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ENEE 661: Nonlinear Systems Electrical and Computer Engineering Department University of Maryland College Park Spring 2004 Professor John S. Baras MID-TERM EXAM Thursday, April 8, 2004 This is a closed book and no notes exam. Please try to be concise. Good Luck! Problem 1 (easy) (10 points) Consider the nonlinear dynamical system 1 1 1 2 2 2 1 2
x x x x x x x
(a) Find all equilibrium points and characterize them with respect to their type and stability. (b) Is there a separatrix? (c) Sketch a rough phase portrait for this system. Problem 2 (easy) (10 points) A model for a chemical oscillator system is described by 1 2 (^1 1 ) 1 2 (^2 1 ) 1
x x x a x x x x bx x
where x 1^ and x 2^ are dimensionless concentrations of certain chemicals and a, b are positive constants. (a) For what values of the parameters a and b does the system have a periodic orbit? Justify your answer completely. (b) Find and classify bifurcations that occur as b varies, while a is fixed.
Problem 3 (medium) (10 points) The nonlinear dynamical equations for an m -link robot take the form M q q ( ) && C q q q ( , & &) Dq & g q ( ) u where q is an m -dimensional vector of generalized coordinates representing joint positions, u is an m -dimensional control (torque) input, and M ( q ) is a symmetric inertia matrix, which is positive definite for all m q R. The term C q q q ( ,^ & &)^ accounts for centrifugal and Coriolis forces. The matrix C has the property that M & ^2 C is a skew symmetric matrix for all , m q q & R , where (^) M &^ is the total derivative of M q ( )^ with respect to t. The term Dq &, accounts for viscous damping, where D is positive semidefinite symmetric matrix. The term g ( q ) , which accounts for gravity forces, is given by (^) ( ) [ ( ) / ] T g q P q q , where^ P ( q )^ is the total potential energy of the links due to gravity. Assume that P ( q ) is a positive definite function of q and that g ( q ) = 0 has an isolated root at q = 0. (a) With u = 0, use the total energy ( ,^ )^12 ( )^ ( ) T V q q &^ q M q q &^ & ^ P q as a Lyapunov function candidate to show that the origin (^ q^ ^ 0,^ q &^ 0)is stable. (b) With u^ ^ K qd &, where K^ d is a positive diagonal matrix, show that the origin is asymptotically stable.
