## Transcription

References, Nonlinear systems Khalil Prentice Hall 2002. Nonlinear Probably the best book to start with nonlinear control. Nonlinear systems S Sastry Springer Verlag 1999, N Marchand. Good general book a bit harder than Khalil s, References Mathematical Control Theory E D Sontag Springer 1998. Mathematically oriented Can be downloaded at, http www math rutgers edu sontag FTP DIR sontag mathematical control theory springer98 pdf. Introduction, Linear Nonlinear, Constructive nonlinear control Sepulchre et al Springer 1997 More.

The X4 example focused on passivity and recursive approaches. Linear Nonlinear control systems A Isidori Springer Verlag 1995. approaches, A reference for geometric approach, Antiwindup. Linearization Applied Nonlinear control J J Slotine and W Li Prentice Hall 1991. Gain scheduling, An interesting reference in particular for sliding mode. Research on Gain Scheduling W J Rugh and J S Shamma. approaches, Automatica 36 10 1401 1425 2000, CLF Survey of gain scheduling analysis and design D J Leith and W E. Sliding mode, Geometric Leithead Int Journal of Control 73 1001 1025 1999. Surveys on Gain Scheduling The second reference can be downloaded at. techniques http citeseer ist psu edu leith99survey html. X4 stabilization, N Marchand gipsa lab Nonlinear Control Master PSPI 2009 2010 2 174.

1 Introduction, Control Linear versus nonlinear, N Marchand The X4 example. References 2 Linear control methods for nonlinear systems. Outline Antiwindup, Introduction Linearization, Linear Nonlinear. The X4 example Gain scheduling, approaches 3 Stability. Antiwindup, Linearization, Gain scheduling, 4 Nonlinear control methods. Stability Control Lyapunov functions, Nonlinear Sliding mode control.

approaches, CLF State and output linearization, Sliding mode. Backstepping and feedforwarding, techniques, Stabilization of the X4 at a position. X4 stabilization, 5 Observers, N Marchand gipsa lab Nonlinear Control Master PSPI 2009 2010 3 174. 1 Introduction, Control Linear versus nonlinear, N Marchand The X4 example. References 2 Linear control methods for nonlinear systems. Outline Antiwindup, Introduction Linearization, Linear Nonlinear.

The X4 example Gain scheduling, approaches 3 Stability. Antiwindup, Linearization, Gain scheduling, 4 Nonlinear control methods. Stability Control Lyapunov functions, Nonlinear Sliding mode control. approaches, CLF State and output linearization, Sliding mode. Backstepping and feedforwarding, techniques, Stabilization of the X4 at a position.

X4 stabilization, 5 Observers, N Marchand gipsa lab Nonlinear Control Master PSPI 2009 2010 4 174. Introduction, N Marchand, Some preliminary vocable and definitions. References, What is control, Introduction A very short review of the linear case properties. Linear Nonlinear, The X4 example, design of control laws etc. approaches, Why nonlinear control, Antiwindup, Linearization Formulation of a nonlinear control problem model.

Gain scheduling, representation closed loop stability etc. Nonlinear Some strange possible behaviors of nonlinear systems. approaches, CLF Example The X4 helicopter, Sliding mode. techniques, X4 stabilization, N Marchand gipsa lab Nonlinear Control Master PSPI 2009 2010 5 174. 1 Introduction, Control Linear versus nonlinear, N Marchand The X4 example. References 2 Linear control methods for nonlinear systems. Outline Antiwindup, Introduction Linearization, Linear Nonlinear.

The X4 example Gain scheduling, approaches 3 Stability. Antiwindup, Linearization, Gain scheduling, 4 Nonlinear control methods. Stability Control Lyapunov functions, Nonlinear Sliding mode control. approaches, CLF State and output linearization, Sliding mode. Backstepping and feedforwarding, techniques, Stabilization of the X4 at a position.

X4 stabilization, 5 Observers, N Marchand gipsa lab Nonlinear Control Master PSPI 2009 2010 6 174. Some preliminary vocable, N Marchand, We consider a dynamical system. References, Outline System, Introduction x t, Linear Nonlinear. The X4 example, approaches, Antiwindup, Linearization. y is the output represents what is visible from outside. Gain scheduling the system, Stability x is the state of the system characterizes the state of the.

Nonlinear system, approaches, CLF u is the control input makes the system move. Sliding mode, techniques, X4 stabilization, N Marchand gipsa lab Nonlinear Control Master PSPI 2009 2010 7 174. The 4 steps to control a system, Control yd u y, N Marchand. References Controller, Introduction, Linear Nonlinear Observer. The X4 example, approaches, 1 Modelization, Antiwindup.

Linearization, To get a mathematical representation of the system. Gain scheduling Different kind of model are useful Often. Stability a simple model to build the control law, Nonlinear a sharp model to check the control law and the observer. approaches, Sliding mode, 2 Design the state reconstruction in order to reconstruct. the variables needed for control, techniques, X4 stabilization. 3 Design the control and test it, Observers 4 Close the loop on the real system.

N Marchand gipsa lab Nonlinear Control Master PSPI 2009 2010 8 174. Linear dynamical systems, Some properties of linear system 1 2. Control Definition Systems such that if y1 and y2 are the. N Marchand outputs corresponding to u1 and u2 then R. References y1 y2 is the output corresponding to u1 u2. Outline Representation near the operating point, Introduction Transfer function. Linear Nonlinear, The X4 example, y s h s u s, approaches. Antiwindup State space representation, Linearization. Gain scheduling, approaches, The model can be obtained.

Sliding mode, physical modelization eventually coupled with. identification, techniques, X4 stabilization identification black box approach. Observers both give h s or A B C D and hence the model. N Marchand gipsa lab Nonlinear Control Master PSPI 2009 2010 9 174. Linear dynamical systems, Some properties of linear system 2 2. Control x Ax Bu, N Marchand y Cx Du, References Nice properties. Outline Unique and constant equilibrium, Introduction.

Controllability resp observability directly given by. Linear Nonlinear rank B AB An 1 B resp rank C CA CAn 1. The X4 example Stability directly given by the poles of h s or the eigenvalues. Linear of A asymptotically stable 0, approaches, Local properties global properties like stability. Antiwindup, Linearization stabilizability etc, Gain scheduling. The time behavior is independent of the initial condition. Stability Frequency analysis is easy, Nonlinear Control is easy simply take u Kx with K such that. approaches eig A BK 0 the closed loop system x Ax BKx will. Sliding mode, be asymptotically stable, techniques. Mathematical tool linear algebra, X4 stabilization This is a caricature of the reality of course problems due to.

Observers uncertainties delays noise etc, N Marchand gipsa lab Nonlinear Control Master PSPI 2009 2010 10 174. Why nonlinear control, Why nonlinear control if linear control is so easy. All physical systems are nonlinear because of, N Marchand. Actuators saturations, References Viscosity proportional to speed2. Outline Sine or cosine functions in robotics, Introduction Chemical kinetic in exp temperature.

Linear Nonlinear, The X4 example Friction or hysteresis phenomena. approaches, Antiwindup More and more the performance specification requires. Linearization, Gain scheduling nonlinear control eg automotive. More and more controlled systems are deeply nonlinear. approaches eg nano systems where hysteresis phenomena friction. Sliding mode discontinuous behavior, Nonlinear control is sometimes necessary oscillators. techniques, X4 stabilization cyclic systems, N Marchand gipsa lab Nonlinear Control Master PSPI 2009 2010 11 174.

Nonlinear dynamical systems, How to get a model, Control Representation. N Marchand State space representation, ODE Ordinary differential equation. References, In this course only, Introduction x t f x t u t. Linear Nonlinear, The X4 example y t h x t u t, approaches PDE Partial differential equation traffic flow etc. Antiwindup, Linearization 0 g x t x t f x, Gain scheduling.

0 h y t x t u t, Algebraic differential equations implicit hybrid with. approaches discrete or event based equations etc, Sliding mode The model can be obtained. physical modelization and then nonlinear identification of. techniques, X4 stabilization, the parameters identifiability problems. N Marchand gipsa lab Nonlinear Control Master PSPI 2009 2010 12 174. Nonlinear dynamical systems, Open loop control versus closed loop control. Control u y, N Marchand, References Controller, Introduction.

Linear Nonlinear, The X4 example, Open loop control. approaches find u t such that limt ky t yd t k 0, Antiwindup. Linearization, Gain scheduling Widely used for path planning problems robotics. Closed loop control, approaches, find u x such that limt ky t yd t k 0. Sliding mode Better because closed loop control can stabilize systems and is robust w r t. control perturbation, techniques, X4 stabilization.

In this course, Only closed loop control problems are treated. N Marchand gipsa lab Nonlinear Control Master PSPI 2009 2010 13 174. Nonlinear dynamical systems, Aim of control, Nonlinear xd. Control u y, N Marchand, References Controller, Introduction. Linear Nonlinear, The X4 example, Tracking problem. approaches find u x such that limt kx t xd t k 0, Antiwindup.

Linearization, Gain scheduling Stabilization problem. Stability find u x such that limt kx t xd t k 0 for xd t constant. approaches Null stabilization problem, Sliding mode. find u x such that limt kx t xd t k 0 for xd t 0, control In any cases a null stabilization problem of z t y t yd t. techniques, X4 stabilization In this course, Observers Only the stabilization problem will be treated. N Marchand gipsa lab Nonlinear Control Master PSPI 2009 2010 14 174. Some strange behaviors of nonlinear systems, The undersea vehicle.

Control Simplified model of the undersea vehicle in one. N Marchand direction, References, Introduction, Linear Nonlinear Step answer. The X4 example, approaches, Antiwindup 1, Linearization No proportionality 1 u s s. Gain scheduling, between the input, and the output 0 5. approaches, Sliding mode 0 1 2 3 4 5 6 7 8 9 10, techniques. X4 stabilization, N Marchand gipsa lab Nonlinear Control Master PSPI 2009 2010 15 174.

4 Close the loop on the real system N Marchand gipsa lab Nonlinear ControlMaster PSPI 2009 20108 174 Nonlinear Control N Marchand References Outline Introduction Linear Nonlinear The X4 example Linear approaches Antiwindup Linearization Gain scheduling Stability Nonlinear approaches CLF Sliding mode Geometric control Recursive techniques X4 stabilization Observers Linear dynamical