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.


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.


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.


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.


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.


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.


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

}


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.


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.


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.


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


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


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


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


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


Motivating Example

B

O

B ∩ cone(S) 6= 0

S ∩ O = ∅ [B10]

B ∩ cone(S) 6= 0 [B10]

[AB13] or [BG14]


Final Design

−4 −3 −2 −1 0 1 2 3 4

−4

−3

−2

−1

0

1

2

3

4


dr. broucke talk

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