Control barrier functionals: Safety‐critical control for time delay systems

This work presents a theoretical framework for the safety-critical control of time delay systems. The theory barrier functions, that provides formal safety guarantees delay-free systems, is extended to systems with state delay. notion functionals introduced, attain by enforcing forward invariance safe sets defined in infinite dimensional space. proposed able handle multiple delays and distributed both dynamics condition, an affine constraint on input yields provable safety. can be incorporated into optimization problems synthesize pointwise optimal controllers. applicability method demonstrated numerical simulation examples. Modern place high priority safety, which frequently precursor other goals such as performance, efficiency, sustainability. There exist numerous examples—from self-driving autonomous vehicles,1 through robotic systems2-4 human-robot collaboration5-7—where plays key role reliable autonomy or sustainable operation. Safety crucial even outside engineering, including biological applications epidemiological models describe pandemics.8, 9 Therefore, establishing techniques significance wide-spread application domains. To formally address dynamical one define set over space, framed set: system must evolve within all time. Rigorous necessitate ensuring invariance. Barrier functions (or functions) have been established certify systems,10-14 while (CBFs) enable controller synthesis CBFs was first introduced Reference 15 later refined 16. A comprehensive review found 17 references therein. While most works are applied often occur many applications. For example, reflex human operators affects human-machine interactions; vehicular traffic include reaction drivers delays;18 wheel-shimmy motion – experienced vehicles due elastic contact between tires road captured using delay;19 manufacturing processes like metal cutting may suffer from vibrations delayed regenerative effect chip formation;20 hydraulic caused wave propagation pipes;21, 22 contain incubation period infectious diseases.23, 24 Time also important population dynamics,25 neural networks,26 brain dynamics,27 sensory system,28, 29 systems.30 Such render unsafe if controllers designed without considering main challenge controlling originates nature dynamics. Namely, function period, implies As such, described functional differential equations (FDEs),43-47 scalar measures constructed state. few instances literature context verification partial 48 state-constrained integral Lyapunov discussed References 49 50. control,51 concept functionals, has investigated further 52 means discretization. relationship discretization 53, domains interpreted basin attraction were 54. These works, however, do not contribution this establishment includes, instance, form (2) requirement (3), we discuss much broader class conditions. achieve introduce tool synthesizing controllers, building existing notions functionals51 functions.16 corresponding filters illustrated Figure 1. We use retarded neutral prove underlying guarantees. remark recent papers approached problem parallel our work. 55 considers disturbances while56-58 investigate combination stability Razumikhin- Krasovskii-type functionals. Although these share some ideas presented paper, establish in-depth study covered previous works. includes wider exhaustive discussion how calculate derivatives FDEs, relative degree However, questions related disturbances. rest paper organized follows. In Section 2, revisited respectively. Then, Sections 3 4 present major contributions work: establishes foundations discusses 5, demonstrate illustrative examples more practical case regulated predator-prey problem. Finally, conclude results future research directions 6. section, revisit certification ordinary (ODEs). Specifically, focus functions. 3, extend frameworks consider (4) when solution x ( t ) $$ x(t) evolves S ⊂ ℝ n S\subset {\mathbb{R}}^n , given following definition. Definition (safety invariance)System w.r.t. invariant 0 ∈ ⇒ x(0)\in S\kern3.0235pt \Rightarrow \kern3.0235pt x(t)\in ∀ ≥ \forall t\ge (4). 2. function)A continuously differentiable h : → h:{\mathbb{R}}^n\to \mathbb{R} (5) there exists α