Lyapunov-like functions and Lie brackets Franco Rampazzo Monica - PowerPoint PPT Presentation

Brockett’s nonholonomic integrator: TRY the ( smooth!) distance function V ( x ) = d ( x , 0) as Lyapunov function: Level sets of V(x)=d(x,0) f 2 (=Spheres) f 1 f 2 f 1 Does the distance V ( x ) = d ( x , 0) verify � � < 0 ? H ( x , DV ( x )) = inf DV ( x ) , u 1 f 1 ( x ) + u 2 f 2 ( x ) u No, it doesn’t! Lyapunov-like functions and Lie brackets Franco RampazzoMonica Motta

Brockett’s nonholonomic integrator: TRY the ( smooth!) distance function V ( x ) = d ( x , 0) as Lyapunov function: Level sets of V(x)=d(x,0) f 2 (=Spheres) f 1 f 2 f 1 Does the distance V ( x ) = d ( x , 0) verify � � < 0 ? H ( x , DV ( x )) = inf DV ( x ) , u 1 f 1 ( x ) + u 2 f 2 ( x ) u No, it doesn’t! In fact, on the vertical axis one has � � H ( x , DV ( x )) = inf DV ( x ) , u 1 f 1 ( x ) + u 2 f 2 ( x ) =0 u Lyapunov-like functions and Lie brackets Franco RampazzoMonica Motta

Brockett’s nonholonomic integrator: So the distance V ( x ) = d ( x , 0) is not a Lyapunov function Lyapunov-like functions and Lie brackets Franco RampazzoMonica Motta

Brockett’s nonholonomic integrator: So the distance V ( x ) = d ( x , 0) is not a Lyapunov function because the system dynamics is in the kernel of DV ( x ) . Lyapunov-like functions and Lie brackets Franco RampazzoMonica Motta

Brockett’s nonholonomic integrator: So the distance V ( x ) = d ( x , 0) is not a Lyapunov function because the system dynamics is in the kernel of DV ( x ) . Maybe another smooth function would work? Lyapunov-like functions and Lie brackets Franco RampazzoMonica Motta

Brockett’s nonholonomic integrator: So the distance V ( x ) = d ( x , 0) is not a Lyapunov function because the system dynamics is in the kernel of DV ( x ) . Maybe another smooth function would work? NO! Lyapunov-like functions and Lie brackets Franco RampazzoMonica Motta

Brockett’s nonholonomic integrator: So the distance V ( x ) = d ( x , 0) is not a Lyapunov function because the system dynamics is in the kernel of DV ( x ) . Maybe another smooth function would work? NO! Actually, by algebraic topological arguments (essentially the hairy ball theorem ) one can prove that No (smooth) Lyapunov functions exist Lyapunov-like functions and Lie brackets Franco RampazzoMonica Motta

”What to do?” Lyapunov-like functions and Lie brackets Franco RampazzoMonica Motta

”What to do?” Nonsmooth Answer: Lyapunov-like functions and Lie brackets Franco RampazzoMonica Motta

”What to do?” Nonsmooth Answer: ”Avoid bad points where H ( x , DV ) = 0 by allowing ... Lyapunov functions which are nonsmooth at those bad points” Lyapunov-like functions and Lie brackets Franco RampazzoMonica Motta

”What to do?” Nonsmooth Answer: ”Avoid bad points where H ( x , DV ) = 0 by allowing ... Lyapunov functions which are nonsmooth at those bad points” (this rules out the distance function in the example) Lyapunov-like functions and Lie brackets Franco RampazzoMonica Motta

Formal definition of nonsmooth Lyapunov Function: (replace DV with D ∗ V ) Lyapunov-like functions and Lie brackets Franco RampazzoMonica Motta

Formal definition of nonsmooth Lyapunov Function: (replace DV with D ∗ V ) A map V : R n → R + is a Control Lyapunov Function ( CLF ), if V is continuous, locally semiconcave, proper; V ( x ) > 0 if x / ∈ C and V ( x ) = 0 if x ∈ C ; It verifies the partial differential inequality : H ( x , D ∗ V ( x )) = min u ∈ U � D ∗ V ( x ) , f ( x , u ) � < 0 Lyapunov-like functions and Lie brackets Franco RampazzoMonica Motta

Formal definition of nonsmooth Lyapunov Function: (replace DV with D ∗ V ) A map V : R n → R + is a Control Lyapunov Function ( CLF ), if V is continuous, locally semiconcave, proper; V ( x ) > 0 if x / ∈ C and V ( x ) = 0 if x ∈ C ; It verifies the partial differential inequality : H ( x , D ∗ V ( x )) = min u ∈ U � D ∗ V ( x ) , f ( x , u ) � < 0 Here D ∗ V ( x ) denotes the set of limiting gradients of V at x : D ∗ V ( x ) . � � = w : w = lim k DV ( x k ) , lim k z k = z . Remark: In general D ∗ V ( x ) is not convex. Lyapunov-like functions and Lie brackets Franco RampazzoMonica Motta

H ( x , D ∗ V ( x )) = min u ∈ U � D ∗ V ( x ) , f ( x , u ) � < 0 Lyapunov-like functions and Lie brackets Franco RampazzoMonica Motta

H ( x , D ∗ V ( x )) = min u ∈ U � D ∗ V ( x ) , f ( x , u ) � < 0 a non-homogeneous special case: H ℓ ( x , D ∗ V ( x )) = min u ∈ U ( � D ∗ V ( x ) , f ( x , u ) � + p 0 ℓ ( x , u )) < 0 for some p 0 ≥ 0, where ℓ ( x , u ) ≥ 0 is a current cost . V is called p 0 -Minimum Restraint Function : if p 0 > 0 its existence guarantees (Motta-Rampazzo 2013): Global Asymptotic Controllability, � T x A bound on the value function W = inf ℓ ( x ( t ) , u ( t )) dx 0 W ≤ V / p 0 . Lyapunov-like functions and Lie brackets Franco RampazzoMonica Motta

H ( x , D ∗ V ( x )) = min u ∈ U � D ∗ V ( x ) , f ( x , u ) � < 0 a non-homogeneous special case: H ℓ ( x , D ∗ V ( x )) = min u ∈ U ( � D ∗ V ( x ) , f ( x , u ) � + p 0 ℓ ( x , u )) < 0 for some p 0 ≥ 0, where ℓ ( x , u ) ≥ 0 is a current cost . V is called p 0 -Minimum Restraint Function : if p 0 > 0 its existence guarantees (Motta-Rampazzo 2013): Global Asymptotic Controllability, � T x A bound on the value function W = inf ℓ ( x ( t ) , u ( t )) dx 0 W ≤ V / p 0 . I am NOT speaking of this special case today Lyapunov-like functions and Lie brackets Franco RampazzoMonica Motta

In the case of Brockett’s nonholonomic integrator �� � � x 2 1 + x 2 x 2 1 + x 2 one can try V = max 2 , | x 3 | − , 2 which has singularities on the vertical axis : V(x)= cost. f 2 f 1 Lyapunov-like functions and Lie brackets Franco RampazzoMonica Motta

In the case of Brockett’s nonholonomic integrator �� � � x 2 1 + x 2 x 2 1 + x 2 one can try: V = max 2 , | x 3 | − , 2 which has singularities on the vertical axis H = 0 avoided! D*V V(x)= cost. Notice: NO VERTICAL f 2 GRADIENTS f 1 Lyapunov-like functions and Lie brackets Franco RampazzoMonica Motta

Why nonsmooth Lyapunov control functions are important? Lyapunov-like functions and Lie brackets Franco RampazzoMonica Motta

Why nonsmooth Lyapunov control functions are important? Because they are useful, namely we can extend the smooth Lyapunov-like theorem: Lyapunov-like functions and Lie brackets Franco RampazzoMonica Motta

Why nonsmooth Lyapunov control functions are important? Because they are useful, namely we can extend the smooth Lyapunov-like theorem: Nonsmooth Lyapunov-like Theorem: If there exists a Control Lyapunov Function then the system is Globally Asymptotically Controllable Lyapunov-like functions and Lie brackets Franco RampazzoMonica Motta

Why nonsmooth Lyapunov control functions are important? Because they are useful, namely we can extend the smooth Lyapunov-like theorem: Nonsmooth Lyapunov-like Theorem: If there exists a Control Lyapunov Function then the system is Globally Asymptotically Controllable Many important results since the 80’s, with various notions of nonsmooth gradients and/or generalized notions of ODE solutions. Quite incomplete list of authors includes : Sontag, Artstein,Bacciotti,Clarke, Subbotin,Malisoff, Rifford Lyapunov-like functions and Lie brackets Franco RampazzoMonica Motta

What if we insist with smooth functions ? Lyapunov-like functions and Lie brackets Franco RampazzoMonica Motta

What if we insist with smooth functions ? For instance, some function V such that it is useful as a Lyapunov function (i.e., a Lyapunov-likeTheorem holds true) it has more chances to be smooth ??? Lyapunov-like functions and Lie brackets Franco RampazzoMonica Motta

IDEA: USE NON-COMMUTATIVITY Lyapunov-like functions and Lie brackets Franco RampazzoMonica Motta

A movie on non-commutativity, in R 3 : the ”Nonholonomic integrator” Lyapunov-like functions and Lie brackets Franco RampazzoMonica Motta

A movie on non-commutativity, in R 3 : the ”Nonholonomic integrator”     1 0 f 1 = 0 f 2 = 1     − x 2 x x = u 1 f 1 + u 2 f 2 ˙ Lyapunov-like functions and Lie brackets Franco RampazzoMonica Motta

A movie of non-commutativity, in R 3 : the ”Nonholonomic integrator” f 2 f 1 Lyapunov-like functions and Lie brackets Franco RampazzoMonica Motta

A movie of non-commutativity, in R 3 : the ”Nonholonomic integrator” f 2 f 1 Φ t f 1 ( x ) Lyapunov-like functions and Lie brackets Franco RampazzoMonica Motta

A movie of non-commutativity, in R 3 : the ”Nonholonomic integrator” f 2 f 1 Φ t f 2 ◦ Φ t f 1 ( x ) Lyapunov-like functions and Lie brackets Franco RampazzoMonica Motta

A movie of non-commutativity, in R 3 : the ”Nonholonomic integrator” -f 1 f 1 f 2 Φ − t f 1 ◦ Φ t f 2 ◦ Φ t f 1 ( x ) Lyapunov-like functions and Lie brackets Franco RampazzoMonica Motta

A movie of non-commutativity, in R 3 : the ”Nonholonomic integrator” -f 2 -f 1 f 1 f 2 Φ − t f 2 ◦ Φ − t f 1 ◦ Φ t f 2 ◦ Φ t f 1 ( x ) � = x Lyapunov-like functions and Lie brackets Franco RampazzoMonica Motta

Lie brackets Definition Lie bracket of C 1 vector fields f , g : [ f , g ] := Dg · f − Df · g Lyapunov-like functions and Lie brackets Franco RampazzoMonica Motta

Lie brackets Definition Lie bracket of C 1 vector fields f , g : [ f , g ] := Dg · f − Df · g Basic properties: [ f , g ] is a vector field (i.e. it is an intrinsic object) 1 [ f , g ] = − [ g , f ] (antisymmetry) ( = ⇒ [ f , f ] = 0) 2 [ f , [ g , h ]] + [ g , [ h , f ]] + [ h , [ f , g ]] = 0 (Jacobi identy) 3 Lyapunov-like functions and Lie brackets Franco RampazzoMonica Motta

Remind: The asymptotics of a Lie bracket: Lyapunov-like functions and Lie brackets Franco RampazzoMonica Motta

Remind: The asymptotics of a Lie bracket: Set Ψ [ f 1 , f 2 ] ( t )( x ) := Φ − t f 2 ◦ Φ − t f 1 ◦ Φ t f 2 ◦ Φ t f 1 ( x ) Lyapunov-like functions and Lie brackets Franco RampazzoMonica Motta

Remind: The asymptotics of a Lie bracket: Set Ψ [ f 1 , f 2 ] ( t )( x ) := Φ − t f 2 ◦ Φ − t f 1 ◦ Φ t f 2 ◦ Φ t f 1 ( x ) Asymptotic formula: Ψ [ f 1 , f 2 ] ( t )( x ) − x = t 2 [ f 1 , f 2 ]( x ) + o ( t 2 ) Lyapunov-like functions and Lie brackets Franco RampazzoMonica Motta

Remind: The asymptotics of a Lie bracket: Set Ψ [ f 1 , f 2 ] ( t )( x ) := Φ − t f 2 ◦ Φ − t f 1 ◦ Φ t f 2 ◦ Φ t f 1 ( x ) Asymptotic formula: Ψ [ f 1 , f 2 ] ( t )( x ) − x = t 2 [ f 1 , f 2 ]( x ) + o ( t 2 ) Lyapunov-like functions and Lie brackets Franco RampazzoMonica Motta

Sophus Lie Figure: Continuous (Lie!) groups, geometry, ODEs Lyapunov-like functions and Lie brackets Franco RampazzoMonica Motta

Observe: Lie brackets show up in higher order necessary conditions for minima controllability boundary conditions of HJ equations, to guarantee continuity of time optimal functions BUT Lyapunov-like functions and Lie brackets Franco RampazzoMonica Motta

Observe: Lie brackets show up in higher order necessary conditions for minima controllability boundary conditions of HJ equations, to guarantee continuity of time optimal functions BUT they are not included in HJ equations or HJ inequalities Lyapunov-like functions and Lie brackets Franco RampazzoMonica Motta

A new PDI ( defining a Lyapunov’like function ) Lyapunov-like functions and Lie brackets Franco RampazzoMonica Motta

A new PDI ( defining a Lyapunov’like function ) For simplicity we limit our attention to control-linear systems: m � x = ˙ u i g i ( x ) u = ± e 1 , . . . , ± e m i =1 Lyapunov-like functions and Lie brackets Franco RampazzoMonica Motta

A new PDI ( defining a Lyapunov’like function ) For simplicity we limit our attention to control-linear systems: m � x = ˙ u i g i ( x ) u = ± e 1 , . . . , ± e m i =1 Set F (1) := � � ± f 1 , . . . , ± f m Lyapunov-like functions and Lie brackets Franco RampazzoMonica Motta

A new PDI ( defining a Lyapunov’like function ) For simplicity we limit our attention to control-linear systems: m � x = ˙ u i g i ( x ) u = ± e 1 , . . . , ± e m i =1 Set F (1) := � � F (2) := F (1) ∪ � � ± f 1 , . . . , ± f m [ f i , f j ] i , j = 1 , . . . , m . . . Lyapunov-like functions and Lie brackets Franco RampazzoMonica Motta

A new PDI ( defining a Lyapunov’like function ) For simplicity we limit our attention to control-linear systems: m � x = ˙ u i g i ( x ) u = ± e 1 , . . . , ± e m i =1 Set F (1) := � � F (2) := F (1) ∪ � � ± f 1 , . . . , ± f m [ f i , f j ] i , j = 1 , . . . , m . . . F ( j ) := F ( j − 1) ∪ { Lie brackets of degree j } Lyapunov-like functions and Lie brackets Franco RampazzoMonica Motta

A new PDI ( defining a Lyapunov’like function ) For simplicity we limit our attention to control-linear systems: m � x = ˙ u i g i ( x ) u = ± e 1 , . . . , ± e m i =1 Set F (1) := � � F (2) := F (1) ∪ � � ± f 1 , . . . , ± f m [ f i , f j ] i , j = 1 , . . . , m . . . F ( j ) := F ( j − 1) ∪ { Lie brackets of degree j } H ( j ) ( x , p ) := v ∈ F ( j ) ( x ) � p , v � inf Lyapunov-like functions and Lie brackets Franco RampazzoMonica Motta

A new PDI ( defining a Lyapunov’like function ) For simplicity we limit our attention to control-linear systems: m � x = ˙ u i g i ( x ) u = ± e 1 , . . . , ± e m i =1 Set F (1) := � � F (2) := F (1) ∪ � � ± f 1 , . . . , ± f m [ f i , f j ] i , j = 1 , . . . , m . . . F ( j ) := F ( j − 1) ∪ { Lie brackets of degree j } H ( j ) ( x , p ) := v ∈ F ( j ) ( x ) � p , v � inf Notice that H (1) = H and Lyapunov-like functions and Lie brackets Franco RampazzoMonica Motta

A new PDI ( defining a Lyapunov’like function ) For simplicity we limit our attention to control-linear systems: m � x = ˙ u i g i ( x ) u = ± e 1 , . . . , ± e m i =1 Set F (1) := � � F (2) := F (1) ∪ � � ± f 1 , . . . , ± f m [ f i , f j ] i , j = 1 , . . . , m . . . F ( j ) := F ( j − 1) ∪ { Lie brackets of degree j } H ( j ) ( x , p ) := v ∈ F ( j ) ( x ) � p , v � inf Notice that H (1) = H and H = H (1) ≥ H (2) ≥ . . . H ( k − 1) ≥ H ( k ) Lyapunov-like functions and Lie brackets Franco RampazzoMonica Motta

A new PDI ( defining a Lyapunov’like function ) Lyapunov-like functions and Lie brackets Franco RampazzoMonica Motta

A new PDI ( defining a Lyapunov’like function ) Definition Let U : R n \ C → R be continuous function, locally semiconcave, positive definite, and proper. If H ( h ) ( x , D ∗ U ( x )) < 0 we say that U is a degree-h Control Lyapunov Function Lyapunov-like functions and Lie brackets Franco RampazzoMonica Motta

A new PDI ( defining a Lyapunov’like function ) Definition Let U : R n \ C → R be continuous function, locally semiconcave, positive definite, and proper. If H ( h ) ( x , D ∗ U ( x )) < 0 we say that U is a degree-h Control Lyapunov Function Remark: if h 1 ≤ h 2 , U is a degree- h 1 CLF = ⇒ U is a degree- h 2 CLF Lyapunov-like functions and Lie brackets Franco RampazzoMonica Motta

A new PDI ( defining a Lyapunov’like function ) Definition Let U : R n \ C → R be continuous function, locally semiconcave, positive definite, and proper. If H ( h ) ( x , D ∗ U ( x )) < 0 we say that U is a degree-h Control Lyapunov Function Remark: if h 1 ≤ h 2 , U is a degree- h 1 CLF = ⇒ U is a degree- h 2 CLF Indeed: H = H (1) ≥ H (2) ≥ . . . H ( k − 1) ≥ H ( k ) Lyapunov-like functions and Lie brackets Franco RampazzoMonica Motta

Why degree-h control Lyapunov functions are useful? ( h ≥ 1 ) Lyapunov-like functions and Lie brackets Franco RampazzoMonica Motta

Why degree-h control Lyapunov functions are useful? ( h ≥ 1 ) Because a ”Lyapunov-like” theorem holds true! Lyapunov-like functions and Lie brackets Franco RampazzoMonica Motta

Why degree-h control Lyapunov functions are useful? ( h ≥ 1 ) Because a ”Lyapunov-like” theorem holds true! degree-h Lyapunov-like Theorem: If there exists a degree- h Control Lyapunov Function then the system is Globally Asymptotically Controllable Lyapunov-like functions and Lie brackets Franco RampazzoMonica Motta

Why degree-h control Lyapunov functions are useful? ( h ≥ 1 ) Because a ”Lyapunov-like” theorem holds true! degree-h Lyapunov-like Theorem: If there exists a degree- h Control Lyapunov Function then the system is Globally Asymptotically Controllable Moreover, degree-h Control Lyapunov Functions are likely smoother than stan- dard CLFs Lyapunov-like functions and Lie brackets Franco RampazzoMonica Motta

Application to the nonholonomic integrator Lyapunov-like functions and Lie brackets Franco RampazzoMonica Motta

Application to the nonholonomic integrator x = u 1 f 1 + u 2 f 2 ˙ Lyapunov-like functions and Lie brackets Franco RampazzoMonica Motta

Application to the nonholonomic integrator x = u 1 f 1 + u 2 f 2 ˙       1 0 0  ,  , f 1 = 0 f 2 = 1 [ f 1 , f 2 ] = 0     − x 2 x 1 2 Lyapunov-like functions and Lie brackets Franco RampazzoMonica Motta

Application to the nonholonomic integrator x = u 1 f 1 + u 2 f 2 ˙       1 0 0  ,  , f 1 = 0 f 2 = 1 [ f 1 , f 2 ] = 0     − x 2 x 1 2 Trivial calculations give: ( p 1 − x 2 p 3 ) 2 + ( p 2 + x 2 p 3 ) 2 H 1 ( x , p ) = H ( x , p ) = − � � � ( p 1 − x 2 p 3 ) 2 + ( p 2 + x 2 p 3 ) 2 , − | p 3 | H 2 ( x , p ) = min � − 16 Lyapunov-like functions and Lie brackets Franco RampazzoMonica Motta