Exercises Solution Nonlinear System Khalil

Exercises Solution Nonlinear System Khalil Exercises Solutions for Nonlinear Systems by Hassan K Khalil This document provides solutions to selected exercises from the textbook Nonlinear Systems by Hassan K Khalil a widely recognized text in the field of nonlinear control systems The solutions are intended to serve as a learning resource for students and professionals working with nonlinear systems analysis and design This document is structured according to the chapters of the textbook providing solutions for representative exercises Each chapter section will follow this format 1 Chapter Title The title of the chapter from the textbook 2 Exercise Numbers List of exercise numbers covered in this section 3 Solution Approach A brief explanation of the key concepts and techniques used to solve the exercises 4 Detailed Solutions Stepbystep solutions for each exercise including relevant equations diagrams and explanations Note This document does not cover all exercises in the textbook The selection is based on their representative nature and their contribution to understanding key concepts Solutions are provided in a concise manner emphasizing the core principles and steps Readers are encouraged to refer to the textbook for a more indepth explanation of the theoretical background Chapter 1 to Nonlinear Systems Exercise Numbers 11 13 15 Solution Approach Exercise 11 This exercise introduces the concept of equilibrium points and their stability The solution involves analyzing the behavior of the system near the equilibrium point using linearization Exercise 13 This exercise focuses on the phase plane analysis of a simple nonlinear system The solution involves constructing the phase portrait and analyzing the behavior of trajectories based on the system dynamics 2 Exercise 15 This exercise explores the concept of Lyapunov stability where the solution involves constructing a Lyapunov function and analyzing its time derivative to determine the stability of the system Detailed Solutions Exercise 11 a Find the equilibrium points of the system x x1x xy y xy 2y Set both derivatives to zero x1x xy 0 xy 2y 0 Solving these equations we get two equilibrium points 0 0 and 1 0 b Linearize the system around the equilibrium point 0 0 The linearized system is obtained by approximating the nonlinear terms with their firstorder Taylor series expansion The Jacobian matrix evaluated at 0 0 gives the linearized system A 1 0 0 2 The eigenvalues of A are 1 and 2 Since one eigenvalue is positive and the other is negative the equilibrium point 0 0 is a saddle point c Linearize the system around the equilibrium point 1 0 The Jacobian matrix evaluated at 1 0 gives the linearized system A 1 1 0 2 The eigenvalues of A are 1 and 2 Since both eigenvalues are negative the equilibrium point 1 0 is a stable node 3 Exercise 13 a Construct the phase portrait for the system x y y x y3 The solution involves finding the nullclines where x 0 and y 0 and analyzing the direction of trajectories in different regions of the phase plane b Determine the stability of the equilibrium point 0 0 The equilibrium point 0 0 is a stable focus This can be observed from the phase portrait where trajectories spiral inwards towards the equilibrium point Exercise 15 a Consider the system x x y3 y x y Construct a Lyapunov function candidate A suitable Lyapunov function candidate is Vx y x y b Analyze the time derivative of the Lyapunov function V x y 2xx y3 2yx y 2x 2y4 2y Since V x y is negative definite the equilibrium point 0 0 is asymptotically stable Chapter 2 Linear Systems Exercise Numbers 21 23 25 Solution Approach Exercise 21 This exercise deals with the analysis of linear timeinvariant LTI systems in statespace form The solution involves finding the eigenvalues eigenvectors and state transition matrix 4 Exercise 23 This exercise focuses on the controllability and observability of LTI systems The solution involves applying the controllability and observability rank tests Exercise 25 This exercise explores the concept of pole placement using state feedback The solution involves determining the feedback gain matrix that places the closedloop system poles at desired locations Detailed Solutions Exercise 21 a Find the eigenvalues and eigenvectors of the system matrix A A 1 2 1 3 The eigenvalues are 2 and 2 The corresponding eigenvectors are v 1 1 and v 2 1 b Determine the state transition matrix t t eAt e2t 2te2t te2t e2t Exercise 23 a Determine the controllability matrix of the system A 1 2 1 3 B 1 0 The controllability matrix is C B AB 1 1 0 1 Since the rank of C is 2 the system is controllable b Determine the observability matrix of the system C 1 0 5 The observability matrix is O C CA 1 0 1 2 Since the rank of O is 2 the system is observable Exercise 25 a Design a state feedback control law to place the closedloop system poles at 1 and 2 The desired characteristic polynomial is ps s 1s 2 s 3s 2 The feedback gain matrix K can be calculated using the Ackermanns formula K 0 1 A 3A 2I B AB 2 3 b Verify the closedloop system poles are at the desired locations The closedloop system matrix is A BK 1 4 1 0 The eigenvalues of this matrix are 1 and 2 confirming that the poles are placed at the desired locations Chapter 3 Stability of Nonlinear Systems Exercise Numbers 31 33 35 Solution Approach Exercise 31 This exercise deals with the concept of Lyapunov stability and its application to analyze the stability of nonlinear systems The solution involves finding a Lyapunov function and analyzing its time derivative Exercise 33 This exercise focuses on the LaSalles Invariance Principle which provides a way to analyze the stability of systems with nonnegative time derivatives of Lyapunov functions Exercise 35 This exercise explores the concept of inputtostate stability ISS where the solution involves analyzing the boundedness of system states in the presence of bounded inputs 6 Detailed Solutions Exercise 31 a Consider the system x x y y x y Construct a Lyapunov function candidate Vx y x y b Analyze the time derivative of the Lyapunov function V x y 2xx y 2yx y 2x 2y 0 Since V x y is negative definite the equilibrium point 0 0 is asymptotically stable Exercise 33 a Consider the system x x y y y xy Construct a Lyapunov function candidate Vx y x y b Analyze the time derivative of the Lyapunov function V x y 2xx y 2yy xy 2x 2y 0 V x y is not negative definite but it is nonpositive Using LaSalles Invariance Principle we can conclude that the trajectories converge to the set where V x y 0 which is the origin 0 0 Therefore the equilibrium point 0 0 is asymptotically stable Exercise 35 7 a Consider the system x x uy where u is a bounded input u M Show that the system is ISS Using the Lyapunov function candidate Vx x we get V x 2xx uy 2x 2xuy 2x 2xMy Using Youngs inequality we can bound the second term 2xMy x My Therefore V x x My Choosing a classK function r Mr we can show that the system is ISS This means that the state x remains bounded for any bounded input u Conclusion This document provides solutions to selected exercises from the textbook Nonlinear Systems by Hassan K Khalil covering key concepts in the analysis and design of nonlinear control systems The solutions are meant to serve as a learning resource and guide readers through the process of solving such exercises It is crucial to note that this document is not a replacement for the textbook and readers are encouraged to refer to the text for a more in depth understanding of the theoretical concepts and their applications 8