On global stabilization of Burgers’ equation by boundary control

Although often referred to as a one-dimensional “cartoon” of Navier–Stokes equation because it does not exhibit turbulence, the Burgers equation is a natural first step towards developing methods for control of flows. Recent references include Burns and Kang [Nonlinear Dynamics 2 (1991) 235–262], Choi et al. [J. Fluid Mech. 253 (1993) 509–543], Ito and Kang [SIAM J. Control Optim. 32 (1994) 831–854], Ito and Yan [J. Math. Anal. Appl. 227 (1998) 271–299], Byrnes et al. [J. Dynam. Control Systems 4 (1998) 457–519] and Van Ly et al. [Numer. Funct. Anal. Optim. 18 (1997) 143–188]. While these papers have achieved tremendous progress in local stabilization and global analysis of attractors, the problem of global asymptotic stabilization has remained open. This problem is non-trivial because for large initial conditions the quadratic (convective) term – which is negligible in a linear/local analysis – dominates the dynamics. We derive nonlinear boundary control laws that achieve global asymptotic stability. We consider both the viscous and the inviscid Burgers’ equation, using both Neumann and Dirichlet boundary control. We also study the case where the viscosity parameter is uncertain, as well as the case of stochastic Burgers’ equation. For some of the control laws that would require the measurement in the interior of the domain, we develop the observer-based versions.     

Exponential stability of variable coefficients Rayleigh beams under boundary feedback controls: a Riesz basis approach

In this paper, we study the boundary stabilizing feedback control problem of Rayleigh beams that have non-homogeneous spatial parameters. We show that no matter how non-homogeneous the Rayleigh beam is, as long as it has positive mass density, stiffness and mass moment of inertia, it can always be exponentially stabilized when the control parameters are properly chosen. The main steps are a detail asymptotic analysis of the spectrum of the system and the proving of that the generalized eigenfunctions of the feedback control system form a Riesz basis in the state Hilbert space. As a by-product, a conjecture in Guo (J. Optim. Theory Appl. 112(3) (2002) 529) is answered.     

Stabilization of the Euler–Bernoulli equation via boundary connection with heat equation

In this paper, we are concerned with the stabilization of a coupled system of Euler–Bernoulli beam or plate with heat equation, where the heat equation (or vice versa the beam equation) is considered as the controller of the whole system. The dissipative damping is produced in the heat equation via the boundary connections only. The one-dimensional problem is thoroughly studied by Riesz basis approach: The closed-loop system is showed to be a Riesz spectral system and the spectrum-determined growth condition holds. As the consequences, the boundary connections with dissipation only in heat equation can stabilize exponentially the whole system, and the solution of the system has the Gevrey regularity. The exponential stability is proved for a two dimensional system with additional dissipation in the boundary of the plate part. The study gives rise to a different design in control of distributed parameter systems through weak connections with subsystems where the controls are imposed.     

Boundary feedback exponential stabilization of a Timoshenko beam with both ends free

In the present paper we consider the boundary feedback stabilization of a Timoshenko beam with both ends free. We propose boundary feedback control law that makes the closed loop system dissipative. Using asymptotic analysis techniques, we give explicit asymptotic formula of eigenvalues of the closed loop system, and prove the Riesz basis property of eigenvectors and generalized eigenvectors. By a detailed analysis of spectrum of the closed loop system, we show that the closed system is exponentially stable.     

Asymptotic characterizations of input–output-to-state stability for discrete-time systems

We establish the equivalence between global detectability and output-to-state stability for difference inclusions with outputs, and we present equivalent asymptotic characterizations of input–output-to-state stability for discrete-time nonlinear systems. These new stability characterizations for discrete-time systems parallel what have been developed for continuous-time systems in Angeli et al. [Uniform global asymptotic stability of differential inclusions, J. Dynamical Control Systems 10 (2004) 391–412] and Angeli et al. [Seperation principles for input–output and integral-input-to-state stability, SIAM J. Control Optim. 43 (2004) 256–276].     

The first real eigenvalue of a one-dimensional linear thermoelastic system

In this note, we show, for a one-dimensional linear thermoelastic equation with Dirichlet-Dirichlet boundary conditions, that there is at least one real eigenvalue which is greater than the dominant eigenvalue of the “pure” heat equation with the same boundary conditions. The result concludes the spectrum-determined growth condition for the system by virtue of a result of Renardy [1]. Moreover, this property is shown to be preserved for the same system with boundary vibration control.     

Language-Emptiness Checking of Alternating Tree Automata Using Symbolic Reachability Analysis

Alternating tree automata and AND/OR graphs provide elegant formalisms that enable branching- time logics to be verified in linear time. The seminal work of Kupferman et al. [Orna Kupferman, Moshe Y. Vardi, and Pierre Wolper. An automata-theoretic approach to branching-time model checking. J. ACM, 47(2):312–360, 2000] showed that 1) branching-time model checking is reducible to the language non-emptiness checking of the product of two alternating automata representing the model and property under verification, and 2) the non-emptiness problem can be solved by performing a search on an AND/OR graph representing this product. Their algorithm, however, can only be implemented in an explicit-state model checker because it needs stacks to detect accept and reject runs. In this paper, we propose a BDD-based approach to check the language non-emptiness of the product automaton. We use a technique called “state recording” from Schuppan and Biere [Viktor Schuppan and Armin Biere. Efficient reduction of finite state model checking to reachability analysis. Int. Journal on Software Tools for Technology Transfer (STTT), 5(2–3):185–204, 2004] to emulate the stack mechanism from explicit-state model checking. This technique allows us to transform the product automaton into a well-defined AND/OR graph. We develop a BDD-based reachability algorithm to efficiently determine whether a solution graph for the AND/OR graph exists and thereby solve the model-checking problem. While “state recording” increases the size of the state space, the advantage of our approach lies in the memory saving BDDs can offer and the potential it opens up for optimisation of the reachability analysis. We remark that this technique always detects the shortest counter-example.     

On the -semigroup generation and exponential stability resulting from a shear force feedback on a rotating beam

In this paper, we show that a linear unbounded operator associated with an Euler–Bernoulli beam equation under shear boundary feedback generates a C₀-semigroup in the underlying state Hilbert space. This provides an answer to a long time unsolved problem due to the lack of dissipativity for the operator. The main steps are a careful estimation of the Green's function and the verification of the Riesz basis property for the generalized eigenfunctions. As a consequence, we show that this semigroup is differentiable and exponentially stable, which is in sharp contrast to the properties possessed by most feedback controlled beams based on a passive design principle.     

Optimal design of two interconnected enzymatic bioreactors

This paper deals with the optimal design of two interconnected continuous stirred bioreactors in which a single enzymatic reaction occurs. The term “optimal” should be understood here as the minimum of the total volume of the reactors required to perform a given conversion rate, given a quantity of matter to be converted per time unit. In determining the optimal volume, it is considered that the input flow may be distributed among the tanks and also that a recirculation loop may be used. The optimal design problem is solved for a wide class of kinetics functions including, in particular, the well-known Michaelis–Menten kinetics function. The analysis of the optimal configurations is investigated, and it is shown that the concept of “Steady State Equivalent Biological System” (SSEBS) first introduced by Harmand et al. [AIChE J., in press] for microbial reactions only applies to enzymatic systems which have non-monotonic kinetics. In addition, a stability as well as a sensitivity analysis of the optimal configurations are performed.     

Pricing fuzzy vulnerable options and risk management

The assumption of unrealistic “identical rationality” in classic option pricing theory is released in this article to amend Klein’s [Klein, P. (1996). Pricing Black–Scholes options with correlated credit risk. Journal of Banking Finance, 1211–1129] vulnerable option pricing formula. Through this formula, default risk and liquidity risk are both well-explained when the investment behaviors and market expectations of the participants are heterogeneous. The numerical results show that when the investing decisions of each market participant come from their individual rationality and use their own subjective price to trade, the option price becomes a boundary. The upper boundary becomes an absolutely safe line and the lower boundary becomes an absolutely unsafe line for investors who want to invest in some financial securities with default risk. The proposed model suggests a more realistic pricing mechanism for the issuers and traders who want to value options with default risk.     

On the Relative Soundness of the Free Algebra Model for Public Key Encryption

Formal systems for cryptographic protocol analysis typically model cryptosystems in terms of free algebras. Modeling the behavior of a cryptosystem in terms of rewrite rules is more expressive, however, and there are some attacks that can only be discovered when rewrite rules are used. But free algebras are more efficient, and appear to be sound for “most” protocols. In [J. Millen, “On the freedom of decryption”, Information Processing Letters 86 (6) (June 2003) 329–333] Millen formalizes this intuition for shared key cryptography and provides conditions under which it holds; that is, conditions under which security for a free algebra version of the protocol implies security of the version using rewrite rules. Moreover, these conditions fit well with accepted best practice for protocol design. However, he left public key cryptography as an open problem. In this paper, we show how Millen's approach can be extended to public key cryptography, giving conditions under which security for the free algebra model implies security for the rewrite rule model. As in the case for shared key cryptography, our conditions correspond to standard best practice for protocol design.     

Methods for parallel computation of complex flow problems

This paper is an overview of some of the methods developed by the Team for Advanced Flow Simulation and Modeling (TAFSM) [http://www.mems.rice.edu/TAFSM/] to support flow simulation and modeling in a number of “Targeted Challenges”. The “Targeted Challenges” include unsteady flows with interfaces, fluid–object and fluid–structure interactions, airdrop systems, and air circulation and contaminant dispersion. The methods developed include special numerical stabilization methods for compressible and incompressible flows, methods for moving boundaries and interfaces, advanced mesh management methods, and multi-domain computational methods. We include in this paper a number of numerical examples from the simulation of complex flow problems.     

Sturm–Liouville problems with discontinuities at two points

In this paper we extend some spectral properties of regular Sturm–Liouville problems to those which consist of a Sturm–Liouville equation with piecewise continuous potentials together with eigenparameter-dependent boundary conditions and four supplementary transmission conditions. By modifying some techniques of [C.T. Fulton, Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A 77 (1977) 293–308; E. Tunç, O.Sh. Muhtarov, Fundamental solutions and eigenvalues of one boundary-value problem with transmission conditions, Appl. Math. Comput. 157 (2004) 347–355; O.Sh. Mukhtarov, E. Tunç, Eigenvalue problems for Sturm–Liouville equations with transmission conditions, Israel J. Math. 144 (2004) 367–380] and [O.Sh. Mukhtarov, M. Kadakal, F.Ş. Muhtarov, Eigenvalues and normalized eigenfunctions of discontinuous Sturm–Liouville problem with transmission conditions, Rep. Math. Phys. 54 (2004) 41–56], we give an operator-theoretic formulation for the considered problem and obtain asymptotic formulae for the eigenvalues and eigenfunctions.     

New criteria on exponential stability of neutral stochastic differential delay equations

Neutral stochastic differential delay equations (NSDDEs) have recently been studied intensively (see e.g. [V.B. Kolmanovskii, V.R. Nosov, Stability and Periodic Modes of Control Systems with Aftereffect, Nauka, Moscow, 1981; X. Mao, Exponential stability in mean square of neutral stochastic differential functional equations, Systems Control Lett. 26 (1995) 245–251; X. Mao, Razumikhin type theorems on exponential stability of neutral stochastic functional differential equations, SIAM J. Math. Anal. 28(2) (1997) 389–401; X. Mao, Stochastic Differential Equations and Their Applications, Horwood Publishing, Chichester, 1997]). More recently, Mao [Asymptotic properties of neutral stochastic differential delay equations, Stochastics and Stochastics Rep. 68 (2000) 273–295] provided with some useful criteria on the exponential stability for NSDDEs. However, the criteria there require not only the coefficients of the NSDDEs to obey the linear growth condition but also the time delay to be a constant. One of our aims in this paper is to remove these two restrictive conditions. Moreover, the key condition on the diffusion operator associated with the underlying NSDDE will take a much more general form. Our new stability criteria not only cover many highly non-linear NSDDEs with variable time delays but they can also be verified much more easily than the known criteria.     

Robust SVC controller design and analysis for uncertain power systems

A Static Var Compensator (SVC) installed in a power transmission network can be effectively used to enhance the damping of electromechanical oscillations [Schweickardt, H. E., Romegialli, G., & Reichert, K. (1978). Closed loop control of static VAR sources (SVS) on EHV transmission lines. IEEE Pes winter power meeting, (paper no A78, pp. 135–136), New York, Jan. 29–Feb. 3]. An adequately designed robust controller, which takes into account variations in the operating conditions, can help to achieve the desired damping control. The proposed approach described in this paper is aimed to achieve damping of electromechanical oscillations by considering a systematic approach, based on interval systems theory and Kharitonov's Theorem. The method presented allows for the design of a fixed-parameter, low-order controller, given a supposed stability degree of the system. The synthesis of a robust SVC controller is divided into two tasks. The first is the determination of the region of stability in the controller parameter plane by plotting the stability boundary locus. The second task is the optimization of the selected controller parameters from the obtained solutions to the first task. Examples of eigenvalue analysis and time simulation demonstrate the effectiveness and robustness of the designed controller.     

弹性梁点控制、点测量与指数镇定

运用近年来发展的基理论对弹性梁点测量、点控制做一个系统的处理, 得到了用速度线性反馈可实现系统的指数镇定的条件且指数镇定当且仅当系统精确可控, 在指数镇定的情况下闭环系统的广义本征函数生成能量空间的Riesz基且谱确定增长条件成立.     

Investigation of engine fault diagnosis using discrete wavelet transform and neural network

An investigation of a fault diagnostic technique for internal combustion engines using discrete wavelet transform (DWT) and neural network is presented in this paper. Generally, sound emission signal serves as a promising alternative to the condition monitoring and fault diagnosis in rotating machinery when the vibration signal is not available. Most of the conventional fault diagnosis techniques using sound emission and vibration signals are based on analyzing the signal amplitude in the time or frequency domain. Meanwhile, the continuous wavelet transform (CWT) technique was developed for obtaining both time-domain and frequency-domain information. Unfortunately, the CWT technique is often operated over a longer computing time. In the present study, a DWT technique which is combined with a feature selection of energy spectrum and fault classification using neural network for analyzing fault signal is proposed for improving the shortcomings without losing its original property. The features of the sound emission signal at different resolution levels are extracted by multi-resolution analysis and Parseval’s theorem [Gaing, Z. L. (2004). Wavelet-based neural network for power disturbance recognition and classification. IEEE Transactions on Power Delivery 19, 1560–1568]. The algorithm is obtained from previous work by Daubechies [Daubechies, I. (1988). Orthonormal bases of compactly supported wavelets. Communication on Pure and Applied Mathematics 41, 909–996.], the“db4”, “db8” and “db20” wavelet functions are adopted to perform the proposed DWT technique. Then, these features are used for fault recognition using a neural network. The experimental results indicated that the proposed system using the sound emission signal is effective and can be used for fault diagnosis of various engine operating conditions.     

Nonlinear stabilization of a thermal convection loop by state feedback

A nonlinear feedback control law that achieves global asymptotic stabilization of a 2D thermal convection loop (widely known for its “Lorenz system” approximation) is presented. The loop consists of viscous Newtonian fluid contained in between two concentric cylinders standing in a vertical plane. The lower half of the loop is heated while the upper half is cooled, which makes the no-motion steady state for the uncontrolled case unstable for values of the non-dimensional Rayleigh number R_a>1. The objective is to stabilize that steady state using boundary control of velocity and temperature on the outer cylinder. We discretize the original nonlinear PDE model in space using finite difference method and get a high order system of coupled nonlinear ODEs in 2D. Then, using backstepping design, we transform the original coupled system into two uncoupled systems that are asymptotically stable in l²-norm with homogeneous Dirichlet boundary conditions. The resulting boundary controls actuate velocity and temperature in the original coordinates. The control design is accompanied by an extensive simulation study which shows that the feedback control law designed on a very coarse grid (using just a few measurements of the flow and temperature fields) can successfully stabilize the actual system for a very wide range of the Rayleigh number.     

A “mixed” small gain and passivity theorem in the frequency domain

We show that the negative feedback interconnection of two causal, stable, linear time-invariant systems, with a “mixed” small gain and passivity property, is guaranteed to be finite-gain stable. This “mixed” small gain and passivity property refers to the characteristic that, at a particular frequency, systems in the feedback interconnection are either both “input and output strictly passive”; or both have “gain less than one”; or are both “input and output strictly passive” and simultaneously both have “gain less than one”. The “mixed” small gain and passivity property is described mathematically using the notion of dissipativity of systems, and finite-gain stability of the interconnection is proven via a stability result for dissipative interconnected systems.     

Averaging with disturbances and closeness of solutions

We establish that, under appropriate conditions, the solutions of a time-varying system with disturbances converge uniformly on compact time intervals to the solutions of the system's average as the rate of change of time increases to infinity. The notions of “average” used for systems with disturbances are the “strong” and “weak” averages introduced in Ne i and Teel (D. Ne i , A.R. Teel, Input-to-state stability for time-varying nonlinear systems via averaging, 1999, submitted for publication. See also: On averaging and the ISS property, Proceedings of the 38th Conference on Decision and Control, Phoenix, AZ, December 1999, pp. 3346–3351.)