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• W4381946688 abstract "This work presents a theoretical framework for the safety-critical control of time delay systems. The theory of control barrier functions, that provides formal safety guarantees for delay-free systems, is extended to systems with state delay. The notion of control barrier functionals is introduced, to attain formal safety guarantees by enforcing the forward invariance of safe sets defined in the infinite dimensional state space. The proposed framework is able to handle multiple delays and distributed delays both in the dynamics and in the safety condition, and provides an affine constraint on the control input that yields provable safety. This constraint can be incorporated into optimization problems to synthesize pointwise optimal and provable safe controllers. The applicability of the proposed method is demonstrated by numerical simulation examples. Modern control systems place high priority on safety, which is frequently a precursor to other control goals such as performance, efficiency, and sustainability. There exist numerous examples—from self-driving autonomous vehicles,1 through robotic systems2-4 to human-robot collaboration5-7—where safety plays a key role for reliable autonomy or sustainable operation. Safety is crucial even outside engineering, including biological applications and epidemiological models that describe pandemics.8, 9 Therefore, establishing safety-critical control techniques is of high significance with wide-spread application domains. To formally address safety in dynamical and control systems, one can define a safe set over the state space, and safety can be framed as the forward invariance of that set: the system must evolve within the safe set for all time. Rigorous guarantees of safety necessitate a theory for ensuring forward set invariance. Barrier functions (or safety functions) have been established to certify set invariance in dynamical systems,10-14 while control barrier functions (CBFs) enable safe controller synthesis in control systems. The framework of CBFs was first introduced in Reference 15 and later refined in Reference 16. A comprehensive review of safety-critical control can be found in Reference 17 and the references therein. While most works in safety-critical control are applied to delay-free systems, time delays often occur in many applications. For example, the reflex delay of human operators affects human-machine interactions; models of vehicular traffic include the reaction time of drivers as delays;18 wheel-shimmy motion – experienced on vehicles due to the elastic contact between tires and the road – can be captured using models with distributed delay;19 manufacturing processes like metal cutting may suffer from vibrations due to the delayed regenerative effect of chip formation;20 hydraulic systems include time delays caused by wave propagation in pipes;21, 22 and epidemiological models contain delays due to the incubation period of infectious diseases.23, 24 Time delay also plays important role in population dynamics,25 neural networks,26 brain dynamics,27 human sensory system,28, 29 and robotic systems.30 Such time delays may render control systems unsafe if controllers are designed without considering the delay. The main challenge of controlling systems with state delay originates from the infinite dimensional nature of delayed dynamics. Namely, the state of the system is a function over the delay period, that implies an infinite dimensional state space. As such, time delay systems are often described by functional differential equations (FDEs),43-47 while scalar safety measures can be constructed as functionals of the state. There exist a few instances of using functionals in the literature in the context of safety. Safety verification for partial differential equations using barrier functionals can be found in Reference 48 and state-constrained control considering integral barrier Lyapunov functionals are discussed in References 49 and 50. For autonomous time delay systems without control,51 introduced the concept of safety functionals, which has been investigated further in Reference 52 by means of discretization. The relationship between discretization and functionals is discussed in Reference 53, while safe domains interpreted as basin of attraction were investigated for delayed dynamical systems in Reference 54. These works, however, do not address control systems with time delay. The main contribution of this work is the establishment of a framework for control synthesis with formal safety guarantees for control systems with state delay. While this includes, for instance, systems of the form (2) with safety requirement (3), we discuss a much broader class of time delay systems and safety conditions. To achieve safety, we introduce the concept of control barrier functionals as a tool for synthesizing safety-critical controllers, through building on the existing notions of safety functionals51 and control barrier functions.16 The corresponding controllers can be constructed as safety filters illustrated in Figure 1. We use the theory of retarded and neutral functional differential equations to prove the underlying formal safety guarantees. We remark that a few recent papers have also approached this problem parallel to our work. Namely, Reference 55 considers delays with disturbances while56-58 investigate the combination of stability and safety by the application of Razumikhin- and Krasovskii-type control Lyapunov and control barrier functionals. Although these recent works share some of the ideas presented in this paper, we establish a comprehensive in-depth study that is not covered by previous works. This includes a wider class of control barrier functionals, an exhaustive discussion on how to calculate the derivatives of these functionals, the safety of neutral FDEs, the notion of relative degree for time delay systems, and numerous application examples. However, we do not address questions related to stability or disturbances. The rest of the paper is organized as follows. In Section 2, safety is revisited for delay-free dynamical and control systems through the notions of safety functions and control barrier functions, respectively. Then, Sections 3 and 4 present the major contributions of this work: formal guarantees of safety for time delay systems. Section 3 establishes the theoretical foundations of safety functionals that certify the safety of autonomous delayed dynamical systems, while Section 4 discusses safety-critical control with state delay by means of control barrier functionals. In Section 5, we demonstrate safety-critical control on illustrative examples and a more practical case study through the regulated delayed predator-prey problem. Finally, we conclude our results and discuss future research directions in Section 6. In this section, we revisit safety certification for delay-free dynamical systems and safety-critical control for delay-free control systems, that are described by ordinary differential equations (ODEs). Specifically, we focus on the notions of safety functions and control barrier functions. Then, in Section 3, we extend these frameworks to time delay systems. We consider system (4) safe when the solution x ( t ) $$ x(t) $$ evolves within a safe set S ⊂ ℝ n $$ Ssubset {mathbb{R}}^n $$ , as given by the following definition. Definition 1. (safety and forward invariance)System (4) is safe w.r.t. set S ⊂ ℝ n $$ Ssubset {mathbb{R}}^n $$ , if S $$ S $$ is forward invariant w.r.t. (4) such that x ( 0 ) ∈ S ⇒ x ( t ) ∈ S $$ x(0)in Skern3.0235pt Rightarrow kern3.0235pt x(t)in S $$ , ∀ t ≥ 0 $$ forall tge 0 $$ for the solution of (4). Definition 2. (safety function)A continuously differentiable function h : ℝ n → ℝ $$ h:{mathbb{R}}^nto mathbb{R} $$ is a safety function for (4) on S $$ S $$ defined by (5) if there exists α ∈ 𝒦 ∞ e (see footnote1) such that ∀ x ∈ ℝ n $$ forall xin {mathbb{R}}^n $$ : Further technical details with discussion about α $$ alpha $$ can be found in Reference 59. With this definition, the main result of Reference 16 establishes the safety of dynamical systems. Theorem 1. ([16])Set S $$ S $$ in (5) is forward invariant w.r.t. (4) if h $$ h $$ is a safety function for (4) on S $$ S $$ , that is, (6) is satisfied. The proof can be found in Appendix A and will be used as basic idea to establish safety for systems with time delay. Safety functions and Theorem 1 provide a useful tool for certifying the safety of dynamical systems: one needs to verify that (6) holds. A similar concept can be used in control systems to design controllers that enforce the safety of the closed-loop dynamics, which is discussed next. Definition 3. (control barrier function, CBF[16])A continuously differentiable function h : ℝ n → ℝ $$ h:{mathbb{R}}^nto mathbb{R} $$ is a control barrier function (CBF)2 for (7) on S $$ S $$ defined by (5), if there exists α ∈ 𝒦 ∞ e such that ∀ x ∈ ℝ n $$ forall xin {mathbb{R}}^n $$ : With this definition, control systems can be rendered safe with the following extension of Theorem 1. Theorem 2. ([16])If h $$ h $$ is a CBF for (7) on S $$ S $$ defined by (5), then any locally Lipschitz continuous controller k : ℝ n → ℝ m $$ k:{mathbb{R}}^nto {mathbb{R}}^m $$ , u = k ( x ) $$ u=k(x) $$ satisfying: Proof. ([16])Definition 3 ensures that controller k $$ k $$ exists. Then, considering the closed-loop dynamics in (8), the set S $$ S $$ is forward invariant according to Theorem 1. This result provides systematic means to safety-critical controller synthesis: one needs to satisfy condition (11) when designing the controller. Condition (11) can be incorporated into optimization problems as constraint to find pointwise optimal safety-critical controllers. For example, one can modify a desired controller k des $$ {k}_{mathrm{des}} $$ in a minimally invasive fashion to a safe controller k $$ k $$ by solving a quadratic program, as stated formally below. Corollary 1.Given a CBF h $$ h $$ and a locally Lipschitz continuous desired controller k des : ℝ n → ℝ m $$ {k}_{mathrm{des}}:{mathbb{R}}^nto {mathbb{R}}^m $$ , u des = k des ( x ) $$ {u}_{mathrm{des}}={k}_{mathrm{des}}(x) $$ , the following quadratic program (QP) yields a controller k : ℝ n → ℝ m $$ k:{mathbb{R}}^nto {mathbb{R}}^m $$ , u = k ( x ) $$ u=k(x) $$ that renders set S $$ S $$ in (5) forward invariant w.r.t. (8): Definition 4. (relative degree[62])Function h : ℝ n → ℝ $$ h:{mathbb{R}}^nto mathbb{R} $$ has relative degree r $$ r $$ (where r ∈ ℤ $$ rin mathbb{Z} $$ , r ≥ 1 $$ rge 1 $$ ) w.r.t. (7) if it is r $$ r $$ times continuously differentiable and the following holds ∀ x ∈ ℝ n $$ forall xin {mathbb{R}}^n $$ : This highlights that the relative degree is an important property in safety-critical control. We will show that the relative degree may be significantly affected when time delay is included in the system.67 Furthermore, we will accommodate the above extension method to time delay systems in Section 4.1 and apply it in examples in Section 5.1 (cases 3 and 4) and Section 5.2. In what follows, we extend the above concepts to certify safety and provide safety-critical controllers for time delay systems. We focus on so-called retarded and neutral type systems. If the rate of change of state depends on the past states of the system, then the corresponding mathematical model is a retarded functional differential equation (RFDE). If the rate of change of state depends on its own past values as well, then the corresponding equation is a neutral functional differential equation (NFDE). Since the state space of (17) is ℬ $$ mathcal{B} $$ , which is infinite dimensional, one needs the state x t $$ {x}_t $$ to evolve within an infinite dimensional safe set 𝒮 ⊂ ℬ to certify safety as given by the following definition. Definition 5. (safety and forward invariance of RFDE)System (17) is safe w.r.t. set 𝒮 ⊂ ℬ , if 𝒮 is forward invariant w.r.t. (17) such that x 0 ∈ 𝒮 ⇒ x t ∈ 𝒮 , ∀ t ≥ 0 $$ forall tge 0 $$ for the solution of (17). Definition 6. (safety functional[51])A continuously Fréchet differentiable functional ℋ : ℬ → ℝ $$ mathscr{H}:mathcal{B} to mathbb{R} $$ is a safety functional for (17) on 𝒮 defined by (20) if there exists α ∈ 𝒦 ∞ e such that ∀ x t ∈ ℬ $$ forall {x}_tin mathcal{B} $$ : Note that the derivative of ℋ $$ mathscr{H} $$ depends both on x t $$ {x}_t $$ and x ˙ t $$ {dot{x}}_t $$ in (19), and its calculation is detailed below. The following theorem from Reference 51 certifies safety, that is, the forward invariance of 𝒮 via safety functionals. Theorem 3. ([51])Set 𝒮 in (20) is forward invariant w.r.t. (17) if ℋ $$ mathscr{H} $$ is a safety functional for (17) on 𝒮 , that is, (21) is satisfied. Proof. ([51])The proof follows from the comparison lemma. We set up the initial value problem (with y ∈ ℝ $$ yin mathbb{R} $$ ): The left-hand side of (21) is the time derivative of ℋ $$ mathscr{H} $$ along the solution of (17). While for finite dimensional delay-free systems the derivative h ˙ $$ dot{h} $$ is given by the directional derivative or Lie derivative,43, 70 the derivative ℋ ˙ $$ dot{mathscr{H}} $$ in the presence of time delay and infinite dimensional dynamics has an intricate representation which we break down below. Recall that in the delay-free case, the derivative h ˙ $$ dot{h} $$ of the safety function is calculated by a scalar product as h ˙ ( x ) = ∇ h ( x ) x ˙ $$ dot{h}(x)=nabla h(x)dot{x} $$ . That is, h ˙ $$ dot{h} $$ is given by a linear function of x ˙ $$ dot{x} $$ where x ˙ = f ( x ) $$ dot{x}=f(x) $$ . For time delay systems, the derivative ℋ ˙ $$ dot{mathscr{H}} $$ of the safety functional is calculated by an integral and it is given by a linear functional of x ˙ t $$ {dot{x}}_t $$ in (19). This is stated by the theorem below. Theorem 4.Consider system (17) and let ℋ : ℬ → ℝ $$ mathscr{H}:mathcal{B} to mathbb{R} $$ be a continuously Fréchet differentiable functional. Then there exists a unique η : ℬ × [ − τ , 0 ] → ℝ 1 × n $$ eta :mathcal{B} times left[-tau, 0right]to {mathbb{R}}^{1times n} $$ that is of bounded variation47, 69 in its second argument such that the time derivative of ℋ $$ mathscr{H} $$ along (17) can be expressed as: Proof.Here we provide the main steps of the proof and the remaining details (definitions and a lemma) are in Appendix B. The derivative of ℋ $$ mathscr{H} $$ along (17) can be expressed by the Gâteaux derivative (see definition at (B1)) along x ˙ t $$ {dot{x}}_t $$ as: The expressions of η $$ eta $$ in (25) and w $$ w $$ in (29) depend on the specific form of ℋ $$ mathscr{H} $$ (just as the expression of ∇ h $$ nabla h $$ depends on the form of h $$ h $$ ). We demonstrate the calculation of these expressions and the time derivative of the functional ℋ $$ mathscr{H} $$ by an example below, which covers most of the scenarios that appear in practical applications. Example 1.Consider the system (17) with functional ℋ $$ mathscr{H} $$ that contains both delay-free states, discrete (point) delays τ j ∈ [ − τ , 0 ] $$ {tau}_jin left[-tau, 0right] $$ , j ∈ { 1 , … , l } $$ jin left{1,dots, lright} $$ , and a continuous (distributed) delay over [ − σ 1 , − σ 2 ] ⊆ [ − τ , 0 ] $$ left[-{sigma}_1,-{sigma}_2right]subseteq left[-tau, 0right] $$ , defined as: Corollary 2.When ℋ ( x t ) $$ mathscr{H}left({x}_tright) $$ involves both delay-free states, point delays τ j ∈ [ − τ , 0 ] $$ {tau}_jin left[-tau, 0right] $$ , j ∈ { 1 , … , l } $$ jin left{1,dots, lright} $$ , and a distributed delay over [ − σ 1 , − σ 2 ] ⊆ [ − τ , 0 ] $$ left[-{sigma}_1,-{sigma}_2right]subseteq left[-tau, 0right] $$ , the time derivative of ℋ $$ mathscr{H} $$ in (25) along (19) is of the form: This leads to two important properties about the derivative of ℋ $$ mathscr{H} $$ , given by the following remarks. We will rely on these properties when discussing the relative degree of control barrier functionals in Section 4.1. Remark 1. (( ℋ ( x t ) $$ mathscr{H}left({x}_tright) $$ contains x ( t ) $$ x(t) $$ ))When ℋ ( x t ) $$ mathscr{H}left({x}_tright) $$ contains the present state x ( t ) = x t ( 0 ) $$ x(t)={x}_t(0) $$ , it is indicated by w 0 ( x t ) ≠ 0 $$ {w}_0left({x}_tright)ne 0 $$ , ∀ x t ∈ ℬ $$ forall {x}_tin mathcal{B} $$ . Then, the time derivative ℋ ˙ ( x t , x ˙ t ) $$ dot{mathscr{H}}left({x}_t,{dot{x}}_tright) $$ is directly affected by the right-hand side ℱ ( x t ) $$ mathcal{F}left({x}_tright) $$ . This will be a necessary requirement for enforcing safety via control (i.e., when we maintain safety by designing ℱ $$ mathcal{F} $$ for a closed control loop). Remark 2. ( ℒ ℱ ℋ $$ {mathcal{L}}_{mathcal{F}}mathscr{H} $$ independent of x ˙ t $$ {dot{x}}_t $$ )If w d $$ {w}_{mathrm{d}} $$ is continuously differentiable in ϑ $$ vartheta $$ , with derivative denoted by w d ′ $$ {w}_{mathrm{d}}^{prime } $$ , then the integral ∫ − σ 1 − σ 2 w d ( x t , ϑ ) x ˙ t ( ϑ ) d ϑ $$ {int}_{-{sigma}_1}^{-{sigma}_2}{w}_{mathrm{d}}left({x}_t,vartheta right){dot{x}}_tleft(vartheta right)mathrm{d}vartheta $$ can be simplified. Since ∂ ∂ t x t ( ϑ ) = ∂ ∂ ϑ x t ( ϑ ) $$ frac{partial }{partial t}{x}_tleft(vartheta right)=frac{partial }{partial vartheta }{x}_tleft(vartheta right) $$ , integration by parts eliminates x ˙ t $$ {dot{x}}_t $$ and leads to an expression that depends on x t $$ {x}_t $$ only: While Example 1 covers most practical choices of safety functionals, it does not include all possibilities for ℋ $$ mathscr{H} $$ . A more general example with a double integral can be found in Appendix C, and one could also include triple, quadruple, integrals. The safety of (39) can be formulated the same way as it was done for retarded systems in Section 3.1. Safety means that the state x t $$ {x}_t $$ evolves within the safe set 𝒮 ⊂ ℬ , which is constructed by the safety functional ℋ : ℬ → ℝ $$ mathscr{H}:mathcal{B} to mathbb{R} $$ (see Definitions 5 and 6). Then, the safety of the neutral system (39) is formally certified by the following corollary. Corollary 3.Set 𝒮 in (20) is forward invariant w.r.t. (39) if ℋ $$ mathscr{H} $$ is a safety functional for (39) on 𝒮 , that is: Building upon the framework establishing safety for dynamical systems with time delays, we now extend this approach to control systems with time delays. Similarly to how safety functions were extended to control barrier functions in Section 2, in this section we extend safety functionals to control barrier functionals, and use them as tool for safety-critical controller synthesis. To design a control input that guarantees the system to be safe motivates the introduction of control barrier functionals. Definition 7. (control barrier functional, CBFal)A continuously Fréchet differentiable functional ℋ : ℬ → ℝ $$ mathscr{H}:mathcal{B} to mathbb{R} $$ is a control barrier functional (CBFal) for (44) on 𝒮 defined by (20), if there exists α ∈ 𝒦 ∞ e such that ∀ x t ∈ ℬ $$ forall {x}_tin mathcal{B} $$ : With the definition of CBFal we state our main result to ensure safety for systems with state delay by extending Theorem 3. Theorem 5.If ℋ $$ mathscr{H} $$ is a CBFal for (44) on 𝒮 defined by (20), then any locally Lipschitz continuous controller 𝒦 : ℬ × 𝒬 → ℝ m , u = 𝒦 ( x t , x ˙ t ) satisfying: Proof.Definition 7 ensures that controller 𝒦 exists. Then considering the closed-loop dynamics in (46), the set 𝒮 is forward invariant according to Theorem 3. This result motivates the construction of pointwise optimal safety-critical controllers that use the nearest safe action to a nominal but potentially unsafe controller. Corollary 4.Given a CBFal ℋ $$ mathscr{H} $$ and a locally Lipschitz continuous desired controller 𝒦 des : ℬ × 𝒬 → ℝ m , u des = 𝒦 des ( x t , x ˙ t ) under the condition (40), the following quadratic program (QP) yields a controller 𝒦 : ℬ × 𝒬 → ℝ m , u = 𝒦 ( x t , x ˙ t ) that renders set 𝒮 in (20) forward invariant w.r.t. (46): The derivation of (52) is detailed in Appendix D. This allows operating the nominal controller when it is safe ( ϕ ≥ 0 $$ phi ge 0 $$ ), and modifies the input to keep the system safe otherwise ( ϕ < 0 $$ phi <0 $$ ). Remark 3. (control of neutral systems)Note that while the control system (44) contains retarded terms (functionals of x t $$ {x}_t $$ ) only, the closed-loop system (46) is neutral (includes x ˙ t $$ {dot{x}}_t $$ ). Therefore, one may extend the theory to neutral control systems of the form: Corollary 5.If ℋ $$ mathscr{H} $$ is a CBFal for (53) on 𝒮 defined by (20), then any locally Lipschitz continuous controller 𝒦 : ℬ × 𝒬 → ℝ m , u = 𝒦 ( x t , x ˙ t ) satisfying (50) ∀ x t ∈ 𝒮 renders set S $$ S $$ forward invariant w.r.t. (54). In the presence of time delay, the relative degree is affected by the delay itself, which is detailed next. For simplicity, we omit further discussions on neutral control systems, and the rest of the paper addresses the retarded control system (44). The CBFal condition (47) sufficiently holds if ℒ 𝒢 ℋ ( x t ) ≠ 0 for all ∀ x t ∈ ℬ $$ forall {x}_tin mathcal{B} $$ . We refer to this as ℋ $$ mathscr{H} $$ having relative degree 1. Relative degree is an important concept for time delay systems, too, which motivates the extension of Definition 4. Definition 8. (relative degree of functional)Functional ℋ : ℬ → ℝ $$ mathscr{H}:mathcal{B} to mathbb{R} $$ has relative degree r $$ r $$ (where r ∈ ℤ $$ rin mathbb{Z} $$ , r ≥ 1 $$ rge 1 $$ ) w.r.t. (44) if it is r $$ r $$ times continuously Fréchet differentiable and satisfies the two conditions below. Condition I is specific to time delay systems and does not have a delay-free counterpart in Definition 4. This condition is imposed because otherwise for higher relative degree, r ≥ 2 $$ rge 2 $$ , higher time derivatives of ℋ $$ mathscr{H} $$ could include higher derivatives of x t $$ {x}_t $$ . Synthesizing a controller in this case could lead to a closed-loop system where the rate of change of state depends on past values of higher derivatives of the state (e.g., systems of the form x ˙ ( t ) = ℱ ( x t , x ˙ t , x ¨ t ) $$ dot{x}(t)=mathcal{F}left({x}_t,{dot{x}}_t,{ddot{x}}_tright) $$ ), which is called advanced functional differential equation. We demonstrate this by an example in Section 5.1. Advanced type equations are rarely used in engineering applications due to their inverted causality problem,75 hence Condition I excludes this possibility. In conclusion, distributed delays in ℋ $$ mathscr{H} $$ may induce higher relative, whereas point delays may lead to no valid relative degree, if the present state does not explicitly appear in ℋ $$ mathscr{H} $$ . Definition 9. (extended control barrier functional)Let ℋ : ℬ → ℝ $$ mathscr{H}:mathcal{B} to mathbb{R} $$ be a twice continuously Fréchet differentiable functional with continuously differentiable α ∈ 𝒦 ∞ e , ℒ 𝒢 ℋ ( x t ) = 0 and continuously differentiable ℒ ℱ ℋ $$ {mathcal{L}}_{mathcal{F}}mathscr{H} $$ excluding terms of x ˙ t $$ {dot{x}}_t $$ . Then functional ℋ e : ℬ → ℝ $$ {mathscr{H}}_{mathrm{e}}:mathcal{B} to mathbb{R} $$ defined by (58) is an extended control barrier functional (extended CBFal) for (44) on 𝒮 ∩ 𝒮 e defined by (20) and (59), if there exists α e ∈ 𝒦 ∞ e such that ∀ x t ∈ ℬ $$ forall {x}_tin mathcal{B} $$ : Note that condition (60) sufficiently holds if ℒ 𝒢 ℒ ℱ ℋ ( x t ) ≠ 0 , ∀ x t ∈ ℬ $$ forall {x}_tin mathcal{B} $$ , that is, in case ℋ $$ mathscr{H} $$ has relative degree 2. With this definition we can state the theorem to ensure safety for systems with (potentially delay-induced) relative degree 2. Theorem 6.If ℋ e $$ {mathscr{H}}_{mathrm{e}} $$ is an extended CBFal for (44) on 𝒮 ∩ 𝒮 e defined by (20) and (59), then any locally Lipschitz continuous controller 𝒦 : ℬ × 𝒬 → ℝ m , u = 𝒦 ( x t , x ˙ t ) satisfying: Proof.We prove that x 0 ∈ 𝒮 ∩ 𝒮 e ⇒ x t ∈ 𝒮 ∩ 𝒮 e , ∀ t ≥ 0 $$ forall tge 0 $$ . Note that x 0 ∈ 𝒮 ∩ 𝒮 e yields both x 0 ∈ 𝒮 and x 0 ∈ 𝒮 e . By Theorem 5, we have x 0 ∈ 𝒮 e ⇒ x t ∈ 𝒮 e , ∀ t ≥ 0 $$ forall tge 0 $$ . This means ℋ e ( x t ) ≥ 0 $$ {mathscr{H}}_{mathrm{e}}left({x}_tright)ge 0 $$ holds, that is, ℋ ˙ ( x t ) ≥ − α ( ℋ ( x t ) ) $$ dot{mathscr{H}}left({x}_tright)ge -alpha left(mathscr{H}left({x}_tright)right) $$ based on (58). Therefore, applying Theorem 3 (or more precisely, Corollary 5) yields x 0 ∈ 𝒮 ∩ 𝒮 e ⇒ x t ∈ 𝒮 ∩ 𝒮 e , ∀ t ≥ 0 $$ forall tge 0 $$ . With this theorem, one can design a pointwise optimal controller similarly as in (52). Corollary 6.Given an extended CBFal ℋ e $$ {mathscr{H}}_{mathrm{e}} $$ and a locally Lipschitz continuous desired controller 𝒦 des : ℬ × 𝒬 → ℝ m , u des = 𝒦 des ( x t , x ˙ t ) , the following quadratic program (QP) yields a controller 𝒦 : ℬ × 𝒬 → ℝ m , u = 𝒦 ( x t , x ˙ t ) that renders set 𝒮 ∩ 𝒮 e in (20) and (59) forward invariant w.r.t. (46): The detailed derivation of (64) is similar to (52) which is discussed in Appendix D. Now we apply the theoretical constructions of this article on demonstrative examples and a relevant practical application. First, we address systems with point delay, second, we discuss a scalar control system with different types of delay. Example 2.Based on the motivating example (2)–(3), consider the following affine control system with point delay τ > 0 $$ tau >0 $$ : System (65) and the corresponding safe set can be rewritten in the form as (44) and (20) with the functionals ℱ : ℬ → ℝ n $$ mathcal{F}:mathcal{B} to {mathbb{R}}^n $$ , 𝒢 : ℬ → ℝ n × m and ℋ : ℬ → ℝ $$ mathscr{H}:mathcal{B} to mathbb{R} $$ defined by: Example 3.Consider the scalar control system: Case 1: First, we intend to keep the solution x ( t ) $$ x(t) $$ within the safe range of [ − 1 , 1 ] $$ left[-1,1right] $$ , defined by − 1 ≤ x ( t ) ≤ 1 $$ -1le x(t)le 1 $$ . To enforce this, we construct the CBFal with delay-free term as: We apply Theorem 5 to ensure safety and implement the QP-based controller (52) using the desired controller 𝒦 des ( x t , x ˙ t ) = 0 and a linear class- 𝒦 ∞ e function α ( r ) = γ r $$ alpha (r)=gamma r $$ with γ > 0 $$ gamma >0 $$ , that results in: The performance of this controller is demonstrated by numerically integrating (75). Simulation results are plotted in the first column of Figure 3 A,D,G,J for τ = 1 $$ tau =1 $$ , γ = 1 $$ gamma =1 $$ and initial condition x 0 ( ϑ ) = 0 . 4 $$ {x}_0left(vartheta right)=0.4 $$ , ϑ ∈ [ − τ , 0 ] $$ vartheta in left[-tau, 0right] $$ . The panel (A) shows the evolution of the state, and the panel (D) indicates the corresponding control input. As the state gets close to the safe set boundary (indicated by green line), the controller needs to intervene by deviating from the desired zero input (see around t = 2 $$ t=2 $$ ) and forces the system to evolve within the safe set. Intervention starts when the state reaches the switching surface in (75): − 2 x 4 ( t ) + γ 1 − x 2 ( t ) = 0 $$ -2{x}^4(t)+gamma left(1-{x}^2(t)right)=0 $$ (see the dashed line in panel (A)). Panels (G), (J) at the bottom indicate that safety is successfully maintained as ℋ $$ mathscr{H} $$ is positive for all time while the trajectory in the corresponding phase portrait is kept within − 1 ≤ x ( t ) ≤ 1 $$ -1le x(t)le 1 $$ . Case 2: Next, we intend to keep the squared mean of the solution x ( t" @default.
• W4381946688 created "2023-06-26" @default.
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• W4381946688 date "2023-05-08" @default.
• W4381946688 modified "2023-10-15" @default.
• W4381946688 title "Control barrier functionals: Safety‐critical control for time delay systems" @default.
• W4381946688 cites W1976166148 @default.
• W4381946688 cites W1980569135 @default.
• W4381946688 cites W1995877162 @default.
• W4381946688 cites W2025434683 @default.
• W4381946688 cites W2044294889 @default.
• W4381946688 cites W2062373349 @default.
• W4381946688 cites W2079871064 @default.
• W4381946688 cites W2122227959 @default.
• W4381946688 cites W2124760969 @default.
• W4381946688 cites W2129992593 @default.
• W4381946688 cites W2134590937 @default.
• W4381946688 cites W2139824906 @default.
• W4381946688 cites W2143805583 @default.
• W4381946688 cites W2152981451 @default.
• W4381946688 cites W2318055712 @default.
• W4381946688 cites W231975775 @default.
• W4381946688 cites W2344321026 @default.
• W4381946688 cites W2400848186 @default.
• W4381946688 cites W2428098056 @default.
• W4381946688 cites W2505714040 @default.
• W4381946688 cites W2525143841 @default.
• W4381946688 cites W2593108859 @default.
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