Control Theory, Networks, and Life ItselfLynn Margulis and Dorion Sagan, What Is Life?, University of California Press (August 31, 2000), 303 pages ISBN-10: 0520220218 Prior outrageously

111

Control Theory, Networks, and Life Itself

John BaillieulC.I.S.E. and

Intelligent Mechatonics Laboratory

[email protected]://people.bu.edu/johnb

http://iml.bu.eduhttp://www.bu.edu/systems

Boston UniversityMarch 5, 2008

222

John Baillieul’s Research GroupIntelligent Mechatronics Laboratory

333

John Baillieul’s Hiking Friends

444

Boston University---from humble beginnings

555

Prior outrageously titled lectures (and books)

Statutory Lectures, 1943, TrinityCollege, Dublin• February 5, 1943• February 12, 1943• February 19, 1943

“How can the events in space and time which take place within the boundary of a living organism be accounted for by physics and chemistry?”

666

Professor Lynn MargulisBoston UniveristyUniversity Lecture 1978The Early Evolution of Life

Lynn Margulis and DorionSagan, What Is Life?,University of California Press (August 31, 2000), 303 pagesISBN-10: 0520220218

Prior outrageously titled lectures (and books)

777

Control Theory: Talk Outline

I. The hidden technology--where is it hidingII. Information-based control - the basis of engineered and

natural systems regulated by feedbackIII. Networked Control Systems

- Technologies enabled by information-based control- Complex networks of natural and engineered systems

IV. The emerging theory of complexity-based risk and the rational management of failure

V. Controlling highly structured motions of groups of agents1) The theory of rigid formations2) Spatial and dynamic information patterns

VI. Conclusions

888

People in Control at B.U.

Professor David A. Castañon42-nd President of the IEEE Control Systems Society

Professor Christos G. CassandrasEditor-in-Chief IEEE Transactions on Automatic Control

999

SIAM J. on Control andOptimization

World’s leading journals at B.U.

101010

The Hidden Technology

• Pervasive• Very successful• Seldom talked aboutExcept when there is an accident!Rare occasions!• Why?Easier to discuss devices than ideas(feedback)We have not done our job!

K. J. Åström ECC August 31, 1999

111111

121212

The World’s largest physics experiment is enabled by control tehnologies

• 27 km. (18 mi.)circumference• Beam steered, collimated by superconducting magnets• Beam comprised of 2835 bunches of 1011 protons

• Experiment operates at ~0oK• 12M litres liq. N• 0.7M litres liq. He

• Time constants• Thermal = 0.5 years• Beam control = nano sec. and 10 hours

131313

The control of machines

From simple origins to life-saving enhancements. . .

141414

151515

“Electronic Stability Program (ESP) is a newsafety system which guides cars through wetor icy bends with more safety. ...The key is a yaw-rate sensor, which detectsvehicle movement around its vertical axis,and software which recognizes critical drivingconditions and responds accordingly. Inan instant, instructions are sent to the engine,transmission and brakes, thereby encounteringa skid at its onset. …”

K. J. Åström ECC August 31, 1999

161616

The Position of Control as a Discipline

• Respected

• Coupled to a vast array of engineered systems

• Lacks distinct identity (CERN example)• Lacks an identifiable industrial base (cf.

computers and mobile comm. devices)

•Academic positioning

171717

A Brief History of the FieldClosely tied to emerging technologies(steam, power, electricity, telephone, aerospace ...)

Telecommunications• Blacks invention 1927• Nyquist 1932• Bode 1940• Servomechanism theoryConsequences

The second wave• Recursive estimation• Maximum principle

181818

The Third Wave

Driving forces• New challenges• New applications• Mathematics• ComputersBasic paradigm• State SpaceRapid expansion• Subspecialities

Optimal ControlNonlinear ControlStochastic ControlComputer ControlRobust ControlRoboticsAdaptive ControlCACECooperative and decentralized control

191919

The Next Wave

202020

By 2015, 1/3 of all deployed military vehicles will be autonomous.

MURI 2007: Behavioral Dynamics in the Cooperative Control of Mixed Human/Robotic Teams

212121

QuickTime™ and aDV/DVCPRO - NTSC decompressor

are needed to see this picture.



QuickTime™ and aVideo decompressor


Autonomous vehicles operate in land, air,

and sea…

222222

Science . . .

Science Fiction

232323

Plant

ControllerNetworked Control

Classical Feedback Control

p21 p22 p23

Plant

MUX

S2S1

S3

PROCESS MODEL

CONTROLLER

R1

R2

-+

inout

p11 p12 p13

Router

242424

Control of Networked Devices

H. N

yqui

stRe

gene

ratio

n Th

eory

Bo s

ch G

mbH

Inte

l 825

26

IBM

701

Mot

e in

trodu

ced

252525

Current problems: • Develop a control theory for effectively managing wireless communications capability for automatically configuring ad hoc networks.

• Understand the relationship between standard network objectives (such as maximizing traffic capacity) and control objectives.

When feedback loops are closed using packet-switched wireless communication links, data-rates become an issue.

262626

Ad Hoc Networks of Mobile Robots



272727

The most important paper of the decade

• W.S. Wong & R.W. Brockett, 1995, 1999, “Systems with finite communication bandwidth constraints, II: Stabilization with limited Information Feedback”IEEE Trans. AC, May, 1999.

282828

The Data-Rate TheoremTheorem: Suppose the system G(s) (previous slide) is controlled using a data-rate constrained feedback channel. Suppose, moreover, G has k right half-plane poles λ1,…,λk. Then there is a critical data-rate

such that the system can be stabilized if and only if the channel capacity R>Rc.

Rc = log2 e ⋅ (Re(λ1) +L+ Re(λk ))

---Baillieul, 1999, (2002CDC), Nair and Evans, 2000, Tatikonda and Mitter, 2002, . . .

292929

Suppose there are only two (finitely many) actions that can be taken:

• Jerk the cart left or right one centimeter

• Under what circumstances can one keep the pendulum upright using this very coarse type of “control?”

• Ans: If and only if there is a sufficiently high actiel rate.

u vs t

t

303030

Brief “Partial” History of the Data-Rate Theorem

• D. Delchamps, “Stabilizing a linear system with quantized state feedback,” IEEE Trans. AC, 1990.•W.S. Wong & R.W. Brockett, 1995, 1999, “Systems with finite communication bandwidth constraints, II: Stabilization with limited Information Feedback” IEEE Trans. AC.• S. Tatikonda, A. Sahai, & S.K. Mitter, 1998, “Control of LQG systems under communication constraints,” CDC• John B., 1999, “Feedback designs for controlling device arrays with communication channel bandwidth constraints,” ARO Workshop.• G.N. Nair & R.J. Evans, 2000, “Stabilization with data-rate-limited feedback: tightest attainable bounds,” Sys, & Control Lett.• F. Fagnani and S. Zampieri, 2001, “Stability analysis and synthesis for scalar systems with a quantized feedback,” Tech. Rept., Politechnico diTorino.

313131

Some References• Keyong Li and John Baillieul,

“Robust and Efficient Quantization and Coding for Control of Multidimensional Linear Systems Under Data-Rate Constraints,” to appear in the Int’l J. of Robust and Nonlinear Control.

• K. Li and J. Baillieul, “Robust Quantization for Digital Finite Communication Bandwidth (DFCB) Control,” IEEE Trans. Automatic Control, Special Issue on Networked Control Systems, September, 2004.

323232

Control of Networked Systems

• Communication constrained and information enabledcontrol systems

Coding for robustness to time-varying data-ratesNoise, bit-errors, and riskFailure-sensitive controlCommunication requirements with decentralized

sensing and actuation• Patterns and constraints on information flow in networked control systems

Sensing patterns for stable motionsConsensus problems for groups of autonomous agentsShaping formations in motionCoverage problems for groups of autonomous agents

333333

RISK AND FAILURE IN ENGINEERED SYSTEMS

• Probabilistic models of risk• Information-constraint risk• Complexity-based risk• Model-mismatch risk• Unplanned for events

UncertaintyU

ncertaint yUnc

erta

inty

Uncertainty

343434

COMPLEXITY-BASED RISK

Sensitivity to trajectory variations.

353535

Complexity-Based Risk vs. Statistical Risk

• Statistical risk is assumed to be stationary.• There are typically important transient effects in complexity-based risk.• The distinctions are not always sharp.

363636

Simplified Roulette: The rotating pendulum

373737

The rotating pendulum: Dissipation enhanced multi-stability

383838

Dissipation enhanced multi-stability brings the risk of falling into the wrong potential well

393939

Highly Complex Systems

Amino acids are linked by peptide bonds of various types to form proteins. Suppose there are three types of bonds between each amino acid in a chain of 101 molecules. There are

conformations (=stable configurations).

404040

The Complex Energy Landscape of Protein Docking

414141

RISK AS AN ENGINEERING DESIGN PARAMETER

Risk level # of Failures

1-sigma Less than 4 in 10



4-sigma Less than 7 in 100,000

5-sigma Less than 6 in 10,000,000

6-sigma Less than 2 in 1000,000,000

424242

Networks Are Everywhere

Enzyme

Enzyme

Mitogen-activated protein kinase (MAPK)http://www.animalport.com

---C. Conradi, J. Saez-Rodriguez, E.-D. Gilles and J. Raisch

434343

Craig C. Mello

Networked Biological Control Systems

444444

Biological Paradigns for Networked Control Systems

---C.V. Rao & A.P. Arkin

454545

Gene Networks as Networked Control Systems

464646

Computer vision

Differential GPS

Sonar

Laser range and bearing sensor

Hall effect direction sensors Wheel encoders

Inertial navigation

IEEE 802.11

Typical real-time data available to control the motion of an autonomous mobile robot

474747





What are the design principles for distributed, real-time, situation-dependent real-time control with localized sensing and information processing?

484848

Strategy: Reason about Formation Control Using Point Robots - Realize Formations in Models of

Nonholonomic Vehicles

Multiply redundant distributed arrays of sensors support flexible design of robust formation control strategies.

Beware of information-induced instability…

A B

C

dAB

dAC dCAdBC

dCB

dBA

494949

Graphs and Frameworks are Tools for Patterns of Information Use

Definition 1: A formation framework (S;E;q) consists of a formation graph (S;E) and a function q(.) from the vertex set S into a squadron configuration space

Definition 2: Motions which preserve all pairwise relative distances are called rigid.

= typical point.

position orientation

505050

Rigid Formations---Creation and Motion Control Strategy

• Sensor data should be used in a maximally parsimonious fashion,

• There will always be a leader-first-follower pair.

We say that a formation framework( S, E, q) is isostatic if the removal of any edgeε∈Ε results in a framework which is not rigid.

Yes No

515151

Main Results in Planar Rigidity Theory

LAMAN’S THEOREM: A planar graph (V,E) is isostatic ″1. |Ε| = 2|V|−3,2. |E(U)| ≤ 2|U| - 3 U⊆V with |U| ≥ 2.

HENNEBERG’S THEOREM (1911): A planar graph (V,E) is isostatic ″ it can be constructed from a single edge by a sequence of vertex extensions and edge splits.

Vertex extension

525252

Rigid Planar Graphs on 2,3,4, and 5 Vertices

• Can also be constructed by an edge-split.

6?

• Can also be constructed by an edge-split.

• Cannot be constructed by an edge-split.

535353

Real-time Stabilized “Rigid” Formations

A B

C

A B

C

• The soft real-time question “Is the formation congruent to the prescribed triangle?” can be answered?• Theorem No triangular formation is asymptotically stable under this peer-following control strategy.

Assume each robot can acquire and process “simultaneous” real-time information on line-of-sight distances to two peers. Then there is a control law based on the sensor data acquired as depicted which will stably configure the robots into any prescribed triangle.

545454

1

2

Rigid Vertex Extension of a Formation Framework

As long as the rest point is not on the line determined by the leader (1) and first follower (2), it is an asymptotically stable rest point for the motion:

This is a semi-global result.

555555

Distributed Relative Distance Control of Point Robots

Definition: Distributed relative distance control of a multi vehicle formation takes the form

where dij is the set-point distance between the i-th and j-thvehicles and ρij is the sensed distance. Thus the control is a function of the prescribed relative distance and the measured relative distance.

Ý x iÝ y i

⎛

⎝ ⎜

⎞

⎠ ⎟ = u(dij , ρij )

xi − x j

yi − y j

⎛

⎝ ⎜

⎞

⎠ ⎟

j ∈Si

∑

Examples: 1. Scaled difference, 2. Normalized difference, 3. Lennard-Jones, 4. Etc.

565656

Proof: Based on Laman’s Theorem and the figure on the left.

Necessary and Sufficient Conditions for Stable Rigidity

Theorem: An acyclic formation graph corresponding to a stably rigid formation under a distributed relative distance control law is isostatic if and only if

1. one vertex (the leader) has out-valence 02. one vertex (the first-follower) has out-valence 1 and

is adjacent to the leader vertex; and3. all other vertices have out-valence equal to 2

575757

The Rigidity Theory of Directed Graphs Differs from the Undirected Case

12

3 4

12

3 4

(a) (b)

Both graphs are rigid (and isostatic by Laman’s theorem) as undirected graphs, but as directed graphs they are not rigid.

585858

All Constructively Rigid Formations Look Like This

Leader

First-follower

Using dual peer sensing control, a stepwise sequence of motions can be carried out to have a set of planar robots assemble themselves into any constructively rigid formation.

595959

Leader - First-Follower - Subsequent Follower Formations

3 Modulo a left-right symmetry, this formation is unique on three vertices.

4

A B C

Formation B can be constructed in one step - once the leader and first-follower are in place. Followers must join in sequence for A and C.

606060

Leader - First-Follower - Subsequent Follower Formations

5

Five (5) of the thirteen (13) formations with five nodes. These five are the formations which are created purely sequentially.

616161

Data-Structures Associated with Dual Peer-Sensing Decentralized Formation Control

1. Logical formation graphs: the set of vertices and their adjacency relationships,

2. The set of possible stable configurations corresponding to a given logical formation graph together with the prescribed distance of the first follower from the leader and prescribed pair of distances of each node from the two nodes to which it is adjacent.

A B C

626262

The enumerative theory of logical isostatic directed graphs remains to be explored

“The Combinatorial Graph Theory of Structured Formations,” In Pro-ceedings of the 46-th IEEE Conference on Decision and Control, New Orleans, December 12-14, 2007.

• Currently, closed form expressions for this enumeration do not exist.• Many stratification classes do have closed-form enumerations.• This is a new sequence, not previously in Sloan’s sequence list.

636363

Counting Isostatic Planar Formations

646464

Interesting Problem

Develop a clear mapping between algebraic representations of information patterns and the motions they support.

Example:

Consider point robot dynamics of the following form:

Ý x Ý y

⎛

⎝ ⎜

⎞

⎠ ⎟ = u1

x − x1

y − y2

⎛

⎝ ⎜

⎞

⎠ ⎟ + u2

x − x2

y − y2

⎛

⎝ ⎜

⎞

⎠ ⎟ .

656565

Information Flow Must Be Localized and Dynamically Reconfigured





Given the geometry of a formation, how many distinct “stable information-flow patterns” will support it?

666666

Formation Equations for Large-Scale Group Motions

For an n-node formation, there are n pairs of equations like these. The system has 2n-2 asymptotically stable equilibria.

676767

The Stability and Bifurcation Theory of Formation Equilibria

(0,0) (0,1)

d1 d2

d12

Proposition: Let d12= and assume:(x1 − x2)2 + (y1 − y2)2

d1 + d2 > d12, d12 + d2 > d1, d1 + d12 > d2 .Then the equilibrium solutions satisfying the equations

(x − x j )2 + (y − y j )

2dj - = 0, (j=1,2)

are locally asymptotically stable.

686868

The Singularity Theory of the Critical Line

For all d1 and d2>0, there are equilibrium solutions of position equations on the critical line. Assume d1≥1. Write r=d1. When d1+d2> d12, (xe,ye)=(r,0) is a solution for all r >1. There are others as depicted:

696969

Ý x Ý y

⎛

⎝ ⎜

⎞

⎠ ⎟ = (d1 − ρ1)

x − x1

y − y2

⎛

⎝ ⎜

⎞

⎠ ⎟ + (d2 − ρ2)

x − x2

y − y2

⎛

⎝ ⎜

⎞

⎠ ⎟

The bifurcation and stability characteristics are quite rich.

707070

For each of the formation-types on n-nodes, there are 2n-2

locally asymptotically stable equilibria.

I.C.’s (x2(0),y2(0))=(8,0), (x3(0),y3(0))=(3,-1), (x3(0),y3(0))=(1,-1).F.C.’s (x2(∞),y2(∞))=(1,0,), (x3(∞),y3(∞))=(0.705, -1.97812), (x3(∞),y3(∞))=(-0.0906396, -0.143197).

I.C.’s (x2(0),y2(0))=(9,0), (x3(0),y3(0))=(3,-1), (x3(0),y3(0))=(1,-1).F.C.’s (x2(∞),y2(∞))=(1,0,), (x3(∞),y3(∞))=(0.705, -1.97812), (x3(∞),y3(∞))=(1.99511, -0.45236).

Careful Planning of Group Motions Is Needed to Deal with Extraordinary Sensitivity to Initial Conditions

StartFinish

717171

• Control is and probably will remain a hidden technology• Networked systems are ubiquitous, and network control systems are essential to understand in both the natural and engineered world• The role of information in relation to the physical world remains to be understood.• The essence of robustness in network dynamics remains to be understood.

Conclusions and Future Work

727272

Theorem (Brockett). Consider the nonlinear system x = f(x, u)with f(x0, 0) = 0 and f(. , .) continuously differentiable in a neighborhood of (x0, 0). Necessary conditions for the existence of a continuously differentiable control law for asymptotically stabilizing (x0, 0) are:(i) The linearized system has no uncontrollable modes associated with eigenvalues with positive real part.(ii) There exists a neighborhood N of (x0, 0) such that for each ξ∈ Nthere exists a control uξ (t) defined for all t > 0 that drives the solution of x = f(x, uξ ) from the point x = ξ at t = 0 to x = x0 at t = ∞.(iii) The mapping γ: N×ℜm→ℜn, N a neighborhood of the origin,defned by γ : (x; u) → f(x; u) should be onto an open set of theorigin.

Sensor-Based Feedback Laws for Nonholonomic Vehicles Are Not Stabilizing

737373

For Nonholonomic Motion Control, Critical Points -> Critical Varieties

Ý x Ý y Ý θ

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

=uCos(θ)uSin(θ)

ω

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

u = λ(ρ − d), andω = ρSin(ϕ) − d.

φ

747474

The Laman conditions in dimension m, m=1,2,3

.

757575

LF

3

4 5

Rigid Formations with Cycles Cannot Be Constructed by this Procedure

The formation is stably rigid.

767676

Using the Basic Point Robot Model to Coordinate Large-Scale Group Motions

777777

Chemical Reaction Networks (CNR’s)

• The basis of cell biology• The understanding of cell functions at the level of aggregate behavior in terms of cascades and interactions of networks of chemical reactions is a truly formidable task---which will likely never be fully accomplished.

---See Eduardo Sontag, 2005• What is needed are systematic tools for decomposing network dynamics into component motifs.

787878

Networks Are Everywhere

There are 13 three-node control motifs.

797979

Special Section on Biochemical Networks and Cell Regulation, Control Systems Magazine, Volume: 24 Issue: 4, Aug. 2004.

Special Issue on Systems Biology, IEEE Transactions on Automatic Control, AC:53, Jan. 2008.

References on Control and Biological Networks

Documents

Control Theory, Networks, and Life ItselfLynn Margulis and Dorion Sagan, What Is Life?, University of California Press (August 31, 2000), 303 pages ISBN-10: 0520220218 Prior outrageously