Syllabus

Nonlinear Systems and Control 2015/2016
Course syllabus
First Part: Nonlinear systems analysis
1. General system theory
1.1. Definition of abstract dynamical system
1.1.1.
1.1.2.
1.1.3.
1.1.4.
Discrete time and continuous time
Time-invariance
Linearity
Uniformity and connection
1.2. Definition of dynamical system with state variables
1.3. Definition of oriented dynamical system
1.3.1. Parametrization of an oriented dynamical system
1.3.2. Causal oriented dynamical system
1.4. Definition of state-space representation (X,ϕ,η)
1.4.1. Existence and uniqueness of a state-space representation of
a given system
1.4.2. Equivalence
1.5. Elements of classification of systems and state-space representation
1.5.1.
1.5.2.
1.5.3.
1.5.4.
Discrete time and continuous time
Finite dimensional
Explicit and implicit
Stationarity
1.6. Linear state-space representation and linear systems
1.6.1. Finite dimension
1.6.2. Stationarity
1.6.3. Φ,H,Ψ,W and properties
1
1.6.4. Discrete time and continuous time
1.7. Explicit and implicit representation - Realization problem
Reference text:
• S.Monaco, ’Notes for the course 2014/2015’, file: ’lessons’abstract’,’ns&c.2’,’ns&c.3’,’ns&c.4’,’ns&c.5’
• S.Monaco, M.D. Normand-Cyrot, ’Teoria dei Sistemi II - Appunti delle lezioni 2004-2005’ (Italian only), Chapter 1, Chapter 2
(until 2.6) (http://www.diag.uniroma1.it/~monaco/PDF_NEW/
Dispense/TdS.II.pdf)
2. Nonlinear systems representations
2.1. Nonlinear differential (o difference) representation of finite dimension system
2.1.1. Classification with respect to the generating and output functions
2.2. From implicit to explicit representation
2.2.1. Volterra kernels and properties
2.2.2. Computability/finiteness of the series
2.2.3. Truncation/approximation of Volterra series
2.3. Nonlinear realization problem
2.3.1.
2.3.2.
2.3.3.
2.3.4.
Definition
Properties versus realization
Propreties at equilibrium
Bilinear realizations of a state affine system
Reference text:
• S.Monaco, ’Notes for the course 2014/2015’, file: ’ns&c.an.2.1’,’ns&c.an.2.2’,’ns&c.an.2.3’
• S.Monaco, M.D. Normand-Cyrot, ’Teoria dei Sistemi II - Appunti
delle lezioni 2004-2005’ (Italian only), Chapter 5 (http://www.
diag.uniroma1.it/~monaco/PDF_NEW/Dispense/TdS.II.pdf)
3. Nonlinear systems: geometric approach
3.1. Linear systems - recalls
3.1.1. Invariant subspaces and decomposition (geometry and structure of the generating functions and A,B,C)
3.1.2. The corresponding geometry of the evolutions in the state
space (properties of internal behaviours)
2
3.1.3. Some properties associated to such a structure (reachability,
observability and decoupling)
3.1.4. How the control can be used to modify the structure
3.2. Nonlinear state affine systems
3.2.1. Elements of differentiable geometry: vector fields, Lie derivative, Lie brackett, distributions
3.2.2. Integrability of vector fields and distributions (Frobenius theorem, necessity and part of sufficiency)
3.2.3. Invariance of a distribution and local decomposition of vector
fields
3.2.4. The corresponding geometry of evolution in state space (properties of internal behaviours)
3.2.5. Some properties associated to such a structure (reachability,
observability and decoupling)
3.2.6. How the control can be used to modify the structure
Reference texts:
• A.Isidori, ’Nonlinear Control Systems’, Springer, 2000, Chapter
1
• S.Monaco, M.D. Normand-Cyrot, ’Teoria dei Sistemi II - Appunti delle lezioni 2004-2005’ (Italian only), Chapter 6 (until 6.1)
(http://www.diag.uniroma1.it/~monaco/PDF_NEW/Dispense/
TdS.II.pdf)
4. Stability theory for linear and nonlinear systems
4.1. Definitions and generality (different stability problem, stability
and equilibria)
4.2. Conditions, peculiarities and criteria for linear time invariant
systems
4.3. Lyapunov theorem (local and global versions)
4.4. Related results (LaSalle, Cetaev, unstability)
4.5. Stability for linear time invariant systems (criteria, robusteness)
4.6. About the construction of Lyapunov function (the variable gradient method), the indirect Lyapunov method (proof)
4.7. Central Manifold theory
Reference texts:
• S.Monaco, ’Notes for the course 2014/2015’, file: ’Notes4.3’,’Notes4.4’
3
• K.H.Khalil, ’Nonlinear Systems’, Prentice Hall, 1996, Chapter 3,
paragraphs 3.1, 3.2, 3.3; Chapter 4, paragraph 4.1 http://wp.
kntu.ac.ir/nobari/Nonlinear/Nonlinear_Systems_2nd-%20Hassan_
K.%20Khalil.pdf
• S.Wiggins, ’Introduction To Applied Nonlinear Dynamical Systems And Chaos’, Springer, 2003, Chapter 18, paragraph 18.1
http://medicinaycomplejidad.org/pdf/soporte/wiggins.pdf
5. Local Behaviours
5.1. Normal forms of vector fields
5.2. Classification of bifurcations
Reference text:
5.1. S. Monaco, ’Notes for the course 2014/2015’, file: ’Notes4.5’
5.2. S. Wiggins, ’Introduction To Applied Nonlinear Dynamical Systems And Chaos’, Springer, 2003, Chapter 19, 20 http://medicinaycomplejidad.
org/pdf/soporte/wiggins.pdf
Exercises:
• S.Monaco, ’Collect of questions’, to be emailed.
• Collection of past exams ’Teoria dei Sistemi II’ avaible at http://www.
mediafire.com/download/4su8c7pypy2vyyo/NCS_OLD_EXAMS.rar (only Italian)
• Partial examination TBA
Second Part: Nonlinear control
• MIMO Linear systems
1. Eigenvalues assignment by state feedback (assignment, collocation and stabilization procedures) and Brunowsy canonical form
2. Observer problem
3. Separation principle (dynamic eigenvalues assignment)
4. The zeros of a control system
5. Disturbance Decoupling by state feedback (with or without disturbance measurement, links to geometry, stability)
6. Input-Output Decoupling (with or without stability)
Reference texts:
– S.Monaco, ’Teoria dei Sistemi - Appunti della lezione 2000-2001’
(Italian only), Chapter 9
– S.Monaco, ’Notes on linear MIMO control systems’, Notes for the
course 2014/2015
4
• Elementary theory of nonlinear Single-Input Single-Output
systems
1. Relative degree and local coordinates transformation (existence
of coordinates transformation, normal form)
2. Input-Output linearization (properties, if r = n, u − x − y full
linearization)
3. Input-State linearization (existence of ϕ, necessary and sufficient
condition, geometric conditions)
4. The concept of Zero Dynamics (y = 0, y = yR )
5. Local asymptotic stabilization (by state feedback, local tracking,
local stabilization and local tracking by output feedback)
6. Disturbance Decoupling Problem (with or without disturbance
measurement, links to geometry, stability)
7. Input-Output Decoupling (with or without stability)
Reference text:
– A.Isidori, ’Nonlinear Control System’, Springer, 2000, Chapter 4
• Elementary theory of nonlinear feedback for Multi-Input MultiOutput systems
1. Vector relative degree and coordinates transformation, (|A(x)|
different form zero), existence of coordinates transformations,
normal form
2. Feedback linearization u − y and u − x (linearization I/O dynamic assignment/r = n, full u − x, u − y feedback linearization),
geometric conditions for u − x full feedback linearization
3. Zero dynamics and elements of local stabilization
Reference text:
– A.Isidori, ’Nonlinear Control System’, Springer, 2000, Chapter 5,
paragraph 5.1, 5.2, 5.4
• Some basic results of Global Asymptotic Stabilization
1. Backstepping techniques (starting from the strict feedback structure)
2. Artstein-Sontag’s Theorem (Control Lyapunov function approach)
Reference text:
– A.Isidori, ’Nonlinear Control System’, Springer, 2000, Chapter 9,
paragraph 9.1 - 9.4
5