Revisiting the regulator problem in the geometric approach,part 1, disturbance localization by dynamic compensation

The aim of this paper is to present a new proof of a well-known fundamental set of necessary and sufficient conditions for disturbance localization by dynamic compensation. Due to a more convenient selection of basic definitions and properties, the arguments herein developed are much more concise and straightforward than those already available in the literature. In Part 2 of the paper, the same approach will be extended to the more general problem of regulator synthesis with plant stability, for which necessary and sufficient conditions significantly different from those already known in the literature will be given.This research has been supported in part by MPI, Rome, Italy.     

Stability without eigenspaces in the geometric approach: The regulator problem

The regulator problem with the plant stability condition is investigated by using new tools of the geometric approach, i.e., self-bounded controlled and self-hidden conditioned invariants, recently introduced by the authors. In this framework, a constructive procedure which solves the problem without eigenspaces is obtained. This procedure also includes a set of necessary and sufficient conditions which, compared with those previously known in the literature, are significantly simpler in their statement and more easily mechanized.This research was supported by the Italian Ministry of Education. This paper is an extended version of a communication presented at the 8th International Symposium on the Mathematical Theory of Networks and Systems, Phoenix, Arizona, 1987.     

The algebraic regulator problem from the state-space point of view

We study the algebraic aspects of the regulator problem, using some new ideas in the state-space (“geometric”) approach to feedback design problems for linear multi- variable systems. Necessary and sufficient conditions are given for the solvability of a general version of this problem, requiring output stability, internal stability, and disturbance decoupling as well. An algorithm is given by which these conditions can be verified from the system parameters.     

Conservative finite-difference scheme for the problem of propagation of a femtosecond pulse in a photonic crystal with combined nonlinearity

A conservative finite-difference scheme is constructed for the problem of propagation of a light pulse in a one-dimensional nonlinear photonic crystal with combined nonlinearity. The invariants of the corresponding differential problem and their difference analogues are given. The scheme is compared with those based on the widespread splitting method. For combined cubic and quadratic nonlinearity in photonic crystal layers, it is shown that the classical splitting method is ineffective, since it requires time steps that are smaller by one or more orders of magnitude. The finite-difference scheme proposed conserves the propagation invariants, which cannot be achieved for splitting schemes even on considerably finer grids. Nonreflecting conditions substantially improve the efficiency of conservative finite-difference schemes as applied to the simulation of complex nonlinear effects in photonic crystals, which require much smaller steps in space and time than those used in the case of linear propagation. The simulation is based on the approach proposed by the authors for the given class of problems.     

Stability in part of the variables of “partial” equilibria of systems with aftereffect

The problem of stability in part of the variables of a “partial” equilibrium (this means that a given part of the phase vector coordinates is zero) is considered for nonlinear nonstationary systems of functional differential equations with aftereffect. The notions of stability in part of the variables, which admit more general (compared with the known ones) assumptions about the values of the supremum-norm of the components of the initial vector function corresponding to the variables that do not determine the given equilibrium, are introduced. The stability and asymptotic stability conditions of the the type mentioned above are obtained in the context of the method of Lyapunov-Krasovskii functionals; this conditions allow generalization of several well-known results.     

Explicit stability conditions for integro-differential equations with periodic coefficients

New explicit stability conditions are derived for a linear integro-differential equation with periodic operator coefficients. The equation under consideration describes oscillations of thin-walled viscoelastic structural members driven by periodic loads. To develop stability conditions two approaches are combined. The first is based on the direct Lyapunov method of constructing stability functionals. It allows stability conditions to be derived for unbounded operator coefficients, but fails to correctly predict the critical loads for high-frequency excitations. The other approach is based on transforming the equation under consideration in such a way that an appropriate ‘differential’ part of the new equation would possess some reserve of stability. Stability conditions for the transformed equation are obtained by using a technique of integral estimates. This method provides acceptable estimates of the critical forces for periodic loads, but can be applied to equations with bounded coefficients only. Combining these two approaches, we derive explicit stability conditions which are close to the Floquet criterion when the integral term vanishes. These conditions are applied to the stability problem for a viscoelastic bar compressed by periodic forces. The effect of material and structural parameters on the critical load is studied numerically. © 1998 B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Contact linearization of nondegenerate Monge-Ampère equations

The present paper is devoted to the problem of transforming the classical Monge-Ampère equations to the linear equations by change of variables. The class of Monge-Ampère equations is distinguished from the variety of second-order partial differential equations by the property that this class is closed under contact transformations. This fact was known already to Sophus Lie who studied the Monge-Ampère equations using methods of contact geometry. Therefore it is natural to consider the classification problems for the Monge-Ampère equations with respect to the pseudogroup of contact transformations. In the present paper we give the complete solution to the problem of linearization of regular elliptic and hyperbolic Monge-Ampère equations with respect to contact transformations. In order to solve this problem, we construct invariants of the Monge-Ampère equations and the Laplace differential forms, which involve the classical Laplace invariants as coefficients.     

Synthesis of multivariable regulators: The internal model principle

For the multivariable control system described by \(\dot x = Ax + Bu,y = Cx,z = Dx\) (wherey is the measured output andz the output to be regulated) conditions are given for the existence of a controller which preserves output regulation and loop stability, in the presence of small parameter variations in controller and plant. Under mild conditions, such a “strong” synthesis is shown to exist if and only if the regulator problem with internal stability (RPIS) is well-posed. Synthesis is achieved by means of a feedback configuration which in general incorporates an invariant, and suitably redundant, copy of the dynamic model adopted for the exogenous disturbance and reference signals which the system is required to process.     

Stability analysis and stabilization of LPV systems with jumps and (piecewise) differentiable parameters using continuous and sampled-data controllers

Linear Parameter-Varying (LPV) systems with jumps and piecewise differentiable parameters is a class of hybrid LPV systems for which no tailored stability analysis and stabilization conditions have been obtained so far.¹ We fill this gap here by proposing an approach based on a clock- and parameter-dependent Lyapunov function yielding stability conditions under both constant and minimum dwell-times. Interesting adaptations of the latter result consist of a minimum dwell-time stability condition for uncertain LPV systems and LPV switched impulsive systems. The minimum dwell-time stability condition is notably shown to naturally generalize and unify the well-known quadratic and robust stability criteria all together. Those conditions are then adapted to address the stabilization problem via timer-dependent and a timer- and/or parameter-independent (i.e. robust) state-feedback controllers, the latter being obtained from a relaxed minimum dwell-time stability condition involving slack-variables. Finally, the last part addresses the stability of LPV systems with jumps under a range dwell-time condition which is then used to provide stabilization conditions for LPV systems using a sampled-data state-feedback gain-scheduled controller. The obtained stability and stabilization conditions are all formulated as infinite-dimensional semidefinite programming problems which are then solved using sum of squares programming. Examples are given for illustration.     

Problems of the partial stability and detectability of dynamical systems

The conditions under which uniform stability (uniform asymptotic stability) with respect to a part of the variables of the zero equilibrium position of a non-linear non-stationary system of ordinary differential equations signifies uniform stability (uniform asymptotic stability) of this equilibrium position with respect the other, larger part of the variables, which include an additional group of coordinates of the phase vector, are established. These conditions include the condition for uniform asymptotic stability of the zero equilibrium position of the “reduced” subsystem of the original system with respect to the additional group of variables. Since within the conditions obtained the stability with respect to the remaining unmeasured coordinates of the phase vector remains undetermined or is investigated additionally, partial zero-detectability of the original system occurs in this case, and the conditions obtained supplement the series of known results from partial stability theory. The application of the results obtained to problems of the partial stabilization of non-linear controlled systems, particularly to the problem of stabilizing an asymmetric rigid body relative to an assigned direction in an inertial space, is considered. The partial detectability of linear systems with constant coefficients is also investigated.     

On ε-optimal continuous selectors and their application in discounted dynamic programming

A quadratic regulator problem for a class of nonlinear systems is considered in which the control cost is multiplied by a small parameter, which becomes a so-called cheap control problem. Conditions are found under which the minimum cost becomes zero (perfect regulation) and the linear part in the optimal control law becomes dominant as the small parameter goes to zero. Near optimality of control laws truncated from the optimal control law in series form is also found.     

Stability of transonic shock fronts in two-dimensional Euler systems

We study the stability of stationary transonic shock fronts under two-dimensional perturbation in gas dynamics. The motion of the gas is described by the full Euler system. The system is hyperbolic ahead of the shock front, and is a hyperbolic-elliptic composed system behind the shock front. The stability of the shock front and the downstream flow under two-dimensional perturbation of the upstream flow can be reduced to a free boundary value problem of the hyperbolic-elliptic composed system. We develop a method to deal with boundary value problems for such systems. The crucial point is to decompose the system to a canonical form, in which the hyperbolic part and the elliptic part are only weakly coupled in their coefficients. By several sophisticated iterative processes we establish the existence and uniqueness of the solution to the described free boundary value problem. Our result indicates the stability of the transonic shock front and the flow field behind the shock.

     

Well-posedness of a modified initial-boundary value problem on stability of shock waves in a viscous gas. Part II

Asymptotic stability by Lyapunov of the steady-state solution to the linear initial-boundary value problem which is formulated in the first part of this paper [A.M. Blokhin, D.L. Tkachev, L.O. Baldan, Well-posedness of a modified initial-boundary value problem on stability of shock waves in a viscous gas. Part I, J. Math. Anal. Appl. 331 (1) (2007) 408-423 (this issue)] proved under two conditions. The main of these conditions is that zeros of Lopatinsky determinant excluding η=0, s=0 lie in the left half-plane. The problem arises while providing grounds for replacing of thin transitional zones of strong gradients of basic flow parameters for viscous heat-conducting gas with surfaces of strong discontinuity.     

The stability and small oscillations of an unduloidal fluid bridge between two coaxial discs under conditions of weightlessness

A fluid bridge between two identical coaxial discs is considered which, in equilibrium, has the form of a convex unduloid (that is, a wave-like surface). It is shown that the stability of the equilibrium and the existence of small oscillations of the fluid depend on the coercivity of the bilinear form associated with the operator arising in the problem which is determined by the potential of the surface tension forces. The problem reduces to an operator equation in which one of the operators is associated, by virtue of Laplace's law, with the mean curvature of the perturbed free surface. The problem of coercivity reduces to an auxiliary eigenvalue problem. The conditions of stability are found to be satisfied if all of the eigenvalues of the problem are strictly greater than unity. Sufficient conditions for stability are obtained using arguments based on the theory of elliptic functions. The existence of natural frequencies is proved using functional analysis methods.     

Asymptotic Behavior of Infinite Dimensional Stochastic Differential Equations by Anticipative Variation of Constants Formula

We consider the long time behavior of an infinite dimensional stochastic evolution equation with respect to a cylindrical Wiener process. New estimates on the disturbance operator related to the problem are proved using a ``variation of constants'-type formula. Such estimates, under the natural assumption of mean-square stability for the linear part of the equation, lead directly to sufficient conditions for the exponential stability of the problem. In the last part of the paper we prove that, under suitable conditions, the equation admits a unique invariant measure that is strongly mixing. To complete the paper, we present examples of interesting situations where our construction applies. Accepted 28 February 2001. Online publication 9 August 2001.     

Stabilization of nonlinear discrete-time dynamic control systems with a parameter and state-dependent coefficients

A numerical-analytical algorithm for designing nonlinear stabilizing regulators for the class of nonlinear discrete-time control systems is proposed that significantly reduces computational costs. The resulting regulator is suboptimal with respect to the constructed quadratic functional with state-dependent coefficients. The conditions for the stability of the closed-loop system are established, and a stability result is stated. Numerical results are presented showing that the nonlinear regulator designed is superior to the linear one with respect to both nonlinear and standard time-invariant cost functionals. An example demonstrates that the closed-loop system is uniformly asymptotically stable.     

The steady motions of a gyrostat suspended on a rod in a central gravitational field

The problem of the existence, stability and bifurcation of the steady motions of two bodies in an orbital tethered system, when one of the bodies is a symmetrical satellite with a rotor on the axis of symmetry, is considered. One-parameter families of steady motions are indicated, and their stability and bifurcations are investigated. The conditions which relate the parameters of the system for which stabilization of the families obtained is possible using a rotating rotor are obtained.     

Simulation of the Spatial Action of a Medium on a Body of Conical Form

We consider amathematical model of the spatial action of a medium on the axisymmetric rigid body whose external surface has a part that is a circular cone.We present a complete system of equations of motion under the quasistationary conditions. The dynamical part forms an independent system of the sixth order in which the independent subsystems of lower order are distinguished. We study the problem of stability with respect to the part of variables of the key regime—the spatial rectilinear translational deceleration of the body. For a particular class of bodies, we show the inertial mass characteristics under which the key regime is stable. For a plane analog of the problem, we obtain a family of phase portraits in the space of quasivelocities.     

基于积分二次约束混合摄动系统的摄动界

讨论正向通道为线性不确定系统,反馈通道为非线性动态不确定系统组成的不确定混合摄动系统的摄动界问题。假定其线性部分的参数不确定由区间摄动模式描述,非线性部分的动态不确定由积分二次约束(IQC)描述。用Minkowski泛函出给出区间摄动模式下的摄动界的定义,并给出参数空间中混合摄动模式下系统摄动界的估计式。根据双凸函数和凹凸函数的特性把混合摄动系统的无穷稳定检验问题转化为顶点检验和一维检验问题。最后给出例子。     

A bioeconomic model of a ratio-dependent predator-prey system and optimal harvesting

This paper deals with the problem of a ratio-dependent prey-predator model with combined harvesting. The existence of steady states and their stability are studied using eigenvalue analysis. Boundedness of the exploited system is examined. We derive conditions for persistence and global stability of the system. The possibility of existence of bionomic equilibria has been considered. The problem of optimal harvest policy is then solved by using Pontryagin’s maximal principle.