Advances in Combined Architecture, Plant, and Control Design - - PowerPoint PPT Presentation

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Advances in Combined Architecture, Plant, and Control Design

Daniel R. Herber

Final Defense Department of Industrial and Enterprise Systems Engineering University of Illinois at Urbana-Champaign November 17, 2017 Urbana, IL, USA

1

Doctoral Committee: Assistant Professor James Allison Professor Yuliy Baryshnikov Associate Professor Carolyn Beck Professor Harrison Kim

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Outline

• 1. Introduction
• 2. Candidate Architectures through Enumeration
• 3. Co-Design: Combined Plant and Control Design
• 4. Scaling of Dynamic Optimization Formulations
• 5. Direct Transcription and Linear-Quadratic Dynamic Optimization
• 6. Case Study: Design of Passive Analog Circuits
• 7. Case Study: Design of Strain-Actuated Solar Arrays
• 8. Case Study: Design of Vehicle Suspensions
• 9. Conclusions and Future Work

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1

Introduction

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Introductory Example

sample placement path sample return path

Figure: Task description. (a) Link length. (b) Cross-section. Figure: Plant variables.

↷ ↷

end effecter links joints ground actuators

(a)

↷ ↷

(b) Figure: Different architectures.

joint 1 joint 2 torque time

Figure: Joint control trajectories.

3

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Three Design Domains

• Architecture
• Architecture is defined as the elements contained within a system

and their relationships1

• Component is a common alternative term for element
• Architectures with heterogeneous components: electrical

circuits2, hybrid powertrains3, vehicle suspensions4, gear trains5, and biological networks6

• Geometric architecture problem: trusses7, heat spreaders8, and

soft robotics9

• Plant
• Generally defined by variables that are generally regarded as time

independent

• An alternative definition: The plant consists of the quantities

fixed during the control design

1 Crawley, et al., 2004; 2 Macmahon, 1994; 3 Bayrak, et al., 2016; 4 Herber, et al., 2017; 5

Pennestri and Valentini, 2015; 6 Trusina, et al., 2009; 7 Bendsøe and Sigmund, 2004; 8 Lohan, et al., 2016; 9 Cheney, et al., 2013 4

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Three Design Domains

• Control
• Control seeks to govern directly the behavior of a dynamic

system, i.e., one which evolves through time

• Generally speaking, there are two types of control paradigms:

closed loop and open loop

• Ambiguity in these delineations
• Legacy design paradigms that treat certain parts of the design as

separate1

• Groups based on variables types
• Approximations or better representations of design variables in
• ne domain are frequently classified in another domain
• Architecture → plant: the SIMP approximation in structural
• ptimization2)

1 Allison and Herber, 2014; 2 Bendsøe and Sigmund, 2004

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A Design Process for Complete Dynamic System Design

• Here we adopt the design process proposed in Ref. 1 for

complete dynamic system design

• Helps manage the complexity and uncertainty found in

combined architecture, plant, and control design problems

Stage 1 Stage 2 Stage 3 Plant Architecture Design

Co-Design with OLC

Control Architecture Design

Co-Design with CLC

Digital Control Design

↻ ↻ ↻Adjust Formulation

• Controller architectures could include a basic feedback, hybrid2,
• r model predictive3 among others

1 Deshmukh, et al., 2015; 2 Lygeros, et al., 2008; 3 Borrelli, et al., 2017

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Solution Generation Challenges

• Automated candidate architecture generation
• Exploring different architectures requires an appropriate

conceptual framework that allows for modifications to the appropriate elements in the architecture

• However, not all architecture representations are equally useful
• Some might produce many infeasible systems or too many

candidates

• Automated model generation
• Given some architecture specification (e.g., a graph), we need to

create a suitable model for use in the optimization problems

• Certain modeling methodologies support this task such as bond

graph modeling [Borutzky, 2010]

• Automated optimization problem generation
• For different candidate architectures of the same design problem,

the optimization problem that predicts its performance may vary

• For example, different constraints may be present depending on

the components in the architecture

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Summary

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2

Candidate Architectures through Enumeration

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Architectures as Graphs

• A variety of different engineering

systems can be represented by (undirected) labeled graphs

• Labels can be used to represent a

variety of concepts

• Finding candidate architectures

requires the generation of new, useful graphs

• Existing approaches have their

drawbacks

• Generative representations1
• Enumerative approaches2

(a) Mechanism. (b) Vehicle suspension. (c) Hybrid powertrain.

1 Chakrabarti, et al., 2011; Schmidt and Cagan, 1997; Hornby, et al., 2003; Starling and

Shea, 2005; 2 Macmahon, 1994; Ma, et al., 2009; Bayrak et al., 2016 9

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Graph Structure Space Specification with (C, R, P)

• A key issue is how to represent the space of graphs of interest
• A natural description is a component catalog defined by:
• C: labels representing distinct component types (e.g., [M, K, B])
• R: # of replicates for each component type (e.g., [2, 3, 1])
• P: # of ports for each component type (e.g., [1, 2, 2])
• The desired graph structure space (a set of all graphs that fulfill a

certain set of conditions) is captured by enumerating all perfect matchings (PMs)

• Each vertex is a port from (C, R, P)
• A PM of a graph is where every vertex is incident to exactly one

edge

Figure: 3!! perfect matchings for K4 (general growth is (np − 1)!!)

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Tree Search Enumeration Algorithm

• Many of the graphs in a pure PM approach are not unique

(i.e., some graphs are isomorphic to each other)

• New algorithm is based on the idea that for simple components,

the port ordering does not matter, so we are free to always choose the first port of a component when making edges

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An Example: Quarter-Car Vehicle Suspension

• Many network structure constraints (NSCs) are included to

restrict the graphs to ones that are useful

• The sprung and unsprung masses must not be directly connected
• 4.7 × 1021 adjacency matrices; 2.1 × 1014 PMs; 13,727 unique

graphs with the newly developed methods

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Enhancements

• A number of enhancements have been made beyond the initial

publication (see Appendix A)

• Includes both general improvements and new network structure

constraints

• For example, we can break the graph generation procedure into

subtasks with appropriate subcatalogs

• Improves parallelizability of the generation procedure
• Improvements in the vehicle suspension example
• Both produce the same set of 13,727 unique, feasible graphs
• 1,943,862 (original) vs. 48,408 (enhancements) feasible graphs
• 17,903 s (original) vs. 688 s (enhancements) total generation time
• Investigating additional enhancements such as a level-order

approach where after each edge is created, only the unique graphs are kept (rather than checking for isomorphisms only when all the edges are added)

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3

Co-Design: Combined Plant and Control Design

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General Co-Design

• Focus on combined plant and control design or co-design
• With a sequential approach, the plant is optimized initially,

followed by the control1

• Many authors have shown the benefit of a combined strategy

rather than a sequential approach2

• General co-design problems are dynamic optimization problems:

min

xp,xc

Ψ = tf

t0

L (t, ξ, xc, xp) dt + M (ξ(t0), ξ(tf ), xc, xp) (1a) s.t. ˙ ξ − f (t, ξ, xc, xp) = 0 (1b) C (t, ξ, xc, xp) ≤ 0 (1c) φ (ξ(t0), ξ(tf ), xc, xp) ≤ 0 (1d)

• Much of the previous co-design research has focused on specific

problem formulations with varying assumptions3

1 Fathy, et al., 2001; 2 Fathy, et al., 2003; Allison, et al., 2014; Yan and Yan, 2009; 3 Fathy,

et al., 2001; Reyer, et al., 2001; Sunar and Rao, 1993; Peters, et al., 2011 14

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Two Solution Strategies

• Simultaneous strategy
• Solve Prob. (1) directly
• Nested (two-level) strategy
• Outer loop optimizes w.r.t the plant
• Inner loop optimizes w.r.t the control
• The two approaches are only equivalent if

for every candidate x†

p, there exists a

feasible solution to the inner-loop problem

(a) Simultaneous. (b) Nested.

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The Optimality Conditions

• Simultaneous approach optimality conditions, derived using

Pontryagin’s minimum principle:

˙ λ∗ = − ∂H ∂ξ ∗ , 0 = ∂H ∂u ∗ (2a) 0 =

• µTC

∗ , 0 =

• νTφ

∗ , µ∗ ≥ 0, ν∗ ≥ 0 (2b) 0 =

• λ + ∂M

∂ξ + νT ∂φ ∂ξ ∗

t0

, 0 =

• λ − ∂M

∂ξ − νT ∂φ ∂ξ ∗

tf

(2c) 0 = ∂M ∂xp + νT ∂φ ∂xp ∗

t0

+ tf

t0

∂L ∂xp + λT ∂f ∂xp + µT ∂C ∂xp ∗ dt (2d)

• Nested approach optimality conditions, derived using the KKT

conditions:

0 = dL dxp = tf

t0

dL dxp dt + dM dxp + χT dφo dxp + ηT dF dxp (3a) dL dxp = ∂L ∂xp + ∂L ∂ξ∗ dξ∗ dxp + ∂L ∂u∗ du∗ dxp + ∂L ∂p∗ dp∗ dxp (3b)

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Practical Solution Considerations

• It can be challenging to find solutions using the optimality

conditions, both analytically and numerically

• An alternative method, known as direct transcription, creates a

finite-dimensional mathematical program that approximates the

• riginal problem1
• Certain problem forms can be solved efficiently such as

linear-quadratic dynamic optimization (LQDO) approximated as a quadratic program (QP)2

• Other forms are used such as infinite-horizon linear-quadratic

regulator (LQR), but the underlying assumptions limit the types

• f problems can be be posed3
• In the literature, some have stated that the nested strategy is

better suited for their particular problem4

• However, all the these problems have specific inner-loop forms

1 Betts, 2010; Biegler, 2010; 2 Herber, et al., 2017; 3 Fathy, et al., 2003; Sunar and Rao,

1993; Belvin and Park, 1990; 4 Chilan, et al., 2017; Belvin and Park, 1990; Onoda and Haftka, 1987 17

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An Example: Co-Design Transfer

• Consider the following simple co-design problem:

min

k,u(t)

tf u2dt (4a) s.t. ˙ ξ = 1 −k

• ξ +

1

• u

(4b) ξ1(0) = x0, ξ2(0) = v0, ξ1(tf ) = 0, ξ2(tf ) = 0 (4c)

• Inner-loop solution (found using scaling):

u∗(t, k) = − 2k kt2

f − sin2(

√ ktf )

• c1(t, k)x0 + c2(t, k) v0

√ k

• (5)

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4

Scaling of Dynamic Optimization Formulations

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Motivating for Scaling

• The primary goal of design studies is to find solutions and gain a

general understanding of the design trade-offs

• Here we show how scaling can facilitate finding accurate,

generalizable, and intuitive information for the design problem at hand

• There have been many uses of scaling
• Buckingham’s Pi theorem and characteristic properties1
• Numerical reasons2
• In design studies3
• The mechanics of scaling are fairly straightforward
• Proper utilization of scaling relies heavily on the creativity and

intuition of the designer

1 C

¸ engel and Cimbala, 2006; Holmes, 2009; Groesen and Molenaar, 2007; 2 Papalambros and Wilde, 2017; Rao, 2010; 3 Kittirungsi, 2008; Ghanekar, et al., 1997; Brennan and Alleyne, 2001 19

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Theory

• Here we consider linear scaling:

x = αx¯ x + βx (6a) y(x) = αy¯ y(x) + βy (6b)

where x is an independent variable, y is a dependent variable, and {¯ x,¯ y} are the new dimensionless variables

• Higher-order derivatives and integrals:

dny dxn = αy αn

x

dn¯ y d¯ xn xf

x0

f(x)dx = αx ¯

xf ¯ x0

f(αx¯ x + βx)d¯ x (7)

• Consider scaling all parts of the dynamic
• ptimization problem (variables, objective,

constraints) in a holistic manner

• Optimal solutions can be generated from
• ptimal scaled solutions:

x∗(ρ2) = S(¯ x∗( ¯ ρ1), ρ2) (8)

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Example 1: SASA Problem

• Better understand the observed trends from an existing design

study on strain-actuated solar arrays (SASAs) [Chilan, et al., 2017]

• riginal system

simplified system

bus solar array solar array

Figure: Original and simplified SASA systems.

• Original problem:

min

k,u(t)

− θ(tf ) s.t. J¨ θ(t) + kθ(t) = u(t) θ(0) = 0 ˙ θ(0) = ˙ θ(tf ) = 0 |u(t)| ≤ umax

• Scaled problem:

min

¯ ρ1,¯ u(¯ t)

− ¯ ρ2¯ θ(2π) s.t. ¯ θ′′ = −¯ ρ1¯ θ + ¯ u ¯ θ(0) = 0 ¯ θ′(0) = ¯ θ′(2π) = 0 |¯ u(¯ t)| ≤ 1

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Example 1: SASA Problem (continued)

• The scaled problem has two

dimensionless quantities:

¯ ρ1 = kt2

f

4π2J ≡ t2

f

T2 , ¯ ρ2 = umaxt2

f

4π2J (11)

where T is the natural period

• All solutions can be obtained from

the scaled problem

• Optimize w.r.t. ¯

ρ1

• Temporarily remove ¯

ρ2

• Simple SASA results agree well

with the original study

time control

(a) Optimal control. (b) Natural period vs. tf .

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5

Direct Transcription and Linear-Quadratic Dynamic Optimization

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Linear-Quadratic Dynamic Optimization

• Linear-quadratic dynamic optimization (LQDO) where certain

elements of the formulation are limited to quadratic and linear functions1

• We present a unified framework for LQDO that can be solved

as quadratic programs (QPs) using direct transcription

• Limited amount of general purpose, easy to use procedures for

LQDO (e.g., MPT32)

• For example, the Lagrange term form is:

[x1, x2, x3, x4, x5] ≡ [u, ξ, p, ξ(t0), ξ(tf )] L = LQP :=

5

• i=1

5

• j=1

xT

i Lij(t)xj

HL = ˜ xTL˜ x +

5

• j=1

lT

j (t)xj

FL = lT˜ x + cL(t) (12)

1 Bryson and Ho, 1075; Anderson and Moore, 2007; 2 Herceg, et al., 2013

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Direct Transcription

• Defect constraints are constructed to ensure accurate dynamics
• For example, for the single-step methods:

ξ(tk+1) = ξ(tk) + tk+1

tk

f

• s, ξ(s), u(s), p
• ds

(13)

• A large number of direct transcription methods are

implemented

• Defects: pseudospectral (PS, both Legendre and Chebyshev),

Euler forward (EF), trapezoidal rule (TR), Hermite-Simpson (HS), classical 4th-order Runge-Kutta (RK4), zero-order hold (ZOH)

• Quadrature: Gaussian(G), Clenshaw-Curtis (CC), composite

Euler forward (CEF), composite trapezoidal rule (CTR), composite quadratic Hermite-Simpson (CQHS)

• CQHS is a new method (specifically for LQDO) that uses linear

interpolation between node points for each term in the quadratic

• bjective function

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Automated Problem Generation Procedure

• Create the appropriate QP

from a structured-based problem description

(integer) left (integer) right (matrix) matrix

• bjective

(structure)

• Matrices in the QP

problem are large sparse matrices with a specific structure

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Example 1: Smooth Solution

Problem [Bryson and Ho]:

min

u(t)

1 2 tf u2dt subject to: ˙ ξ = 1 −1

• ξ +

1

• u

ξ1(0) = x0, ξ2(0) = v0 ξ1(tf ) = 0, ξ2(tf ) = 0

Figure: Optimal states. Figure: Convergence results for different methods.

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Example 2: Nonsmooth Solution

Problem [Bryson and Ho]:

min

u

1 2 1 u2dt subject to: ˙ ξ = ξ2 u

• ξ1(0) = 0,

ξ1(1) = 0 ξ2(0) = 1, ξ2(1) = −1 ξ1(t) ≤ 1/9

Figure: Optimal states. Figure: Convergence results for different methods.

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6

Case Study: Design of Passive Analog Circuits

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Circuit-Level Synthesis

• Circuit-level synthesis involves two

major attributes: 1) the topology and 2) sizing

• Formal design methodologies have

been developed for specific problem classes1

• Evolutionary-based methods have

received significant attention2

• An infrequently utilized approach is

to actually test all topologies

1 3 4 5 2 6 7 I R R R C C L L L O

(a) Circuit schematic.

I R L L L R R N4 N3 C C G O

(b) Practical circuit.

I N4 Z Z Z Z G O N3

(c) Primitive circuit.

1 Wanhammar, 2009; 2 Gan, et al., 2010; Lohn and Colombano, 1999; Grimbleby, 1995;

Das and Vemuri, 2007 28

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Enumeration-Based Synthesis Methodology

Key: Input node Output node Impedance

(a) One circuit structure space.

C L R C R L R

Z

C L R C L

(b) Subcircuits considered.

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Synthesis Example: Frequency Response Matching

• Match |F(jω)| =
• 2π

10ω over 0.2 ≤ ω 2π ≤ 5 [Grimbleby, 1995]

• Test “all topologies that have up to 6 impedance subcircuits and

a required connection to the ground” (43,249 circuits tested)

0.5815 0.6841 0.2395 0.2356 0.6149 0.3047 0.5569 1.0000 0.3673 0.9492 0.9315 1.0000 0.8127 0.0765 0.1697 0.9955 0.4986 0.4299 0.2165 0.4673 0.6586 0.7488 0.3878 0.0609

(b) (c) (d)

0.2724 0.4583 0.6276 0.0787 0.3579 0.8292 0.2650

(f)

0.0374 0.2747 0.6818 0.3379 0.5225 0.3793 0.5574

(e) (g) (a)

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Synthesis Example: Low-Pass Filter Realizability

• A low-pass filter attenuates

signals above a certain frequency and passes all other signals

• Test “all topologies that have

up to 7 passive components with an optional connection to the ground” (123,156 circuits tested)

Figure: Specifications.

# fp (Hz) fs (Hz) Kp (dB) Ks (dB) L bounds (H) C bounds (F) 1 925 3200 3.01 22.0 [0.1m, 1.5] [0.1p, 200µ] 2 1000 2000 1.00 60.0 [0.01m, 10] [0.1p, 100µ] 3 800 2000 0.60 68.0 [0.1m, 1] [100p, 1µ] 4 1000 2000 0.01 63.5 [0.1m, 1.5] [0.1p, 200µ]

1 Lohn and Colombano, 1999; 2 Gan, et al., 2010; 3 Das and Vemuri, 2007; 4 Goh and Li,

2001 31

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Synthesis Example: Low-Pass Filter Realizability

Table: Results.

# # feasible % feasible min nc 1 38172 30.99 3 2 280 0.23 6 3 197 0.16 6 4 > 7

242.5 152.0 282.3 241.0 44.5 129.9 281.9

(e) Kp = 0.0875, Ks = 63.08.

128.4 28.3 264.2 246.4 282.4 154.0 202.7

(f) Kp = 0.1222, Ks = 62.92.

Figure: Task #4 best circuits.

177.7 31.5 518.3 258.7 321.1 277.2

(a)

355.4 198.9 484.3 217.6 304.1 28.1

(b)

201.9 280.4 23.1 238.1 528.7 361.3

(c)

345.0 453.3 311.4 233.5 194.4 12.3

(d)

Figure: Task #3 attenuation responses.

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7

Case Study: Design of Strain-Actuated Solar Arrays

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SASA Introduction

• Faster reorientation and ultra-quiet jitter-free operation of

spacecraft has the potential to transform the rate and quality of the data obtained1

• The proposed strain-actuated solar array (SASA) architecture

uses internal actuation on the solar arrays to perform small-scale reorientations and suppress jitter disturbances

• Fundamentally governed by the conservation of angular

momentum

• This is a detailed co-design problem with a comprehensive

treatment of the plant design

1 Arnon, et al., 1998; Ruiter, et al., 2103

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Modeling of the Strain-Actuated Solar Arrays

(a) Beam theory model. (b) Lumped parameter model.

• For (a), coupled ODE-PDE

dynamic model with a structural model of composite array utilizing a Galerkin formulation (linearized)

• For (b), pseudo-rigid body

dynamic model (PRBDM)

Figure: Piecewise linear distributed array thickness.

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Co-Design Problem Formulation

• Two phase problem: 1) slew the bus from θ(t0) to the origin; 2)

hold the bus fixed about the origin starting at tm

• Maximize θ(t0) (slew amount)
• Nested co-design strategy used
• Inner-loop problem is a LQDO problem
• Outer-loop handles the nonlinear plant design
• All matrices updated for each candidate plant design

Linear w.r.t. Name ξ xc xp Level

• Max. Slew Amount

Yes Both Dynamics Yes Yes No Inner Initial Conditions Yes Inner Pointing Yes Inner Voltage Limits Yes No Inner Strain Limits Yes No Inner Geometry Bounds Yes Outer Array Volume No Outer Planform Area No Outer 35

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Maximum Slewing Bounds

Figure: Array displacement in the slewing phase for various values of tm.

• Due to the strain limits and

conservation of angular momentum, an upper bound on θ(t0) can be found analytically

• The numerical results are

consistent with this analysis

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Optimal Design Tradeoff for Array Structure

Figure: Optimal array designs.

• Observed synergy between

natural passive dynamics and active dynamics

• Observed trends explained by

the scaled simple SASA problem

37

8

Case Study: Design of Vehicle Suspensions

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A Problem Class with Linear Physical Elements

• A useful framework for describing linear physical elements is

bond graph modeling with power nodes [Borutzky, 2010]

• They can be classified as source nodes, storage nodes, resistive

nodes, reversible transducer, and junction nodes

• The key property is representable by a linear descriptor model
• Linear state-space when there are no algebraic loops
• A trilevel solution approach is used to obtain solutions

candidate architecture candidate plant

• ptimal control
• ptimal dynamics
• ptimal architecture
• ptimal plant

level a level c level p

problem definition component catalog network structure constraints

⟲ ⟲

nested co-design

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A Problem Class with Linear Physical Elements

• Solve architecture design problems of the following form:

min

xa

Ψa(xa) + Ψd (fp(xa), fu(xa)) (14a) subject to: fa(xa) = a ∈ Fa (14b)

• Co-design problem for Ψd for the ith architecture:

min

x(i)

p

,u(i)

Ψd = tf

t0

LP t, y, x(i)

p

• dt + MP

y(t0), y(tf ), x(i)

p

• (15a)

s.t.

• ˙

ξ = f P (t, ξ, u, xp) (i) (15b) hP t, y, x(i)

p

• = 0

(15c) gP t, y, x(i)

p

• ≤ 0

(15d) where: y = yP t, ξ(i), u(i), x(i)

p

• (15e)
• y are the architecture-independent outputs
• P indicates that the problem element is in the LQDO problem

class when x(i)

p is fixed 39

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Different Vehicle Suspension Architectures

(a) Canonical passive1. (b) Pure active2. (c) Canonical active3. (d) Active with dynamic absorber4. (e) Active candidate. (f) Passive candidate.

1 Gobbi and Mastinu, 2001; 2 Hrovat, 1997; 3 He and McPhee, 2005; Alyaqout, et al.,

2007; Ulsoy, et al., 1994; Allison, et al., 2014; 4 Hrovat, 1997 40

1 2 3 4 5 6 7 8 9

Problem Formulation

• The vehicle traverses a rough road input
• Four outputs in the co-design problem:

y =

• zU

¨ zS zS u T (16a)

• The co-design objective is the sum of several metrics:

Ψd = tf

t0

• w1 (y1 − z0)2 + w2y2

2 + w3y2 4

• dt

(16b)

• Rattlespace constraint path constraint:

|y3 − y1| ≤ rmax (16c)

• The outer-loop specific plant constraints are simple bounds:

mmin ≤ x(i)

m ≤ mmax

(16d) bmin ≤ x(i)

b ≤ bmax

(16e) kmin ≤ x(i)

k

≤ kmax (16f)

41

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Results

(a) Maximum control and optimal plant variables.

# Complexity Ψd w1 (y1 − z0)2 w2y2

2

w3y2

4

(a) Canonical passive 2 10.96 6.60 4.36 0.00 (b) Pure active 1 7.79 3.10 1.51 3.18 (c) Canonical active 3 7.52 3.25 1.99 2.28 (d) Active w/ dynamic absorber 4 7.79 3.09 1.51 3.19 (e) Active candidate 7 6.58 2.48 2.43 1.68 (f) Passive candidate 7 9.69 5.27 4.42 0.00

(b) Maximum control and optimal plant variables.

# max |u| k1 k2 k3 b1 b2 m1 m2 (a) 1.77E4 − − 1.88E3 − − − (b) 598 − − − − − − − (c) 634 1.47E4 − − 1.00E3 − − − (d) 611 6.89E6 − − 6.04E5 − 1.02E-3 − (e) 478 9.55E4 6.67E3 8.83E4 1.46E4 2.01E3 3.11E-3 − (f) 7.28E4 8.54E3 2.51E5 1.00E3 2.27E3 3.21E0 1.10E0 42

9

Conclusions and Future Work

1 2 3 4 5 6 7 8 9

Summary

43

1 2 3 4 5 6 7 8 9

Contributions

1 A method for representing and enumerating all architectures

described by colored graphs under certain assumptions was developed

2 Previous work in co-design theory imposed restrictions on the

type of problems that could be posed; the work in this dissertation lifted many of those restrictions

3 A unified approach to the scaling of dynamic optimization

formulations was developed with a particular focus on how to leverage scaling in design studies

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Contributions

4 A unified framework for solving linear-quadratic dynamic

• ptimization problems was developed. An automated problem

generation procedure was created that generates the matrices for the quadratic program given a natural structure-based description

5 Several engineering design examples were presented. Each one

• f the case studies handles a complex, relevant design problem

with some combination of architecture, plant, and control design decisions

45

1 2 3 4 5 6 7 8 9

Future Work

• Design process for complete dynamic system design
• Effective utilization of enumerative methods in architecture

design

• Choosing between nested and simultaneous co-design solution

strategies

• Future work in linear-quadratic dynamic optimization
• Combined architecture, plant, and control design with linear

physical elements

• Further development of the case studies

46

1 2 3 4 5 6 7 8 9

Publication Status

As first author:

1 Architecture generation; status: published in ASME Journal of Mechanical

Design1

2 Circuits case study; status: submitted to IEEE Transactions on

Computer-Aided Design of Integrated Circuits and Systems

3 Co-design theory; status: IDETC 20172, to be submitted to ASME Journal of

Mechanical Design

4 Scaling in dynamic optimization; status: IDETC 20173 5 LQDO and DT; status: submitted to International Journal for Numerical

Methods in Engineering As coauthor:

1 SASA case study; status: published in AIAA Journal4 2 Design process: ACC 20155, submitted to IEEE/ASME Transactions on

Mechatronics

1 Herber, et al., 2017; 2 Herber and Allison, 2017; 3 Herber and Allison, 2017; 4 Chilan, et

al., 2017; 5 Deshmukh, et al., 2015 47

Thank you.

Advances in Combined Architecture, Plant, and Control Design

Daniel R. Herber