# 2002

**20 December 2002****Antoni Ras**, Department of Applied Mathematics IV, UPC**Hamiltonian formulation of distributed-parameter systems**- Dep. de Matemàtica Aplicada IV, Campus Nord UPC, edifici C3, 204a (biblioteca de Matemàtica); 10.15 h
**Abstract:**As we already know, port-controlled hamiltonian models and control by interconnection give successful techniques to deal with nonlinear finite-dimensional systems. Now, we will present an extension of such methods to distributed-parameter systems with varying boundary conditions inducing energy exchange through the boundary. The cornerstone of the new framework will be the so-called Dirac-Stokes structures.

In order to clarify further definitions, the talk will begin with an example: the interconnection of a RCL circuit (lumped-parameter system) with a controller through a transmission line (distributed-parameter system).Basic references are:

[1] van der Schaft, A. and Maschke, B.: Hamiltonian formulation of distributed parameter systems with boundary energy flow. Memorandum 1586, Fac. of Math. Sci., University of Twente, 2001.

[2] Rodrï¿½uez, H., van der Schaft, A. and Ortega, R.: On stabilization of nonlinear distributed parameter port-controlled hamiltonian systems via energy-shaping. Proc. 40th IEEE CDC, Orlando, 2001.

**22 November 2002****Sonia Martínez Díaz**, Department of Applied Mathematics IV, UPC**Design of oscillatory control systems**- Dep. de Matemàtica Aplicada IV, Campus Nord UPC, edifici C3, 204a (biblioteca de Matemàtica); 10.15 h
**Abstract:**In this talk we review a differential-geometric setting in which we study a type of control systems subject to a kind of high amplitude and high frequency oscillatory forcing. The main analysis result characterizes the averaged system by means of a Volterra-like series expansion in which iterated Lie brackets of the original input vector fields appear. This allows us to obtain some new stabilization and trajectory-tracking results for a class of underactuated mechanical systems.

**8 November 2002****Gerard Olivar**, Department of Applied Mathematics IV, UPC**EU project SICONOS: modelling, SImulation and COntrol of NOnsmooth dynamical Systems**- Dep. de Matemàtica Aplicada IV, Campus Nord UPC, edifici C3, 204a (biblioteca de Matemàtica); 10.15 h
**Abstract:**The purpose of SICONOS is to develop algorithms and software for the simulation and feedback control of dynamical systems which are nonsmooth, and more specifically so-called complementarity dynamical systems. Nonsmoothness is usually introduced into the system either by some nonsmooth control action or by the presence of nonsmooth events at macroscopic level (such as impacts or switchings). Nonsmooth models abound in many engineering systems such as sliding mode or hybrid control and rigid body mechanics such as rattle of automotive components and other mechanical freeplay, and switching circuits in power electronics.

Complementarity systems are chosen as the mathematical framework for studying nonsmooth nonlinear systems. This framework is large in terms of the range of potential applications, yet specific enough to allow for deep investigation. The research will tackle head on the fundamental issue that smooth numerical methods fail on nonsmooth systems. Algorithms need to be developed that deal with hit crossings, impacts, complementarity problems, sliding and chatter in a robust and easily applicable way.

**25 October 2002****Carles Batlle**, Department of Applied Mathematics IV, UPC**Electromechanical systems: the PCHD formulation**[slides Pdf 167 KB]**Abstract:**

**11 July 2002****Sonia Martínez Díaz**, Institute of Applied Mathematics and Fundamental Physics, CSIC**Skinner and Rusk formalism for optimal control problems**- Dep. de Matemàtica Aplicada IV, Campus Nord UPC, edifici C3, 204a (biblioteca de Matemàtica); 15.30 h
**Abstract:**In this talk we review how the Skinner and Rusk formalism can be used to geometrically formalize optimal control problems. We treat several aspects of the formalism such as the autonomous and time-dependent cases, symmetry properties, and the proposal of a transition principle for a type of optimal control problems.

**22 May 2002****Carles Batlle**, Department of Applied Mathematics IV, UPC**Interconnection and control of Hamiltonian systems**[slides PS 552 KB]- Dep. de Matemàtica Aplicada IV, Campus Nord UPC, edifici C3, 204a (biblioteca de Matemàtica); 15.45 h
**Abstract:**Control systems are presented as a plant (system to be controlled) and a controller interchanging energy through a power preserving interconnection. Stabilization is achieved by shaping the total energy and using the so called passivity property. This general idea is applied to systems modeled as port controlled Hamiltonian systems (PCHS).

### Seminar on geometric techniques in control theory

Barcelona, 12-15 February 2002

Venue: Dep. de Matemàtica Aplicada IV, Campus Nord UPC, edifici C3, aula 005

**12 February 2002**, 15.30 h**José Cariñena**, Department of Theoretical Physics, University of Zaragoza**Applications of Lie systems in control theory I**[slides PS 115 KB]**Abstract:**The two talks in this series are devoted to show the usefulness in control theory of the geometric approach to a special kind of systems of differential equations which we refer to as Lie systems. These systems have the property that they admit a superposition function allowing us to write their general solution in terms of some particular solutions and several constants determining each particular solution.

We will devote this first talk to the introduction and exposition of the geometric theory of Lie systems. The main idea consists of associating a Lie system defined on a given carrier space*M*with a right-invariant (or left-invariant) system on a Lie group*G*of transformations of the manifold*M*, such that its Lie algebra is that associated to the original system.

Based on the geometric structure of these systems, we develop two main methods to deal with them: the first is a generalization of the method proposed by Wei and Norman for linear systems, and the second is a reduction theory which enables us to reduce the problem of solving a given Lie system to an associated one in a homogeneous space*G/H*of the original Lie group, and another problem into the chosen Lie subgroup*H*.

In particular, the theory will be shown to be very appropriate for dealing with control systems on Lie groups and homogeneous spaces, which is the subject of the second talk.[1] Cariñena J.F., Grabowski J. and Marmo G.,

*Lie-Scheffers systems: a geometric approach*, (Bibliopolis, Napoli, 2000).

[2] Cariñena J.F., Grabowski J. and Ramos A., "Reduction of time-dependent systems admitting a superposition principle",*Acta Appl. Math.***66**67-87 (2001).

[3] Cariñena J.F., Marmo G. and Nasarre J., "The nonlinear superposition principle and the Wei-Norman method",*Int. J. Mod. Phys. A***13**3601-27 (1998).

[4] Cariñena J.F. and Ramos A., "Integrability of the Riccati equation from a group theoretical viewpoint",*Int. J. Mod. Phys. A***14**1935-51 (1999).

[5] Cariñena J.F. and Ramos A., "A new geometric approach to Lie systems and physical applications",*Acta Appl. Math.***70**43-69 (2002).

[6] Lie S.,*Vorlesungen über continuierliche Gruppen mit geometrischen und anderen Anwendungen*, (Teubner, Leipzig, 1893).

[7] Wei J. and Norman E., "Lie algebraic solution of linear differential equations",*J. Math. Phys.***4**575-81 (1963).

[8] Wei J. and Norman E., "On global representations of the solutions of linear differential equations as a product of exponentials",*Proc. Amer. Math. Soc.***15**327-34 (1964).

**12 February 2002**, 17.30 h**Arturo Ramos**, Department of Theoretical Physics, University of Zaragoza and Dipartimento di Matematica Pura ed Applicata, Università di Padova**Applications of Lie systems in control theory II****Abstract:**In the first talk, it has been shown the geometric structure of Lie systems. In this second talk we will take advantage of the structure and methods therein developed in order to deal with systems from the control theory literature. That will allow us to illustrate the use of the generalized Wei-Norman Method and the reduction theory on specific examples:- Brockett control systems and related ones
- Trailers and chained forms
- Kinematics of the generalized elastic problem of Euler
- Lie systems in SO(3) and in SE(3)

[1] Brockett R.W., "System theory on group manifolds and coset spaces",

*SIAM J. Control***10**265-284 (1972).

[2] Brockett R.W., "Control theory and singular Riemannian geometry", in*New Directions in Applied Mathematics*, Hilton P.J. and Young G.S. eds., (Springer-Verlag, New York, 1982).

[3] Brockett R.W. and Dai L., "Nonholonomic kinematics and the role of elliptic functions in constructive controllability", in*Nonholonomic motion planning*, Li Z.X. and Canny J.F. eds., (Kluwer, Norwell, 1993).

[4] Crouch P.E., "Spacecraft attitude control and stabilization: applications of geometric control theory to rigid body models",*IEEE Trans. Autom. Control***29**321-331 (1984).

[5] Jurdjevic V.,*Geometric Control Theory*, (Cambridge University Press, New York, 1997).

[6] Lafferriere G. and Sussmann H.J., "A differential geometric approach to motion planning", in*Nonholonomic motion planning*, Li Z.X. and Canny J.F. eds., (Kluwer, Norwell, 1993).

[7] Leonard N.E. and Krishnaprasad P.S., "Motion control of drift-free, left-invariant systems on Lie groups",*IEEE Trans. Autom. Control***40**1539-1554 (1995).

[8] Murray R.M., "Control of nonholonomic systems using chained form", in*Dynamics and control of mechanical systems, the falling cat and related problems*, Enos M.J. ed., Fields Institute Communications 1, (Amer. Math. Soc., Providence, 1993).

[9] Murray R.M. and Sastry S.S., "Steering nonholonomic systems in chained form",*Proc. IEEE Conf. Decision and Control*, pp. 1121-1126 (IEEE Publications, New York, 1991).

[10] Nijmeijer H. and van der Schaft A.J.,*Nonlinear dynamical control systems*(Springer-Verlag, New York, 1990).

[11] Sørdalen O.J., "Conversion of the kinematics of a car with*n*trailers into chained form",*Proc. IEEE Conf. Robotics and Automation*, pp. 382-387 (IEEE Publications, New York, 1993).

[12] Yang R. and Krishnaprasad P.S. and Dayawansa W., "Optimal control of a rigid body with two oscillators", in*Mechanics Day*, Fields Institute Communications 7 (Amer. Math. Soc., Providence, 1996).

**13 February 2002**, 10 h**Jesús Clemente Gallardo**, Technical University of Delft**A brief introduction to Dirac structures and port controlled Hamiltonian systems****Abstract:**The aim of this first seminar is to serve as an introduction to the topic. We will consider two aspects:- The geometrical nature of Dirac structures: Dirac structures were introduced by the end of the eighties by two different groups: Courant and Weinstein [CW:86] searching for a generalization of the usual Poisson structures and I. Dorfman [Dor:87] in the context of field theories. The main geometrical issues of the finite dimensional case were discussed later by Courant [Cou:90]. We will consider here the main geometrical properties of Dirac structures: definition, alternative formulations, relations with Poisson and presymplectic structures, Lie algebroid structure, Darboux theorem, etc.
- Port controlled Hamiltonian systems were introduced in the early nineties by van der Schaft and Maschke [MSB:92] [MS:92] [SM:95]. They have proved to be very useful from the system theoretical and control theoretical points of view, in order to describe systems defined by power preserving interconnections of different subsystems. We will present their main properties in the finite dimensional case, putting special emphasis on the examples and applications.

[CW:86] T.J. Courant and A. Weinstein. "Beyond poisson structures". Technical report, U.C.B., 1986.

[Cou:90] T.J. Courant. "Dirac manifolds".*Trans. AMS***319**631-661 (1990).

[Dor:87] I. Dorfman. "Dirac structures of integrable evolution equations".*Phys. Lett. A***125**(1987).

[MS:92] B. Maschke and A.J. van der Schaft. "Port controlled hamiltonian systems: Modelling origins and system-theoretic properties". Pp. 282-288 in*Proceedings of the Second IFAC NOLCOS*, Bordeaux, 1992.

[MSB:92] B. Maschke, A.J. van der Schaft and P. Breedveld. "An intrinsic hamiltonian formulation of network dynamics: Non standard poisson structure and gyrators".*J. Franklin Inst.***329**923-966 (1992).

[SM:95] A.J. van der Schaft and B. Maschke. "The hamiltonian formulation of energy conserving physical systems with external ports".*Arch. Elek. Übertr.***49**362-371 (1995).

**13 February 2002**, 12 h**Jesús Clemente Gallardo**, Technical University of Delft**Dirac structures: some new results and work in progress****Abstract:**After the general introduction of the previous talk we will discuss now a few recent results of the theory. We will focus in these aspects:- The relation with passive systems: the use of passivity based control for PCHS [OSME:99].
- The ``Lagrangian'' formalism for Dirac structures: a very simple Lie algebroid [MCS:01]. Applications to LC circuits.
- The Stokes-Dirac structure: a distributed parameter PCHS [SM:01]. Discretization of infinite dimensional Dirac structures.

[MCS:01] B. Maschke J. Clemente-Gallardo and A.J. van der Schaft. "Kinematical constraints and algebroids".

*Rep. Math. Phys.***47**(2001) 413-429.

[OSME:99] R. Ortega, A.J. van der Schaft, B. Maschke, and G. Escobar. "Interconnection and damping assignement passivity-based control of port controlled hamiltonian systems". Submitted, 1999.

[SM:01] A.J. van der Schaft and B. Maschke. "Hamiltonian formulation of distributed parameter system with boundary energy flow".*J. Geom. Phys***775**1-29 (2001).

**13 February 2002**, 15.30 h**Javier Yániz**, Department of Applied Mathematics IV, UPC**Exterior differential systems in control theory and robotics I****Abstract:**This talk deals with exterior differential systems. Pfaffian systems will be introduced and some normal forms (Engel, Goursat and extended Goursat) will be described in order to solve the nonholonomic motion planning problem.[1] G. Pappas, J. Lygeros, D. Tilbury and S. Sastry, "Exterior differential systems in control and robotics", pp. 271-372 in

*Essays on Mathematical Robotics*, edited by Baillieul, Sastry and Sussmann. IMA Volumes in Mathematics and its Applications, 104; Springer-Verlag, 1998. [Also available here]

**13 February 2002**, 17.30 h**Jaume Franch**, Department of Applied Mathematics IV, UPC**Exterior differential systems in control theory and robotics II**[slides PS 52 KB]**Abstract:**We discuss some of the connections between the exterior differential systems formalism, specialized to the case of control systems, and the vector field approach in nonlinear control. The most recent theoretical advances will be given and some open problems will be highlighted.[1] G. Pappas, J. Lygeros, D. Tilbury and S. Sastry, "Exterior differential systems in control and robotics", pp. 271-372 in

*Essays on Mathematical Robotics*, edited by Baillieul, Sastry and Sussmann. IMA Volumes in Mathematics and its Applications, 104; Springer-Verlag, 1998. [Also available here]

[2] M. van Nieuwstadt, M. Rathinam and R. Murray, "Differential flatness and absolute equivalence", pp. 326-332 in*Proceedings IEEE CDC 1994*.

[3] M. Rathinama and W. Sluis, "A test for differential flatness by reduction to single-input systems", pp. 257-262 in*Proceedings 13th IFAC Worl Congress, vol. E*(1996).

[4] W. Sluis and D. Tilbury, "A bound on the number of integrators needed to linearize a control system",*Systems Control Lett.***29**(1996) 43-50.

**14 February 2002**, 15.30 h**Jorge Cortés Monforte**, Systems, Signals and Control Department, University of Twente, and**Sonia Martínez Díaz**, Institute of Applied Mathematics and Fundamental Physics, CSIC**Motion planning of underactuated mechanical systems: oscillatory control and kinematic controllability**[slides PDF 458 KB]**Abstract:**In this two-part talk we study mechanical control systems under the affine connection framework. This formalism, which has experimented a strong development in recent years, takes advantage of the special geometric structure of mechanical systems to tackle a variety of problems ranging from controllability to series expansions and motion planning. Here, we present two recent contributions to address the motion planning and trajectory tracking problems for underactuated systems: the use of oscillatory controls and the introduction of the notion of kinematic controllability. Examples are presented throughout the talk to illustrate the results.[1] Francesco Bullo and Kevin M. Lynch, "Kinematic controllability for decoupled trajectory planning in underactuated mechanical systems",

*IEEE Trans. Robotics Automation***17**:4 (2001) 402-412.

[2] Andrew D. Lewis and Richard M. Murray, "Configuration controllability of simple mechanical control systems",*SIAM Review***41**(1999) 555-574.

[3] Sonia Martínez, Jorge Cortés and Francesco Bullo, "On analysis and design of oscillatory controls systems", submitted to*IEEE Trans. Automatic Control*(2001).

**14 February 2002**, 17.30 h**David Martín de Diego**, Institute of Applied Mathematics and Fundamental Physics, CSIC**Optimal control with applications to economics**[slides PDF 181 KB]**Abstract:**Optimal control theory can be considered as a mathematical theory with applications extending to all of human activities, in particular, economic activities. The purpose of this talk is to illustrate optimal control theory with some applications to economy. Special attention will be payed to optimal growth theory. This area of economics is concerned with analysing the level of economic growth which maximizes social welfare.

**15 February 2002**, 10 h**Carlos López**, Department of Applied Mathematics, University of Zaragoza**Geometric formalism in optimal control theory for ordinary differential equations****Abstract:**

**15 February 2002**, 11 h**Jesús Marín Solano**, Department of Economical and Financial Mathematics, University of Barcelona**Geometric formalism in optimal control theory for partial differential equations****Abstract:**