A Fixed-Time Stable Adaptation Law for Safety-Critical Control under Parametric Uncertainty
aa r X i v : . [ m a t h . O C ] N ov A Fixed-Time Stable Adaptation Law for Safety-Critical Control underParametric Uncertainty
Mitchell Black Ehsan Arabi Dimitra Panagou
Abstract — We present a novel technique for solving the prob-lem of safe control for a general class of nonlinear, control-affinesystems subject to parametric model uncertainty. InvokingLyapunov analysis and the notion of fixed-time stability (FxTS),we introduce a parameter adaptation law which guaranteesconvergence of the estimates of unknown parameters in thesystem dynamics to their true values within a fixed-timeindependent of the initial parameter estimation error. We thensynthesize the adaptation law with a robust, adaptive controlbarrier function (RaCBF) based quadratic program to computesafe control inputs despite the considered model uncertainty. Tocorroborate our results, we undertake a comparative case studyon the efficacy of this result versus other recent approachesin the literature to safe control under uncertainty, and closeby highlighting the value of our method in the context of anautomobile overtake scenario.
I. I
NTRODUCTION
Safe control design in our increasingly technology-reliantsociety is important, yet difficult even under idealized con-ditions. In theory, it may be performed by a variety oftechniques, including control barrier function (CBF)-basedstrategies [1], [2]. As a set-theoretic method for enforcingthat the states of a system remain safe, i.e. within a specifiedset, CBFs have proven to be effective both in correcting somepotentially unsafe control action [3], [4] and as a constraintin optimization-based control [5], [6]. In fact, control designusing quadratic programs (QP) as a means to synthesizecontrol Lyapunov function (CLF)-based performance spec-ifications, which can guarantee convergence of the closed-loop trajectories to some goal set, and CBF-based safety hasbecome a staple in the literature [7], [8].While CLFs encode a goal-reaching performance spec-ification, the rate of convergence depends on the form ofthe condition imposed on its time-derivative. For example,while an exponentially stable CLF [9] guarantees that systemtrajectories reach a goal set as time tends toward infinity,the principle of fixed-time stability (FxTS) is used to definea fixed-time stable CLF [6] which drives the closed-looptrajectories to the goal within a fixed-time independent ofinitial conditions.Robust approaches to safety [10], which often shrink thesafe set in order to preserve set-invariance in the presenceof worst-case disturbances, tend to be overly conservative.Typically, conservatism degrades the ability of the controllerto meet performance objectives. In contrast, adaptive safety
The authors would like to acknowledge the support of the NationalScience Foundation award number 1931982.The authors are with the Department of Aerospace Engineering, Uni-versity of Michigan, Ann Arbor, MI, USA; { mblackjr, earabi,dpanagou } @umich.edu . techniques [11] are prone to chattering control solutions,especially as the state approaches the boundary of the safeset. While a highly oscillatory control signal may suffice insimulation, it can destabilize dynamical systems in practice.In an approach which combines these two regimes, however,the authors of [12] demonstrate that robust, adaptive controlbarrier functions (RaCBF) can mitigate these shortcom-ings. In addition, they utilize set-membership identification(SMID) to reduce the uncertainty surrounding a set ofunknown parameters in the system dynamics, which resultsin improved performance. In a learning-based approach, [13]seeks to improve the performance of a Segway controllerby learning unstructured uncertainty in the dynamics of thecontrol barrier function itself. And while many of thesetechniques have been demonstrably effective in improvingcontroller performance, the issue of navigating the domainnear the boundary of the safe region remains problematicwithout formal guarantees of deciphering model uncertainty.In considering some structured, parametric uncertainty fora class of nonlinear, control-affine systems, we introducea novel parameter adaptation law which provides such aguarantee to drive the parameter estimates to their truevalues within fixed-time, independent of the initial estimates.Using Lyapunov analysis, we leverage this result to define anupper bound on the parameter estimation error as an explicitfunction of time. We then introduce a new condition onthe time derivative of an RaCBF that guarantees forward-invariance of a shrunken safe set, which approaches the nominal safe set within the fixed-time horizon. In a casestudy on the ability of our proposed controller to toleratemarginally safe regions of the state space versus a selectionfrom the literature, we compare the works of [5], [11], [12]and [14] on a simple problem with static obstacles. Finally,we use an automobile overtaking scenario to highlight an-other advantage of the proposed method: with the uncertainparameters in the system dynamics guaranteed to be knownwithin fixed-time, our controller can accomplish a maneuvereven under high levels of uncertainty.The paper is organized as follows. Section II reviews setinvariance via CBFs, finite-time stability (FTS), FxTS, andan FTS parameter estimation scheme from the literature. InSection III we formalize the problem at hand. Section IVcontains our novel FxTS adaptation law and its implicationson safe control under uncertainty. Section V highlights asimple case study used to compare other recent work to ourproposed method, and its application on a highway overtakeexample. We conclude with a summary of contributions anddirections for future work in Section VI.I. M ATHEMATICAL P RELIMINARIES
In the rest of the paper, R denotes the set of real numbers.The ones vector of size n × m is denoted n × m . We use k · k to denote the Euclidean norm and k · k ∞ to denotethe L ∞ norm. As convention, we denote the minimumand maximum eigenvalue of a matrix M as λ min ( M ) and λ max ( M ) respectively. We write ∂S for the boundary of theclosed set S , and int ( S ) for its interior. The Lie derivativeof a function V : R n → R along a vector field f : R n → R n at a point x ∈ R n is denoted as L f V ( x ) , ∂V∂x f ( x ) .We now review the preliminaries, including a finite-timeparameter adaptation law from the literature which inspiredour fixed-time adaptation law. A. Set Invariance
In this paper, we consider the following class of nonlinear,control-affine systems subject to parametric uncertainty: ˙ x ( t ) = f ( x ( t )) + g ( x ( t )) u ( t ) + ∆( x ( t )) θ, x ( t ) = x , (1) where x ( t ) ∈ R n denotes the state, u ( t ) ∈ U ⊂ R m thecontrol input, and θ ∈ Θ ⊂ R p some constant, bounded,unknown parameters. We assume that f : R n → R n and g : R n → R n × m are locally Lipschitz, that ∆ : R n → R n × p is a known regressor matrix, and that f , g , and ∆ arebounded for bounded inputs. The regressor ∆ may capture,for example, the effects of sensing or modelling errors in thesystem dynamics, whereas the θ vector parameterizes sucherrors. We also define ˆ θ as the estimated parameter vector,and ˜ θ = θ − ˆ θ as the parameter estimation error vector. Assuch, the parameter error dynamics are described by ˙˜ θ = ˙ θ − ˙ˆ θ = − ˙ˆ θ, ˜ θ (0) = ˜ θ . (2)Let us also define a set of safe states as S = { x ∈ R n : h ( x ) ≥ } (3a) ∂S = { x ∈ R n : h ( x ) = 0 } (3b)int ( S ) = { x ∈ R n : h ( x ) > } (3c)where h : R n → R is continuously differentiable. Then,the following Lemma, known as Nagumo’s Theorem [15],provides a necessary and sufficient condition on the forwardinvariance of the set (3) under the system dynamics (1). Lemma 1.
Let a unique closed-loop solution to (1) exist inforward time. We say that the set S can be rendered forward-invariant if and only if there exists a control u ∈ U such that sup u ∈ U { L f h ( x ) + L g h ( x ) u + L ∆ h ( x ) θ } ≥ , ∀ x ∈ ∂S. (4) Remark 1.
The closed-loop system (1) admits a uniquesolution if u ( t ) ∈ U is Lipschitz. The authors of [12] ensure S is safe with respect to theuncertain dynamics (1) by enforcing that a shrunken set, S r ,is safe. Before formally defining S r we make the followingassumption. Assumption 1.
The set Θ to which the unknown parameters θ belong is known, compact, and convex. Remark 2.
Assumption 1 imples that we can also restrict ˆ θ ∈ Θ . Thus, we can define an upper bound ϑ :=sup θ ,θ ∈ Θ ( k θ − θ k ∞ ) on the norm k ˜ θ k ∞ of the parameterestimation error ˜ θ , so that k ˜ θ k ∞ ≤ ϑ. We let ϑ = ϑ · p × , and are now ready to present thedefinition for the shrunken set, S r , as defined in [12]. S r = (cid:26) x ∈ R n : h ( x ) ≥ ϑ T Γ − ϑ (cid:27) (5a) ∂S r = (cid:26) x ∈ R n : h ( x ) = 12 ϑ T Γ − ϑ (cid:27) (5b)int ( S r ) = (cid:26) x ∈ R n : h ( x ) > ϑ T Γ − ϑ (cid:27) (5c)where Γ is a constant, positive-definite matrix such that h ( x (0)) ≥ ϑ T Γ − ϑ . A new CBF may be defined as h r ( x, ϑ ) = h ( x ) − ϑ T Γ − ϑ , (6)and the sufficient condition that renders S r safe is ˙ h r = ∂h∂x ˙ x − ϑ T Γ − ˙ ϑ ≥ − α ( h r ) , (7)where α : [0 , a ) → [0 , ∞ ) is some class K function, i.e.,strictly increasing with α (0) = 0 . Remark 3. If S r ⊂ S in (5) is safe, then S in (3) is safe.Thus, satisfaction of (7) implies that (4) holds.B. Finite- and Fixed-Time Stability We now address performance criteria: finite- and fixed-time stability of an equilibrium point of the dynamical system ˙ x ( t ) = f ( x ( t )) , x ( t ) = x , (8)for which it is assumed that a unique solution exists, where x ∈ R n , f : R n → R n is continuous, and f (0) = 0 . Definition 1 ([16]) . The origin of (8) is globally finite-timestable (FTS) if the following conditions hold: • The origin of (8) is stable in the sense of Lyapunov • Any solution x ( t, x ) of (8) reaches the origin in finitesettling-time, T ( x ) , i.e. x ( t, x ) = 0 , ∀ t ≥ T ( x ) . Definition 2 ([17]) . The origin of (8) is fixed-time stable(FxTS) if it is globally FTS and any solution x ( t, x ) of (8) reaches the origin in finite settling-time, T , independent of x , i.e. x ( t, x ) = 0 , ∀ t ≥ T . We refer the interested reader to [16] and [17] respectivelyfor Lyapunov conditions which guarantee FTS and FxTS ofthe origin of (8).
C. Finite-Time Parameter Estimation
The parameter estimation scheme in the discussion tofollow is predicated on Assumption 2.
Assumption 2.
The state, x , and control input u , of (1) arebounded, and x is accessible for measurement. Let us now review the notion of persistent excitation (PE). efinition 3 ([18]) . A vector or matrix function, φ , ispersistently excited (PE) if there exist T > , ǫ > , suchthat R t + Tt φ ( r ) φ T ( r ) dr ≥ ǫI, ∀ t ≥ . Now we introduce the following assumption on ∆( x ) . Assumption 3.
The transpose of the regressor matrix in (1) , ∆ T ( x ) , is persistently excited. Remark 4.
Positive-definiteness of ∆ T ∆ is sufficient for ∆ T to satisfy the PE condition. We now review a FTS parameter estimation scheme whichforms the basis for our FxTS adaptation law and wasfirst proposed for nonlinear dynamical systems in [19], andextended for robotic applications in [20]. First, we note thatwe may re-write (1) as: ˙ x ( t ) = ϕ ( x, u ) + Φ( x ) θ, (9)where ϕ ( x, u ) = f ( x ) + g ( x ) u and Φ( x ) = ∆( x ) . Theauthors of [19] and [20] introduce x f , ϕ f , and Φ f to filter x , ϕ , and Φ as follows: k ˙ x f + x f = x, x f (0) = 0 , ˙ x f (0) = 0 (10) k ˙ ϕ f + ϕ f = ϕ, ϕ f (0) = 0 , ˙ ϕ f (0) = 0 (11) k ˙Φ f + Φ f = Φ , Φ f (0) = 0 , ˙Φ f (0) = 0 (12)where k > is a design parameter. Mimicking the form ofthe system dynamics, the filtered system dynamics are: ˙ x f = ϕ f + Φ f θ, (13)which serves as the basis for estimating θ . We now observethat ˙ x f − ϕ f = Φ f θ and define an auxiliary and integratedregressor matrix P and vector Q such that: ˙ P = − ℓ e P + Φ Tf Φ f , P (0) = 0 (14) ˙ Q = − ℓ e Q + Φ Tf ( ˙ x f − ϕ f ) , Q (0) = 0 (15)where ℓ e > is another design parameter. The solutions are: P ( t ) = Z t e − ℓ e ( t − r ) Φ Tf ( r )Φ f ( r ) dr (16) Q ( t ) = Z t e − ℓ e ( t − r ) Φ Tf ( r )( ˙ x f ( r ) − ϕ f ( r )) dr (17)and from them it may be discerned that Q = P θ . Now, definean additional auxiliary vector as W = P ˆ θ − Q = − P ˜ θ. (18)Then, the authors of [20] introduce their adaptation law as ˙ˆ θ = − Λ P T W k W k , (19)where Λ is a constant, positive definite, gain matrix. Finite-time (FT) convergence of the estimated parameters to theirtrue values is guaranteed by the following result. Theorem 1 ([20]) . For system (9) with parameter adaptationlaw (19) and λ min ( P ) > σ > , the parameter estimationerror ˜ θ converges to zero in finite-time t a , satisfying t a ≤k ˜ θ (0) k λ max (Γ − ) σ . While in general it is not required that Λ from (19) and Γ from (5) be equivalent, we consider this to be true for the restof the paper. In Section IV, we advance the FT parameterestimation result by proposing a FxTS adaptation law for theclass of systems described by (1).III. P ROBLEM F ORMULATION
We now formalize the problem under consideration.
Problem 1.
Consider a nonlinear, control-affine dynamicalsystem subject to parametric uncertainty as in (1) . Giventhat Assumptions 1-3 hold, design an adaptation law, ˙ˆ θ , andcontroller, u , such that the following conditions are satisfied:i) The parameter estimation error converges to zero withinfixed-time, T θ , i.e. ˜ θ ( t ) → as t → T θ < ∞ ,independent of ˜ θ (0) .ii) The system trajectories remain safe for all time, i.e. x ( t ) ∈ S , ∀ t ≥ t .iii) The system trajectories converge to a goal set withinfixed-time, T g , i.e. x ( t ) ∈ S g , ∀ t ≥ T g . The following section contains our proposed method.IV. M
AIN R ESULTS
Before introducing one of the main results of the paper,we address modifications to the filtering scheme (10)-(12).In place of the first-order scheme of [19], [20], we use thefollowing second-order filters: k e ¨ x f + 2 k e ˙ x f + x f = x (20) k e ¨ ϕ f + 2 k e ˙ ϕ f + ϕ f = ϕ (21) k e ¨Φ f + 2 k e ˙Φ f + Φ f = Φ (22)where all initial conditions are zero, i.e. β f (0) = 0 , ˙ β f (0) =0 , ¨ β f (0) = 0 , ∀ β ∈ { x, ϕ, Φ } . This second-order system isstable, strictly proper, and minimum-phase similarly to (10)-(12), and in addition, it is critically damped with a naturalfrequency of ω n = 1 /k e . This is desirable, as criticallydamped systems exhibit the smallest settling time, t s , withoutoscillations [21]. A. FxTS Adaptation Law
We now introduce one of the main results of the paper,an adaptation law which renders the trajectories ˜ θ ( t ) of theparameter estimation error fixed-time stable to zero, and thusguarantees convergence of the parameters to their true valueswithin fixed-time. Theorem 2.
Consider a nonlinear, control-affine system withparametric uncertainty as in (1) . If (20) - (22) filter x , ϕ , and Φ , and the auxiliary matrix P and vectors Q, W are definedby (14) - (18) , then, under the ensuing adaptation law ˙ˆ θ = Γ W (cid:0) W T P − T W (cid:1) − ( − c e ν γ e − c e ν γ e ) , (23) the estimated parameters, ˆ θ ( t ) , converge to the true param-eters, θ , in fixed-time, T θ , i.e., ˜ θ ( t ) → and ˆ θ ( t ) → θ as t → T θ , where T θ ≤ T b = 1 c e (1 − γ e ) + 1 c e ( γ e − . (24) ith ν = W T P − T Γ − P − W , c e > , c e > , <γ e < , γ e > , and Γ ∈ R p × p being a constant, positive-definite, gain matrix.Proof. Consider the Lyapunov function candidate V ˜ θ = ˜ θ T Γ − ˜ θ for the system of the parameter-error dynamics(2). Since ˙ θ = 0 , its time derivative along the trajectories of(2) reads ˙ V ˜ θ = − ˜ θ T Γ − ˙ˆ θ . Applying the adaptation law (23)yields ˙ V ˜ θ = − ˜ θ T W (cid:0) W T P − T W (cid:1) − ( − c e ν γ e − c e ν γ e ) .Then, by substituting (18), we obtain ˙ V ˜ θ = − c e V γ e ˜ θ − c e V γ e ˜ θ , (25)i.e., the fixed-time stability condition from [17, Lemma1]. Hence, the origin of (2) is fixed-time stable, and thetrajectories ˜ θ ( t ) reach the origin within a settling time T θ ,given by (24). Consequently, the estimated parameter vector, ˆ θ ( t ) , converges to the true parameter vector, θ , within a fixedtime, T θ , i.e., ˆ θ ( T θ ) = θ .Whereas previous studies (e.g. [22]) use (14) and (15) inthe design of a FxT adaptation scheme that converges tosome bounded set, (23) guarantees fixed-time convergenceof the estimated parameters to their true values. With thisknowledge, we derive an expression for the upper bound onthe infinity norm of the parameter error as a function of time. Corollary 1.
Let ˜ θ = θ − ˆ θ be the parameter estimationerror vector associated with a system of the form (1) . If allof the following conditions holdi) The estimated parameter update law, ˙ˆ θ , is given by (23) ii) u ( x ( t )) is any locally Lipschitz controlleriii) Γ is constant, positive-definite, and diagonal iv) γ e = 1 − µ e and γ e = 1 + µ e for some µ e > then the following expression constitutes an upper bound on k ˜ θ ( t ) k ∞ , ∀ t ∈ [0 , T θ ] : k ˜ θ ( t ) k ∞ ≤ s M (cid:18) N tan (cid:20) Ξ − N c e µ e t (cid:21)(cid:19) µ e := η ( t ) , (26) where M = 2 λ max (Γ) , N = q c e c e , and Ξ = tan − (cid:18) N η T (0)Γ − η (0) (cid:19) (27) with η ( t ) = η ( t ) · p × and T θ ≤ µ e Ξ √ c e c e ≤ T b (28) Proof.
We consider the Lyapunov function candidate V ˜ θ = ˜ θ T Γ − ˜ θ , whose time derivative along the trajectories of (2)reads dV ˜ θ dt = − c e V − /µ e ˜ θ − c e V /µ e ˜ θ .By separation of variables and integration, we now solvefor t as a function of V (0) and V ( t ) . The change of variables x = V /µ e ˜ θ and dx = µ e V − /µ e ˜ θ dV allows us to obtain: t = Z V ( t ) V (0) µ e x µ e − dx − c e x µ e − − c e x µ e +1 = (29) − µ e √ c e c e h tan − (cid:16) NV /µ e ˜ θ ( t ) (cid:17) − tan − (cid:16) NV /µ e ˜ θ (0) (cid:17)i where N = q c e c e . This leads to V ˜ θ ( t ) = (cid:18) N tan (cid:20) tan − ( NV (0)) − N c e µ e t (cid:21)(cid:19) µ e (30) where V (0) = ˜ θ (0) T Γ − ˜ θ (0) ≤ η (0) T Γ − η (0) . Now,since V ˜ θ = ˜ θ T Γ − ˜ θ , then with Γ diagonal we can express V ˜ θ = (Γ − ˜ θ + ... + Γ − pp ˜ θ p ) , and observe that V ˜ θ ≥ λ − max (Γ) k ˜ θ k ≥ λ − max (Γ) k ˜ θ k ∞ . Then, we substitute (30)in this inequality and rearrange terms to recover (26).Then, for ≤ t ≤ µ e c e N tan − (cid:0) N η T Γ − η (cid:1) we havethat (30) decreases monotonically to zero. As such, we let Ξ = tan − (cid:0) N η T Γ − η (cid:1) and obtain (28), which places atighter upper bound on the settling time, T θ , than (24). Remark 5.
As a consequence of (26) , we may tighten theset of admissible parameters at time t as Θ( t ) = { θ ∈ R p : k θ − ˆ θ ( t ) k ∞ ≤ η ( t ) } , ∀ t ∈ [0 , T θ ] , and as Θ( t ) = { ˆ θ } , ∀ t ∈ ( T θ , ∞ ] . We now formalize the new RaCBF condition for forward-invariance of sets S and S r , defined in (3) and (5). Theorem 3.
Let η ( t ) = η ( t ) · p × where η ( t ) is givenby (26) . Under the premises of Corollary 1, the followingcondition is sufficient for forward-invariance of S r , i.e. (5) : sup u ∈ U { L f h ( x ) + L g h ( x ) u + Ψ }≥ − α (cid:18) h ( x ) − η ( t ) T Γ − η ( t )) (cid:19) + Tr(Γ − ) η ( t ) ˙ η ( t ) (31) where ˙ η ( t ) = Z tan µe − (cid:18) Ξ − N c e µ e t (cid:19) sec (cid:18) Ξ − N c e µ e t (cid:19) (32) Ψ = p X i =1 min n C i P Θ (ˆ θ i − η ) , C i P Θ (ˆ θ i + η ) o (33) for Z = − c e √ M N − µ e , where C i denotes the i th columnof L ∆ h ( x ) for i ∈ { , . . . , p } , and P Θ the vector projectiononto Θ , as defined in [23].Proof. We recall (6) and now replace the constant ϑ with thetime-varying η ( t ) , where henceforth we drop the argument.Thus, h r ( x, η ) = h ( x ) − η T Γ − η , and ˙ h r ( x, η ) = ∂h r ∂x ˙ x + ∂h r ∂ η ˙ η . We will show that if (31) holds then (7) holds and thus(5) is forward-invariant.First, ∂h r ∂x ˙ x = ∂h∂x ˙ x = L f h ( x ) + L g h ( x ) u + L ∆ h ( x ) θ ,then with Γ diagonal we can express ∂h r ∂ η ˙ η = − η T Γ − ˙ η = − Tr(Γ − ) η ˙ η , where we obtain ˙ η ( t ) in (32) by differentiating(26) from Corollary 1.Next, we consider the case where ( η + ˆ θ ) ∈ Θ , forwhich P Θ ( η + ˆ θ ) = ( η + ˆ θ ) . By (26) we have that ˆ θ i − η ≤ θ i ≤ ˆ θ i + η , ∀ i ∈ { , . . . , p } , where Θ isconvex by Assumption 1. Thus, the solution of the followingminimization problem represents the worst admissible effectof the unknown parameters on safety: φ ∗ = arg min φ ∈ Θ L ∆ h ( x ) φ (34)This is a constrained linear program. As such, a uniqueminimizer, φ ∗ = [ φ ∗ . . . φ ∗ p ] T exists, where by the facthat L ∆ h ( x ) φ = P pi =1 C i φ i with C i as the i th column of L ∆ h ( x ) , we have that φ ∗ , . . . , φ ∗ p are the minimizers of thefollowing p constrained linear programs: φ ∗ i = arg min ˆ θ i − η ≤ φ i ≤ ˆ θ i + η C i φ i , ∀ i ∈ { , . . . , p } (35)Furthermore, the solutions of constrained linear programs areguaranteed to be on the boundary of the solution domain,which in this case implies that either φ ∗ i = ˆ θ i − η or φ ∗ i =ˆ θ i + η . Thus, we denote Ψ = L ∆ h ( x ) φ ∗ and recover (33)so that Ψ ≤ L ∆ h ( x ) θ , ∀ θ ∈ Θ .For ( η + ˆ θ ) / ∈ Θ , (34) is again solved, but now P Θ ( η + ˆ θ ) reduces η when necessary to enforce that φ ∈ Θ . We againobtain that a unique solution exists and that Ψ ≤ L ∆ h ( x ) θ , ∀ θ ∈ Θ .For both cases we have that L f h ( x )+ L g h ( x ) u +Ψ ≤ ∂h∂x ˙ x and thus (31) implies (7), and (5) is forward-invariant.The use of the projection operator in Theorem 3 reducesthe conservatism of the approach without compromising therobustness of the forward-invariance condition.V. C ASE S TUDY
A. Comparing Controllers
In the first numerical study, we investigate how ourapproach compares to other recent results in the literature,namely the adaptation laws from [11], [12], and [14], and theworst-case disturbance consideration of [5]. As a basis forcomparison, we consider a 2D single-integrator system sub-ject to parametric uncertainty and challenge the controllersto safely achieve convergence to the origin by avoiding staticobstacles separated by a small gap (Fig. 1).
X (m) − − Y ( m ) BarrierInitial ConditionGoal
Fig. 1:
Problem setup for the first numerical case study, ”Shoot theGap.” The controller must determine what actions, u x and u y , totake in order to realize safe trajectories from the Initial Conditionto the Goal.
1) Dynamics:
We denote z = [ x y ] T as the state, where x and y are the lateral and longitudinal position coordinateswith respect to an inertial frame. The system dynamics are ˙ z = (cid:20) (cid:21) (cid:20) u x u y (cid:21) + ∆( z ) (cid:20) θ θ (cid:21) , (36)where the known regressor matrix is given by ∆( z ) = K ∆ (cid:20) (2 πf x ) 00 1 + cos (2 πf y ) (cid:21) with θ , θ as constant parameters that are unknown a priori,and K ∆ , f , f given in Table I. Assumption 1 is enforcedby defining lower and upper bounds ¯ θ and ¯ θ , respectively,and imposing ¯ θ ≤ θ , θ ≤ ¯ θ . The choice of ∆ is such that ∆ T ∆ is positive-definite for all z ∈ R , thus satisfying thePE condition and Assumption 3.
2) Control Formulation:
To encode the goal-convergencecriterion we define the CLF: V ( z ) = K V ( x + y ) , (37)The safe states are those residing outside of the two ellipsesshown in Figure 1, which results in the following two CBFs: h ( z ) = ( x − x ) a + ( y − y ) b − (38) h ( z ) = ( x − x ) a + ( y − y ) b − (39)where x , x , y , y , a , and b are parameters that define thelocation, size, and shape of the ellipses.We choose the CLF-CBF-QP framework ([7], [10]) forcomputing the control inputs. While we simulated the con-trollers from the literature both in their original form andwith standardized FxT-CLFs to more fairly assess theirabilities, no meaningful differences were observed in theirability to ”shoot the gap.” As such, we present results forthe latter case. Our control framework is then: min u,δ ,δ ,...,δ q u T Qu + p δ + q X i =1 p i δ i (40a)s.t. − ¯ u ≤ u ≤ ¯ u (40b) − ¯ u ≤ u ≤ ¯ u (40c) ≤ δ i (40d) L f V ( z ) + L g V ( z ) u + φ ( x, ∆( x, t ) , ˆ θ, η ) ≤ δ − c V ( z ) γ − c V ( z ) γ (40e) L f h i ( z ) + L g h i ( z ) u + ψ ( x, ∆( x, t ) , ˆ θ, η ) ≥ − δ i h i ( z ) (40f) ∀ i ∈ { , . . . , q } , where generally u = [ u u ] T and for thisproblem u = u x and u = u y , δ is a relaxation parameteron the performance objective whose inclusion guaranteesfeasibility of the QP, δ i allows for larger negative valuesof ˙ h i ( z ) away from the boundary of the safe set, and p i penalizes values of δ i , ∀ i ∈ { , ..., q } . The functions φ : R n × R n × p × R p × R p → R and ψ : R n × R n × p × R p × R p → R represent the terms specific to the way each respectivecontroller handles the uncertainty in the system dynamics.While all of (40b)-(40f) are linear in the decision variables,(40b) and (40c) enforce input constraints, (40d) preventsover-conservatism in enforcing safety, (40e) encodes FxTconvergence to the goal, and safety is guaranteed by (40f).
3) Results:
The full set of parameters for this numericalcase study are provided in Table I.We endeavor to demonstrate that by learning the truevalues of the uncertain parameters in the system dynamicsof (36), our method is capable of approaching the boundaryof the safe set more closely than previous results in the Simulation code is accessible at Github: https://tinyurl.com/y3xhylug. x − − y BarrierGoal z TAYBLALOPLSMZHAPRO (a) State trajectories − − − u x TAYBLALOPLSMZHAPRO ± ¯ u x Time (sec) − − − u y TAYBLALOPLSMZHAPRO ± ¯ u y (b) Control inputs Time (sec) . . . . . . . . . h ( x ) Boundary
TAYBLALOPLSMZHAPRO (c) Value of level curve for safe set
Fig. 2:
State trajectories, control inputs, and control barrier function evolutions in time for the Shoot the Gap example.
TABLE I: Shoot the Gap Parameters ˙ x Val QP Val CBF Val CLF Val ˙ˆ θ Val f Q I × a K V k e f p b T T e θ -1 p x µ µ e θ p x c c e ¯ θ ¯ u y -6 c c e ¯ θ ¯ u y γ γ e K ∆ γ γ e ℓ e literature and, as a consequence, able to reach a goal whichmay require such an approach despite uncertainty. Table IIprovides the legend codes used to refer to these other works.TABLE II: Controllers from the Literature Authors Citation Legend CodeTaylor et al. [11] TAYBlack et al. [5] BLALopez et al. [12] (w/o SMID) LOPLopez et al. [12] (w/ SMID) LSMZhao et al. [14] ZHAProposed Method PRONote: [12] presents RaCBF-based control formulations with and withoutSMID for parameter estimation. We have considered both cases.
First, we observe that in accordance with Theorem 2,Figure 3 highlights that the parameter estimates, ˆ θ , do in factconverge to their true values within fixed-time T θ given by(28). Figure 2a shows that our proposed method ”shoots thegap” where the others do not; that is, our method can tolerateregions of the state space which exist in close proximityto the boundary of the safe region. As such, it fulfills itsspecification of FxT convergence to the origin. In this sense,our synthesized adaptation law and RaCBF-based controlleris less restrictive than the existing literature. B. Highway Overtake
We now consider an automobile highway overtake prob-lem, similarly to [5], and show how our control formulation − − θ Fixed-Time θ ,bounds θ ,true ˆ θ ,LSM ˆ θ ,h ,TAY ˆ θ ,h ,TAY ˆ θ ,PRO Time (sec) − − θ Fixed-Time θ ,bounds θ ,true ˆ θ ,LSM ˆ θ ,h ,TAY ˆ θ ,h ,TAY ˆ θ ,PRO Fig. 3:
Estimates of the unknown model parameters, θ . can guarantee success of the overtake maneuver under un-certainty, where the robust CBF approach cannot.
1) Dynamics:
Just as in [5], we model the vehicles askinematic bicycles using the model from [24]. Accordingly,the state vectors are z i = [ x i y i θ i v i ] T , where x and y areplanar Euclidean coordinates (longitudinal and transverse), θ is the heading angle, v is velocity, and the subscript i ∈{ e, l } denotes belonging to the Ego or Lead vehicle. Thecorresponding dynamical system is described by: ˙ z i = v i cos( θ i ) v i sin( θ i )00 + /M (cid:20) ω i a i (cid:21) + ∆ i ( z ) θ, (41) where M is the mass of the vehicle in kg, and ω i and a i represent the angular velocity and heading accelerationcontrol inputs, for which the bounds ¯ ω ≤ ω i ≤ ¯ ω in rad − and ¯ a ≤ a i ≤ ¯ a in m/s hold. For reference, all overtakearameter values may be found in Table III . We elect tomodel erratic, or distracted, driver behavior by the additionof the uncertain term ∆ i ( x ) θ , where ∆ i : R → R × R isthe known regressor matrix. As such, we let ∆ e = n × p and ∆ l = n × p with the exception of ∆ l, (0 , ( z ) = 1 + (1 − cos(2 πf l, x l )) and ∆ l, (1 , ( z ) = + (1 − sin(2 πf l, x l )) .
2) Problem Formulation:
We define the safe sets as: S i = { z | h i ( z ) ≥ } , ∀ i ∈ { , , } (42)where h ( z ) = K s (( y e − E R ( z ))( E L ( z ) − y e )) (43) h ( z ) = L − v (44) h ( z ) = (cid:18) x e − x l s x (cid:19) + (cid:18) y e − y l s y (cid:19) − (45)and E R ( z ) = e r + θ e v e sin( θ e )¯ ω − θ e ω (cid:18) ¯ a sin( θ e ) M + v e ¯ ω cos( θ e ) (cid:19) (46) E L ( z ) = e l − θ e v e sin( θ e )¯ ω − θ e ω (cid:18) ¯ a sin( θ e ) M − v e ¯ ω cos( θ e ) (cid:19) (47) where e r and e l denote the physical edges of the right andleft side of the road such that (46) and (47) imply that (43)encodes that the Ego vehicle remain on the road despitebounded steering control. We also have that (44) enforcesthe road speed limit, L , in m/s, and (45) ensures that safetymargins s x = τ v e cos( θ e ) + l c and s y = w c + 0 . betweenvehicles are observed, where l c and w c are the length andwidth of the vehicles in m. Then, S = ∩ S i , ∀ i ∈ { , , } .In addition, Oncoming vehicles are known to obey thefollowing pattern: the first vehicle has a time-headway of 24swith the Lead Vehicle, and subsequent Oncoming vehiclesarrive in 30s intervals. Consequently, the Ego vehicle mustcomplete the overtake within 24s to proceed at the outset,and within 30s to proceed after the first Oncoming vehicle.We now formally define the overtake problem. Problem 2.
Given the initial states, z e (0) , z l (0) , the timeheadway of an oncoming vehicle, T h , and the set Θ to whichthe unknown parameter vector, θ , belongs, determine whetherit is safe for the Ego vehicle to overtake the Lead vehicle, i.e.whether there exist z ( t ) , u e ( t ) ∈ U = { ( ω e , a e ) | ¯ ω ≤ ω e ≤ ¯ ω, ¯ a ≤ a e ≤ ¯ a } such that z e ( t ) ∈ S , ∀ t ∈ [0 , T ] , where T is the upper bound on time to complete the overtake. If safeand T ≤ T h , design a control input, u e ( t ) ∈ U for the given z (0) such that the Ego vehicle overtakes the Lead vehicle.3) Control Formulation: Just as in [5], we partition theproblem into the following sub-problems:i) Ego Vehicle approaches Lead Vehicleii) Ego Vehicle merges into overtake lane M , l c , and w c taken from the2020 Ford Mustang Shelby GT: https://tinyurl.com/yxhn63of. iii) Ego Vehicle advances beyond Lead Vehicleiv) Ego Vehicle merges back into original laneWe use the CLF-CBF-QP control framework presented in(40) to compute the control inputs, u = ω e and u = a e ,pointwise-in-time where q = 3 in accordance with h ( z ) , h ( z ) , and h ( z ) in (43)-(45). Our CLF is: V ( z ) = K V ( k x ¯ x + k y ¯ y + k θ ¯ θ + k v ¯ v − (48)where ¯ x = x − x d , ¯ y = y − y d , ¯ θ = θ − θ d , and ¯ v = v − v d ,and z d = [ x d y d θ d v d ] T is the desired state. We define thefixed-time convergence times for the four sub-problems as T = 3 , T = 5 , T = 7 , and T = 5 respectively.TABLE III: Overtake Parameters ˙ x Val QP Val CBF Val CLF Val ˙ˆ θ Val M Q , ¯ ω e r K V − k e f l, Q , ¯ a e l µ T e f l, p × L γ µ e θ p l c γ c e θ p w c k x c e p τ k y c e ¯ ω k θ γ e ¯ a k v γ e ℓ e Q i,j denotes the value of the row i column j entry for the Q matrix. Non-specified entries are uniformly zero.
4) Results:
The scenario was initialized as x e (0) = − . , y e (0) = 1 . , θ e (0) = 0 , v e (0) = 24 , x l (0) = 0 , y l (0) = 1 . , y θ (0) = 0 , and v l (0) = 19 . For all consideredsets of admissible parameters, Θ , we set ¯ θ = ¯ θ = − ¯ θ =¯ θ = − ¯ θ , and chose ¯ θ = 1 , , , , , . Table IV showshow the fixed-time horizon grows for BLA as ¯ θ increases.TABLE IV: Overtake Fixed-Time Horizons ¯ θ T PRO T BLA t oncoming, t oncoming,i As such, the PRO technique completes the overtake with-out delay for all parameter bounds, whereas the BLA con-troller appropriately proceeds immediately with the overtakewhen ¯ θ = 1 , , , proceeds after the first oncoming vehiclehas passed when ¯ θ = 6 , , and cannot guarantee a safeovertake when ¯ θ = 10 . This is precisely the advantageof our proposed controller. Because it is guaranteed toadaptively learn the true parameters within fixed-time, it isable to successfully complete the overtake maneuver for allconsidered sets, Θ . VI. C ONCLUSION
In this study on the efficacy of various techniques for safecontrol under parametric model uncertainty, we presenteda novel adaptation law that learns the uncertain parametersassociated with a class of nonlinear, control-affine dynamicalsystems in fixed-time. We synthesized our parameter adapta-tion law with a robust, adaptive CBF-based controller in the
200 400 600 800 1000
X (m) Y ( m ) Road EdgeLane Divider
BLA : ¯ θ = 1 PRO : ¯ θ = 1 BLA : ¯ θ = 4 PRO : ¯ θ = 4 BLA : ¯ θ = 8 PRO : ¯ θ = 8 X (m) Y ( m ) Road EdgeLane Divider
BLA : ¯ θ = 2 PRO : ¯ θ = 2 BLA : ¯ θ = 6 PRO : ¯ θ = 6 BLA : ¯ θ = 10 PRO : ¯ θ = 10 − . − . . . . ω e ¯ ω BLA : ¯ θ = 8 PRO : ¯ θ = 8 ± ¯ ω x Time (sec) − . − . . . . a e ¯ a BLA : ¯ θ = 8 PRO : ¯ θ = 8 ± ¯ a x − −
10 0 10 20 30 40
Time (sec) C B F s BLA road : ¯ θ = 8 BLA speed : ¯ θ = 8 BLA car : ¯ θ = 8 PRO road : ¯ θ = 8 PRO speed : ¯ θ = 8 PRO car : ¯ θ = 8 Boundary − . − . . . . ω e ¯ ω BLA : ¯ θ = 10 PRO : ¯ θ = 10 ± ¯ ω x Time (sec) − . − . . . . a e ¯ a BLA : ¯ θ = 10 PRO : ¯ θ = 10 ± ¯ a x − −
10 0 10 20 30 40
Time (sec) C B F s BLA road : ¯ θ = 10 BLA speed : ¯ θ = 10 BLA car : ¯ θ = 10 PRO road : ¯ θ = 10 PRO speed : ¯ θ = 10 PRO car : ¯ θ = 10 Boundary
Fig. 4:
Results for 6 different simulations of the overtake problem. The top row displays state trajectories. The bottom row containscontrol inputs and CBF trajectories for cases where [5] must postpone the overtake maneuver (left) and cannot complete safely completeit (right). form of a quadratic program, and provided an upper boundon the parameter estimation error as an explicit functionof time. We then studied the performance of our methodon a simple, 2D single integrator system in relation toseveral recent works from the literature and demonstratedthat our contribution succeeds in navigating near unsaferegions where the others fail. We further illustrated thepromise of our method in applications where a decision onwhether to initiate a possibly unsafe maneuver is required,using the automobile overtake problem as a case study.In the future, we intend to study cases for which theuncertain parameters are time-varying and an upper boundis not known a priori, as we recognize that these may havebroader applicability to real-world scenarios.R
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