# Alexander ZuyevMax Planck Institute for Dynamics of Complex Technical Systems | MPI · Group of Computational Methods in Systems and Control Theory (CSC)

Alexander Zuyev

Doctor of Sciences, Professor

## About

120

Publications

4,496

Reads

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623

Citations

Introduction

Additional affiliations

December 2014 - January 2016

November 2014 - present

September 2014 - October 2014

Education

November 1997 - September 2000

**Institute of Applied Mathematics and Mechanics, National Academy of Sciences of Ukraine**

Field of study

- Control Theory, Mechanics

September 1992 - June 1997

**Donetsk National University**

Field of study

- Mathematics

## Publications

Publications (120)

This monograph provides a rigorous treatment of problems related to partial asymptotic stability and controllability for models of flexible structures described by coupled nonlinear ordinary and partial differential equations or equations in abstract spaces. The text is self-contained, beginning with some basic results from the theory of continuous...

This work addresses the optimal stabilization problem of a nonlinear control system by using a smooth output feedback. The optimality criterion is the maximization of the decay rate of solutions in a neighborhood of the origin. We formulate this criterion as a minimax problem with respect to non-integral functional. An explicit construction of a Ly...

Symmetric functions of critical Hamiltonians, called symbols, were used in such problems of nonlinear control as the characterization of symmetries and feedback invariants. We derive here a stabilizability condition in the class of almost continuous feedback controls based on symbols. The methodology proposed consists of defining a selector of the...

A mathematical model of a controlled shell structure based on Hamilton’s principle and the generalized Ritz–Galerkin method is proposed in this paper. The problem of minimizing the stress energy is solved explicitly for a static version of this model. For the dynamical system under consideration, a procedure for estimating external disturbances and...

This paper is devoted to the study of attractive sets for dynamical systems in a metric space with a measure. It is assumed that the measure of a set of points in the phase space is increasing along the flow. We prove that an invariant set is an attractor for almost all initial conditions under some extra assumptions. For a system of autonomous ord...

We consider the problem of attitude stabilization for a low Earth orbit satellite having only electromagnetic actuation. Such a satellite is not fully actuated, as the control torque is the cross-product of magnetic moment due to magnetorquers and the geomagnetic field. The aim of this work is to study whether oscillating controls can be designed s...

We consider a mathematical model of a simply supported Euler–Bernoulli beam with an attached springmass system. The model is controlled by distributed piezo-actuators and a lumped force. We address the issue of asymptotic behavior of the solutions of this system driven by a linear feedback law. The precompactness of trajectories is established for...

The paper is devoted to the observability study of a dynamic system, which describes the vibrations of an elastic beam with an attached rigid body and distributed control actions. The mathematical model is derived using Hamilton's principle in the form of the Euler-Bernoulli beam equation with hinged boundary conditions and interface condition at t...

The paper is devoted to the observability study of a dynamic system, which describes the vibrations of an elastic beam with an attached rigid body and distributed control actions. The mathematical model is derived using Hamilton's principle in the form of the Euler-Bernoulli beam equation with hinged boundary conditions and interface condition at t...

This paper focuses on the stability analysis of periodic trajectories for non-autonomous systems whose right-hand sides are time-periodic discontinuous functions. Such systems arise, in particular, in optimal control problems for nonlinear control systems with periodic bang-bang inputs. We obtain sufficient orbital stability conditions of the Andro...

UDC 517.977 A mathematical model of a simply supported Euler – Bernoulli beam with attached spring-mass system is considered. The model is controlled by distributed piezo actuators and a lumped force. We address the issue of asymptotic behavior of solutions of this system driven by a linear feedback law. The precompactness of trajectories is establ...

A mathematical model of a simply supported Euler-Bernoulli beam with attached spring-mass system is considered. The model is controlled by distributed piezo actuators and lumped force. We address the issue of asymptotic behavior of solutions of this system driven by a linear feedback law. The precompactness of trajectories is established for the op...

An isoperimetric optimal control problem with non-convex cost is considered for a class of nonlinear control systems with periodic boundary conditions. This problem arises in chemical engineering as the maximization of the product of non-isothermal reactions by consuming a fixed amount of input reactants. It follows from the Pontryagin maximum prin...

This chapter deals with mathematical models of continuous crystallization described by hyperbolic systems of partial differential equations coupled with ordinary and integro-differential equations. The considered systems admit nonzero steady-state solutions with constant inputs. To stabilize these solutions, we present an approach for constructing...

We study a mathematical model of a hinged flexible beam with piezoelectric actuators and electromagnetic shaker in this chapter. The shaker is modeled as a mass and spring system attached to the beam. To analyze free vibrations of this mechanical system, we consider the corresponding spectral problem for a fourth-order differential operator with in...

We study a mathematical model of a hinged flexible beam with piezoelectric actuators and electromagnetic shaker in this paper. The shaker is modelled as a mass and spring system attached to the beam. To analyze free vibrations of this mechanical system, we consider the corresponding spectral problem for a fourth-order differential operator with int...

This paper deals with mathematical models of continuous crystallization described by hyperbolic systems of partial differential equations coupled with ordinary and integro-differential equations. The considered systems admit nonzero steady-state solutions with constant inputs. To stabilize these solutions, we present an approach for constructing co...

The problem of partial stabilization for nonlinear control systems described by the Ito stochastic differential equations is considered. For these systems, we propose a constructive control design method which leads to establishing the asymptotic stability in probability of the trivial solution of the closed-loop system with respect to a part of st...

In this paper, a class of nonlinear driftless control-affine systems satisfying the bracket generating condition is considered. A gradient-free optimization algorithm is developed for the minimization of a cost function along the trajectories of the controlled system. The algorithm comprises an approximation scheme with fast oscillating controls fo...

This paper focuses on the problem of constructing time-varying feedback laws that asymptotically stabilize a given part of the state variables for nonlinear control-affine systems. It is assumed that the class of systems under consideration satisfies nonlinear controllability conditions with respect to the stabilizable variables. Under these assump...

In this paper, a class of nonlinear driftless control-affine systems satisfying the bracket generating condition is considered. A gradient-free optimization algorithm is developed for the minimization of a cost function along the trajectories of the controlled system. The algorithm comprises an approximation scheme with fast oscillating controls fo...

This paper focuses on the development of stability conditions for systems of nonlinear non‐autonomous ordinary differential equations and their applications to control problems. We present a novel approach for the study of asymptotic stability properties for nonlinear non‐autonomous systems based on considering a parameterized family of sets. The p...

This paper deals with a mathematical model of a non‐isothermal chemical reaction described by nonlinear ordinary differential equations with two‐dimensional control. The control variables correspond to the possibility of manipulating the concentration of the input reactant and the total flow rate. We consider the task of maximizing the reactor perf...

An isoperimetric optimal control problem with non-convex cost is considered for a class of nonlinear control systems with periodic boundary conditions. This problem arises in chemical engineering as the maximization of the product of non-isothermal reactions by consuming a fixed amount of input reactants. It follows from the Pontryagin maximum prin...

This paper addresses the problem of stabilizing a part of variables for control systems described by stochastic differential equations of the Ito type. The considered problem is related to the asymptotic stability property of invariant sets and has important applications in mechanics and engineering. Sufficient conditions for the asymptotic stabili...

A class of nonlinear control-affine systems with bounded time-varying drift is considered. It is assumed that the control vector fields together with their iterated Lie brackets satisfy Hormander's condition in a neighborhood of the origin. Then the problem of exponential stabilization is treated by exploiting periodic time-varying feedback control...

This paper deals with the stabilization problem for nonlinear control-affine systems with the use of oscillating feedback controls. We assume that the local controllability around the origin is guaranteed by the rank condition with Lie brackets of length up to 3. This class of systems includes, in particular, mathematical models of rotating rigid b...

The paper deals with the extremum seeking problem for a class of cost functions depending only on a part of state variables of a control system. This problem is related to the concept of partial asymptotic stability and analyzed by Lyapunov's direct method and averaging schemes. Sufficient conditions for the practical partial stability of a system...

This paper deals with the stabilization problem for nonlinear control-affine systems with the use of oscillating feedback controls. We assume that the local controllability around the origin is guaranteed by the rank condition with Lie brackets of length up to~3. This class of systems includes, in particular, mathematical models of rotating rigid b...

The problem of tracking an arbitrary curve in the state space is considered for underactuated driftless control-affine systems. This problem is formulated as the stabilization of a time-varying family of sets associated with a neighborhood of the reference curve. An explicit control design scheme is proposed for the class of controllable systems wh...

This paper addresses the problem of stabilizing a part of variables for control systems described by stochastic differential equations of the Itô type. The considered problem is related to the asymptotic stability property of invariant sets and has important applications in mechanics and engineering. Sufficient conditions for the asymptotic stabili...

A class of nonlinear control-affine systems with bounded time-varying drift is considered. It is assumed that the control vector fields together with their iterated Lie brackets satisfy Hörmander’s condition in a neighborhood of the origin. Then the problem of exponential stabilization is treated by exploiting periodic time-varying feedback control...

The paper deals with the extremum seeking problem for a class of cost functions depending only on a part of state variables of a control system. This problem is related to the concept of partial asymptotic stability and analyzed by Lyapunov’s direct method and averaging schemes. Sufficient conditions for the practical partial stability of a system...

This paper deals with a mathematical model of the preferential crystallization of enantiomers. The dynamics is governed by two first‐order partial differential equations with controls appearing in the coefficients and in the boundary conditions. We study the problem of stabilizing the equilibrium of this system by a one‐dimensional input. For this...

In this paper, we consider the problem of extremum seeking in the presence of obstacles. The analytical expression of the cost function as well as locations and shapes of the obstacles are assumed to be partially or completely unknown. We describe a broad family of control algorithms for a unicycle type system, which provides a solution of the abov...

This paper deals with an isoperimetric optimal control problem for nonlinear control-affine systems with periodic boundary conditions. As it was shown previously, the candidates for optimal controls for this problem can be obtained within the class of bang-bang input functions. We consider a parametrization of these inputs in terms of switching tim...

Poster for the meeting of the Scientific Advisory Board at the Max Planck Institute for Dynamics of Complex Technical Systems (Magdeburg).

In this paper, we propose a new control design scheme for solving the obstacle avoidance problem for nonlinear driftless control-affine systems. The class of systems under consideration satisfies controllability conditions with iterated Lie brackets up to the second order. The time-varying control strategy is defined explicitly in terms of the grad...

This paper deals with an isoperimetric optimal control problem for nonlinear control-affine systems with periodic boundary conditions. As it was shown previously, the candidates for optimal controls for this problem can be obtained within the class of bang-bang input functions. We consider a parametrization of these inputs in terms of switching tim...

In this paper, we propose and practically evaluate a class of gradient-free control functions ensuring the motion of a unicycle-type system towards the extremum point of a time-varying cost function. We prove that the unicycle is able to track the extremum point, and illustrate our results by numerical simulations and experiments that show that the...

A mechanical system consisting of a rigid body and attached Kirchhoff plates under the action of three independent controls torques is considered. The equations of motion of such model are derived in the form of a system of coupled nonlinear ordinary and partial differential equations. The operator form of this system is represented as an abstract...

In this paper an isoperimetric control problem for the optimization of the performance measure for a nonlinear chemical reaction model with periodic inputs is considered. For this problem, a family of bang-bang controls parametrized by switching times is introduced. The issue of defining these switching times is addressed for periodic boundary cond...

In this paper, we consider a class of controlled population balance equations describing granulation processes in chemical engineering. Such a control system admits an equilibrium which is not asymptotically stable in general. In order to stabilize this equilibrium, we consider the perturbed system and introduce a Lyapunov functional candidate as a...

This paper considers the obstacle avoidance problem for control-linear systems satisfying the Hörmander condition with the first-order Lie brackets. To solve the problem under consideration, we use a family of oscillating periodic control functions with state-dependent coefficients. The proposed approach is shown to be applicable for different shap...

This paper is devoted to the motion planning problem for control-affine systems by using trigonometric polynomials as control functions. The class of systems under consideration satisfies the controllability rank condition with the Lie brackets up to the second order. The approach proposed here allows to reduce a point-to-point control problem to s...

The paper focuses on the development of the navigation function approach for nonlinear systems with fast oscillating controls. This approach allows to solve the obstacle avoidance problem and generate reference trajectories on the state space with obstacles by using the gradient flow of a navigation function. In general, such gradient flow cannot b...

The paper presents a control algorithm that steers a system to an extremum point of a time-varying function. The proposed extremum seeking law depends on values of the cost function only and can be implemented without knowing analytical expression of this function. By extending the Lie brackets approximation method, we prove the local and semi-glob...

In this paper, we study the optimal control problem for a continuous stirred tank reactor (CSTR) that represents a reaction of the type "A → product". The reactor dynamics is described by a nonlinear system of ordinary differential equations controlled by two inputs: the inlet concentration and the inlet temperature. We formulate the problem of max...

In this paper, we describe a broad class of control functions for extremum seeking problems. We show that it unifies and generalizes the existing extremum seeking strategies which are based on Lie bracket approximations, and allows to design new controls with favorable properties in extremum seeking and vibrational stabilization tasks. The second r...

We study a point-to-point control problem for driftless control-affine systems. The class of problems under consideration satisfies controllability conditions with the Lie brackets up to the second order. To solve the control problem, we use trigonometric polynomials whose coefficients are computed by expanding the solutions into the Volterra serie...

In this paper, a nonlinear control-affine system that describes the dynamics of a continuous crystallizer with fines trap is considered. We study the approximate steering problem for this system, i. e. the problem of constructing an admissible open-loop control that steers the system from a given initial state to a neighborhood of the target state....

In this paper, we propose a stabilization scheme for nonlinear control systems whose vector fields satisfy Hörmander's condition with the second-order Lie brackets. This scheme is based on the use of trigonometric controls with bounded frequencies. By using the Volterra series and a modification of Lyapunov's direct method, we reduce the stabilizat...

This paper is devoted to the stabilization problem for nonlinear driftless
control systems by means of a time-varying feedback control. It is assumed that
the vector fields of the system together with their first order Lie brackets
span the whole tangent space at the equilibrium. A family of trigonometric
open-loop controls is constructed to approx...

This chapter is focused on the partial stabilization problem of a rotating rigid body endowed with a number of elastic beams. To stabilize the equilibrium of this mechanical system, we apply results of Chap. 2. In addition, we prove strong (non-asymptotic) stability in the sense of Lyapunov as well as relative compactness of the trajectories for th...

In this chapter, we study a mechanical system consisting of a rotating rigid body and the Kirchhoff plate. We derive the equations of motion of this system with modal coordinates by assuming that the control is the angular acceleration of the body. The linearized control system is shown to be neither controllable nor stabilizable in general case. W...

A class of abstract dynamical systems with multivalued flows of solutions in a metric space is introduced in this chapter. For this class of systems, the property of partial asymptotic stability with respect to a continuous functional is studied. In order to characterize the limit set of a trajectory of a multivalued system, a modification of the i...

The approximate steering problem is considered in this chapter for a linear distributed parameter system with finite-dimensional control. An approach for solving this problem is proposed by using exact solutions of the steering problem for reduced systems and the spillover analysis. This approach allows also to estimate the reachable sets and to st...

This chapter presents a brief survey of basic results from the theory of \(C_0\)-semigroups of operators in a Banach space. Some facts concerning nonlinear semigroups of contractions and their infinitesimal generators are considered as well. The issue of the relative compactness of trajectories is set up for abstract differential equations with acc...

This paper is devoted to the analysis of reachable sets for quasilinear distributed parameter systems by using smooth finite dimensional controls. The class of systems under consideration includes models of a moving bed chromatographic process. We propose sufficient conditions on the states of such quasilinear systems to be approximately reachable...

We consider a dynamical system with distributed parameters that describes controlled vibrations of a Kirchhoff plate without the polar moment of inertia. A class of optimal controls corresponding to finite-dimensional approximations is used to study the reachable set. Analytic estimates for the norm of these control functions are obtained depending...

A mathematical model of a flexible-link manipulator is studied in this chapter within the framework of the Timoshenko beam theory. In contrast to the majority of publications in this area, we consider here the model with two rigid bodies and the beam with non-collocated sensors and actuators. For the mathematical model described by coupled ordinary...

The book is devoted to stability problems for dynamical systems governed by ordinary differential equations, functional differential equations, and impulse systems. New approaches for the stability study are presented by using Lyapunov functions of constant sign and higher order derivatives, as well as density functions.
The stabilization problem...

This paper is devoted to the stability analysis of nonlinear systems whose linear approximation exhibits a pair of purely imaginary eigenvalues. By using the center manifold approach and normalization procedure, we estimate the decay rate of solutions in the critical case considered. Such an estimate is applied for computing the cost of an optimal...

This paper is devoted to the stability problem of a nonlinear system in the critical case of q pairs of purely imaginary eigenvalues. The center manifold theory and the normal form method are exploited in this study. The main result of the paper is a power estimate for the solutions in the case when stability is ensured by third order forms. A Lyap...

In this paper, we consider a mechanical system consisting of a rigid body with a thin elastic plate. The plate vibration is governed by the Kirchhoff plate theory. We derive the equations of motion as a system of ordinary and partial differential equations and consider the angular acceleration as a control parameter. The dynamical equations are tra...

A linear control system in Hilbert space is considered. An estimate of solutions of the system is carried out for the class of optimal control corresponding to a subsystem with finite number of degrees of freedom. It is proved that the family of controllers considered solves the problem of approximate controllability for the infinite dimensional sy...

This paper is devoted to the controllability analysis of a class of linear control systems in a Hilbert space. It is proposed to use the minimum energy controls of a reduced lumped parameter system for solving the infinite dimensional steering problem approximately. Sufficient conditions of the approximate controllability are formulated for a modal...

The controllability property plays a crucial role in mathematical control theory. For linear systems, necessary and sufficient
conditions of controllability are given by the Kalman criterion. This criterion also allows to study the local controllability
by linear approximation [5]. There is a number of necessary as well as sufficient conditions of...

Dynamical models of flexible-link robot manipulators are generally described by a set of coupled ordinary and partial differential Dynamical models of flexible-link robot manipulators are generally described by a set of coupled ordinary and partial differential
equations, which gives rise to series of mathematical control problems in infinite dimen...

The method of oriented manifolds is developed to study geometric properties of the sets of trajectories of nonlinear differential systems with control. This method is conceptually connected with the classical methods of Lyapunov, Poincaré, and Levi-Civita and is a natural extension and development of results of the Donetsk school of mechanics. In t...

We investigate a differential equation in a Hilbert space that describes vibrations of the Euler-Bernoulli elastic beam with
feedback control. The relative compactness of positive semitrajectories of the considered equation is proved. Constructing
a Lyapunov functional in explicit form and using the invariance principle, we obtain representations o...

A system of partial differential equations describing the spatial oscillations of an Euler-Bernoulli beam with a tip mass is considered. The linear system considered is actuated by two independent controls and separated into a pair of differential equations in a Hilbert space. A feedback control ensuring strong stability of the equilibrium in the s...

A control system describing the dynamics of a rotating Timoshenko beam is considered. We assume that the beam is driven by a control torque at one of its ends, and the other end carries a rigid body as a load. The model considered takes into account the longitudinal, vertical, and shear motions of the beam. For this distributed parameter system, we...

In this paper, we consider a control system describing the motion of a flexible-link manipulator with a payload under the action of gravity. The system is actuated by a control torque at the bottom part (hub) of the manipulator. We use Galerkin approximations of an arbitrary order for the Timoshenko beam model in order to take into account any desi...

We consider the problem of partial asymptotic stability with respect to a continuous functional for a class of abstract dynamical
processes with multivalued solutions on a metric space. This class of processes includes finite-and infinite-dimensional dynamical
systems, differential inclusions, and delay equations. We prove a generalization of the B...

A study is made of a controllable mechanical system in the form of a Timoshenko beam with a weight. The system models a flexible-link
robot manipulator. A Galerkin approximation based on the solutions of the corresponding Sturm-Liouville problem is constructed
for the partial differential equations of motion. Conditions of local controllability of...

A control system describing the dynamics of a flexible-link robot manipulator is considered. Observability conditions are derived for the output signal as the raising angle and the strain at a fixed point of the manipulator. An explicit procedure for observer design is proposed based on the Barbashin-Krasovskii-LaSalle invariance principle. This re...

To describe the dynamic behavior of a fire rescue turntable ladder the flexibility of the ladder set has to be analyzed. It turned out, that it can be modelled as a flexible manipulator consisting of an arbitrary number of links governed by the Euler-Bernoulli beam equation. The first link is fixed at one end and is driven by a control torque. The...

The paper is devoted to stability and stabilization of a class of evolution equations arising from mathematical modeling of hybrid mechanical systems with flexible parts. A sufficient condition is obtained for partial strong asymptotic stability of nonlinear, infinite-dimensional dynamic systems in Banach spaces. This result is applied to deriving...

A hybrid control system describing the dynamics of a ∞exible robot manipulator is considered. For the distributed parameter part of this system, we construct a family of Galerkin approximations based on solutions of the homogeneous Timoshenko beam equation. We derive necessary and su-cient conditions for controllability of such flnite dimensional s...

A mathematical model of a wind engine unit has been proposed as a system of coupled rigid bodies. Stability conditions of a stationary motion have been obtained by means of linearization. For the case of small blades deviation and large stiffness coefficients, we derive a stability condition in terms of the angular velocity and inertial parameters....

The problem of partial asymptotic stability with respect to a continuous functional is considered for a class of abstract multivalued systems on a metric space. Such a class includes nonlinear finite and infinite dimensional dynamical systems, differential inclusions, delay equations, etc. An extension of the invariance principle for multivalued dy...

## Projects

Project (1)

The goal of this project is to develop new mathematical methods for process control and to solve the stabilization problem for a class of prototype processes in chemical engineering.