Robust Hybrid Control Systems Andy Teel with Rafal Goebel, Ricardo - - PowerPoint PPT Presentation

Robust Hybrid Control Systems

Andy Teel

with Rafal Goebel, Ricardo Sanfelice, Chaohong Cai

Center for Control, Dynamical Systems, and Computation Department of Electrical and Computer Engineering University of California, Santa Barbara

Outline

“Hybrid” arcs and systems Robustness: illustrations and general statements “Classical” stability analysis tools in the hybrid setting Control examples Smooth “patchy” control Lyapunov functions Conclusions

“Hybrid” arcs

Continuous arc Hybrid arc Not just for bookkeeping or pedagogy. Provides a natural way of characterizing convergence of solutions: graphical convergence (from set-valued analysis). Discrete arc

x(t) x(j)

Domain

Multiple jumps at the same “t” allowed.

• P. Collins, MTNS ’04

USCB, NOLCOS ’04 Goebel/T, Automatica ‘06

Robustness from graphical convergence

2 4 6 8 5 10 5 10 15 j [jumps] tz1 tz2 t [ordinary time] tz3 x1 [m]

Looking for hybrid systems for which each sequence of solutions has a subsequence converging to a solution. This property comes “for free” for continuous ODEs and difference equations. A ball bouncing from a sequence of decreasing heights.

Hybrid arcs generated by data

System data: 1) if 2) if

x j t j t dom ) , ( ), , ( : ∈ > ∃ τ τ

Now looking for conditions on data to guarantee that each sequence of solutions has a subsequence converging to a solution. For simplicity, f and g are single-valued and continuous…

D x x g x C x x f x ∈ = ∈ =

+

) ( ) (

• x may contain variables

taking values in a discrete set, timers, etc.

“Lateral” evolution “Out of the screen” evolution

Bouncing Ball, First Pass

• g

h − =

• )

1 , ( ∈ − =

+

γ γh h

• ?

, < = h h

• {

}

: ) , ( :

1

> = h h h C

• {

}

, : ) , ( :

1

< = = h h h h D

Bouncing Ball, Pass one (cont’d)

2 4 6 8 5 10 5 10 15 j [jumps] tz1 tz2 t [ordinary time] tz3 x1 [m]

Ball bouncing from a sequence of decreasing heights: converges to an instantaneous Zeno solution that remains at zero height. But this is not a solution of the BB since the origin does not belong to D.

Bouncing Ball, Second Pass

• g

h − =

• )

1 , ( ∈ − =

+

γ γh h

• ?

, ≤ = h h

• {

}

: ) , ( :

2

≥ = h h h C

• {

}

, : ) , ( :

2

≤ = = h h h h D

• 1

2

D D =

Admits instantaneous Zeno solution at origin.

1 2

C C =

Conditions on data for convergence property

System data: If C and D are closed then each sequence of solutions has a subsequence converging to a solution. Otherwise, no guarantee. This leads to a natural notion of “generalized solutions” (a la Filippov or Krasovskii for discontinuous ODEs) for hybrid systems:

. ) ( ) ( D x x g x C x x f x ∈ = ∈ =

+

• Hybrid “Krasovskii” solutions satisfy:

Natural from a control point of view since they agree with “hybrid Hermes” solutions, i.e., the zero noise graphical limit of solutions to

. ) ( ) ( D e x e x g x C e x e x f x ∈ + + = ∈ + + =

+

• e

On “generalized” solutions

Biases against (stemming from ODEs): Rejoinders: 1) “I don’t work with discontinuous continuous-time systems, so I have no use for generalized solutions.” 2) “The solution notion is flawed because it is too restrictive; e.g., it prevents point stabilization by state feedback for mobile robots.” 1) For hybrid systems, generalized solutions are relevant even for systems without discontinuities. 2) Every asymptotically controllable nonlinear system (e.g., the mobile robot model) can be stabilized using hybrid feedback with closed flow and jump sets, i.e., using generalized solutions. 3) Value added: Converse Lyapunov theorems, LaSalle’s invariance principle, generic robustness of asymptotic stability become free.

Generalized solutions matter, Example 1

“Objection! I would implement with ‘zero crossing detection’ (in Simulink.)”

) sgn(

1 3 3

x x x ε − = =

+

• {

}

. : ) , ( : \ :

2 2 1 2

= = = x x x D D R C

) , (

2

R C D D = =

1 , ) ( 1 1 1 1 ) ( << <

• −

=

• −

= ε ε x x g x x f Generalized solutions:

• riginal solutions plus:

and combinations thereof…

{ } { }

. : : : :

2 3 2

= = ≥ = x x D x x x C

Reply: Put it in your model, e.g. Benefits: Explicit robustness, Lyapunov function exists, LaSalle available.

sequence of solutions converging to non-solution???

Or: make D “thick” perhaps a sector.

Generalized solutions matter, Example 2

Mode 1: drive to Mode 2: drive away from Target acquisition with

• bstacle avoidance
• n

.

2

Q R C × =

Extra generalized solutions

Simple way to get rid

• f extra solutions:

(preferred to zero cross detection or thickening here)

Benefits as before: Lyapunov, LaSalle,…

+ = + =

2 , 2 1 , 1

: : D D D D

new new

inside of circle it describes.

• utside of circle it describes.

}) 2 { ( }) 1 { ( , \ : \ :

2 1 , 2 2 2 , 1 2 1

× ∪ × = = = C C C D R C D R C

new new

Generalized solutions:

Q R C × =

2

LaSalle’s invariance principle

D x x u x V x g V C x x u x f x V

d c

∈ ≤ ≤ − ∈ ≤ ≤ ∇ ) ( ) ( )) ( ( ) ( ) ( ), (

then each complete, bounded trajectory converges to the largest weakly invariant set contained in

( ) ( )

r u g u u r V M

d d c r

some for ))] ( ( ) ( [ :

1 1 1 1 − − − −

∩ ∪ ∩ =

Theorem: If

x x x V x x g x f

T

=

• −

= = ) ( , 1 1 ) ( ) (

Example:

C D

r

M

Largest weakly invariant set contained in excludes all points in green except the one on the vertical axis. (solutions are not unique)

A Converse Lyapunov theorem

Theorem: If a compact set is asymptotically stable (locally stable + attractive) (using generalized solutions) then there exists a smooth Lyapunov function.

( ) ( )

D x x V x g V C x x V x f x V O x x x V x ∈ − ≤ ∈ − ≤ ∇ ∈ ≤ ≤ ) ( ) 1 exp( )) ( ( ) 3 ) ( ) ( ), ( ) 2 ) ( ) ( ) ( ) 1

2 1

ω α ω α

• positive definite with respect to the compact set,
• proper w.r.t. O, which is an open set related to the basin of attraction,

the latter which is open relative to

. D C ∪

Let be

:

≥

→ R O ω

There exists a smooth function

: , & :

2 1 ∞ ≥

∈ → K R O V α α

Significance: Can be used to show a wide variety of robustness properties. Note:

( ) ( ) ( )

) , ( ( ) exp( ) , (

2 1 1

x j t j t x ω α α ω − − ≤

−

Autonomous vs. exogenous perturbations

Message: make flow and jump sets overlap, or keep measurements out of the switching conditions, via sample and hold or filtering measurements.

( ) { } ( ) { }

∅ ≠ ∩ + = ∅ ≠ ∩ + = + + = + + = D B x x D C B x x C B B x g x G B B x f x F δ δ δ δ δ δ

δ δ δ δ

: : : : ) ( : ) ( ) ( co : ) (

Lyapunov functions can be used to show robustness to “autonomous” perturbations, i.e., inflations of (f,g,C,D):

. ) ( ) ( D e x e x g x C e x e x f x ∈ + + = ∈ + + =

+

• Solutions with exogenous

disturbances covered by those

• f inflated systems:

However, existence is not guaranteed for arbitrary signal e:

C D

) , ( ) , ( > t t e e

Local separation principles attainable.

Thermostat control and 2-measure “stability”

−10 10 20 30 40 50 60 70 80 90 100 19 20 21 22 23 24 25 26

• ff mode (q=−1)

C− C− ∩ D− D− η (engine temperature) ξ (plant temperature)

−10 10 20 30 40 50 60 70 80 90 100 19 20 21 22 23 24 25 26

• n mode (q=+1)

D+ C+ ∩ D+ C+ η (engine temperature) ξ (plant temperature)

( ) ( )

j t x j t , , ) , ( ) , (

2 ] 25 , 20 [

ω β ξ ≤ KLL ∈ β

The set of plant temperatures in the range [20,25] is not forward invariant, and thus not asymptotically stable. Nevertheless:

( )

( )

D x x V x g V C x x V x f x V x x V ∈ − ≤ ∈ − ≤ ∇ ≤ ≤ ) ( ) 1 exp( )) ( ( ) 3 ) ( ) ( ), ( ) 2 ) ( ) ( ) 1

2 2 ] 25 , 20 [ 1

ω α ξ α

A smooth Lyapunov function ensues: and gives several robustness consequences:

singular perturbations (actuators or sensors), slowly-varying parameters, sample & hold, small delays, temporal regularization, etc.

must be positive outside of dashed blue

“Hybrid handoff” from global to local controller

local

C

global

D

Trajectory due to local controller Full state feedback:

• C. Prieur/L. Praly ‘99

Extension to output feedback in the presence of norm observers.

• C. Prieur/T, in preparation

local n local

C R D \ :=

global n global

D R C \ :=

Smooth “patchy” control Lyapunov functions

Theorem: Every asymptotically controllable (to a compact set) nonlinear system admits a smooth “patchy control Lyapunov function” and, in turn, is robustly stabilizable by hybrid feedback. Remark: In a sense, the latter was already known, from results by Sontag and also Clarke et al., on the robustness that results from implementing certain discontinuous controllers with sample and hold: “Sample fast, but not too fast”. Indeed, sample and hold produces a hybrid system:

) , [ ) ( 1 ] , [ ) , ( = ∞ ∈ = = ∈ =

+ +

τ τ τ τ T x g u T u x f x

• Using “patches” instead of sample and hold allows you to keep fast sampling

without sacrificing robustness with respect to measurement noise. ) , [ ] , [

1 2

∞ ∈ ∈ T T τ τ

Time-varying sampling period

Smooth “patchy” control Lyapunov functions

Nonholonomic integrator: 3 patches, Artstein’s circles: 2 patches.

( ) ( ) ( ) ( )

' , ' , ' ' 2 1

\ ) , ( ), ( ) 2 \ ) , ( ), ( ) 2 , ) ( ) ( ) 3 \ ) ( ) 1

r q r q x q q r q r q x q q r q q r r q r q q

x x u x f x n b x x u x f x V a x q r x V x V x x x V x Ω ∪ Ω ∂ ∈ − ≤ Ω ∪ Ω ∈ − ≤ ∇ Ω ∩ Ω ∈ ≤ Ω ∪ Ω ∈ ≤ ≤

• ρ

ρ α α

An ordered index set Q (L.f.) Families of open patches that cover: smooth boundary Smooth functions Vq defined on

← Ω ⊂ Ω

q q '

Ingredients:

q

Ω

Mechanism:

' r

Ω

' q

Ω

q

Ω

Start in stay strictly in until move to new patch. Switch to index r when reach

• Patches without control Lyapunov functions

Any mechanism guaranteeing flow from a patch to a “higher” patch is fine. Inverted pendulum on a cart: 3 patches. Robustness to singular perturbations to smoothen control signal.

Robust decision making in noisy environments

Conclusions

We have emphasized robustness in hybrid systems, especially hybrid control systems. Robustness in hybrid systems relies on closed flow and jump sets,

• r “generalized solutions”.

Robustness provides many of the stability analysis tools our community has exploited for continuous-time control systems.