Controlling Structured Spatially Interconnected Systems

Transcription

1 Controlling tructured patially Interconnected ystems Raffaello D Andrea Mechanical & Aerospace Engineering Cornell University

2 Recent Developments... Proliferation of actuators and sensors Moore s Law Embedded systems, CAN, Bluetooth MORE PERFORMANCE!!!

3 What is, and will be, needed: NEW CONTROL TOOL: LARGE numbers of actuators and sensors Distributed computation Limited connectivity Robustness Performance Flexibility Etc.

4 Modeling Interconnected ystems xt &() = f( xt (), dt ()) z() t = h( x(), t d()) t d M z x(t),d(t), and z(t) live in a Hilbert space: dt () = dts (,, s, L, s) = dt (, s) * dt () : L d(, ts) dt (, s) l = s = s L = L

5 Restrict the model class: local interconnection WHY? Large class of systems, non-trivial behavior: vehicle platoons finite difference approximations of PDEs cellular automata, artificial life, etc. behavior of groups, swarm intelligence, etc.

6 Case tudy: Formation Flight Use upwash created by neighbouring craft to provide extra lift MOTIVATION satellite type of applications (Wolfe, Chichka and peyer 96) MAVs and UAVs, extend range

7 1 EFFICIENCY OF FORMATION, ELLIPTICAL DITRIBUTION 0.9 POWER RATIO WING 5 WING 21 WING 101 WING /(1+) Lissaman and hollenberger 70: Formation Flight of Birds

8 Formation Flight Test-Bed tudent: Jeff Fowler (ME) && y(, t s) = c1y(, t s) c2θ (, t s) && θ = c & θ u c u c & θ + c y y ( t, s) ( t, s) + ( t, s) ( t, s + 1) + ( t, s + 1) ( t, s) ( t, s + 1) ( ) 2

9 Define shift operator : ( u)(, t s): = u(, t s+ 1) yields && y = c1y c2θ && & & θ = c 6( ) 3θ + u c4u + c5θ + c y y 2 In general: d d I t 1I = -1 1 I O -1 L I ( ) x (, t s) = F( x(, t s), d(, t s), s) zt (, s) = H( xt (, s), dt (, s), s) d(t,s), x(t, s), z(t, s) are FD, F and H NL functions on FD space

10 ( ) s s s s pecial cases... x (, t ) = F( x(, t ), d(, t ), ) zt (, s) = H( xt (, s), dt (, s), s) - d = I, dt F() = Ax+ Bd, H( ) = Cx+ Dd x& = Ax+ Bd, z = Cx + Dd - F() = Ax+ Bd, H() = Cx+ Dd Linear, spatial invariant systems - = d dt I Family of completely decentralized systems CONVENIENT FRAMEWORK FOR CAPTURING TRUCTURE

11 Recent Related Work iljak et al: Decentralized control of complex systems Bamieh, Paganini, Dahleh: patially Invariant ystems Cheng, Yang, Zhai, Peterson, avkin, : Decentralized Control of IC systems. tewart, Gorinevski, Dumont: Cross directional control

12 Control Design and Analysis: patially Invariant ystems z y G K d u d(t,s): disturbances z(t,s): errors y(t,s): sensors u(t,s): actuators Closed Loop: x = Ax+ Bd z = Cx + Dd table: 1 ( A) exists and is bounded Contractive: + 1 < D C( A) B 1

13 x& T = ATT xt + AT x+ BT d x = A x + A x + B d T T z = C x + C x + Dd T T Analysis 1I -1 1 I = O -1 L I X1 X X1 X1 = O XL XL XL X L table and Contractive if there exists X T >0 and structured X s.t. X ATTXT XTATT XTAT XTB I + T I 0 0 A ( X ) T A B + TAT -X 0 AT A B 0 0 I ( ) 0 0 I XTBT 0 I I XT A 0 T I 0 0 ( X ) AT A B TAT X 0 AT A B CT C D 0 0 I CT C D < 0 +

14 Theorem: There exists a controller such that the analysis LMI is satisfied if and only if there exists structured Y and X such that AY + YA YC B < * * 1 1 * U C1Y I D 11 U * * B1 D11 I * * A X + XA XB1 C 1 * * * V B1X I D 11 V < C1 D11 I X0 I > I Y

15 Controller implementation: x& = A x + A x + B y T TT T T T x = A x + A x + B y T T u = C x + C x + Dy T T EX: 2D x 2 (s 1,s 2 +1) x -2 (s 1,s 2 ) x 1 (s 1,s 2 ) K(s 1,s 2 ) x -1 (s 1-1,s 2 ) x = ( x, x, x, x ) x 2 (s 1,s 2 ) x -2 (s 1,s 2-1) u(s 1,s 2 ) x 1 (s 1 +1,s 2 ) x -1 (s 1,s 2 ) y(s 1,s 2 )

16 Control Architecture

17 Decentralized Control

18 Decentralized Control Distributed Control

19 COMPARION OF PATIAL 2-NORM, ROLL ANGLE

20 TRONG NONLINEAR COUPLING Nonlinear patially Interconnected ystems: x = f ( x) + g( x) d z = h( x) Feedback linearization Backstepping etc.

21 patially and Time Varying ystems: non-homogeneous properties finite boundary conditions Other ongoing work x = A() s x+ B() s d z = C() s x+ D() s d TOOL: LTI to LTV machinery (GEIR DULLERUD, UIUC) method of images, etc. LPV tools

22 Framework for Robust Control of IC systems ( ) x (, t s) = F( x(, t s), d(, t s), s) zt (, s) = H( xt (, s), dt (, s), s) Delta contains temporal operators, spatial operators, AND uncertainty. tudent: Ramu Chandra (AE) Model Reduction (CAROLYN BECK, UIUC) Cross-Directional Control (GREG TEWART, HONEYWELL)

23 Phased Array Antennas for AFV Communication tudent: ean Breheny (ECE) High data rate comms between AFVs and base station/satellite (video, etc.) Difficult to put a high gain antenna on an AFV (size constraint) ince it may be advantageous to use groups of AFVs anyway, why not investigate whether a formation of AFVs, each carrying a low gain antenna, could form a high gain phased array?

24 What is a Phased Array Antenna? Exploit EM wave interference among several antennas. For the simplest case (where array elements are not strongly coupled to each other), gain increases roughly linearly in N, the number of elements. Channel capacity increases linearly when the maximum bandwidth is used.

25 Example: Endfire Array