A split least-squares characteristic mixed element method for nonlinear nonstationary convection–diffusion problem

In this paper, a split least-squares characteristic mixed finite element method is proposed for solving nonlinear nonstationary convection–diffusion problem. By selecting the least-squares functional property, the resulting least-squares procedure can be split into two independent symmetric positive definite sub-schemes. The first sub-scheme is for the unknown variable u, which is the same as the standard characteristic Galerkin finite element approximation. The second sub-scheme is for the unknown flux σ. Theoretical analysis shows that the method yields the approximate solutions with optimal accuracy in L ²(Ω) norm for the primal unknown and in H(div; Ω) norm for the unknown flux, respectively. Some numerical examples are given to confirm our theory results.     

A local stabilizing control scheme using an approximate feedbacklinearization

We present a method of stabilizing single-input (approximately) feedback linearizable systems in the generic situation where the linearizing feedback and coordinate transformation map are not directly available. In obtaining a stabilizing feedback, we utilize a technique of an approximate feedback linearization which is based on an approximate integrating factor calculated from a recursive algorithm. We construct a feedback controller for the approximately linearized system so that the system is asymptotically stable in a neighborhood of the origin where the one-to-one relation holds with the original coordinate system     

Sliding-mode control design for a class of systems with non-matching nonlinearities and disturbances

A novel sliding-mode control design is proposed for the class of single-input nonlinear systems with non-matching nonlinearities and/or disturbances. This class of systems includes dual-spin spacecraft and rotational proof mass actuators that are used for suppressing translational oscillation. The sliding-mode control law is designed via an L ₂ -norm bounding technique that ensures, under certain conditions, asymptotic stability of the system when the disturbances satisfy the matching condition or are zero; otherwise, it ensures system stability with an a priori bound on the transmission gain of the disturbances through the system for non-zero non-matching disturbances. Simulation results for a rotational proof mass actuator or dual-spin spacecraft example illustrate the excellent responses obtained by applying the control laws derived from this proposed design procedure.     

Stability of nonlinear feedback systems in a Volterra representation

Presented in this paper is a stability condition for a class of nonlinear feedback systems where the plant dynamics can be represented by a finite series of Volterra kernels. The class of Volterra kernels are limited to p‐linear stable operators and may contain pure delays. The stability condition requires that the linear kernel is non‐zero and that the closed loop characteristic equation associated with the linearized system is stable. Next, a sufficient condition is developed to upper bound the infinity‐norm of an external disturbance signal thereby guaranteeing that the internal and output signals of the closed loop nonlinear system are contained in L_∞. These results are then demonstrated on a design example. A frequency domain controller design procedure is also developed using these results where the trade‐off between performance and stability are considered for this class of nonlinear feedback systems. Copyright © 2000 John Wiley & Sons, Ltd.     

Explicit constructions of global stabilization and nonlinear H∞ control laws for a class of nonminimum phase nonlinear multivariable systems

This paper investigates a global stabilization problem and a nonlinear H_∞ control problem for a class of nonminimum phase nonlinear multivariable systems. To avoid the complicated recursive design procedure, an asymptotic time‐scale and eigenstructure assignment method is adopted to construct the control laws for the stabilization problem and the nonlinear H_∞ control problem. A sufficient solvability condition is established onthe unstable zero dynamics of the system for global stabilization problem and nonlinear H_∞ control problem, respectively. Moreover, based on the sufficient solvability condition, an upper bound of the achievable L₂‐gain is estimated for the nonlinear H_∞ control problem. Copyright © 2007 John Wiley & Sons, Ltd.     

Nonlinearizable single-input control systems do not admit stationary symmetries

The main result of the paper states that almost any analytic single-input control system, which is truly nonlinear, that is not feedback linearizable, with controllable linearization at an equilibrium point, does not admit any symmetry preserving that point. By almost any system, we mean that we exclude a small class of odd systems, that admit just one nontrivial symmetry conjugated to minus identity. The obtained results are based on a recent classification of nonlinear single-input systems under formal feedback. We also describe symmetries of feedback linearizable systems.     

Backstepping Robust H ∞ Control for a Class of Uncertain Switched Nonlinear Systems Under Arbitrary Switchings

This paper addresses global robust H _∞ control for a class of switched nonlinear systems with uncertainty under arbitrary switchings. Each subsystem is in lower triangular form. The uncertainties are assumed to be in a known compact set. The backstepping design technique is used to design a smooth state feedback controller that renders the associated closed‐loop switched system globally robustly asymptotically stable and imposes a pre‐specified upper bound to the L ₂‐gain under arbitrary switchings. An example is provided to demonstrate the efficacy of the design approach.     

A split least-squares characteristic mixed finite element method for Sobolev equations with convection term

In this paper, a split least-squares characteristic mixed finite element method for a kind of Sobolev equation with convection term is proposed, in which the characteristic method is based on the approximation of the material derivative term, that is, the time derivative term plus the convection term. The resulting least-squares procedure can be split into two independent symmetric positive definite sub-schemes and does not need to solve a coupled system of equations. Theory analysis shows that the method yields the approximate solutions with optimal accuracy in L²(Ω) norm for the primal unknown and in H(div;Ω) norm for the unknown flux, respectively. Numerical examples in one dimension, which are consistent with the theoretical results, are provided to demonstrate the characteristic behavior of this approach.     

CONTROLLER REDUCTION FOR NONLINEAR PLANTS—AN L2 APPROACH

This paper extends the H_∞ balanced truncation approach of Mustafa–Glover for linear plants to input affine nonlinear plants. Nonlinear H_∞ balanced truncation is used to obtain a reduced order controller. Conditions which ensure that this controller stabilizes the full order plant are derived. This is done by relating the model reduction problem to a robust stabilization problem with unstructured perturbation. In addition an upper bound on the performance of the closed loop system, with respect to the L₂ gain, is obtained. When specialized to linear plants this bound reduces to Mustafa–Glover's result. © 1997 by John Wiley & Sons, Ltd. Int. J. Robust Nonlinear Control, Vol. 7, 475–505 (1997)     

L1 impulse response error bound for balanced truncation

This paper presents an upper bound in L₁ for the impulse response error between a system and its balanced truncation. It is an a priori bound and can be computed easily. Numerical examples are used to illustrate its applications and to compare with other available error bounds.     

Gauss–Newton Multilevel Methods for Least-Squares Finite Element Computations of Variably Saturated Subsurface Flow

We apply the least-squares mixed finite element framework to the nonlinear elliptic problems arising in each time-step of an implicit Euler discretization for variably saturated flow. This approach allows the combination of standard piecewise linear H ¹-conforming finite elements for the hydraulic potential with the H(div)-conforming Raviart–Thomas spaces for the flux. It also provides an a posteriori error estimator which may be used in an adaptive mesh refinement strategy. The resulting nonlinear algebraic least-squares problems are solved by an inexact Gauss–Newton method using a stopping criterion for the inner iteration which is based on the change of the linearized least-squares functional relative to the nonlinear least-squares functional. The inner iteration is carried out using an adaptive multilevel method with a block Gauss–Seidel smoothing iteration. For a realistic water table recharge problem, the results of computational experiments are presented. Received January 4, 1999; revised July 19, 1999     

Rational L∞-suboptimal controllers for SISOcontinuous-time systems

We study the problem of minimizing the weighted amplitude of the time response due to a given fixed input signal for single-input/single-output (SISO) continuous-time systems and focus on obtaining rational suboptimal solutions. An EAS (Euler approximating system)-based method is proposed for designing a rational L_∞ -suboptimal controller for SISO systems. It is shown that this rational approximation is the best one among a set of rational approximations, in the sense of providing the tightest upper bound and that it can approximate the optimal cost arbitrarily close     

A global state feedback output regulating control for uncertain systems in strict feedback form

A family of single-input, single-output, nonlinear systems in strict feedback form with uncertain constant parameters which appear linearly and belong to a known compact set Π is considered. A global robust output regulation problem via state feedback is addressed and solved under the assumptions that the regulator equations are solvable in Π, the output equation does not depend on uncertain parameters, no modelled disturbances affect the system and the output reference signal is generated by a known exosystem whose state is bounded and available for feedback. Robust adaptive techniques are used to guarantee closed loop boundedness and to achieve, without requiring persistency of excitation, global asymptotic output regulation for a class of feedback linearizable systems which may have unbounded uncertain tracking dynamics or may not have a well-defined global relative degree.     

Weighted least-squares finite element methods for the linearized Navier–Stokes equations

We implemented weighted least-squares finite element methods for the linearized Navier-Stokes equations based on the velocity–pressure–stress and the velocity–vorticity–pressure formulations. The least-squares functionals involve the L²-norms of the residuals of each equation multiplied by the appropriate weighting functions. The weights included a mass conservation constant, a mesh-dependent weight, a nonlinear weighting function, and Reynolds numbers. A feature of this approach is that the linearized system creates a symmetric and positive-definite linear algebra problem at each Newton iteration. We can prove that least-squares approximations converge with the linearized version solutions of the Navier–Stokes equations at the optimal convergence rate. Model problems considered in this study were the flow past a planar channel and 4-to-1 contraction problems. We presented approximate solutions of the Navier–Stokes problems by solving a sequence of the linearized Navier–Stokes problems arising from Newton iterations, revealing the convergence rates of the proposed schemes, and investigated Reynolds number effects.     

Adaptive output feedback tracking controller for a class of uncertain strict feedback nonlinear systems in the absence of state measurements

In this article, design of an adaptive control scheme for a class of uncertain single-input single-output systems in strict feedback form via a backstepping technique has been proposed. It is assumed that system output and its derivatives are available. By virtue of the observability concept, it is shown that for this class of systems there exists a one-to-one map, which maps output and its derivatives to system states. By means of this mapping and using linearly parametrised approximators, such as fuzzy logic systems or neural networks, the uncertain nonlinear dynamics and unavailable states are estimated. The proposed adaptive controller guarantees that the closed-loop system is uniformly ultimately bounded and the influence of minimum approximation error on the L ₂-norm of the output tracking error is attenuated arbitrarily. The effectiveness of the proposed scheme has been demonstrated through simulation results.     

L∞ Observer for Uncertain Linear Systems

In this paper, an L_∞ observer design method is proposed for linear system subject to parameter uncertainty and bounded disturbance. The proposed L_∞ observer, which satisfies a peak‐to‐peak disturbances attenuation performance, is designed to overbound the estimation error. Moreover, sufficient conditions for the design of L_∞ observer are derived and expressed in terms of linear matrix inequalities (LMIs). The novelty of the proposed method is that we develop an L_∞ observer that not only can attenuate bounded disturbance but also provides an upper bound of estimation error norm. Simulation results are presented to illustrate the effectiveness of the proposed method.     

Extended time-frequency conditions for the Instability of a class of non-linear time-varying systems

This paper presents new criteria for the L ₂-input, L ₂-output stability of a class of feedback systems containing a time-invariant convolution operator and a time varying non-linearity in cascade, in a negative feedback loop. These are sufficient conditions for the system stability derived using the multiplier concept and involve certain time-domain conditions on the multiplier. The method of derivation draws on the theory of positivity of compositions of operators and time-varying gains and necessitates an upper bound on the rate of variation of the time-varying gain.     

A new algorithm for constructing approximate transformations fornonlinear systems

An algorithm is provided for constructing an approximating transformation for a feedback linearizable system, and for a system that is not feedback linearizable. Instead of approximating the given nonlinear system, the authors approximate the transformation in a proper sense. They assume that the lead variables of a transformation have a certain form with unknown parameters, substitute these variables into the required equations, and determine the unknown parameters through linear algebraic equations. There is a tremendous savings in terms of number of equations and unknowns, and it is sometimes possible to find exact (or best possible) transformations in a surprisingly small number of steps     

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.