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Robust Taylor-Lagrange Control for Safety-Critical Systems
Wei Xiao, Christos Cassandras, Anni Li
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 94%
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Summary
The paper proposes a Robust Taylor-Lagrange Control (rTLC) method to address the feasibility preservation problem (inter-sampling effect) in safety-critical systems. Unlike the standard Taylor-Lagrange Control (TLC) which relies on future time states, rTLC expands the safety function using Taylor's theorem with Lagrange remainder to explicitly include current-time control inputs. This approach guarantees system safety with only one hyper-parameter (discretization time interval), reducing computational complexity and tuning difficulty compared to Control Barrier Functions (CBF) and event-triggered methods. The method is validated on an adaptive cruise control problem.
Entities (8)
Relation Signals (5)
Wei Xiao → authored → Robust Taylor-Lagrange Control
confidence 99% · Robust Taylor-Lagrange Control for Safety-Critical Systems Wei Xiao
Robust Taylor-Lagrange Control → addresses → Inter-sampling Effect
confidence 95% · The rTLC method naturally addresses the feasibility preservation problem with only one hyper-parameter
Robust Taylor-Lagrange Control → extends → Taylor-Lagrange Control
confidence 92% · Specifically, the rTLC method expands the safety function at an order higher than the relative degree... using Taylor's expansion with Lagrange remainder
Robust Taylor-Lagrange Control → appliedto → Adaptive Cruise Control
confidence 90% · we illustrate the effectiveness of the proposed rTLC method through an adaptive cruise control problem
Robust Taylor-Lagrange Control → outperforms → Control Barrier Function
confidence 85% · The existence of a CBF is only a sufficient condition... rTLC... allows the control to explicitly show up at the current time
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Abstract
Abstract:Solving safety-critical control problem has widely adopted the Control Barrier Function (CBF) method. However, the existence of a CBF is only a sufficient condition for system safety. The recently proposed Taylor-Lagrange Control (TLC) method addresses this limitation, but is vulnerable to the feasibility preservation problem (e.g., inter-sampling effect). In this paper, we propose a robust TLC (rTLC) method to address the feasibility preservation problem. Specifically, the rTLC method expands the safety function at an order higher than the relative degree of the function using Taylor's expansion with Lagrange remainder, which allows the control to explicitly show up at the current time instead of the future time in the TLC method. The rTLC method naturally addresses the feasibility preservation problem with only one hyper-parameter (the discretization time interval size during implementation), which is much less than its counterparts. Finally, we illustrate the effectiveness of the proposed rTLC method through an adaptive cruise control problem, and compare it with existing safety-critical control methods.
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- Source: https://arxiv.org/abs/2602.20076v1
- Canonical: https://arxiv.org/abs/2602.20076v1
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Robust Taylor-Lagrange Control for Safety-Critical Systems Wei Xiao, Christos Cassandras and Anni Li W. Xiao is with the Robotics Engineering Department at Worcester Polytechnic Institute and MIT CSAIL, USA wxiao3@wpi.edu, weixy@mit.eduA. Li is with the Electrical and Computer Engineering Department at The University of North Carolina at Charlotte, Charlotte, NC, 28223, USA ali20@charlotte.eduC. Cassandras is with the division of Systems Engineering at Boston University, USA cgc@bu.edu Abstract Solving safety-critical control problem has widely adopted the Control Barrier Function (CBF) method. However, the existence of a CBF is only a sufficient condition for system safety. The recently proposed Taylor-Lagrange Control (TLC) method addresses this limitation, but is vulnerable to the feasibility preservation problem (e.g., inter-sampling effect). In this paper, we propose a robust TLC (rTLC) method to address the feasibility preservation problem. Specifically, the rTLC method expands the safety function at an order higher than the relative degree of the function using Taylor’s expansion with Lagrange remainder, which allows the control to explicitly show up at the current time instead of the future time in the TLC method. The rTLC method naturally addresses the feasibility preservation problem with only one hyper-parameter (the discretization time interval size during implementation), which is much less than its counterparts. Finally, we illustrate the effectiveness of the proposed rTLC method through an adaptive cruise control problem, and compare it with existing safety-critical control methods. I Introduction Stabilizing dynamical systems while optimizing costs and satisfying (nonlinear) safety constraints has received increasing attention in recent years with rising autonomy. Existing methods like optimal control [7] [14] and dynamic programming [5] [10] are primarily designed for linear systems and constraints, limiting their extensions to nonlinear systems. Model Predictive Control (MPC) methods [11] [19] have been applied to receding horizon safety-critical control, but they are generally computationally heavy, especially for nonlinear systems. Although linearization is possible to improve MPC computational efficiency, such approximations may compromise safety guarantees. Reachability analysis [1] [3] is also widely used to verify system safety, but it incurs extensive computation load. In order to address the computational challenge, barrier-based methods have received increasing attention for nonlinear systems. Barrier functions (BFs) are originally used in optimizations with inequality constraints [6] by taking their reciprocal form as part of the cost function, and they are also used in machine learning systems to improve safety, such as in safe reinforcement learning [8]. However, this method cannot guarantee system safety as the safety constraint is taken as part of the cost function or reward, and thus its functionality is similar to a soft constraint. BFs are also used as Lyapunov-like functions [25], and they have been employed in system verification and control [24][4][18][26]. Control BFs (CBFs) are the use of BFs for control (affine) systems. CBFs can trace back to the Nagumo’s theorem [16], and it shows that if the safety function is initially non-negative and its derivative is always non-negative at the safe set boundary, then the safety function remains non-negative for all times. In other words, the safety can be preserved. For safe control synthesis with CBFs, class K functions are considered in reciprocal CBFs [2] and zeroing CBFs [12] [29] to allow the safety function to decrease when system state is far away from the boundary. Both CBF forms are fundamentally the same, and they are conservative due to the introduction of class K functions. In other words, the existence of a CBF is only a sufficient condition for system safety. There are different variations for CBFs. Stochastic CBFs [9] have been proposed for stochastic systems, and finite-time convergence CBFs [20] are for systems that initially violate the safety constraint. Adaptive CBFs [21] [27] are also introduced to deal with system uncertainty and noise. To ensure safety for systems and constraints with high relative degree, exponential CBFs (ECBFs) [17] and high-order CBFs (HOCBFs) [29] are proposed. However, the existence of ECBFs or HOCBFs implies the forward invariance of the intersection of a set of safe sets, which is very restrictive and may potentially limit system performance. The CBF method also introduces additional hyper-parameters, such as those in class K functions, that are non-trivial or difficult to tune. The recently proposed Taylor-Lagrange Control (TLC) method [30] ensures system safety using Taylor’s theorem with Lagrange remainder [23] [15], and the TLC method has been shown to be a necessary and sufficient condition for system safety with much less hyper-parameters (only one) than the CBF method. However, the TLC method is vulnerable to the feasibility preservation problem, such as the notorious inter-sampling effect that specifies the constraint satisfaction in continuous time for sampled-data systems. Although we may employ event-triggered methods [22] [28] to address the inter-sampling effect, these methods may introduce additional hyper-parameters that are non-trivial to tune by hand. In this paper, we propose a robust Taylor-Lagrange Control (rTLC) method for system safety using the Taylor’s theorem with Lagrange remainder. The TLC method expands a safety function at an order higher than the relative degree of the safety function. This allows the control to show up at the current time in stead of the future time in Taylor’s expansion. We quantify the bound of the Lagrange remainder in rTLC using a reachable set analysis or the task space set, which eliminates the calculation of the future time state, control, and the derivative of control in a rTLC. As a results, the proposed rTLC has been shown to naturally address the feasibility preservation problem without additional considerations. Moreover, the rTLC method has only one hyper-parameter, and is much less than existing methods like event-triggered approaches. The contributions of this paper are as follows: 1. We propose a robust Taylor-Lagrange Control (rTLC) method for system safety, and demonstrate its dependence only on the current time instead of the future time in Taylor’s expansion. 2. We show that the proposed rTLC method naturally addresses the feasibility preservation problem with only one hyper-parameter (the discretization time interval size during implementation), and thus guaranteeing system safety without additional parameter tuning. 3. We verify the effectiveness of the proposed rTLC method on an adaptive cruise control problem, and compare it with existing safety-critical control methods. The remainder of the paper is organized as follows. In Section I, we provide preliminaries on the TLC method and briefly discuss its connection with CBFs and HOCBFs. We propose the rTLC method for nonlinear systems control systems in Section I with a subsection on how to solve optimal control problems using the proposed rTLC. The case studies and simulation results are shown in Section IV. We conclude with final remarks and directions for future work in Section V. I PRELIMINARIES In this section, we introduce the background on the Taylor-Lagrange Control (TLC), and briefly discuss its connection with CBFs. Consider an affine control system ˙=f()+g(), x=f( x)+g( x) u, (1) where ∈X∈ℝn x∈ X ^n (X is a compact set), f is as defined above, g:ℝn→ℝn×qg:R^n ^n× q is locally Lipschitz, and ∈U⊂ℝq u∈ U ^q where U denotes a compact control constraint set: U=∈ℝq:min≤max,U=\ u ^q: u_min≤ u≤ u_max\, (2) where min∈ℝq,max∈ℝq u_min ^q, u_max ^q, and the above constraint is interpreted component-wise. Solutions (t) x(t) of (1), starting at (0) x(0), t≥0t≥ 0, are forward complete. Definition 1 (Class K function [13]) A function α:[0,a)→[0,∞),a>0α:[0,a)→[0,∞),a>0 is said to belong to class K if it is strictly increasing and passes the origin. Definition 2 (Forward invariance [2]) A set C is forward invariant for (1) if its solution x(t)∈C,∀t≥0x(t)∈ C,∀ t≥ 0 for any x(0)∈Cx(0)∈ C. Definition 3 (Relative degree [13]) The relative degree of a function h()h( x) (or a constraint h()≥0h( x)≥ 0) w.r.t. (1) is defined as the minimum number of times we need to differentiate h()h( x) along system (1) until any control component of u explicitly shows up in the corresponding derivative. For a safety constraint h()≥0h( x)≥ 0 that has relative degree m w.r.t system (1), h:ℝn→ℝh:R^n , we can define a safe set C based on h: C:=∈ℝn:h()≥0,C:=\ x ^n:h( x)≥ 0\, (3) where h(x)=0h(x)=0 at the boundary of C and h(x)<0h(x)<0 outside C. Then we have the following definition for a Taylor-Lagrange Control (TLC) function: Definition 4 (Taylor-Lagrange Control [30]) The function h:ℝn→ℝh:R^n is a Taylor-Lagrange Control (TLC) function of relative degree m for system (1) if sup(ξ)∈U[∑k=0m−1Lfkh((t0))k!(t−t0)k+Lfmh((ξ))m!(t−t0)m _ u(ξ)∈ U [ _k=0^m-1 L_f^kh( x(t_0))k!(t-t_0)^k+ L_f^mh( x(ξ))m!(t-t_0)^m . (4) +LgLfm−1h((ξ))(ξ)m!(t−t0)m]≥0,t0∈[0,∞),ξ∈(t0,t), .\!+ L_gL_f^m-1h( x(ξ)) u(ξ)m!(t\!-\!t_0)^m ]\!≥\!0,t_0∈[0,∞),ξ∈(t_0,t), for all (t0)∈C x(t_0)∈ C, where Lfh,LghL_fh,L_gh denote the Lie derivative of h along f and g, respectively. The TLC exhibits connection with CBFs [2]. In other words, when h()h( x) has relative degree m=1m=1, then the TLC constraint in (4) degenerates to a CBF with a linear class K function: Lfh((ξ))+Lgh((ξ))(ξ)+1t−t0h((t0))≥0.L_fh( x(ξ))+L_gh( x(ξ)) u(ξ)+ 1t-t_0h( x(t_0))≥ 0. (5) where ξ=t0ξ=t_0 in the zero-order hold discretization implementation and t is the next time instant for the discretization. The TLC also extends HOCBFs [29] to the complex domain (i.e., the class K functions in HOCBFs are complex functions) for high relative degree constraints. Moreover, the existence of a TLC is a necessary and sufficient condition for the safety of the system [30], while the existence of a CBF is only sufficient. Theorem 1 ([30]) Given a TLC function h()h( x) from Def. 4 with the associated safe set defined as in (3), if h((t0))≥0h( x(t_0))≥ 0, then any Lipschitz continuous controller (ξ)∈Ktlc((t0),(ξ)),∀t0∈[0,∞),ξ∈(t0,t),t>t0 u(ξ)∈ K_tlc( x(t_0), x(ξ)),∀ t_0∈[0,∞),ξ∈(t_0,t),t>t_0 renders the set C forward invariant for system (1). The TLC can be used for enforcing both the system safety and stability [30]. We may first discretize the time, and then employ zero-order hold implementation within each time interval. We take t0t_0 as the current time instant, and define t=t1t=t_1 in Def. 4, where t1t_1 is the next implementation time instant. This allows us to transform the original nonlinear constrained control problems into a sequence of quadratic programs. The inter-sampling effect of the TLC can be addressed by event-triggered methods [30]. However, event-triggered methods may have too many hyperparameters that are hard to tune. In this work, we address the inter-sampling effect using the proposed robust TLC method that has few hyperparameters. I Robust Taylor-Lagrange Control In this section, we propose the robust Taylor-Lagrange Control (rTLC) for obtaining state-feedback safe control policies for nonlinear control systems while addressing the inter-sampling effect. We also discuss how to solve constrained optimal control problems using the proposed rTLC. As the feasibility preservation problem is mainly determined by the inter-sampling effect, we refer to the inter-sampling effect in the remainder of this work. I-A Robust Taylor-Lagrange Control Suppose the relative degree of the safety constraint h()≥0h( x)≥ 0 is m with respect to (1), and the function h()h( x) is m+1m+1 times differentiable. Given a TLC as in Def. 4, we notice that the control explicitly shows up in LgLfm−1h((ξ))(ξ)m!(t−t0)m L_gL_f^m-1h( x(ξ)) u(ξ)m!(t\!-\!t_0)^m, where ξ∈(t0,t)ξ∈(t_0,t) is hard to determine. Note that the (ξ) u(ξ) is the control at the time ξ instead of the current time t0t_0, which makes it hard to find a control policy for the current time t0t_0. Although we can employ zero-order implementation such that t=t1t=t_1 (t1t_1 denotes the next implementation time instant) and (t0)=(ξ) u(t_0)= u(ξ), this may introduce some errors. We may further employ event-triggered methods to address the inter-sampling effect due to the zero-order hold error, there are usually too many hyper-parameters that are hard to tune. Higher-order TLC. Let Δt=t−t0 t=t-t_0. In order to avoid the above issue, we propose to expand the function h()h( x) at an order of m+1m+1 instead of m in the Taylor’s theorem: ∑k=0m−1Lfkh((t0))k!Δtk+Lfmh((t0))m!Δtm _k=0^m-1 L_f^kh( x(t_0))k! t^k+ L_f^mh( x(t_0))m! t^m (6) +LgLfm−1h((t0))(t0)m!Δtm \!+ L_gL_f^m-1h( x(t_0)) u(t_0)m! t^m +R((ξ),(ξ),˙(ξ))≥0,t0∈[0,∞),ξ∈(t0,t), +R( x(ξ), u(ξ), u(ξ))\!≥\!0,t_0∈[0,∞),ξ∈(t_0,t), where the Lagrange remainder is defined as R((ξ),(ξ),˙(ξ))=Lfm+1h((ξ))(m+1)!Δtm+1 R( x(ξ), u(ξ), u(ξ))= L_f^m+1h( x(ξ))(m+1)! t^m+1 (7) +LgLfmh((ξ))(ξ)+LfLgLfm−1h((ξ))(ξ)(m+1)!Δtm+1 + L_gL_f^mh( x(ξ)) u(ξ)+L_fL_gL_f^m-1h( x(ξ)) u(ξ)(m+1)! t^m+1 +Lg2Lfm−1h((ξ))(ξ)+LgLfm−1h((ξ))˙(ξ)(m+1)!Δtm+1 + L_g^2L_f^m-1h( x(ξ)) u(ξ)+L_gL_f^m-1h( x(ξ)) u(ξ)(m+1)! t^m+1 In (7), the h()h( x) is m+1m+1 times differentiable, and ∈X,∈U x∈ X, u∈ U, where X,UX,U are compact sets. In order to ensure that R((ξ),(ξ),˙(ξ))R( x(ξ), u(ξ), u(ξ)), we need to check whether ˙(ξ) u(ξ) is bounded. Boundedness of u˙(ξ) u(ξ). The control set U is defined as in (2). We need to ensure that ∈U u∈ U when ˙ u is introduced. Let =(u1.…,uq),min=(u1,min,…,uq,min),max=(u1,max,…,uq,max) u=(u_1.…,u_q), u_min=(u_1,min,…,u_q,min), u_max=(u_1,max,…,u_q,max), where ui,min∈ℝ,ui,max∈ℝ,∀i∈1,…,qu_i,min ,u_i,max ,∀ i∈\1,…,q\. In order to ensure ∈U u∈ U with ˙ u, we introduce two 1st order TLC for each i∈1,…,qi∈\1,…,q\: hi,1((ξ))=ui,max−ui, h_i,1( u(ξ))=u_i,max-u_i, (8) hi,2((ξ))=ui−ui,min, h_i,2( u(ξ))=u_i-u_i,min, In order to make the two functions in the above become TLCs, the following constraints have to be satisfied: Lfhi,1(ui(ξ))+Lghi,1(ui(ξ))u˙i(ξ)+1Δthi,1(ui(ξ))≥0, L_fh_i,1(u_i(ξ))+L_gh_i,1(u_i(ξ)) u_i(ξ)+ 1 th_i,1(u_i(ξ))≥ 0, (9) Lfhi,2(ui(ξ))+Lghi,2(ui(ξ))u˙i(ξ)+1Δthi,2(ui(ξ))≥0, L_fh_i,2(u_i(ξ))+L_gh_i,2(u_i(ξ)) u_i(ξ)+ 1 th_i,2(u_i(ξ))≥ 0, Combining the last two equations, we have that u˙i(ξ) u_i(ξ) is bounded by: −1Δt(ui(ξ)−ui,min)≤u˙i(ξ)≤1Δt(ui,max−ui(ξ)). - 1 t(u_i(ξ)-u_i,min)≤ u_i(ξ)≤ 1 t(u_i,max-u_i(ξ)). (10) The above equation can be rewritten as −1Δt((ξ)−min)≤˙(ξ)≤1Δt(max−(ξ)). - 1 t( u(ξ)- u_min)≤ u(ξ)≤ 1 t( u_max- u(ξ)). (11) Boundedness of R(x(ξ),u(ξ),u˙(ξ))R( x(ξ), u(ξ), u(ξ)). Since ˙(ξ) u(ξ) is bounded as shown in (11), we have that R((ξ),(ξ),˙(ξ))R( x(ξ), u(ξ), u(ξ)) is also bounded. Let SR((ξ),(ξ),˙(ξ))S_R( x(ξ), u(ξ), u(ξ)) denote the reachable set of R((ξ),(ξ),˙(ξ))R( x(ξ), u(ξ), u(ξ)) within the time interval (t0,t)(t_0,t), where ξ∈(t0,t)ξ∈(t_0,t). Then we can find the minimum value RminR_min of R((ξ),(ξ),˙(ξ))R( x(ξ), u(ξ), u(ξ)) by: Rmin=min((ξ),(ξ),˙(ξ))∈SR((ξ),(ξ),˙(ξ))R((ξ),(ξ),˙(ξ)).R_min= _( x(ξ), u(ξ), u(ξ))∈ S_R( x(ξ), u(ξ), u(ξ))R( x(ξ), u(ξ), u(ξ)). (12) Since h()h( x) is m+1m+1 times differentiable, ∈U,∈X u∈ U, x∈ X, where U,XU,X are compact sets, and ˙ u is also bounded as in (11), we have that RminR_min must exist. Let Uu˙U_ u be defined by: Uu˙:=˙(ξ)∈ℝq:−1Δt((ξ)−min)≤˙(ξ)≤ U_ u=\ u(ξ) ^q:- 1 t( u(ξ)- u_min)≤ u(ξ)≤ (13) 1Δt(max−(ξ)). 1 t( u_max- u(ξ)).\ Since (ξ)∈U u(ξ)∈ U, we have that Uu˙U_ u is a subset of Uu˙,sup:=˙(ξ)∈ℝq:−1Δt(max−min)≤˙(ξ)≤ U_ u,sup=\ u(ξ) ^q:- 1 t( u_max- u_min)≤ u(ξ)≤ (14) 1Δt(max−min. 1 t( u_max- u_min.\ Finally, we have that RminR_min in (12) is lower bounded by: Rmin≥min((ξ),(ξ),˙(ξ))∈(X,U,Uu˙,sup)R((ξ),(ξ),˙(ξ)).R_min≥ _( x(ξ), u(ξ), u(ξ))∈(X,U,U_ u,sup)R( x(ξ), u(ξ), u(ξ)). (15) The right-hand side of the above equation can always be obtained as we do not explicitly need to calculate the reachable set SR((ξ),(ξ),˙(ξ))S_R( x(ξ), u(ξ), u(ξ)). Deriving a tight reachable set for SR((ξ),(ξ),˙(ξ))S_R( x(ξ), u(ξ), u(ξ)) could significantly reduce the conservativeness of our proposed robust TLC method, and thus is worth a future study. Now, we are ready to define the robust TLC: Definition 5 (Robust Taylor-Lagrange Control) Let RminR_min be defined as in (12). The function h:ℝn→ℝh:R^n is a Robust Taylor-Lagrange Control (rTLC) function of relative degree m for system (1) if sup(t0)∈U[∑k=0m−1Lfkh((t0))k!Δtk+Lfmh((t0))m!Δtm _ u(t_0)∈ U [ _k=0^m-1 L_f^kh( x(t_0))k! t^k+ L_f^mh( x(t_0))m! t^m . (16) +LgLfm−1h((t0))(t0)m!Δtm+Rmin]≥0, .\!+ L_gL_f^m-1h( x(t_0)) u(t_0)m! t^m+R_min ]\!≥\!0, for t0∈[0,∞),t_0∈[0,∞), and for all (t0)∈C x(t_0)∈ C. Note the difference between the rTLC in Def. 5 and the the TLC as in Def. 4 lies in that there is no ξ∈(t0,t)ξ∈(t_0,t) in Def. 5, and thus we can directly make the constraint be defined over (t0) u(t_0) instead of (ξ) u(ξ). Moreover, (16) only depends on (t0) x(t_0) without dependence on (ξ) x(ξ). This property significantly facilitates the implementation of the rTLC method. Given a rTLC as in Def. 5, we can define a state feedback controller that satisfies (16): Krtlc((t0))=(t0)∈U:∑k=0m−1Lfkh((t0))k!Δtk+Rmin K_rtlc( x(t_0))=\ u(t_0)∈ U: _k=0^m-1 L_f^kh( x(t_0))k! t^k+R_min (17) +Lfmh((t0))m!Δtm+LgLfm−1h((t0))(t0)m!Δtm≥0. + L_f^mh( x(t_0))m! t^m+ L_gL_f^m-1h( x(t_0)) u(t_0)m! t^m≥ 0\. Theorem 2 Given a rTLC function h()h( x) from Def. 5 with the associated safe set defined as in (3), if h((t0))≥0h( x(t_0))≥ 0, then any Lipschitz continuous controller (t0)∈Krtlc((t0)) u(t_0)∈ K_rtlc( x(t_0)) ensures h((t′))≥0h( x(t ))≥ 0 ∀t′∈[t0,t]∀ t ∈[t_0,t]. Moreover, if (t0)∈Krtlc((t0)) u(t_0)∈ K_rtlc( x(t_0)) for all t0∈[0,∞)t_0∈[0,∞), then the set C is forward invariant for system (1). Proof: By (12), we have that ∑k=0m−1Lfkh((t0))k!Δtk+Lfmh((t0))m!Δtm _k=0^m-1 L_f^kh( x(t_0))k! t^k+ L_f^mh( x(t_0))m! t^m (18) +LgLfm−1h((t0))(t0)m!Δtm+R((ξ),(ξ),˙(ξ))≥ \!+ L_gL_f^m-1h( x(t_0)) u(t_0)m! t^m+R( x(ξ), u(ξ), u(ξ))\!≥ ∑k=0m−1Lfkh((t0))k!Δtk+Lfmh((t0))m!Δtm+Rmin _k=0^m-1 L_f^kh( x(t_0))k! t^k+ L_f^mh( x(t_0))m! t^m+R_min +LgLfm−1h((t0))(t0)m!Δtm,t0∈[0,∞),ξ∈(t0,t), + L_gL_f^m-1h( x(t_0)) u(t_0)m! t^m,t_0∈[0,∞),ξ∈(t_0,t), Since (t0)∈Krtlc((t0)) u(t_0)∈ K_rtlc( x(t_0)), we have that ∑k=0m−1Lfkh((t0))k!Δtk+Lfmh((t0))m!Δtm _k=0^m-1 L_f^kh( x(t_0))k! t^k+ L_f^mh( x(t_0))m! t^m (19) +LgLfm−1h((t0))(t0)m!Δtm+R((ξ),(ξ),˙(ξ))≥0, \!+ L_gL_f^m-1h( x(t_0)) u(t_0)m! t^m+R( x(ξ), u(ξ), u(ξ))\!≥\!0, t0∈[0,∞),ξ∈(t0,t), t_0∈[0,∞),ξ∈(t_0,t), As the left side of the above constraint is the Taylor’s expansion of h((t))h( x(t)) with Lagrange remainder, we have that h((t))≥0.h( x(t))≥ 0. (20) Now, let us consider any t′∈(t0,t)t ∈(t_0,t). We expand h((t′))h( x(t )) similarly as the one of h((t))h( x(t)) using the Taylor’s expansion with Lagrange remainder, we can get: h((t′))=∑k=0m−1Lfkh((t0))k!Δtk+Lfmh((t0))m!Δtm h( x(t ))= _k=0^m-1 L_f^kh( x(t_0))k! t^k+ L_f^mh( x(t_0))m! t^m (21) +LgLfm−1h((t0))(t0)m!Δtm+R((ξ′),(ξ′),˙(ξ′)), \!+ L_gL_f^m-1h( x(t_0)) u(t_0)m! t^m+R( x(ξ ), u(ξ ), u(ξ )), t0∈[0,∞),ξ′∈(t0,t′), t_0∈[0,∞),ξ ∈(t_0,t ), Since t′∈(t0,t)t ∈(t_0,t) and ξ∈(t0,t′)ξ∈(t_0,t ), we have that R((ξ′),(ξ′),˙(ξ′))≥Rmin,R( x(ξ ), u(ξ ), u(ξ ))≥ R_min, (22) By (16) and the last equation, we have that h((t′))=∑k=0m−1Lfkh((t0))k!Δtk+Lfmh((t0))m!Δtm h( x(t ))= _k=0^m-1 L_f^kh( x(t_0))k! t^k+ L_f^mh( x(t_0))m! t^m (23) +LgLfm−1h((t0))(t0)m!Δtm+R((ξ′),(ξ′),˙(ξ′))≥0, \!+ L_gL_f^m-1h( x(t_0)) u(t_0)m! t^m+R( x(ξ ), u(ξ ), u(ξ ))≥ 0, t0∈[0,∞),ξ′∈(t0,t′), t_0∈[0,∞),ξ ∈(t_0,t ), Since h((t0))≥0h( x(t_0))≥ 0, therefore, we conclude that h((t′))≥0,∀t′∈[t0,t]h( x(t ))≥ 0,∀ t ∈[t_0,t]. When we move t0t_0 within [0,∞)[0,∞), we can ensure that h((t))≥0,∀t∈[t0,∞)h( x(t))≥ 0,∀ t∈[t_0,∞). Thus, the set C is forward invariant for system (1). ■ Remark 1 (rTLC addresses the inter-sampling effect) As shown in the proof of Thm. 2, any controller (t0) u(t_0) that satisfies the rTLC constraint in (16) ensures h((t′))≥0h( x(t ))≥ 0 for all t′∈[t0,t]t ∈[t_0,t]. Therefore, the controller (t0) u(t_0) obtained at the current state (t0) x(t_0) can naturally address the inter-sampling effect. In (16), the only parameter we need to consider is Δt t, which is much less than other approaches (like event-triggered methods [22] [28]) that address the inter-sampling effect. I-B Optimal Control with rTLC Consider a constrained optimal control problem for system (1) with the cost defined as: J((t))=∫0T(‖(t)‖)tJ( u(t))= _0^TC(|| u(t)||)dt (24) where ||⋅||||·|| denotes the 2-norm of a vector. T denotes the final time, and (⋅)C(·) is a strictly increasing function of its argument (such as the energy consumption function (‖(t)‖)=‖(t)‖2C(|| u(t)||)=|| u(t)||^2). We also wish the state x of system (1) to reach a desired state d∈ℝn x_d ^n at the final time T. Mathematically, we wish to achieve the following: min‖(T)−d‖. _ u|| x(T)- x_d||. (25) Assume a safety constraint h()≥0,h( x)≥ 0, (26) with relative degree m that has to be satisfied by system (1). Formally, we have the following constrained optimal control problems: Problem 1 Our objective is to find a controller ∗ u^* for (1) by solving the following optimization: ∗ u^* =argmin∈U∫0T(‖(t)‖)t+w‖(T)−d‖, = _ u∈ U _0^TC(|| u(t)||)dt+w|| x(T)- x_d||, (27) s.t. (26), where w>0w>0. We may consider multiple safety constraints in the above by simply adding more constraints in the above. Approach: We use a rTLC to enforce the satisfaction of the safety constraint (26) for all times. We may employ a Taylor-Lagrange Stability (TLS) function [30] to enforce the state convergence objective (25), and directly take the cost (24) as the objective to reformulate Problem 1 as a sequance of Quadratic Programs (QPs) that take the control u as the decision variable. Since the rTLC only need to consider the current state (t0) x(t_0) without (ξ),ξ∈(t0,t) x(ξ),ξ∈(t_0,t), we can directly solve the Problem 1 by solving the QPs. IV CASE STUDIES AND RESULTS In this section, we present a case study for the Adaptive Cruise Control (ACC). All the computations were conducted in MATLAB, and we used quadprog to solve QPs and used ode45 to integrate the dynamics. Consider the adaptive cruise control (ACC) problem with the vehicle dynamics in the form: [v˙(t)z˙(t)]=[−Fr(v(t))Mv0−v(t)]+[1M0]u(t), [ array[]c v(t)\\ z(t) array ]= [ array[]c -F_r(v(t))M\\ v_0-v(t) array ]+ [ array[]c 1M\\ 0 array ]u(t), (28) where v(t)v(t) denotes the velocity of the ego vehicle along its lane, z(t)z(t) denotes the along-lane distance between the ego and its preceding vehicles, v0>0v_0>0 denotes the speed of the preceding vehicle, M is the mass of the ego vehicle, and u(t)u(t) is the control input of the controlled vehicle. Fr(v(t))F_r(v(t)) denotes the resistance force, which is expressed [13] as: Fr(v(t))=f0sgn(v(t))+f1v(t)+f2v2(t),F_r(v(t))=f_0sgn(v(t))+f_1v(t)+f_2v^2(t), (29) where f0>0,f1>0f_0>0,f_1>0 and f2>0f_2>0 are scalars determined empirically. The first term in Fr(v(t))F_r(v(t)) denotes the coulomb friction force, the second term denotes the viscous friction force and the last term denotes the aerodynamic drag. Objective 1. We wish to minimize the acceleration of the ego vehicle in the form: minu∫0T(u−Fr(v)M)2t. _u _0^T ( u-F_r(v)M )^2dt. (30) Objective 2. The ego vehicle seeks to achieve a desired speed vd>0v_d>0. Safety. We require that the distance z(t)z(t) between the controlled vehicle and its immediately preceding vehicle be greater than a constant δ>0δ>0 for all the times, i.e., z(t)≥c,∀t≥0,z(t)≥ c, ∀ t≥ 0, (31) where c>0c>0. Control limitation. We also consider a control constraint umin≤u(t)≤umax,umin=−cdMg,umax=caMg,g=9.81m/s2,ca>0,cd>0u_min≤ u(t)≤ u_max,u_min=-c_dMg,u_max=c_aMg,g=9.81m/s^2,c_a>0,c_d>0 for (28). The ACC problem is to find a control policy that achieves Objectives 1 and 2 subject to the safety constraint and control bound. The relative degree of (31) is two, and we use either a second order HOCBF, TLC, event-triggered TLC, and the proposed rTLC to implement it by defining h()=z−c≥0h( x)=z-c≥ 0 and: Lf2h()+LgLfh()u+(p1+p2)Lfh()+p1p2h(x)≥0.HOCBF aligned L_f^2h( x)+L_gL_fh( x)u+(p_1+p_2)L_fh( x)\\ +p_1p_2h(x)≥ 0. aligned (32) Lf2h()+LgLfh()u+2ΔtLfh()+2Δt2h()≥0.TLC aligned L_f^2h( x)+L_gL_fh( x)u+ 2 tL_fh( x)\\ + 2 t^2h( x)≥ 0. aligned (33) The state bound for the event-triggered TLC is defined as S()=∈ℝ2:−lower≤+up.S( x)=\ y ^2: x- x_lower≤ y≤ x+ x_up\. Lf2h((t0))+LgLfh()u(t0)+2ΔtLfh((t0))+2Δt2h((t0))+R((ξ),u(ξ),u˙(ξ))≥0.rTLC aligned L_f^2h( x(t_0))+L_gL_fh( x)u(t_0)+ 2 tL_fh( x(t_0))\\ + 2 t^2h( x(t_0))+R( x(ξ),u(ξ), u(ξ))≥ 0. aligned (34) ξ∈(t0,t)ξ∈(t_0,t), where R((ξ),u(ξ),u˙(ξ))=Δt3((Lf3h((ξ))+LgLf2h((ξ)))u(ξ) R( x(ξ),u(ξ), u(ξ))= t3((L_f^3h( x(ξ))+L_gL_f^2h( x(ξ)))u(ξ) (35) +(LfLgLfh((ξ))+Lg2Lfh((ξ)))u(ξ) +(L_fL_gL_fh( x(ξ))+L_g^2L_fh( x(ξ)))u(ξ) +LgLfh((ξ))u˙(ξ)),ξ∈(t0,t). +L_gL_fh( x(ξ)) u(ξ)),ξ∈(t_0,t). Following (11), the above equation can be rewritten as R((ξ),u(ξ),u˙(ξ))=Δt3(−u˙(ξ)−f1u(ξ)−2f2v(ξ)u(ξ)M) R( x(ξ),u(ξ), u(ξ))= t3(- u(ξ)-f_1u(ξ)-2f_2v(ξ)u(ξ)M) (36) ≥Δt3(−1Δt(umax−u(ξ))+f1u(ξ)+2f2v(ξ)u(ξ)M) ≥ t3( - 1 t(u_max-u(ξ))+f_1u(ξ)+2f_2v(ξ)u(ξ)M) ≥Δt3(−1Δt(umax−umin)+f1umin+2f2v(ξ)uminM) ≥ t3( - 1 t(u_max-u_min)+f_1u_min+2f_2v(ξ)u_minM) ≥Δt3(−1Δt(umax−umin)+f1uminM) ≥ t3( - 1 t(u_max-u_min)+f_1u_minM) +Δt3(2f2(v(t0)+Δt(umax−Fr(v(ξ)))/M)uminM) + t3( 2f_2(v(t_0)+ t(u_max-F_r(v(ξ)))/M)u_minM) ≥Δt3(−1Δt(umax−umin)+f1uminM) ≥ t3( - 1 t(u_max-u_min)+f_1u_minM) +Δt3(2f2(v(t0)+Δtumax/M)uminM). + t3( 2f_2(v(t_0)+ tu_max/M)u_minM). We define the RminR_min by the lower bound in the above equation in the rTLC. We employ a TLS with relative degree one to enforce the desired speed. All the simulation parameters are shown in Table I, in which dtdt denotes the time interval we apply the control to the system dynamics (28) at each discretization time. As long as dt≤Δtdt≤ t, we can easily show that the safety of the system can be guaranteed. TABLE I: Simulation parameters for ACC Parameter Value Units Parameter Value Units v(0)v(0) 24 m/sm/s z(0)z(0) 90 m v0v_0 13.89 m/sm/s vdv_d 24 m/sm/s M 1650 kgkg g 9.81 m/s2m/s^2 f0f_0 0.1 N f1f_1 5 Ns/mNs/m f2f_2 0.25 Ns2/mNs^2/m c 10 m Δt t 0.1 s dtdt 0.1 s cac_a 0.4 unitless cdc_d 0.7 unitless lower x_lower (0.5, 1) unitless up x_up (0.5,1) unitless The simulation results are shown in Fig 1. All the HOCBF, event-driven TLC and rTLC can guarantee the vehicle safety, while the time-driven TLC may violate the safety constraint due to the inter-sampling effect. However, the rTLC may be conservative as the vehicle stays unnecessarily far away from the preceding vehicle, as shown with a larger h()h( x) steady value for rTLC in the last framework of Fig. 1. We found that decreasing Δt t could significantly reduce the conservativeness, as shown in Table I. The event-driven TLC is a little more computational expensive than other methods. Figure 1: Comparison of speed, control and safety function h()h( x) profiles between between time-driven HOCBF, time-driven TLC, event-driven TLC, and the proposed rTLC (Δt=0.85,dt=0.1 t=0.85,dt=0.1). TABLE I: Comparison of minimum CBF values and computation time between time-driven HOCBF, time-driven TLC, event-driven TLC, and the proposed rTLC. Safety in time-driven is violated due to the inter-sampling effect. Method min h()h( x) Compute time (s) Time-driven HOCBF 1.8250e-7 0.0013± 1.8714e-4 Time-driven TLC -0.6414 0.0013± 1.2619e-4 Event-driven TLC 0.6511 0.0017± 6.8994e-4 rTLC Δt=0.85,dt=0.1 t=0.85,dt=0.1 6.2487 0.0014± 1.9303e-4 rTLC Δt=0.5,dt=0.5 t=0.5,dt=0.5 1.1867 0.0013± 2.0533e-4 rTLC Δt=0.1,dt=0.1 t=0.1,dt=0.1 0.0092 0.0014± 3.8113e-4 V Conclusion We proposed a robust Taylor-Lagrange control method for ensuring the safety of nonlinear control systems using a higher-order expansion of the safety function with Taylor’s theorem with Lagrange remainder. The proposed robust Taylor-Lagrange control method can naturally address the inter-sampling effect while introducing much less hyper-parameters than other methods. We applied the proposed method to an adaptive cruise control problem to demonstrate the effectiveness of our methods. In the future, we plan to study the tradeoff between conservativeness and feasibility of the proposed robust Taylor-Lagrange control method. 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