-bounded robust control of nonlinear cascade systems

In this paper, we consider the L^∞-bounded robust control problem for a class of nonlinear cascade systems with disturbances. Sufficient conditions are provided under which a hard bound is imposed on the system performance measure. The backstepping approach is used for controller design. A practical example is provided to illustrate the method.     

On the strict logic foundation of fuzzy reasoning

This paper focuses on the logic foundation of fuzzy reasoning. At first, a new complete first-order fuzzy predicate calculus system K^* corresponding to the formal system L^* is built. Based on the many-sort system K_ms^* corresponding to K^*, the triple I methods of FMP and FMT for fuzzy reasoning and their consistency are formalized, thus fuzzy reasoning is put completely and rigorously into the logic framework of fuzzy logic.The author is indebted to anonymous referee for his useful comments which have helped to improve the paper.     

Mixed Discontinuous Galerkin Finite Element Method for the Biharmonic Equation

In this paper, we first split the biharmonic equation Δ² u=f with nonhomogeneous essential boundary conditions into a system of two second order equations by introducing an auxiliary variable v=Δu and then apply an hp-mixed discontinuous Galerkin method to the resulting system. The unknown approximation v _h of v can easily be eliminated to reduce the discrete problem to a Schur complement system in u _h , which is an approximation of u. A direct approximation v _h of v can be obtained from the approximation u _h of u. Using piecewise polynomials of degree p≥3, a priori error estimates of u−u _h in the broken H ¹ norm as well as in L ² norm which are optimal in h and suboptimal in p are derived. Moreover, a priori error bound for v−v _h in L ² norm which is suboptimal in h and p is also discussed. When p=2, the preset method also converges, but with suboptimal convergence rate. Finally, numerical experiments are presented to illustrate the theoretical results. Supported by DST-DAAD (PPP-05) project.     

Structure-Texture Image Decomposition—Modeling, Algorithms, and Parameter Selection

This paper explores various aspects of the image decomposition problem using modern variational techniques. We aim at splitting an original image f into two components u and ρ, where u holds the geometrical information and ρ holds the textural information. The focus of this paper is to study different energy terms and functional spaces that suit various types of textures. Our modeling uses the total-variation energy for extracting the structural part and one of four of the following norms for the textural part: L², G, L¹ and a new tunable norm, suggested here for the first time, based on Gabor functions. Apart from the broad perspective and our suggestions when each model should be used, the paper contains three specific novelties: first we show that the correlation graph between u and ρ may serve as an efficient tool to select the splitting parameter, second we propose a new fast algorithm to solve the TV − L¹ minimization problem, and third we introduce the theory and design tools for the TV-Gabor model. First online version published in February, 2006     

On nonlinear H ∞ sliding mode control for a class of nonlinear cascade systems

In this work two main robust control strategies, the sliding mode control (SMC) and nonlinear H ^∞ control, are integrated to function in a complementary manner for tracking control tasks. The SMC handles matched L ^∞[0,∞) type system uncertainties with known bounding functions. H ^∞ control deals with unmatched disturbances of L ₂[0,∞) type where the upper-bound knowledge is not available. The new control method is designed for a class of nonlinear uncertain systems with two cascade subsystems. Nonlinear H ^∞ control is applied to the first subsystem in the presence of unmatched disturbances. Through solving a Hamilton-Jacoby inequality, the nonlinear H ^∞ control law for the first subsystem well defines a nonlinear switching surface. By virtue of nonlinear H ^∞ control, the resulting sliding manifold in the sliding phase possesses the desired L ₂ gain property and to a certain extend the optimality. Associated with the new switching surface, the SMC is applied to the second subsystem to accomplish the tracking task, and ensure the L ₂ gain robustness in the reaching phase. Two illustrative examples are given to show the effectiveness of the proposed robust control scheme.     

Stabilization and H control of symmetric systems: an explicit solution

In this paper we address the H^∞ control analysis, the output feedback stabilization, and the output feedback H^∞ control synthesis problems for state-space symmetric systems. Using a particular solution of the Bounded Real Lemma for an open-loop symmetric system we obtain an explicit expression to compute the H^∞ norm of the system. For the output feedback stabilization problem we obtain an explicit parametrization of all asymptotically stabilizing control gains of state-space symmetric systems. For the H^∞ control synthesis problem we derive an explicit expression for the optimally achievable closed-loop H^∞ norm and the optimal control gains. Extension to robust and positive real control of such systems are also examined. These results are obtained from the linear matrix inequality formulations of the stabilization and the H^∞ control synthesis problems using simple matrix algebraic tools.     

UNIQUENESS CONDITIONS OF CL-STRUCTURES IN GL-STRUCTURE EQUIVALENCE CLASSES OF SYSTEM RECONSTRUCTABILITY ANALYSIS

This paper first introduces the basic notions of overall systems with logical relations, their subsystems, structure representation graphs, sets of H-structures and G-structures with logical relations S _H ^L and S _G ^L, immediate refinement and aggregate with logical relations, structure-graph mappings r_V ^L, r_G ^L, etc., and the notions of inclusiveness with logical relations, upper bounding and lower bounding with logical relations. Then it proves the sufficient conditions under which representation graphs R ^L form a lattice. By defining the least upper bound and largest lower bound of (R^L, ) and recommending a lemma on distributivity, this paper proves the sufficient conditions under which G-structures with logical relations form Boolean lattices. Finally after defining the M-structures and C-structures with logical relations, i.e. M-structures and C-structures, this paper proposes the sufficient conditions for the uniqueness of C-structures in G-structure equivalence classes, and proves that the C ^L-structure is the least refined GL-structure in equivalence class S _G ^L/r _G ^L.     

An extension theorem for polynomials on triangles

We present an extension theorem for polynomial functions that proves a quasi-optimal bound for a lifting from L ² on edges onto a fractional-order Sobolev space on triangles. The extension is such that it can be further extended continuously by zero within the trace space of H ¹. Such an extension result is critical for the analysis of non-overlapping domain decomposition techniques applied to the p-and hp-versions of the finite and boundary element methods for elliptic problems of second order in three dimensions. Supported by the FONDAP Programme in Applied Mathematics, Chile.     

Simultaneous Lp-stabilization and internal stabilization of linear systems subject to input saturation — state feedback case

We consider the problem of simultaneous finite gain L_p-stabilization and internal stabilization of linear systems subject to input saturation via linear static state feedback. We show that bounded input finite-gain L_p-stabilization and local asymptotic stabilization can always be achieved simultaneously no matter where the poles of the open-loop system are, and the locations of these poles play a role only when bounded input finite gain L_p-stabilization and global or semi-global stabilization are required simultaneously.     

On Steiner minimal trees withLp distance

LetL _p be the plane with the distanced _p (A ₁ ,A ₂ ) = (¦x ₁ –x ₂¦ ^p + ¦y₁ –y ₂¦^p)^/1p wherex _i andy _i are the cartesian coordinates of the pointA _i . LetP be a finite set of points inL _p . We consider Steiner minimal trees onP. It is proved that, for 1 <p < , each Steiner point is of degree exactly three. Define the Steiner ratio _p to be inf{L _s (P)/L _m (P)¦PL _p } whereL _s (P) andL _m (P) are lengths of the Steiner minimal tree and the minimal spanning tree onP, respectively. Hwang showed ₁ = 2/3. Chung and Graham proved ₂ > 0.842. We prove in this paper that {} = 2/3 and (2/2)_{12 p} 3/2 for anyp.This work was supported in part by the National Science Foundation of China and the President Foundation of Academia Sinica.     

A faster divide-and-conquer algorithm for constructing delaunay triangulations

An easily implemented modification to the divide-and-conquer algorithm for computing the Delaunay triangulation ofn sites in the plane is presented. The change reduces its (n logn) expected running time toO(n log logn) for a large class of distributions that includes the uniform distribution in the unit square. Experimental evidence presented demonstrates that the modified algorithm performs very well forn2¹⁶, the range of the experiments. It is conjectured that the average number of edges it creates—a good measure of its efficiency—is no more than twice optimal forn less than seven trillion. The improvement is shown to extend to the computation of the Delaunay triangulation in theL _p metric for 1<p.This research was supported by National Science Foundation Grants DCR-8352081 and DCR-8416190.     

Tabular Invariants on Graphs and Their Application

A system of tabular invariants on graphs is proposed, and their properties and possibilities of their use for construction of efficient algorithms of solution of some "difficult to solve" problems on graphs are investigated. The notion of an S ^(p)-set is formulated and the theorem on the ordering of such sets is proved. Empirical estimates are obtained for the failure probability of first- and second-order tabular invariants and ordered vectors of degrees of nodes (-invariants) of graphs whose number of nodes varies from 8 to 20.     

<Emphasis Type="Italic">hp</Emphasis>-Version <Emphasis Type="Italic">a priori</Emphasis> Error Analysis of Interior Penalty Discontinuous Galerkin Finite Element Approximations to the Biharmonic Equation

We consider the symmetric formulation of the interior penalty discontinuous Galerkin finite element method for the numerical solution of the biharmonic equation with Dirichlet boundary conditions in a bounded polyhedral domain in . For a shape-regular family of meshes consisting of parallelepipeds, we derive hp-version a priori bounds on the global error measured in the L² norm and in broken Sobolev norms. Using these, we obtain hp-version bounds on the error in linear functionals of the solution. The bounds are optimal with respect to the mesh size h and suboptimal with respect to the degree of the piecewise polynomial approximation p. The theoretical results are confirmed by numerical experiments, and some practical applications in Poisson–Kirchhoff thin plate theory are presented.     

A note on solving linear Diophantine systems by using L 3-reduction algorithm

The main difficulty in solving linear Diophantine systems is the very rapid growth of intermediate results which makes many algorithms, for solving linear Diophantine systems, impractical even for large computers. One way for controlling this growth is to use the L ³-reduction algorithm, introduced by Lenstra et al. [A.K. Lenstra, H.W. Lenstra, and L. Lov?sz, Factoring polynomials with rational coefficients, Math. Ann. 261 (1982), pp. 515–534.]. Esmaeili [H. Esmaeili, How can we solve a linear Diophantine equation by the basis reduction algorithm, Int. J. Comput. Math. 82 (2005), pp. 1227–1234.] proposed a method for obtaining the general integer solution of a linear Diophantine equation by using L ³-reduction algorithm. Here we propose a procedure for generalizing Esmaeili's method, to a method for obtaining the general integer solution of systems of linear Diophantine equations by using L ³-reduction algorithm. Then we consider the complexity issues and show that the generalized algorithm controls the growth of the intermediate results and the number of required arithmetic operations well. Finally, some illustrative numerical examples are given to show the efficiency of the proposed algorithm.     

Inverse NP Problems

The inverse problem relative to a verifier V of proofs of membership for a NP language is the problem of deciding, given a set π of proofs, whether or not there exists a string x having exactly π as its set of proofs. In this paper, we study the complexity of inverse problems. We develop a new notion of reduction which allows one to compare the complexity of inverse problems. Using this notion, we classify as coNP-complete the inverse problems for the “natural” verifiers of many NP-complete problems. We also show that the inverse complexity of a verifier for a language L cannot be predicted solely from the complexity of L, but rather, is highly dependent upon the choice of verifier used to accept L. In this context, a verifier with a Σ₂ ^p -complete inverse problem is exhibited, giving a new and natural example of a Σ₂ ^p -complete problem.      

Achieving nonvanishing stability regions with high-gain cheap control using H techniques: the second-order case

This paper demonstrates how to use an asymptotically H^∞-optimal controller to stabilize a second-order system subject to unknown disturbances such that the stability region does not vanish as the feedback gains increase. The high-gain feedback arises when one attempts to achieve the lowest achievable limit of the disturbance attenuation under the H^∞ design. This type of gain increase can cause the stability region to vanish if the disturbance contains nonlinear terms. The analysis using Lyapunov techniques derives a sufficient condition on the design parameters to prevent the stability region from vanishing. In addition to describing exact solutions for six different cases, the paper provides simulations to illustrate the results.     

Achievable H performance in sampled-data smoothing: beyond the -barrier

The lifting technique is a powerful tool for handling the periodically time-varying nature of sampled-data systems. Yet all known solutions of sampled-data H^∞ problems are limited to the case when the feedthrough part of the lifted system, , satisfies , where γ is the required H^∞ performance level. While this condition is always necessary in feedback control, it might be restrictive in signal processing applications, where some amount of delay or latency between measurement and estimation can be tolerated. In this paper, the sampled-data H^∞ fixed-lag smoothing problem with a smoothing lag of one sampling period is studied. The problem corresponds to the a-posteriori filtering problem in the lifted domain and is probably the simplest problem for which a smaller than performance level is achievable. The necessary and sufficient solvability conditions derived in the paper are compatible with those for the sampled-data filtering problem. This result extends the scope of applicability of the lifting technique and paves the way to the application of sampled-data methods in digital signal processing.     

Simultaneous measurement of Pb, Cd and Zn using differential pulse anodic stripping voltammetry at a bismuth/poly(p-aminobenzene sulfonic acid) film electrode

A novel sensor was developed for simultaneous detection of Pb, Cd and Zn, based on the differential pulse anodic stripping response at a bismuth/poly(p-aminobenzene sulfonic acid) (Bi/poly(p-ABSA)) film electrode. This electrode was generated in situ by depositing simultaneously bismuth and the metals by reduction at −1.40 V on the poly(p-ABSA) modified electrode. Compared with the bismuth film electrode, the Bi/poly(p-ABSA) film electrode can yield a larger stripping signal for Pb, Cd and Zn. Under the optimum conditions, a linear response was observed for Cd and Zn in the range from 1.00 to 110.00 μg L⁻¹ and for Pb in the range from 1.00 to 130.00 μg L⁻¹. The detection limits of Pb(II), Cd(II) and Zn(II) were 0.80, 0.63 and 0.62 μg L⁻¹, respectively. Finally this sensor had been applied to the simultaneous determination of Pb(II), Cd(II) and Zn(II) in river water samples and the results were quite corresponding to the value obtained by atomic absorption spectrometry.     

On pure H reduced-order dynamic controllers: an erratum

There are at least two approaches advocated to obtain a pure H^∞ reduced-order dynamic controller for a given augmented plant. One approach is to eliminate completely the H² aspect from a standard H²/H^∞ setting. A second approach is to equate the H² aspect with the H^∞ aspect in that same setting. This paper invalidates the first approach but affirms the second approach and produces the correct equations resulting therefrom.     

An inequality governing nonlinear H∞ control

This note gives necessary and sufficient conditions for solving a reasonable version of the nonlinear H^∞ control problem. The most objectionable hypothesis is elegant and holds in the linear case, but every possibly may not be forced for nonlinear systems. What we discover in distinction to Isidori and Astolfi (1992) and Ball et al. (1993) is that the key formula is not a (nonlinear) Riccati partial differential inequality, but a much more complicated inequality mixing partial derivatives and an approximation theoretic construction called the best approximation operator. This Chebeshev-Riccati inequality when specialized to the linear case gives the famous solution to the H^∞ control problem found in Doyle et al. (1989). While complicated the Chebeshev-Riccati inequality is (modulo a considerable number of hypotheses behind it) a solution to the nonlinear H^∞ control problem. It should serve as a rational basis for discovering new formulas and compromises. We follow the conventions of Ball et al. (1993) and this note adds directly to that paper.