Nonlinear Systems Khalil Homework Solutions

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There will be two take-home quizzes, to be completed and returned within 24 hours, but no final exam. The quizzes will cover the theory of 6.243J (divided as equally as possible). The questions will be based on the ideas used in the problem set solutions made available at least a week before the test. No homework will be given on the last Wednesdays before the quizzes. No cooperation is allowed on take-home quizzes.

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. 2b1af7f3a8