Authors: Lev Ryashko

Abstract: An exponential mean square stability for the limit cycles of nonlinear stochastic systems is considered. The first approximation linear systems are introduced and a notion of P-stability (projective) is proposed. A spectral criterion for P-stability is obtained. Mean square stabilization of periodic solutions of stochastically forced nonlinear systems is considered. The necessary and sufficient conditions of stabilizability are presented. The possibilities of constructive design of stabilizing regulator for 2D limit cycles are demonstrated.

Keywords: control,limit cycles,stochastic stability,stabilization

Digital Object Identifier (DOI): 10.3182/20070829-3-RU-4912.00025

Conference: 3rd IFAC Workshop on Periodic Control Systems, 2007

Location: Anichkov Palace, Russia

Start Date: Wed Aug 29 2007 - End Date: Fri Aug 31 2007

Page Numbers: 148-152

Authors: G. V. Kostin; V. V. Saurin

Abstract: An approach to modeling and optimization of controlled dynamical systems with distributed elastic and inertial parameters is considered. The general method of integrodifferential relations (IDR) for solving a wide class of boundary value problems is developed and criteria of solution quality are proposed. A numerical algorithm for discrete approximation of controlled motions has been worked out and applied to design the optimal control law moving an elastic system to the terminal position and minimizing the given objective function. The polynomial control for plane motions of a homogeneous cantilever beam is investigated. The optimal control problem of beam transportation from the initial rest position to given terminal states, in which the total mechanical energy of the system reaches its minimal value, is considered.

Keywords: approximate analysis,distributed-parameter systems,optimal control

Digital Object Identifier (DOI): 10.3182/20070829-3-RU-4912.00026

Conference: 3rd IFAC Workshop on Periodic Control Systems, 2007

Location: Anichkov Palace, Russia

Start Date: Wed Aug 29 2007 - End Date: Fri Aug 31 2007

Page Numbers: 153-156

Authors: Andras Varga

Abstract: At the First IFAC Workshop on Periodic Systems (Como, Italy, 2001), a state of the art survey of computational methods for periodic systems has been presented (Varga and Van Dooren, 2001). This contribution continues this survey by presenting the main achievements in this field since 2001. Besides many foreseen developments mentioned in 2001 as open problems, important new developments took place as general algorithms for analysis of periodic descriptor systems, solution of periodic Riccati equations, or computational methods for continuous-time periodic systems.

Keywords: periodic systems,discrete-time systems,time-varying systems,numerical methods,computer aided control system design

Digital Object Identifier (DOI): 10.3182/20070829-3-RU-4912.00027

Conference: 3rd IFAC Workshop on Periodic Control Systems, 2007

Location: Anichkov Palace, Russia

Start Date: Wed Aug 29 2007 - End Date: Fri Aug 31 2007

Page Numbers: 157-162

Authors: Stefan Johansson; Bo Kagstrom; Anton Shiriaev; Andras Varga

Abstract: One-shot methods and recently proposed multi-shot methods for computing stabilizing solutions of continuous-time periodic Riccati differential equations are examined and evaluated on two test problems: (i) a stabilization problem for an artificially constructed time-varying linear system for which the exact solution is known; (ii) a nonlinear stabilization problem for a devil stick juggling model along a periodic trajectory. The numerical comparisons are performed using both general purpose and symplectic integration methods for solving the associated Hamiltonian differential systems. In the multi-shot method a stable subspace is determined using recent algorithms for computing a reordered periodic real Schur form. The results show the increased accuracy achievable by combining multi-shot methods with structure preserving (symplectic) integration techniques.

Keywords: periodic systems,reordered periodic Schur form,Riccati differential equations,stabilizing controllers,linear quadratic regulators

Digital Object Identifier (DOI): 10.3182/20070829-3-RU-4912.00028

Conference: 3rd IFAC Workshop on Periodic Control Systems, 2007

Location: Anichkov Palace, Russia

Start Date: Wed Aug 29 2007 - End Date: Fri Aug 31 2007

Page Numbers: 163-168

Authors: Robert Granat; Bo Kagstrom; Daniel Kressner

Abstract: Software for computing eigenvalues and invariant subspaces of general matrix products is proposed. The implemented algorithms are based on orthogonal transformations of the original data and thus attain numerical backward stability, which enables good accuracy even for small eigenvalues. The prospective toolbox combines the efficiency and robustness of library-style Fortran subroutines based on state-of-the-art algorithms with the convenience of MATLAB interfaces. It will be demonstrated that this toolbox can be used to address a number of tasks in systems and control theory, such as the solution of periodic discrete-time algebraic Riccati equations.

Keywords: linear periodic system,matrix product,periodic eigenvalue problem,periodic Schur form,library software,MATLAB tools

Digital Object Identifier (DOI): 10.3182/20070829-3-RU-4912.00029

Conference: 3rd IFAC Workshop on Periodic Control Systems, 2007

Location: Anichkov Palace, Russia

Start Date: Wed Aug 29 2007 - End Date: Fri Aug 31 2007

Page Numbers: 169-174

Authors: Andras Varga

Abstract: We propose computationally efficient and numerically reliable algorithms to compute minimal realizations of periodic descriptor systems. The main computational tool employed for the structural analysis of periodic descriptor systems (i.e., reachability and observability) is the orthogonal reduction of periodic matrix pairs to Kronecker-like forms. Specializations of a general reduction algorithm are employed for particular type of systems. One of the proposed minimal realization methods relies exclusively on structure preserving manipulations via orthogonal transformations for which the backward numerical stability can be proved.

Keywords: periodic systems,discrete-time systems,time-varying systems,Kalman decomposition,numerical methods

Digital Object Identifier (DOI): 10.3182/20070829-3-RU-4912.00030

Conference: 3rd IFAC Workshop on Periodic Control Systems, 2007

Location: Anichkov Palace, Russia

Start Date: Wed Aug 29 2007 - End Date: Fri Aug 31 2007

Page Numbers: 175-180

Authors: Luca Vigano; Marco Lovera; Andras Varga

Abstract: This paper addresses the problem of optimal static output feedback control of linear periodic systems. The proposed approach, which allows to deal with both stable and unstable open loop systems, relies on a continuous-time formulation of the control problem. A nonlinear programming based approach is used to solve the underlying optimization problem. The evaluation of cost function and its derivative largely benefits of the recent advances in algorithms to solve continuous-time periodic Lyapunov equations.

Keywords: time-varying systems,output feedback,optimal control

Digital Object Identifier (DOI): 10.3182/20070829-3-RU-4912.00031

Conference: 3rd IFAC Workshop on Periodic Control Systems, 2007

Location: Anichkov Palace, Russia

Start Date: Wed Aug 29 2007 - End Date: Fri Aug 31 2007

Page Numbers: 181-186

Authors: Masayoshi Tomizuka

Abstract: Periodic disturbances are common in control of mechanical systems. Such disturbances may be due to rotational elements such as motors and vibratory elements. When the period of a periodic disturbance is fixed and known in advance, repetitive control can be used for attenuating their effect. The most popular repetitive controller is based on the internal model principle. When the period is not fixed and unknown, adaptation capability must be introduced. This paper presents some fundamental issues and new challenges in the design of controllers to deal with periodic disturbances along with applications to mechanical systems.

Keywords: repetitive control,learning control,mechanical systems,periodic disturbances

Digital Object Identifier (DOI): 10.3182/20070829-3-RU-4912.00032

Conference: 3rd IFAC Workshop on Periodic Control Systems, 2007

Location: Anichkov Palace, Russia

Start Date: Wed Aug 29 2007 - End Date: Fri Aug 31 2007

Page Numbers: 187-195

Authors: A. N. Daryin; A. B. Kurzhanski

Abstract: This paper deals with the problem of damping the oscillations of a finite cascade of springs through impulsive controls in finite time. The control problem is treated as one of specifying a closed-loop (feedback) strategy and is solved by applying Hamiltonian technique in its Dynamic Programming version. A numerical algorithm is indicated and the limit case with control time tending to infinity is described.

Keywords: impulse control,control synthesis,dynamic programming,oscillating systems

Digital Object Identifier (DOI): 10.3182/20070829-3-RU-4912.00033

Conference: 3rd IFAC Workshop on Periodic Control Systems, 2007

Location: Anichkov Palace, Russia

Start Date: Wed Aug 29 2007 - End Date: Fri Aug 31 2007

Page Numbers: 196-201

Authors: Erik I. Verriest

Abstract: This note provides the general set-up and solution method for a frictioncoupled system (the 'body') moving in the landscape. This separates the shape (or configuration) space from the state space. The quotient space is the space where the motion of the body can be envisioned as a point mass. In particular we are interested in cases where the inertial forces are small in comparison to applied forces, and propose a simple perturbation expansion. Next we consider a periodic regime for the motion in shape space and pose and solve the optimal control problem via Fourier techniques. This is illustrated on a simple toy system: the two-piece worm with differential friction. Alternative friction models are proposed.

Keywords: locomotion,periodic control,biomimetic worm

Digital Object Identifier (DOI): 10.3182/20070829-3-RU-4912.00034

Conference: 3rd IFAC Workshop on Periodic Control Systems, 2007

Location: Anichkov Palace, Russia

Start Date: Wed Aug 29 2007 - End Date: Fri Aug 31 2007

Page Numbers: 202-207

Authors: Patrizio Colaneri; Riccardo Scattolini

Abstract: This paper proposes a model predictive approach for H∞ control of switched systems in discrete-time. A finite horizon dynamic game is set up for the computation of the control input and a switching strategy guaranteeing robustness at the face of bounded disturbances. A main point investigated in the paper is the existence of an auxiliary law useful to activate the predictive algorithm.

Keywords: model predictive control,switched systems,Riccati inequality

Digital Object Identifier (DOI): 10.3182/20070829-3-RU-4912.00035

Conference: 3rd IFAC Workshop on Periodic Control Systems, 2007

Location: Anichkov Palace, Russia

Start Date: Wed Aug 29 2007 - End Date: Fri Aug 31 2007

Page Numbers: 208-212

Authors: Irina Bashkirtseva; Lev Ryashko; Henri Schurz

Abstract: An analysis of the classical Hopf differential system perturbed by multiplicative and additive noises is carried out. The difference in the response of Hopf systems perturbed by additive and multiplicative random noises is investigated. A delaying shift of the Hopf bifurcation point induced by multiplicative noise is demonstrated. The phenomenon of inverse stochastic bifurcation in which autooscillations are suppressed by multiplicative noise is clearly observed.

Keywords: limit cycle,bifurcation,stochastic disturbances

Digital Object Identifier (DOI): 10.3182/20070829-3-RU-4912.00036

Conference: 3rd IFAC Workshop on Periodic Control Systems, 2007

Location: Anichkov Palace, Russia

Start Date: Wed Aug 29 2007 - End Date: Fri Aug 31 2007

Page Numbers: 213-217

Authors: Zhanybai T. Zhusubaliyev; Sergey Chevychelov; Erik Mosekilde

Abstract: Considering a two-dimensional system of nonautonomous differential equations with discontinuous right-hand sides describing the behavior of a DC/DC converter with pulse-width modulated control, the paper demonstrates how a two-dimensional invariant torus can arise from a stable equilibrium point. We determine the chart of dynamical modes and show that there is a region of parameter space in which the system has a single stable node equilibrium point. Under variation of the parameters, this equilibrium may collide with a discontinuity boundary between two smooth regions in the phase space. When this happens, one can observe a variety of different bifurcation scenarios. One scenario is the continuous transformation of the stable equilibrium into a stable period-1 focus. A second is the transformation of the stable node equilibrium into an unstable period-1 focus, and the associated formation of a two-dimensional (ergodic or resonant) torus.

Keywords: nonlinear theory,control system analysis,stability analysis,chaos,equilibrium

Digital Object Identifier (DOI): 10.3182/20070829-3-RU-4912.00037

Conference: 3rd IFAC Workshop on Periodic Control Systems, 2007

Location: Anichkov Palace, Russia

Start Date: Wed Aug 29 2007 - End Date: Fri Aug 31 2007

Page Numbers: 218-223

Authors: Yu. Kolokolov; A. Melikhov; V. Tsukanov; A. Hamzaoui; N. Essounbouli

Abstract: The parametric stability regions of two induction machines mathematical models with the unsymmetrical connection to the supply-line were compared in the paper. The first model was formed by only Ohm's and Kirchhoff's laws use. The second one was formed through the transformations, which are used usually in the classical theory of the electrical machines. As the research the parameter combinations, at which the model stability can not be treated one-to-one, are revealed. The algorithm of Lyapunov's high-order characteristic index calculation is used for the formation of the induction machine model parametrical stability domains.

Keywords:

Digital Object Identifier (DOI): 10.3182/20070829-3-RU-4912.00038

Conference: 3rd IFAC Workshop on Periodic Control Systems, 2007

Location: Anichkov Palace, Russia

Start Date: Wed Aug 29 2007 - End Date: Fri Aug 31 2007

Page Numbers: 224-227

Authors: J. Hatonen; F. Zhang; S. Daley; D. H. Owens

Abstract: In this paper a well-known adjoint SISO Repetitive Control algorithm is extended to the multivariable case and its convergence properties are analysed. The new adjoint algorithm is validated using a simulation of an experimental facility consisting of a passive Naval vibration isolation mount that is combined with six active control channels located in a Stewart platform style arrangement. The algorithm utilises a multivariable FIR system description that is derived from frequency response function measurements. The adjoint repetitive control algorithm is used to eliminate a harmonic disturbance where the first harmonic coincides with the fundamental mount resonance of the passive component. The simulation results show that the repetitive control algorithm ultimately achieves good vibration isolation but due to the wide spread of system eigenvalues at the harmonic frequency, convergence is slow.

Keywords: learning control,repetitive control,signal processing,real-time control width (maximum),left justified

Digital Object Identifier (DOI): 10.3182/20070829-3-RU-4912.00039

Conference: 3rd IFAC Workshop on Periodic Control Systems, 2007

Location: Anichkov Palace, Russia

Start Date: Wed Aug 29 2007 - End Date: Fri Aug 31 2007

Page Numbers: 228-233

Authors: Alexander Churilov; Alexander Medvedev; Alexander Shepeljavyi

Abstract: Periodic solutions of a recently suggested parsimonious mathematical model of pulse-modulated regulation of non-basal testosterone secretion in the male are studied. The model is of third order, reflecting the three most significant hormones in the regulation loop, but yet is shown to be capable of sustaining periodic solutions with one or two pulses of gonadotropin-releasing hormone on each period. Lack of stable periodic solutions is otherwise a main shortcoming of existing low order hormone regulation models.

Keywords: biomedical systems,impulse signals,oscillation,limit cycles

Digital Object Identifier (DOI): 10.3182/20070829-3-RU-4912.00040

Conference: 3rd IFAC Workshop on Periodic Control Systems, 2007

Location: Anichkov Palace, Russia

Start Date: Wed Aug 29 2007 - End Date: Fri Aug 31 2007

Page Numbers: 234-239

Authors: Volker Reitmann

Abstract: We consider a control problem for the heating process of an elastic plate. The heat flux within the plate is modeled by the heat equation with nonlinear Neumann boundary conditions according to Newton's law. As input at a part of the boundary we take the nonlinearly transformed and modulated heat production of a separate heater which is given by a nonlinear Duffing-type ODE. This ODE depends on measurements of the temperature within the plate and on Bohr resp. Stepanov almost periodic in time forcing terms. The physical problem is generalized to a bifurcation problem for non-autonomous evolution systems in rigged Hilbert spaces. Using Lyapunov functionals, invariant cones and monotonicity properties of the nonlinearities in certain Sobolev spaces, we derive frequency domain conditions for the existence and uniqueness of an asymptotically stable and almost periodic in time temperature field.

Keywords: periodic motion,stability analysis,partial differential equations,frequency domains,control closed-loop

Digital Object Identifier (DOI): 10.3182/20070829-3-RU-4912.00041

Conference: 3rd IFAC Workshop on Periodic Control Systems, 2007

Location: Anichkov Palace, Russia

Start Date: Wed Aug 29 2007 - End Date: Fri Aug 31 2007

Page Numbers: 240-244

Authors: Vera B. Smirnova; Alexander I. Shepeljavyi

Abstract: Asymptotic behavior of a class of multidimensional discrete control systems with periodic nonlinearities and denumerable set of equilibria is investigated. By means of discrete version of Yakubovich-Kalman theorem and certain modification of Lur'e-Postnikov function a frequency-domain criterion which guarantees that every solution of a system tends to an equilibrium is obtained.

Keywords: discrete phase systems,frequency-domain criteria,gradient-like behavior,Yakubovich-Kalman theorem,second Lyapunov method

Digital Object Identifier (DOI): 10.3182/20070829-3-RU-4912.00042

Conference: 3rd IFAC Workshop on Periodic Control Systems, 2007

Location: Anichkov Palace, Russia

Start Date: Wed Aug 29 2007 - End Date: Fri Aug 31 2007

Page Numbers: 245-249

Authors: Dmitry A. Altshuller

Abstract: The paper, describes the problem of stability of oscillations in nonlinear feedback systems. The concept of stability is defined in a way that makes the problem tractable using the absolute stability approach. The result is formulated in frequency domain and has the form of the Zames-Falb multiplier, which makes it amenable to geometric interpretation. Numerical examples are given to illustrate the application of the new result to cases, where the Circle Criterion is not applicable. The advantage of the new criterion is that only the period of the oscillations needs to be known, not the complete expression of the oscillatory solution.

Keywords: absolute stability,stability analysis,nonlinear control systems,oscillators

Digital Object Identifier (DOI): 10.3182/20070829-3-RU-4912.00043

Conference: 3rd IFAC Workshop on Periodic Control Systems, 2007

Location: Anichkov Palace, Russia

Start Date: Wed Aug 29 2007 - End Date: Fri Aug 31 2007

Page Numbers: 250-253

Authors: Sergei V. Gusev; Anton S. Shiriaev; Leonid B. Freidovich

Abstract: The paper presents a new method for numerical solution of matrix Riccati equation with periodic coefficients. The method is based on approximation of stabilizing solution of the Riccati equation by trigonometric polynomials.

Keywords: Riccati equation,periodic,numerical solution,Yakubovich Lemma,linear matrix inequality

Digital Object Identifier (DOI): 10.3182/20070829-3-RU-4912.00044

Conference: 3rd IFAC Workshop on Periodic Control Systems, 2007

Location: Anichkov Palace, Russia

Start Date: Wed Aug 29 2007 - End Date: Fri Aug 31 2007

Page Numbers: 254-256

Authors: Ichiro Jikuya; Ichijo Hodaka

Abstract: In this note, structural decomposition of linear periodic continuous-time systems is discussed. A fundamental problem to decompose a state of a periodic system into controllable and uncontrollable parts is conjectured to be achieved by a continuously differentiable and periodic coordinate transformation with the same period of the system, however there is a counterexample to this conjecture. Hence we derive a condition for the existence of such a coordinate transformation. We also prove that, by relaxing a class of coordinate transformation, it is always possible to construct a periodic coordinate transformation with the double period of the periodic system.

Keywords: periodic systems,Kalman canonical decomposition,decomposition theorems,coordinate transformations,periodic structures,linear systems,continuous time systems,time-varying systems,controllability,stabilizability

Digital Object Identifier (DOI): 10.3182/20070829-3-RU-4912.00045

Conference: 3rd IFAC Workshop on Periodic Control Systems, 2007

Location: Anichkov Palace, Russia

Start Date: Wed Aug 29 2007 - End Date: Fri Aug 31 2007

Page Numbers: 257-262

Authors: Dimitri Peaucelle; Christophe Farges; Denis Arzelier

Abstract: Linear matrix inequality based techniques, most often used for robust analysis of linear systems, are applied to the stability analysis of periodic equilibrium trajectories of nonlinear systems. Results are derived by linear-fractional representation of the nonlinearities and taking into account parametric uncertainties in the same time. Numerically exploitable formulas are obtained by discretization. An academic example illustrates the entire methodology.

Keywords: robustness,periodic systems,LMI,LFR

Digital Object Identifier (DOI): 10.3182/20070829-3-RU-4912.00046

Conference: 3rd IFAC Workshop on Periodic Control Systems, 2007

Location: Anichkov Palace, Russia

Start Date: Wed Aug 29 2007 - End Date: Fri Aug 31 2007

Page Numbers: 263-267

Authors: Tomomichi Hagiwara; Yasuhiro Ohara

Abstract: This paper is concerned with robust stability analysis of discrete-time systems. We first consider linear periodically time-varying (LPTV) nominal systems, for which we apply the discrete-time lifting to have their equivalent linear time-invariant (LTI) representations. Applying the conventional but general scaling approach to the LTI representations leads to the notion of noncausal LPTV scaling when the scaling is interpreted in the original time axis without lifting. Regarding this discrete-time noncausal LPTV scaling, we confirm its effectiveness over causal LPTV scaling and (causal) LTI scaling theoretically as well as with a numerical example. We next consider LTI nominal systems, for which we again apply noncausal LPTV scaling by regarding the LTI systems as a special case of LPTV systems and thus applying the discrete-time lifting in the same way as in the LPTV nominal systems. We then study the relationship of such an approach with the conventional LTI scaling applied directly to the LTI nominal systems without lifting treatment. In particular, we show that even static noncausal LPTV scaling yields dynamic (frequency-dependent) LTI scaling if it is interpreted in the context of lifting-free treatment, and an advantage of noncausal LPTV scaling for LTI nominal systems is investigated from this viewpoint. A numerical example is also provided that supports the advantage.

Keywords: robust control,quadratic separator,scaling,lifting

Digital Object Identifier (DOI): 10.3182/20070829-3-RU-4912.00047

Conference: 3rd IFAC Workshop on Periodic Control Systems, 2007

Location: Anichkov Palace, Russia

Start Date: Wed Aug 29 2007 - End Date: Fri Aug 31 2007

Page Numbers: 268-273

Authors: Bernhard P. Lampe; Efim N. Rosenwasser

Abstract: The paper considers the stabilization problem for linear continuous-time T-periodic processes with pure delay by a T-periodic digital controller. Necessary and sufficient conditions for the stabilizability are formulated, and an algorithm for constructing the set of stabilizing controllers is provided. Its application is demonstrated by a simple example.

Keywords: periodic processes,sampled-data control,time delay,stabilizing controllers,parametrization

Digital Object Identifier (DOI): 10.3182/20070829-3-RU-4912.00048

Conference: 3rd IFAC Workshop on Periodic Control Systems, 2007

Location: Anichkov Palace, Russia

Start Date: Wed Aug 29 2007 - End Date: Fri Aug 31 2007

Page Numbers: 274-278

Authors: Lukasz Hladowski; Krzysztof Galkowski; Eric Rogers; Paul L. Lewin; Christopher T. Freeman

Abstract: In this paper we use a repetitive process setting to develop a new iterative learning control algorithm for plants modelled by a discrete linear time-varying state space model. As the next step in evaluating its performance, the results of a simulation based study are given where the plant model used is that obtained from frequency responses tests on a gantry robot.

Keywords: linear repetitive process,iterative learning control

Digital Object Identifier (DOI): 10.3182/20070829-3-RU-4912.00049

Conference: 3rd IFAC Workshop on Periodic Control Systems, 2007

Location: Anichkov Palace, Russia

Start Date: Wed Aug 29 2007 - End Date: Fri Aug 31 2007

Page Numbers: 279-282