Input Design for Nonlinear Model Discrimination via Affine Abstraction⁎

Proceedings, 6th 6th IFAC IFAC Conference Conference on on Analysis Analysis and and Design Design of of Proceedings, Hybrid Proceedings, 6th Hybrid Systems Systems Proceedings, 6th IFAC IFAC Conference Conference on on Analysis Analysis and and Design Design of of Oxford, UK, 11-13, 2018 Hybrid Available online at www.sciencedirect.com Oxford, UK, July July 2018 Hybrid Systems Systems Proceedings, 6th 11-13, IFAC Conference on Analysis and Design of Oxford, UK, Oxford,Systems UK, July July 11-13, 11-13, 2018 2018 Hybrid Oxford, UK, July 11-13, 2018

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IFAC PapersOnLine 51-16 (2018) 175–180

Input Design for Nonlinear Model Input Design for Nonlinear Model Input Design for Nonlinear Model Input Design for Nonlinear Model  Discrimination via Affine Abstraction Discrimination via Affine Abstraction Input Design for Nonlinear Model  Discrimination via Affine Abstraction Discrimination via Affine Abstraction ∗ ∗∗ ∗∗  Kanishka Ding Necmiye Ozay Discrimination via Affine Abstraction Kanishka Singh Singh ∗∗ Yuhao Yuhao Ding ∗∗ Necmiye Ozay ∗∗ ∗∗ ∗ ∗∗

∗ Yuhao ∗∗ ∗ Kanishka Ding Necmiye Zheng Yong Kanishka Singh SinghSze Yuhao Ding Necmiye Ozay Ozay ∗∗ Sze Zheng Yong ∗ ∗ Sze Zheng Yong ∗ ∗∗ Zheng Yong Necmiye Ozay ∗∗ Kanishka SinghSze Yuhao Ding ∗ ∗ School for EngineeringSze of Matter, Transport Zheng Yong ∗ and Energy, Arizona ∗ School for Engineering of Matter, Transport and Energy, Arizona ∗ School forState Engineering of Matter, Transport and Energy, Arizona Arizona University, AZ, (emails: School forState Engineering of Matter, Energy, University, Tempe, Tempe,Transport AZ, 85287 85287and (emails: State University, Tempe, AZ, 85287 (emails: ∗ {krsingh1,szyong}@asu.edu) State University, Tempe, AZ, 85287 (emails: {krsingh1,szyong}@asu.edu) School for Engineering of Matter, Transport and Energy, Arizona ∗∗ {krsingh1,szyong}@asu.edu) ∗∗ Electrical Engineering and Computer Science Department, {krsingh1,szyong}@asu.edu) Electrical Engineering and Computer Science Department, State University, Tempe, AZ, 85287 (emails: ∗∗ ∗∗ Electrical Computer Science University of Michigan, Ann Arbor, 48109 (emails: Electrical Engineering and Computer Science Department, UniversityEngineering of{krsingh1,szyong}@asu.edu) Michigan,and Ann Arbor, MI, MI, 48109Department, (emails: of Ann Arbor, 48109 (emails: ∗∗ University {yuhding,necmiye}@umich.edu) UniversityEngineering of Michigan, Michigan,and Ann Arbor, MI, MI, 48109Department, (emails: {yuhding,necmiye}@umich.edu) Electrical Computer Science {yuhding,necmiye}@umich.edu) University {yuhding,necmiye}@umich.edu) of Michigan, Ann Arbor, MI, 48109 (emails: {yuhding,necmiye}@umich.edu) Abstract: This This paper paper considers considers the design design of of separating separating input input signals signals in in order order to to discriminate discriminate Abstract: the Abstract: This paper considers the design input signals order to among a finite number of uncertain models. nonlinear model to Abstract: Thisnumber paper considers the nonlinear design of of separating separating input signals in in ordercorresponds to discriminate discriminate among a finite of uncertain nonlinear models. Each Each nonlinear model corresponds to aa among a finite number of uncertain nonlinear models. Each nonlinear model corresponds to system operating mode, unobserved intents of other drivers or robots, or to fault types or attack among a finite of uncertain nonlinear models. Each model corresponds to aa system operating mode,considers unobserved drivers ornonlinear robots, toorder fault types or attack Abstract: Thisnumber paper theintents design of ofother separating input signalsorin to discriminate system operating mode, unobserved intents of other drivers or robots, or to fault types or attack strategies, etc., and the separating inputs are designed such that the output trajectories of system operating mode, unobserved intents of other drivers robots, to fault types or attack strategies, etc., number and the of separating are designed suchorthat the or output trajectories oftoall all among a finite uncertaininputs nonlinear models. Each nonlinear model corresponds a strategies, etc., and the separating are designed such the output trajectories of the models are guaranteed to from each other any strategies, etc., and the separating inputs are designed such that the output trajectories of all all the nonlinear nonlinear models areunobserved guaranteedinputs to be be distinguishable distinguishable from each other under any realization realization system operating mode, intents of other drivers orthat robots, or tounder fault types or attack the nonlinear models are guaranteed to distinguishable from other any of in condition, discrepancies or noise. We propose two-step the nonlinear models areinitial guaranteed to be bemodel distinguishable from each other under any aarealization realization of uncertainties uncertainties in the the initial condition, model discrepancies oreach noise. Weunder propose two-step strategies, etc., and the separating inputs are designed such that the output trajectories of all of uncertainties in the initial condition, model discrepancies or noise. We propose a two-step approach. First, using an optimization-based approach, we over-approximate nonlinear dynamics of uncertainties in the condition, discrepancies oreach noise. Weunder propose two-step approach. First, using aninitial optimization-based approach, we from over-approximate nonlinear dynamics the nonlinear models are guaranteed to bemodel distinguishable other any arealization approach. First, using aninitial optimization-based approach, weall over-approximate nonlinear dynamics by uncertain affine models, as abstractions that preserve its such that any approach. First, using an optimization-based approach, we over-approximate by uncertainties uncertain affine models, as condition, abstractions that preserve all itsorsystem system behaviors such that any of in the model discrepancies noise. behaviors We nonlinear propose adynamics two-step by uncertain affine models, as abstractions that preserve all its behaviors such that any discrimination guarantees for affine also the nonlinear system. by uncertain affine models, as the abstractions that preserve all itsfor system behaviors suchdynamics that any discrimination guarantees for the affine abstraction abstraction alsowehold hold forsystem the original original nonlinear system. approach. First, using an optimization-based approach, over-approximate nonlinear discrimination guarantees affine abstraction hold the original nonlinear system. Then, we a novel solution in form of mixed-integer program (MILP) to discrimination guarantees for the affine abstraction also hold forsystem thelinear original nonlinear system. Then, we propose propose amodels, novel for solution in the the form of aaalso mixed-integer linear program (MILP) to by uncertain affine as the abstractions that preserve all itsfor behaviors such that any Then, we propose a novel solution in the form of a mixed-integer linear program (MILP) to the active model discrimination problem for uncertain affine models, which includes the affine Then, we propose a novel for solution in the form of aalso mixed-integer (MILP) to the active model discrimination for uncertain affine models, whichprogram includes the affine discrimination guarantees theproblem affine abstraction hold for thelinear original nonlinear system. the active model discrimination problem for uncertain affine models,linear which includes(MILP) theof affine abstraction and the Finally, we demonstrate the effectiveness our the active model discrimination problem uncertain affine models, which includes the abstraction and thus, thus, the nonlinear nonlinear models. Finally, we demonstrate the program effectiveness ofaffine our Then, we propose a novel solution inmodels. thefor form of a mixed-integer to abstraction and thus, the models. Finally, demonstrate the effectiveness our approach identifying the intention of vehicles in aa highway scenario. abstraction and thus, the nonlinear models. Finally, we demonstrate thechanging effectiveness ofaffine our approach for identifying thenonlinear intention of other other vehicleswe in highway lane changing scenario. the activefor model discrimination problem for uncertain affine models,lane which includes theof approach for identifying the intention of vehicles in aa highway lane changing scenario. approach forand identifying the intention of other otherFinally, vehicles in highway lane changing scenario. abstraction thus, the nonlinear demonstrate theLtd. effectiveness of our © 2018, IFAC (International Federation ofmodels. Automatic Control)we Hosting by Elsevier All rights reserved. Keywords:for Input design, the model discrimination, nonlinear affine approach identifying intention of other vehicles in systems, a highway laneabstraction changing scenario. Keywords: Input design, model discrimination, nonlinear systems, affine abstraction Keywords: Input Input design, design, model model discrimination, discrimination, nonlinear nonlinear systems, systems, affine affine abstraction abstraction Keywords: 1. INTRODUCTION INTRODUCTION Passive Keywords: Input design, model discrimination, nonlinear systems, affine techniques abstraction seek 1. Passive discrimination discrimination techniques seek the the separation separation of of 1. INTRODUCTION INTRODUCTION Passive discrimination techniques seek the separation of the models regardless of the input [Lou and Si (2009); Rosa 1. Passive discrimination techniques seek the separation of the models regardless of the input [Lou and Si (2009); Rosa Recently, there is much public interest in the integration of Recently, there is much public interest in the integration of and the models regardless of the input [Lou and Si (2009); Rosa Silvestre (2011); Yong et al. (2014); Harirchi et al. the models regardless of the input [Lou and Si (2009); Rosa 1. INTRODUCTION and Silvestre (2011); Yong et al. (2014); Harirchi et al. Passive discrimination techniques seek the separation of Recently, there is much public interest in the integration of smart systems systems into everyday lives. These systems that ininRecently, there into is much publiclives. interest in the integration of (2017a)], smart everyday These systems that and Silvestre (2011); Yong et al. (2014); Harirchi et al. while active methods design aa separating input and Silvestre (2011); Yong et al. (2014); Harirchi et al. (2017a)], while active methods design separating input the models regardless of the input [Lou and Si (2009); Rosa smart systems into everyday lives. These systems that include smart homes, smart grids, intelligent transportation smart systems into everyday lives. These systems that include smart homes, smart grids, intelligent transportation Recently, there is much public interest in the integration of such (2017a)], active methods design aa separating input that the behaviors of models are (2017a)], while active Yong methods design separating input such Silvestre thatwhile the (2011); behaviors of different different models are distinct. distinct. et al. (2014); Harirchi et al. clude smart homes, smart grids, intelligent transportation and smart smart cities, are essentially complex, integrated and clude smart homes, smart grids, intelligent transportation and cities, essentially complex, integrated smart systems intoare everyday lives. These systems thatand in- and such that the behaviors of different models are distinct. Specifically in the area of input design for active model such that the behaviors of different models are distinct. Specifically in the area of input design for active model (2017a)], while active methods design a separating input and smart cities, are essentially complex, integrated and interconnected engineered systems with multiple operating and smart aresmart essentially complex, integrated and discrimination, interconnected engineered systems with multiple operating clude smartcities, homes, grids, intelligent transportation Specifically in the area of design for active model many approaches have been with Specifically in behaviors the area ofof input input design forproposed active model discrimination, many approaches have been proposed with that the different models are distinct. interconnected with multiple operating modes thatcities, areengineered often not systems directlycomplex, observed or measured; measured; interconnected engineered systems with multiple operating modes that are often not directly observed or and smart are essentially integrated and such discrimination, many approaches have been proposed with the goal of finding a small excitation that has a minimal discrimination, many approaches have been proposed with the goal of finding a small excitation that has a minimal Specifically in the area of input design for active model modes that are often not directly observed or measured; thus, they can be modeled as hidden mode hybrid systems. modes thatcan arebe often not systems directly observed or measured; thus, they modeled as hidden mode hybrid systems. effect interconnected engineered with multiple operating the goal of finding a small excitation that has a minimal on the desired behavior of the system, while guarthe goal of finding a small excitation that has a minimal effect on the desired behavior of the system, while guardiscrimination, many approaches have been proposed with thus, they can be modeled as hidden mode hybrid systems. For example, autonomous vehicles/robots have no access thus, they beoften modeled hiddenobserved mode hybrid systems. For example, autonomous vehicles/robots have no access effect modes thatcan are not as directly or measured; on thefinding desired behavior of the system, while guaranteeing isolation of different different fault models [Cheong effect on desired behavior of the system, while guaranteeing the isolation of fault models [Cheong the goal of a small excitation that has a minimal For example, autonomous vehicles/robots have no access to the intentions or decisions of other vehicles/humans For example, vehicles/robots have no access anteeing to the intentions or decisions of other thus, they can autonomous be modeled as hidden modevehicles/humans hybrid systems. ˇˇdifferent isolation of fault models and Manchester Manchester (2015); Simandl and Punˇ och´ aˇ ˇrr[Cheong (2009); anteeing the desired isolation ofSimandl different fault models [Cheong on the behavior of the system, while guarto the intentions or Yong decisions of otherDing vehicles/humans [Sadigh et al. al. (2016); (2016); Yong et al. of (2014); Ding et al. al. (2018)], and (2015); Punˇ ccoch´ a (2009); to the intentions or decisions other vehicles/humans [Sadigh et et al. (2014); et For example, autonomous vehicles/robots have no(2018)], access effect ˇˇ (2006);and and Manchester (2015); Simandl and Punˇ c och´ a ˇ rr[Cheong (2009); Nikoukhah and Campbell Scott et al. (2014); anteeing the isolation of different fault models [Sadigh et al. (2016); Yong et al. (2014); Ding et al. (2018)], and Manchester (2015); Simandl and Punˇ c och´ a ˇ (2009); while smart infrastructures are prone to different fault Nikoukhah and Campbell (2006); Scott et al. (2014); [Sadigh et al. (2016); et al. (2014); et al. (2018)], while infrastructures areof prone to different fault Harirchi to thesmart intentions or Yong decisions otherDing vehicles/humans Nikoukhah and Campbell Scott al. et (2017b); Ding et (2018)]. However, these ˇ (2006); while smart infrastructures are prone to different fault Nikoukhah and Campbell (2006); Scott et al. (2014); types [Harirchi [Harirchi and Ozay Ozay (2015); Cheong and Manchester Harirchi et al. al. (2017b); Ding et al. al. and (2018)]. However, these Manchester (2015); Simandl Punˇ cet och´ aˇr (2014); (2009); while smart infrastructures are(2014); prone toand different fault and types and Cheong [Sadigh et al. (2016); Yong(2015); et al. Ding etManchester al. (2018)], Harirchi et al. (2017b); Ding et al. (2018)]. However, these methods are only applicable for known linear or affine types [Harirchi and Ozay (2015); Cheong and Manchester Harirchi et al. (2017b); Ding et al. (2018)]. However, these (2015)] or attack modes [Pasqualetti et al. (2013); Yong methods are only applicable for known linear or affine Nikoukhah and Campbell (2006); Scott et al. (2014); types andmodes Ozay [Pasqualetti (2015); Cheong and Manchester (2015)][Harirchi or attack ettoal. (2013); Yong while smart infrastructures are prone different fault methods are only for known linear affine models, and and not forapplicable nonlinear or uncertain affineor models (2015)] or attack modes [Pasqualetti et al. (2013); Yong methods are only applicable for known linear or affine et al. (2015); Harirchi et al. (2017b)]. In these scenarmodels, not for nonlinear or uncertain affine models Harirchi et al. (2017b); Ding et al. (2018)]. However, these (2015)] or attack et In al. (2013); Yong that et al. [Harirchi (2015); Harirchi al. (2017b)]. scenartypes andmodes Ozayet[Pasqualetti (2015); Cheong andthese Manchester models, and not for nonlinear or uncertain affine models we consider in this work. et al. (2015); Harirchi et al. (2017b)]. In these scenarmodels, and not for nonlinear or uncertain affine models ios, approaches for discriminating among these operating that we consider in this work. methods are only applicable for known linear or affine et (2015); Harirchi et[Pasqualetti al. (2017b)]. these scenarios,al. approaches formodes discriminating among these operating (2015)] or attack et In al. (2013); Yong that we consider in this work. ios, approaches for discriminating among these operating that we consider in work. or pertains modes (or more more generally, models of system system behaviors) and not forthis nonlinear uncertaintoaffine models Another set of of relevant literature abstractions ios, approaches for discriminating among operating modes (or generally, models of behaviors) et al. (2015); Harirchi et al. (2017b)]. Inthese these scenar- models, Another set relevant literature pertains to abstractions modes (or more generally, models of system behaviors) based on noisy observed measurements can have a signifithat we consider in this work. Another set of relevant literature pertains to abstractions of nonlinear systems as linear or affine models that modes (or more generally, models of system behaviors) based on noisy observed measurements canthese haveoperating a signifi- Another ios, approaches for discriminating among set of relevant pertains to abstractions nonlinear systems as literature linear or affine models that overoverbased on noisy observed measurements can have have signifi- of cant impact impact on observed broad range range of applications applications inbehaviors) robotics, of nonlinear systems as linear or affine models that overapproximates all possible original system behaviors, which based on measurements can aa significant on aa generally, broad of in robotics, modes (ornoisy more models of system of nonlinear systems as linear or affine models that overapproximates all possible original system behaviors, which Another set of relevant literature pertains to abstractions cant impact on observed broad range range of fault applications in robotics, robotics, process control, medical devices, fault detection, etc. approximates all possible original system behaviors, which a common systematic approximation approach in the cant impact on aamedical broad of applications in process devices, detection, based oncontrol, noisy measurements can have aetc. signifi- is approximates all possible original system behaviors, which is a common systematic approximation approach in the of nonlinear systems as linear or affine models that overprocess control, medical devices, fault detection, etc. is aa common systematic approximation approach in the literature on hybridization hybridization [Asarin et al. al.behaviors, (2007); Althoff Althoff process control, devices, detection, etc. cant impact on amedical broad range of fault applications in robotics, is common systematic approximation approach in the Literature Review: The problem of discriminating among literature on [Asarin et (2007); approximates all possible original system which Literature Review: The problem of discriminating among et literature on hybridization [Asarin et al. (2007); Althoff al. (2008); Dang et al. (2010)]. This is typically achieved process control, medical devices, fault detection, etc. Literature Review: The problem of discriminating among literature on hybridization [Asarin et al. (2007); Althoff a set of models appears in a plethora of research areas such et al. (2008); Dang et al. (2010)]. This is typically achieved is a common systematic approximation approach in the Literature Review: Theinproblem of discriminating among a set of models appears a plethora of research areas such by et al. (2008); Dang et al. (2010)]. This is typically achieved linear interpolation and over-approximating the intera set of models appears in a plethora of research areas such et al. (2008); Dang et al. (2010)]. This is typically achieved as fault detection, input-distinguishability and mode disby linear interpolation and over-approximating the interliterature on hybridization [Asarin et al. (2007); Althoff aasset of models appears a plethora of research such fault detection, input-distinguishability and areas mode dis- polation Literature Review: Theinproblem of discriminating among linear interpolation and over-approximating the intererrors as additive bounded term, which as fault detection, input-distinguishability and areas mode disby linear interpolation and over-approximating the intercernibility of hybrid hybrid systems, where the the approaches insuch the by polation errors as an an additive bounded noise term, which et al. (2008); Dang et al. (2010)]. This is noise typically achieved as fault detection, input-distinguishability and mode discernibility of systems, where approaches in the a set of models appears in a plethora of research polation errors as additive bounded noise term, which may at times be aa an crude approximation. cernibility of hybrid systems, where the approaches in the polation errors as an additive bounded noise term, which literature can be grouped into passive and active methods. may at times be crude approximation. by linear interpolation and over-approximating the intercernibility of hybrid systems, approaches in disthe may at times be a crude approximation. literature can be grouped into where passivethe and active methods. as fault detection, input-distinguishability and mode literature can be grouped into passive and active methods. may at times be a crude approximation. polation errors as an additive bounded noise term, which  Contributions: literature be grouped passive and active(“TRI”). methods. cernibility of systems, where the approaches in the This work can washybrid supported by into Toyota Research Institute TRI  Contributions: We We propose propose aa novel novel two-step two-step approach approach to to This work was supported by Toyota Research Institute (“TRI”). TRI may atmodel times discrimination beWe a crude approximation. provided funds to assist the authors withpassive their research but this(“TRI”). article solely  Contributions: propose aa novel two-step approach to active among a set of uncertain nonliterature can be grouped and active methods. This work was supported by into Toyota Research Institute TRI provided funds to assist the authors with their research but this article solely  Contributions: We propose novel two-step approach to active model discrimination among a set of uncertain nonThis the work was supported by Toyota Research Institute (“TRI”). TRI reflects opinions and conclusions of its authors and not TRI or any other providedthe funds to assist the authors with their research but TRI this article solely reflects opinions and conclusions of its authors and not or any other active model discrimination among a set of uncertain nonlinear models, consisting of the affine abstraction/overprovided funds to assist the authors with their research but this article solely  Toyota entity. active model discrimination among a set of uncertain nonlinear models, consisting of the affine abstraction/overContributions: We propose a novel two-step approach to This work was supported by Toyota Research Institute (“TRI”). TRI reflects entity. the opinions and conclusions of its authors and not TRI or any other Toyota reflects the opinions and conclusions of its authors and not TRI or any other linear models, consisting the provided funds to assist the authors with their research but this article solely Toyota entity. linear model models, consisting of ofamong the affine affine abstraction/overactive discrimination a set abstraction/overof uncertain nonToyota entity. reflects the opinions and conclusions of its authors and not TRI or any other Copyright © 2018 IFAC 175 linear models, consisting of the affine abstraction/overToyota entity.

2405-8963 © 2018 2018, IFAC IFAC (International Federation of Automatic Control) Copyright 175 Hosting by Elsevier Ltd. All rights reserved. Copyright 2018 175 Peer review© responsibility of International Federation of Automatic Copyright © under 2018 IFAC IFAC 175 Control. 10.1016/j.ifacol.2018.08.030 Copyright © 2018 IFAC 175

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approximation and the corresponding input design problem. To the best of our knowledge, this problem is relatively unexplored in the literature. First, to address the potential conservativeness of conventional abstraction/over-approximation methods, we propose an optimization-based approach to over-approximate nonlinear dynamics by uncertain affine models that are compactly described using interval-valued matrices/vectors, in contrast to only having a interval-valued affine vector. In particular, this uncertain affine model must preserve all the system behaviors of the original nonlinear dynamics such that any model discrimination guarantees for uncertain affine abstraction also hold for the original nonlinear models. Next, using the resulting set of uncertain affine models, we propose a novel solution to the active model discrimination problem for uncertain affine models, which includes the affine abstraction. We show that this problem can be casted as a mixed-integer linear program (MILP), for which off-the-shelf optimization tools are readily available. To our knowledge, this input design problem for uncertain affine models (an important problem on its own) has not been considered in the literature. Finally, we demonstrate the effectiveness of our approach for identifying the intention of other vehicles in a lane changing scenario. 2. PRELIMINARIES 2.1 Notation and Definitions Let x ∈ Rn denote a vector and M ∈ Rn×m a matrix, with transpose M  and M ≥ 0 denotes element-wise nonnegativity. The vector norm of x is denoted by xi with i ∈ {1, 2, ∞}, while 0, 1 and I represent the vector of zeros, the vector of ones and the identity matrix of appropriate dimensions. The diag and vec operators are defined for a collection of matrices Mi , i = 1, . . . , n and matrix M as:     M1

 diagn i=1 {Mi } = 

M1

..

.



Mi 0 diagi,j {Mk } = , 0 Mj diagN {M } = IN ⊗ M,

Mn

 , vecni=1 {Mi } =  ..  , . Mn

veci,j {Mk } =





Mi , Mj

vecN {M } = 1N ⊗ M,

where ⊗ is the Kronecker product. The set of positive integers up to n is denoted by Z+ n , and the set of nonnegative integers up to n is denoted by Z0n . Note also the definition of Special Ordered Set of degree 1 (SOS-1) constraints in [Gurobi Optimization (2015)]. 2.2 Nonlinear Modeling Framework Consider N discrete-time affine time-invariant models Gin , #” ∈ Rn , outputs z ∈ Rp , inputs each with states x i i #” ∈ Rm , process noise u wi ∈ Rmw and measurement noise i vi ∈ Rmv . The models evolve according to the following autonomous nonlinear state equation: #” , u #” , w ), #”+ = f ( x x i i i i i

(1)

#”+ where fi is differentiable. For discrete-time systems, x denotes the state at the next time instant while for #” #”+ = x ˙ is the time derivative continuous-time systems, x of the state. Moreover, the output equation is: #” + D u #” zi = Ci x i i i + Dv,i vi + gi .

(2)

#” = x (0), The initial condition for model i, denoted by x i,0 i is constrained to a polyhedral set with c0 inequalities: #” ∈ X = { x #” ≤ p }, ∀i ∈ Z+ . #” ∈ Rn : P x x 0 0 0 i,0 N

(3)

176

#” are controlled inputs, The first mu components of u i #” , while mu denoted as u ∈ R , which are equal for all u i #” , denoted the other md = m − mu components of u i as di ∈ Rmd , are uncontrolled inputs that are model#” are partitioned into dependent. Further, the states x i nx ny xi ∈ R and yi ∈ R , where ny = n − nx , as follows:     #” = u i

u #” = xi . ,x i di yi

(4)

#” that The states xi and yi represent subsets of the states x i are the ‘responsibilities’ of the controlled and uncontrolled inputs, u and di , respectively. The term ‘responsibility’ in this paper is to be interpreted as u and di , respectively, having to independently satisfy the following polyhedral state constraints with cx and cy inequalities: xi ∈ Xx,i = {x ∈ Rnx : Px,i x ≤ px,i }, yi ∈ Xy,i = {y ∈ Rny : Py,i y ≤ py,i },

(5) (6)

subject to constrained inputs described by polyhedral sets with cu and cd inequalities, respectively: u ∈ U = {u ∈ Rmu : Qu u ≤ qu }, di ∈ Di = {d ∈ Rmdi : Qd,i d ≤ qd,i }.

(7) (8)

wi ∈ Wi = {w ∈ Rmw : Qw,i w ≤ qw,i }, vi ∈ Vi = {v ∈ Rmv : Qv,i v ≤ qv,i },

(9) (10)

#” (k + 1) ∈ C (k), x i i #” (k) + D u #” zi (k) = Ci x i i i (k) + Dv,i vi (k) + gi ,

(11) (12)

Further, X i  Xx,i × Xy,i and U i  U × Di . On the other hand, the process noise wi and measurement noise vi are also polyhedrally constrained with cw and cy inequalities:

and have no responsibility to satisfy any state constraints. The readers are referred to [Remark 1, Ding et al. (2018)] for a description of the well-posedness of the formulation. 2.3 Uncertain Affine Modeling Framework We also consider the uncertain affine modeling framework, that we will show in Section 4 to be a good representation for abstracting/over-approximating nonlinear models. Specifically, consider N discrete-time uncertain affine models Gia = (Ai , Bi , Bw,i , Ci , Di , Dv,i , Fi , gi ), each with #” ∈ Rn , outputs z ∈ Rp , inputs u #” ∈ Rm , process states x i i i noise wi ∈ Rmw and measurement noise vi ∈ Rmv . The models evolve according to the state and output equations:

where Ci (k) is the cover of xi (k + 1) in given domains #” ∈ X , u #” ∈ U , w ∈ W , defined by x i i i i

#” (k + 1) ∈ X | Ci (k)  { x i #” #” (k)+B u #” Ai x i u,i i (k)+B d,i di (k)+B w,i wi (k)+f i ≤ x i (k+1) (13) #” (k) + B u #” (k) + B d (k) + B w (k) + f , ≤ Ai x i u,i i w,i i d,i i i #” ∀ x i ∈ X , ui ∈ U , wi ∈ Wi }.

For compactness, we will represent the above model with interval matrices Ai = [Ai , Ai ], Bi = [B i , B i ], Bw,i = [B w,i , B w,i ] and an interval vector Fi = [f i , f i ]. This uncertain model can be obtained by construction, as described in Section 4, and we will use (13) directly for active model discrimination in Section 5. Based on the partitions of inputs in (4), the corresponding partitioning of the matrices B i , B i and Di are:       B i = B u,i B d,i , B i = B u,i B d,i , Di = Du,i Dd,i .

The states xi and yi represent the subset of the states #” that are the ‘responsibilities’ of the controlled and x i uncontrolled inputs, u and di , respectively. The term ‘responsibility’ in this paper is to be interpreted as u and di , respectively, having to independently satisfy the following polyhedral state constraints (for k ∈ Z+ T ) with

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cx and cy inequalities, as is given in (5) and (6), subject to constrained inputs described by polyhedral sets (for k ∈ Z0T −1 ) with cu and cd inequalities, respectively, given by (7) and (8). Similarly, the process noise wi and measurement noise vi are also polyhedrally constrained with cw and cy inequalities, as given in (9) and (10). In addition, we will consider a time horizon of length T and introduce some time-concatenated notations. The timeconcatenated states and outputs are defined as #” #” T x i,T = veck=0 { x i (k)},

zi,T = vecT k=0 {zi (k)},

while the time-concatenated inputs and noises are #” (k)}, u =vecT −1{u(k)}, d =vecT −1{d (k)}, #” =vecT −1{ u u i i i,T T i,T k=0 k=0 k=0 −1 {wi (k)}, vi,T =vecT wi,T =vecT k=0{vi (k)}. k=0

Using the above time-concatenated inputs, noise, states and outputs, the corresponding time-concatenated state and output equations can be written as: #” Mi x i,T + Γu,i uT + Γd,i di,T + Γw,i wi,T + Fi ≤ 0, #” zi,T =Ei x i,T + Fu,i uT + Fd,i di,T + Fv,i vi,T + Gi ,

(14)

where the matrices and vectors Mi , Γu,i , Γd,i , Γw,i , Fi,T , Ei , Fu,i , Fd,i , Fd,i and Gi are defined in the appendix.   Given N discrete-time affine models, there are I = N2 model pairs and let the mode ι ∈ {1, · · · , I} denote the #” , x , y , pair of models (i, j). Then, concatenating x i,T i,T i,T di,T , zi,T , wi,T and vi,T for each model pair, we define #” }, z ι = vec {z }, dι = vec {d }, #”ι = vec { x x i,j i,j i,j i,T i,T i,T T T T ι ι = vec {v wT = veci,j {wi,T }, vT i,j i,T }.

The states and outputs over the entire time horizon for each mode ι can be written as simple functions of the state #”ι , input vectors u , dι , and noise wι , v ι : x T T T T #”ι + Γι u + Γι dι + Γι wι + F ι ≤ 0, Mι x u T w T T T d T ι #”ι + F ι uι + F ι dι + F ι v ι + Gι . zT =E ι x u v T T T d T

(15)

The matrices and vectors M ι , Γιu , Γιd , Γιw , FTι , E ι , Fuι , Fdι , Fvι and Gι are defined in the appendix. Moreover, the uncertain variables for each mode ι are concatenated #”ι T dι T wι T v ι T ]T . We then concatenate the as xι = [ x T T T T polyhedral state constraints in (5) and (6). First, let ι

P x = diagi,j diagT {Px,i },

where Px,i =



Px,i 0 0 Py,i



pιx = veci,j vecT {px,i },

and px,i =



px,i py,i



. Further, we

concatenate the initial state constraint in (3): ι

Similarly, let

P 0 = diag2 {[P0 0]},

pι0 = vec2 {p0 }.

ι

Qu = diagT {Qu }, Q† = diagi,j diagT {Q†,i },

q u = vecT {qu }, q ι† = veci,j vecT {q†,i }, † ∈ {d, w, v}.

Then, the input constraints in (7) and (9) for all k are: ι Qu uT ≤ q u and Q† †ιT ≤ q ι† . Hence, in terms of xι , we have a polyhedral constraint of the form Hx¯ι x ¯ι ≤ hιx¯ , with    ι    Hx¯ι =

P¯0 0 ι, ι, ι p¯ι, q¯dι, q¯w q¯vι, x ¯ι , Q ¯ι , Q ¯ ι } , hx¯ = p¯0 diag{P¯xι , Q w v d



.

Note that the above definitions of matrices and vectors for the uncertain affine model are different from the ones in Ding et al. (2018) that were defined for affine models. 3. PROBLEM FORMULATION We consider the active model discrimination problem among a finite number of uncertain nonlinear models, i.e., to design a separating admissible input for the system 177

177

such that when the system is excited with this input, any observed trajectory is consistent with only one model, regardless of any realization of uncertain parameters. In addition, we minimize a cost function J(u), as follows: Problem 1. (Active Model Discrimination). Given N wellposed nonlinear models Gi , and state, input and noise constraints, (3),(6),(8)-(10), find an optimal input sequence u∗T to minimize a given cost function J(uT ) such that #” , uncontrolled input d , for all possible initial states x 0 T process noise wT and measurement noise vT , only one model is valid, i.e., the output trajectories of any pair of models have to differ in at least one time instance. The above problem is in general computationally intractable with no optimality guarantees, as it would result in a mixed-integer nonlinear program (MINLP). Thus, we propose a two-step approach to tackle this problem. The first subproblem (Problem 1.1) is to over-approximate the nonlinear dynamics in (1) by uncertain affine models (for each i ∈ Z+ N ; subscript i omitted below for brevity), as abstractions that preserve (or “contain”) all its system behaviors. This is a common systematic approximation approach in the literature on hybridization [Asarin et al. (2007); Althoff et al. (2008); Dang et al. (2010)]. Problem 1.1. (Affine Abstraction). Given a nonlinear n#” u, #” w) with (polytopic) dodimensional vector field f( x, #” #” main x ∈ X , u ∈ U , w ∈ W, find two n dimensional affine planes (i.e., upper and lower planes defined by matrices A, A, B, B, B w , B w and vectors f , f ) such that they “contain” (i.e., upper- and lower bound) the given vector field with minimum separation, as expressed by the following:  + λ1 B  + λ2 B w + λ3 f min A A,A,B,B,B w ,B w ,f ,f A ,B,Bw ,f subject to #” +B u #” +B w+f ≤ f( x, #” u, #” w), Ax #” #” w #” +B u #” +B w+f , ∀( x ∈ X , u ∈ U , w ∈ W), #” u, #” w) ≤ A x f( x,

(16a)

w

 B − B ≤ B,  B w − B w  ≤ B w , A − A ≤ A,  f − f  ≤ f , A ≥ A, B ≥ B, B w ≥ B w , f ≥ f ,

(16b)

where λ1 , λ2 and λ3 are tuning weights and suitable norms are chosen based on the application at hand. Then, using the resulting set of uncertain affine models that conform to the modeling framework in Section 2.3, we consider the following second subproblem (Problem 1.2): Problem 1.2. (Active Affine Model Discrimination). Given N well-posed affine models Gia , and state, input and noise constraints, (3),(6),(8)-(10), find an optimal input sequence u∗T to minimize a given cost function J(uT ) such #” , uncontrolled input that for all possible initial states x 0 dT , process noise wT and measurement noise vT , only one model is valid, i.e., the output trajectories of any pair of models have to differ in at least one time instance. The optimization problem can be formally stated as follows: min J(uT ) s.t.

uT ,xT

∀k ∈ Z0T −1 : (7) hold,

, i < j, ∀k ∈ Z0T , ∀i, j ∈ Z+ #” N, y , d , w , v : ∀x T T T T T (3),(5),(6),(8)-(10),(15) hold



:

∃k ∈ Z0T , zi (k) = zj (k).

(17a)

(17b)

Using the two-step design, the discrimination guarantees in Problem 1.2 also hold for the original nonlinear models because the affine abstraction in Problem 1.1 is designed such that any properties that hold for the abstraction also hold for the original nonlinear models. Our approach has

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the advantage of “divorcing” the difficulty associated with optimization problems with nonlinear constraints from the active model discrimination problem, providing insight into whether the nonlinearity is the bottleneck or if the active discrimination problem itself is ill-posed. However, the solution is, by construction, suboptimal. Nonetheless, the designed separating input guarantees that all output trajectories of all the nonlinear models are distinguishable from each other under any realization of uncertainties in the initial condition, model discrepancies or noise. 4. AFFINE ABSTRACTION In this section, we address Problem 1.1. Since the robust formulation given in Problem 1.1 cannot be directly implemented using standard optimization packages, we will convert the problem into a more amenable form such that offthe-shelf optimization tools can be applied. For simplicity, we first describe our approach for a 1-dimensional vector field with interval domains, before describing how this can be extended to higher dimensional systems. Moreover, we will assume in the following discussion that there is at most one local optimum in the domain X , U , W. This can also be extended to the case with multiple local optima. Theorem 1. (Affine Abstraction). Given a nonlinear 1#” u, #” w) with indimensional differentiable vector field f( x, #” ∈ X , u #” ∈ U , w ∈ W, two affine planes terval domains x (i.e., upper and lower planes defined by A, A, B, B, B w , B w and vectors f , f ) that contain the given vector field with minimum separation are solutions to:  + λ1 B  + λ2 B w + λ3 f min A  B,  Bw , f, A, A, B, B, B w , B w , f , f A, #” , w , x #” , u #” , w #” , u x u u u b b b

subject to #” u, #” w), #” +B u #” +B w+f ≤ f( x, Ax w #” ∈ U #”  , u  , w ∈ W),  (18a) #” +B u #” +B w+f , ∀( x ∈ X #” u, #” w) ≤ A x f( x, w

#” , u #” , w ) = 0, B − ∇ #” f( x #” , u #” , w ) = 0, #” f( x A − ∇x u b b b b b b #” , u #” , w ) = 0, B w − ∇w f( x b b b (18b) #” ∈ X ∧ u #” + B u #” #” ∈ U ∧ w ∈ W ∧ (A x (( x b b b b b #” #” #” #” +B w wb + f ≤ f( x b , u b , wb ))) ∨ x b ∈ / X ∨ ub ∈ / U ∨ wb ∈ / W), #” , u #” , w ) = 0, B − ∇ #” f( x #” , u #” , w ) = 0, #” f( x A − ∇x u u u u u u u #” , u #” , w ) = 0, B w − ∇w f( x u u u (18c) #” ∈ X ∧ u #” ∈ U ∧ w ∈ W ∧ (A x #” + B u #” (( x u u u u u #” #” #” #” +B w + f ≥ f( x , u , w ))) ∨ x ∈ /X ∨ u ∈ /U ∨ w ∈ / W), w

u

u

u

u

u

u

u

 B − B ≤ B,  B w − B w  ≤ B w , A − A ≤ A, (18d)  f − f  ≤ f , A ≥ A, B ≥ B, B w ≥ B w , f ≥ f , where λ1 , λ2 , λ3 are tuning weights (chosen based on the , U , W  are the sets of endpoints of the application), X #” and w #” , u interval domains X , U , W, respectively, x u u u are the local optimum for the difference between the upper #” , u #” and w for plane and function f, and similarly, x b b b the bottom plane, while ∧ and ∨ are logical AND and OR operators 1 and ∇∗ denotes the derivative operator with #” u, #” w}. respect to variable i ∈ { x, Proof. The optimization formulation above is as in Problem 1.1 except that the semi-infinite constraints (16a) are replaced by readily implementable constraints (18a), (18c), (18b). This is possible because the maximum of a differentiable (thus, continuous) function over a closed domain is the maximum of its local optima (i.e., minima, maxima or saddle points) in its interior (i.e., (18b)) and the maximum over its boundaries (i.e., its endpoints; (18a)). 1 Logical operators are implementable by off-the-shelf softwares, e.g., YALMIP [L¨ ofberg (2004)], and if needed, converted to mixed-integer constraints.

178

The same holds for minimization over a closed domain, resulting in (18a) and (18c) for the lower plane. Eq. (18a) is required such that the upper and lower planes upper- and lower-bound each boundary/endpoint of the domains. On the other hand, Eqs. (18b) and (18c) are such that the local optima, which are given by their first order necessary condition (i.e., their first order derivative is set to zero), are also upper- and lower-bounded by the two planes, respectively. The constraints with logical operators are to be understood to be in conjunction with the other constraints, and has the interpretation that only the local optima in the given domain needs to be upper- or lower-bounded by the two planes. Finally, Eq. (18d) along with the objective function ensures that the upper plane remains above the lower plane and that the separation between the two planes is as small as possible.  To extend the above formulation to higher dimensions, the key difference from the 1-dimensional case is that the boundaries of the domain are higher dimensional facets as opposed to line segments. To deal with this, the above procedure of replacing the maximization or minimization over a domain with the maximizing or minimizing over the local optima (using first order optimality condition for constrained optimization) and its boundaries as in (18a), (18c), (18b) can be repeated to recursively reduce the dimension of the boundaries by one until we obtain line segments. This procedure may be tedious to implement for systems with high dimensions and domains with many facets, hence, in practice, we may start with only constraints on the vertices and iteratively add facets of the domain of interest when the resulting planes are found to intersect them. Further effort to reduce the complexity of this procedure is the subject of ongoing research. 5. ACTIVE MODEL DISCRIMINATION APPROACH We now extend and modify the approach in Ding et al. (2018) to solve Problem 1.2 with given uncertain affine models from Section 4. This approach relies on formulating the problem as a bi-level optimization problem which is then converted to a single level MILP using KKT conditions, for which off-the-shelf softwares are readily available, e.g., [Gurobi Optimization (2015)]. For the sake of clarity, we will defer the definitions of certain matrices in the following results to the appendix. Moreover, for brevity, the proofs of this approach are omitted, as they follow similar steps to the proofs in [Ding et al. (2018)]. Theorem 2. ( Discriminating Input Design as an MILP ). Given a separability index , the active model discrimination problem (Problem 1.2) is equivalent (up to ) to: min

uT ,δ ι ,,¯ xι ,µι1 ,,µι2 ,µι3

J(uT )

(PDID )

¯ u uT ≤ q¯u , s.t. Q : δ ι (uT ) ≥ , 0 = 1 − µι3 T 1, ∀ι ∈ Z+ I i=κ ι ι j=ξ ι ι 0= µ H ¯ (i, m) + µ R (j, m) i=1 1,i x j=1 2,j 1

k=ρ

µι Rι (k, m), ∀m = 1, · · · , η, k=1 3,k 2 Hx¯ι ,i x ¯ι − hιx¯,i ≤ 0, µι1,i ≥ 0, ∀i = 1, . . . κ, ι x ι − S ι u ≤ 0, µι R1,j ¯ι − r1,j 1,j T 2,j ≥ 0, ∀j = 1, . . . ξ, ι x ι ι u ≤ 0, µι R2,k ¯ι − δ ι − r2,k − S2,k T 3,k ≥ 0, ∀k = 1, . . . ρ, + + ι ι ι ∀ι ∈ ZI , ∀i ∈ Zκ : SOS-1 : {µ1,i , Hx¯,i x ¯ − hιx¯,i }, + + ι ι ι −S ˜ι uT }, ¯ι − r1,j ∀ι ∈ ZI , ∀j ∈ Zξ : SOS-1 : {µ2,j , R1,j x 1,j ρ + ι ι ι ι ι ι u }, ∀ι ∈ ZI , ∀j ∈ Zρ : SOS-1 : {µ3,k , R2,k x ¯ − δ − r2,k − S˜2,k T +

  

  

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 ι is where µι1,i , µι2,j and µι3,j are dual variables, while H x ¯,i ι and Sι are the j-th row of Rι the i-th row of Hx¯ι , R 1 1,j 1,j ι and Sι are the k-th row of Rι and S1ι , respectively, R 2 1,k 2,k and S2ι , respectively, η = IT (n + md + mw + mv ) is the number of columns of Hxι , κ = 2IT (c0 + cd + cw + cv ) is the number of rows of Hxι , ξ = 2IT (cx + cy ) is the number of rows of R1ι and ρ = 2IT p is the number of rows of R2ι . Note that the result above is an extension of Ding et al. (2018). Although the notations of matrices and vectors look similar, their definitions are rather different.

179

Fig. 1. Affine abstraction/over-approximation of Dubins vehicle dynamics such that the true nonlinear system behavior is contained/enveloped by the abstraction.

6. APPLICATION EXAMPLE: ACTIVE INTENTION IDENTIFICATION IN LANE CHANGE SCENARIO In this section, we apply the active model discrimination approach proposed in the previous sections to design a separating control input that, in conjunction with a modified model invalidation algorithm (e.g., [Harirchi and Ozay (2015); Harirchi et al. (2017a, 2018)]), can be used for active intention identification in a highway lane change scenario (cf. [Figure 1, Ding et al. (2018)]). 6.1 Vehicle and Intention Models The dynamics of the ego vehicle is of the Dubins car [Dubins (1957)] with acceleration input: x˙ e = ve cos φe , y˙ e = ve sin φe , v˙ e = u1 + w1 , φ˙ e = u2 + w2 ,

while the other car dynamics is described by: x˙ o = vo , v˙ o = di + w3 ,

where xe and ye are the longitudinal and lateral coordinates of the ego car, ve is its speed and φe is its heading angle, while xo is the longitudinal coordinate of the other car with its speed given by vo (no lateral movement), with process noise signals wj , j ∈ {1, 2, 3}. u1 ∈ Ua  [−7.848, 3.968] and u2 ∈ Us  [−0.44, 0.44] are the acceleration and steering inputs of the ego car, whereas di is the (uncontrolled) acceleration input of the other car for each intention i ∈ {I, C, M }, corresponding to an Inattentive, Cautious or Malicious driver. The Inattentive driver is unaware of the ego car and maintains his speed using an acceleration input which lies in a small range dI ∈ DI  [−0.392, 0.198]. The Cautious driver tends to yield the lane to the ego car with the input equal to dC  −Kd,C (ve − vo ) + Lp,C φe + Ld,C φ˙ e + d˜C , where φ˙ e = u2 + w2 , Kd,C = 1, Lp,C = 12 and Ld,C = 14 are PD controller parameters and the input uncertainty is d˜C ∈ DC = DI . Finally, the Malicious driver does not want to yield the lane and attempts to cause a collision with input equal to dM  Kd,M (ve − vo ) − Lp,M φe − Ld,M φ˙ e + d˜M , if provoked, where φ˙ e = u2 +w2 , Kd,C = 0.9, Lp,C = 12 and Ld,C = 14 are PD controller parameters and the input uncertainty satisfies d˜M ∈ DM = DI . Without loss of generality, we assume that the initial position and heading angle of the ego car are 0, while the initial velocities match typical speed limits of the highway. Moreover, both cars are close to the center of their lanes that are 3.2 m wide. Thus, the initial conditions are: ve (0) ∈ vo (0) ∈

[23, 27] m , s [23, 27] m , s

ye (0) ∈ [1.5, 1.7]m, xo (0) ∈ [10, 12]m.

Further, the velocity of the ego vehicle is constrained to be between [20, 30] m s at all times (in order to obey the 179

Fig. 2. Effect of different choices of objective functions on the resulting separating inputs. (Note that the control inputs with 1- and 2-norms as objective functions are near identical.)

speed limit of a highway), its heading angle is between [−0.44, 0.44]rad and its lateral position is constrained between [0.3, 2.5]m. Process and measurement noise signals are bounded with a range of [−0.01, 0.01] and the separability threshold is set to  = 0.5 m s . Moreover, we assume the extreme scenario where only noisy observation of other car’s velocity is observed z = vo + v. 6.2 Affine Abstraction The Dubins vehicle and intention models above are nonlinear. Hence, we resort to the approach in Section 4 to obtain affine abstractions of the models. Since the nonlinearity only affects the speed ve and the heading angle φe , we first define a suitable domain that is appropriate for the lane changing scenario. Specifically, we consider the speed of the ego car to be between 20 m/s to 30 m/s (72 to 108 km/h) and a heading angle range of −25◦ to 25◦ ([−0.44, 0.44]rad). Using this domain, we can obtain an affine abstraction for the reduced 2-dimensional system with λ1 = 1000, λ2 = λ3 = 0 and ∞-norms, as illustrated by Figure 1, and using the compact interval matrix representation, the abstracted open-loop model is given by: A = [A, A] = [(1, 3) : [0.9947, 0.9956]; (1, 4) : 2.3821; (2, 3) : [−0.1028, −0.0928]; (2, 4) : [23.0325, 25.2673]; (5, 6) : 1]6×6 , B = Bw = B = [(3, 1) : 1; (4, 2) : 1; (6, 3) : 1]6×3 , = [(1, 1) : [−3.7471, 0.2859]; F = [f , f ] (1, 2) : [0.4413, 4.4754]]6×1 ,

where we used a sparse matrix notation with the size indicted in the subscript. Combining the abstraction with the intention models and using Euler method for time-discretization with sampling time δt = 0.4 s, we have the following intention models: Inattentive Driver (i = I): AI = I + δtA, AI = I + δtA, B I = B I = B w,I = B w,I = δtB, CI = [(1, 6) : 1]1×6 , DI = 0, Dv,I = 1, f = δtf , f I = δtf . I

Cautious Driver (i = C): ˜C = [(6, 3) : −Kd,C ; (6, 4) : Ld,C ; (6, 6) : Kd,C ]6×6 , A ˜C = [(6, 2) : Lp,C ]6×3 , B ˜C ), AC = I + δt(A + A ˜C ), AC = I + δt(A + A ˜C ), B C = B C = B w,C = B w,C = δt(B + B CC = CI , DC = 0, Dv,C = 1, f = δtf , f C = δtf . C

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Malicious Driver (i = M ): ˜M = [(6, 3) : −Kd,M ; (6, 4) : Ld,M ; (6, 6) : Kd,M ]6×6 , A ˜M = [(6, 2) : Lp,M ]6×3 , B ˜M ), AM = I + δt(A − A ˜M ), AM = I + δt(A − A ˜M ), B M = B M = B w,M = B w,M = δt(B − B CM = CI , DM = 0, Dv,M = 1, f = δtf , f M = δtf . M

6.3 Active Nonlinear Model Discrimination Next, we apply the active model discrimination approach in Section 5 to the above uncertain affine models for the three intentions of the other driver in a highway lane changing scenario. Figure 2 shows the active inputs of the ego car to discern the other car’s intention based on its response when using various objective functions. When comparing the solutions with different norms as the objective functions, we observe that u1 reduces the sum of absolute values, thus keeping all the data points as close to zero as possible, u2 minimizes energy and may be desirable to reduce fuel consumption, and u∞ ensures comfort by minimizing the maximum absolute input values. In all solutions, the ego car accelerates and turns towards the other car’s lane, and then either maintains its speed or decelerates while re-aligning to its lane, although now laterally shifted toward the other car’s lane. Throughout the maneuver, the inattentive car coasts with small accelerations, the cautious car slows down when it observes the ego car turning to its lane and the malicious car tries to match the ego car’s position, causing a collision. The obtained optimal separating input is then applied to the ego vehicle in real-time. After measurements are recorded for T = 3, the passive model discrimination approach based on model invalidation of each intention model (e.g., [Harirchi and Ozay (2015); Harirchi et al. (2018)]) can be applied to identify the intention of the other vehicle. This is guaranteed to find the true intention by design (see above definition of separating input). 7. CONCLUSION This work considered the novel design of separating input signals in order to discriminate among a finite number of uncertain nonlinear models, using a two-step approach. First, we developed an optimization-based approach that is less conservative than existing methods to over-approximate nonlinear dynamics by uncertain affine models that contain all the system behaviors of the original nonlinear system. Then, we proposed one of the first active model discrimination algorithms for uncertain affine models, which includes the affine abstraction, hence, the nonlinear models. Finally, we demonstrated our approach on an example of intent estimation/identification in a lane changing scenario on a highway. For future work, we are interested to overcome the complexity of our affine abstraction approach and to reduce its optimality gap. REFERENCES Althoff, M., Stursberg, O., and Buss, M. (2008). Reachability analysis of nonlinear systems with uncertain parameters using conservative linearization. In IEEE CDC, 4042–4048. Asarin, E., Dang, T., and Girard, A. (2007). Hybridization methods for the analysis of nonlinear systems. Acta Inf., 43(7), 451–476. Cheong, S. and Manchester, I. (2015). Input design for discrimination between classes of LTI models. Automatica, 53, 103–110. Dang, T., Maler, O., and Testylier, R. (2010). Accurate hybridization of nonlinear systems. In ACM HSCC, 11–20.

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Ding, Y., Harirchi, F., Yong, S.Z., Jacobsen, E., and Ozay, N. (2018). Optimal input design for affine model discrimination with applications in intention-aware vehicles. In ACM/IEEE ICCPS, 5161–5167. Dubins, L.E. (1957). On curves of minimal length with a constraint on average curvature, and with prescribed initial and terminal positions and tangents. American J. of Math., 79, 497–516. Gurobi Optimization, I. (2015). Gurobi optimizer reference manual. Harirchi, F. and Ozay, N. (2015). Model invalidation for switched affine systems with applications to fault and anomaly detection. IFAC-PapersOnLine, 48(27), 260–266. Harirchi, F., Yong, S.Z., and Ozay, N. (2017a). Guaranteed fault detection and isolation for switched affine models. In IEEE CDC. Harirchi, F., Yong, S.Z., Jacobsen, E., and Ozay, N. (2017b). Active model discrimination with applications to fraud detection in smart buildings. IFAC-PapersOnLine, 50(1), 9527–9534. Harirchi, F., Yong, S.Z., and Ozay, N. (2018). Passive diagnosis of hidden-mode switched affine models with detection guarantees via model invalidation. In Diagnosability, Security and Safety of Hybrid Dynamic and Cyber-Physical Systems, 227–251. Springer. L¨ ofberg, J. (2004). YALMIP: A toolbox for modeling and optimization in MATLAB. In CACSD. Taipei, Taiwan. Lou, H. and Si, P. (2009). The distinguishability of linear control systems. Nonlinear Analysis: Hybrid Systems, 3(1), 21–38. Nikoukhah, R. and Campbell, S. (2006). Auxiliary signal design for active failure detection in uncertain linear systems with a priori information. Automatica, 42(2), 219–228. Pasqualetti, F., D¨ orfler, F., and Bullo, F. (2013). Attack detection and identification in cyber-physical systems. IEEE Trans. on Aut. Cont., 58(11), 2715–2729. Rosa, P. and Silvestre, C. (2011). On the distinguishability of discrete linear time-invariant dynamic systems. In IEEE CDC-ECC, 3356– 3361. Sadigh, D., Sastry, S., Seshia, S., and Dragan, A. (2016). Information gathering actions over human internal state. In IEEE/RSJ IROS, 66–73. Scott, J., Findeisen, R., Braatz, R.D., and Raimondo, D. (2014). Input design for guaranteed fault diagnosis using zonotopes. Automatica, 50(6), 1580–1589. ˇ Simandl, M. and Punˇ coch´ aˇr, I. (2009). Active fault detection and control: Unified formulation and optimal design. Automatica, 45(9), 2052–2059. Yong, S.Z., Zhu, M., and Frazzoli, E. (2014). Generalized innovation and inference algorithms for hidden mode switched linear stochastic systems with unknown inputs. In IEEE CDC, 3388–3394. Yong, S.Z., Zhu, M., and Frazzoli, E. (2015). Resilient state estimation against switching attacks on stochastic cyber-physical systems. In IEEE CDC, 5162–5169.

APPENDIX M ι = diag{Mi }, Γιu = vec{Γu,i }, Γιd = diag{Γd,i }, i,j

i,j

i,j

Γιw = diag{Γw,i }, FTι = vec{Fi }, E ι = diag{Ei }, i,j

i,j

i,j

Fuι = vec{Fu,i }, Fdι = diag{Fd,i }, Fvι = diag{Fv,i }, Gι = vec{Gi }, i,j



i,j

i,j



i,j

Mi 0 Γd,i 0 Γw,i 0 0 0 = , 0 Mj 0 Γd,j 0 Γw,j 0 0   Ei −Ej Fd,i −Fd,j 0 0 Fv,i −Fv,i R2ι = ,    −Ei Ej −F  d,i Fd,j 0 0 −Fv,i  Fv,i Fu,j − Fu,i −Fi −Γu,i G j − Gi r1ι = . , S1ι = , r2ι = , S2ι = G i − Gj − Fu,j −Fj −Γu,j   Fu,i  fi Ai −I 0 0 · · · 0 0 0 ··· 0 −Ai I −fi   0  fi  Ai −I 0 · · · 0      0 −Ai I 0 · · · 0    Mi =   , Fi = −fi  , .. .. .. ..   ..  ..  ..  .  .  . . . . .   0  f  0 · · · · · · Ai −I i 0 0 · · · · · · −Ai I −fi Ei = diag{Ci }, Gi = vec{gi }. R1ι

T

For = {d, w, u} : Γ,i





T

B ,i = diag{ }, F,i = diag{D,i }. −B ,i T T