Modeling & Simulation 2019 Lecture 8. Bond graphs

Claudio Altafini Automatic Control, ISY Linköping University, Sweden 1 / 45 Summary of lecture 7

• General modeling principles • Physical modeling: dimension, dimensionless quantities, scaling • Models from physical laws across different domains • Analogies among physical domains 2 / 45 Lecture 8. Bond graphs

Summary of today • Analogies among physical domains • Bond graphs • Causality

In the book: Chapter 5 & 6 3 / 45 Basic physics laws: a survey

Electrical circuits Hydraulics

Mechanics – translational Thermal systems

Mechanics – rotational 4 / 45 Electrical circuits

Basic quantities: • Current i(t) (ampere) • u(t) (volt)

• P (t) = u(t) · i(t) 5 / 45 Electrical circuits

Basic laws relating i(t) and u(t) • d 1 Z t L i(t) = u(t) ⇐⇒ i(t) = u(s)ds dt L 0 • capacitance

d 1 Z t C u(t) = i(t) ⇐⇒ u(t) = i(s)ds dt C 0 • resistance (linear case) u(t) = Ri(t) 6 / 45 Electrical circuits

Energy storage laws for i(t) and u(t) • electromagnetic 1 T (t) = Li2(t) 2 • electric field energy 1 T (t) = Cu2(t) 2 • energy loss in a resistance

Z t Z t Z t 1 Z t T (t) = P (s)ds = u(s)i(s)ds = R i2(s)ds = u2(s)ds 0 0 0 R 0 7 / 45 Electrical circuits

Interconnection laws for i(t) and u(t) • Kirchhoff law for On a loop: ( X +1, σk aligned with loop direction σkuk(t) = 0, σk = −1, σ against loop direction k k

• Kirchhoff law for currents On a node: ( X +1, σk inward σkik(t) = 0, σk = −1, σ outward k k 8 / 45 Electrical circuits

Transformations laws for u(t) and i(t) •

u1 = ru2 1 i = i 1 r 2 u1i1 = u2i2 ⇒ no power loss

•

u1 = ri2 1 i = u 1 r 2 u1i1 = u2i2 ⇒ no power loss 9 / 45 Electrical circuits

Example

State space model: d 1 i = (u − Ri − u ) dt L s C d 1 u = i dt C C 10 / 45 Mechanical – translational

Basic quantities: • v(t) (meters per second) • F (t) (newton)

• Power P (t) = F (t) · v(t) 11 / 45 Mechanical – translational

Basic laws relating F (t) and v(t) • Newton second law d 1 Z t m v(t) = F (t) ⇐⇒ v(t) = F (s)ds dt m 0 • Hook’s law (elastic bodies, e.g. spring)

1 d Z t F (t) = v(t) ⇐⇒ F (t) = k v(s)ds k dt 0

• (e.g. dry )

F (t) = bv(t) 12 / 45 Mechanical – translational

Energy storage laws for F (t) and v(t) • kinetic energy 1 T (t) = mv2(t) 2 • potential energy 1 T (t) = F 2(t) 2k • energy loss due to friction

Z t Z t Z t 1 Z t T (t) = P (s)ds = F (s)v(s)ds = b v2(s)ds = F 2(s)ds 0 0 0 b 0 13 / 45 Mechanical – translational Interconnection laws for F (t) and v(t) • sum of ; same velocity (series connection)

F = F1 + F2

v1 = v2

• sum of ; same force (parallel connection)

F1 = F2

v = v1 + v2 14 / 45 Mechanical – translational

Transformations laws for F (t) and v(t) • levers

`2 F1 = − F2 `1 `1 v1 = − v2 `2 • pulleys

1 F = F 1 2 2 1 v = − v 1 2 2 15 / 45 Mechanical – rotational

Basic quantities: • ω(t) (radians per second) • M(t) (newton · meter)

• Power P (t) = M(t) · ω(t) 16 / 45 Mechanical – rotational

Basic laws relating M(t) and ω(t) • Newton second law d 1 Z t J ω(t) = M(t) ⇐⇒ ω(t) = M(s)ds dt J 0 • torsion of a body

1 d Z t M(t) = ω(t) ⇐⇒ M(t) = k ω(s)ds k dt 0

• rotational friction (typically nonlinear)

M(t) = h(ω(t)) 17 / 45 Mechanical – rotational

Energy storage laws for M(t) and ω(t) • rotational energy 1 T (t) = Jω2(t) 2 • torsional energy 1 T (t) = M 2(t) 2k • energy loss due to rotational friction

Z t Z t T (t) = P (s)ds = M(s)ω(s)ds 0 0 18 / 45 Mechanical – rotational Interconnection laws for M(t) and ω(t) • sum of ; same angular velocity (series connection)

M = M1 + M2

ω1 = ω2

• sum of velocities; same force (parallel connection)

M1 = M2

ω = ω1 + ω2 19 / 45 Mechanical – rotational

Transformations laws for M(t) and ω(t) • gears

M1 = rM2 1 ω = − ω 1 r 2

• “gyrator”

My = rωz 1 ω = − M y r z 20 / 45 Flow system

Basic quantities: • Flow Q(t) (cubic meters per second) • p(t) (newton per square meter)

• Power P (t) = p(t) · Q(t) 21 / 45 Flow system Basic laws relating p(t) and Q(t) • Newton second law ρ` d 1 Z t Q(t) = p(t) ⇐⇒ Q(t) = p(s)ds A dt Lf 0 |{z} Lf =inertance

• pressure of a liquid column

A d 1 Z t p(t) = Q(t) ⇐⇒ p(t) = Q(s)ds ρg dt Cf 0 |{z} Cf =capacitance

• flow resistance (laminar flow)

p(t) = Rf Q(t) |{z} flow resistance 22 / 45 Flow system

Energy storage laws for p(t) and Q(t) • kinetic energy (fluid in a tube) 1 T (t) = L Q2(t) 2 f • potential energy (fluid in a tank) 1 T (t) = C p2(t) 2 f • energy loss due to flow resistance

Z t Z t T (t) = P (s)ds = p(s)Q(s)ds 0 0 23 / 45 Flow system

Interconnection laws for p(t) and Q(t) • series connection

p = p1 + p2

Q1 = Q2

• parallel connection

p1 = p2

Q = Q1 + Q2 24 / 45 Flow system

Transformations laws for p(t) and Q(t) • flow transformer

p1 = rp2 1 Q = Q 1 r 2

where r = A2/A1 25 / 45 Analogies among physical domains

Electrical Mechanical Mechanical Hydraulic Thermal translational rotational flow current velocity angular volume heat flow velocity flow effort voltage force torque pressure temperature power power power power power power · temperature inductive inertia of −− element inertia fluid capacitive spring torsional tank heat element spring storage resistive friction friction friction thermal element resistance transformer transformer lever gears transducer −− gyrator gyrator −− gyro −− −− 26 / 45 Conversions among domains

• Mechanical (translational) – Hydraulics

transformer:

1 p = F A Q = A v • Mechanical (rotational) – Electrical

gyrator:

M = k i 1 ω = u k 27 / 45 Analogies among physical domains

• two "power" variables 1. flow f =  i, v, ω, Q, q 2. effort e =  u, F, M, p, T

• their product: power

e · f =  u · i, F · v, M · ω, p · Q, T · q 28 / 45 Analogies, cont’d

• three energy "exchange" relationships 1. inductance =⇒ effort storage

df 1 1 Z t = e =⇒ f(t) = e(s)ds dt α α 0 2. capacitance =⇒ flow storage

de 1 1 Z t = f =⇒ e(t) = f(s)ds dt β β 0 3. resistance =⇒ loss of energy

e = γf (possibly nonlinear: e = h(f)) 29 / 45 Bond graphs

Bond graph theory: an exchange of energy is a bond

−−−e* f

• line = connection between parts of the system • above the line (harpoon side): effort • below the line: flow • direction of half arrow = direction of energy flow (i.e., direction in which power p = e · f is positive)

−−u* −−−F* −−−M* −−−p* −−T* i v ω Q q 30 / 45 Bond graphs for energy exchanges

• for effort storage: I-type element 1 Z t f(t) = e(s)ds ⇐⇒ −−−e* I : α α 0 f • bond graph for flow storage: C-type element 1 Z t e(t) = f(s)ds ⇐⇒ −−−e* C : β β 0 f • bond graph for resistive elements: R-type element

e(t) = γ f(t) ⇐⇒ −−−e* R : γ f 31 / 45 Bond graphs for sources

• bond graph for effort source e  Se −−−−* system ] f

• bond graph for flow source e  Sf −−−−* system ] f 32 / 45 Bond graphs for junctions

bond graph for series connection: s - junction

e2 f2 ...... e1 ...... s f . 1 ......

en fn

f1 = f2 = ... = fn =⇒ common flow

e1 + e2 + ... + en = 0

outgoing harpoon at ej: change sign to ej in the summation 33 / 45 Bond graphs for junctions

bond graph for parallel connection: p - junction

e2 f2 ...... e .. 1 ...... p f . 1 ......

en fn

e1 = e2 = ... = en =⇒ common effort

f1 + f2 + ... + fn = 0

outgoing harpoon at fj: change sign to fj in the summation 34 / 45 Bond graphs for transformer and

• bond graph for transformer: TF - junction

−−−−e1 * TF −−−−e2 * f1 f2 1 e = ne , f = f 2 1 2 n 1 • bond graph for gyrator: GY - junction

−−−−e1 * GY −−−−e2 * f1 f2

1 e = rf , f = e 2 1 2 r 1 Both conserve power 35 / 45 Bond graphs and ports

Edges (bonds) of the bond graph are connecting points (or ports) of the system. The number of ports can be

• one : Se, Sf , C, I, R • two ports: TF , GY • multiport: s-junction, p-junction 36 / 45 Example: electrical circuit / hydraulic system 37 / 45 Example: electrical circuit 37 / 45 Example: electrical circuit 38 / 45 Example: an hydraulic system 38 / 45 Example: an hydraulic system 39 / 45 State space description

Is it possible to obtain a state space description out of a bond graph?

Information flow in a state space model

x˙ = f(x, u)

• for given x och u it is possible to compute x˙?

• in a bond graph: causality marking 40 / 45 Information flow

Information flow for bond graphs of C , I, and Se, Sf types:

| {z } integral causality 41 / 45 Causality

Causal strokes

• “A sets e and B sets f” (i.e., A is of C and Se types)

• “B sets e and A sets f” (i.e., A is of I and Sf types) 42 / 45 Causality, cont’d

• Bond graphs with fixed causality strokes: 1. sources

2. C and I type

3.

4. gyrators 43 / 45 Causality, cont’d

• Causality at junctions 1. s-junctions

2. p-junctions 44 / 45 Causality, cont’d

• Elements with free causality 1. R-type – e can be computed from a given f – f can be computed from a given e – certain nonlinearities may constitute an exception

e = φ(f) f = φ−1(e) 45 / 45 Bond graphs

Meaning of bond graphs: • describes various physical domains in the same way • expresses simple addition rules for effort and flow variables • illustrates the structure of a system from its components • translates physical laws into graphical interactions • formalism prone to object-oriented programming Claudio Altafini www.liu.se