Least-squares integration of one-dimensional codistributions with application to approximate feedback linearization

We study the problem of approximating one-dimensional nonintegrable codistributions by integrable ones and apply the resulting approximations to approximate feedback linearization of single-input systems. The approach derived in this paper allows a linearizable nonlinear system to be found that is close to the given system in a least-squares (L ₂) sense. A linearly controllable single-input affine nonlinear system is feedback linearizable if and only if its characteristic distribution is involutive (hence integrable) or, equivalently, any characteristic one-form (a one-form that annihilates the characteristic distribution) is integrable. We study the problem of finding (least-squares approximate) integrating factors that make a fixed characteristic one-form close to being exact in anL ₂ sense. A given one-form can be decomposed into exact and inexact parts using the Hodge decomposition. We derive an upper bound on the size of the inexact part of a scaled characteristic one-form and show that a least-squares integrating factor provides the minimum value for this upper bound. We also consider higher-order approximate integrating factors that scale a nonintegrable one-form in a way that the scaled form is closer to being integrable inL ₂ together with some derivatives and derive similar bounds for the inexact part. This allows a linearizable nonlinear system that is close to the given system in a least-squares (L ₂) sense together with some derivatives to be found. The Sobolev embedding techniques allow us to obtain an upper bound on the uniform (L _∞) distance between the nonlinear system and its linearizable approximation. This research was supported in part by NSF under Grant PYI ECS-9396296, by AFOSR under Grant AFOSR F49620-94-1-0183, and by a grant from the Hughes Aircraft Company.     

LTI approximation of nonlinear systems via signal distribution theory

L₂ and L₁ optimal linear time-invariant (LTI) approximation of discrete-time nonlinear systems, such as nonlinear finite impulse response (NFIR) systems, is studied via a signal distribution theory motivated approach. The use of a signal distribution theoretic framework facilitates the formulation and analysis of many system modelling problems, including system identification problems. Specifically, a very explicit solution to the L₂ (least squares) LTI approximation problem for NFIR systems is obtained in this manner. Furthermore, the L₁ (least absolute deviations) LTI approximation problem for NFIR systems is essentially reduced to a linear programming problem. Active LTI modelling emphasizes model quality based on the intended use of the models in linear controller design. Robust stability and LTI approximation concepts are studied here in a nonlinear systems context. Numerical examples are given illustrating the performance of the least squares (LS) method and the least absolute deviations (LAD) method with LTI models against nonlinear unmodelled dynamics.     

Squared and absolute errors in optimal approximation of nonlinear systems

Optimal squared error and absolute error-based approximation problems for static polynomial models of nonlinear, discrete-time, systems are studied in detail. These problems have many similarities with other linear-in-the-parameters approximation problems, such as with optimal approximation problems for linear time-invariant models of linear and nonlinear systems. Nonprobabilistic signal analysis is used.Close connections between the studied approximation problems and certain classical topics in approximation theory, such as optimal L₂(−1,1) and L₁(−1,1) approximation, are established by analysing conditions under which sample averages of static nonlinear functions of the input converge to appropriate Riemann integrals of the static functions. These results should play a significant role in the analysis of corresponding system identification and model validation problems. Furthermore, these results demonstrate that optimal modelling based on the absolute error can offer advantages over squared error-based modelling. Especially, modelling problems in which some signals possess heavy tails can benefit from absolute value-based signal and error analysis.     

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.     

L2–L∞ fuzzy control for Markov jump systems with neutral time-delays

The L₂–L_∞ fuzzy control problem is considered for nonlinear stochastic Markov jump systems with neutral time-delays. By means of Takagi–Sugeno fuzzy models, the fuzzy controller systems and the overall closed-loop fuzzy dynamics are constructed. A sufficient condition is firstly established on the stochastic stability using stochastic Lyapunov–Krasovskii functional. Then in terms of linear matrix inequalities techniques, the sufficient conditions on the existence of mode-dependent state feedback L₂–L_∞ fuzzy controller are presented and proved respectively for constant and time varying case. Finally, the design problems are formulated as optimization algorithms. Simulation results are exploited to illustrate the effectiveness of the developed techniques.     

A duality-based approach to the multiobjective H 2/H ∞ optimisation problem

In this article, a duality approach to multiobjective H ₂/H _∞ problems is pursued in which real-rational, para-Hermitian multipliers and real-valued ones are associated to H _∞ and (as usual) H ₂ constraints, respectively. It is shown that the maximisation of a dual functional over all such multipliers yields the optimal value of the original multiobjective H ₂/H _∞ problem. To compute lower bounds on the latter and the corresponding approximate solutions to the original problem, the maximisation of the dual functional over linearly-parameterised, finite-dimensional classes of real-rational multipliers is shown to be equivalent to semi-definite, linear programming problems – once the optimal multipliers in such a class are obtained, the corresponding approximate solutions can be computed from an unconstrained H ₂ problem. Iterative modification of such classes is discussed to obtain increasing sequences of lower bounds on the optimal value of the original problem. This is done on the basis of (locally) increasing directions for the dual functional which go beyond the finite-dimensional class of multipliers considered in a given step. Finally, a numerical example is presented to illustrate the way the presented results can lead to approximate solutions to the multiobjective H ₂/H _∞ problem together with tight estimates of the corresponding deviation from its optimal value.     

Almost disturbance decoupling of MIMO nonlinear systems and application to chemical processes

The problem of almost disturbance decoupling is considered for a class of nonlinear systems with unmeasurable time-varying disturbances. A structure, called nested lower triangular form, is introduced, which contains lower triangular form as a special case. The backstepping design technique is applied to construct an H_∞ feedback controller which achieves internal stability of the closed-loop system and renders a bounded L₂ gain from the disturbance input to the output. The application of the developed design method is illustrated through a two continuous stirred tank reactor example, which can be put into the nested lower triangular form.     

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.     

L2 disturbance attenuation for highly dissipative nonlinear spatially distributed processes via HJI approach

For many practical industrial spatially distributed processes (SDPs), their dynamics are usually described by highly dissipative nonlinear partial differential equations (PDEs). In this paper, we address the L₂ disturbance attenuation problem of nonlinear SDPs using the Hamilton–Jacobi–Isaacs (HJI) approach. Firstly, by collecting an ensemble of PDE states, Karhunen–Loève decomposition (KLD) is employed to compute empirical eigenfunctions (EEFs) of the SDP based on the method of snapshots. Subsequently, these EEFs together with singular perturbation (SP) technique are used to obtain a finite-dimensional slow subsystem of ordinary differential equation (ODE) that accurately describes the dominant dynamics of the PDE system. Secondly, based on the slow subsystem, the L₂ disturbance attenuation problem is reformulated and a finite-dimensional H_∞ controller is synthesized in terms of the HJI equation. Moreover, the stability and L₂-gain performance of the closed-loop PDE system are analyzed. Thirdly, since the HJI equation is a nonlinear PDE that has proven to be impossible to solve analytically, we combine the method of weighted residuals (MWR) and simultaneous policy update algorithm (SPUA) to obtain its approximate solution. Finally, the simulation studies are conducted on a nonlinear diffusion-reaction process and a temperature cooling fin of high-speed aerospace vehicle, and the achieved results demonstrate the effectiveness of the developed control method.     

Numerical existence proofs and explicit bounds for solutions of nonlinear elliptic boundary value problems

For elliptic boundary value problems of the form ?ΔU+F(x, U, U _x )=0 on Ω,B[U]=0 on ?Ω, with a nonlinearityF growing at most quadratically with respect to the gradientU _x and with a mixed-type linear boundary opeatorB, a numerical method is presented which can be used to prove the existence of a solution within a “close”H _1,4(Ω)-neighborhood of some approximate solution ω∈H ₂(Ω) satisfying the boundary condition, provided that the defect-norm ∥?Δω +F(·, ω, ω _x )∥₂ is sufficiently small and, moreover, the linearization of the given problem at ω leads to an invertible operatorL. The main tools are explicit Sobolev imbeddings and eigenvalue bounds forL or forL*L. All kinds of monotonicity or inverse-positivity assumptions are avoided.     

Unified Set Membership theory for identification, prediction and filtering of nonlinear systems

The problem of making inferences from data measured on nonlinear systems is investigated within a Set Membership (SM) framework and it is shown that identification, prediction and filtering can be treated as specific instances of the general presented theory. The SM framework presents an alternative view to the Parametric Statistical (PS) framework, more widely used for studying the above specific problems. In particular, in the SM framework, a bound only on the gradient of the model regression function is assumed, at difference from PS methods which assume the choice of a parametric functional form of the regression function. Moreover, the SM theory assumes only that the noise is bounded, in contrast with PS approaches, which rely on noise assumptions such as stationarity, uncorrelation, type of distribution, etc. The basic notions and results of the general inference making theory are presented. Moreover, some of the main results that can be obtained for the specific inferences of identification, prediction and filtering are reviewed. Concluding comments on the presented results are also reported, focused on the discussion of two basic questions: what may be gained in identification, prediction and filtering of nonlinear systems by using the presented SM framework instead of the widely diffused PS framework? why SM methods could provide stronger results than the PS methods, requiring weaker assumptions on system and on noise?     

On ultimate boundedness around non-assignable equilibria of linear time-invariant systems

In this note we investigate the following questions: given a (finite-dimensional) linear time-invariant (LTI) multivariable system and a constant desired value for its output, say y_?. Assume there is no assignable equilibrium point corresponding to y_?. How “close” to y_? can we ultimately keep the output using LTI static state-feedback stabilizing controllers? Can this neighborhood of y_? be reduced with dynamic, nonlinear, time-varying controllers? Our main contributions are the proof that the optimal ultimate boundedness neighborhood is achieved with LTI static state-feedback, the explicit computation of the neighborhood's size and the proof, under some reasonable rank assumptions, that the system has non-assignable values for the output if and only if it has a transmission zero at zero. Interestingly, there is no connection between this problem and the more familiar concepts of controllability and observability.     

Decentralized adaptive stabilization for interconnected systems with dynamic input-output and nonlinear interactions

This paper considers the problem of robust decentralized adaptive output feedback stabilization for a class of interconnected systems with dynamic input and output interactions and nonlinear interactions by using MT-filters and the backstepping design method. It is shown that the closed-loop decentralized system based on MT-filters is globally uniformly bounded, all the signals except for the parameter estimates can be regulated to zero asymptotically, and the L₂ and L_∞ norms of the system outputs are also be bounded by functions of design parameters. The scheme is demonstrated by a simulation example.     

Online H∞ control for completely unknown nonlinear systems via an identifier–critic-based ADP structure

ABSTRACT

In this paper, we propose an identifier–critic-based approximate dynamic programming (ADP) structure to online solve H∞ control problem of nonlinear continuous-time systems without knowing precise system dynamics, where the actor neural network (NN) that has been widely used in the standard ADP learning structure is avoided. We first use an identifier NN to approximate the completely unknown nonlinear system dynamics and disturbances. Then, another critic NN is proposed to approximate the solution of the induced optimal equation. The H∞ control pair is obtained by using the proposed identifier–critic ADP structure. A recently developed adaptation algorithm is used to online directly estimate the unknown NN weights simultaneously, where the convergence to the optimal solution can be rigorously guaranteed, and the stability of the closed-loop system is analysed. Thus, this new ADP scheme can improve the computational efficiency of H∞ control implementation. Finally, simulation results confirm the effectiveness of the proposed methods.     

Control of 1-D parabolic PDEs with Volterra nonlinearities, Part I: Design

Boundary control of nonlinear parabolic PDEs is an open problem with applications that include fluids, thermal, chemically-reacting, and plasma systems. In this paper we present stabilizing control designs for a broad class of nonlinear parabolic PDEs in 1-D. Our approach is a direct infinite dimensional extension of the finite-dimensional feedback linearization/backstepping approaches and employs spatial Volterra series nonlinear operators both in the transformation to a stable linear PDE and in the feedback law. The control law design consists of solving a recursive sequence of linear hyperbolic PDEs for the gain kernels of the spatial Volterra nonlinear control operator. These PDEs evolve on domains T_n of increasing dimensions n+1 and with a domain shape in the form of a “hyper-pyramid”, 0≤ξ_n≤ξ_n−1?≤ξ₁≤x≤1. We illustrate our design method with several examples. One of the examples is analytical, while in the remaining two examples the controller is numerically approximated. For all the examples we include simulations, showing blow up in open loop, and stabilization for large initial conditions in closed loop. In a companion paper we give a theoretical study of the properties of the transformation, showing global convergence of the transformation and of the control law nonlinear Volterra operators, and explicitly constructing the inverse of the feedback linearizing Volterra transformation; this, in turn, allows us to prove L² and H¹ local exponential stability (with an estimate of the region of attraction where possible) and explicitly construct the exponentially decaying closed loop solutions.     

Practical L2 disturbance attenuation for nonlinear systems

In this work, a notion of generalized L₂-gain for nonlinear systems, where the gain is considered as a function of the state instead of a (global) constant, is presented. This new notion seems to be adequate to characterize the gain properties of several nonlinear systems which do not possess a uniform L₂-gain property (i.e. the L₂-gain depends on the operating point). Moreover, a notion of practical L₂-gain attenuation, which extends the standard definition and parallels (mutatae mutandis) the concepts of practical stability, is also proposed.     

Robust tracking control design for uncertain robotic systems with persistent bounded disturbances

In this study, a robust nonlinear L_∞‐gain tracking control design for uncertain robotic systems is proposed under persistent bounded disturbances. The design objective is that the peak of the tracking error in time domain must be as small as possible under persistent bounded disturbances. Since the nonlinear L_∞ ‐gain optimal tracking control cannot be solved directly, the nonlinear L_∞ ‐gain optimal tracking problem is transformed into a nonlinear L_∞ ‐gain tracking problem by given a prescribed disturbance attenuation level for the L_∞ ‐gain tracking performance. To guarantee that the L_∞ ‐gain tracking performance can be achieved for the uncertain robotic systems, a sliding‐mode scheme is introduced to eliminate the effect of the parameter uncertainties. By virtue of the skew‐symmetric property of the robotic systems, sufficient conditions are developed for solving the robust L_∞ ‐gain tracking control problems in terms of an algebraic equation instead of a differential equation. The proposed method is simple and the algebraic equation can be solved analytically. Therefore, the proposed robust L_∞ ‐gain tracking control scheme is suitable for practical control design of uncertain robotic systems. Copyright © 2008 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

L 2-optimization and fixed poles for sampled-data systems with generalized hold

This paper considers the L ₂-optimization problem for a standard sampled-data system with generalized hold of an arbitrary order. The original problem is reduced to the minimization problem of a degenerate quadratic functional, where solution appears nonunique. And finally, the authors propose a polynomial procedure for constructing a set of optimal discrete causal controllers and establish some relevant properties of L ₂-optimal systems in the context of applications.     

Sulla soluzione dell'equazione biarmonica con metodi di programmazione lineare

The solution of biharmonic equation is studied with linear programming methods to obtain an a posteriori estimate of truncation errors. For this the algorithm proposed by Barrodale and Young has been adopted, which adjusts to the peculiar problem's structure the simplex method, and a lowest memory occupation requires, avoiding any constraints duplication. Carrying out some different boundary points distribution, a comparison is made among the results obtained by minimizing the error according to two different norms. It's pointed out that withL ₁ norm, we can reach a simpler and shorter problem formulation, and a solution which offers a better punctual approximation than we could obtainwithL _∞ norm, which supplies at contrary a better approximation. It's important to point out that such algorithm, when we try to approximate withL _∞ norm the biharmonic equation solution generates a cyclic procedure; so the ‘lexicographic method’ variant has been introduced. Numerical tests carried out are exposed in the following tables.