Robust Mini-Max regulator for uncertain nonlinear polynomial systems

This paper presents a solution to a robust optimal regulation problem for a nonlinear polynomial system affected by parametric and matched uncertainties, which is based only on partial state information. The parameters describing the dynamics of the nonlinear polynomial plant depend on a vector of unknown parameters, which belongs to a finite parametric set, and the application of a certain control input is associated with the worst or least favorable value of the unknown parameter. A high-order sliding mode state reconstructor is designed for the nonlinear plant in such a way that the previously designed control can be applied for a system with incomplete information. Additionally, the matched uncertainty is also compensated by means of the same output-based regulator. The obtained algorithm is applied to control a mechanical pendulum of the third order, successfully verifying the effectiveness of the developed approach.

Adaptive signal processing algorithms are often used in order to track an unknown time-varying parameter vector. This work develops an upper bound on the mean of the norm-squared error between the unknown parameter vector being tracked and the value obtained by the algorithm. The results require very mild covariance decay rate conditions on the training data and a bounded algorithm. The upper bound illustrates the relationship between the algorithm step size and the maximum rate of variation in the parameter vector being tracked.

The iterative method, which can be applied to estimation of the unknown vector of discrete parameters in combination with filtering of the unknown vector of continuous signal parameters, is proposed. It is demonstrated that this method makes it possible to construct highly efficient algorithms for joint demodulation and filtering of communications channel parameters in code-division multiple-access and orthogonal frequency-division multiplexing systems.

This paper presents the central finite‐dimensionalH∞filter for nonlinear polynomial systems, which is suboptimal for a given threshold γ with respect to a modified Bolza–Meyer quadratic criterion including the attenuation control term with the opposite sign. In contrast to the previously obtained results, the paper reduces the originalH∞filtering problem to the corresponding optimalH2filtering problem, using the technique proposed in (IEEE Trans. Automat. Control1989;34:831–847). The paper presents the central suboptimalH∞filter for the general case of nonlinear polynomial systems based on the optimalH2filter given in (Int. J. Robust Nonlinear Control2006;16:287–298). The central suboptimalH∞filter is also derived in a closed finite‐dimensional form for third (and less) degree polynomial system states. Numerical simulations are conducted to verify performance of the designed central suboptimal filter for nonlinear polynomial systems against the central suboptimalH∞filter available for the corresponding linearized system. Copyright © 2008 John Wiley & Sons, Ltd.

A robust output feedback control which globally stabilizes a class of SISO minimum-phase nonlinear systems with known and constant relative degree containing a vector of unknown parameters is developed. The constant parameter vector is not restricted to enter linearly in the state equations but it is assumed to belong to a known compact set and an imprecise knowledge of the nonlinearities (e.g., lookup tables) is allowed. The class of nonlinear systems is determined by geometric conditions; an additional assumption, which generalizes the knowledge of the sign of high-frequency gain for linear systems, is also required. The nonlinearities are restricted to depend, in suitable coordinates, on the output only: no growth conditions, such as sector or Lipschitz, are required. The robust output feedback control stabilizes the system for every value of the parameter vector in a known compact set. The order of the compensator is equal to the relative degree minus one. >

Let \( \{ ({x'_i},{y_i})\} _{i = 1}^N \) be an iid sample drawn from a known distribution F(x i,y i, s), where s is a k × 1 vector of unknown parameters. Let f y|x (y, β) denote the likelihood function of y | x, which is the density function of y | x if y |x is continuous or the probability of y | x if y | x is discrete. Define f x (x) analogously, which is not a function of s.

The paper considers the problem of estimating the parameters in a continuous time regression model with a non-Gaussian noise of pulse type. The vector of unknown parameters is assumed to belong to a compact set. The noise is specified by the Ornstein–Uhlenbeck process driven by the mixture of a Brownian motion and a compound Poisson process. Improved estimates for the unknown regression parameters, based on a special modification of the James–Stein procedure with smaller quadratic risk than the usual least squares estimates, are proposed. The developed estimation scheme is applied for the improved parameter estimation in the discrete time regression with the autoregressive noise depending on unknown nuisance parameters.

On best unbiased prediction and its relationships to unbiased estimation

: The general multiplicative model represents the hazard function as the product of an 'underlying hazard rate, gamma (t), of unspecified form and a certain function of known form, g(z; beta), where z is a vector of concomitant variables, and Beta is a vector of unknown parameters. Assuming that gamma (t) can be approximated by a constant between any two consecutive failures, the general forms of likelihood function are derived. The likelihood utilizes the available information on the time of exposure to risk of each individual (until failure or withdrawal). Special cases, when the z's do not depend on t are discussed in some detail. Multiple failures are handled in a simple manner - no ordering of failures is required. Estimation of empirical survival function when there are no covariates is discussed. An example using heart transplant data, is given (for illustrative purpose only).

SELF-TUNING OUTPUT FEEDBACK STABILIZATION OF A CLASS OF NONLINEAR SYSTEMS

We consider a generalized partially linear model E(Y|X, T) = G{X T β + m(T)}, where G is a known function, β is an unknown parameter vector, and m is an unknown function. We introduce a test statistic that allows one to decide between a parametric and a semiparametric model: (a) m is linear (i.e., m(t) = t T γ for a parameter vector γ), and (b) m is a smooth (nonlinear) function. Under linearity (a), we show that the test statistic is asymptotically normal. Moreover, we prove that the bootstrap works asymptotically. Simulations suggest that (in small samples) the bootstrap outperforms the calculation of critical values from the normal approximation. The practical performance of the test is demonstrated in applications to data on East–West German migration and credit scoring.

Stability analysis of high-order Hopfield type neural networks with uncertainty

We consider estimation in a semiparametric generalized linear model for clustered data using estimating equations. Our results apply to the case where the number of observations per cluster is finite, whereas the number of clusters is large. The mean of the outcome variable μ is of the form g(μ) = XTβ + θ(T), where g(·) is a link function, X and T are covariates, β is an unknown parameter vector, and θ(t) is an unknown smooth function. Kernel estimating equations proposed previously in the literature are used to estimate the infinite-dimensional nonparametric function θ(t), and a profile-based estimating equation is used to estimate the finite-dimensional parameter vector β. We show that for clustered data, this conventional profile-kernel method often fails to yield a √n-consistent estimator of β along with appropriate inference unless working independence is assumed or θ(t) is artificially undersmoothed, in which case asymptotic inference is possible. To gain insight into these results, we derive the semiparametric efficient score of β, which is found to have a complicated form, and show that, unlike for independent data, the profile-kernel method does not yield a score function asymptotically equivalent to the semiparametric efficient score of β, even when the true correlation is assumed and θ(t) is undersmoothed. We illustrate the methods with an application to infectious disease data and evaluate their finite-sample performance through a simulation study.

The convexity of reachable sets plays an essential role in the development of algorithms for solving optimal control problems and problems of feedback control. For nonlinear control systems the reachable sets are generally not convex and may have a rather complicated structure. However, for systems with integral quadratic constraints on the control B. Polyak showed that the reachable sets are convex if the linearization of the system is controllable and control inputs are restricted from above in L2 norm by a sufficiently small number. In the present paper we use this result to prove sufficient conditions for the convexity of reachable sets of a nonlinear control-affine system on small time intervals, assuming that control resources are limited by a given (not necessarily small) value. These conditions are based on the asymptotics for the minimal eigenvalue of the controllability Gramian of system linearization as a function of the length of the time interval. We prove the asymptotics for a linear time-invariant system containing a small parameter that implies the convexity of small-time reachable sets for some classes of two-dimensional nonlinear control systems. The results of numerical simulations for illustrative examples are discussed.

ABSTRACTThis paper presents a solution to the optimal control problem for discrete-time stochastic nonlinear polynomial systems confused with white Poisson noises subject to a quadratic criterion. The solution is obtained in the following way: a nonlinear optimal controller is first developed for polynomial systems, considering the state vector completely available for control design. Then, based on the solution of the state estimation problem for polynomial systems with white Poisson noises, the state estimate vector is used in the control law to obtain a closed-form solution. Performance of this controller is compared to that of the controller employing the extended Kalman filter and the linear-quadratic regulator and the controller designed for polynomial systems confused with white Gaussian noises.