Re: Nonlinear Systems Khalil Homework Solutions
Violetta Wagganer
Graduate students with interests in control and dynamical systems, robotics, artificial intelligence and machine learning, signal and image processing, communications, computer science and engineering, optimization, power systems, systems biology, and financial engineering.
Introduction and examples of nonlinear systems. State-space models. Equilibrium points. Linearization. Range of nonlinear phenomena: finite escape time, multiple isolated equilibria, limit cycles, chaos. Bifurcations. Phase portraits. Bendixson and Poincare-Bendixson criteria. Mathematical background: existence and uniqueness of solutions, continuous dependence on initial conditions and parameters, normed linear spaces, comparison principle, Bellman-Gronwall Lemma. Lyapunov stability. Lyapunov's direct method. Lyapunov functions. LaSalle's invariance principle. Estimating region of attraction. Control Lyapunov functions. Center manifold theory. Stability of time-varying systems. Gradient algorithm for estimation of unknown parameters. Uniform observability and persistency of excitation. Input-output and input-to-state stability. Small gain theorem. Positive real transfer functions. Kalman-Yakubovich-Popov Lemma. Passivity. Circle and Popov criteria for absolute stability. Theory of integral quadratic constraints. Feedback and input-output linearization. Relative degree and zero dynamics. Model reference adaptive control. Integrator backstepping. Adaptive backstepping design.
Homework policy
Homework is intended as a vehicle for learning, not as a test. Moderate collaboration with your classmates is encouraged. However, I urge you to invest enough time alone to understand each homework problem, and independently write the solutions that you turn in. Homework is generally handed out every other Thursday, and it is due at the beginning of the class a week later. Late homework will not be accepted. Start early!
CDS 140b is a continuation of CDS 140a. A large part of the course will focus on tools from nonlinear dynamics, such as perturbation theory and averaging, advanced stability analysis, the existence of periodic orbits, bifurcation theory, chaos, etc. In addition, guest lecturers will give an introduction to current research topics in dynamical systems theory. There will be 8 homeworks throughout the term but no exams.
Homeworks are to be done and handed in individually. To improve the learning process, students are encouraged to discuss the problems with, provide guidance to and get help from other students, the TAs and instructors. However, to make sure each student understands the concepts, solutions must be written independently and should reflect your understanding of the subject matter at the time of writing. Copying solutions, using solutions from previous years, having someone else type or dictate any part of the solution manual or using publicly available solutions (from the Internet) are not allowed.
Introduction. Examples of nonlinear systems. State-space models. Equilibriumpoints. Linearization. Range of nonlinear phenomena: finite escape time,multiple isolated equilibria, limit cycles, chaos. Bifurcations. Phaseportraits. Bendixson and Poincare-Bendixson criteria. Mathematical background:existence and uniqueness of solutions, continuous dependence on initialconditions and parameters, normed linear spaces, comparison principle,Bellman-Gronwall Lemma. Lyapunov stability. Lyapunov's direct method. Lyapunovfunctions. LaSalle's invariance principle. Estimating region of attraction.Center manifold theory. Stability of time-varying systems. Input-output andinput-to-state stability. Small gain theorem. Passivity. Circle and Popovcriteria for absolute stability. Perturbation theory and averaging. Singularperturbations. Feedback and input-output linearization. Zero dynamics.Backstepping design. Control Lyapunov functions.
Homework policy
Homework is intended as a vehicle for learning, not as a test. Moderatecollaboration with your classmates is allowed. However, I urge you to investenough time alone to understand each homework problem, and independently writethe solutions that you turn in. Homework is generally handed out every Thursday,and it is due at the beginning of the class a week later. Late homework will notbe accepted. Start early!
Even though I plan to cover everything from scratch, the students wouldbenefit from a solid background in linear systems (EE 5231 or an equivalentcourse). Those interested should contact the instructor.
In mathematics and science, a nonlinear system (or a non-linear system) is a system in which the change of the output is not proportional to the change of the input.[1][2] Nonlinear problems are of interest to engineers, biologists,[3][4][5] physicists,[6][7] mathematicians, and many other scientists since most systems are inherently nonlinear in nature.[8] Nonlinear dynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems.
Typically, the behavior of a nonlinear system is described in mathematics by a nonlinear system of equations, which is a set of simultaneous equations in which the unknowns (or the unknown functions in the case of differential equations) appear as variables of a polynomial of degree higher than one or in the argument of a function which is not a polynomial of degree one.In other words, in a nonlinear system of equations, the equation(s) to be solved cannot be written as a linear combination of the unknown variables or functions that appear in them. Systems can be defined as nonlinear, regardless of whether known linear functions appear in the equations. In particular, a differential equation is linear if it is linear in terms of the unknown function and its derivatives, even if nonlinear in terms of the other variables appearing in it.
As nonlinear dynamical equations are difficult to solve, nonlinear systems are commonly approximated by linear equations (linearization). This works well up to some accuracy and some range for the input values, but some interesting phenomena such as solitons, chaos,[9] and singularities are hidden by linearization. It follows that some aspects of the dynamic behavior of a nonlinear system can appear to be counterintuitive, unpredictable or even chaotic. Although such chaotic behavior may resemble random behavior, it is in fact not random. For example, some aspects of the weather are seen to be chaotic, where simple changes in one part of the system produce complex effects throughout. This nonlinearity is one of the reasons why accurate long-term forecasts are impossible with current technology.
Additivity implies homogeneity for any rational α, and, for continuous functions, for any real α. For a complex α, homogeneity does not follow from additivity. For example, an antilinear map is additive but not homogeneous. The conditions of additivity and homogeneity are often combined in the superposition principle
Solving systems of polynomial equations, that is finding the common zeros of a set of several polynomials in several variables is a difficult problem for which elaborated algorithms have been designed, such as Grbner base algorithms.[11]
For the general case of system of equations formed by equating to zero several differentiable functions, the main method is Newton's method and its variants. Generally they may provide a solution, but do not provide any information on the number of solutions.
A nonlinear recurrence relation defines successive terms of a sequence as a nonlinear function of preceding terms. Examples of nonlinear recurrence relations are the logistic map and the relations that define the various Hofstadter sequences. Nonlinear discrete models that represent a wide class of nonlinear recurrence relationships include the NARMAX (Nonlinear Autoregressive Moving Average with eXogenous inputs) model and the related nonlinear system identification and analysis procedures.[12] These approaches can be used to study a wide class of complex nonlinear behaviors in the time, frequency, and spatio-temporal domains.
One of the greatest difficulties of nonlinear problems is that it is not generally possible to combine known solutions into new solutions. In linear problems, for example, a family of linearly independent solutions can be used to construct general solutions through the superposition principle. A good example of this is one-dimensional heat transport with Dirichlet boundary conditions, the solution of which can be written as a time-dependent linear combination of sinusoids of differing frequencies; this makes solutions very flexible. It is often possible to find several very specific solutions to nonlinear equations, however the lack of a superposition principle prevents the construction of new solutions.
and the left-hand side of the equation is not a linear function of u \displaystyle u and its derivatives. Note that if the u 2 \displaystyle u^2 term were replaced with u \displaystyle u , the problem would be linear (the exponential decay problem).
Second and higher order ordinary differential equations (more generally, systems of nonlinear equations) rarely yield closed-form solutions, though implicit solutions and solutions involving nonelementary integrals are encountered.
The most common basic approach to studying nonlinear partial differential equations is to change the variables (or otherwise transform the problem) so that the resulting problem is simpler (possibly linear). Sometimes, the equation may be transformed into one or more ordinary differential equations, as seen in separation of variables, which is always useful whether or not the resulting ordinary differential equation(s) is solvable.
Another common (though less mathematical) tactic, often exploited in fluid and heat mechanics, is to use scale analysis to simplify a general, natural equation in a certain specific boundary value problem. For example, the (very) nonlinear Navier-Stokes equations can be simplified into one linear partial differential equation in the case of transient, laminar, one dimensional flow in a circular pipe; the scale analysis provides conditions under which the flow is laminar and one dimensional and also yields the simplified equation.
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