Iliya V. Miroshnik

Physics and Control

During decades the interest of the physics community in control theory was not substantial. However, the situation changed dramatically in the 90s after it was discovered that even small feedback introduced into a chaotic system can change its behavior significantly, e.g., turn chaotic motion into periodic one [221]. The seminal paper [221] gave rise to a variety of publications demonstrating metamorphoses of numerous systems — both simple and complicated — under the action of feedback. However, the potential of modern nonlinear control theory (see [113, 206, 165]) still was not seriously appreciated although the key role of nonlinearity definitely was. On the other hand, new problems seem not traditional for control theorists: the desired position or the desired trajectory of the system is not specified whilst the “small feedback” requirement is imposed instead. It took some time to realize that such kind of problems is typical for control of more general oscillatory behavior and to work out a unified approach to nonlinear control of oscillations and chaos [88].

PARTIAL STABILIZATION AND GEOMETRIC PROBLEMS OF NONLINEAR CONTROL

Abstract The paper addresses problems of analysis and control of spatial behavior of dynamical systems associated with properties of invariance and attractivity of smooth geometric objects (goal sets). A comparative study of the main concepts of geometric theory, coordinating and synergetic approaches is given and the problems of attractivity of invariant sets are considered in their connection with the partial stability analysis. Design procedures, static and dynamic control laws ensuring the desired properties of spatial dynamics are proposed, simplified local conditions of attractivity of the goal sets and partial stabilization of the system are represented.

Stabilization of Spatial Attractors and Control of Nonlinear Systems

Abstract The paper discusses the problems of nonlinear plant controlled spatial motion along given multidimensional hypersurfaces (submanifolds). The analysis of the system dynamics in the vicinity of the regular hypersurface with respect to relevant movable frame is proposed in order to derive, under an appropriate assumption, weakly connected models of the controlled internal dynamics and that of the transversal one, On that basis, the dynamic control law is designed, providing the local stability of the hypersurface as a system attractor and the desired internal behavior prescribed hy given reference trajectories on the hypersurface.

Trajectory motion control of underactuated manipulators

Abstract A problem of control of trajectory motion of manipulators with two DOF and one controlling input is considered. Techniques of analysis of these underactuated systems and a procedure of the control design are represented. The relevant solution of the control problem is based on the differential-geometric approach and is reduced to the systems stabilization about the required smooth curve (trajectory). New simulation results are represented.

Stabilization of pendulum oscillations around upper position 1

Abstract The paper addresses problems of control of pendulum oscillations under high-frequency vertical vibration of the pivot. By using a model of oscillations and virtual energy concept, an energy-based feedback control law with the observer of slow state variables is designed to provide desired stable oscillations of the pendulum with respect to the upright position.

Attractors and Partial Stability of Nonlinear Dynamical Systems

Abstract The paper represents a comparative study of key concepts of attracting sets (submanifolds) and partially stable systems aimed at establishing relations of the local properties and the unification of the analysis methodologies. The use of recent results of stability theory and techniques of geometric control allow one to design quadratic Lyapunov functions of the partially stable system and to find simplified conditions of attractivity (and partial stability) based on the analysis of partially linearized models.

Functional decomposition and nonlinear control of MIMO systems

Abstract The paper is devoted to nonlinear MIMO problems of coordinating control which fail to be directly solved by using techniques of decoupling and decentralized control but admit functional decomposition, i.e. splitting the main control problem into several conventional subproblems of stabilization and tracking. An approach to the task-oriented conversion of the original MIMO system and its partial decoupling by means of the relevant transformation of the control variables is proposed. This leads to a set of weakly connected subsystems corresponding to the separate subproblems and enables one to make use of standard techniques of decentralized control.

Trajectory Motion Control and Coordination of Multi-Link Robots

Abstract The trajectory motion control problem for multi-link dexterous robots is analyzed on the basis of the Coordinating Control Principle. In addition to the trajectory description introducing the conventional coordinating relations of the endpoint coordinates, auxiliary holonomic relations of the robot variables are involved to define the configuration of the robot kinematic chain and overcome the problem redundancy. The resultant system is designed on the basis of standard nonlinear control techniques and includes a nonlinear converter and a set of linear controllers providing the execution of the local stabilization tasks.

Speed-Gradient Method and Partial Stabilization

In this Chapter we present a unified approach, developed in the 70s, to solving nonlinear control problems: the so-called Speed-Gradient method [67, 69]. This approach is closely related to both stability and passivity. In order to apply the Speed-Gradient method the initial control goal should be reformulated via some goal function employed to construct a Lyapunov function for the closed loop system. Several kinds of Speed-Gradient algorithms are presented and the conditions ensuring achievement of the control goal are given. Other properties like parametric convergence and robustness are studied. In case the goal function is not positive definite, the method allows us to achieve partial stabilization of the system: approaching the goal surface. The special cases of Lagrangian and Hamiltonian description of the controlled plant, which are important for control of mechanical systems, are considered separately.

Control of oscillations and state estimation of Kapitza pendulum

Abstract In this paper we introduce the concept of virtual energy of Kapitza pendulum and propose an energy-based approach to problem of stabilization of the given periodic motions around the upright position. The problem is reduced to control of the virtual energy and implies estimating the pendulums slow motions provided by a nonlinear observer of the slow oscillations and virtual energy. New simulation results are represented.