dr. broucke talk

Reach Control Problem

Mireille E. Broucke

Systems Control Group

Department of Electrical and Computer Engineering

University of Toronto

March 31, 2014

Acknowledgements

• Past Students

– Dr. Mohamed Helwa, PhD

– Dr. Elham Semsar-Kazerooni, Postdoc

– Krishnaa Mehta, MASc

– Graeme Ashford, MASc

– Marcus Ganness, MASc

– Dr. Zhiyun Lin, Postdoc

– Bartek Roszak, MASc

• Current Students

– Zachary Kroeze, PhD

– Matthew Martino, MASc

– Melkior Ornik, PhD

– Mario Vukosavljev, MASc

0

Logic Specifications in Control

Motivating Example 1

Linear Temporal Logic

Boolean operators: ¬, ∧, ∨

Temporal operators: � - always, ♦ - eventually, U - until

• The power in the resistor never exceeds P.

• The power through load 1 reaches P1 before the power through load 2

reaches P2.

• Two loads must be turned on but not at the same time.

• The pendulum swings through θ = 0 at least two times.

• In a multi-agent system, if a vehicle visits a target, then no vehicle will

visit the same target ever again.

• For a walking robot, eventually always the right leg touches the ground

and the left knee has an angle between θ1 and θ2.

Motivating Example 2

Cart and Conveyor Belts

Control Specifications:

1. Safety: |x| ≤ 3, |x| ≤ 3, |x+ x| ≤ 3.

2. Liveness: |x|+ |x| ≥ 1.

3. Temporal behavior: Every box arriving on conveyor 1 is picked up

and deposited on conveyor 2.

Motivating Example 3

Stabilization-based Approach

• Equilibrium stabilization gives no guarantee of safety or liveness.

• Difficult to design for trade-off between safety and liveness.

• Operationally inefficient - system must be run unnaturally slowly.

Motivating Example 4

Literature on LTL and Control

• A. Bemporad and M. Morari. Control of systems integrating logic,

dynamics, and constraints. Automatica, 1999.

• P. Tabuada and G. Pappas. Model checking LTL over controllable

linear systems is decidable. LNCS, 2003.

• M. Kloetzer and C. Belta. A fully automated framework for control of

linear systems from temporal logic specifications. IEEE TAC, 2008.

• T. Wongpiromsarn, U. Topcu, and R.M. Murray. Receding horizon

temporal logic planning for dynamical systems. CDC, 2009.

• S. Karaman and E. Frazzoli. Sampling-based motion planning with

deterministic mu-calculus specifications. CDC, 2009.

• A. Girard. Synthesis using approximately bisimilar abstractions:

state-feedback controllers for safety specifications. HSCC, 2010.

Motivating Example 5

Preferred Approach

• Safety is built in up front and is provably robust.

• Liveness can be traded off with safety by adjusting the size of the hole.

• Triangulation is used in algebraic topology, physics, numerical solution of

PDE’s, computer graphics, etc. It has a role in control theory.

Motivating Example 6

Reach Control Problem

Reach Control Problem 7

What is Given

1. An affine system

x = Ax+Bu+ a , x ∈ Rn, u ∈ R

m . (1)

2. An n-dimensional simplex S = conv{v0, . . . , vn}.

3. A set of restricted facets {F1, . . . ,Fn}.

4. One exit facet F0.

Reach Control Problem 8

Setup

v0

v1 v2C(v1) C(v2)

h1h2

B = ImB

F0

S

O = {x | Ax+ a ∈ B}

cone(S)

Notation: C(vi) :={

y ∈ Rn | hj · y ≤ 0 , j 6= i

}

Reach Control Problem 9

Problem Statement

Problem. (RCP) Given simplex S and system (1), find u(x) such

that: for each x0 ∈ S there exist T ≥ 0 and γ > 0 such that

• x(t) ∈ S for all t ∈ [0, T ],

• x(T ) ∈ F0, and

• x(t) /∈ S for all t ∈ (T, T + γ).

Notation: SS

−→ F0 by feedback of class U.

Reach Control Problem 10

Affine Feedback

Theorem. SS

−→ F0 by affine feedback u(x) = Kx+ g if and only

if

(a) The invariance conditions hold:

Avi + a+Bu(vi) ∈ C(vi), i ∈ {0, . . . , n}.

(b) The closed-loop system has no equilibrium in S.

[HvS06] L.C.G.J.M. Habets and J.H. van Schuppen. IEEE TAC 2006.

[RosBro06] B. Roszak and M.E. Broucke. Automatica 2006.

Reach Control Problem 11

Numerical Example

x =

0 1

0 0

x+

0

1

u+

4

1

S determined by: v0 = (−1,−3), v1 = (4,−1), and v2 = (3,−6).

Invariance conditions give:

• hT1 (Av0 +Bu0 + a) ≤ 0 ⇒ u0 ≥ −1.75

• hT2 (Av0 +Bu0 + a) ≤ 0 ⇒ u0 ≤ −0.6

• hT2 (Av1 +Bu1 + a) ≤ 0 ⇒ u1 ≤ 0.2

• hT1 (Av2 +Bu2 + a) ≤ 0 ⇒ u2 ≥ 0.5.

Reach Control Problem 12

Numerical Example

Choose u0 = −1.175, u1 = 0.2, u2 = 0.5.

h1

h2

v0 = (−1,−3)

v1 = (4,−1)

v2 = (3,−6)

F0

equilibrium

Reach Control Problem 13

Numerical Example

The affine feedback is: u = [0.325 − 0.125]x− 1.225

h1

h2

v0 = (−1,−3)

v1 = (4,−1)

v2 = (3,−6)

F0

equilibrium

Reach Control Problem 14

Geometric Theory

Geometric Theory 15

Equilibrium Set and Triangulations

Let B = ImB. The equilibrium set is

O = {x | Ax+ a ∈ B} .

Define

G := S ∩ O .

Assumption. If G 6= ∅, then G is a κ-dimensional face of S, where

0 ≤ κ < n. Reorder indices so that

G = conv{v1, . . . , vκ+1} .

Note: v0 6∈ G, otherwise there’s a trivial solution to RCP.

Can be achieved using the placing triangulation.

Geometric Theory 16

Continuous State Feedback

Theorem. Suppose the triangulation assumption holds. TFAE:

(a) SS

−→ F0 by affine feedback.

(b) SS

−→ F0 by continuous state feedback.

Proof. Fixed point argument using Sperner’s lemma, M -matrices.

[B10] M.E. Broucke. SIAM J. Control and Opt. 2010.

Geometric Theory 17

Limits of Continuous State Feedback

S

v0

v1

v2

y0

b1

b2

F0

F1

F2

Let u(x) be a continuous state feedback satisyfing the invariance conditions. If B = sp{b}

and G = v1v2, then

y(x) := c(x)b , x ∈ v1v2 c : Rn

→ R continuous ,

where c(v1) ≥ 0 and c(v2) ≤ 0. By Intermediate Value Theorem there exists x s.t. c(x) = 0.

Geometric Theory 18

Conditions for a Topological Obstruction

1. B ∩ cone(S) = 0 nontriviality condition

2. 6 ∃ lin. indep. set {b1, . . . , bκ+1 | bi ∈ B ∩ C(vi)} system is “underactuated”

v0

v1

v2

v3

b1b2

h3

O

B

G

cone(S)

Geometric Theory 19

Reach Control Indices

Theorem. There exist integers r1, . . . , rp ≥ 2 and control directions

bi ∈ B ∩ C(vi) such that w.l.o.g.

B ∩ C(vi) ⊂ sp{bm1 , . . . , bm1+r1−1} , i = m1, . . . ,m1 + r1 − 1 ,

......

B ∩ C(vi) ⊂ sp{bmp , . . . , bmp+rp−1} , i = mp, . . . ,mp + rp − 1 ,

where mk := r1 + · · ·+ rk−1 + 1 , k = 1, . . . , p. Moreover,

0 ∈ conv{

bmk, . . . , bmk+rk−1

}

, k = 1, . . . , p .

{r1, . . . , rp} are called the reach control indices of system (1).

[BG14] M.E. Broucke and M. Ganness. SIAM J. Control and Opt. 2014.

Geometric Theory 20

Reach Control Indices

For k = 1, . . . , p define

Gk := conv{

vmk, . . . , vmk+rk−1

}

.

Theorem. Let u(x) be a continuous state feedback satisfying the

invariance conditions. Then each Gk contains an equilibrium of the

closed-loop system.

[B10] M.E. Broucke. SIAM J. Control and Opt. 2010.

Geometric Theory 21

Piecewise Affine Feedback

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−0.2

0

0.2

0.4

0.6

0.8

1

F0

O

(a) Affine feedback

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−0.2

0

0.2

0.4

0.6

0.8

1

1.2

F0’

F0

(b) PWA feedback

Geometric Theory 22

Piecewise Affine Feedback

Theorem. Suppose the triangulation assumption holds. For a

somewhat stronger version of RCP, TFAE:

1. SS

−→ F0 by piecewise affine feedback.

2. SS

−→ F0 by open-loop controls.

[BG14] M.E. Broucke and M. Ganness. SIAM J. Control and Opt. 2014.

Geometric Theory 23

Time-varying Affine Feedback

Equilibria are such a drag...

v0v0

v1 v1v2v2xx

Geometric Theory 24

Time-varying Affine Feedback

Theorem. Suppose the triangulation assumption holds and the

invariance conditions are solvable. There exists c > 0 sufficiently small

such that SS

−→ F0 using the time-varying affine feedback

u(x, t) = e−ctu0(x) + (1− e−ct)u∞(x)

where u0(x) = K0x+ g0 places closed-loop equilibria at

vm1 , . . . , vmp

and u∞(x) = K∞x+ g∞ places closed-loop equilibria at

vm1+r1−1, . . . , vmp+rp−1 .

[AB13] G. Ashford and M. Broucke. Automatica 2013.

Geometric Theory 25

Lyapunov Theory

Lyapunov Theory 26

Revisiting RCP on Simplices

Theorem. SS

−→ F0 by the affine feedback u(x) = Kx+ g, if and

only if

(a) The invariance conditions hold.

(b) Linear Flow Condition:

(∃ξ ∈ Rn) ξ · (Ax+ Bu(x) + a) < 0, ∀x ∈ S.

Under affine feedback, the flow condition is necessary and sufficient for

leaving S in finite time.

[HvS06] Habets and van Schuppen. IEEE TAC 2006.

[RosBro06] Roszak and Broucke. Automatica 2006.

Lyapunov Theory 27

Lyapunov Theory for RCP

• For a given feedback u(x) solving RCP on a polytope P , the

invariance conditions can be easily verified, but a linear flow

condition may not exist.

• In these cases RCP is verified to be solved via exhaustive

simulation.

• Urgent need for a verification tool for RCP, analogous to

Lyapunov theory for stability.

We propose a Lyapunov-like analysis tool.

Lyapunov Theory 28

Flow Functions

Let u(x) be a locally Lipschitz state feedback satisfying the invariance

conditions of P .

To verify trajectories leave a compact set P :

• V (x) is C1, V (x) 6= 0, x ∈ P [RHL77]

• V (x) = h0 · x, V (x) = h0 · (Ax+Bu(x) + a) 6= 0, x ∈ P [HvS04]

• V (x) = ξ · x, V (x) = ξ · (Ax+Bu(x) + a) 6= 0, x ∈ P [RB06,HB11]

Definition. Let P ⊂ D. A flow function is any scalar function

V : D → R that is strictly decreasing along solutions.

Lyapunov Theory 29

Locally Lipschitz Flow Functions

Theorem. Consider system (1) on P ⊂ D. Let u(x) be a c.s.f. such

that the closed-loop vector field f(x) is locally Lipschitz on D. If there

exists a locally Lipschitz V : D → R such that

D+f V (x) ≤ −1 , x ∈ P

then all trajectories leave P in finite time.

[HB14] M. Helwa and M. Broucke. Automatica 2014.

Lyapunov Theory 30

LaSalle Principle for RCP

Theorem. Consider system (1) on P ⊂ D. Let u(x) be a c.s.f. such

that the closed-loop vector field f(x) is locally Lipschitz on D.

Suppose there exists a locally Lipschitz V : D → R such that

D+f V (x) ≤ 0, x ∈ P . Let

M := {x ∈ P | D+f V (x) = 0} .

If M does not contain an invariant set, then all trajectories starting in

P leave it in finite time.

Lyapunov Theory 31

Flow Functions on Simplices

v1 v2

v3 v4

S1

S2

F0

Theorem. Let T = {S1,S2} be a triangulation of P , and u(x) a

PWA feedback defined on T. If PP−→ F0 by u(x), then there exist

affine functions V1(x) and V2(x) such that

V (x) := max{V1(x), V2(x)}

satisfies D+f V (x) < 0 for all x ∈ P .

Lyapunov Theory 32

Analogies with Stability Theory

Stability RCP

Lyapunov function: V (x) C1 posi-

tive definite

Flow function: V (x) locally Lips-

chitz

Neg. def. condition: V (x) < 0 Flow condition: D+f V (x) < −1

LaSalle theorem: V (x) ≤ 0, trajec-

tories tend to inv. set in M

LaSalle theorem: D+f V (x) ≤ 0, no

inv. sets in M

Linear systems: V (x) = xTPx Affine systems: V (x) = ξ · x

Piecewise linear systems: V (x) =

max{Vi(x)}

PWA feedback: V (x) =

max{Vi(x)}

Lyapunov Theory 33

Emerging Applications

Emerging Applications 34

Automated Anesthesia

Emerging Applications 35

Aggressive Maneuvers of Mechanical

Systems

−1 −0.5 0 0.5 1

−1.5

−1

−0.5

0

0.5

1

1.5

S1

S2

S3

S4

S5

S6

S7

S8

S9

S10

S11

S12

S13

Trolley Position (m)

CraneAngle

(rad)

State Space Polytope

−1 −0.5 0 0.5 1−0.3

−0.2

−0.1

0

0.1

Trolley Position (m)

VerticalPosition(m

)

Crane Problem Animation

Emerging Applications 36

Bumpless Transfer of Robotic Manipulators

x0

y0

x1y1

x2

y2

x1 x2

θ1

θ2l1

l2

ks, rs

m2

ρe

ξ1

ξ2

ρ0 ρ1 ρ2ρ3

ρemin

ρemax

−µ1

µ1

µ2µsw

P1

P2

P3

P4

P5

F0

F1F2

F3

O

Emerging Applications 37

Motivating Example

B

O

B ∩ cone(S) 6= 0

S ∩ O = ∅ [B10]

B ∩ cone(S) 6= 0 [B10]

[AB13] or [BG14]

Emerging Applications 38

Final Design

−4 −3 −2 −1 0 1 2 3 4

−4

−3

−2

−1

0

1

2

3

4

Emerging Applications 39