Finite Dimensional Linear Systems

Roger W. Brockett

Society for Industrial and Applied Mathematics, 2015 / xvi + 244 pagesISBN 978-1-611973-87-7.

siamC L A S S I C S

In Applied Mathematics

EC/ME/SE 501 – Dynamic System Theory

EC/ME/SE 501 – Dynamic System Theory

Control System Design: An Introduction to State-Space Methods (Dover Books on Engineering)

Bernard Friedland Dover Publications (March 24, 2005), 528 pagesISBN-10: 0486442780, ISBN-13: 978-0486442785

Control Theory, Networks, and Life Itself

John BaillieulC.I.S.E. and

Intelligent Mechatonics Laboratory

johnb@bu.eduhttp://people.bu.edu/johnb

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

Boston UniversityME 501 First Lecture

John Baillieul’s Research GroupIntelligent Mechatronics Laboratory

John Baillieul’s Hiking Friends

Boston University---from humble beginnings

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?”

Professor Lynn Margulis Boston UniveristyUniversity Lecture 1978The Early Evolution of Life

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

Prior outrageously titled lectures (and books)

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 systemsIV. The emerging theory of complexity-based risk and the

rational management of failureV. Controlling highly structured motions of groups of agents

1) The theory of rigid formations2) Spatial and dynamic information patterns

III. Conclusions

People in Control at B.U.

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

Professor Christos G. Cassandras46-th President of the IEEE Control Systems Society

People in Control at B.U.

Professor Theodore E. Djaferis41-st President of the IEEE Control Systems Society

Emeritus Dean of EngineeringUniversity of Massachusetts

SIAM J. on Control andOptimization

World’s leading journals at B.U.

The Hidden Technology• Pervasive • Very successful • Seldom talked about Except 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

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

The control of machines

From simple origins to life-saving enhancements. . .

“Electronic Stability Program (ESP) is a new safety system which guides cars through wet or icy bends with more safety. ... The key is a yaw-rate sensor, which detects vehicle movement around its vertical axis, and software which recognizes critical driving conditions and responds accordingly. In an instant, instructions are sent to the engine, transmission and brakes, thereby encountering a skid at its onset. …”

K. J. Åström ECC August 31, 1999

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

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

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

The Next Wave

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

The two ages of joint cognitive transport systems

4,000 BCFirst domesticateddraught animals

The First Age of Joint Cognitive Transport

1900 AD 2010 ADThe age of mechanized transport

The Second Age

Autonomous vehicles operate in land, air,

and sea…

Science . . .

Science Fiction

Plant

Controller

Classical Feedback Control

Networked Control

p21 p22 p23

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Control of Networked Devices

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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.

Ad Hoc Networks of Mobile Robots

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.

The Data-Rate Theorem

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

Theorem: 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.

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

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 di Torino.

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.

Control of Networked Systems

• Communication constrained and information enabled control systems

§ Coding for robustness to time-varying data-rates§ Noise, bit-errors, and risk§ Failure-sensitive control§ Communication requirements with decentralized sensing and actuation

• Patterns and constraints on information flow in networked control systems

§ Sensing patterns for stable motions§ Consensus problems for groups of autonomous agents§ Shaping formations in motion§ Coverage problems for groups of autonomous agents

RISK AND FAILURE IN ENGINEERED SYSTEMS

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

UncertaintyU

ncertaintyUnc

erta

inty

Uncertainty

COMPLEXITY-BASED RISK

Sensitivity to trajectory variations.

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.

Simplified Roulette: The rotating pendulum

The rotating pendulum: Dissipation enhanced multi-stability

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

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).

The Complex Energy Landscape of Protein Docking

RISK AS AN ENGINEERING DESIGN PARAMETER

Risk level # of Failures1-sigma Less than 4 in 102-sigma Less than 5 in 1003-sigma Less than 3 in 10004-sigma Less than 7 in

100,0005-sigma Less than 6 in 10,000,0006-sigma Less than 2 in 1000,000,000

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

Craig C. Mello

Networked Biological Control Systems

Biological Paradigns for Networked Control Systems

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

Gene Networks as Networked Control Systems

Computer vision

Differential GPS

Sonar

Laser range and bearing sensor

Hall effect direction sensors

Inertial navigation

Wheel encoders

IEEE 802.11

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

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

Asymptotic Stability Must Be Rethought

Asymptotic Stability Must Be Rethought

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

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dBA

dAC dCAdBC

dCB

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

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

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

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

• Can also be constructed by an edge-split.

• Can also be constructed by an edge-split.

• Cannot be constructed by an edge-split.

6?

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.

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.

Distributed Relative Distance Control of Point Robots

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

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-th vehicles and ρij is the sensed distance. Thus the control is a function of the prescribed relative distance and the measured relative distance.

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

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.

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.

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.

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.

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

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.

Counting Isostatic Planar Formations

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:

Information Flow Must Be Localized and Dynamically Reconfigured

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

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.

The Stability and Bifurcation Theory of Formation Equilibria

(0,0) (0,1)

d1 d2

d12

The Stability and Bifurcation Theory of Formation Equilibria

(0,0) (0,1)

d1 d2

d12

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:

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

What happens when humans are nodes in a control network?

• People act on the basis of perceptions.

• Peoples’ perceptions of any situation will typically differ.

• Perceptions may change due to social interactions.

• For a perception to be actionable, it must be sufficiently clear.

A broader theory of perception based control and mixed initiative teams

A fundamental distinction between humans and robots in mixed teams is that robots react rapidly to real-time sensor data, whereas humans react to more complex perceptions of the environment in which cognitive processes and prior experience play a role.

Why do we need to think about perception based control?

• Future operating environments will involve big data (in real time);

• Allocation of effort between human and machine must play to the strengths of each for optimal synergy;• Cognitive and perceptual processes must be explicitly taken into account.

Cf: OODA loops (Col. John Boyd, USAF)

S/He who maximizes throughput wins.

Understanding the perceptual basis for observed actions in the natural world - the case of looming obstacles

Time-to-impact perception based on optical flow

Time-to-impact perception based on optical flow

Action attributable to perception of tau

D.N. Lee and P.E. Reddish, 1981. “Plummeting gannets: a paradigm of ecological optics,” Nature, 293:293-294.

Motion control based on perception of tau

Challenge:Can perception of looming obstacles be used to transit an obstacle field?

How do they do it?

Motion control based on perception of tau

• 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 dynamics of perception are important when humans are nodes in a control network.• The role of information in relation to the physical world remains to be understood.• The essence of robustness and resilience in network dynamics remains to be understood.

Concluding Remarks – The state of the art

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∈ N there 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

For Nonholonomic Motion Control, Critical Points -> Critical Varieties

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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.

Networks Are Everywhere

There are 13 three-node control motifs.

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