Worst case optimal observer design for linear discrete-time systems with unknown bounded square summable disturbances

The problem of the existence and design of an optimal observer is considered for linear discrete-time systems with unknown disturbances additive to the output. The optimal observer reconstructs the best estimate of the state at a given time with respect to the worst disturbances constrained to a ball in l ². It is proved that an observability condition is necessary and sufficient for the existence of such an observer. An explicit formula for the optimal observer is derived (it includes the degenerate case when some of the outputs are disturbance free).     

Variable structure controller design for linear systems with bounded inputs

This paper studies the design of variable structure systems with saturation inputs. Sliding mode domain, reaching domain, and unescapable reaching domain of linear systems with variable structure are defined and investigated. When the state matrix of the linear system is Hurwitz, the stability of the variable structure systems is proven by using passivity theory. Moreover, variable structure systems with novel nonlinear switching surfaces are proposed for second order systems. Two strategies for designing variable structure control for high order linear systems are also proposed, such as stepby-step variable structure control and moving-surface variable structure control, which were found to guarantee that the reaching condition of the variable structure control is always satisfied. Finally, an illustrative example pertaining to the attitude control of a flexible spacecraft demonstrates the effectiveness of the proposed methods.     

Ultimate bounded stability and stabilization of linear systems interconnected with generalized saturated functions

This paper proposes some ultimate bounded stability analysis and stabilization conditions for systems involving actuators with different nonlinear elements, like for instance both saturation and dead-zone or both saturation and stick–slip. Results are based on the use of a convex differential inclusion approach. Indeed, an adequate property allowing to upper-bound some product terms related to the nonlinearity is provided. Thus, constructive conditions associated to convex optimization schemes are developed to determine suitable regions of the state space in which the closed-loop trajectories can be captured.     

Scheduled controllers for linear systems with bounded actuators

State and output feedback controllers are designed for disturbance attenuation in linear systems with bounded actuators. The controller is scheduled according to the proximity to the origin of the state of the plant in the state-feedback case and the compensator state in the case of output feedback. This procedure yields a linear parameter-varying structure for the controller that allows higher-gain and hence higher-performance controllers as the states move closer to the origin. The main results give sufficient conditions for the satisfaction of a parameter-dependent performance measure, without violating the saturation bounds. Linear splines are used to obtain solutions that can be obtained by standard LMI software. Examples—including an open-loop unstable system—highlight the application of the results.     

Systems of linear difference equations with bounded and robustly bounded solutions

In this paper, we investigate higher-order systems of linear difference equations where the associated characteristic matrix polynomial is self-inversive. We consider classes of equations with bounded solutions. It is known that stability properties of higher-order systems of linear difference equations are determined by the characteristic values of the corresponding matrix polynomials. All solutions are bounded (in both time directions) if the spectrum of the corresponding matrix polynomial lies on the unit circle, and moreover if the characteristic values of modulus 1 are semisimple. If the corresponding matrix polynomial is self-inversive, then one can use the inner radius of the numerical range to obtain a criterion for boundedness of solutions. We show that all solutions are bounded if the inner radius is greater than 1. In the case of matrix polynomials with positive definite coefficient matrices, we derive a computable lower bound for the inner radius and we obtain a criterion for robust boundedness.     

Control design for chained-form systems with bounded inputs

Discontinuous, time-invariant controllers have been recently proposed in the literature as an alternative method to stabilize nonholonomic systems. These control laws are not continuous at the origin and although they provide exponential rates of convergence, they may use significant amount of control effort, especially if the initial conditions are close to an equilibrium manifold. We seek to remedy this situation by constructing bounded controllers (with exponential convergence rates) for nonholonomic systems in chained form.     

Quasi-trefftz spectral method for separable linear elliptic equations

A new numerical method QTSM which was earlier suggested for solving boundary value problems with elliptic operators with constant coefficients is applied to separable, second-order elliptic equations with varying coefficients. The problems in irregular domains are considered. The method combines the properties of the boundary methods with the spectral representation of the solution in the form of expansion over the eigenfunctions of some Sturm-Liouville problem. The method is tested on several one- and two-dimensional problems with exact analytic solution. The possibilities of further developments of the method are discussed.     

Adaptive suboptimal control of a linear system with bounded disturbance

An adaptive suboptimal control of a linear discrete system with unknown parameters is proposed. An additive disturbance v_t acting on the system is supposed to be uniformly bounded. The criterion is sup_vtI(y^∞₁, u^∞₁), where y_t is the output, u_t is the control. The adaptive control law gives almost the same guaranteed value of the criterion as the optimal linear feedback does for a system with known parameters.     

Optimization with few violated constraints for linear bounded errorparameter estimation

In the context of linear constrained optimization, we study the problem of finding an optimal solution satisfying all but k of the given n constraints. A solution is obtained by means of an algorithm of the complexity min{O(n·k^d), O(n·d^k+1)}, where d is the dimension of the problem. We then use these results to solve the problem of robust identification in the presence of outliers in the setting of bounded error parameter identification. Finally, we show that the estimate obtained converges to the true but unknown parameter in the presence of outliers     

Neural network for solving linear programming problems with bounded variables

A new neural network for solving linear programming problems with bounded variables is presented. The network is shown to be completely stable and globally convergent to the solutions to the linear programming problems. The proposed new network is capable of achieving the exact solutions, in contrast to existing optimization neural networks which need a suitable choice of the network parameters and thus can obtain only approximate solutions. Furthermore, both the primal problems and their dual problems are solved simultaneously by the new network.     

Tube-based robust MPC for linear time-varying systems with bounded disturbances

In this paper, a novel synthesis approach for tube-based robust MPC is proposed. A novel feature is the fact that the proposed tube-based robust MPC algorithm can deal with both time-varying parameter and bounded disturbance. The proposed control law is the linear combination of two components. The first component steers the state of the nominal linear time-varying system to the origin. The second component keeps the state at each time step of the linear time-varying system with bounded disturbance within a tube whose center is the state of the nominal linear time-varying system. Finally, the state of the linear time-varying system with bounded disturbance is kept within a tube whose center is at the origin so robust stability is guaranteed. Satisfaction of the state and control constraints is ensured by employing tighter constraint sets for the nominal linear time-varying system. The computational burdens are the same as those in the case of the nominal linear time-varying system.     

On infinity norms as Lyapunov functions for linear systems

The note presents a construction of the Lyapunov function defined by l_∞ⁿ vector norm for both continuous- and discrete-time linear systems. In contrast to the previous results, no restrictions on the positions or multiplicity of system matrix eigenvalues are required     

Time-varying discrete-time linear systems with bounded rates of variation: Stability analysis and control design

This paper investigates the problems of robust stability analysis and state feedback control design for discrete-time linear systems with time-varying parameters. It is assumed that the time-varying parameters lie inside a polytopic domain and have known bounds on their rate of variation. A convex model is proposed to represent the parameters and their variations as a polytope and linear matrix inequality relaxations that take into account the bounds on the rates of parameter variations are proposed. A feasible solution provides a parameter-dependent Lyapunov function with polynomial dependence on the parameters assuring the robust stability of this class of systems. Extensions to deal with robust control design as well as gain-scheduling by state feedback are also provided in terms of linear matrix inequalities. Numerical examples illustrate the results.     

Settling time in a linear dynamic system with bounded external disturbances

The article deals with the problem of estimating the settling time for a linear dynamic system subjected to the action of nonrandom bounded external disturbances. The suggested approach is based on the method of invariant ellipsoids and the technique of linear matrix inequalities. Both the continuous and the discrete version of the problem are considered. As an example the problem of control of a two-mass system is investigated.     

Design of linear systems with saturating linear control and bounded states

A theorem is developed, based on the circle criterion, which gives sufficient conditions for the stability of a saturating linear control for a linear system with bounded states. This theorem can be applied to determine a range of gains for the linear control which ensures stability. It is shown by example that this approach results in an increased speed of response over that obtained by previous methods which do not account for state bounds.     

Sparsity-preserving sor algorithms for separable quadratic and linear programming

The main purpose of this work is to give explicit sparsity-preserving SOR (successive overrelaxation) algorithms for the solution of separable quadratic and linear programming problems. The principal and computationally-distinguishing feature of the present SOR algorithms is that they preserve the sparsity structure of the problem and do not require the computation of the product of the constraint matrix by its transpose as is the case in earlier SOR algorithms for linear and quadratic programming.     

Influence functions for a linear subspace method

A linear subspace method, which is one of discriminant methods, was proposed as a pattern recognition method and was studied. Because the method and its extensions do not encounter the situation of singular covariance matrix, we need not consider extensions such as generalized ridge discrimination, even when treating a high dimensional and sparse dataset. In addition, classifiers based on a multi-class discrimination method can function faster because of the simple decision procedure. Therefore, they have been widely used for face and speech recognition. However, it seems that sufficient studies have not been conducted about the statistical assessment of training data performance for classifier in terms of prediction accuracy. In statistics, influence functions for statistical discriminant analysis were derived and the assessments for analysis result were performed. These studies indicate that influence functions are useful for detecting large influential observations for analysis results by using discrimination methods and they contribute to enhancing the performance of a target classifier.     

Controllable regions of linear systems with bounded inputs

It is known that the controllable region of a general unstable system with bounded control is the Cartesian product of the controllable region of its subsystem with antistable modes and that of its subsystem with stable and marginally stable modes. While the controllable region of a system with only stable and marginally stable modes is well known to be the whole state space, that of an antistable system is studied in this paper. A necessary and sufficient condition for a state of an antistable system to be controllable is given. The boundary of the controllable region is characterized.     

Cryptanalysis of RSA for a special case with d>e

In this paper,we study the RSA public key cryptosystem in a special case with the private exponent d larger than the public exponent e. When N0.258 e N0.854,d > e and satisfies the given conditions,we can perform cryptanalytic attacks based on the LLL lattice basis reduction algorithm. The idea is an extension of Boneh and Durfee's researches on low private key RSA,and provides a new solution to finding weak keys in RSA cryptosystems.     

Goal-oriented state control of a cognitive linear model with a bounded state space

Representing the basic concept of this paper, cognitive map is used to construct a cognitive linear dynamic model with a bounded state space. We consider the problem of transferring this model from an arbitrary initial state to some asymptotically stable state belonging to a neighborhood of a given state. We suggest two classes of controls and introduce the transfer “quality” as the proximity of the resulting steady state to the desired state. And finally, an illustrative example is provided.