ps4sol_6243_2003

ps4sol_6243_2003 - Massachusetts Institute of Technology...

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Unformatted text preview: Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.243j (Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS by A. Megretski Problem Set 4 Solutions 1 Problem 4.1 ¯ Find a function V : R 3 ≥ R + which has a unique minimum at x = 0 , and is strictly monotonically decreasing along all non-equilibrium trajecto- ries of system x ˙ 1 ( t ) = − x 1 ( t ) + x 2 ( t ) 2 , x ˙ 2 ( t ) = − x 2 ( t ) 3 + x 3 ( t ) 4 , x ˙ 3 ( t ) = − x 3 ( t ) 5 . Let us begin with collecting storage function and compatible quadratic supply rate pairs for the system. Naturally, positive definite functions of system states are a good starting point. For V 1 ( x ) = x 2 we have 1 2 2 2 ˙ V 1 = − 2 x 1 + 2 x 1 x 2 − x 1 + w 2 = 1 , 1 2 where w 1 = x 2 , and the classical inequality 2 ab a 2 + b 2 2 was used. For V 2 = x 2 we have 4 4 2 2 V ˙ 2 = − 2 x 2 + 2 x 2 x 3 − w 1 + 2 w 2 = 2 , 8 / 3 where w 2 = x 3 and the inequality 2 ab 3 a 4 + 2 b 4 1 Version of October 11, 2003 2 (a weakened version of a classical inequality) was used. Finally, for V 3 = 3 x 4 / 3 we have 3 16 / 3 2 V ˙ 3 = − 4 x 3 = − 4 w 2 . Now, for V = c 1 V 1 + c 2 V 2 + c 3 V 3 we have 2 2 2 V ˙ c 1 1 + c 2 2 + c 3 3 = − c 1 x 1 + ( c 1 − c 2 ) w 1 + (2 c 2 − 4 c 3 ) w 2 . Taking c 1 = 1, c 2 = c 3 = 2 yields a continuously differentiable Lyapunov function 2 4 / 3 V ( x ) = x 2 + 2 x 2 + 6 x 3 1 for which the derivatives along system trajectories are bounded by 4 16 / 3 V ˙ ( x ) − x 2 − x 2 − 4 x 3 ....
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ps4sol_6243_2003 - Massachusetts Institute of Technology...

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