Bridging the Gap between Open-Loop and Closed-Loop Control ...drherber/files/Deshmukh2015a.pdf · with traditional CLC with assumed control architectures. Co-design based on CLC limits

Bridging the Gap between Open-Loop and Closed-Loop Control inCo-Design: A Framework for Complete Optimal Plant and Control

Architecture Design*

Anand P. Deshmukh1†, Daniel R. Herber2†, and James T. Allison3†

Abstract— Here we propose a novel framework for thecombined plant and controller architecture design based ona set of systematic studies that culminate in an optimalplant architecture and associated realizable control law. Thisframework bridges the inherent gap between open-loop opti-mal trajectories provided by particular co-design studies andpractically implementable control laws. This is accomplishedthrough a step-by-step process where a series of optimizationproblems are solved that provide important system insights suchas maximum system performance limits, controller architecture,actuator selection, etc. at appropriate design phases. Eachoptimization problem thus informs subsequent formulations.This methodology is applied to semi-active suspension design.

I. INTRODUCTION

Optimal design of complex dynamic systems is typicallyan iterative process involving plant and associated con-troller design. The strong interdependence between plantand control design must be addressed to obtain system-optimal solutions. This can be achieved through integratedplant and control design (co-design) methods, which can leadto significant performance improvements over traditionalsequential methods [1], [2]. Moreover, recent advances inco-design can handle nonlinear dynamic, path, control, andplant constraints efficiently [1]–[4] using open-loop control(OLC) where no assumptions are made on control structure.Co-design has been used traditionally in early-stage designstudies where optimal plant design and optimal controlinput trajectories are sought for a specified physical-systemarchitecture. OLC trajectories can provide great insightsinto system performance limits, but may not be easilytranslated into practical control laws. Furthermore, optimalOLC trajectories can be highly sensitive to plant or otheruncertainties. Current co-design formulations address onlya small part of complete dynamic system development. Toaddress this shortcoming we propose an extended frameworkfor dynamic system design, supported by co-design, that laysout a full process from system architecture to implementablecontrol laws. Our long-term objective is to bring integrateddesign methods into practice to support new levels of systemperformance and reduce costly design iterations.

In our proposed framework, we consider the completearchitecture design of a dynamic system that is defined by the

*This work was supported in part by MathWorks R©, Inc.† University of Illinois at Urbana-Champaign, Department of Industrial

and Enterprise Systems Engineering, 104 S Mathews Ave, Urbana, IL 618011Anand Deshmukh is a Ph.D. pre-candidate, [email protected] Herber is a Ph.D. pre-candidate, [email protected] Allison is an Assistant Professor, [email protected]

selection of the specific components and their connections.For example, candidate physical architectures for automotivehybrid powertrains include series, parallel, power-split, andother configurations. Many system architecture design prob-lems have a large number of possible components and con-nections, making enumeration of architecture candidates im-practical as a design strategy. OLC can instead replace somecomponents or interfaces with optimal trajectories, reducingthe design problem size while still providing an optimalsolution [3]. In addition, OLC-based studies are particularlyvaluable at early design stages for gaining insights into uppersystem performance limits and into dynamic behaviors andinteractions that lead to system-optimal performance [1], [3].

OLC techniques used in co-design include both indirectmethods (dynamic programming [5], Pontryagin’s maxi-mum principle [6]) and direct methods (shooting [1], directtranscription [7]–[9]). In OLC, optimal control trajectoriesare sought without assuming control architecture, whereasclosed-loop control (CLC) design requires specification ofa control structure (e.g., state/output feedback) that mayimplicitly limit performance or ability to satisfy system con-straints. While OLC provides important insights at early de-sign stages, CLC is normally required for actual implemen-tation to provide stability, robustness, disturbance rejection,and other desirable properties of feedback systems (explain-ing the vast range of CLC methods, including LQR/LQG[10], adaptive control [11], and H∞ control [12]). CLCcan produce optimal solutions if specific (but potentiallyrestrictive) conditions are met such as with LQR/LQG (lineardynamics, quadratic objective, no path or plant constraints).The trade-offs between OLC and CLC has motivated a fewtechniques such as model predictive control (MPC) [13] andfeedfoward control [14].

Useful design insights can be extracted from optimal OLCsolutions when employing co-design. Since the resultingsolutions have the optimal dynamics, the desired natural dy-namics of the system can emerge [3], [15]–[20]. Furthermore,the resulting trajectories can serve as a basis for develop-ing implementable feedback control systems and physical-system/architecture design if the OLC was used in early-stage design [3], [15]–[18]. If the significant assumption ofuni-directional plant and control coupling is made, controlproxy functions may be used in the plant design formulationto yield a sequential design process that approximates (or inlimited cases equals) the co-design solution [21].

Finally and perhaps most critically, OLC supports solution

2015 American Control ConferencePalmer House HiltonJuly 1-3, 2015. Chicago, IL, USA

978-1-4799-8686-6/$31.00 ©2015 AACC 4916

MechanicalOwnership

Control/Hardware/SoftwareOwnership

ControllerArchitecture

Design

Co-Design withCLC, xc

a⇤ c⇤ xp⇤xc⇤

Adjust Formulation

�Co-Design withOLC, u(t)

PlantArchitecture

DesignDigital

ControllerDesign

31 2

�

�(a) Proposed stages.

Inputs• Environment

• System objectives• System requirements• Candidate components

Outer-Loop Discrete Topology Optimization1

opti

miz

eroptimizer

a

Inner-Loop Continuous Optimization

opti

miz

eroptimizer

xip,ui

Outputs∗ Optimal System• Architecture• Plant design• Control design

Candidate architectureai

Optimized performanceΦ∗

(b) Stage 1 details.

Fig. 1: Proposed stages for complete dynamic system design.

of sophisticated co-design optimization problems that involvenonlinear dynamics, nonlinear plant constraints, path con-straints, hybrid dynamics, and other challenging elements ofrealistic system design problems. OLC often is successful atcapturing complex control solutions beyond what is possiblewith traditional CLC with assumed control architectures. Co-design based on CLC limits creative plant design explorationat early design stages. While OLC is an important toolfor early studies, a significant gap exists between OLCand implementable CLC. An ideal design framework willsupport identification of system-optimal plant designs whilesupporting the development of CLC systems that yieldapproximately optimal system performance based on anunderstanding of how a system should behave dynamicallythat is gained through early-stage OLC co-design studies.

Here we propose a novel process for dynamic systemdesign that integrates OLC and CLC studies (illustratedin Fig. 1a). Stage 1 finds the optimal plant architecture(a∗), employing established co-design and OLC methods tocompare architecture candidates. Stage 2 finds the optimalcontroller architecture, i.e. control laws, (c∗) and optimalplant design (xp∗ ) again employing co-design but with CLC.Stage 3 realizes the digital controller design. Stage 1will be typically handled by mechanical engineers whileStages 2 and 3 will be solved by control engineers. Iter-ations between the stages should be performed as necessary,including modification of problem formulations as insightsare gained. Stages 1 and 2 will be further expanded uponin the following sections.

II. FORMULATION CONSIDERATIONS FOR CO-DESIGNAND SYSTEM ARCHITECTURE DESIGN

Effective implementation of the design process outlined inFig. 1a requires consideration of several optimization prob-lem formulations. The outer-loop nonlinear programming(NLP) formulation for optimal physical architecture designin Stage 1 is:

mina

Φ∗ (a) (1a)

where: a ∈ Fa (1b)

where a represents plant architecture design (e.g., an ad-jacency matrix). Architecture design is a discrete designproblem. Fair comparison between architecture candidates in

the set of possible architectures Fa requires evaluation of thebest possible performance for each candidate. This requiresoptimization with respect to continuous design variables(plant design variables xp and control trajectories u(t)) foreach candidate architecture, through solution of the followinginner-loop problem (co-design with OLC) to obtain Φ∗ (·):

minxip,u

i(t)Φi(t, ξi(t),ui(t),xi

p

)(2a)

subject to: ξi − f id(t, ξi(t),ui(t),xi

p

)= 0 (2b)

Ci(t, ξi(t),ui(t),xi

p

)≤ 0 (2c)

φi(t0, ξ

i(t0), tf , ξi(tf ),xi

p

)≤ 0 (2d)

where: {ξi(t),ui(t),xip, f

id,C

i,φi,Φi} ∈ ai (2e)

where t is time and ξ(t) is the state trajectory vector[3]. System dynamics are enforced through the dynamicconstraints in Eqn. (2b). The collection of states, time deriva-tives [fd(·)], control trajectories, plant variables, objectivefunction, and constraints

[Ci(·),φi(·)

]are all well-defined

for a given candidate architecture ai (ith architecture denotedwith a superscript i). These inner-loop problem formulationelements could be decidedly different for each ai consideredby the outer loop.

Figure 1b illustrates the Stage 1 problem in more detail.Linking the outer- and inner-loop formulation is an importantaspect that needs to be done efficiently to facilitate a thor-ough exploration of the physical architecture design space.While a human designer could formulate and evaluate theoptimized performance of each candidate architecture, thistask would be quite time consuming and limit the number ofcandidate architectures that could be evaluated. Automatingthis process is an ongoing task. An automated interfacebetween the outer- and inner-loops requires a number oftasks to be performed in an automated fashion includingmodel creation, optimization problem generation, and modelparameterization. Developing such an interface with theproper inputs to Stage 1 would allow a truly diverse setof candidate architectures to be explored efficiently.

Some existing approaches for physical-system architecturedesign (other than the enumeration of possible architectures)include agent-based synthesis for mechatronic systems [22],graph grammar for general engineering systems [23], andgenerative representations (for robotics [24] and static trusses

4917

[25]). In these approaches, the feasible set of architecturesare implicitly ensured through the underlying algorithmsand rules as suggested in Eqn. (1b), rather than throughtraditional constraints.

Physical architecture design traditionally has involved sim-plified dynamic models. The above design strategy accountsfully for system dynamics, including passive and activeelements, through the solution of the inner-loop co-designproblem. Co-design with OLC will be utilized primarily inStage 1 as an infinite-dimensional problem where we seekoptimal control trajectories, u(t) [1], [3]. In Stage 2 weseek a finite set of optimal control variables (e.g., feedbackgains) for the CLC system, xc [26].

With the formulation in place, additional solution consid-erations will be discussed.

A. Classifications of Information Horizons

OLC typically has been used on complete horizon systemswhere the environment is completely known for the timehorizon. This, however, is a strong assumption. Consideringsmaller horizons is more realistic.

1) Complete Horizon: The control (either OLC or CLC)problem is solved with complete information about envi-ronment (e.g., exogenous inputs, physical model). Examplesinclude robotic manipulator path planning [19] and spaceshuttle reentry trajectory design [7]. These problems aretypically offline optimization problems.

2) Instantaneous: This class of problems deal with in-stantaneous control problem, where the environment infor-mation is available only at the current instant of time. CLCfor tracking and regulation are well-known examples. Withinstantaneous information, providing an acceptable level ofperformance over the entire horizon (such as satisfying pathconstraints) can be challenging.

3) Limited Horizon: This control problem is solved foronly a small portion of the horizon. This solution is repeatedat regular intervals during operation. Some examples includelook ahead control of wind turbines [27] and fuel optimalhauling trucks or railways [28]. MPC is an important solutionstrategy since it can provide comparable performance tocomplete horizon solutions and the desirable properties ofinstantaneous ones [13]. These problems can be offline oronline optimization problems. Limited horizon is a middleground between complete and instantaneous.

B. Early-Stage Design Considerations

Multiple special formulations in Stage 1 can be usedto make better informed plant and control design decisions.Utilizing co-design with OLC maintains an unrestrictive (andpotentially system optimal) formulation while performingthese studies. Conventional Stage 1 design strategies in-volve physical-system design without complete considera-tion of the interaction between physical- and control-systemdesign, leading to suboptimal results.

Unstructured OLC can be used replace select componentsor interfaces in a system to simplify early design studies.

Electric subsystems may be approximated as OLC trajecto-ries in mechatronic systems [19], particularly if fast electricalsystem dynamics support a direct mapping from OLC toCLC with tracking. Co-design with OLC temporarily forgoessome architecture design decisions for this problem, such aselectric machine or gearbox selection. Another example isdeciding between semi-active/active control strategies [29].At a fundamental level, OLC formulations can model ide-alized versions of these components (e.g., constraints onpower: P ≤ 0 for semi-active, P ∈ R for active) andguide the selection process based insights extracted fromOLC results (e.g., Does the performance benefit of an activesystem outweigh the cost penalty compared to a semi-activeone?).

Another recent example from the literature involvespower-take off (PTO) design for a wave energy converter(WEC) that extracts (or inputs) power through a force [3].Several WEC PTO types have been investigated, includinglinear generators, rotary electric machines, and hydraulicssystems, each of which have widely different dynamics. Ifthe PTO type is ignored at early design stages, and theforce trajectory provided by the PTO is optimized usingOLC, important insights about how the system should beoperated can be extracted before some architecture decisionsare made. These insights can then inform both physicaland control architecture design decisions. Distinct behaviors,such as latching (holding the WEC in place for a shorttime), emerge via OLC studies. Physical- and control-systemarchitecture decisions can then be made to achieve the typeof behavior exhibited in the OLC co-design results [3]. Thisstrategy helps engineers develop implementable systems thatapproach the fundamental maximum system performancelimit, and offers a path toward integrated mechatronic systemdesign processes than can be adopted in engineering practice.

Similarly, for the transition from computational models tofabricated plants, structured studies addressing the commonissues associated with reconfigurability of the plant such ascost estimation and selection can be performed in a completeway [30]. Co-design with OLC enables the proper study ofreconfigurability of controlled dynamic systems.

III. EXTRACTING IMPLEMENTABLE CONTROLSOLUTIONS FROM OPEN-LOOP STUDIES

The objective of the framework presented here is to sup-port the development of integrated, implementable activelycontrolled systems. Recent advances in co-design based onoptimal OLC methods make possible the design of systemsthat account fully for the interaction between physical-and control-system design, including detailed and realisticphysical-system design considerations. These methods arehighly effective for early-stage design, generating physicalsystems with natural dynamics that interact with an activecontrol system in a way that yields maximal system perfor-mance. The associated optimal control trajectories can leadto new insights, and help engineers discover what the trueperformance limits are on the system without constraintsimposed by control architecture assumptions. These optimal

4918

control trajectories, however, cannot be used directly in acontrol-system implementation.

Elements of optimal OLC may be used in realizablecontrol systems. For example, open-loop optimization canbe repeated online based on feedback of measured/estimatedvariables [9], [31], [32]. Robustness may be improved usingopen-loop multi-objective optimization that aims to improvenominal performance and reduce variance [33]. Alternatively,classical feedback methods may be used in well-modeledregimes, but is complemented through the use of open-looptrajectories in poorly modeled regions [34], [35].

While the above strategies enhance the utility of OLCin practice, in many cases a feedback control architectureis required. There is a significant gap that exists betweenthe output of OLC co-design methods that are appropriatefor early-stage design, and implementable control-system de-sign. This article presents a first effort to formalize this gap inthe context of co-design and integrated system development,and presents a first approach for addressing this gap. Severalapproaches may be used to extract CLC designs from OLCco-design results with varying levels of rigor. Optimal OLCtrajectories may be analyzed for patterns, spectral properties,or other characteristics that can guide control architecturedevelopment. Given sufficient data from optimal controltrajectories, system identification [36] or trajectory matchingstrategies [37] might be used to determine a CLC systemdesign that approximates OLC performance.

Bridging the gap between OLC and CLC in integrated dy-namic system design is an opportunity to make possible newlevels of design integration. The existing co-design literatureoffers significant advancements in design integration betweenphysical- and control-system design, capitalizing on synergyto improve system performance, but does not by itself offer ameans for integrated design in a realistic system developmentprocess. Connecting co-design to CLC architecture designis an important step toward incorporating co-design intodesign practice, and toward a more comprehensive integrateddesign framework for designing systems with new levels ofperformance in shorter time periods.

The study presented here begins with an early-stage co-design problem that makes no assumptions regarding actua-tor or control-system architecture. A sequence of problems issolved, each informed by the results of the previous problem,moving toward greater levels of system specificity, eventuallyresulting in an optimal system design based on a detailedCLC architecture. While adding detail brings us closer to animplementable design, it also constrains the design problem.It is important to use a multi-step approach where early-stagedesign problems entail few assumptions and support a broaddesign space search.

IV. CO-DESIGN STUDIES FOR A SEMI-ACTIVESUSPENSION

The co-design and early-stage architecture design processspecified in the previous sections is applied to a trailing-arm type suspension (shown in Fig. 2). The objectives are to1) minimize the sprung mass acceleration, zs, (i.e., improve

↵ �

Ms

Mus

`

CtKt

3 Active Suspension Case StudyIn this section we introduce a new model for an active

suspension system, in fully reproducible detail, that includes amodel of important physical system design considerations in ad-dition to a dynamic model of the suspension. Effort was madeto maintain linearity of system dynamics to preserve the use-fulness of this model in other studies that are limited to lineartime-invariant systems.

Consider the quarter-car model of a vehicle suspension il-lustrated in Fig. 4. The the sprung mass ms (325 kg) and theunsprung mass mus (65 kg) vertical positions are given by zs andzus, respectively. The system is excited by variations in road ele-vation z0 as the vehicle travels at speed v.

v

ks cs

kt ct

z0

zus

zs

ms/4

mus/4

Figure 4: Quarter-car vehicle suspension model.

The passive dynamic response of this system can be charac-terized by the following system of linear differential equations:

xxx = Axxx+

2664

�14ctmus00

3775 z0, (11)

where xxx =

2664

zus � z0zus

zs � zuszs

3775 and A =

26664

0 1 0 0� 4kt

mus� 4(cs+ct )

mus4ksmus

4csmus

0 �1 0 10 4cs

ms� 4ks

ms� 4cs

ms

37775

The tire and sprung mass spring stiffnesses are kt (232.5 ·103

N/m) and ks, respectively, and the tire and spring mass damp-ing rates are ct and cs, respectively. Here we assume ct = 0.This canonical model has been used as an example in numer-ous design studies [34–36], including the design of active con-

trol systems [37–39], where an additional control input term Buis appended to Eqn. (11). Often ks and cs are treated as inde-pendent design variables [36, 37, 40], but are in fact dependenton geometric design and are subject to stress, fatigue, packaging,thermal, and other constraints. Here we introduce an extensionto the basic quarter-car model that treats ks and cs as dependentvariables, and incorporates a plant model that computes stiffnessand damping coefficients as a function of independent geomet-ric spring and damper design variables. The detailed spring anddamper models are presented, followed by a demonstration ofactive suspension co-design using DT.

3.1 Spring DesignThe vehicle suspension in this model utilizes a helical com-

pression spring with squared and ground ends (Fig. 5). The sus-pension has a coil-over configuration; the coil spring surroundsthe damper and they share the same axis. The model presentedhere is derived from [41]. See also [42] and [43] for alterna-tive spring design optimization formulations. The independentspring design variables here are the helix diameter D, wire diam-eter d, spring pitch p, and the number of active coils Na, whichis relaxed to a continuous variable. These are components of thecomplete vector of plant design variables xp, along with othervariables yet to be discussed. The formula for stiffness and acollection of spring design constraints are presented below.

L0

p

D

d

Ls

Fs

Figure 5: Helical compression spring with squared ground ends.

The free length of the spring is L0 = pNa +2d, and the solidheight is Ls = d(Na + Q� 1), where Q = 1.75 for squared andground ends. Fs is the axial force at the solid height, and thespring constant is:

ks =d4G

8D3Na

⇣1+ 1

2C2

⌘ (12)

where G is the shear modulus (ASTM A401, G = 77.2 MPa),

5 Copyright c� 2011 by ASME

v

zs

zus

z0

Step

3Active Suspension Case Study

In this section we introduce a new model for an active

suspension system, in fully reproducible detail, that includes a

model of important physical system design considerations in ad-

dition to a dynamic model of the suspension. Effort was made

to maintain linearity of system dynamics to preserve the use-

fulness of this model in other studies that are limited to linear

time-invariant systems.

Consider the quarter-car model of a vehicle suspension il-

lustrated in Fig. 4. The the sprung mass ms (325 kg) and the

unsprung mass mus (65 kg) vertical positions are given by zs and

zus , respectively. The system is excited by variations in road ele-

vation z0 as the vehicle travels at speed v.

v

ks

cs

kt

ct

z0

zus

zs

ms /4

mus /4


The passive dynamic response of this system can be charac-

terized by the following system of linear differential equations:

˙xxx = Axxx+

2664

�14ctmus00

3775 z0 ,

(11)

where xxx =

2664

zus � z0zuszs � zuszs

3775 and A =

26664

01

00

� 4ktmus � 4(cs +ct )mus 4ksmus 4csmus

0�1

01

04csms � 4ksms � 4csms

37775

The tire and sprung mass spring stiffnesses are kt (232.5 ·10 3

N/m) and ks , respectively, and the tire and spring mass damp-

ing rates are ct and cs , respectively. Here we assume ct = 0.

This canonical model has been used as an example in numer-

ous design studies [34–36], including the design of active con-

trol systems [37–39], where an additional control input term Bu

is appended to Eqn. (11). Often ks and cs are treated as inde-

pendent design variables [36, 37, 40], but are in fact dependent

on geometric design and are subject to stress, fatigue, packaging,

thermal, and other constraints. Here we introduce an extension

to the basic quarter-car model that treats ks and cs as dependent

variables, and incorporates a plant model that computes stiffness

and damping coefficients as a function of independent geomet-

ric spring and damper design variables. The detailed spring and

damper models are presented, followed by a demonstration of

active suspension co-design using DT.

3.1Spring Design

The vehicle suspension in this model utilizes a helical com-

pression spring with squared and ground ends (Fig. 5). The sus-

pension has a coil-over configuration; the coil spring surrounds

the damper and they share the same axis. The model presented

here is derived from [41]. See also [42] and [43] for alterna-

tive spring design optimization formulations. The independent

spring design variables here are the helix diameter D, wire diam-

eter d, spring pitch p, and the number of active coils Na , which

is relaxed to a continuous variable. These are components of the

complete vector of plant design variables xp , along with other

variables yet to be discussed. The formula for stiffness and a

collection of spring design constraints are presented below.L0

p

D

d

Ls

Fs


The free length of the spring is L0 = pNa +2d, and the solid

height is Ls = d(Na +Q� 1), where Q = 1.75 for squared and

ground ends. Fs is the axial force at the solid height, and the

spring constant is:

ks =d 4G8D 3Na⇣

1+ 12C 2

⌘

(12)


5

Copyright c� 2011 by ASME

Klin




v

ks cs

kt ct

z0

zus

zs

ms/4

mus/4



xxx = Axxx+

2664

�14ctmus00

3775 z0, (11)

where xxx =

2664

zus � z0zus

zs � zuszs

3775 and A =

26664

0 1 0 0� 4kt

mus� 4(cs+ct )

mus4ksmus

4csmus

0 �1 0 10 4cs

ms� 4ks

ms� 4cs

ms

37775






L0

p

D

d

Ls

Fs



ks =d4G

8D3Na

⇣1+ 1

2C2

⌘ (12)



Ms

Mus




v

ks cs

kt ct

z0

zus

zs

ms/4

mus/4



xxx = Axxx+

2664

�14ctmus00

3775 z0, (11)

where xxx =

2664

zus � z0zus

zs � zuszs

3775 and A =

26664

0 1 0 0� 4kt

mus� 4(cs+ct )

mus4ksmus

4csmus

0 �1 0 10 4cs

ms� 4ks

ms� 4cs

ms

37775






L0

p

D

d

Ls

Fs



ks =d4G

8D3Na

⇣1+ 1

2C2

⌘ (12)



F

1 2


In this section we introduce a newmodel for an active




to maintain linearity of systemdynamics to preserve the use-








v

ks

cs

kt

ct

z0

zus

zs

ms /4

mus /4




xxx = Axxx+

2664

�14ctmus0

0

3775 z0 ,

(11)

where xxx =

2664


3775 and A

=

26664

0

1

00

� 4ktmus � 4(cs +ct )m

us 4ksmus 4csm

us

0

�1

01

04csms

� 4ksms � 4csm

s

37775



ing rates are ct and cs , respectively. Here we assume ct =0.














3.1Spring Design





here is derived from[41]. See also [42] and [43] for alterna-







collection of spring design constraints are presented below.

L0

p

D

d

Ls

Fs


The free length of the spring is L0 =pN

a +2d, and the solid

height is Ls =d(N

a +Q� 1), where Q= 1.75 for squared and


spring constant is:

ks =

d 4G8D 3N

a⇣1+

12C 2⌘

(12)

where Gis the shear modulus (ASTM

A401, G=

77.2 MPa),

5


F


In this section we introduce a newmodel for an active




to maintain linearity of systemdynamics to preserve the use-








v

ks

cs

kt

ct

z0

zus

zs

ms /4

mus /4




xxx = Axxx+

2664

�14ctmus0

0

3775 z0 ,

(11)

where xxx =

2664


3775 and A

=

26664

0

1

00

� 4ktmus � 4(cs +ct )m

us 4ksmus 4csm

us

0

�1

01

04csms

� 4ksms � 4csm

s

37775



ing rates are ct and cs , respectively. Here we assume ct =0.














3.1Spring Design





here is derived from[41]. See also [42] and [43] for alterna-







collection of spring design constraints are presented below.

L0

p

D

d

Ls

Fs


The free length of the spring is L0 =pN

a +2d, and the solid

height is Ls =d(N

a +Q� 1), where Q= 1.75 for squared and


spring constant is:

ks =

d 4G8D 3N

a⇣1+

12C 2⌘

(12)

where Gis the shear modulus (ASTM

A401, G=

77.2 MPa),

5


3 4

P < 0 I K⇠

1D simplification

xp4

xp1

xp2

xp3

Ks

Fig. 2: Trailing-arm type suspension schematic.

passenger comfort) and 2) minimize tire deflection, zus−z0,(improve road handling). The gray box in Fig. 2 indicatesan unknown component in the system that we need todetermine through optimal OLC studies. The performancecan be improved further by optimizing the plant designvariables: xp = [xp1, xp2, xp3, xp4, α,Klin] where xpi and αare geometric variables, and Klin the stiffness of the physicalspring that is assumed to be linear. The 1D simplification ismade accurate by modeling geometric nonlinearities for bothKs(xp, ·) and the unknown component.

The process outlined in Fig. 1a is completed using fourlinked co-design problems that are solved using direct tran-scription with trapezoidal collocation [2], [3], [7], [8]. Solu-tion accuracy was verified using high-order simulation. Theinsights from each problem inform the subsequent problem,with the final study culminating in a realizable feedback con-troller. The differences in each problem (additional structureon F (u) and control bounds) are shown in Table I. Theunderlying problem formulation is:

minxp,u(t)

Φ(xp, u(t)) =

∫ tF

0

(r1ξ

21 + r2ξ

24

)dt (3a)

subject to: ξ = A (xp, ξ) ξ + B1q + B2F (u) (3b)Apxp ≤ 0 (3c)xp ≥ 0 (3d)

where: A =

0 1 0 0

− KtMus

− CtMus

Ks(xp,ξ)

Mus0

0 −1 0 1

0 0 −Ks(xp,ξ)

Ms0

(3e)

B1=

−1Ct

Mus00

,B2=

0− 1

Mus01

Ms

, ξ=

[zus − z0

zus

zs − zus

zs

](3f)

Ap =[1 0 0 −1 0 00 1 −1 0 0 0

], q = z0 (3g)

where the spring rate Ks is a nonlinear function of xp

and states ξ. Mus,Ms, Ct, and Kt are the unsprung mass,sprung mass, tire damping, and tire stiffness, respectively.The road disturbance and disturbance velocity are z0 and

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TABLE I: Sequence of problem descriptions and optimal solutions for the semi-active suspension study.

Problem Stage Type Control u F (u) Bounds Φ∗ Klin (N/m)P0 – Passive – – – – 5.80× 10−4 26810

P1 1 Active OLC Force u – 2.06× 10−4 20094

P2 1 Semi-active OLC Damping −u · (zs − zus) 0 ≤ u(t) 3.72× 10−4 27634

P3 1 Semi-active OLC Current −F (u, zs − zus) 0 ≤ u(t) ≤ 1 9.58× 10−4 26262

P4 2 Semi-active CLC Feedback gains −F (Kξ, zs − zus) 0 ≤ Kξ(t) ≤ 1 2.00× 10−3 12507

z0,z

s(m

)

−0.01

0

0.01

z0 zs: P0 zs: P1zs(m

/s2)

−3

0

3P0P1

F(u)(N

)

−700

0

700 P0P1

z0,z

s(m

)

−0.01

0

0.01

z0 zs: P2

zs(m

/s2)

−3

0

3

P2

F(u)(N

)

−900

0

900 P2

t (s)

z0,z

s(m

)

−0.010

0.05

0 1 2 3 4 5

z0 zs: P3 zs: P4

t (s)

zs(m

/s2)

−3

0

3

0 0.5 1 1.5 2

P3P4

ξ4 − ξ2 (m/s)F(u)(N

)

−1200

0

1200

−0.5 −0.25 0 0.25 0.5

P3P4

0A

0A

1A

1A(c)

(b)

(a)

(f)

(e)

(d)

(i)

(h)

(g)

Fig. 3: Solutions for each of the problems outlined in Table I: (a)–(c) demonstrate the sprung mass displacement vs. roaddisturbance, (d)–(f) demonstrate the passenger comfort, and (g)–(i) demonstrate the resulting optimal forces and velocitiesat the damper location.

z0. The road data has an IRI of 7.37, corresponding to amaintained unpaved road [38, p. 170]. The weights on roadhandling and passenger comfort objectives are r1 and r2.

Problem 1 (P1) is solved first to obtain the actuator forcetrajectory that minimizes Φ∗(·) (a Stage 1 study). No as-sumptions are made yet on actuator structure or force bounds(unrestricted OLC). The P1 solution serves as a benchmarkfor maximum system performance (see its minimal objectivevalue Φ∗(·) in Table I).

The open-loop optimal actuator force trajectory obtainedin P1 may be realized using electric, pneumatic, or hydraulicactuators [29], which make it an actively controlled suspen-sion. These actuators typically have prohibitively high powerrequirements that hinder their widespread use in practice.An alternative is a semi-active suspension using magneto–rheological (MR) damping [39]. MR dampers can achievecomparable performance (to active suspensions) with near-zero power consumption, and are inherently BIBO stable[40]. To make an informed decision on the active/semi-activeactuator selection, P2 was solved assuming an ideal semi-active actuator. This was accomplished by constraining thecontrol force such that the energy is always dissipated in

the system, and assuming that we can achieve any dampercoefficient at a given velocity zs − zus. This structures theOLC and force as F (u) = −u(t) · (zs − zus) , u(t) ≥ 0.

The Φ∗(·) value for P2 is only ∼1.80× worse thanfor P1. Since an ideal semi-active component has similarperformance to the active system and previously mentionedadvantages, this actuation strategy was chosen. Selection andsizing of the specific semi-active actuator can also be guidedby the P2 solution. Dampers are typically characterized bythe performance in the velocity vs. force space (which isshown in Figs. 3g–3i). Using the maximum force from P2(about 900 N) and regions of attained velocities and forces(see Fig. 3h), we can quantify force demands required ofthe damper to produce optimal semi-active system perfor-mance. Additionally, the sprung mass displacement and roaddisturbance vs. time plots for all the problem are shown inFigs. 3a–3c and sprung mass acceleration vs. time plots areshown in Figs. 3d–3f.

At this step, we select an MR damper sized using theP2 solution (specifically, a Lord 8041-1). The continuouscurrent operation range for this damper is 0A–1A, and it hasa maximum stroke of 74 mm. The damper behavior was then

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characterized in the laboratory to obtain the data needed toconstruct a smooth surrogate model that estimates damperforce as a function of damper velocity and input current:F (I, zs − zus) (see Fig. 3i). MR damper use is challengingdue to its inherently hysteretic and nonlinear dynamics. P3seeks the optimal OLC damper input current trajectory thatis within saturation bounds. The Φ∗(·) value for P3 is about2.57× worse than P2, which can be expected since weare moving from an ideal damper to a highly structuredMR damper model (constraining available force for givenvelocities). This concludes Stage 1 studies, yielding aspecific damper architecture and knowledge of maximumsystem performance.

Finally, in P4, we move toward a more realizable CLCbased on a full-state feedback controller with optimizedgains, K (now a Stage 2 study). We assume that allthe states are measurable. Specifically, mass accelerations(zs, zus) are measured using accelerometers, displacements(zus − z0, zs − zus) are obtained using linear encoders, andvelocities (zs, zus) are estimated by integrating accelerationtrajectories. A feedback control law is determined for damperinput current: I(t), and a path constraint was added toenforce damper saturation limits: 0 ≤ Kξ(t) ≤ 1.

At this final step, the Φ∗(·) value for P4 is 2.08× worsethan for P3. There may exist a better performing imple-mentable control law that could be extracted from the P4solution using the techniques described in Sec. III, whichis left as a future task. We would then expect the Φ∗(·)value for P4 to be closer to the optimal objective for P3, butnever better. A new architecture design (or different damper)would need to be selected to surpass this performance limitand attempt to arrive at the ideal performance points definedby Φ∗(·) for P1 and for P2. Lastly, for comparison, we alsoprovide a solution to P0 in which we solve a dynamic sys-tem design optimization problem with an idealized passivelinear damping coefficient as a design variable. The effectivepassive damping in the system however is nonlinear due togeometric nonlinearities of the system. The idealized damper(P0) exhibits slightly better performance than P3 and P4 asexpected. It should be noted however that such an idealizeddamper can not be realized in practice.

The optimal objective function value Φ∗(·) degrades aswe move from P1 to P4 (Table I). This is congruent withthe intuition that as we gradually add detail and refine theconstraints and structure of the problem, we increase therealism at the cost of performance degradation. However, theend result of this process is an implementable control law(with corresponding system-optimal physical design) that weknow is only 9.70× worse than the maximum performancepredicted by the unstructured problem P1.

V. CONCLUSIONS

In traditional co-design there exists a gap between OLCtrajectories and implementable CLC. Here we proposed aframework to bridge this gap. In this framework we laidout a step-by-step approach to the design of controlled dy-namic systems, wherein well-informed design and controller

xp2

Klin

xp3

xp1

xp4

MR Damper

z0

Fig. 4: Fabricated reconfigurable trailing-arm type suspen-sion testbed (xp1 through xp4 are reconfigurable geometricvariables).

architecture decisions are made throughout the process,culminating in an implementable control law and optimalplant design. Proceeding with this framework necessitatesnovel formulation considerations to properly address thechallenges associated with dynamic system design. Fourlinked optimization problems were solved sequentially indesigning a trailing-arm type quarter car suspension, wheresolutions from previous problems inform the subsequent one.The results show the potential of this framework in helpingsystematic selection of optimal plant design variables, con-troller architecture, and implementable control laws.

Further work is required to bring more rigor and automa-tion to this design process, making possible the use of co-design in design practice. In the co-design paradigm, thechoice of design variables xp directly modifies the systemmatrices A, B1 and B2, hence a deeper analysis of struc-tural properties of dynamic systems such as stability, con-trollability and robustness under uncertainties is necessary.Moreover a more exhaustive review of available techniquesfor moving from OLC solutions to CLC in Sec. III is stillrequired and new approaches suitable for this task need to beinvestigated. In addition, formalizing the trade-offs betweenthe system performance and implementability of each of thedesign studies is desired. Specifically the implementabilityconsiderations of co-design solutions for nonlinear systems(for example using standard feedback linearization) need tobe investigated. Lastly, the validation of our control lawswill be performed using a physical reconfigurable trailing-arm suspension testbed (Fig. 4).

ACKNOWLEDGMENTS

We would like to thank the following University of Illinoisstudents for their contributions toward the construction ofthe semi–active suspension testbed: Adam Cornell, JohnnyHo, Dhruv Kanwal, Danny Lohan, Kevin Lohan, JasonMcDonald, and Insuck Suh.

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Documents

Bridging the Gap between Open-Loop and Closed-Loop Control ...drherber/files/Deshmukh2015a.pdf · with traditional CLC with assumed control architectures. Co-design based on CLC limits