# Project

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## Iterative Subspace Decomposition for Lyapunov-Based Controller Design and Stability Analysis

 Applicant Müllhaupt Philippe 119816 Project funding (Div. I-III) Laboratoire d'automatique EPFL - STI - IGM - LA1 EPF Lausanne - EPFL Mathematics 01.04.2008 - 30.09.2009 69'488.00
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### All Disciplines (2)

Discipline
 Mathematics
 Mechanical Engineering

### Keywords (7)

Nonlinear Control; Lyapunov Method; Stability; Symbolic Computation; Lyapunov Stability; Formal Calculus; Polynomial Systems

### Lay Summary (English)

Lay summary
It is proposed to address the Lyapunov function construction from the perspective of iteratively selecting, deforming, and transforming equivalence sets of level surfaces of sub-Lyapunov functions.
Stated differently, we proceed from the simple to the more complex, starting with a small-sized state space (say 1 dimensional) and gradually increasing the synthesis until the full-sized original system is attained.
Iteratively reducing the state space is not a new idea, however, and it appears at the core of many numerical methods applied to linear systems, such as for either pole placement (Miminis-Paige algorithm1 and the Nichols-Van Dooren), or finding linear time invariant Lyapunov functions (using the Hessenberg-Shur).
In the nonlinear setting this idea of reducing the system to a staircase form and then constructing the Lyapunov function from this representation is not commonplace.
There is nevertheless a formal calculus approach based on algebraic geometry that, to a certain extent, goes along this line of thought. The idea is cast in the polynomial multivariable setting. The reason for this is that for linear systems, only quadratic polynomial forms need to be considered.
Therefore, restricting the class of systems to those for which multivariate polynomials are used (as building blocks for the Lyapunov function construction) is a good start. This handles a relatively large class of nonlinear systems. Moreover, it is well adapted through the use of algebraic ideal basis construction such as Gröbner basis. The link to our proposal comes from the fact that Gröbner basis construction heavily relies on Dikson’s Lemma. This lemma is used for testing that a particular monomial belongs to a monomial ideal. The test is a staircase comparison between the particular monomial and those defining the ideal, i.e. if it lies below the staircase it does not belong to the ideal. Nevertheless, there is yet no clear way on how to use such tools so as to address the level set modification required. Especially, the main difficulty, as opposed to classical Gröbner basis usage for systems of polynomial equations is that, in this case, we deal with nonlinear polynomial inequalities, instead of equalities.
 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

Name Institute

### Associated projects

Number Title Start Funding scheme
 117753 Assessment of European emissions of non-CO2 greenhouse gases by a combination of continuous measurements, transport models and RN-222 emission maps 01.06.2008 Project funding (Div. I-III)
 126916 Iterative Methods for Nonlinear Control 01.10.2009 Project funding (Div. I-III)
 68059 Poursuite pour la stabilisation de systèmes non linéaires 01.03.2004 Project funding (Div. I-III)

### Abstract

In this project, we propose to address the Lyapunov function construction from the perspective of iteratively selecting, deforming, and transforming equivalence sets of level surfaces of sub-Lyapunov functions. Stated differently, we proceed from the simple to the more complex, starting with a small-sized state space (say 1 dimensional) and gradually increasing the synthesis until the full-sized original system is attained. Iteratively reducing the state space is not a new idea, however, and it appears at the core of many numerical methods applied to linear systems, such as for either pole placement (Miminis-Paige algorithm (a Hessenberg reduction) and the Nichols-Van Dooren algorithm (based on canonical matrix-pencil decomposition leading to a staircase-like system), or finding linear time invariant Lyapunov functions (using the Hessenberg-Shur decomposition, again leading to a staircase-like representation). In the nonlinear setting this idea of reducing the system to a staircase form and then constructing the Lyapunov function from this representation is not commonplace. There is nevertheless a formal calculus approach based on algebraic geometry that, to a certain extent, goes along this line of thought. The idea is cast in the polynomial multivariable setting. The reason for this is that for linear systems, only quadratic polynomial forms need to be considered. Therefore, restricting the class of systems to those for which multivariatepolynomials are used (as building blocks for the Lyapunov function construction) is a good start. This handles a relatively large class of nonlinear systems. Moreover, it is well adapted through the use of algebraic ideal basis construction such as Groebner basis. The link to our proposal comes from the fact that Gr\"obner basis construction heavily relies on Dikson's Lemma. This lemma is used for testing that a particular monomial belongs to a monomial ideal. The test is a staircase comparison between the particular monomial and those defining the ideal, i.e. if it lies below the staircase it does not belong to the ideal. Nevertheless, there is yet no clear way on how to use such tools so as to address the level set modification required. Especially, the main difficulty, as opposed to classical Groebner basis usage for systems of polynomial equations is that, in this case, we deal with nonlinear polynomial inequalities, instead of equalities. The goal of the current research project is to reduce the gap between necessary and sufficient conditions for establishing stability using the Lyapunov method, that is, finding a positive function V which decreases along a trajectory for both stability assessment and control Lyapunov methods (i.e. also designing a nonlinear feedback control law). The method should rely on iterative reduction-elimination techniques that use a high degree of information about the structure of the original problem at hand (contrary to most existing techniques that simply convert the problem into an optimization one). However, we will concentrate on small to medium-sized problems.
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