Funnel control for fully actuated systems under a fragment of signal temporal logic specifications

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Abstract

Temporal logics have lately proven to be a valuable tool for various control applications by providing a rich specification language. Existing temporal logic-based control strategies discretize the underlying dynamical system in space and/or time. We will not use such an abstraction and consider continuous-time systems under a fragment of signal temporal logic specifications by using the associated robust semantics. In particular, this paper provides computationally-efficient funnel-based feedback control laws for a class of systems that are, in a sense, feedback equivalent to single integrator systems, but where the dynamics are partially unknown for the control design so that some degree of robustness is obtained. We first leverage the transient properties of a funnel-based feedback control strategy to maximize the robust semantics of some atomic temporal logic formulas. We then guarantee the satisfaction for specifications consisting of conjunctions of such atomic temporal logic formulas with overlapping time intervals by a suitable switched control system. The result is a framework that satisfies temporal logic specifications with a user-defined robustness when the specification is satisfiable. When the specification is not satisfiable, a least violating solution can be found. The theoretical findings are demonstrated in simulations of the nonlinear Lotka–Volterra equations for predator–prey models.

Introduction

Temporal logics allow to express temporal properties in a logical framework that may represent specifications imposed on a dynamical system. Formal verification techniques, such as model checking [1], can then be used to check whether or not the system satisfies these specifications. Motivated by the trade-off between complexity and expressivity, linear temporal logic (LTL) has widely been used in formal verification. More expressive temporal logics are metric temporal logic (MTL) and signal temporal logic (STL) that allow to impose quantitative time constraints, as opposed to qualitative time constraints in LTL. Formal methods-based control has emerged due to the need of more complex system specifications in areas such as robotics and intelligent transportation systems. In contrast to formal verification, the task is to design the control input so that the dynamical system satisfies the given specifications. Recent advances in this area, with a focus on LTL, have been reported in [2] and [3]. Formal methods-based control strategies for multi-agent systems have also appeared in [4] and [5]. The multi-agent case is special in the sense that global and/or local, i.e., individual, specifications may be assigned to the agents, potentially requiring collaboration and communication among them. At the same time, additional agent couplings such as connectivity maintenance and/or collision avoidance need to be taken care of. All previously mentioned approaches rely on automata-based formal verification techniques. This implies that the temporal logic specification is translated into a language equivalent Büchi automaton, while the dynamical system is abstracted into a discrete transition system that needs to satisfy some equivalence properties. If the abstracted system is non-deterministic, e.g., modeled as a Markov decision process, then a Rabin automaton instead of a Büchi automaton is needed. A search algorithm is then run on their product automaton to find a specification-satisfying discrete path. These automata-based approaches, however, may be subject to the state–space explosion problem. The works in [6] and [7] are based on game theoretical results and consider a special fragment of LTL, called the generalized reactivity(1) fragment, which explicitly accounts for adversarial and dynamical environments, and hence establish reactive control synthesis frameworks. This fragment allows for computationally more efficient control synthesis methods similarly to [8] and [9] where sampling-based methods are introduced. Robustness of temporal logics has been introduced in [10] where MTL specifications have been interpreted over continuous-time signals. In particular, the robustness degree measures the distance of a signal to the set of signals where the boolean evaluation of the MTL specification changes. The authors in [10] also introduce robust semantics, which are easier to compute than the robustness degree and which are an under-approximation of the robustness degree. STL is a predicate logic interpreted over continuous-time signals [11] entailing space robustness [12], a special form of the robust semantics. Formal methods-based control of dynamical systems under STL specifications is a difficult task due to the nonlinear, nonconvex, noncausal, and nonsmooth semantics. This has been considered for discrete-time systems by means of model predictive control in [13] where space robustness is incorporated into a mixed integer linear program. Robust extensions of this approach have been reported in [14], [15], [16], while the extension to multi-agent systems with a special focus on communication is discussed in [17]. Optimization-based approaches to this problem are summarized in [18]. Reinforcement learning-based control strategies have been derived in [19]. Our recent work in [20] neither involves learning nor optimization and embeds a funnel-based feedback control law into a hybrid system framework to satisfy STL specifications. Despite losing optimality guarantees, our framework applies directly to continuous-time systems and is computationally tractable and robust. An extension to coupled multi-agent systems under local and possibly conflicting STL specifications can be found in [21]. A learning-based extension of these funnel-based feedback control laws to account for optimality can be found in [22]. Prescribed performance control (PPC) [23], [24] is a funnel-based feedback control strategy that explicitly takes the transient and steady-state behavior of a tracking error into account. A user-defined performance function prescribes a desired temporal behavior to this error that is subsequently achieved by a continuous feedback control law. In other words, the performance function defines a funnel and the task of the continuous feedback control law is to keep the error within this funnel. In this work, we leverage this funnel and replace, in a suitable way, the tracking error by the robust semantics of the STL specification at hand.

We consider nonlinear continuous-time systems that are, in a sense, feedback equivalent to single integrator systems with, however, partially unknown dynamics. For a fragment of STL specifications, we cast the control problem into a PPC control problem. We first leverage the transient properties of the performance function to derive a continuous feedback control law that maximizes the robust semantics of some atomic temporal logic formulas. Subsequently, a switched control system is presented to satisfy specifications consisting of conjunctions of such atomic temporal logic formulas with overlapping time intervals. For the case of an unsatisfiable specification, a least violating solution can directly be found due to the use of the robust semantics. To the best of the authors’ knowledge, the approach presented in this paper is the first approach deriving a continuous-time feedback control law for temporal logic specifications, while discretizing neither the system dynamics nor the environment in space or time and without resorting to automata representations of the temporal logic specification. The advantages of our approach are low computational complexity and inherent robustness properties of the feedback control laws. Compared with [13] and its robust extensions, the approach presented in this paper considers continuous-time systems and also admits nonlinear predicates. This paper extends [20] by allowing specifications with overlapping time intervals, providing all technical proofs, and presenting an extension to the case when specifications are not satisfiable.

Section 2 states preliminaries, while Section 3 illustrates the underlying main idea. Section 4 proposes a feedback control law that satisfies atomic temporal logic formulas, while Section 5 proposes a switched control system for a set of atomic temporal logic formulas. Section 6 presents simulations of a predator–prey system using the Lotka–Volterra equations, followed by conclusions in Section 7.

Section snippets

Notation and preliminaries

True and false are denoted by and with B{,}; Rn is the n-dimensional vector space over the real numbers R and 0nRn consists of n zeros. The non-negative and positive real numbers are R0 and R>0, respectively. Let x denote the Euclidean norm of xRn. All technical proofs of Lemmas and Theorems, derived in this paper, are provided in the Appendix.

At time t, let x(t)Rn, u(t)Rm, and w(t)WRn be the state, input, and additive noise of the system ẋ(t)=f(x(t))+g(x(t))u(t)+w(t),x(0)x0

Problem formulation and approach

A fragment of STL is considered in this paper. Considering the predicate μ, we first define the STL formulas ψμ¬μψ1ψ2ϕF[a,b]ψG[a,b]ψF[ā,b̄]G[ā,b̄]ψ where ψ in (3b) and ψ1,ψ2 in (3a) are formulas of class ψ given in (3a) and where a,b,ā,b̄,ā,b̄R0 with ab, āb̄ and āb̄. Disjunctions are not considered here as they can, in general, not be handled by continuous feedback control laws unless the formula is trivially simplified as illustrated next. Consider the system ẋ(t)=u(t)

A funnel-based feedback control law for atomic temporal formulas

Define first the one-dimensional error, the normalized error, and the transformed error as e(x)ρψ(x)ρmaxξ(x,t)e(x)γ(t)ϵ(x,t)S(ξ(x,t))=ln(ξ(x,t)+1ξ(x,t)), respectively. When referring to the solution x:R0Rn to (1), we use the notation e(t)e(x(t)), ξ(t)ξ(x(t),t), and ϵ(t)ϵ(x(t),t), while we use e(x), ξ(x,t), and ϵ(x,t) when we want to emphasize dependence on the state xRn. Eq. (5) can now be written as γ(t)<e(t)<0, which resembles (2) by setting M0 and can further be written as 1<ξ(t

A switched control system for temporal formulas with overlapping time intervals

Let us first consider θψU[a,b]ψ where ψ and ψ are as in (3a). The main idea is to combine two funnel-based control laws as derived in the previous section. Hence, let us decompose ψU[a,b]ψ into G[0,t]ψ and F[t,t]ψ where t[a,b]. Let ρmax(max(ρoptψ,ρoptψ),) and let γ(t)(γ0γ)exp(lt)+γ and γ(t)(γ0γ)exp(lt)+γ be performance functions for ρψ(x) and ρψ(x), respectively. Our goal is to achieve γ(t)+ρmax<ρψ(x(t))<ρmaxγ(t)+ρmax<ρψ(x(t))<ρmax for

Simulations

A modified version of the Lotka–Volterra equations for predator–prey models is considered. In particular, consider three species x1, x2, and x3 where x3 is a predator hunting x1 and x2, while x2 is a predator hunting x1. The dynamics of this system are given by ẋ1=β1x1βˆ1,2x1x2βˆ1,3x1x3+u1ẋ2=β2,1x1x2βˆ2,3x2x3+u2ẋ3=β3,1x1x3+β3,2x2x3βˆ3x3+u3 where β1 is the growth rate of x1, while β2,1x1 and β3,1x1+β3,2x2 are the growth rates of x2 and x3, respectively. The terms βˆ1,2x1x2 and βˆ1,3x1x3

Conclusion

We considered the control of fully actuated systems under a fragment of signal temporal logic specifications without discretizing neither the system dynamics nor the environment in space or time. A continuous feedback control law for atomic temporal logic formulas was derived by considering a funnel-based feedback control strategy where the transient behavior of the funnel was exploited. This control law is robust in two ways. First, bounded additive noise does not affect the satisfaction of

CRediT authorship contribution statement

Lars Lindemann: Visualization, Writing - original draft, Writing - review & editing, Data curation, Investigation, Formal analysis, Validation, Software, Methodology, Conceptualization. Dimos V. Dimarogonas: Funding acquisition, Project administration, Supervision, Resources, Writing - review & editing, Conceptualization.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Lars Lindemann was born in Lübbecke, Germany, in 1989. He received the B.Sc. degree in Electrical and Information Engineering and the B.Sc. degree in Engineering Management both from the Christian-Albrechts-University (CAU), Kiel, Ger- many, in 2014 and the M.Sc. degree in Systems, Control and Robotics from the KTH Royal Institute of Technology, Stockholm, Sweden, in 2016. Since June 2016, he is pursuing the Ph.D. degree at KTH Royal Institute of Technology, Stockholm, Sweden. His current

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    Lars Lindemann was born in Lübbecke, Germany, in 1989. He received the B.Sc. degree in Electrical and Information Engineering and the B.Sc. degree in Engineering Management both from the Christian-Albrechts-University (CAU), Kiel, Ger- many, in 2014 and the M.Sc. degree in Systems, Control and Robotics from the KTH Royal Institute of Technology, Stockholm, Sweden, in 2016. Since June 2016, he is pursuing the Ph.D. degree at KTH Royal Institute of Technology, Stockholm, Sweden. His current research interests include control theory, formal methods, multi-agent systems, and autonomous systems. He was a Best Student Paper Award Finalist at the 2018 American Control Conference and is a recipient of the Outstanding Student Paper Award at the 58th IEEE Conference on Decision and Control.

    Dimos V. Dimarogonas was born in Athens, Greece, in 1978. He received the Diploma in Electrical and Computer Engineering in 2001 and the Ph.D. in Mechanical Engineering in 2007, both from National Technical University of Athens (NTUA), Greece. Between 2007 and 2010, he held postdoctoral positions at the KTH Royal Institute of Technology, Department of Automatic Control and MIT, Laboratory for Information and Decision Systems (LIDS). He is currently Professor at the Division of Decision and Control Systems, School of Electrical Engineering and Computer Science, at KTH. His current research interests include multi-agent systems, hybrid systems and control, robot navigation and manipulation, human-robot-interaction and networked control. He serves in the Editorial Board of Automatica and the IEEE Transactions on Control of Network Systems and is a Senior Member of IEEE. He is a recipient of the ERC Starting Grant in 2014, the ERC Consolidator Grant in 2019, and the Knut och Alice Wallenberg Academy Fellowship in 2015.

    This work was supported in part by the Swedish Research Council (VR), the European Research Council (ERC), the Swedish Foundation for Strategic Research (SSF), the EU H2020 Co4Robots project, and the Knut and Alice Wallenberg Foundation (KAW) .

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