Harvard - Boston University - University of Maryland Patterns and Control Eric Justh, George Kantor, and P.S. Krishnaprasad Institute for Systems Research

Harvard - Boston University - University of Maryland

Patterns and Control

Eric Justh, George Kantor, and P.S. KrishnaprasadInstitute for Systems Research and

Department of Electrical & Computer EngineeringUniversity of Maryland

College Park, MD [email protected]

[email protected]

ARO Review Meeting, Harvard University, Nov. 15-16, 1999.

Presentation for Dr. Randy Zachery, ARO May 25, 2004, Harvard University


Focus of this Talk• Patterns of communication and computation for approximate decoupling of distributed systems of actuators and sensors - exploitation of distributed computation.

• Pattern formation via diffusive coupling and nonlinear activation for distributed control - exploitation of nonlinearity.

• In the first part of this talk, we make essential use of linear system structure (e.g. spatial invariance), basis selections (e.g. via wavelets) and matrix diagonalization techniques.

• In the second part of this talk, we make essential use of nonlinear partial differential equation and dynamical systems theory.

• An emerging communications and hierarchical control theory for controlling very large arrays of sensors and actuators


References

• G. Kantor and P. S. Krishnaprasad (2001), “An application of Lie groups in distributed control”, Systems and Control Letters, 43:43-52.

• E. Justh and P. S. Krishnaprasad (2001), “Pattern forming systems for control of large arrays of actuators”, Journal of Nonlinear Science, 11: 239-277


Plant Diagonalizationu

Py

u~ y~uP

y2Q̂ 1

1ˆ Q

111~u 1

~y

222~u 2

~y

nnnu~ ny~

Given the system

is approximately equal to

find fast, distributed, scalable transforms and so that the system1Q̂ 2Q̂


Motivation: Recent Developments• Microelectromechanical Systems (MEMS)

– makes it possible to fit large numbers of sensors and actuators into small areas

• Smart Structures– distributed sensors and actuators are incorporated into

the natural structure of an object• strain sensors and PZT actuators embedded in a composite

panel for reducing vibrations, monitoring fatigue, or controlling the shape of a flight surface

• Inexpensive Microprocessors– makes it feasible to distribute microprocessors over a

large control network, giving sensors and actuators access to local computing power


• C is a low order, low pass FIR filter.

• D is a low order, high pass FIR filter.

• , called downsampling, removes every other element of its input vector

Discrete Wavelet Transform

The DWT is achieved by iterating the filter bank on the low pass channel

C

D 2

2C

D 2

2C

D 2

2

xQx

C

D 2

2

This filter bank is the basic building block for the DWT

2


Distributed Implementation of DWT (1 of 4)

2)( 81 yy

2)( 23 yy

2)( 45 yy

2)( 67 yy

2)( 12 yy

2)( 34 yy

2)( 56 yy

2)( 78 yy

the Haar wavelet transform:

2)1()(

2)1()(1

1

zzD

zzC

1y

2y

3y

4y

5y

6y

7y

8y

implementation of low pass filter



2)( 12 yy

2)( 34 yy

2)( 56 yy

2)( 78 yy


2)1()(

2)1()(1

1

zzD

zzC

1y

2y

3y

4y

5y

6y

7y

8y

low pass filter after downsample



2)( 12 yy

2)( 34 yy

2)( 56 yy

2)( 78 yy

2)( 12 yy

2)( 34 yy

2)( 56 yy

2)( 78 yy


2)1()(

2)1()(1

1

zzD

zzC

1y

2y

3y

4y

5y

6y

7y

8y

first level of filter bank




2)1()(

2)1()(1

1

zzD

zzC

1y

2y

3y

4y

5y

6y

7y

8y

2)( 34 yy

2)( 56 yy

2)( 78 yy

2)( 34 yy

2)( 56 yy

2)( 78 yy

2)( 12 yy

2)( 12 yy 2))()(( 1234 yyyy

2))()(( 1234 yyyy

2))()(( 5678 yyyy

2))()(( 5678 yyyy

22)))()(())()((( 12345678 yyyyyyyy

22)))()(())()((( 12345678 yyyyyyyy

denotes elements of transformed vector, Qy

full Haar transform


iterating the filter bank on the high pass channel gives another transform

The Wavelet Packet

the wavelet transform iterated the filter bank on the low pass channel

iterating on any combination of high and low pass filter banks at each level

yields many additional transforms (3 examples shown)

The collection of all possible filter bank transforms is called the wavelet packet. Each wavelet packet transform has the following properties:


Wavelet Packet Transforms as Matrix Operations

Let Q be the Haar-Walsh WPT given by iterating both channels at every level:

21

2100

002

12

12

12

100

002

12

1

21

2100

21

2100

002

12

1

002

12

1

Q

21

2100

21

2100

002

12

1

002

12

1

1000

0010

0100

0001

21

2100

21

2100

002

12

1

002

12

1


Distributed Transform Implementation (1 of 3)

Step 1: local rotations

)2/(1

12

11

1

00

0

0

00

n

xa 1

2x

nx

3x

4x

1x

1a

2a

1nx

5x

6x

2nx

11

12

13

)12/(1 n

)2/(1 n

3a

4a

5a

6a

3na

2na

1na

na



Step 2: even permutationand local rotations

1

2

1

)2/(2

22

21

00

0

0

00

n

n

na

a

a

a

b

1b

2b21

22

23

)2/(2 n

3b

4b

5b

6b

1nb

nb

xa 1step 1:

2x

nx

3x

4x

1x

1nx

5x

6x

2nx

(to top)

1a

2a

3a

4a

5a

1na

na

2na

na

xPb e 12

0100

0

010

0001

1000

ePdefine

then



Step 3: even permutationand local rotations

1

3

2

)2/(3

32

31

00

0

0

00

b

b

b

b

c

nn

xPb e 12 step 1

2x

nx

3x

4x

1x

1nx

5x

6x

2nx

1b

2b

31

32

)12/(3 n

1c

2c

3c

4c

3nc

2nc

(to bottom)

xPPc eo 123

0001

1

000

0100

0010

oPdefine

then

step 2

)2/(3 n 1nc

nc

3b

4b

5b

1nb

nb

1b


Recursive Orthogonal Transform

LLPPP 2211

~

km

k

k

k

k

00

0

0

00

2

1

)( kjkj nSO

nnkm

jkj

1

Motivated by the wavelet packet transforms and the ad hoc distributed transform implementation presented in the previous slides, we define the recursive orthogonal transform (ROT) to be a transformation whichcan be written as the product

where, for k = 1,2,…,L,

and is a given, fixed permutation of the identity matrix.kP

The elements of the L-tuple are called the ROT variables. L ,,, 21


Spatially Invariant Systems

• Brockett and Willems [1971,1974] found optimal feedback laws for discretized PDEs.

• Melzer and Kou [1971] developed a Riccatti equation for SI systems and applied it to infinite strings of vehicles.

• El-Sayed and Krishnaprasad [1981] developed optimal feedback laws to control the depth of a seismic cable.

• Bamieh [1997] and Bamieh, Paganini, and Dahleh [1998] proposed these ideas for use in large scale sensor/actuator arrays.

Many potential sensor/actuator pair applications result in systems which exhibit a spatial invariance property, i.e. the coupling between any two nodes depends only on their relative positions. As a result, many of the familiar tools for LTI systems can be applied.


ROT for Approximate Diagonalization of Circulant Plants

LLPPP 2211

~

km

k

k

k

k

00

0

0

00

2

1

)( kj

kj nU

nnkm

jkj

1

)()()( 21 kkmkkkk nUnUnUM

LL MMMM 2121 ,,,,

We can search for an ROT which most nearly approximates the action of the unitary matrix .

~

nF


The Cost FunctionLet be a set of orthogonal basis vectors for the vector space of circulant matrices.

110 ,, nEEE nn

The task of diagonalizing any circulant matrix can be posed asthe task of simultaneously diagonalizing each of the basis vectors.

Let (each is diagonal).1,,1,0 , niFEFN Hnini iN

Then a suitable cost function is:

1

0

2~~n

ii

Hi ENJ

Minimizing J is equivalent to maximizing the “diagonalness” function:

1

0

~~retr

n

ii

HHi EN


Finding the Gradient FlowThe gradient of the function on M is defined by

1. (tangency)2. (compatibility)

MMT MTXXXD ,

Using the trace inner product, these two properties can be used tofind a (unique) expression for the gradient vector field.

The resulting gradient flow is, for k = 1,2,…,L:

1

0

~~ ,

n

iki

Hk

Hk

Hikkkk EN

where , ,

k

lllk P

1

~

n

klllk P

1

kek MTn u:and the projection operator


Impulse Response Matrix

The (i,j)th pixel represents the energy of the response to an input applied To the ith actuator measured by the jth sensor.


Level 3 ROT Level 5 ROT Level 7 ROT

Impulse Response Matrices for ROT Transformed Systems


Pattern Generation for Control• Goal is to coordinate the action of large arrays of actuators.

• Our pattern-formation approach:

• Generating patterns involves “stressing” the interconnections by changing a parameter, which results in the crossing of stability thresholds.

• Making such parametric adjustments via feedback on a slower time scale solves the problem of communicating with a large array: communication is achieved through the interconnection template.

• Interconnection templates are created in analog or digital circuitry.• Passing to the continuum limit, we obtain dynamical systems which support interesting spatio-temporal patterns.• The spatio-temporal patterns determine the actuation patterns.


Pattern-Forming Systems• Systems of differential equations having the property that as a bifurcation (or control) parameter passes through a critical value, a stable spatially uniform solution gives way to a stable pattern solution, which may have spatial variation, time variation, or both.

• Arise in many diverse physical contexts.

• Extensively studied (both theoretically and experimentally) for many years.

• Only recently starting to receive attention in a controls context.

• Biology: population dynamics, animal coloration patterns, nervous systems (e.g., visual hallucination patterns)• Chemistry: certain catalyzed reactions • Physics: shaken collections of particles, gas discharge tubes, semiconductor electron-hole plasmas, Josephson-junction arrays


Actuator Arrays• Potential applications for large two-dimensional arrays of small actuators include

• As the number of actuators increases, so does the bandwidth required to command each actuator individually.

• Goals of using pattern-forming systems for large actuator arrays:

• Adaptive optics (e.g., micromirror arrays) - joint work with ARL• Control of flow separation• Micropositioning small parts• Manipulating small quantities of chemical reactants

• Enable external control inputs to be lower bandwidth• Allow actuator control signals to be computed in parallel at each actuator site using mostly local information• Nonlinear dynamical systems theory can be used


MEMS Arrays for Peristaltic Flows• Peristaltic flows are driven by transversely oscillating boundaries.

• Applications: mean flow (directed transport), small-scale mixing.

• Analyzed by Kiril Selverov and Howard Stone (1998) for a closed, two-dimensional rectangular geometry.

• The lower wall of the cavity can be actuated by closely-spaced piezo-electric elements or membrane-covered piston-type actuators.

• It is natural to drive such actuators using pattern-forming systems.


Activator-Inhibitor System• We consider the activator-inhibitor system of coupled PDEs with a cubic nonlinearity:

• A spatially uniform equilibrium is given by the intersection of a cubic curve with a line (shown for C = -.54)

where is the Laplacian, is the activator, is the inhibitor, and are time constants, l and L are diffusion lengths, and C is the control (or bifurcation) parameter.


Activator-Inhibitor System Key Parameters

• Dynamics:

• Key parameters which determine the pattern-forming properties:

• We consider the case << 1 and > 1, so that either spatially periodic pattern equilibria or other interesting equilibria can be stable, depending on the value of C.

• = / (ratio of time constants)

• = l / L (ratio of diffusion lengths)


Two-Dimensional Equilibria

• Spike equilibrium (C = -.54):

• Pattern equilibria (C = 0):

A symmetric pattern An irregular pattern


Electronic Circuit Realization• A VLSI realization of the (spatially discretized) dynamics could be collocated with a MEMS actuator array.

• The voltages could drive, for example, electrostatic piston actuators.

k


Lyapunov Function• We have gradient dynamics with respect to the energy function

• The matrix of second partial derivatives is

• The Lyapunov function (when > 1) is

(where = [1 1 ... 1] )

T


Exciting Spikes (Creating Patterns)• A spike can be excited by locally raising the control parameter into the pattern regime:

• Multiple spikes can be excited if the control parameter is raised over a larger region:

C locally raised Resulting spike solution

C locally raised Resulting group of spikes


Translating Spikes (Moving Patterns)• Spikes (or patterns) can be made to translate by the addition of an advective term to the dynamics:

• If the advective term is sufficiently small, the spike shape is slightly distorted, and the spike translates.

Initial excitation Traveling spike


Laboratory Implementation• Combining MEMS with VLSI circuitry on a single chip is an order of magnitude more expensive than MEMS or VLSI alone.

• To try out control schemes based on pattern-forming system concepts, we need a practical laboratory implementation.

• We have (analytically) investigated a block diagram on which a practical laboratory realization could potentially be based:

Documents

Harvard - Boston University - University of Maryland Patterns and Control Eric Justh, George Kantor, and P.S. Krishnaprasad Institute for Systems Research