On the approximation of the boundary layers for the controllability problem of nonlinear singularly perturbed systems

A new systematic approach to the construction of approximate solutions to a class of nonlinear singularly perturbed feedback control systems using the boundary layer functions especially with regard to the possible occurrence of the boundary layers is proposed. For example, problems with feedback control, such as the steady-states of the thermostats, where the controllers add or remove heat, depending upon the temperature registered in another place of the heated bar, can be interpreted with a second-order ordinary differential equation subject to a nonlocal three--point boundary condition. The $O(\epsilon)$ accurate approximation of behavior of these nonlinear systems in terms of the exponentially small boundary layer functions is given. At the end of this paper, we formulate the unsolved controllability problem for nonlinear systems.Comment: 13 pages with 1 figure; The final publication is available at Elsevier via doi.org/10.1016/j.sysconle.2012.01.00

Stabilization and state trajectory tracking problem for nonlinear control systems in the presence of disturbances

In this paper we consider the problem of stabilization and tracking of desired state trajectory for a wide range of nonlinear control problems with disturbances. We present the sufficient conditions for the existence of $C^k$ state feedback controllers and the process of their mathematical designing is described.Comment: 12 pages; This is an Author's Original Manuscript (AOM) of an article published by Taylor & Francis in International Journal of Control on 14 August 2017, available online: http://www.tandfonline.com/10.1080/00207179.2017.136104

Feedback stabilization of double pendulum: Application to the crane systems with time-varying rope length

In the present paper we focus our attention on the design of the feedback-based feed-forward controller asymptotically stabilizing the double-pendulum-type crane system with the time-varying rope length in the desired end position of payload (the origin of the coordinate system). In principle, we will consider two cases, in the first case, the sway angle of payload is uncontrolled and the second case, when the sway angle of payload is controlled by an external force. Mathematical modelling in the framework of Lagrange formalism and numerical simulation in the Matlab environment indicate the substantial reduction of the transportation time to the desired end position. Another principal novelty of this paper lies in deriving and analysis of a complete mathematical model without approximating the nonlinear terms and without neglecting some structural parameters of systems for the reasons described in the Remark 4.2 and Remark 5.1

Design of the state feedback-based feed-forward controller asymptotically stabilizing the overhead crane at the desired end position

The problem of feed-forward control of overhead crane system is discussed. By combining the Kalman's controllability theory and Hartman-Grobman theorem from dynamical system theory, a linear, continuous state feedback-based feed-forward controller that stabilizes the crane system at the desired end position of payload is designed. The efficacy of proposed controller is demonstrated by comparing the simulation experiment results for overhead crane with/without time-varying length of hoisting rope.Comment: 12 pages,5 figures. arXiv admin note: text overlap with arXiv:1805.0388

Singularly perturbed linear Neumann problem with the characteristic roots on the imaginary axis. A non-resonant case

In this note we are dealing with the problem of existence and asymptotic behavior of solutions for the non-resonant singularly perturbed linear Neumann boundary value problem \begin{eqnarray*} \epsilon y"+ky=f(t),\quad k>0,\quad 0<\epsilon<<1,\quad t\in\langle a,b\rangle \end{eqnarray*} \begin{equation*} y'(a)=0,\quad y'(b)=0. \end{equation*} Our approach is based on the analysis of an integral equation equivalent to this problem.Comment: 6 pages; So far unpublished manuscript was originally written in 201

Local null controllability of the control-affine nonlinear systems with time-varying disturbances. Direct calculation of the null controllable region

The problem of local null controllability for the control-affine nonlinear systems $\dot x(t)=f(x(t))+Bu(t)+w(t),$ $t\in[0,T]$ is considered in this paper. The principal requirements on the system are that the LTI pair $\left((\partial f/\partial x)(0), B\right)$ is controllable and the disturbance is limited by the constraint $|f(0)+w(t)|\leq M_d\left(1-\frac{t}{T}\right)^\eta,$ $M_d\geq0$ and $\eta>0.$ These properties together with one technical assumption yield a complete answer to the problem of deciding when the null controllable region have a nonempty interior. The criteria obtained involve purely algebraic manipulations of vector field $f,$ input matrix $B$ and bound on the disturbance $w(t).$ To prove the main result we have derived a new Gronwall-type inequality allowing the fine estimates of the closed-loop solutions. The theory is illustrated and the efficacy of proposed controller is demonstrated by the examples where the null controllable region is explicitly calculated. Finally we established the sufficient conditions to be the system under consideration (with $w(t)\equiv 0$) globally null controllable.Comment: 13 pages, 2 figures; We value your feedback and welcome any comments and suggestions you may have to help improve our manuscrip

Frequency control of singularly perturbed forced Duffing's oscillator

We analyze the dynamics of the forced singularly perturbed differential equation of Duffing's type. We explain the appearance of the large frequency nonlinear oscillations of the solutions. It is shown that the frequency can be controlled by a small parameter at the highest derivative. We give some generalizations of results obtained recently by B.S. Wu, W.P. Sun and C.W. Lim, Analytical approximations to the double-well Duffing oscillator in large amplitude oscillations, Journal of Sound and Vibration, Volume 307, Issues 3-5, (2007), pp. 953-960. The new method for an analysis of the nonlinear oscillations which is based on the dynamic change of coordinates is proposed.Comment: 12 pages with 7 figures; The final publication is available at Springer via doi.org/10.1007/s10883-011-9125-

Criterion for robustness of global asymptotic stability to perturbations of linear time-varying systems

In this brief note, we establish a novel criterion for robustness of global asymptotic stability of zero solution of LTV system $\dot x=A(t)x$ in the presence of possibly unbounded perturbations (external disturbances). To prove the result, logarithmic norm will be used under which the stability becomes a topological notion depending on the chosen vector norm in the state-space $\mathbb{R}^n.$Comment: Some inaccuracies of previous version were fixed and the title is change

A note on uniform exponential stability of linear periodic time-varying systems

In this paper we derive new criterion for uniform stability assessment of the linear periodic time-varying systems $\dot x=A(t)x,$ $A(t+T)=A(t).$ As a corollary, the lower and upper bounds for the Floquet characteristic exponents are established. The approach is based on the use of logarithmic norm of the system matrix $A(t).$ Finally we analyze the robustness of the stability property under external disturbance.Comment: 6 pages, 3 figure

Robust adaptive state-feedback control of linear time-varying systems under both potentially unbounded system's modeling uncertainty and external disturbance

The conceptually new approach based on the logarithmic norm to design of robust adaptive state-feedback controller for linear time-varying (LTV) systems under system's modeling uncertainty and nonlinear external disturbance is proposed. This controller, consisting of two independent parts - adaptive and robust ones - globally asymptotically stabilizes every LTV system regardless how large the disturbance is.Comment: 15 pages, 2 figure