Guaranteed Reachability for Systems with Unknown Dynamics
aa r X i v : . [ m a t h . O C ] O c t Guaranteed Reachability for Systems with UnknownDynamics
Melkior Ornik ∗ Abstract
The problem of computing the reachable set for a given system is a quintessential questionin nonlinear control theory. While previous work has yielded a plethora of approximate andanalytical methods for determining such a set, these methods naturally require the knowledgeof the controlled system dynamics throughout the state space. In contrast to such classicalmethods, this paper considers the question of estimating the reachable set of a control systemusing only the knowledge of local system dynamics at a single point and a bound on the rateof change of dynamics. Namely, motivated by the need for safety-critical planning for systemswith unknown dynamics, we consider the problem of describing the guaranteed reachabilityset: the set of all states that are guaranteed to be reachable regardless of the true systemdynamics, given the current knowledge about the system. We show that such a set can beunderapproximated by a reachable set of a related known system whose dynamics at everystate depend on the velocity vectors that are guaranteed to all control systems consistentwith the assumed knowledge. Complementing the theory, numerical examples of a single-dimensional control system and a simple model of an aircraft in distress verify that such anunderapproximation is meaningful in practice, and may indeed equal the desired guaranteedreachability set.
Following an adverse event affecting flight safety, such as partial loss of actuation, significant changein system dynamics due to damage, or sensor malfunction, a pilot operating an aircraft in distressneeds to determine whether to divert to an alternate airport, and if so, which airport to choose.While it may be possible for the aircraft to continue to its original destination, the pilot is requiredto determine a diversion airport where the aircraft can certainly land [Ekstrand and Pandey, 2003,DeSantis, 2013]. At the time of decision, the pilot might not have a correct model of flight dynamics;in extreme cases such as loss of a wing [Aloni, 2006], there may be little prior experience or availableknowledge about the system. Thus, it is imperative to immediately provide the pilot, or the flightcontroller for an autonomous vehicle, with a set of landing points that the aircraft is guaranteed tobe able to reach, given the current information about the system.Abstracted from its motivation, the above example describes the classical problem of reachabil-ity : given the system’s initial state, we wish to describe all states for which there exists a controlinput that drives the system to that state [Brockett, 1976, Isidori, 1985, Lewis, 2012]. Such a prob-lem has been explored in many related flavors, such as reachability at a given time [Ding et al., 2011], ∗ University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA. e-mail: [email protected] local controlled dynamics of a nonlinearsystem at a given state using solely the information from a single trajectory until the time that thesystem reached that state, we assume that the only available knowledge at the time of computingthe reachable set consists of (i) local dynamics at a single point and (ii) Lipschitz bounds on therate of change of system dynamics in the state space. As we do not know anything else about thesystem dynamics, we wish to determine the set of states that are guaranteed to be reachable fromthat single point regardless of the true system dynamics, as long as they are consistent with theabove knowledge.The primary contribution of this paper is to provide an underapproximation of such a guaranteedreachability set (GRS) . Our approach for determining this underapproximation relies on interpretinga control system as a differential inclusion with the available velocity set at every state in the statespace describing the possible tangents to the system’s trajectory when leaving that state. Localdynamics at each state determine the available velocity set at that state, and, while exact availablevelocity sets are not known anywhere except for a single point, we can determine the family of allvelocity sets that are consistent with our knowledge about the system dynamics. The intersectionof the elements of such a family provides the guaranteed velocity set at every state in the statespace; a set of velocities for which we know there exists a control input generating them at a givenstate, even if we do not know the exact dynamics at that state. Since such a set may be difficult tocompute or express in closed form, we underapproximate it by a ball of maximal possible radius.Moving back from differential inclusions to controlled differential to differential equations, such anunderapproximation generates a control system with known dynamics; the reachable set of such acontrol system is a subset of the GRS.The outline of this paper is as follows. In Section 2 we connect our work with previous effortson reachability and estimation of dynamics, discuss the realism of the knowledge that we assumeabout the system, and provide a broad discussion of the paper’s approach and results. Section3 provides the formal problem statement and assumptions about the system dynamics and prioravailable knowledge. Section 4 introduces the guaranteed velocity set, expresses it as an intersectionof a family of simply computable sets, provides its underapproximation by a ball, and shows thatsuch an underapproximation is maximal. Section 5 describes an underapproximation of the GRS asa reachable set of a known control system, discusses the currently available methods of computingsuch a set, and proposes a strategy for determining, in real time, a control input to drive thesystem to a state guaranteed to be reachable. In Section 6 we provide two numerical examples toillustrate and validate the developed theory. Section 6.1 considers a nonlinear 1D control system2otivated by a phenomenon of gene autoregulation and discusses the relationship between the truereachable set, the guaranteed reachable set, and its previously described underapproximation; itsresults serve to both confirm the preliminary theory established in this paper and inform futureavenues of research. Section 6.2 builds upon the motivating example by discussing a simple modelof an aircraft in need to divert. Finally, Section 7 discusses open questions for future work.
Notation.
The set of all matrices with n rows and m columns is denoted by R n × m . Notation B m ( x ; r ) denotes a closed ball in R m with the center at x ∈ R m and radius r ≥
0. Set GL ( n )denotes the general linear group of all matrices R n × n , i.e., all invertible n × n matrices. For avector v , k v k denotes its Euclidean norm. For a matrix M , M T denotes its transpose and k M k denotes its 2-norm: k M k = max k v k =1 k M v k . For a matrix M , vector v , and set S of vectors ofappropriate size, notation v + M S denotes the set { v + M s | s ∈ S} . Notation diag( λ , . . . , λ n )denotes a diagonal n × n matrix with elements λ , . . . , λ n on the diagonal, in that order. Vector e i ∈ R n denotes the i -th coordinate vector, i.e., a vector of all zeros, except a 1 in position i . Matrix I denotes the identity matrix. The assumptions and objectives of this work come from the context of online learning and controlof a nonlinear system with unknown dynamics. Such a framework, described by Ornik et al. [2017,2019], is motivated by the desire to successfully control a system with entirely unknown dynamicsby learning as much as possible about the dynamics “on the fly”, i.e., solely from the system’sbehavior during a single system run; it differs from classical work on adaptive or robust control[Ioannou and Sun, 1996, Dullerud and Paganini, 2000] by not assuming almost any knowledge aboutthe magnitude or the structure of uncertainty about the system dynamics, and from work on data-driven learning and control synthesis by not allowing collection of information on system dynamicsby way of repeated system runs [Brunton et al., 2016, Chen et al., 2018].Under some technical assumptions, the results of Ornik et al. [2017, 2019] show that, for anunknown control-affine system ˙ x = f ( x ) + G ( x ) u , the local dynamics u f ( x ) + G ( x ) u can beobtained for any state x lying on a system trajectory, with an arbitrarily small error and usingonly the knowledge of the trajectory prior to the time at which x is visited, as well as the boundson Lipschitz constants and magnitudes of functions f and G . Thus, along with the assumptionof perfect observations of system state at every time, the only “outside” knowledge required toestablish these local dynamics are the bounds on Lipschitz constants and magnitudes of functions f and G . These bounds are not required to be sharp, and may thus come from basic knowledge ofthe physical laws and the system’s environment.The work of Ornik et al. [2017, 2019] considers a predetermined control objective, i.e., a targetstate that should be reached. The proposed approach is to design a utility function which, whenmaximized, appears to lead the system in a desired direction, given the learned local system dy-namics. However, a control signal driving the system to a desired state may not exist and, evenif it does, it may be impossible to learn online. Consequently, this paper aims to take a differentapproach. While it may be impossible to know whether a system can be driven to a desired state,the learned dynamics, coupled with the bounds on the rate of change of the underlying vector fields,naturally yield the two following sets:(i) states that may be possible to reach using admissible control signals (optimistic reachabilityset) , 3ii) states that are guaranteed to be reachable using admissible control signals (guaranteed reach-ability set — GRS) .The optimistic reachability set and the GRS are clearly a superset and a subset, respec-tively, of the true reachable set. While computing over- and underapproximations of the reach-able set has been the subject of substantial previous research [Kurzhanski and Varaiya, 2000,Mitchell and Tomlin, 2003], including the setting of computing the optimistic reachability set forsystems with uncertain parameters [Althoff et al., 2008], prior work has not — to the best of ourknowledge — considered the framework of an unknown control system with knowledge of localdynamics at a single state.Without discussing the reachable sets, the work of Ornik et al. [2017, 2019] has implicitly con-centrated on optimistic reachability, i.e., attempting to reach the original objective while thereappears to be any chance of reaching it. The contribution of this paper is to consider the latteroption, that is, to provide a set of states that — using the knowledge of local dynamics at the sys-tem’s intital state and Lipschitz constants of the underlying vector fields — are certifiably reachable using an admissible control signal. While the motivation for our work stems from Ornik et al. [2017,2019], our contribution is entirely standalone: we do not concern ourselves with how local dynamicswere found, but assume that they were obtained, and seek to exploit them to find a reachable setof states.Motivated by considerations similar to the ones of this paper, the work of Ahmadi et al. [2017]follows the same broad idea of guaranteed reachability, deeming an unknown dynamical system“safe” if it is guaranteed that its trajectory will not enter a particular unsafe set. However, unlikeour paper, the dynamical system in that work has no control input. Hence, the reachable set issolely a single trajectory; the focus of Ahmadi et al. [2017] is thus on certifying the safety of adynamical system. Our results focus on finding a certifiably reachable set for a control system;applied to the narrative of Ahmadi et al. [2017], our paper provides a certificate of existence of acontrol input that renders the resulting dynamical system safe.The key idea of this paper is the following: if dynamics u f ( x ) + G ( x ) u are known at aninitial state x for the set of admissible controls U , then the set of system velocities V x availableat x is also known: V x = { f ( x ) + G ( x ) u | u ∈ U} = f ( x ) + G ( x ) U . While V x may not beknown for any other x , x
7→ V x is a Lipschitz continuous function under some appropriate metric,given the Lipschitz bounds on f and g . In other words, while we may not know V x , we can computethe bound on the difference between V x and V x dependent on the distance k x − x k and Lipschitzbounds on f and g . Consequently, we are able to determine the set of guaranteed velocities V G x ,i.e., the intersection of all sets that can possibly equal V x . Hence, regardless of the true systemdynamics — as long as they are consistent with our prior knowledge about the dynamics — theset of velocities V G x will be available at x . Such a set will satisfy V G x → V x as x → x , with someappropriate notion of convergence.As V G x ⊆ V x for all x , any trajectory that satisfies the differential inclusion ˙ x ∈ V G x for all x willbe a trajectory of ˙ x = f ( x )+ G ( x ) u produced by some admissible control signal; an analogous claimholds for all other dynamics ˙ x = ˆ f ( x ) + ˆ G ( x ) u consistent with the assumed knowledge. Hence, thereachable set of the differential inclusion ˙ x ∈ V G x , with x (0) = x , is a subset of the guaranteedreachablility set — the intersection of reachable sets for all control systems consistent with theassumed knowledge. As the reachable set of ˙ x ∈ V G x may be difficult to characterize analytically,we will also provide a method of its underapproximation by a reachability set of a simpler controlsystem, based on underapproximating each set V G x by a ball of maximal radius.4he remainder of this paper serves to describe, formalize, and quantify the notions outlined inthe above paragraphs. We begin by posing a formal problem statement. Throughout the paper, we consider a control system M ( f, G ) given by the following control-affinesystem dynamics: ˙ x ( t ) = f ( x ( t )) + G ( x ( t )) u ( t ), x (0) = x , (1)where t ≥ x ( t ) ∈ R n for all t , admissible control inputs u ( t ) lie in the set U ⊆ R m , and functions f : R n → R n and G : R n → R n × n are globally Lipschitz-continuous, i.e., there exist L f ≥ L G ≥ k f ( x ) − f ( y ) k ≤ L f k x − y k and k G ( x ) − G ( y ) k ≤ L G k x − y k for all x, y ∈ R n .While the framework of this paper, and many of its results, could be analogously applied to controlsystems on more general state spaces, we consider the Euclidean state space for ease of exposition.Without loss of generality, we assume x = 0; the system state can be shifted by − x to ensuresuch a property. For technical reasons we also make the following assumption. Assumption 1.
The system described by (1) is fully actuated at x = 0 , i.e., m = n and G (0) ∈ GL ( n ) . Set U satisfies U = B n (0; 1) . The assumption of full actuation makes estimation of available system velocities at points x around x = 0 significantly simpler, as the estimates will depend on k G (0) − k . We will brieflydiscuss the case of a non-invertible G (0) later in the paper: such an estimation can still be performedby using the reciprocal of the smallest non-zero singular value of matrix G (0) instead of k G (0) − k .The assumption of U = B n (0; 1) parallels an assumption of Ornik et al. [2017, 2019] that U =[ − , m ; in our motivating example, the inputs available to the aircraft are clearly bounded, andtaking U = B n (0; 1) renders technical work significantly easier.Building upon the assumptions and results of Ornik et al. [2017, 2019], we place the followinglimitations on our knowledge about the system dynamics: we assume that bounds L f and L G areknown, along with values f (0) and G (0). We assume that nothing else is known about functions f and G . Remark 1.
It can be trivially shown that knowing the values f (0) and G (0) is equivalent to knowingthe local system dynamics u f (0) + G (0) u . We will denote the set of all functions ( ˆ f , ˆ G ) consistent with the above knowledge by D con , bydefining D con = { ( ˆ f , ˆ G ) | ˆ f (0) = f (0), ˆ G (0) = G (0), L f is a Lipschitz bound for ˆ f , L G is a Lipschitz bound for ˆ G } .Our goal is to describe the set of states that are reachable from x = 0, regardless of the truedynamics of system (1), as long as they are consistent with the assumed knowledge. Computing5his set, or its underapproximation, serves to enable the planner to determine an objective that isguaranteed to be reachable regardless of the unknown system dynamics.We define the (forward) reachable set R ˆ f, ˆ G ( T ; x ) as the set of all states reachable by a system M ( ˆ f , ˆ G ) from x at time T using some control signal u : [0 , T ] → U . In other words, if φ ˆ f, ˆ Gu ( T ; x )denotes a controlled trajectory of system M ( ˆ f , ˆ G ) with control signal u , R ˆ f, ˆ G ( T, x ) = { φ ˆ f, ˆ Gu ( T ; x ) | u : [0 , T ] → U} .Throughout the paper, we assume that the existence and uniqueness of all trajectories φ u ( T ; x )are guaranteed. We can now state the following central problem of this paper. Problem 1.
Let T ≥ . Describe the guaranteed reachability set (GRS) defined by R G ( T,
0) = \ ( ˆ f, ˆ G ) ∈D con R ˆ f, ˆ G ( T, . (2)While we pose Problem 1 in terms of finding the GRS at time T , we could analogously askfor the set of guaranteed reachability before time T , i.e., ∪ t ∈ [0 ,T ] R G ( t, ∪ t ∈ [0 , + ∞ ) R G ( t, how to land at that airport. The interest of this paper is primarily in solving Problem 1.However, Section 5 also provides a method of determining, under technical assumptions, a controlinput to reach the desired state x end if x end is certified to be reachable. As such a control inputwill depend on system dynamics, the proposed method exploits previously mentioned work inOrnik et al. [2017, 2019] on real-time estimation of local system dynamics.Naturally, in order for Problem 1 to be meaningful, we wish to describe R G ( T,
0) as simply aspossible. The primary contribution of this paper is to provide an underapproximation of R G ( T, x = a + g ( k x k ) u , where g : [0 , + ∞ ) → [0 , + ∞ ) is a rampfunction. We begin with a discussion of a relationship of controlled system dynamics given by anordinary differential equation and its induced differential inclusion. Ordinary differential equations with control inputs have been interpreted as ordinary differentialinclusions in numerous previous works; see, e.g., Clarke [1983], Kurzhanski and Varaiya [2000],Bressan and Piccoli [2007]. Following the same approach, we define a differential inclusion˙ x ∈ V x = f ( x ) + G ( x ) U , x (0) = x . (3)where in the future we denote V x = f ( x ) + G ( x ) U . Clearly, any solution φ u ( · ; x ) to the differentialequation (1) also satisfies (3), while any trajectory that satisfies (3) at all times t by definitionsatisfies (1). Let us define the reachable set of a differential inclusion (3) as a set of all possiblevalues, at a given time, of trajectories that satisfy (3). Then, the reachable set of (3) is naturally R ( T, x ). 6iven our assumptions about the available knowledge of the system, set V x = V is assumedto be entirely known, but no other set V x is known. Our goal in this section is to provide anunderapproximation for the sets V x using solely the available knowledge of V , as well as Lipschitzconstants L f and L G .Set V x describes all the available velocities for a control system that satisfies (1), i.e., (3), at atime t when its state is φ ( t ; x ) = x . If f ( x ) and G ( x ) were known, computing such a set wouldbe simple. However, by our assumptions, we do not know their exact values, but do know that( f, G ) ∈ D con . Then, analogously to the GRS, we can define the guaranteed velocity set , i.e., the setof all velocities that will certainly be available at x , regardless of which element in D con representsthe true system dynamics. Such a set is given by V G x = \ ( ˆ f, ˆ G ) ∈D con ˆ f ( x ) + ˆ G ( x ) U ⊆ V x . (4)Let us first relate the reachable set of the differential inclusion˙ x ∈ V G x , x (0) = 0, (5)to R G ( T, Proposition 1.
Let T ≥ . If a function φ : [0 , + ∞ ) → R n satisfies (5) at all times t ≤ T , then φ ( T ) ∈ R G ( T, .Proof. If dφ ( t ) /dt ∈ V G φ ( t ) , then φ satisfies ˙ x ∈ ˆ f ( x ) + ˆ G ( x ) U for all ( ˆ f , ˆ G ) ∈ D con . Thus, by adiscussion analogous to that under (3), φ ( T ) ∈ R ˆ f, ˆ G ( T,
0) for all ( ˆ f , ˆ G ) ∈ D con . By (2), φ ( T ) ∈ R G ( T, V G x may be empty for certain x . Such a property does not disable us fromdiscussing the reachable set of (5); if a trajectory reaches a state x where V G x = ∅ , we will considerby convention that it ceases to exist at that time. Remark 2.
Proposition 1 states that the reachable set of (5) may be used as an underapproximationof R G ( T, . We do not claim that those sets are in general equal: the intersection of reachable setsof ˙ x ∈ F i ( x ) may not equal the reachable set of ˙ x ∈ ∩ i F i ( x ) . To illustrate this fact, consider F ( x ) = { k [1 2] T | k ∈ [0 , } and F ( x ) = { k [1 x ] T | k ∈ [0 , } with x (0) = (0 , . Then, ˙ x ∈ F ∩ F gives ˙ x = 0 at x (0) , i.e., the reachable set of ˙ x ∈ F ∩ F at any time is just { x (0) } .On the other hand, the reachable set of ˙ x ∈ F ( x ) at time T includes the state ( T, T + 1) , andthe reachable set of ˙ x ∈ F ( x ) at time T includes the state ( T, e T ) . Thus, at time T that satisfies e T = 2 T + 1 , the intersection of reachable sets of ˙ x ∈ F ( x ) and ˙ x ∈ F ( x ) contains at least onestate other than { x } .While this paper contains no theoretical discussion of equality between the reachable set of (5) and R G ( T, , the example in Section 6.1 shows that the two sets may in fact be equal — the abovecounterexample may not be applicable as R G ( T, is not an intersection of any general reachablesets, but of reachable sets R ˆ f, ˆ G ( T, , where ˆ f and ˆ G satisfy the additional structure ( ˆ f , ˆ G ) ∈ D con . Motivated by the result of Proposition 1, we seek to underapproximate R G ( T,
0) by consideringthe reachable set of (5). To do that, let us first discuss the geometric properties of V G x . For a given7 ∈ R n it is simple to show that { ( ˆ f ( x ) , ˆ G ( x )) | ( ˆ f , ˆ G ) ∈ D con } = B n ( f (0); L f k x k ) × B n ( G (0); L G k x k ). (6)Consequently, since U = B n (0; 1) and, for a matrix M , M B n (0; 1) is an ellipsoid [Barvinok, 2002], V G x is an intersection of a parametrized, infinite set of ellipsoids. Unfortunately, such an intersectionmay be difficult to express in a simple form; as noted by Kurzhanskiy and Varaiya [2006], anintersection of any number of ellipsoids is generally not an ellipsoid. In particular, Figure 1 showsthat, while easy characterization may be possible in special cases such as when G (0) = I , V G x cannotbe expected to be an ellipsoid.Figure 1: Guaranteed velocity set V G x , with x = (1 , L f = 0 . L G = 0 . f (0) = [0 0] T , and G (0) = diag(3 , V G x together provide ∪ ( ˆ f, ˆ G ) ∈D con ˆ f ( x ) + ˆ G ( x ) U , i.e., the optimistic set of velocities that are availablefor at least one control system M ( ˆ f , ˆ G ). These sets, and sets in most subsequent figures, werecomputed by Monte Carlo methods using the software of Shilon [2006].Our next step is thus to provide a simply computable underapproximation of V G x . Indeed, asset D con is a Cartesian product of balls and U is a ball, set V G x can be underapproximated in ageometrically appealing fashion. We derive such an approximation using the following results. Lemma 1.
Let a ∈ R n , B ∈ GL ( n ) , and U = B n (0; 1) . Then the following holds:(i) a + B U ⊇ B n ( a ; k B − k − ) , ii) B n ( a ; k B − k − ) is a ball of maximal radius contained in a + B U .Proof. For (i), it clearly suffices to prove that, for every vector δ , k δ k ≤ k B − k − , there exists u ∈ R n , k u k ≤
1, such that a + δ = a + Bu . (7)Since B is invertible, setting u = B − δ satisfies (7). Additionally, k u k = k B − δ k ≤ k δ kk B − k bythe definition of the matrix 2-norm. Since k δ k ≤ k B − k − , we have k u k ≤ a + B U using singular values of B . Inorder to not drift from our paper’s central narrative, we omit a detailed discussion of singularvalues and direct the reader to ˇSirca and Horvat [2012], Ben-Israel and Greville [2003]. Combiningthe results in the above works, we note that the length of shortest principal semi-axes of theellipsoid a + B U equals the smallest singular value of B , which equals exactly k B − k − . Thus,through a change of coordinates, we can assume without loss of generality that a + B U is given by { x | α x + α x + . . . + α n x n ≤ } , where 0 < α ≤ α ≤ · · · ≤ α n = k B − k .Consider now a ball of radius δ > k B − k − around any point x ′ = ( x ′ , . . . , x ′ n ) ∈ R n . Then,at least one of the following inequalities hold: x ′ n + δ > k B − k − or x ′ n − δ < −k B − k − . Hence, B n ( x ′ ; δ ) ∋ x ′ + δe n a + B U or B n ( x ′ ; δ ) ∋ x ′ − δe n a + B U .Lemma 1 provides the maximal underapproximation of V x by a ball B n ( f ( x ); k G ( x ) − k − ),assuming that G ( x ) is invertible; Figure 2 provides an illustration. Hence, by (4) and (6), V G x canbe underapproximated by \ ˆ a ∈ B n ( f (0); L f k x k )ˆ B ∈ B n × n ( G (0); L G k x k ) B n (ˆ a ; k ˆ B − k − ). (8)The next two results build on Lemma 1 to provide a simpler approximation of V G x . Lemma 2.
Let a ∈ R n , B ∈ GL ( n ) , ≤ r < k B − k − , and U = B n (0; 1) . Define ε =min ˆ B ∈ B n × n ( B ; r ) k ˆ B − k − . Then \ ˆ B ∈ B n × n ( B ; r ) a + ˆ B U ⊇ B n ( a ; ε ) .Proof. Assume first that all matrices ˆ B ∈ B n × n ( B ; r ) are invertible. For all such matrices ˆ B ,the inequality k ˆ B − k − ≥ min ˆ B ∈ B n × n ( B ; r ) k ˆ B − k − = ε obviously holds, so the claim holds byLemma 1.It thus remains to note that all matrices ˆ B ∈ B n × n ( B ; r ) are indeed invertible. Such a propertyfollows directly from the Eckart-Young-Mirski theorem; we again omit the details and point thereader to Ben-Israel and Greville [2003] and Bj¨orck [2014] for a longer discussion.Lemma 2 dealt with the intersection of a + ˆ B U for different ˆ B , while vector a was fixed; Figure 3provides an illustration. Let us now consider a varying vector a . Lemma 3.
Let a ∈ R n , r ≥ , and R ≥ r . Then \ ˆ a ∈ B n ( a ; r ) B n (ˆ a ; R ) = B n ( a ; R − r ) . Figure 2: Set f (0) + G (0) U , bounded in blue, and its underapproximation B n ( f (0); k G (0) − k − ),drawn in red, for f (0) = [0 0] T and G (0) = diag(3 , Proof.
Let v ∈ B n ( a ; R − r ). Then, k v − a k ≤ R − r . Thus, for every ˆ a ∈ B n ( a ; r ), k v − ˆ a k ≤k v − a k + k a − ˆ a k ≤ R − r + r ≤ R . Hence, v ∈ B n (ˆ a ; R ).Now, let v B n ( a ; R − r ). Hence, k v − a k > R − r . Now, choose ˆ a = a + r ( a − v ) / k a − v k . Clearly,ˆ a ∈ B n ( a ; r ). On the other hand, k v − ˆ a k = k v − a − r ( a − v ) / k a − v kk = k ( a − v ) k (1 + r/ k a − v k ) = r + k a − v k > R . Thus, v B n (ˆ a ; R ).Finally, we can combine the results of the above lemmas to obtain an approximation for V G x . Theorem 1.
Let U , L f , L G , f (0) , and G (0) be as above. Let x ∈ R n satisfy the inequality ( L f + L G ) k x k ≤ k G (0) − k − . Define V G x = B n (cid:0) f (0); k G (0) − k − − L f k x k − L G k x k (cid:1) .Then, V G x ⊆ V G x .Proof. As noted in (8), V G x = \ ˆ a ∈ B n ( f (0); L f k x k ) \ ˆ B ∈ B n × n ( G (0); L G k x k ) ˆ a + ˆ B U .10igure 3: Intersection of sets f (0) + ˆ B U , ˆ B ∈ B n × n ( G (0); L G k x k ), with x = (1 , L G = 0 . f (0) = [0 0] T , and G (0) = diag(3 , B n ( f (0); min ˆ B ∈ B n × n ( G (0); L G k x k ) k ˆ B − k − ) = B n (0; 0 .
7) is drawn in red.Then, by Lemma 2, V G x ⊇ \ ˆ a ∈ B n ( f (0); L f k x k ) B n ˆ a ; min ˆ B ∈ B n × n ( G (0); L G k x k ) k ˆ B − k − ! .By Weyl’s inequality for singular values [Stewart, 1990] and a characterization of k ˆ B − k − as an appropriate singular value of matrix ˆ B [Ben-Israel and Greville, 2003], k G (0) − k − − r ≤ min ˆ B ∈ B n × n ( G (0); r ) k ˆ B − k − for any r ≥
0. Hence, V G x ⊇ \ ˆ a ∈ B n ( f (0); L f k x k ) B n (ˆ a ; k G (0) − k − − L G k x k ).The claim of the theorem now holds by Lemma 3, since we assume L f k x k ≤ k G (0) − k − − L G k x k .Theorem 1 is the central result of the current section. Assuming L f + L G >
0, for each x in a ball around x = 0 of radius 1 / (( L f + L G ) k G (0) − k ) it generates a nonempty set V G x ofguaranteed velocities, i.e., velocities v for which there certainly exist a control input u ∈ U suchthat f ( x ) + G ( x ) u = v . The case of L f + L G = 0 is uninteresting as it results in dynamics (1)11eing entirely known from f (0) and G (0), and the true reachable set can thus be computed usingstandard methods described at the end of subsequent section.Theorem 1 is illustrated by Figure 4, with the underapproximation V G x added to the illustrationof V G x from Figure 1. While one might expect that consecutive underapproximations in the threelemmas preceding Theorem 1 would result in an exceedingly bad approximation V G x , the followingproposition shows that V G x is indeed the best approximation of V G x by a ball.Figure 4: Set V G x , with x = (1 , L G = 0 . f (0) = [0 0] T , and G (0) = diag(3 , V G x = B n (0; 0 .
6) is drawn in red.
Proposition 2.
Let V G x and V G x be as above. Then, V G x is a ball of maximal radius that is containedin V G x .Proof. The proof follows the same steps as the proof of part (ii) of Lemma 1, which proves theproposition in the special case of L f = 0 and L G = 0. As the detailed proof is technically inelegant,if straightforward, we provide its outline.As in part (ii) of Lemma 1, we may assume without loss of generality that f (0) + G (0) U is anellipsoid with principal semi-axes parallel to coordinate axes e i and its shortest principal semi-axesgiven by ±k G (0) − k − e n . Now consider two ellipsoids centered around L f k x k e n and − L f k x k e n ,respectively, such that their principal semi-axes are parallel with the coordinate axes and such theirshortest principal semi-axes are given by ( k G (0) − k − − L G k x k ) e n . It can be shown that theseellipsoids can be written as ˆ a + ˆ B U , where ˆ a = ± L f k x k e n and ˆ B ∈ B n × n ( G (0); L G k x k ). By (6),12here thus exist two control systems M ( ˆ f , ˆ G ) consistent with assumed knowledge such that theconstructed ellipsoids are their available velocity sets at x .For any x ′ ∈ R n and any δ > k G (0) − k − − L G k x k − L f k x k , set B n ( x ′ ; δ ) will contain at leastone point y = ( y , . . . , y n ) with | y n | ≥ δ . On the other hand, any such point is in at most one of thetwo above ellipsoids ˆ a + ˆ B U , and is thus not in their intersection, and consequently not in V G x .While V G x is the best approximation of V G x by a ball, such an underapproximation still potentiallydiscards a large part of the intersection. In the case x = 0, the quality of the approximation, i.e.,the volume of the set being discarded, depends on the magnitude of k G (0) − k − (i.e., the length ofshortest principal semi-axes of f (0) + G (0) U ) relative to the lengths of other principal semi-axes of V = V G . By considering solely the ratio of the length of the longest principal semi-axes and theshortest principal semi-axes, we obtain that the volume being discarded roughly depends on the condition number of matrix G (0) [Ben-Israel and Greville, 2003]. Such a number, representing theratio of the largest and smallest singular value of a matrix, is commonly used to determine “distance”from matrix singularity. In other words, the closer G (0) is to being singular, i.e., the closer thecontrol system is to being underactuated at x = 0, the worse the provided underapproximationwill be. Remark 3.
If system M ( f, G ) satisfied rank( G (0)) = k < n , i.e., Assumption 1 did not hold, allour work could be performed by considering the smallest non-zero singular value σ k of G (0) insteadof k G (0) − k − ; the set f (0)+ G (0) U would be a k -dimensional ellipsoid with shortest principal semi-axes of length σ k , and we would obtain a k -dimensional underapproximation for V G x . In that case,the quality of the underapproximation would depend on the ratio between the largest and smallest nonzero singular value of matrix G (0) . In the remainder of the text, we assume that L f + L G > L f = L G = 0 results in the system dynamics being known immediately from ourassumed knowledge. Its reachable set can thus be computed using the methods described in thesubsequent section.We now proceed to exploit the geometrically simple form of V G x to obtain a control system withknown dynamics whose reachable set is an underapproximation of the GRS R G ( T, Using the result of Theorem 1, we can underapproximate the set of trajectories to the differentialinclusion (3), i.e., the controlled dynamics (1), by the trajectories of the differential inclusion˙ x ∈ V G x , x (0) = x . (9)Set V G x is only defined for x such that k G (0) − k − − L G k x k − L f k x k ≥
0, and we only care aboutthe reachability of (9) within the ball B know = B n (0; 1 / (( L f + L G ) k G (0) − k )). The interpretationof set B know is that any state x B know is so far away from x = 0 that the local dynamics at x do not provide any information about local dynamics at x .In order to exploit previous work on computing reachable sets, we continuously extend differ-ential inclusion (9) to the entire R n by defining V G x = { f (0) } outside of the ball B n (0; 1 / (( L f +13 G ) k G (0) − k )). We emphasize that such an extension is purely technical, i.e., f (0) is not a guar-anteed velocity at states x B know .Let R ( T, x ) be the reachable set of (9) at time T , i.e., the set of all states achieved by trajectories φ : [0 , + ∞ ) → R n that satisfy φ (0) = x and (9) for all t ≤ T . It can be shown that any solution of(9) connecting two points in B know needs to entirely lie within B know . Thus, underapproximation R ( T, x ) of the guaranteed reachable set R G ( T, x ) is given by R ( T, x ) = R ( T, x ) ∩ B know ⊆ R G ( T, x ). (10)The question that remains to be considered is, thus, determining the set R ( T, x ). The prob-lem of computing reachable sets of differential inclusions is generally difficult [Asarin et al., 2001,Quincampoix and Veliov, 2002]. However, we will show that the fact that each set V G x is a ballallows us to describe the set R ( T, x ) as a solution to a partial differential equation given in closedform.Going back to inclusion (9), we note that — analogously to our conversion of (1) into inclusion(3) — inclusion (9) can be interpreted as, now entirely known , ordinary differential equation˙ x = a + g ( k x k ) u , x (0) = 0, (11)with a = f (0), u ∈ U = B n (0; 1), and where g : [0 , + ∞ ) → R is a ramp function given by g ( s ) = k G (0) − k − − ( L G + L f ) s if s ≤ k G (0) − k − / ( L G + L f ) and g ( s ) = 0 otherwise.The set R ( T, x ) is now the reachable set of the system with dynamics (11), in the sensediscussed in Section 3. As mentioned in the preliminary sections, there exist multiple approaches toapproximating or computing reachable sets for nonlinear control systems [Chen, 2017, Vinter, 1980,Scott and Barton, 2013, Baier et al., 2013]; we follow the Hamilton-Jacobi approach described byChen [2017] and references therein.The approach in Chen [2017] relies on defining any sufficiently smooth function l : R n → R with { } = { x | l ( x ) ≤ } . Applied to our setting and switching between backward reachablesets described by Chen [2017] and forward reachability that is of interest to us, the reachable set R ( T, x ) is then given by R ( T, x ) = { x ∈ R n | V ( T, x ) ≤ } .Function V ( t, x ) is the viscosity solution [Crandall and Lions, 1983] of the Hamilton-Jacobi partialdifferential equation V t ( t, x ) + H ( x, V x ( t, x )) = 0 for all t ∈ [ − T, , x ∈ R n , V (0 , x ) = l ( x ) for all x ∈ R n , (12)where H ( x, λ ) = min u ∈U λ T ( − a + g ( k x k ) u ). Due to the simplicity of the dynamics (11), theHamiltonian H ( x, λ ) can be written in closed form. The following theorem presents such a result. Theorem 2.
Let l : R n → R with { x | l ( x ) ≤ } = { } and T > . Then, R ( T, x ) = { x ∈ B know | V ( T, x ) ≤ } ,where V is the viscosity solution of V t ( t, x ) − V x ( t, x ) T a − k V x ( t, x ) k g ( k x k ) = 0 for all t ∈ [ − T, , x ∈ R n , V (0 , x ) = l ( x ) for all x ∈ R n . (13)14 roof. The result follows simply from (10) and (12), noticing that H ( x, λ ) = min u ∈U λ T ( − a + g ( k x k ) u ) = − λ T a + g ( k x k ) min u ∈U λ T u = − λ T a − g ( k x k ) λ T λ/ k λ k = − λ T a − g ( k x k ) k λ k .Solving (13) thus yields the desired underapproximation R ( T, x ) of the set R G ( T, x ). We re-mark that the above method for computing R ( T, x ) is not the only possible one; for instance, as x G x easily satisfies the one-sided Lipschitz and upper semicontinuity conditions of Donchev et al.[2003], it is possible to use the exponential formula derived therein to obtain an expression for R ( T, x ) as a limit of sets, with a guaranteed order of convergence.Having found R ( T, x ), let us briefly return to the question of determining a control signal thatdrives the system satisfying the original dynamics (1) from x to an element of this set, first raisedin Section 3. Such a control signal clearly depends on system dynamics. However, the results ofChen [2017] provide an expression for a control signal u on (11) that drives the system from x to the desired objective. By taking v = a + g ( k x k ) u , such a signal yields a function of desiredvelocity vectors v ( t ) that will drive the system to the desired objective. As discussed in the proofof Lemma 2, G ( x ) is invertible for all x ∈ B know . Thus, if f ( x ( t )) and G ( x ( t )) were known, theappropriate control input u ( r ) for (1) would be given by u ( t ) = ( G ( x ( t ))) − ( v ( t ) − f ( x ( t ))) .Combining v with the algorithm of Ornik et al. [2017, 2019] — technical assumptions notwithstand-ing — to determine f ( x ( t )) and G ( x ( t )) at time t with arbitrarily small error, we can thus obtaina control input that takes the system from x to any desired state in R ( T, x ).We now proceed to illustrate the results obtained in this section and preceding sections by wayof numerical examples. We investigate two scenarios. In the first one, computing the GRS of a 1D system will validate ourtheory and show that, while our current theoretical results do not establish equality between R ( T, R G ( T, R G ( T,
0) and the reachable set of inclusion ˙ x ∈ V G x , is a meaningfuldirection for future work.The second example models the motivational scenario of this paper’s narrative. We will modelan aircraft with deteriorating actuation capabilities and use our theory — slightly modified to allowfor time-varying dynamics — to determine the set of states that it is guaranteed to safely reach. We will first illustrate the developed results on a simple single-dimensional system motivated by asystems biology narrative. The system state is given by x ∈ R , and the input is given by u ∈ [ − , f (0) = 0, G (0) = 1 / L f = 0 .
1, and L G = 0 .
7. The true — but unknown — dynamics of the system are given by˙ x ( t ) = ( x ( t ) + 1) x ( t ) + 1) u , x (0) = 0. (14)15ynamics (14) are a slight modification of dynamics describing the phenomenon of gene au-toregulation — the ability of a gene product to determine its own rate of production by binding tothose elements of the gene that regulate its production [Alon, 2006, Jablonka et al., 1992]. However,in the context of this paper, our interest in equation (14) is purely mathematical.Based on the above available knowledge of the system dynamics, we seek to determine theunderapproximation R ( T,
0) from (10), and compare it to the GRS R G ( T, R ( T, L f = 0 . L G = 0 . f equals 0 (i.e., f ( x ) = 0 for all x ). Consequently,the guaranteed reachable set is smaller than the one that would have been obtained with tighterbounds L f and L G .In order to determine R ( T, f (0), G (0), L f and L G , we obtain V G x = B (0; 0 . − . k x k ) = [ − . . | x | , . − . | x | ] for | x | ≤ /
8. By (10) and (11), the set R ( T,
0) is thus givenby intersecting [ − / , /
8] with the reachable set of˙ x = (0 . − . | x | ) u , x (0) = 0 , (15)with u ∈ [ − , u ; it can consequently be easily shownthat the reachable set R ( T,
0) of (15) equals R ( T,
0) = (cid:20)
58 ( e − . T − ,
58 (1 − e − . T )] (cid:21) .Hence, R ( T, ⊆ [ − / , /
8] and thus R ( T,
0) = R ( T, R ( T, R G ( T, ⊇ R ( T, R G ( T, ⊆ R ( T, R G ( T,
0) = R ( T, x = 0 . | x | + (0 . − . | x | ) u , x (0) = 0. These dynamics are consistentwith the available knowledge about the system, i.e., if ˆ f ( x ) = 0 . | x | and ˆ G ( x ) = 0 . − . | x | , then( ˆ f , ˆ G ) ∈ D con . By considering the dynamics when x ≤
0, the reachable set R ˆ f, ˆ G ( T,
0) can be shownto equal [ ( e − . T − , α T ] for some α T ≥ x = − . | x | + (0 . − . | x | ) u analogously. By looking at the case of x ≥
0, we obtain that the reachable set of a system with these dynamics equals [ β T , (1 − e − . T )]for some β T ≤ R G ( T, ⊆ R ( T, R G ( T,
0) = R ( T, R ( T,
0) — or, more likely, the reachable set of inclusion (5) — may indeed equalthe GRS.Finally, we note that the true reachable set R ( T,
0) of dynamics (14) equals R ( T,
0) = " √ T + 4 − T − , √ T + 4 + T − .16hile such a set is impossible to find only from the available knowledge of f (0), G (0), L f , and L g , it is indeed not difficult — if algebraically onerous — to verify that R G ( T, ⊆ R ( T,
0) forall T ≥
0. Additionally, the relative difference between R G ( T,
0) and R ( T,
0) vanishes as T → l ( I ) represents the length of interval I , thenlim T → l ( R G ( T, l ( R ( T, T → − e − . T ) / T = 1.This feature is in line with our intuition that, the closer the system state is to x = 0, the betterwe can predict its dynamics. We now proceed to a slightly more involved example. In this section, we will consider a rudimentary model of an aircraft exposed to unexpected actuatordeterioration and environmental effects on dynamics (e.g., wind), and determine a set of states(“diversion airports”) that are guaranteed to be reachable by the aircraft.We assume that the aircraft dynamics are known to satisfy the following equation:˙ x = f ( t ) + G ( t ) u , x (0) = 0, (16)where x ∈ R and u ∈ U = B (0; 1). Function f models the time-dependent drift caused by theenvironment, while G models the actuator deterioration. We will use G ( t ) = 1 − t/
10 for t ≤ G ( t ) = 0 for t >
10. In other words, the vehicle’s actuators deteriorate linearly until theyentirely stop working at time t = 10. We choose f ( t ) = [0 . t ) + 1) 0] T to model the effect ofa non-zero drift on the GRS — the chosen f ( t ) represents the wind flowing in a constant directionwith varying strength. We emphasize that those functions, apart from the values of f (0) and G (0),are considered unknown when determining the GRS.While it is clear that model (16) does not accurately represent the dynamics of any existingaircraft and the described environmental dynamics are simplistic, versions of the underlying singleintegrator model ˙ x = u have been extensively used in planning, e.g., by Bertsimas and van Ryzin[1991], Oh and Ahn [2014], Nikou et al. [2016].We note that dynamics (16) do not formally fall within the class of dynamics given by (1);however, we can bring them into that class by the standard method of appending a variable t to thestate vector x = ( x , x ) ∈ R as a state x with ˙ x = 1, x (0) = 0. Such a method has been used insimilar circumstances, e.g., by Ornik et al. [2017]. While such modified dynamics still slightly differfrom the assumptions of this paper — namely, the system has three states, but only two inputs,functions f and G are known to depend on only one state, and the dynamics of one of the states arefully known — the theory of previous sections can be directly adapted to the new circumstances.We can thus proceed with dynamics (16). Our interest is in determining the set of all states thatcan eventually be reached, i.e., the guaranteed eventual reachability set ∪ T ≥ R G ( T, f (0) = [0 . T , G (0) = 1, L f = 0 .
1, and L G = 1 /
5. Asin the previous example, such knowledge is pessimistic: while the bound on the rate of change ofdrift is correct, the maximal deterioration rate of the actuators is assumed to be twice as large asthe true rate.By adapting Theorem 1 to the framework of (16), we obtain a subset of the guaranteed velocityset at time t equaling V G t = B ((0 . , − . t ), with t ≤ /
3. At time t ≥ / V G t = B ((0 . , − . t ) we obtain differentialequation ˙ x = a + (1 − . t ) u , (17)where a = [0 . T with a reachable set denoted by R ( T, R (0) defined by R (0) = [ T ∈ [0 , / R ( T, ⊆ [ T ∈ [0 , / R G ( T, z ( t ) = x − at , equation (17) becomes ˙ z = (1 − . t ) u . Itsreachable set at time T can be easily computed to equal B (0; T − T / R ( T,
0) = B ( aT ; T − T / R (0) of the set of guaranteed eventual reachability is comparedwith true reachable sets ∪ T ∈ [0 , / R ( T, ∪ T ∈ [0 , R ( T, ∪ T ≥ R ( T,
0) in Figure 5; the lastthree sets can be computed from (16) by making the substitution z = x − [0 . t + sin( t )) 0] T andintegrating both sides of the resulting equation.Figure 5: Set R (0) = ∪ T ∈ [0 , / R ( T,
0) is drawn in red. The sets of true reachable states, ∪ T ∈ [0 , / R ( T, ∪ T ∈ [0 , R ( T, ∪ T ≥ R ( T,
0) are drawn in increasingly light shades of blue.We note that set ∪ T ≥ R ( T,
0) is unbounded as the vehicle can continue “gliding” even after losingall actuation capabilities at time t = 10.As shown in Figure 5, our results indeed provide an underapproximation of the set of states thatthe aircraft can reach. While the true reachable set is larger than R (0), such a property naturallyfollows from the lack of available accurate knowledge of true dynamics; only the dynamics at time18 = 0 are known, and the rate of change of dynamics is assumed to be larger than the true rate ofchange. Motivated by the problem of choosing an appropriate desired state for a control system with un-known dynamics, this paper establishes preliminary results on estimating the GRS — a set of statesthat are guaranteed to be reachable by such a system, given our current knowledge of the system.The mathematical framework of the paper follows the work of Ornik et al. [2017, 2019]; our paperexploits information on local system dynamics that can be collected using an algorithm proposedtherein to obtain an underapproximation of the GRS.The results of this paper represent an initial effort in estimating the GRS. Notably, while weshow that the produced set R ( T, x ) is indeed its subset, we provide only a preliminary discussionof the quality of such an underapproximation.The difference between the GRS and R ( T, x ) depends on the quality of two intermediate ap-proximations: (i) the approximation of the GRS by the reachable set of the differential inclusion˙ x ∈ V G x , where V G x are the velocities guaranteed to be available at state x , and (ii) the approxima-tion of V G x by a ball V G x . In (i), the current paper only shows that the latter set is a subset of theformer. However, the numerical results of Section 6.1, along with the intuition described in Section4, strongly imply that the two sets may be equal, potentially under some additional conditions.Proving such a claim would significantly improve the theoretical underpinning of the estimationmethod provided by this paper.In (ii), the current paper shows that V G x is a ball with maximal radius of all the balls containedin V G x . Nonetheless, as discussed at the end of Section 4, V G x may have a significantly smaller volumethan V G x , thus resulting in an underapproximation of the GRS of significantly lower volume thanthe true set. Approximating V G x by a geometrical object that more closely fits the complex shape of V G x would yield a better approximation of the GRS. A possible candidate, potentially simple enoughto enable computational work, is a general ellipsoid considered for approximation of related sets byKurzhanskiy and Varaiya [2006].Finally, let us move from the problem of computing the GRS, given the assumed system knowl-edge, to the problem of obtaining a GRS that is closer to the true reachable set of a control system.In that vein, a possible area for theoretical improvement of the results in the paper is in consideringa different class of available prior knowledge of system dynamics. While we currently consider andexploit only local dynamics at a single point, the algorithm proposed in Ornik et al. [2017, 2019]can produce local dynamics at arbitrarily many points on a single trajectory. Possibly combinedwith additional prior knowledge about system dynamics (e.g., bounds on higher partial derivativesof f and G ), exploiting such information will result in larger sets of guaranteed velocities comparedto current work, and thus in guaranteed reachability sets that are closer underapproximations ofthe true reachable sets. The primary obstacle to successfully improving the estimates using suchadditional information, even if it may be naturally available, is computational. Namely, with alarge number of constraints describing all the information available about the system dynamics, set { ( ˆ f ( x ) , ˆ G ( x )) | ( ˆ f , ˆ G ) ∈ D con } , crucial in approximating V G x , may be geometrically more complexthan the one described in this paper. 19 cknowledgments This work was supported by an Early Stage Innovations grant from NASA’s Space TechnologyResearch Grants Program, grant no. 80NSSC19K0209. The author wishes to thank Ufuk Topcuand Franck Djeumou for discussions on related topics.
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