# Project

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## Iterative Methods for Nonlinear Control

 English title Iterative Methods for Nonlinear Control Müllhaupt Philippe 126916 Project funding Laboratoire d'automatique EPFL - STI - IGM - LA1 EPF Lausanne - EPFL Mechanical Engineering 01.10.2009 - 31.03.2012 251'307.00
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### All Disciplines (2)

Discipline
 Mechanical Engineering
 Mathematics

### Keywords (13)

Nonlinear Control; Lyapunov Method; Stability; Symbolic Computation; Feedback Linearization; Differential Geometry; Polynomial Algebra; Lyapunov Stability; Differential Flatness; Dynamic Feedback Linearization; Polynomial Systems; Groebner Bases; Quotient Manifolds

### Lay Summary (English)

Lead
Lay summary
The aim of this project is to investigate two different yet complementary iterative methods of decomposing the original nonlinear system into smaller simpler parts from which a control design is performed. This control design proceeds then from this decomposition either directly, or by proceeding by another iterative process starting from the transformed system back to the original system. The project splits into two subparts.In Subpart A), iterative methods are used to determine successive approximations of the original system and to a suitable set of Lyapunov functions. Power series generated from the data of the polynomial system are resorted to, together with iterative algorithms for determining their coefficients. When truncated, a power series becomes a polynomial and the classical tools of computer algebra can be used. As for the Lyapunov construction, both sum-of-squares decomposition-like and Groebner bases based techniques are investigated. A somewhat more specific class of systems, namely differentially flat system and approximately differentially flat systems are investigated using this polynomial paradigm. If a system is flat, it can be put in correspondence with a linear system. From this linear system, one can either directly determine a stabilizing control law, or a Control Lyapunov Function (CLF) for the original system. From this CLF other control laws can be found with sometimes better qualities compared with the direct approach.Subpart B) A nonlinear control design scheme using quotient-manifold methods is investigated. The design scheme keeps as much knowledge as necessary on both the initial manifold and the vectorfields and does not require them to be polynomial. Taking a quotient along an input vector field gives the directions that are not directly affected by the inputs. This allows reducing the dynamical system step by step through the generalization of the classical orthogonal projection used in linear algebra. The challenge lies in the manner in which both the reduction should be undertaken and the partial quotients should be used so as to design a stabilizing feedback.
 Direct link to Lay Summary Last update: 21.02.2013

### Responsible applicant and co-applicants

Name Institute
 Müllhaupt Philippe Laboratoire d'automatique EPFL - STI - IGM - LA1
 Longchamp Roland Laboratoire d'automatique EPFL - STI - IGM - LA1
 Bonvin Dominique Laboratoire d'automatique EPFL - STI - IGM - LA1

### Employees

Name Institute
 Sudarsandhari Shibani Willson Department of Electrical & Electronic Engineering Imperial College of London

### Publications

Publication
Graf B. and Mullhaupt Ph. (2011), Feedback Linearizability and Flatness in Restricted Control Systems, in Proceedings of the 18th IFAC World Congress, Milano.
Ingram D. Willson S.S. Mullhaupt Ph. and Bonvin D. (2011), Stabilization of an experimental cart-pendulum system through approximate manifold decomposition, in Proceedings of the 18th IFAC World Congress, Milano.
(2011), A quotient method for designing nonlinear controllers, in Proceedings of the 50th IEEE CDC Conference, Orlando.
(2010), Modeling and flatness of rigid and flexible cable suspended underactuated robots, in IEEE Int. Conf on Control Applications, Yokohama.
(2009), Quotient method for controlling the acrobot, in Proceedings of the IEEE Conference on Decision and Control, 1770-1775.

### Associated projects

Number Title Start Funding scheme
 117573 Quotient Methods for Approximate Linearization of Nonlinear Dynamical 01.10.2007 Project funding
 119816 Iterative Subspace Decomposition for Lyapunov-Based Controller Design and Stability Analysis 01.04.2008 Project funding

### Abstract

The general problem of automatic control can roughly be described as trying to assign the behavior of a dynamical system. Typically, a physical system is described mathematically by a set of ordinary differential equations (ODE) involving variables related to the state of the system and some other free variables, called the control inputs, which one can use to influence the time-evolution in time of the state variables. The goal is to find a controller that stabilizes the original system and achieves good transient behavior of the transformed system.The aim of this project is to investigate two different yet complementary iterative methods of decomposing the original nonlinear system into smaller simpler parts from which a control design is performed. This control design proceeds then from this decomposition either directly, or by proceeding by another iterative process starting from the transformed system back to the original system. The project splits into two subparts, the first part A) addresses polynomial systems and Lyapunov functions. It uses iterative techniques for determining power series representing successive approximation to either the system or the Lyapunov function. Subpart B) addresses systems on manifolds and iterative techniques are resorted to by constructing quotient submanifolds. It handles the control design for both feedback linearizable and non feedback linearizable systems (i.e. approximate feedback linearization).Subpart A)For Subpart A), Lyapunov functions are one of the most general tools to study stability or stabilisation, and the iterative methods are used to determine successive approximations of the original system. Here, power series generated from the data of the polynomial system are resorted to, together with iterative algorithms for determining their coefficients. When truncated, a power series becomes a polynomial and the classical tools of computer algebra can be used. As for the Lyapunov construction,both sum-of-squares decomposition-like and Groebner bases based techniques are investigated. It has also been decided, in a first attempt, to concentrate on a somewhat more specific class of systems, namely differentially flat system and approximately differentially flat systems. To summarize shortly, if a system is flat, it can be put into correspondance with a linear system. From this linear system, one can either directly determine a stabilizing control law, or a Control Lyapunov Function (CLF) for the original system. From this CLF other control laws can be found with sometimes more or less interesting qualities compared to the direct approach.Subpart B)A nonlinear control design scheme using quotient-manifold methods should be undertaken. The system is not restricted to be polynomial and the approximation is not taken through the truncation of power series. Its purpose is to keep as much knowledge that the initial manifold and vectorfields provide and proceed with the approximation while taking advantage of the manifold structure. Since taking a quotient along an input vector field gives the directions that are not directly affected by the inputs, the manifold spanned by these directions is not affected directly by the inputs. This allows reducing the dynamical system step by step through the generalization of the classical orthogonal projection used in linear algebra. The purpose of this projection is to decompose the system into a chain of single-dimensional systems. The central algorithm for single-input systems has been successfully developed. It should however be generalized to the multi-input case which necessitates a dynamical extension, the size of which is until now still unknown. We strongly believe that quotients methods can shed some light on this difficult topic. Moreover, we still need some efficient numerical scheme for both the single-input case and the multi-input case. For single-input systems, the algorithm found has two steps. The first step is an iterative forward sweep that reduces the state space successfully, one by one, using quotient methods. Then the second step consists in building intermediate control laws that stabilizes the system starting from the reduced system back to the orginial full-fledged initial system. The rigorous justification uses suitable Lyapunov functions constructed iteratively. This resembles the backstepping approach. However, the proposed method for control design has additional benefit, in particular it avoids dragging the unnecessary terms that appear in the backstepping design.
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