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Course Outline

• Quick Sneak Preview• I. Assembling Models from Physical Problems

– I.1. Sample problems– I.2. Assembling models from conservation/constit. laws – I.3. Assembling models from PDE Solvers

• II. Simulating Models– II.1. Steady state analysis of linear & non-linear models – II.2. Time domain integration of dynamical models– II.3. Important system properties (stability/dissipativity etc.)

• III. Model Order Reduction for Linear Systems– III.1. Compressing LTI with modal analysis & point match– III.2. Compressing LTI with projection framework

– POD Proper Orthogonal Decomposition– TBR Trancated Balance Realizations– PRIMA Krylov moment matching with passivity

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Course Outline

• IV. Model Order Reduction for Non-Linear Systems– IV.1. Compressing weakly non-linear systems– IV.2. Compressing strongly non-linear systems (TPWL)– IV.3. Generating compact models from input/output data

• V. Parameterized Model Order Reduction– V.1. Compressing linear parameterized systems– V.2. Compressing non-linear parameterized systems– V.3. Examples of problems solved by PMOR

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Introduction toCompact Dynamical Modeling

I.1 - Sample Problems

Luca DanielMassachusetts Institute of Technology

NSF & NIH

Power delivery network for Integrated Circuit (IC) or city/state

Many engineering systems can be viewed asnetworks of dynamical components

Droop

Blood/Oil delivery network Building/Mechanical frames

Heat sink for 3D IC

Group Activity 1 and 2 (previews)

• During the next 30min of my lecture think of two examples of networks of interconnected components:– Examples should be ideally from your own field of

studies/experience/hobby/interests– If you are short of ideas you can use and modify one of the examples I will

present (generated by other students like you last year)

• At the end of my 30min you will form groups:– 2 to 3 students per group;– “different” technical background in each group (e.g. different departments);

• Each student will INTERACTIVELY discuss HER/HIS OWN two examples with his/her group, identifying common network concepts:– What do the “nodes” of the network represent? – What are the quantities associated with the nodes?– What do the “edges” of the network represent? – What are the quantities associated with the edges?– What are the conditions that specify “which” nodes are connected by

edges?– What are the conditions that specify the “behavior” of the edges?

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From www.maxim.com

• Nodes: e.g. gates• Edges: e.g. wires• Quantities associated with nodes: e.g. voltages • Quantities associated with edges: e.g. currents on the wires• Conditions that specify which nodes are connected: Kirchhoff’s Current

Law (KCL), Kirchhoff’s Voltage Law (KVL)• Conditions that specify what kind of connections: Current-voltage

relations (e.g. resistance, inductance, impedance of the wires etc…)

Courtesy of Harris semiconductor

Networks of electrical components on Integrated Circuits(e.g. cell phone…)

Source•Coal: 42%•Natural Gas: 25%•Nuclear: 19%•Hydropower: 8%•Other Renewable 5%•Petroleum: 1%

POWER PLANTS

US Power Grid, from http://www.npr.org/templates/story/story.php?storyId=110997398

• Objective: Compute state of power grid (voltages at nodes, power flow across lines, energy loss, …) given network topology + power consumed and generated at every node.

• Used in planning, monitoring, etc… of power grid

Electrical power transportation networksHamza Fawzi, Lab. for Information and Decision Systems (http://lids.mit.edu)

Electrical power transportation networksHamza Fawzi, Lab. for Information and Decision Systems (http://lids.mit.edu)

US Power Grid, from http://www.npr.org/templates/story/story.php?storyId=110997398

TRANSMISSIONLINES

• Objective: Compute state of power grid (voltages at nodes, power flow across lines, energy loss, …) given network topology + power consumed and generated at every node.

• Used in planning, monitoring, etc… of power grid

Generator

Power consumption (load)

Admittance of line

• Nodes: e.g. power plants, cities, (or buildings etc..)• Edges: e.g. power transmission lines between cities and power plants• Nodal quantities: e.g. voltages • Edge quantities: e.g. power flow• Conditions that specify which nodes are connected: conservation of

energy at each node• Conditions that specify what kind of connections: Current-voltage

relations (e.g. admittance of the transmission lines etc…)

Electrical power transportation networksHamza Fawzi, Lab. for Information and Decision Systems (http://lids.mit.edu)

Traffic flow networksDilip Thekkoodan, PhD student mechanical Eng. at MIT

• Reference:– “A Circuit Simulation technique for Congested Network Traffic Assignment”, Cho,H.J. & Huang,H.

AIP Conference Proceedings, 963, 993(2007)• Images from :

– ZwahLenImages.com, Redbubble.com, dreamstime.com, maps.google.com, circuitlab.com, pixabay.com

Traffic flow networksDilip Thekkoodan, PhD student mechanical Eng. at MIT

• Reference:– “A Circuit Simulation technique for Congested Network Traffic Assignment”, Cho,H.J. & Huang,H.

AIP Conference Proceedings, 963, 993(2007)• Images from :

– ZwahLenImages.com, Redbubble.com, dreamstime.com, maps.google.com, circuitlab.com, pixabay.com

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• Nodes: e.g. crossings (or cities)• Edges: e.g. roads (or highways)• Nodal quantities: e.g. time• Edge quantities: e.g. traffic load (# of cars on that road)• Conditions that specify which nodes are connected: conservation of

“cars” at each crossing• Conditions that specify what kind of connections: relation between

travel time and traffic load (can be interpreted as a nonlinear resistor)

e.g. Cambridge traffic flow map

Traffic flow networksDilip Thekkoodan, PhD student mechanical Eng. at MIT

“Shortest Path” problem on as a resistor networkAustin Nicholas, PhD student MIT

• Shortest Path Problems (SPP) have many real world applications– Driving directions– Factory floor layouts– Robot arm path planning– Solving Rubik’s Cubes

• Why I Care: Electromagnetic Satellite Formation Flight

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• Intuition:– Electrical current “knows” how to flow from source to ground– Water “knows” how to flow from mountains to valleys

• Problem: current and water will spread into several paths…• Solution: iterate and penalizing paths with low current flow until

achieving a single path

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Shortest Path Problem Resistor Network

“Shortest Path” problem on as a resistor networkAustin Nicholas, PhD student MIT

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Natural gas pipeline networkParthsarathi Trivedi, Laboratory for Aviation and the Environment [LAE.mit.edu]

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U.S. Natural Gas Pipeline Network (2009)Parthsarathi Trivedi, Laboratory for Aviation and the Environment [LAE.mit.edu]

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Isolate Primary Flow PathwaysParthsarathi Trivedi, Laboratory for Aviation and the Environment [LAE.mit.edu]

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Simplify to a Network of Resistors & CapacitorsParthsarathi Trivedi, Laboratory for Aviation and the Environment [LAE.mit.edu]

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Simplify to a Network of Resistors & CapacitorsParthsarathi Trivedi, Laboratory for Aviation and the Environment [LAE.mit.edu]

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• Nodes: e.g. storage facilities, pipe junctions etc..• Edges: e.g. pipes • Nodal quantities: e.g. pressures • Edge quantities: e.g. gas flow• Conditions that specify which nodes are connected: conservation of

gas at each node• Conditions that specify what kind of connections: pressure – gas flow

relations (e.g. flow impedance of the pipes etc…)

Natural gas pipeline networkParthsarathi Trivedi, Laboratory for Aviation and the Environment [LAE.mit.edu]

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Blood floow networkMohammad M. Ghassemi, Lab. for Computational Physiology http://lcp.mit.edu/

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What will the system output?Mohammad M. Ghassemi, Lab. for Computational Physiology http://lcp.mit.edu/

Nodes

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Answering Case 1: A Blood Clot.Mohammad M. Ghassemi, Lab. for Computational Physiology http://lcp.mit.edu/

A blood clot is suspected by the doctors in either the legs, or digestive tract of a patient. Where should they measure to find the clot?

Nodes

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Answering Case 2: A Bleeding PatientMohammad M. Ghassemi, Lab. for Computational Physiology http://lcp.mit.edu/

This time our clinicians would like to estimate how much blood was lost in the patient by measuring the blood pressure, they would also like to know where on the patient’s body is the best place to measure the pressure for the purposes of this estimate.

Nodes

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• Nodes: e.g. vessel bifocartions, heart, organs• Edges: e.g. arteries, capillaries, veins• Nodal quantities: e.g. pressures • Edge quantities: e.g. blood flow• Conditions that specify which nodes are connected: conservation of

blood• Conditions that specify what kind of connections: pressure – blood

flow relations (e.g. flow impedance of the vessels etc…)

Blood floow networkMohammad M. Ghassemi, Lab. for Computational Physiology http://lcp.mit.edu/

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Cerebro-spinal fluid networkAditya Kalluri, undergraduate student MIT

• Cerebrospinal Fluid surrounds the brain and fills its internal spaces

• Generated by capillaries, passes through fluid-filled cavities (ventricles), reabsorbed into venous system

Kashif (2011) Wikimedia Commons

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• Cerebrospinal Fluid surrounds the brain and fills its internal spaces

• Generated by capillaries, passes through fluid-filled cavities (ventricles), reabsorbed into venous system

Kashif (2011) Wikimedia Commons

Cerebro-spinal fluid networkAditya Kalluri, undergraduate student MIT

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Kashif (2011), originally Sorek et al.

Cerebro-spinal fluid networkAditya Kalluri, undergraduate student MIT

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Kashif (2011), originally Sorek et al.

Cerebro-spinal fluid networkAditya Kalluri, undergraduate student MIT

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• Nodes: e.g. ventricles (cavities)• Edges: e.g. canals• Nodal quantities: e.g. pressures • Edge quantities: e.g. fluid flow• Conditions that specify which nodes are connected: conservation of fluid

at ventricle• Conditions that specify what kind of connections: pressure – flow

relations (e.g. flow impedance of the canals etc…)

Cerebro-spinal fluid networkAditya Kalluri, undergraduate student MIT

Kashif (2011)

Kashif (2011), originally Sorek et al.

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• Nodes: e.g. joints• Edges: e.g. thermal connectors• Nodal quantities: temperature • Edge quantities: heat flow• Conditions that specify which nodes are connected: conservation of heat

flow• Conditions that specify what kind of connections: voltage – heat relations

(i.e. thermal resistance)

Heat Flow in SpacecraftFarah Alibay MIT Space Systems Laboratory (ssl.mit.edu)

Need to control temperature of small spacecraft and instruments passively using a radiator and thermal strap connected to a part that we wish to keep cool. The maximum temperature on a radiator is the minimum temperature

that the part/instrument connected to it can be.

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• Nodes: e.g. squares on a bacteria culture biofilm

• Edges: e.g. ecological connections• Nodal quantities: concentration of

bacteria species• Edge quantities: nutrient flow

Ecological nutrient flow networksSarah Spencer, Alm Lab, Biological Engineering [almlab.mit.edu]

• Conditions that specify which nodes are connected: conservation of nutrients

• Conditions that specify what kind of connections: relation between species mortality and nutrients availability

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• Nodes: e.g. small volumes in combustion chamber• Edges: e.g. chemical reactions• Nodal quantities: e.g. concentration of chemical species• Edge quantities: energy and mass

Network of chemical reactions in a Jet Engine CombustionMitch Withers, Lab. for Aviation and the Environment [http://lae.mit.edu]

• Conditions that specify which nodes are connected: mass and energy balances

• Conditions that specify what kind of connections: relation between concentration of species and mass/energy in chemical reactions

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Cargo

Droop

Network of Structural-Mechanical Systems(e.g. space frames, bridges, buildings...)

• Nodes: mechanical joints• Edges: beams (or struts)• Nodal quantities: joint displacements• Edge quantities: forces• Conditions that specify which nodes are connected: balance of forces• Conditions that specify what kind of connections: Force-displacement

relationships (elasto-mechanical) for structural elements

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Structural network modeling of Hydrogel in SyringeChristopher N. Lam, Olsen Lab, Chemical Engineering, MIT

• Nodes: copolymer domains• Edges: proteins• Nodal quantities: copolymer

displacements• Edge quantities: protein forces• Conditions that specify which nodes are connected: balance of forces• Conditions that specify what kind of connections: Force-displacement

relationships (elasto-mechanical)

Group Activity 1

• During the next 30min of my lecture think of two examples of networks of interconnected components:– Examples should be ideally from your own field of

studies/experience/hobby/interests– If you are short of ideas you can use and modify one of the examples I will

present (generated by other students like you last year)

• At the end of my 30min you will form groups:– 3 to 4 students per group;– “different” technical background in each group (e.g. different departments);

• Each student will INTERACTIVELY discuss HER/HIS OWN two examples with his/her group, identifying common network concepts:– What do the “nodes” of the network represent? – What are the quantities associated with the nodes?– What do the “edges” of the network represent? – What are the quantities associated with the edges?– What are the conditions that specify “which” nodes are connected by

edges?– What are the conditions that specify the “behavior” of the edges?