
Nonlinear System Theory
Credits 
6 credit points 
Instructors 
Jeltsema, D. (Technische Universiteit Delft), Schaft, A.J. van der (Rijksuniversiteit Groningen), Scherpen, J.M.A. (Rijksuniversiteit Groningen) 
Email 
d.jeltsema@tudelft.nl, a.j.van.der.schaft@rug.nl, j.m.a.scherpen@rug.nl 
Prerequisites 
The course is aimed at students at the comprehensive as well as the technical universities. Solid knowledge of linear algebra, calculus (in particular, the inverse and implicit function theorem) and ordinary differential equations, all at the bachelor level, is essential. Furthermore, some knowledge of linear control systems (as e.g. provided in the course Systems & Control) is beneficial, although not necessary. 
Aim 
The course aims at students in mathematics (or having a strong background in mathematics) with an interest in the analysis and design of nonlinear dynamics. The purpose of the course is to introduce the students to basic concepts and results in the theory of nonlinear control systems and interconnected nonlinear dynamical systems. The course differs from, and is complementary to, more traditional courses on nonlinear dynamical systems, by emphasizing i) the possibility of influencing the nonlinear dynamics by the addition of suitable feedback loops and controller systems, and ii) analyzing complex, largescale, nonlinear systems by decomposing the system in interconnected subsystems. The availability of control provides fascinating possibilities to influence and regulate the nonlinear dynamics. Furthermore, the passivity, smallgain, and portHamiltonian framework yield powerful tools for inferring stability properties of the interconnected system. 
Description 
In general, mathematical systems and control theory is concerned with the analysis and control of dynamical systems in interaction with their environment. In contrast with the Mastermath course Systems & Control, which is mainly devoted to linear control systems, the present course deals with nonlinear control systems, modeled by nonlinear differential equations depending on a set of input variables, and additional nonlinear output mappings, relating the state variables to the output variables. As such, the course combines tools from nonlinear dynamical systems with fundamental concepts of systems and control theory. The first fundamental topics to be treated in the course are the controllability and observability properties of nonlinear control systems. A system is state controllable if for any pair of initial and final states there exists an input (as function of time) that steers the system from the initial to the final state. The key ingredients to analyze controllability are the Lie brackets of the vector fields of the nonlinear control system. Observability, which is the property to uniquely determine the state based on a timerecord of the input and output variables, can be analyzed by considering the (repeated) Lie derivatives of the output mapping with respect to the system vector fields. Next we will discuss the problem of transforming a nonlinear control system by feedback transformations and the choice of state space coordinates into a linear control system. It turns out that for controllable systems an elegant ‘if and only if’ condition can be given, formulated in terms of involutivity of certain Lie bracket expressions of the system vector fields. Applications to control problems such as tracking of desired output trajectories will be given. Another main topic concerns the extension of Lyapunov stability theory to systems with inputs and outputs, by the introduction of the concept of dissipative systems. The two main examples are passive systems and nonlinear control systems having finite L2induced norm (from the inputs to the outputs). We will show under which conditions the interconnection of such systems are (asymptotically) stable, i.e., the celebrated passivity and smallgain theorems. This has important implications towards the stability analysis of largescale physical systems, as well as to the robustness of stability with respect to unmodeled dynamics. As a further structured case of passive systems we will deal with the theory of portHamiltonian systems. Portbased network modeling of physical systems directly leads to portHamiltonian descriptions, and we analyze their main properties, such as energybalance and the existence of additional conserved quantities, and its implications for control. All topics will be illustrated by examples from various application domains, in particular actuated mechanical systems (robotics, animal locomotion), electromechanical systems (nonlinear circuits, power converters, mechatronics), and biological systems. 
Organization 
This is a regular course, with course meetings in weeks 814, 16, 17 (9 course meetings in total), with Written Exam on Monday May 14. 
Examination 
There will be three Homework Assignments (with average grade HA). At the end of the course there will be a Written Exam (with grade WE). Final grade will be the average of HA and WE. 
Literature 
Lecture Notes will be provided. Background material (optional reading): H. Nijmeijer, A.J. van der Schaft, Nonlinear Dynamical Control Systems, Springer, 1990. A.J. van der Schaft, L2Gain and Passivity Techniques in Nonlinear Control, Springer 2000 S. S. Sastry, Nonlinear Systems, IAM, Springer, 1999. 
