This document discusses a lecture on nonlinear control systems. It begins with announcements about guidance sessions and assignment deadlines. It then summarizes the previous lecture on nonlinear systems and previews the goals of the current lecture on fundamental properties and phase plane analysis. The lecture will cover existence and uniqueness of solutions, the comparison principle, constructing phase portraits, classifying equilibrium points, limit cycles, and criteria for detecting periodic solutions like the Poincare-Bendixson criterion.
This document discusses a lecture on nonlinear control systems. It begins with announcements about guidance sessions and assignment deadlines. It then summarizes the previous lecture on nonlinear systems and previews the goals of the current lecture on fundamental properties and phase plane analysis. The lecture will cover existence and uniqueness of solutions, the comparison principle, constructing phase portraits, classifying equilibrium points, limit cycles, and criteria for detecting periodic solutions like the Poincare-Bendixson criterion.
This document discusses a lecture on nonlinear control systems. It begins with announcements about guidance sessions and assignment deadlines. It then summarizes the previous lecture on nonlinear systems and previews the goals of the current lecture on fundamental properties and phase plane analysis. The lecture will cover existence and uniqueness of solutions, the comparison principle, constructing phase portraits, classifying equilibrium points, limit cycles, and criteria for detecting periodic solutions like the Poincare-Bendixson criterion.
This document discusses a lecture on nonlinear control systems. It begins with announcements about guidance sessions and assignment deadlines. It then summarizes the previous lecture on nonlinear systems and previews the goals of the current lecture on fundamental properties and phase plane analysis. The lecture will cover existence and uniqueness of solutions, the comparison principle, constructing phase portraits, classifying equilibrium points, limit cycles, and criteria for detecting periodic solutions like the Poincare-Bendixson criterion.
Lecture 2 Fundamental properties and Phase plane analysis 2 Announcement Guidance: Guidance on the assignments is every Thursday between 16.15 and 18.00 in EL4. Handing in: You can hand in your solutions manually at Room D238 (over the Department's office), or as a file through it's:learning - preferably as a pdf document, scanned pages (gif/jpg/png) or a Word .doc file. Assignments that are handed in after deadlines will not be accepted. The deadline is at 13.00 at the date found in its learning. The first deadline is on 02.09.08 (next Tuesday). 3 Previous lecture Nonlinear Control System -- What is it? Examples of nonlinear systems and nonlinear phenomena The basic differences between linear and nonlinear systems The need for new analysis and control design methods How to calculate equilibrium points 4 Todays goals Fundamental properties Be able to validate a mathematical model by ensuring the existence and uniqueness of solutions of the initial value problem The first analysis tool Be able to use the comparison principle to find an upper bound for the solution x(t) without computing the solution itself 5 Todays goals cont. Phase plane analysis: analysis of 2D systems Phase portraits: graphical analysis tools Know how to construct phase portraits and interpret them Be able to describe a periodic solution and a limit cycle Be able to tell whether a periodic solution may or may not exist for a 2D system 6 Literature Khalil Chapter 3, Sections 3.1 and 3.4 Chapter 2, Sections 2.1 - 2.6 7 Fundamental properties 8 Existence and uniqueness of solutions For each initial condition there is a unique behaviour of the physical system Process/System/Mechanism 9 Existence and uniqueness of solutions For each initial condition there is a unique behaviour of the physical system Process/System/Mechanism System model modelling ( , ) x f t x = & Does there exist a unique solution of the initial value problem? ! x(t 0 ) = x 0 Initial condition initial value problem 10 Rudolph Lipschitz (1832-1903) (Germany) Lipschitz condition ! f (x, t) " f (y, t) # L x " y (the slope of is bounded) ! f (x, t) ! Lipschitz on a set W ! Locally Lipschitz on a domain D ! Globally Lipschitz 11 The first analysis tool: Comparison Principle 12 Graphical analysis of 2D systems: Phase plane analysis 13 Todays goals continued Phase plane analysis Be able to answer: what is the phase plane and what are phase portraits? Know how to construct phase portraits and interpret these Analytic method Vector field diagram Computer simulations Know how to do a local phase plane analysis by classifying equilibria into nodes, foci, saddle points and center points Be able to describe what a limit cycle is (and how it differs from the periodic solutions about a center point) Be able to tell whether a limit cycle may or may not exist for a two-dimensional system 14 Phase Plane Analysis Computer simulations 1. Select a region of interest in the plane (Typically a bounding box x 1,min ! x 1 ! x 1,max x 2,min ! x 2 ! x 2,max ) 2. Select a number of initial values inside this region 3. Calculate and plot the corresponding trajectories, i.e. the solutions of the initial value problems For unstable equilibrium points and limit cycles, the only way to obtain a phase portrait close to these is to simulate backwards in time, i.e. to solve the IVP x = - f(x) x(0)=x 0 15 Vector field diagram 1. Select a region of interest in the plane, and assign a grid 2. To each point (x 1 ,x 2 ) in the grid, assign the vector f(x 1 ,x 2 ), i.e. assign a directed line segment from (x 1 ,x 2 ) to (x 1 ,x 2 ) + f(x 1 ,x 2 ) Matlab: pplane6 (From Rice University) 16 17 18 How to do phase plane analysis 1) Find the equilibria of the system 2) Classify the (isolated) equilibria in order to obtain qualitative knowledge about the system behavior locally around the equilibria (this will guide you in the next step) 3) Construct a phase portrait using a) the analytical method b) a vector field diagram c) computer simulations 4) Try to find possible periodic solutions and limit cycles 19 Periodic solutions and limit cycles 20 Limit cycles (non-trivial isolated periodic orbits) Examples: - ship with a deadzone in actuator, electronic oscillator Criteria for detecting the presence or absence of periodic orbits: 1. Poincar-Bendixson criterion 2. The Bendixson (negative) criterion 3. The index method (read yourself in Khalil) 21 Limit cycles (non-trivial isolated periodic orbits) Electronic oscillator: 2 2 1 2 1 1 2 1 2 2 2 1 2 1 2 2 (1 ) ( ) (1 ) ( ) x x x x x f x x x x x x f x = + ! ! = = ! + ! ! = & & 22 Poincar-Bendixson Criterion (Lemma 2.1) Consider the system Let M be a bounded, closed subset of the plane such that M contains no equilibrium points of the system, or it contains only one equilibrium point with the property that the eigenvalues of the Jacobian matrix at this point have positive real parts (unstable focus or unstable node) Every trajectory starting in M stays in M for all future time (i.e. M is a positively invariant set) Then M contains a periodic orbit of the system. 2 ) ( ! " = x x f x& 23 Poincar-Bendixson Criterion Henri Poincare (1854-1912) (France) Ivar Otto Bendixson (1861-1935) (Sweeden) 24 Limit cycles (non-trivial isolated periodic orbits) Electronic oscillator: 2 2 1 2 1 1 2 1 2 2 2 1 2 1 2 2 (1 ) ( ) (1 ) ( ) x x x x x f x x x x x x f x = + ! ! = = ! + ! ! = & & 25 Bendixson (negative) Criterion (Lemma 2.2) Consider the system If, on a simply connected region D of the plane, the expression is not identically zero and does not change sign, then the system has no periodic orbits lying entirely in D 2 ) ( ! " = x x f x& ! "f 1 "x 1 (x) + "f 2 "x 2 (x)