The motion planning problem and exponential stabilisation of a heavy chain. Part I

A model of a heavy chain system with a punctual load (tip mass) in the form of a system of partial differential equations is interpreted as an abstract semigroup system on a Hilbert state space. Our aim is to solve the output motion planning problem of the same nature as in the case of an unloaded heavy chain (Grabowski, P. (2003), ‘Abstract Semigroup Model of Heavy Chain System with Application to a Motion Planning Problem’, in Proceedings of 9th IEEE International Conference: Methods and Models in Automation and Robotics, 25–28 August, Międzyzdroje, Poland, pp. 77–86 (IS1-2-3.PDF)). In order to solve this problem we first analyse its well-posedness and some basic properties. Next, we solve the output motion planning problem using a substitute of the inverse of the input–output operator represented in terms of the Laplace transforms. A problem of exponential stabilisation is also formulated and solved using a stabiliser of the colocated type. The exponential stabilisation is proved using the method of Lyapunov functionals combined with some frequency-domain tools. The method of Lyapunov functionals can be replaced by the spectral or exact controllability approach as shown in the second part (Grabowski, P. (2008), ‘The Motion Planning Problem and Exponential Stabilisation of a Heavy Chain. Part II’, Opuscula Mathematica, 28 (2008) (Special issue dedicated to the memory of Professor Andrzej Lasota), 481–505) of the present article. A laboratory setup which allows verification of the results in practice is described in detail. Its dynamical model is used as an example to illustrate the theoretical results. †Dedicated to Frank M. Callier on the occasion of his 65th birthday.

[1]  H. Logemann,et al.  The Circle Criterion and Input-to-State Stability , 2011, IEEE Control Systems.

[2]  Pierre Rouchon,et al.  Flatness of Heavy Chain Systems , 2001, SIAM J. Control. Optim..

[3]  L. A. Li︠u︡sternik,et al.  Elements of Functional Analysis , 1962 .

[4]  Milton Abramowitz,et al.  Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , 1964 .

[5]  Bao-Zhu Guo,et al.  Riesz Basis Property of Evolution Equations in Hilbert Spaces and Application to a Coupled String Equation , 2003, SIAM J. Control. Optim..

[6]  R. Dingle.  Asymptotic expansions : their derivation and interpretation , 1975 .

[7]  A. Kugi,et al.  Application of a combined flatness- and passivity-based control concept to a crane with heavy chains and payload , 2006, 2006 IEEE Conference on Computer Aided Control System Design, 2006 IEEE International Conference on Control Applications, 2006 IEEE International Symposium on Intelligent Control.

[8]  Piotr Grabowski,et al.  Stability of a heat exchanger feedback control system using the circle criterion , 2007, Int. J. Control.

[9]  P. Grabowski.  On the Spectral-Lyapunov Approach to Parametric Optimization of Distributed-Parameter Systems , 1990 .

[10]  Brigitte d'Andréa-Novel,et al.  Control of an overhead crane: Stabilization of flexibilities , 1992 .

[11]  Piotr Grabowski,et al.  CORRIGENDUM TO: "ON THE CIRCLE CRITERION FOR BOUNDARY CONTROL SYSTEMS IN FACTOR FORM: LYAPUNOV STABILITY AND LUR'E EQUATIONS" , 2009 .

[12]  R. Young,et al.  An introduction to nonharmonic Fourier series , 1980 .

[13]  Yu. I. Lyubich,et al.  Asymptotic stability of linear differential equations in Banach spaces , 1988 .

[14]  Brigitte d'Andréa-Novel,et al.  Exponential stabilization of an overhead crane with flexible cable via a back-stepping approach , 2000, Autom..

[15]  Piotr Grabowski,et al.  Duality of observation and control using factorizations , 1998 .

[16]  A. Erdélyi,et al.  Tables of integral transforms , 1955 .

[17]  V. C. L. Hutson,et al.  Applications of Functional Analysis and Operator Theory , 1980 .

[18]  Andreas Kugi,et al.  Infinit-dimensionale Regelung eines Brückenkranes mit schweren Ketten (Infinite-dimensional Control of a Gantry Crane with Heavy Chains) , 2005, Autom..

[19]  D. F. Hays,et al.  Table of Integrals, Series, and Products , 1966 .

[20]  Ruth F. Curtain,et al.  Exponential stabilization of well-posed systems by colocated feedback , 2006, SIAM J. Control. Optim..

[21]  R. Triggiani,et al.  Lack of uniform stabilization for noncontractive semigroups under compact perturbation , 1989 .

[22]  Mark R. Opmeer,et al.  Distribution semigroups and control systems , 2006 .

[23]  J. Desanto.  Mathematical and numerical aspects of wave propagation , 1998 .

[24]  J. Ball.  Strongly continuous semigroups, weak solutions, and the variation of constants formula , 1977 .

[25]  B. Rao.  Decay estimates of solutions for a hybrid system of flexible structures , 1993, European Journal of Applied Mathematics.

[26]  Eduard Feireisl,et al.  Stabilization of a hybrid system with a nonlinear nonmonotone feedback , 1999 .

[27]  Ruth F. Curtain,et al.  Stability Results of Popov‐Type for Infinite‐Dimensional Systems with Applications to Integral Control , 2003 .

[28]  A. Ingham.  Some trigonometrical inequalities with applications to the theory of series , 1936 .

[29]  Piotr Grabowski,et al.  On the circle criterion for boundary control systems in factor form , 2000 .

[30]  R. Datko.  Extending a theorem of A. M. Liapunov to Hilbert space , 1970 .

[31]  Brigitte d'Andréa-Novel,et al.  Feedback stabilization of a hybrid PDE-ODE system: Application to an overhead crane , 1994, Math. Control. Signals Syst..

[32]  A. Haraux.  Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps , 1989 .

[33]  R. Triggiani,et al.  L2(Σ)-regularity of the boundary to boundary operator B∗L for hyperbolic and Petrowski PDEs , 2003 .

[34]  Seppo Hassi,et al.  A general realization theorem for matrix-valued Herglotz–Nevanlinna functions , 2005 .

[35]  F. Gesztesy,et al.  On Matrix–Valued Herglotz Functions , 1997, funct-an/9712004.

[36]  A. G. Greenhill,et al.  Handbook of Mathematical Functions with Formulas, Graphs, , 1971 .

[37]  A. V. Balakrishnan.  Superstability of systems , 2005, Appl. Math. Comput..

[38]  Pierre Rouchon,et al.  Motion planning for heavy chain systems , 2001 .

[39]  Francis Conrad,et al.  Strong stability of a model of an overhead crane , 1998 .

[40]  Keum-Shik Hong,et al.  Anti-sway control of container cranes as a flexible cable system , 2004, Proceedings of the 2004 IEEE International Conference on Control Applications, 2004..

[41]  Amnon Pazy,et al.  Semigroups of Linear Operators and Applications to Partial Differential Equations , 1992, Applied Mathematical Sciences.

[42]  Wolfgang Arendt,et al.  Tauberian theorems and stability of one-parameter semigroups , 1988 .