems are behaviors with two manifest variables. Passive SISO systems can, therefore, be viewed as J-dissipative behaviors with two manifest variables. Here the special matrix J defines a QDF that captures the passivity property of SISO systems. In this paper, we investigate more general QDFs Q_Φs induced by some operator Φ. These QDFs define some relation between the input, the output and their derivatives of a SISO system. We characterize all behaviors that are dissipative with respect to the prescribed QDF Q_Φ. In fact, we parametrize all the behaviors dissipative with respect to Q_Φ in terms of those dissipative with respect to the special QDF Q_J induced by the matrix J. Similar results can also be given for lossless systems.     

Stability and L∞‐gain analysis for a class of nonlinear positive systems with mixed delays

This paper addresses the asymptotic stability and L_∞‐gain analysis problem for a class of nonlinear positive systems with both unbounded discrete delays and distributed delays. With the assumption that the nonlinear function is strictly increasing, we first give a characterization on the positivity of the nonlinear system. Then, with some mild assumptions on the delays, a necessary and sufficient condition to ensure the asymptotic stability is presented. Moreover, an explicit expression of the L_∞‐gain of such nonlinear positive systems is given in terms of the system matrices. Finally, a numerical example is given to illustrate the theoretical results. Copyright © 2016 John Wiley & Sons, Ltd.     

Singularity crossing phenomena in DAEs: A two-phase fluid flow application case study

This paper analyzes the existence of smooth trajectories through singular points of differential algebraic equations, or DAEs, arising from traveling wave solutions of a degenerate convection-diffusion model. The DAE system can be written in the quasilinear form A(x)x′ = b(x). In this setting, singularities are displayed when the matrix A(x) undergoes a rank change. The singular hypersurface may be smoothly crossed by trajectories in a finite time if x* is a geometric singularity satisfying certain directional conditions. The basis of our analysis is a two-phase fluid flow model in one spatial dimension with dissipative mechanism involved.     

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.     

The zero divisor problem of multivariable stochastic adaptive control

Stochastic adaptive minimum variance control algorithms require a division by a function of a recursively computed parameter estimate at each instant of time. In order that the analysis of these algorithms is valid, zero divisions must be events of probability zero. This property is established for the stochastic gradient adaptive control algorithm under the condition that the initial state of the system and all finite segments of its random disturbance process have a joint distribution which is absolutely continuous with respect to Lebesgue measure. This result is deduced from the following general result established in this paper: a non-constant rational function of a finite set of random variables {x₁},x_n} is absolutely continuous with respect to Lebesgue measure if the joint distribution function of {x₁,…,x_n} has this property.     

Dissipative control for a class of uncertain nonlinear control systems

A general dissipative controller is proposed to achieve robust tracking control performance for a class of uncertain single‐input single‐output (SISO) nonlinear systems. The feedback linearization technique is employed to transform the nonlinear system into an assignable inner linear system with a differential control input so that the relationship of the external (input) power and the stored energy of system can be shown clearly. Then, a dissipative controller with an assignable attenuation level is proposed to make the system energy dynamics fit a required dissipative inequality. The unstable factors of the system can then be attenuated accordingly. The system stability is guaranteed even if the system has permanent unavoidable uncertainties. The proposed design can be achieved without the use of traditional means, i.e. optimal control, which requires solution of a Hamilton inequality (or Riccati equation). The Lyapunov stable condition is assured in our approach when the system uncertainties belong to L _∞. Moreover, due to the compatibility of the proposed controller, the controller can be embedded into the designs of other controllers. In those designs, knowledge of the system functions is not required. Basically, the proposed dissipative controller is independent of the system functions. The use of the bounds of the system function is considered to prove the system stability only. Two simulations are given to illustrate the effectiveness of the proposed controller. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Decentralized robust disturbance attenuation for a class of large-scale nonlinear systems

We solve the problem of decentralized H_∞ almost disturbance decoupling for a class of large-scale nonlinear uncertain systems in the absence of matching conditions. The method combines ideas from decentralized adaptive control and centralized nonlinear H_∞ control. We relax earlier assumptions on the uncertain time-varying interconnections which are demanded to be only bounded by general nonlinear functions in this work.     

Coprime factorization over a class of nonlinear systems

In this paper, we consider the problem of generalizing elements of linear coprime factorization theory to a nonlinear context. The idea is to work with a suitably wide class of nonlinear systems to cover many practical situations, yet not cope with so broad a class as to disallow useful generalizations to the linear results. In particular, we work with nonlinear systems characterized in terms of (possibly time-varying) state-dependent matrices A(x), B(x), C(x), D(x) and an initial state x₀. (This class clearly does contain the class of finite-dimensional linear (time-varying) systems.) We achieve first right coprime factorizations for idealized situations. To achieve stable left factorizations we specialize to the case where the matrices are output-dependent. Alternatively, we work with systems, perhaps augmented by a direct feedthrough term, where the input is reconstructible from the output. For nonlinear feedback control systems, with plant and controller having stable left factorizations, then under appropriate regularity-conditions earlier results have allowed the generation of the class of stabilizing controllers for a system in terms of an arbitrary stable system (parameter). Plant uncertainties, including unknown initial conditions are modelled by means of a Yula–Kucera-type parametrization approach developed for nonlinear systems. Certain robust stabilization results are also shown, and simulations demonstrate the regulation of nonlinear plants using the techniques developed. All the results are presented in such a way that specialization for the case of linear systems is immediate.     

Observability and observer design for hybrid multicell choppers

Multicell choppers are part of a class of hybrid systems in which the continuous state vector is always unobservable, in the sense that the observability matrix never has full rank. Due to their hybrid behaviour, the recent concept of Z(T _N )-observability can be applied and analysed in the context of multicell choppers, which allows to give conditions, in terms of the switching sequence, under which the voltage across each capacitor can be reconstructed, not instantly, but after some number of switchings. The case when a DC-motor is coupled to the multicell chopper is also considered. It is shown that, under certain admissible assumptions, the voltages across the capacitors and the motor speed can be acceptably estimated. Two observers, one based on the super-twisting algorithm and the other one based on an adaptive approach, are designed. Additionally, we design an observer for the partial state observation. Simulations are given where the proposed observers are compared and the effectiveness of both is shown.     

Iterative Methods for Linear Complementarity Problems with Interval Data

A ] and an interval vector [b]. If all A∈[A] are H-matrices with positive diagonal elements, these methods are all convergent to the same interval vector [x ^*]. This interval vector includes the solution x of the linear complementarity problem defined by any fixed A∈[A] and any fixed b∈[b]. If all A∈[A] are M-matrices, then we will show, that [x ^*] is optimal in a precisely defined sense. We also consider modifications of those methods, which under certain assumptions on the starting vector deliver nested sequences converging to [x ^*]. We close our paper with some examples which illustrate our theoretical results. Received October 7, 2002; revised April 15, 2003 Published online: June 23, 2003 RID="*" ID="*" Dedicated to U. Kulisch on the occasion of his 70th birthday. We are grateful to the referee who has given a series of valuable comments.     

Control of Selective Catalytic Reduction Systems Using Feedback Linearisation

This paper discusses the identification and control of a selective catalytic reduction (SCR) system. SCR after‐treatment systems form an important technology for reducing the nitrogen oxides, NO_x, produced by diesel engines. To be able to control the system, i.e. reducing the output NO_x, good models of the after‐treatment system are essential. In this paper a nonlinear black‐box model is identified using a recursive prediction error method. The nonlinear model is applied for design of a controller using feedback linearization techniques including an adaptive strategy. A linear quadratic Gaussian controller is used for the control of the linearized system. A total of 17 parameters were estimated for the nonlinear model. The results indicate that output NO_x control using feedback linearization based on a second order black‐box nonlinear model is feasible, provided that identification or adaptivity is used for model tuning. The latter requirement is a result of a study of the robustness. In summary, the paper indicates that significant improvements as compared to linear control can be obtained with the proposed strategy.     

Exact finite-dimensional filters via systems realization for a class of discrete-time nonlinear systems

We obtain necessary and sufficient conditions for the existence of a finite-dimensional filter for the discrete-time nonlinear system (ε x_k+1 =φ(x_k), y_k = h(x_k)+η(x_k)ν_k, K=0, 1,…. This system is distinguished by the absence of noise in the dynamic and by the correlation between the state and the intensity of noise in the observations.The necessary and sufficient condition provides an explicit formula for the minimal filter and various system-theoretic properties of (ε) and of the minimal filter.     

Design of finite dimensional robust H ∞ distributed consensus filters for dissipative PDE systems with sensor networks

This paper presents a design method of finite dimensional robust H _∞ distributed consensus filters (DCFs) for a class of dissipative nonlinear partial differential equation (PDE) systems with a sensor network, for which the eigenvalue spectrum of the spatial differential operator can be partitioned into a finite dimensional slow one and an infinite dimensional stable fast complement. Initially, the modal decomposition technique is applied to the PDE system to derive a finite dimensional ordinary differential equation system that accurately describes the dynamics of the dominant (slow) modes of the PDE system. Then, based on the slow subsystem, a set of finite dimensional robust H _∞ DCFs are developed to enforce the consensus of the slow mode estimates and state estimates of all local filters for all admissible nonlinear dynamics and observation spillover, while attenuating the effect of external disturbances. The Luenberger and consensus gains of the proposed DCFs can be obtained by solving a set of linear matrix inequalities (LMIs). Furthermore, by the existing LMI optimization technique, a suboptimal design of robust H _∞ DCFs is proposed in the sense of minimizing the attenuation level. Finally, the effectiveness of the proposed DCFs is demonstrated on the state estimation of one dimensional Kuramoto–Sivashinsky equation system with a sensor network. Copyright © 2014 John Wiley & Sons, Ltd.     

H∞ nonlinear filtering

This paper investigates the problem of H_∞ estimation of nonlinear processes. An estimator, which may be nonlinear, is looked for so that a given bound on the ratio between the energy of the estimation error and the energy of the oxogeneous inputs to the estimated process is achieved. Conditions for the existence of such an estimator and formulas for its derivation are obtained using both the game theory approach and the theory of dissipative systems. The results of the paper extend the recent results on H_∞ nonlinear control. They are demonstrated by a simple example of a linear system with a nonlinear measurement rule and compared with corresponding results that are obtained by the extended Kalman filter.     

Flight controller design for uncertain and nonlinear plants with pilot compensation

Nonlinear quantitative feedback theory (QFT) is used to design a flight control system for the nonlinear model of the YF-16 aircraft (A/C) with C* as the controlled output. The resulting closed loop stability augmentation system (SAS), P_e(S), becomes part of the outer loop containing the pilot. The Neal-Smith pilot model for a compensatory tracking task is used to develop a technique which allows the designer to synthesize compensation in the outer loop, which includes a free compensator F_p(S). The latter is chosen to minimize pilot workload, increase system bandwidth, and improve handling qualities ratings as per the Neal–Smith criteria, for the tracking task. The available pilot compensation abilities are then available for further increasing of system bandwidth to improve overall capabilities. This approach can be used at the early stages of flight control design, thus saving time and money over the current practice. Simulations in the time and frequency domains demonstrate that the desired performance is attained.