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The design problem of an adaptive state observer for linear time-varying systems is considered. Some parameters of a time-varying plant are assumed to be unknown numbers multiplied by known functions of time. The approach proposed below is based on identification methods of adaptation. In other words, the main idea is to transform a mathematical model in the form of a linear time-varying differential equation to a static linear regression model containing unknown parameters. In this case, both the unknown parameters of the plant and its initial conditions are the unknown parameters of the regression model under consideration. Further, standard gradient methods or other parametric identification approaches are used to estimate the unknown parameters of the regression model and to construct the observer. The results of computer simulations illustrate the effectiveness of the new state observer design method. [ABSTRACT FROM AUTHOR]

We consider the output stabilization problem for a linear time-independent system. We show that if the system output is zero, y( t) ≡ 0, then, for the stabilization of the entire state vector, the sufficient condition of stability of the zero dynamics can be replaced by the boundedness of the system input (provided that the characteristic polynomial of the zero dynamics has no eigenvalues on the imaginary axis). [ABSTRACT FROM AUTHOR]

We derive a linear system of equations governing the interaction of water waves with partially or totally submerged freely floating structures in a two-layer fluid. We establish conditions for the stability of equilibrium and, by considering time-harmonic motions, rewrite the problem as a spectral boundary-value problem consisting of a differential equation and an algebraic system, coupled through boundary conditions. We give also a suitable variational formulation for the problem and provide examples of configurations where the problem admits only the trivial solution. [ABSTRACT FROM AUTHOR]

An analytical solution of the B.V. Bulgakov problem of maximum of the norm of finite state of the stationary linear system with one control (perturbation) function taking values over an interval was proposed. The switching function of the worst relay perturbation was shown to depend on time and coordinates and be the product of the phase system vector, its transition matrix transposed to it, and the transposed column of the coefficients at perturbation. [ABSTRACT FROM AUTHOR]

Regularization is necessary for inversion of ill-posed geophysical problems. Appraisal of inverse models is essential for meaningful interpretation of these models. Because uncertainties are associated with regularization parameters, extra conditions are usually required to determine proper parameters for assessing inverse models. Commonly used techniques for assessment of a geophysical inverse model derived (generally iteratively) from a linear system are based on calculating the model resolution and the model covariance matrices. Because the model resolution and the model covariance matrices of the regularized solutions are controlled by the regularization parameter, direct assessment of inverse models using only the covariance matrix may provide incorrect results. To assess an inverted model, we use the concept of a trade-off between model resolution and covariance to find a proper regularization parameter with singular values calculated in the last iteration. We plot the singular values from large to small to form a singular value plot. A proper regularization parameter is normally the first singular value that approaches zero in the plot. With this regularization parameter, we obtain a trade-off solution between model resolution and model covariance in the vicinity of a regularized solution. The unit covariance matrix can then be used to calculate error bars of the inverse model at a resolution level determined by the regularization parameter. We demonstrate this approach with both synthetic and real surface-wave data. [ABSTRACT FROM AUTHOR]

We present methods for computing the local null controllability time in the class of linear control systems (not necessarily autonomous) whose families of null controllability sets are monotone with respect to inclusion. [ABSTRACT FROM AUTHOR]

We establish a relationship between the growth rates of algebroid solutions and coefficients of nth-order linear differential equations. [ABSTRACT FROM AUTHOR]

In this paper, necessary and sufficient conditions for null-controllability of a linear system under geometric constraints on the control are given, without the assumption that the origin is an equilibrium point of the system. The criterion for controllability uses the concept of a return condition on an interval which is introduced in the paper. This condition generalizes the existence of an equilibrium point. [ABSTRACT FROM AUTHOR]

We present sufficient conditions for the asymptotic equivalence of a nonlinear impulsive system and a nonlinear system without pulses. We also consider the case of the asymptotic equivalence of a weakly nonlinear impulsive system and a linear system with pulses. [ABSTRACT FROM AUTHOR]

The article discusses the nonegativity and positivity concepts and associated attributes for the state and output trajectory results of active linear time-invariant systems depicted by functional differential equations subject to point time delays. It is suggested that the respective nonnegativities and positivities introduced hierarchically from the weakest to the strongest while distinguishing the matching properties when used to the state space or to the output space including the zero-initial state or zero-input responses.

The article discusses several topics on mathematical theory of differential equations reported in the spring semester 2006 seminar at Moscow State University in Russia. Most of this report were partially supported by the Russian Foundation for Basic Research. These topics aims to provide support for engineering sciences applications of mostly high order integral-differential approach.

This paper proposes a nonlinear multi-input control law using sliding mode concepts for continuous-time, uncertain, linear systems. The control law introduces a state-dependent layer around the sliding mode plane to remove chattering. This layer combines two types of boundary layers: a constant layer and a sector-shaped layer. The states will always enter the state-dependent boundary layer and the choice of the sliding mode will be seen to determine the ultimate system performance. A proof of stability shows ultimate boundedness. The controller is applied to a nonlinear simulation model of a cart-pendulum and exhibits a high degree of robustness. The new boundary layer in connection with a novel dynamically changing, state-dependent gain can be used to obtain a narrow boundary-layer shape in the operating region of interest. This permits rejection of disturbances without chattering of the control and improves on the performance expected of a sliding-mode control with constant boundary layer. [ABSTRACT FROM AUTHOR]

Backtracking adaptive search is a simplified stochastic optimization procedure which permits the acceptance of worsening objective function values. It generalizes hesitant adaptive search, which in turn is a generalization of pure adaptive search. In this paper, we use ideas from the theory of stochastic processes to determine the full distribution of the number of iterations to convergence for backtracking adaptive search. [ABSTRACT FROM AUTHOR]

The design of control laws for systems subject to complex state constraints still presents a significant challenge. This paper explores a dynamic programming approach to a specific class of such problems, that of reachability under state constraints. The problems are formulated in terms of nonstandard minmax and maxmin cost functionals, and the corresponding value functions are given in terms of Hamilton-Jacobi-Bellman (HJB) equations or variational inequalities. The solution of these relations is complicated in general; however, for linear systems, the value functions may be described also in terms of duality relations of convex analysis and minmax theory. Consequently, solution techniques specific to systems with a linear structure may be designed independently of HJB theory. These techniques are illustrated through two examples. [ABSTRACT FROM AUTHOR]

In this paper, by the KAM method, under weaker small denominator conditions and nondegeneracy conditions, we prove a positive measure reducibility for quasi-periodic linear systems close to constant: Ẋ = ( A(λ) + F( ϕ, λ)) X, $$ \dot{\varphi } = \omega $$ where the parameter λ ∈ ( a, b), ω is a fixed Diophantine vector, which is a generalization of Jorba & Simó’s positive measure reducibility result. [ABSTRACT FROM AUTHOR]

An algorithm is proposed to synthesize a reliable controller with a given stability margin for linear and periodic systems optimized with respect to a quadratic performance criterion. A reliable controller synthesized by the algorithm guarantees the stability margin and is close to the linear-quadratic requlator. The importance of ensuring the stability margin is demonstrated. The proposed algorithm is based on methods of linear matrix inequalities and can be implemented using standard MATLAB routines. As an example, a reliable controller that stabilizes the program motion of a hopping machine is synthesized [ABSTRACT FROM AUTHOR]

Model reduction is of fundamental importance in many control applications. We consider model reduction methods for linear time-invariant continuous-time descriptor systems. These methods are based on the balanced truncation technique and closely related to the controllability and observability Gramians and Hankel singular values of descriptor systems. The Gramians can be computed by solving generalized Lyapunov equations with special right-hand sides. Numerical examples are given. [ABSTRACT FROM AUTHOR]

In this paper the problem of system equivalence is tackled for a rather general class of linear time-invariant systems. We consider AR-systems described by linear continuous shift-invariant operators with finite memory, acting on Fréchet-signal spaces, containing the space {\cal E} ({\open R}) of infinitely differentiable functions on {\open R}. This class is in one–one correspondence with matrices of suitable sizes over the convolution algebra {\cal E}′ ({\open R}) of all compactly supported distributions. Using some deep results from the theory of Fréchet spaces, various necessary and sufficient conditions for system equivalence and system inclusion are formulated. It is shown that a surjectivity demand on the system defining convolution operator matrix is necessary and sufficient for being able to translate the problem of system equivalence into division properties over the convolution algebra {\cal E}′({\open R}). This surjectivity condition is guaranteed if the system defining matrix over {\cal E}′({\open R}) has a right-inverse over {\cal D}′({\open R}), the space of all Schwartz distributions. [ABSTRACT FROM AUTHOR]