NETWORK MODELING OF PHYSICAL SYSTEM DYNAMICS a.k.a. lumped parameter modeling Discussion GOAL: Systematic approach to synthesizing multi-domain physical system models MODEL TYPES: Purpose of the model is to promote understanding facilitate machine design and analysis perhaps be part of a real-time control system Hence our models are minimal (i.e., simple) low-order (i.e., simple) lumped-parameter Mod. Sim. Dyn. Sys. Network modeling discussion page 1
IMPORTANT DISTINCTIONS: There are at least three distinct entities of interest the physical system our model its (many) mathematical representation(s) DON T CONFUSE THEM! Mod. Sim. Dyn. Sys. Network modeling discussion page 2
STATE-DETERMINED REPRESENTATIONS of physical system dynamics MAIN CONCEPT: STATE a minimal set of variables to characterize all that changes in a system (knowing the state variables, every other variable can be computed) GENERAL FORM dx dt := f(x,u) y := g(x,u) x: state variables u: input variables y: output variables Mod. Sim. Dyn. Sys. Network modeling discussion page 3
A CAUSAL (INPUT/OUTPUT) REPRESENTATION prediction operator is (time-) integration suits control system analysis and design also suits many other applications NETWORKS OF LUMPED-PARAMETER ELEMENTS YIELD STATE-DETERMINED REPRESENTATIONS Mod. Sim. Dyn. Sys. Network modeling discussion page 4
MAJOR (MIS)PERCEPTIONS: Modeling is commonly (mis)perceived as just writing down the equations Prematurely writing down the equations can be misleading confusing inefficient standard forms of engineering equations are often based on implicit, non-general assumptions standard forms of engineering equations assume particular choices of input and output This frequently results in incompatible boundary conditions when multi-domain system models are assembled. DEFER WRITING DETAILED EQUATIONS AS LONG AS POSSIBLE Mod. Sim. Dyn. Sys. Network modeling discussion page 5
FURTHER IMPORTANT DISTINCTIONS: There are at least four distinct steps in the modeling process model synthesis (formulation) choice of state variables (state and output) equation derivation analysis of behavior DON T CONFUSE THEM! Mod. Sim. Dyn. Sys. Network modeling discussion page 6
ENERGY BASED MODELING... beware of mathematicians and all who make empty prophecies. St. Augustine, Bishop of Hippo, circa 400 A.D. * ALL MODELS ARE WRONG (to some degree). but it makes a difference what kind of errors are allowed. The energy-based approach ensures satisfaction of the first law of thermodynamics. * The full quotation: The good Christian should beware of mathematicians and all who make empty prophecies. The danger already exists that the mathematicians have made a covenant with the devil to darken the spirit and confine man in the bonds of Hell. Mod. Sim. Dyn. Sys. Network modeling discussion page 7
LUMPED-PARAMETER MODELS one element one physical phenomenon inertia kinetic energy storage elasticity deformational potential energy storage resistivity (free) energy dissipation Conversely one object (component) may embody multiple phenomena e.g., energy and dissipation in a real capacitor or inductor NETWORK MODEL ELEMENTS DESCRIBE PHENOMENA NOT OBJECTS Mod. Sim. Dyn. Sys. Network modeling discussion page 8
BOND GRAPHS AND BLOCK DIAGRAMS Block diagrams depict signal transmission Signal transmission implies uni-lateral interaction one-way influence Bond graphs depict power flow Power flow implies bi-lateral interaction two-way influence Mod. Sim. Dyn. Sys. Network modeling discussion page 9
BASIC BOND GRAPH NOTATION Port for power exchange: depicted by a bond (line) Energy storage elements: C and I Dissipative elements: R Junction elements: 0, 1, GY, TF Boundary elements (or power sources): S e and S f Mod. Sim. Dyn. Sys. Network modeling discussion page 10
THE FOREMOST PURPOSE OF MODELING IS TO DEVELOP INSIGHT (the following belongs after the multi-port section) SYSTEMATIC APPROACH TO PHYSICAL SYSTEM DYNAMICS MODELING: Interaction between (sub)systems is port concept Energy-based formalism unambiguous network notation: bond graph causal (input-output) analysis network models Energy-based network theory assumptions are physically based rather than mathematically based equilibrium energy storage steady-state (free) energy dissipation power-continuous inter-connections Common (mis)perception The energy-based modeling formalism dictates a single choice of variables and a single equation derivation procedure Energy-based physical system modeling is not an equation derivation procedure Behavior-based model synthesis Mod. Sim. Dyn. Sys. Network modeling discussion page 11
network primitives are elements of behavior, not physical components. Goals: present condensed review of systematic, energy-based approach to modeling introduce basic bond-graph notation un-teach some common misconceptions Mod. Sim. Dyn. Sys. Network modeling discussion page 12