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Backstepping From simple designs to take-off Ola Hrkegrd Control & Communication Linkpings universitet Ola Hrkegrd Internal seminar AUTOMATIC CONTROL COMMUNICATION SYSTEMS Backstepping: From simple designs to take-off January

  1. Backstepping From simple designs to take-off Ola Härkegård Control & Communication Linköpings universitet Ola Härkegård Internal seminar AUTOMATIC CONTROL COMMUNICATION SYSTEMS Backstepping: From simple designs to take-off January 27, 2005 LINKÖPINGS UNIVERSITET Backstepping Constructive (=systematic) control design for nonlinear systems � Applies to systems of lower triangular form � ( ) x f x , x = � 1 1 1 2 (essentially same as for ( ) x f x , x , x = � feedback linearization) 2 2 1 2 3 � ( ) x f x , x , x , , x , u = � � n n 1 2 3 n Can be used to avoid cancellation of "useful nonlinearities" � (unlike feedback linearization) Different flavours: adaptive , robust and observer backstepping � Ola Härkegård Internal seminar AUTOMATIC CONTROL COMMUNICATION SYSTEMS Backstepping: From simple designs to take-off January 27, 2005 LINKÖPINGS UNIVERSITET 1

  2. Paper statistics IEEE Explore 1990-2003 Backstepping in title Conference papers: 194 30 � Conference papers 25 Journal papers Not adaptive: 108 20 � 15 Applied 2003: 16 of 25 � 10 5 0 92 93 94 95 96 97 98 99 00 01 02 03 Ola Härkegård Internal seminar AUTOMATIC CONTROL COMMUNICATION SYSTEMS Backstepping: From simple designs to take-off January 27, 2005 LINKÖPINGS UNIVERSITET References � Books � Nonlinear and Adaptive Control Design, 1995 (Krstic, Kanellakopolous, Kokotovic ). � Constructive Nonlinear Control, 1997 (Sepulchre, Jankovic , Kokotovic ). � Any recent textbook on nonlinear control. � Papers � The joy of feedback: nonlinear and adaptive, 1991 Bode lecture (Kokotovic). � Constructive nonlinear control: a historical perspective, Automatica, 2001 (Kokotovic , Arcak). Ola Härkegård Internal seminar AUTOMATIC CONTROL COMMUNICATION SYSTEMS Backstepping: From simple designs to take-off January 27, 2005 LINKÖPINGS UNIVERSITET 2

  3. Lyapunov stability (geometric interpretation) ( ) x = f x Dynamics: � � 3 ( ) Lyapunov function: V x 2 � 1 V � V f 0 For stability: = x ≤ � 0 x2 ( ) ( ) x f x g x u Dynamics: = + -1 � � -2 ( ) f V is a CLF if x V � x -3 -4 -3 -2 -1 0 1 2 3 4 V V f V gu 0 � = + < x1 x x for some u Ola Härkegård Internal seminar AUTOMATIC CONTROL COMMUNICATION SYSTEMS Backstepping: From simple designs to take-off January 27, 2005 LINKÖPINGS UNIVERSITET Example 1 (backstepping) ~ Step 2: x x x x x 2 x = − = + + x x 2 x = + � 2 2 2 , d 2 1 1 1 1 2 ~ x u φ = � x x x  = − + �  2 1 1 2  ~ ) ( ~ ) ( �  x u 2 x 1 x x = + + − +  2 1 1 2 x x 2 x Step 1: = − − ~ 2 , d 1 1 V = x 2 + x 2 1 1 2 1 2 2 2 V = x 2 1 ( ~ ) ~ ( ) 1 1 V � x x x x u 2 = − + + + φ 2 1 1 2 2 V � x x x 2 0 = = − ≤ � ~ ( ) 1 1 1 1 = − x 2 + x x + u + φ 1 2 1 x = x if ~ 2 2 , d x 2 x 2 0 = − − ≤ 1 2 ~ u x x = − − φ − if 1 2 Ola Härkegård Internal seminar AUTOMATIC CONTROL COMMUNICATION SYSTEMS Backstepping: From simple designs to take-off January 27, 2005 LINKÖPINGS UNIVERSITET 3

  4. Example 1 (feedback linearization) x x 2 x = + � 1 1 2 x u = � Which control law Which control law 2 � should I choose? should I choose? y = x = z 1 1 z x 2 x z = + = � 1 1 2 2 Same control law with z 2 z z u = + Same control law with � 2 1 2 k = k = 2 � 1 2 u = − 2 z z − k z − k z gives stability 1 2 1 1 2 2 Ola Härkegård Internal seminar AUTOMATIC CONTROL COMMUNICATION SYSTEMS Backstepping: From simple designs to take-off January 27, 2005 LINKÖPINGS UNIVERSITET "Conclusion" Backstepping with linearizing virtual control laws and quadratic Lyapunov functions Feedback linearization Easier to tune Ola Härkegård Internal seminar AUTOMATIC CONTROL COMMUNICATION SYSTEMS Backstepping: From simple designs to take-off January 27, 2005 LINKÖPINGS UNIVERSITET 4

  5. Example 1 (backstepping) ~ Step 2: x = x − x = x + x 2 + x x x 2 x = + � 2 2 2 , d 2 1 1 1 1 2 ~ x u φ = � x x x  = − + �  2 1 1 2  ~ ) ( ~ ) ( �  x u 2 x 1 x x = + + − +  2 1 1 2 x x 2 x Step 1: = − − ~ 2 , d 1 1 V = x 2 + x 2 1 1 2 1 2 2 2 V = x 2 1 ( ~ ) ~ ( ) 1 1 V � x x x x u 2 = − + + + φ 2 1 1 2 2 V � x x x 2 0 = = − ≤ � ~ ( ) 1 1 1 1 = − x 2 + x x + u + φ 1 2 1 x = x if ~ 2 2 , d x 2 x 2 0 = − − ≤ 1 2 ~ u x x = − − φ − if 1 2 Ola Härkegård Internal seminar AUTOMATIC CONTROL COMMUNICATION SYSTEMS Backstepping: From simple designs to take-off January 27, 2005 LINKÖPINGS UNIVERSITET Example 2 (adaptive backstepping) ~ Step 2: x x x x x 2 x = − = + + x x 2 x = + � 2 2 2 , d 2 1 1 1 1 2 ~ x u x 2 = + θ � x x x  = − + �  2 2 1 1 2  ~ ) ( ~ ) ( �  x u 2 x 1 x x x 2 = + + − + + θ  2 1 1 2 2 ( ) x x 2 x Step 1: = − − ~ ˆ 2 V x 2 x 2 2 , d 1 1 = 1 + 1 + 1 θ − θ 2 2 1 2 2 2 ( ) V = x 2 1 ~ ( ) ˆ � ˆ V � x 2 x x u x 2 1 1 = − + + + φ + θ − θ − θ θ 2 2 1 2 1 2 V � x x x 2 0 = = − ≤ � ~ ˆ u x x x 1 1 1 1 = − − φ − − θ 2 gives 1 2 2 ( ) ~ ~ ˆ  � ˆ  V � = − x 2 − x 2 + θ − θ x x 2 − θ 0   ≤ 2 2 2 1 2   ~ � ˆ x 2 x 2 if θ = − 2 Ola Härkegård Internal seminar AUTOMATIC CONTROL COMMUNICATION SYSTEMS Backstepping: From simple designs to take-off January 27, 2005 LINKÖPINGS UNIVERSITET 5

  6. Example 3 (useful nonlinearity) ~ x = x − x = x + x x x 3 x x Step 2: = − + + � 2 2 2 , d 2 1 1 1 1 2 ~ x u = �  x = − x 3 + x �  2 1 1 2  ~ ~ �  x = u − x 3 + x  2 1 2 6 x 3 + ~ ( ) − x 4 V = W x + x 1 2 1 2 2 2 ( ) ( ) ( ) ~ ~ ~ ′ V � = W x − x 3 + x + x u − x 3 + x 0 2 1 1 2 2 1 2 ( ) -2 ′ ~ ′ ~ ( ) ( ) W x x x W x u x x = − 3 + + − 3 + -4 1 1 2 1 1 2 -6 -2 -1 0 1 2 ( ) With we can select W x = x 4 1 x = − x Step 1: 1 1 4 2 , d 1 ~ ( ) ~ u 3 x V = W x = − to achieve V � x 6 2 x 2 0 = − − ≤ 1 1 2 2 1 2 Ola Härkegård Internal seminar AUTOMATIC CONTROL COMMUNICATION SYSTEMS Backstepping: From simple designs to take-off January 27, 2005 LINKÖPINGS UNIVERSITET Example 3 (control law properties) Cascaded control implementation � ~ x x u − 2 , d x 2 = u x x 3 x x 2 3 = − + + 0 � � + − + − x 1 1 1 2 x 2 1 No cancellations ⇒ gain margin (1/3, ∞ ) � Inverse optimal, minimizes � ( ) ∞ ( ) ∫ 2 x 6 + x + x + u 2 dt 1 1 1 1 2 2 6 0 Ola Härkegård Internal seminar AUTOMATIC CONTROL COMMUNICATION SYSTEMS Backstepping: From simple designs to take-off January 27, 2005 LINKÖPINGS UNIVERSITET 6

  7. Design paths backstepping System CLF, control law backstepping Sontag’s formula control law System CLF (inverse optimal) ( ) ( ( ) ( ) ) ( ) 2 4 z cz z 2 z z z cz z z cz z cz + + + + + + + Ex 1: u 2 z z 1 2 1 1 2 2 1 2 1 2 2 1 = − − 1 2 cz + z 1 2 control law H ∞ backstepping System CLF (locally optimal, (Ezal et al. 2000) globally inv. opt.) Ola Härkegård Internal seminar AUTOMATIC CONTROL COMMUNICATION SYSTEMS Backstepping: From simple designs to take-off January 27, 2005 LINKÖPINGS UNIVERSITET Backstepping control of a rigid body � Plug-and-play flight controller � Plug-and-play flight controller � Use vector description of dynamics for control design � Use vector description of dynamics for control design (Extension of CDC 2002 paper by Glad & Härkegård.) Ola Härkegård Internal seminar AUTOMATIC CONTROL COMMUNICATION SYSTEMS Backstepping: From simple designs to take-off January 27, 2005 LINKÖPINGS UNIVERSITET 7

  8. Control problem ω u � 6 states: V, ω F � 4 inputs: u F , u M V u M Ola Härkegård Internal seminar AUTOMATIC CONTROL COMMUNICATION SYSTEMS Backstepping: From simple designs to take-off January 27, 2005 LINKÖPINGS UNIVERSITET Dynamics Gravity Aerodynamics ω Engine u ( ) F ˆ m V mV f V , t u n � = − ω × + + F J J u ω = − ω × ω + � M ˆ γ V Stationary motion: 0 V u � M V V = 0 ( ) ˆ ω = g V + γ V 0 0 Ola Härkegård Internal seminar AUTOMATIC CONTROL COMMUNICATION SYSTEMS Backstepping: From simple designs to take-off January 27, 2005 LINKÖPINGS UNIVERSITET 8

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