MIT16_30F10_lec23

# MIT16_30F10_lec23 - Topic #23 16.30/31 Feedback Control...

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Topic #23 16.30/31 Feedback Control Systems Analysis of Nonlinear Systems Anti-windup Notes developed in part from 16.30 Estimation and Con- trol of Aerospace Systems, Lecture 21: Lyapunov Stability Theory by Prof. Frazzoli

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Fall 2010 16.30/31 23–2 Nonlinear System Analysis Example: Car cruise control 1 g F mg F Equation of motion in the direction parallel to the road surface: dv m = F eng + F aero + F frict + F g . dt where 1 F aero = ρC d Av ·| v | , 2 F g = mg sin( θ ) , F frict = mgC r cos( θ ) sgn ( v ) . Engine model 200 200 Torque T [Nm] 0 20 40 60 n =1 n =2 n =3 n =4 n =5 180 180 [Nm] 160 140 160 140 120 120 0 100 200 400 600 100 Angular velocity [rad/s] Velocity v [m/s] ± ² ³ 2 Engine torque (at full throttle): T ω = T m 1 β ω ω m 1 , where ω = n r v = α n v , n is gear ratio, and r wheel radius. The engine driving force can hence be written as F eng = α n T ( α n v ) u, 0 u 1 . 1 The example is taken from ˚ om and Murray: Feedback Systems, 2008 Astr¨ November 21, 2010
± ± ± ± ² ³ ´µ ³ ´µ Fall 2010 16.30/31 23–3 Jacobian Linearization Any (feasible) speed corresponds to an equilibrium point. Choose a reference speed v ref > 0 , and solve for dv/dt =0 with respect to u , assuming a horizontal road ( θ =0 ). 1 0= α n T ( α n v ¯)¯ u ρC d Av ¯ 2 mgC r 2 i.e., 1 ρC d Av ¯ 2 + mgC r u ¯ = 2 . α n T ( α n v ¯) Linearized system ( ξ = v v ¯ , η = u u ¯ ): d 1 ∂T ( α n v ) 1 ξ = α n u ¯ ρC d Av ¯ ξ + α n T ( α n v ¯) η dt ∂v m m v ¯ A dyn B dyn Example: numerical values Let us use the following numerical values (all units in SI): T m = 190 =0 . 4 m = 420 5 =10 ,C r =0 . 01 , m = 1500 ,g =9 . 81 =1 . 2 ,C d A =0 . 79 . For v ¯=25 (90 km/h, or 55 mph), we get u ¯=0 . 2497 . The linearization yields: A dyn = 0 . 0134 , B dyn =1 . 1837 1 . 1837 G ( s )= s +0 . 0134 November 21, 2010

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Fall 2010 16.30/31 23–4 Cruise control design A proportional controller would stabilize the closed-loop system. Assume
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## MIT16_30F10_lec23 - Topic #23 16.30/31 Feedback Control...

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