Introduction to System Optimization: Part 1

Introduction to System Optimization: Part 1 James Allison ME 555 March 7, 2007

System Optimization Subsystem optimization results optimal system? Objective: provide tools for developing system optimization strategy

System Optimization Subsystem optimization results optimal system? Objective: provide tools for developing system optimization strategy

Overview Systems and Interactions 1 Systems and Interactions Definitions Interaction Examples 2 3 AiO IDF ATC

System Definition Systems and Interactions Definitions Interaction Examples What makes something a system?

System Definition Systems and Interactions Definitions Interaction Examples What makes something a system? Comprised of several components (or subsystems)

System Definition Systems and Interactions Definitions Interaction Examples What makes something a system? Comprised of several components (or subsystems) Interactions exist between the components

System Definition Systems and Interactions Definitions Interaction Examples What makes something a system? Comprised of several components (or subsystems) Interactions exist between the components Some aspect of one component influences the effect of changes in another component

Explicit Analysis Interactions Definitions Interaction Examples Analysis: evaluate f, g, and h in design problem for a given x y ij : analysis output passed from subsystem j to i (coupling variable) Example: deflection and pressures in aeroelastic analysis

Interaction Example (First Type) Definitions Interaction Examples System objective function: f (y 12, y 13 ) = c 1 (y 12 c 2 ) 2 + c 3 (y 13 c 4 ) 2 + c 5 y 12 y 13 Inputs to subsystem 1: y 12 : output (response) of subsystem 2 y 13 : output (response) of subsystem 3 Three objective function terms: 1 Depends only on SS2 response 2 Depends only on SS3 response 3 Depends on a combination of the responses (interaction)

Interaction Example (First Type) Case I: No Interaction c = (1, 1, 1, 1, 0) T Definitions Interaction Examples 6 (x 1-1) 2 +(x 2-1) 2 5 4 x 2 3 2 1 0 0 1 2 3 4 5 6 x 1

Interaction Example (First Type) Definitions Interaction Examples Case II: Interaction Present c = (1, 1, 1, 1, 1) T 6 (x 1-1) 2 +(x 2-1) 2 -x 1 x 2 5 4 x 2 3 OFAT solution 2 optimum 1 0 0 1 2 3 4 5 6 x 1

Definitions Interaction Examples Interaction Example (Second Type) Governing Equations of Motion: [ ] { } m1 0 z1 + 0 m 2 z 2 [ k1 + k 2 k 2 k 2 k 2 ] { z1 z 2 } = { 0 0 } m 1 k 1 Coupled in stiffness: Motion of m 1 (i.e. x 1 (t)) is dependent of the value of k 2 ( z 1 (t) depends on the state of z 2 ), and visa-versa. m 2 k 2 z 1 z 2

Interaction Representation Definitions Interaction Examples Example system: a 1 (x 1, x 2, x 4 ) a 2 (x 6, y 21 ) a 3 (x 2, x 3, x 4, y 31, y 34 ) a 4 (x 4, x 5, y 41 )

Interaction Identification Definitions Interaction Examples Example: Interaction between automotive subsystems structure powertrain suspension steering braking cabin (interior geometry, HVAC, etc.)

Analysis Interactions Definitions Interaction Examples What might result if interactions are ignored in system design?

Analysis Interactions Definitions Interaction Examples What might result if interactions are ignored in system design? Missed opportunity to improve performance (not system optimal) Incompatibility between subsystems (inconsistent system)

Analysis Interactions Definitions Interaction Examples What are some interactions between your subsystems? Specific effects of ignoring interactions?

System Consistency A system has coupling variable consistency if for every coupling variable, y ij a j (x j, y j ) = 0 is satisfied.

System Consistency A system has coupling variable consistency if for every coupling variable, y ij a j (x j, y j ) = 0 is satisfied. System Analysis: task of solving system analysis equations for y p, given x.

System Optimality Subsystem Optimality System Optimality

From Subsystem to System Formulation Subsystem i optimization formulation: min x i f i (x i, y p ) subject to g i (x i, y p ) 0 System optimization formulation: h i (x i, y p ) = 0 min x f (x, y p ) subject to g(x) = [g 1, g 2,..., g N ] 0 h(x) = [h 1, h 2,..., h N ] = 0 where: f k select one subsystem objective f = N i=1 w if i (x, y p ) weighted sum of subsystem objectives f (x, y p ) define new function

AiO IDF ATC All-in-One (AiO) Design Approach Nest system analysis within system optimization but: Can identify system optimum Can be very computationally expensive Requires a complete system analysis for every design iteration May not converge System Optimizer x f, g, h System Analysis

AiO IDF ATC Example Problem Single Element Aeroelasticity Aeroelasticity: Requires both aerodynamic and structural analysis Application: Air-flow sensor design

Air-flow Sensor Analysis Structural Analysis: M = kθ = 1 2 F l cos θ Given a design (l, w) and a drag force F, solve for the corresponding deflection θ. AiO IDF ATC Aerodynamic Analysis: F = CA f v 2 = Clw cos θv 2 Given a design (l, w) and a deflection θ, find the drag force F. System Analysis: Given a design (l, w), find the equilibrium values F and θ. v F 1 2 l cos θ θ l k

Air-flow Sensor Design AiO IDF ATC (Sensor Calibration Problem) AiO Formulation: min l,w (θ ˆθ) 2 subject to F F max 0 Design Parameters: k, A, F max, C, v lw A = 0

AiO IDF ATC Solution by Monotonicity Analysis For cases where meeting the deflection target requires a drag force greater than F max, the target cannot be met, and the inequality constraint will be active.

AiO IDF ATC Solution by Monotonicity Analysis For cases where meeting the deflection target requires a drag force greater than F max, the target cannot be met, and the inequality constraint will be active. F max = Clw cos θv 2 kθ 1 2 F maxl cos θ = 0 ( ) θ = cos 1 Fmax CAv 2 l = 2k cos 1 ( F max CAv 2 ) CAv 2 F 2 max

AiO IDF ATC Solution by Monotonicity Analysis For cases where meeting the deflection target requires a drag force greater than F max, the target cannot be met, and the inequality constraint will be active. F max = Clw cos θv 2 kθ 1 2 F maxl cos θ = 0 ( ) θ = cos 1 Fmax CAv 2 l = 2k cos 1 ( F max CAv 2 ) CAv 2 F 2 max F, θ and l known solve for w using w = A/l

AiO IDF ATC Shared Quantities and System Analysis θ Structural Analysis l F Aerodynamic Analysis l, w Shared variable: l Coupling variables: θ and F System analysis requires finding θ and F such that: θ = θ(f ) F = F (θ)

System Analysis Options AiO IDF ATC Solve the system analysis equations at each optimization iteration (AiO) Could we use the optimization algorithm to solve these equations?

System Analysis Options AiO IDF ATC Solve the system analysis equations at each optimization iteration (AiO) Could we use the optimization algorithm to solve these equations? Yes! Make the system analysis equations constraints Let the optimizer choose values for the coupling variables, in addition to design variables

AiO IDF ATC Why Use the Optimizer for Analysis? No longer have to completely solve system analysis when you are far from the system optimum Breaks feedback loops (eliminating nested iterations) Enables coarse-grained parallel computation AiO can overlook global optimum in some cases, or even fail

Air Flow Sensor Reformulation AiO IDF ATC AiO Formulation: min l,w (θ ˆθ) 2 subject to F F max 0 lw A = 0 System analysis is implicitly required in the calculation of F and θ. New Formulation: min l,w,θ,f (θ ˆθ) 2 subject to F F max 0 lw A = 0 θ θ(l, F ) = 0 F F (l, w, θ) = 0

AiO IDF ATC Individual Disciplinary Feasible (IDF) Method IDF: simplest formal approach to combining analysis and optimization tasks. IDF General Formulation: min x,y f (x, y) subject to g(x, y) = [g 1, g 2,..., g N ] 0 h(x, y) = [h 1, h 2,..., h N ] = 0 h aux (x, y) = Sa(x, y) y = 0

IDF Architecture Systems and Interactions AiO IDF ATC System Optimizer g 1, h 1, y i1 f, g 2, h 2, y i2 x l1, x s1, y 1j SS1 Analysis x l2, x s2, y 2j SS2 Analysis

AiO IDF ATC IDF Formulation Example: Electric Water Pump min x subject to P e = VI P P min = 100 kpa T T max = 428 K L + l c 0.2 m Q = 1.55 10 3 m 3 /sec Analysis Functions T = a 1 (I, ω, d, d 2, d 3, L, l c ) Mot. winding temp. (K) I = a 2 (τ, T, d, d 2, d 3, L) Motor current (amps) ω = a 3 (I, T, d, d 2, d 3, L, l c ) Motor speed (rad/sec) τ = a 4 (ω, D 2, b, β 1, β 2, β 3 ) Pump drive torque (Nm) P = a 5 (ω, D 2, b, β 1, β 2, β 3 ) Pressure differential (kpa)

Other Options Systems and Interactions AiO IDF ATC IDF seems helpful for many problems, but what if I have more design variables than my optimization algorithm can handle?

Other Options Systems and Interactions AiO IDF ATC IDF seems helpful for many problems, but what if I have more design variables than my optimization algorithm can handle? Multilevel Methods: Use multiple optimization algorithms to share the load Distributed decision making reduces individual problem dimension

Multi-level Methods AiO IDF ATC Optimization algorithm coupled with every element of the system Local optimizers make local decisions (distributed decision making) Can utilized specialized optimization algorithms Best for sparse problem structures Many local decisions required, but relatively few subsystem connections Possible to reduce individual problem dimension w/ multilevel methods

AiO IDF ATC Analytical Target Cascading (ATC) Multi-level system design method Developed based on needs in the automotive industry Intended for hierarchical problems with object-based decomposition (covered in detail in a later lecture)

AiO IDF ATC Individual Disciplinary Feasible (IDF) Method R 1A i = 1 j = A x1a, y1a R 2B R 2C Levels i i = 2 j = B x2b, y2b j = C x2c, y2c R 3D R 3E R 3F R 3G i = 3 j = D j = E j = F j = G x3d, y3d x3e, y3e x3f, y3f x3f, y3g