Non Linear Control Systems

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Please see HW3.pdf for questions. The other 2 attachments may be needed for some question they are MATLAB files to assist in solving the questions.

Homework 3 ECE6552 Nonlinear Control Systems Magnus Egerstedt Due: March 13, 2018 (and Mar. 20 for DL students – sections Q, Q3, QSZ) 1 Consider a so-called double pendulum (a pendulum with another pendulum attached to its end). Let the angles of the two pendula be θ1 and θ2 , which gives θ̇1 = θ̇2 = 6 2pθ1 − 3 cos(θ1 − θ2 )pθ2 m`2 16 − 9 cos2 (θ1 − θ2 ) 6 8pθ2 − 3 cos(θ1 − θ2 )pθ1 , m`2 16 − 9 cos2 (θ1 − θ2 ) where m, ` are the masses and lengths of the two pendula (both pendula have the same dimensions), and where the two momenta pθ1 , pθ2 satisfy ṗθ1 ṗθ2   1 g = − m`2 θ̇1 θ̇2 sin(θ1 − θ2 ) + 3 sin(θ1 ) + u 2 `  1 2 g = − m` −θ̇1 θ̇2 sin(θ1 − θ2 ) + sin(θ2 ) , 2 ` where u is the control input (roughly equal to a scaled torque applied at the pivot point of the first pendulum) Linearize this model around the doubly-upright position, i.e., θ1 = θ2 = π, pθ1 = pθ2 = 0, to get ẋ = Ax + Bu. a What are A, B? b Is the resulting linear system completely controllable? What are the stability properties of the uncontrolled linear system? Hint: The Matlab commands eig, ctrb and rank may come in handy. 2 a Setting m = 1, ` = 1, and g = 10, using your favorite method, design a stabilizing control law for the linear system in Question 1. 1 b Plot θ1 as a function of time under your proposed control  0.1  −0.1 x(0) =   0 0 law from the initial condition   .  Attach the plot (only the plot - no m-files please) to the homework submission. Hint: The m-files pendulum.m and dist.m found under T-square/resources/m-files might come in handy. 3 Using the controller designed in Question 2, but deployed on the full nonlinear system, plot once again θ1 as a function of time under your proposed control law from the same initial conditions as before. Is the system doing what you were expecting it to? (Attach the plot (only the plot - no m-files please) to the homework submission.) Hint: The m-files pendulum.m and dist.m found under T-square/resources/m-files might yet again come in handy. 4 Let ẋ1 = x2 ẋ2 = −x1 + a(1 − x21 )x2 + u. a If y = x1 , what is the relative degree of the system? b If y = x2 , what is the relative degree of the system? 5 The idea behind Control Lypapunov Functions is to, given a system ẋ = f (x, u), pick a Lyapunov function candidate V (x) and then pick u such that V̇ (x) < 0, ∀x ∈ D. Why is the notion of relative degree relevant to this control design method? 6 Using only material covered in class (no outside results are allowed), complete the following theorem (replace ???? with actual math/text) based on what you know about the connections between a nonlinear system and its linearization: Theorem: Consider the system ẋ = f (x), with f (0) = 0, and where f is locally Lipschitz continuous everywhere. If ???? then there exists a quadratic Lyapunov function V (x) such that V̇ (x) < 0, ∀x ∈ D\{0}, for some domain D. 2
Homework 3 ECE6552 Nonlinear Control Systems Magnus Egerstedt Due: March 13, 2018 (and Mar. 20 for DL students – sections Q, Q3, QSZ) 1 Consider a so-called double pendulum (a pendulum with another pendulum attached to its end). Let the angles of the two pendula be θ1 and θ2 , which gives θ̇1 = θ̇2 = 6 2pθ1 − 3 cos(θ1 − θ2 )pθ2 m`2 16 − 9 cos2 (θ1 − θ2 ) 6 8pθ2 − 3 cos(θ1 − θ2 )pθ1 , m`2 16 − 9 cos2 (θ1 − θ2 ) where m, ` are the masses and lengths of the two pendula (both pendula have the same dimensions), and where the two momenta pθ1 , pθ2 satisfy ṗθ1 ṗθ2   1 g = − m`2 θ̇1 θ̇2 sin(θ1 − θ2 ) + 3 sin(θ1 ) + u 2 `  1 2 g = − m` −θ̇1 θ̇2 sin(θ1 − θ2 ) + sin(θ2 ) , 2 ` where u is the control input (roughly equal to a scaled torque applied at the pivot point of the first pendulum) Linearize this model around the doubly-upright position, i.e., θ1 = θ2 = π, pθ1 = pθ2 = 0, to get ẋ = Ax + Bu. a What are A, B? b Is the resulting linear system completely controllable? What are the stability properties of the uncontrolled linear system? Hint: The Matlab commands eig, ctrb and rank may come in handy. 2 a Setting m = 1, ` = 1, and g = 10, using your favorite method, design a stabilizing control law for the linear system in Question 1. 1 b Plot θ1 as a function of time under your proposed control  0.1  −0.1 x(0) =   0 0 law from the initial condition   .  Attach the plot (only the plot - no m-files please) to the homework submission. Hint: The m-files pendulum.m and dist.m found under T-square/resources/m-files might come in handy. 3 Using the controller designed in Question 2, but deployed on the full nonlinear system, plot once again θ1 as a function of time under your proposed control law from the same initial conditions as before. Is the system doing what you were expecting it to? (Attach the plot (only the plot - no m-files please) to the homework submission.) Hint: The m-files pendulum.m and dist.m found under T-square/resources/m-files might yet again come in handy. 4 Let ẋ1 = x2 ẋ2 = −x1 + a(1 − x21 )x2 + u. a If y = x1 , what is the relative degree of the system? b If y = x2 , what is the relative degree of the system? 5 The idea behind Control Lypapunov Functions is to, given a system ẋ = f (x, u), pick a Lyapunov function candidate V (x) and then pick u such that V̇ (x) < 0, ∀x ∈ D. Why is the notion of relative degree relevant to this control design method? 6 Using only material covered in class (no outside results are allowed), complete the following theorem (replace ???? with actual math/text) based on what you know about the connections between a nonlinear system and its linearization: Theorem: Consider the system ẋ = f (x), with f (0) = 0, and where f is locally Lipschitz continuous everywhere. If ???? then there exists a quadratic Lyapunov function V (x) such that V̇ (x) < 0, ∀x ∈ D\{0}, for some domain D. 2

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