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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]

This paper is concerned with the problem of robust $\mathcal{H}_{2}$ and $\mathcal{H}_{\infty}$ filter design for discrete-time linear time-invariant systems with polytopic parameter uncertainties. Less conservative robust $\mathcal{H}_{2}$ and $\mathcal{H}_{\infty}$ filter design procedures are proposed in terms of single-parameter minimization problems with linear matrix inequality constraints. To this end, we generalize the filter structures available in the literature to date in such a way that the filter's next state is built by summing the filter's states over several samples from the past to the present. For stability of the filtering error system, the homogeneous polynomial parameter-dependent Lyapunov functions are employed. Finally, illustrative examples are given to demonstrate the merits of the proposed methods. [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]

This paper addresses the problem of discretization of fractional differintegrators and fractional systems in general. Two new discretization schemes are proposed-optimal direct method, and computationally inexpensive indirect method. The proposed methods are highly flexible and suitable for obtaining both general purpose and specialized, application-specific approximations. Both proposed methods outperform recently proposed approximations, as it is shown in several illustrative examples. [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]

In Cheb-Terrab and Roche (Comput Phys Commun 130(1-2):204-231, 2000) a classification of the Abel equations known as solvable in the literature was presented. In this paper, we show that all the integrable rational Abel differential equations that appear in Cheb-Terrab and Roche (Comput Phys Commun 130(1-2):204-231, 2000) and consequently in Cheb-Terrab and Roche (Eur J Appl Math 14(2):217-229, 2003) can be reduced to a Riccati differential equation or to a first-order linear differential equation through a change with a rational map. The change is given explicitly for each class. Moreover, we have found a unified way to find the rational map from the knowledge of the explicitly first integral. [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]

The problem of modal sampled-data control for continuous-time linear time-invariant plant with delay is considered. The characteristic matrix of the system is constructed. An algorithm is given for generating the set of causal discrete-time controllers that place eigenvalues of the characteristic matrix at specified points of the complex plane. [ABSTRACT FROM AUTHOR]

We give several linear time algorithms for the continuous quadratic knapsack problem. In addition, we report cycling and wrong-convergence examples in a number of existing algorithms, and give encouraging computational results for large-scale problems. [ABSTRACT FROM AUTHOR]

Abstract   We propose and study the Maximum Constrained Agreement Subtree (MCAST) problem, which is a variant of the classical Maximum Agreement Subtree (MAST) problem. Our problem allows users to apply their domain knowledge to control the construction of the agreement subtrees in order to get better results. We show that the MCAST problem can be reduced to the MAST problem in linear time and thus we have algorithms for MCAST with running times matching the fastest known algorithms for MAST. [ABSTRACT FROM AUTHOR]

Abstract   We show efficient algorithms for edge-coloring planar graphs. Our main result is a linear-time algorithm for coloring planar graphs with maximum degree Δ with max {Δ,9} colors. Thus the coloring is optimal for graphs with maximum degree Δ≥9. Moreover for Δ=4,5,6 we give linear-time algorithms that use Δ colors. These results improve over the algorithms of Chrobak and Yung (J. Algorithms 10:35–51, 1989) and of Chrobak and Nishizeki (J. Algorithms 11:102–116, 1990) which color planar graphs using max {Δ,19} colors in linear time or using max {Δ,9} colors in time. [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]

Static slicing has shown itself to be a valuable tool, facilitating the verification of hardware designs. In this paper, we present a sharpened notion, antecedent conditioned slicing that provides a more effective abstraction for reducing the size of the state space. In antecedent conditioned slicing, extra information from the antecedent is used to permit greater pruning of the state space. In a previous version of this paper, we applied antecedent conditioned slicing to safety properties of the form G( antecedent ⇒ consequent) where antecedent and consequent were written in propositional logic. In this paper, we use antecedent conditioned slicing to handle safety and bounded liveness property specifications written in linear time temporal logic. We present a theoretical justification of our technique. We provide experimental results on a Verilog RTL implementation of the USB 2.0 functional core, which is a large design with about 1,100 state elements (10331 states). The results demonstrate that the technique provides significant performance benefits over static program slicing using state-of-the-art model checkers. [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.

We consider robust adaptive control designs for relative degree one, minimum phase linear systems of known high frequency gain. The designs are based on the dead-zone and projection modifications, and we compare their performance w.r.t. a worst case transient cost functional with a penalty on the $$\mathcal{L}$$ ∞ norm of the output, control and control derivative. We establish two qualitative results. If a bound on the $$\mathcal{L}$$ ∞ norm of the disturbance is known and the known a priori bound on the uncertainty level is sufficiently conservative, then it is shown that a dead-zone controller outperforms a projection controller. The complementary result shows that the projection controller is superior to the dead-zone controller when the a priori information on the disturbance level is sufficiently conservative. [ABSTRACT FROM AUTHOR]

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]

Neural networks consist of highly interconnected and parallel nonlinear processing elements that are shown to be extremely effective in computation. This paper presents an architecture of recurrent neural networks that can be used to solve several classes of optimization problems. More specifically, a modified Hopfield network is developed and its internal parameters are computed explicitly using the valid-subspace technique. These parameters guarantee the convergence of the network to the equilibrium points, which represent a solution of the problem considered. The problems that can be treated by the proposed approach include combinatorial optimization problems, dynamic programming problems, and nonlinear optimization problems. [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]

We present a unified approach to establishing the existence of global minima of a (non)convex constrained optimization problem. Our results unify and generalize previous existence results for convex and nonconvex programs, including the Frank-Wolfe theorem, and for (quasi) convex quadratically constrained quadratic programs and convex polynomial programs. For example, instead of requiring the objective/constraint functions to be constant along certain recession directions, we only require them to linearly recede along these directions. Instead of requiring the objective/constraint functions to be convex polynomials, we only require the objective function to be a (quasi)convex polynomial over a polyhedral set and the constraint functions to be convex polynomials or the composition of coercive functions with linear mappings. [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]

This paper solves two problems raised in [HSK]. The first is a uniform norm time-varying linear estimation problem and the second is a general time-varying linear control problem. [ABSTRACT FROM AUTHOR]

Singular $$\mathcal{H}$$ 2-optimization problems are considered for the standard discrete-time control system. Two types of singularity (type I and type II) are distinguished. A detailed treatment of problems with singularity of type II, which leads to nonuniqueness of solution, is presented. New algorithms for design of optimal controllers are presented both in frequency domain and state space, which generalize standard procedures onto the case of singular $$\mathcal{H}$$ 2-problems. A parameterization of the set of optimal controllers is given. [ABSTRACT FROM AUTHOR]

Novel constructions of empirical controllability and observability gramians for nonlinear systems are proposed for subsequent use in a balanced truncation style of model reduction. The new gramians are based on a generalisation of the fundamental solution for a Linear Time-Varying system. Relationships between the given gramians for nonlinear systems and the standard gramians for both Linear Time-Invariant and Linear Time-Varying systems are established as well as relationships to prior constructions proposed for empirical gramians. Application of the new gramians is illustrated through a sample test-system. [ABSTRACT FROM AUTHOR]

In an earlier paper, Awerbuch presented an innovative distributed algorithm for solving minimum spanning tree (MST) problems that achieved optimal time and message complexity through the introduction of several advanced features. In this paper, we show that there are some cases where his algorithm can create cycles or fail to achieve optimal time complexity. We then show how to modify the algorithm to avoid these problems, and demonstrate both the correctness and optimality of the revised algorithm. [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]

This work presents an effective procedure devised to implement the time discontinuous Galerkin method for linear dynamics. In particular, the method with piecewise linear time interpolation is considered. The procedure is based on a simple and low-cost iterative scheme, which is designed not as a mere solution algorithm, but rather as a method to generate improved approximations to the exact solution. The corrected solutions inherit the desired stability and dissipative properties from the target solution, while accuracy is improved by iterations. Indeed, no more than two iterations are shown to be needed. The resultant algorithm leads to remarkable computational savings and can be easily implemented into existing finite element codes. Numerical tests confirm that the present procedure possesses many attractive features for applications to dynamic analysis. [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]

This paper deals with the class of uncertain continuous-time linear systems with Markovian jumps, time delay, and saturating actuators. Under norm-bounded uncertainties and based on the Lyapunov method, sufficient conditions on stochastic stability and stochastic stabilizability are developed. A design algorithm for a stabilizing observer- based robust output feedback controller is proposed in terms of the solutions of linear matrix inequalities. [ABSTRACT FROM AUTHOR]

Technical stability allowing quantitative estimation of trajectory behavior of a dynamical system over a given time interval was considered. Based on a differential comparison principle and a basic monotonicity condition, technical stability relative to certain prescribed state constraint sets of a class of nonlinear time-varying systems with small parameters was analyzed by means of vector Liapunov function method. Explicit criteria of technical stability are established in terms of coefficients of the system under consideration. Conditions under which the technical stability of the system can be derived from its reduced linear time-varying (LTV) system were further examined, as well as a condition for linearization approach to technical stability of general nonlinear systems. Also, a simple algebraic condition of exponential asymptotic stability of LTV systems is presented. Two illustrative examples are given to demonstrate the availability of the presently proposed method. [ABSTRACT FROM AUTHOR]

An amendment to this paper has been published and can be accessed via a link at the top of the paper. [ABSTRACT FROM AUTHOR]