SYSTEM INTERCONNECTION

Transcription

1 p. 1/60 Elgersburg Lectures March 2010 Lecture IV SYSTEM INTERCONNECTION

2 p. 2/60 Outline Motivation Modeling by tearing, zooming, and linking An example Control as interconnection Pole assignment and stabilization

3 p. 3/60 Theme How are systems interconnected? How are interconnected systems modeled? How does control fit in?

4 p. 3/60 Theme How are systems interconnected? How are interconnected systems modeled? How does control fit in? We deal with very simple examples, mainly electrical circuits and 1-dimensional mechanical systems. Other applications: hydraulic systems chemical systems thermal systems,...

5 Systems p. 4/60

6 p. 5/60

7 p. 6/60 Features Open Interconnected Modular Dynamic The ever-increasing computing power allows to model such complex interconnected systems accurately by tearing, zooming, and linking. Simulation, model based design,...

8 p. 7/60 Open systems System Environment Systems are open, they interact with their environment. In the previous lectures, we have seen that thinking of systems in terms of their behavior captures the open nature of systems very well.

9 p. 8/60 Interacting systems Environment System 1 System 2 Environment Interconnected systems interact. How is interaction formalized?

10 p. 9/60 Motivation The ever-increasing computing power allows to model complex interconnected systems accurately. Requires the right mathematical concepts for dynamical system (the behavior), for interconnection (this lecture), for interconnection architecture (this lecture).

11 Classical view p. 10/60

12 p. 11/60 Input/output systems inputs System outputs Oliver Heaviside ( ) Norbert Wiener ( ) Rudy Kalman (1930- )

13 p. 11/60 Input/output systems inputs System outputs Input/output thinking is inappropriate for describing the functioning of physical systems. A physical system is not a signal processor. Better concept: the behavior.

14 Signal flow graphs p. 12/60

15 p. 12/60 Signal flow graphs Signal flow graphs are inappropriate for describing the interaction architecture of physical systems. A physical system is not a signal processor. Better concept: graph with leaves.

16 p. 13/60 Interconnection Interconnection as output-to-input assignment.

17 p. 13/60 Interconnection Interconnection as output-to-input assignment. Examples: series parallel feedback

18 p. 13/60 Interconnection Interconnection as output-to-input assignment. Output-to-input assignment is inappropriate for describing the interconnection of physical systems. A physical system is not a signal processor. Better concept: variable sharing.

19 Examples p. 14/60

20 p. 15/60 Electrical circuit terminals 1 V 1 V 2 N 2 V N I N I 1 I 2 Electrical circuit Electrical circuit k I k V k At each terminal: a potential (!) and a current (counted > 0 into the circuit), The relation between potentials of the terminals and voltages across the terminals is discussed in Exercise IV.3.

21 p. 15/60 Electrical circuit terminals 1 V 1 V 2 N 2 V N I N I 1 I 2 Electrical circuit Electrical circuit k I k V k At each terminal: a potential (!) and a current (counted > 0 into the circuit), behavior B ( R N R N) R. (V 1,V 2,...,V N,I 1,I 2,...,I N ) B means: this potential/current trajectory is compatible with the circuit architecture and its element values.

22 p. 16/60 Mechanical device pins 1 F 1 N 2 F N q N q 1 q 2 F 2 Mechanical system Mechanical system k F k q k At each terminal: a position and a force. position/force trajectories (q,f) B ((R ) 2N ) R. More generally, a position, force, angle, and torque.

23 p. 17/60 Other domains Thermal systems: At each terminal: a temperature and a heat flow. Hydraulic systems: At each terminal: a pressure and a mass flow. Multidomain systems: Systems with terminals of different types, as motors, pumps, etc....

24 Interconnection p. 18/60

25 p. 19/60 Connection of terminals System 1 System 2 By interconnecting, the terminal variables are equated.

26 p. 20/60 Connection of circuit terminals Interconnection = connecting terminals, like soldering wires together. V 1 I 1 I 1 I 2 V 1 V N I N I 2 V 2 V I V 2 V N 1 I N 1 Electrical circuit V I Electrical circuit V N 2 I N 2 V N 2 I N 2 I k I k V k V k Connecting terminals N 1 and N leads to V N 1 = V N, I N 1 + I N = 0. After interconnection the terminals share the variables V N 1,V N, and I N 1,I N (up to a sign).

27 Connection of circuit terminals V 1 I 1 I 1 I 2 V 1 V N I N I 2 V 2 V I V 2 V N 1 I N 1 Electrical circuit V I Electrical circuit V N 2 I N 2 V N 2 I N 2 I k I k Connecting terminals N 1 and N leads to V k V k V N 1 = V N, I N 1 + I N = 0. The interconnected circuit has N 2 terminals. Its behavior = B = {(V 1,I 1,V 2,I 2,...,V N 2,I N 2 ) : R R 2(N 2) V,I such that (V 1,I 1,V 2,I 2,...,V N 2,I N 2, V,I,V, I ) B}. p. 20/60

28 p. 21/60 Interconnection of circuits Electrical N Electrical circuit 1 circuit 2 N N 1 N 1 V N = V N and I N + I N = 0. Behavior after interconnection: B 1 B 2 := {(V 1,...,V N 1,V 1,...,V N 1,I 1,...,I N 1,I 1,...,I N 1) V, I such that ( V1,...,V N 1, V,I 1,...,I N 1, I ) B 1 and ( V1,...,V N 1, V,I 1,...,I N 1, I ) B 2 }.

29 p. 21/60 Interconnection of circuits more terminals and more circuits connected Electrical Electrical Circuit 1 Circuit 2 Electrical Circuit 3

30 p. 22/60 Connection of mechanical terminals Interconnection = connecting terminals, like screwing pins together. q 1 F N 1 q N 1 q N F N F 1 Mechanical device F 2 q 2 F q F Mechanical device q N 2 F N 2 q k F k Connecting terminals N 1 and N leads to q N 1 = q N, F N 1 + F N = 0. After interconnection the terminals share the variables q N 1,q N, and F N 1,F N (up to a sign).

31 Connection of mechanical terminals q 1 F N 1 q N 1 q N F N F 1 Mechanical device F 2 q 2 F q F Mechanical device q N 2 F N 2 Connecting terminals N 1 and N leads to q k F k q N 1 = q N, F N 1 + F N = 0. The interconnected circuit has N 2 terminals. Its behavior = B = {(q 1,F 1,q 2,F 2,...,q N 2,F N 2 ) : R R 2(N 2) q,f such that (q 1,F 1,q 2,F 2,...,q N 2,F N 2, q,f,q, F ) B}. p. 22/60

32 p. 23/60 Interconnection of mechanical systems N Mechanical Mechanical system 1 system 2 N N 1 N 1 q N = q N and F N + F N = 0.

33 p. 24/60 Other terminal types Thermal systems: At each terminal: a temperature and a heat flow. T N = T N and Q N + Q N = 0. Hydraulic systems: At each terminal: a pressure and a mass flow. p N = p N and f N + f N = 0....

34 p. 25/60 Sharing variables V N = V N and I N + I N = 0, q N = q N and F N + F N = 0, T N = T N and Q N + Q N = 0, p N = p N and f N + f N = 0,. Interconnection means variable sharing.

35 Tearing, zooming, and linking p. 26/60

36 p. 27/60 Tearing Model the behavior of selected variables!!

37 p. 27/60 Tearing Model the behavior of selected variables!! Tear

38 p. 28/60 Zooming Zoom

39 p. 28/60 Zooming Hierarchically Proceed until subsystems ( modularity ) are obtained whose model is known, from first principles, or stored in a database.

40 Linking p. 29/60

41 p. 29/60 Linking Link

42 Interconnection architecture p. 30/60

43 p. 31/60 Graph with leaves A graph with leaves : V a finite set of vertices, E a finite set of edges, L a finite set of leaves, f E the edge incidence map, f L the leaf incidence map. G = (V,E,L, f E, f L ) f E maps each element e E into an unordered pair [v 1,v 2 ], with v 1,v 2 V, f L is a map from L to V, it maps each element l L into an element v V.

44 p. 31/60 Graph with leaves Example: l 1 v e 1 e 2 v 2 v 3 e 3 e 4 v l 2 f E : e 1 [v 1,v 2 ],e 2 [v 1,v 3 ],e 3 [v 2,v 4 ],e 4 [v 3,v 4 ]. f L : l 1 v 1,l 2 v 4.

45 p. 32/60 Formalization of interconnected system An interconnected system is identified with a graph with leaves G = (V,E,L, f E, f L ). The vertices subsystems The edges connections, The leaves external terminals.

46 p. 33/60 A model is obtained as follows. Model specification For each subsystem, specify the behavior of the variables on its terminals, i.e. on the edges and the leaves that are incident to the vertex corresponding to the subsystem. For each connection, specify the sharing variable conditions on the connected terminals. I.e., for each edge, specify the interconnection constraints on the variables of the subsystem terminals that correspond to the edges. Specify the manifest variables. Subsystems in the vertices. Connections in the edges. External terminals in the leaves.

47 Example p. 34/60

48 p. 35/60 A transmission line Consider the transmission line shown below. source load The aim is to model the relation between the voltage of the source on the left and the voltage across the load on the right.

49 p. 36/60 A transmission line View the system as an interconnection of 4 subsystems.

50 p. 36/60 A transmission line View the system as an interconnection of 4 subsystems. The interconnection architecture the graph with leaves

51 p. 37/60 A transmission line l 1 e 1 e 3 e 5 v 1 v 2 v 3 v l 2 e 2 e 4 e 6

52 p. 37/60 A transmission line l 1 e 1 e 3 e 5 v 1 v 2 v 3 v l 2 e 2 e 4 e 6 In vertices v 1,v 2,v 3 we have identical subsystems. We deal with them later. In vertex v 4 there is a resistor v 4,1 v 4,2 V v4,1 V v4,2 = R I v4,1, I v4,1 + I v4,2 = 0.

53 p. 38/60 A transmission line section In each of the vertices v 1,v 2,v 3 we have: This system can be viewed as the interconnection of 6 subsystems:

54 p. 38/60 A transmission line section The associated interconnection architecture is

55 p. 38/60 A transmission line section The associated interconnection architecture is l 1 l 2 e e 2 v e 3 v 2 v 3 v 4 e 5 v 5 e 6 v 6 e l 3 l 4

56 p. 39/60 Modeling the transmission line section Subsystems: 2 resistors, 1 inductor, 1 capacitor, 2 connectors. The inductor in vertex v 5, for example, v 5,1 v 5,2 V v5,1 V v5,2 = L d dt I v 5,1, I v5,1 + I v5,2 = 0.

57 p. 39/60 Modeling the transmission line section Subsystems: 2 resistors, 1 inductor, 1 capacitor, 2 connectors. The inductor in vertex v 5, for example, v 5,1 v 5,2 V v5,1 V v5,2 = L d dt I v 5,1, I v5,1 + I v5,2 = 0. The connector in vertex v 1, for example, l 1 v 1,4 v 1,2 v 1,3 V l1 = V v1,2 = V v1,3 = V v1,4, I l1 + I v1,2 + I v1,3 + I v1,4 = 0. For each vertex we obtain a set of equations.

58 p. 40/60 Modeling the transmission line section The connection equations for edge e 5, for example, V v1,4 = V v5,1, I v1,4 + I v5,1 = 0. For each edge we obtain two such equations.

59 p. 41/60 Modeling the transmission line section For each vertex and for each edge we obtain a set of equations. The manifest variables for this subsystem are The latent variables are V l1,i l1,v l2,i l2,v l3,i l3,v l4,i l4. V v1,1,i v1,1,...,v v6,1,i v6,1.

60 p. 41/60 Modeling the transmission line section For each vertex and for each edge we obtain a set of equations. The manifest variables for this subsystem are The latent variables are V l1,i l1,v l2,i l2,v l3,i l3,v l4,i l4. V v1,1,i v1,1,...,v v6,1,i v6,1. Eliminating the latent variables, a LTIDS with (4) ODEs in the variables V l1,i l1,v l2,i l2,v l3,i l3,v l4,i l4.

61 Modeling the transmission line section Denote these equations as V l1 I l1 V R ( l2 ) d I l2 dt V l3 I l3 V l4 I l4 = 0. Note: For RLC-circuits, as is the case here, there are more efficient ways to arrive at these equations (see Lecture V). p. 42/60

62 p. 43/60 A transmission line Going back to the transmission line yields the subsystem equations R ( ) d dt V l1 I l1 V v1,2 I v1,2 V v1,3 I v1,3 V v1,4 I v1,4 V v2,1 I v2,2 V v2,2 I v2,2 V v2,3 I v2,3 V v2,4 I v2,4 V v3,1 I v3,2 V v3,2 I v3,2 V = 0, v3,3 I v3,3 V v3,4 I v3,4 V v4,1 V v4,2 = R I v4,1, I v4,1 + I v4,2 = 0,

63 p. 43/60 A transmission line and the interconnection equations V v1,3 = V v2,1, I v1,3 + I v2,1 = 0, V v1,4 = V v2,2, I v1,4 + I v2,2 = 0, V v2,3 = V v3,1, I v2,3 + I v3,1 = 0, V v2,4 = V v3,2, I v2,4 + I v3,2 = 0, V v3,3 = V v4,1, I v4,3 + I v4,1 = 0, V v3,4 = V v4,2, I v3,4 + I v4,2 = 0.

64 p. 43/60 A transmission line and the interconnection equations V v1,3 = V v2,1, I v1,3 + I v2,1 = 0, V v1,4 = V v2,2, I v1,4 + I v2,2 = 0, V v2,3 = V v3,1, I v2,3 + I v3,1 = 0, V v2,4 = V v3,2, I v2,4 + I v3,2 = 0, V v3,3 = V v4,1, I v4,3 + I v4,1 = 0, V v3,4 = V v4,2, I v3,4 + I v4,2 = 0. Finally, there is the manifest variable assignment w 1 = V l1 V l2, w 2 = V v4,1 V v4,2.

65 p. 44/60 A transmission line After elimination of the latent variables, we obtain the desired differential equation that describes the behavior of (w 1,w 2 ) r 1 ( ddt ) w1 = r 2 ( d dt) w2. In practice, all these steps need to be carried out more explicitly, faster, better, and more reliably with the help of software and a computer toolbox.

66 p. 45/60 Input/output versus tearing, zooming, and linking

67 p. 46/60

68 Control as interconnection p. 47/60

69 p. 48/60 Feedback control to be controlled variables exogenous inputs Actuators Plant Sensors to be controlled outputs control inputs Controller measured outputs control variables

70 p. 49/60 Behavioral control to-be-controlled terminals Plant control terminals Controller

71 p. 49/60 Behavioral control to-be-controlled terminals Plant control terminals Controller control = interconnection. Plant Controller controlled system

72 p. 50/60 A quarter car load chassis damper axle road wheel

73 p. 50/60 A quarter car load chassis damper axle road wheel

74 p. 50/60 A quarter car load chassis damper axle road wheel controller

75 p. 50/60 A quarter car load chassis damper axle measurement Controller Actuator road wheel active controller

76 p. 50/60 A quarter car load chassis damper axle road wheel passive controller

77 p. 51/60 Suspension controllers in Formula 1 Nigel Mansell victorious in 1992 with an active damper suspension. Active dampers were banned in 1994 to break the dominance of the Williams team.

78 p. 51/60 Suspension controllers in Formula 1 Renault successfully uses a passive tuned mass damper in 2005/2006. Banned in 2006, under the movable aerodynamic devices clause.

79 p. 51/60 Suspension controllers in Formula 1 inerter Kimi Räikkönen wins the 2005 Grand Prix in Spain with McLaren s J-damper (see Lecture V).

80 Pole placement and stabilization p. 52/60

81 p. 53/60 LTIDSs as controllers We consider only full control. Plant Controller Controlled system and controllers such that the controlled system is autonomous.

82 p. 53/60 LTIDSs as controllers Consider a LTIDS P L w, the plant, and a LTIDS C L w the controller. [[C is called a regular controller for P]] : [[(i) p(c ) =m(p) and (ii) P C is autonomous ]].

83 p. 53/60 LTIDSs as controllers Consider a LTIDS P L w, the plant, and a LTIDS C L w the controller. [[C is called a regular controller for P]] : [[(i) p(c ) =m(p) and (ii) P C is autonomous ]]. In terms of minimal kernel representations P ( d dt) w = 0 C ( d dt ) w = 0 [ ] for P and C, regularity P C square and nonsingular.

84 p. 54/60 The characteristic polynomial Let B L w be autonomous, and R ( d dt) w = 0 be a minimal kernel representation of B. Assume thatdeterminant(r) is monic (otherwise change R αr with 0 α R suitably chosen). The characteristic polynomial of B, χ B R[ξ], is defined as χ B :=determinant(r). Note: χ B is independent of the R chosen to represent B.

85 The characteristic polynomial Let B L w be autonomous, and R ( d dt) w = 0 be a minimal kernel representation of B. The minimal polynomial of B, µ B R[ξ], is defined as the monic polynomial of least degree that annihilates B, i.e. ( ) d µ B B = 0, dt ( ) d [[p B B = 0,0 p R[ξ]]] [[degree(p) degree(µ dt B )]]. Note: µ B is a factor of χ B. For B described by d dt x = Ax,w = Cx with (A,C) observable, χ B and µ B are equal to the characteristic and minimal polynomial of A. p. 54/60

86 p. 55/60 Pole placement Theorem Let P L w be controllable. Then for all monic polynomials π R[ξ], there exists a regular controller C L w such that χ P C = π.

87 p. 55/60 Pole placement Theorem Let P L w be controllable. Then for all monic polynomials π R[ξ], there exists a regular controller C L w such that χ P C = π. Proof: Let P ( ) d dt w = 0 be a minimal kernel representation of ] P. The Smith form yields P = U [0 p(p) m(p) I p(p) p(p) V, with U and V unimodular. Take for C the system C ( ) d dt w = 0, ] with C = [diag(π,1,...,1) 0 m(p) p(p) V.

88 p. 56/60 Pole placement The above proof can be refined in numerous directions (see Exercise IV.5). In particular, we obtain Theorem Let P L w be controllable. Then for all monic polynomials ν R[ξ], there exists a regular controller C L w such that µ P C = ν.

89 p. 57/60 Stabilization and Theorem Let P L w be stabilizable. Then there exists a regular controller C L w such that the autonomous system P C is stable.

90 Recapitulation p. 58/60

91 p. 59/60 Summary Complex systems consist of interconnections of subsystems.

92 p. 59/60 Summary Complex systems consist of interconnections of subsystems. Input/output thinking and signal flow graphs provide a very limited view of interconnected physical systems.

93 p. 59/60 Summary Complex systems consist of interconnections of subsystems. Input/output thinking and signal flow graphs provide a very limited view of interconnected physical systems. Modeling of interconnected systems proceeds by tearing, zooming, and linking.

94 p. 59/60 Summary Complex systems consist of interconnections of subsystems. Input/output thinking and signal flow graphs provide a very limited view of interconnected physical systems. Modeling of interconnected systems proceeds by tearing, zooming, and linking. A graph with leaves is a useful formalization of the interconnection architecture of an interconnected system.

95 p. 59/60 Summary Complex systems consist of interconnections of subsystems. Input/output thinking and signal flow graphs provide a very limited view of interconnected physical systems. Modeling of interconnected systems proceeds by tearing, zooming, and linking. A graph with leaves is a useful formalization of the interconnection architecture of an interconnected system. Control as interconnection is an effective way of thinking about control, with feedback control as a very useful special case.

96 Summary Complex systems consist of interconnections of subsystems. Input/output thinking and signal flow graphs provide a very limited view of interconnected physical systems. Modeling of interconnected systems proceeds by tearing, zooming, and linking. A graph with leaves is a useful formalization of the interconnection architecture of an interconnected system. Control as interconnection is an effective way of thinking about control, with feedback control as a very useful special case. Controllable LTIDSs allow regular controllers that achieve an arbitrary characteristic polynomial for the controlled system. Stabilizable LTIDSs can be stabilzed by regular controllers. p. 59/60

97 End of Lecture IV p. 60/60