Control Synthesis for Stability and Safety by Differential Complementarity Problem Yinzhuang Yi Shumon Koga Bogdan Gavrea and Nikolay Atanasov

2025-05-06 0 0 833.75KB 6 页 10玖币

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Control Synthesis for Stability and Safety by Differential

Complementarity Problem

Yinzhuang Yi, Shumon Koga, Bogdan Gavrea, and Nikolay Atanasov

Abstract—This paper develops a novel control synthesis

method for safe stabilization of control-afﬁne systems as

a Differential Complementarity Problem (DCP). Our design

uses a control Lyapunov function (CLF) and a control bar-

rier function (CBF) to deﬁne complementarity constraints in

the DCP formulation to certify stability and safety, respec-

tively. The CLF-CBF-DCP controller imposes stability as a

soft constraint, which is automatically relaxed when the

safety constraint is active, without the need for parameter

tuning or optimization. We study the closed-loop system

behavior with the CLF-CBF-DCP controller and identify con-

ditions on the existence of local equilibria. Although in cer-

tain cases the controller yields undesirable local equilibria,

those can be conﬁned to a small subset of the safe set

boundary by proper choice of the control parameters. Then,

our method can avoid undesirable equilibria that CLF-CBF

quadratic programming techniques encounter.

Index Terms—Constrained control, Stability of nonlinear

systems, Differential-algebraic systems

I. INTRODUCTION

STABILITY veriﬁcation and stabilizing control design are

fundamental problems in control theory that have im-

pacted numerous industrial systems. One of the main tools

for constructing control laws that stabilize nonlinear systems

is a control Lyapunov function (CLF) [1], [2]. As control sys-

tems are increasingly deployed in less structured, yet safety-

critical settings, in addition to stability, control designs need

to guarantee safety. Inspired by the property of CLFs to yield

invariant level sets, control barrier functions (CBFs) [3] have

been developed to enforce that a desired safe subset of the state

space is invariant. While various techniques for guaranteeing

safety and stability exist, important challenges remain when

both requirements are considered simultaneously. They include

conditions under which it is possible to obtain a single

control policy that guarantees CLF stability and CBF safety

simultaneously (compatibility [4]), existence and uniqueness

of the closed-loop system trajectories (well-posedness), con-

vergence to points other than the origin (undesired equilibria),

identiﬁcation of initial conditions that ensure joint stability and

safety (region of attraction).

The design of stabilizing control with guaranteed safety is

studied in [7]. The authors deﬁne a Control Lyapunov Barrier

Function (CLBF), whose existence enables joint stabilization

We gratefully acknowledge support from NSF RI IIS-2007141.

Y. Yi, S. Koga, and N. Atanasov are with the Department of Electrical

and Computer Engineering, UC San Diego, 9500 Gilman Drive, La Jolla,

CA, 92093, USA (e-mails: {yiyi,skoga,natanasov}@ucsd.edu).

B. Gavrea is with the Department of Mathematics, Technical

University of Cluj-Napoca, Cluj-Napoca, 400114, Romania (e-mail:

bogdan.gavrea@math.utcluj.ro).

Fig. 1: Comparison of the closed-loop trajectories resulting from our

CLF-CBF-DCP control approach (orange), two recent CLF-CBF-QP

techniques (blue [4] and red [5]) and Lyapunov shaping (brown [6])

with a concave obstacle (green) for three initial conditions (black

dots). Existing CLF-CBF-QP techniques converge to an undesired

equilibrium on the obstacle boundary (black cross). Lyapunov shap-

ing converges to a different undesired equilibrium on the obstacle

boundary near the black cross for different initial conditions. Our

approach stabilizes the system at the origin.

and safety. However, constructing a CLBF may be challenging

and may require modiﬁcation of the safe set. A comprehensive

overview of CBF techniques and their use as safety constraints

in quadratic programs (QPs) for control synthesis is provided

in [3]. Garg and Panagou [8] propose a ﬁxed-time CLF and

analyze conditions for ﬁnite-time stabilization to a region of

interest with safety guarantees. Local asymptotic stability for

a particular CLF-CBF QP was proven in [9]. Reis et al. [6]

show that a CLF-CBF-QP controller introduces equilibria

other than the origin. The authors develop a new formulation

by introducing a new parameter in the CLF constraint and

an additional CBF constraint, aimed at avoiding undesired

equilibria on the safe set boundary. In [5], it is shown that

minimizing the distance to a nominal controller satisfying the

CLF condition as the objective of a CLF-CBF QP ensures

local stability of the origin without any additional assumptions.

However, the region of attraction of this controller has not

be characterized yet. Compatibility between CLF and CBF

constraints is considered in [10]. The authors deﬁne a CBF-

stabilizable sublevel set of the CLF and characterize the con-

ditions under which the closed-loop system is asymptotically

stable with respect to the origin. Mestres and Cort´

es [4]

develop a penalty method to incorporate the CLF condition

as a soft constraint in a QP, identify conditions on the penalty

parameter that eliminate undesired equilibria, and provide an

inner approximation of the region of attraction, where joint

stability and safety is ensured. As illustrated in Fig. 1, however,

the regions of attraction guaranteed by recent CLF-CBF-QP

techniques remain conservative, and initial conditions inside

arXiv:2210.01978v3 [math.OC] 9 Dec 2022

the safe set may still converge to undesired equilibria.

Our ﬁrst contribution is a formulation of control synthesis

with CLF stability and CBF safety constraints as a Differential

Complementarity Problem (DCP). A DCP is composed of an

ODE subject to complementarity constraints:

y=ˆ

f(y,z)

s.t.0≤z⊥C(y,z)≥0,(1)

where a⊥bmeans that aand bare orthogonal vectors,

i.e., a>b= 0. Complementarity problems are used to model

combinatorial constraints in nonlinear optimization [11] or

hybrid dynamics in mechanical systems with contact [12].

We formulate a DCP in which the closed-loop dynamics are

subject to constraints that certain control terms are activated

only when the state may violate the CLF or CBF conditions.

Our second contribution is an analysis of the closed-loop

system equilibria, showing that our CLF-CBF-DCP controller

can eliminate undesired equilibria on the safe set boundary that

CLF-CBF-QP methods encounter (see Fig. 1). We prove that

our controller is Lipschitz continuous, ensuring existence and

uniqueness of solutions, and show that the set of undesired

equilibria can be restricted to a small subset of the safe

set boundary by choosing a sufﬁciently large control gain

parameter. A related work in the DCP literature is Camlibel

et al. [13], which derives conditions for Lyapunov stability of

linear DCP. While DCP formulations have potential to capture

switching behavior in dynamical systems, they have not yet

been used for safe control design. This paper introduces a new

idea to model the mode switch between stability and safety

satisfaction as a DCP.

II. PRELIMINARIES

Consider a nonlinear control-afﬁne system:

x=f(x) + G(x)u(2)

with state x∈Rnand control input u∈Rm, where f:Rn→

Rnand G:Rn→Rn×mare locally Lipschitz continuous.

Assume that f(0) = 0so that the origin x=0is the desired

equilibrium of the unforced system. A typical objective is to

design a stabilizing controller for the system. Stability of the

closed-loop system is often veriﬁed by a Lyapunov function,

while a stabilizing controller can be obtained using a control

Lyapunov function.

Deﬁnition 1. Given an open and connected set D ⊆ Rnwith

0∈ D, a continuously differentiable positive-deﬁnite function

V:Rn→Ris a control Lyapunov function (CLF) on Dfor

system (2) if for each x∈ D \ {0}it satisﬁes:

inf

u∈Rm[LfV(x) + LGV(x)u]≤ −αl(V(x)),(3)

where αl:R≥0→R≥0is a class Kfunction [14].

The set of stabilizing control inputs at state x∈ D \ {0},

corresponding to a valid CLF Von D, is:

Kclf(x) = {u∈Rm:LfV(x) + LGV(x)u≤ −αl(V(x))},

in the sense that any Lipschitz continuous control law r:

D 7→ Rmsuch that r(x)∈Kclf(x)makes the origin of the

closed-loop system asymptotically stable [15].

Beyond stability, it is often necessary to ensure that the

system trajectories remain within a safe set C ⊂ D, in the

sense that Cis forward invariant [16]. We consider a closed

set C:= {x∈Rn|h(x)≥0}, deﬁned as the zero-superlevel

set of a continuously differentiable function h:Rn→R. One

way to guarantee that Cis forward invariant is to require that

his a control barrier function. Details can be found in the

comprehensive work by Ames at el. [16].

Deﬁnition 2. A continuously differentiable function h(x) :

Rn→Ris a control barrier function (CBF) of Con Dfor

system (2) if it satisﬁes:

sup

u∈Rm

[Lfh(x) + LGh(x)u]≥ −αh(h(x)),∀x∈ D,(4)

where αh:R→Ris an extended class Kfunction.

The set of safe control inputs at state x, corresponding to

a valid CBF hon D, is:

Kcbf(x) = {u∈Rm:Lfh(x) + LGh(x)u≥ −αh(h(x))},

in the sense that any Lipschitz control law r:D 7→ Rmsuch

that r(x)∈Kcbf(x)renders the set Cforward invariant [17].

Finding a single control input uachieving both stability and

safety may be infeasible. In order to guarantee safety when

(3) and (4) are not compatible [18], a popular approach is to

modify a stabilizing controller minimally so as to guarantee

safety [16]. Given a locally Lipschitz stabilizing control law

r(x)∈Kclf(x), the following CBF QP obtains the minimum

control perturbation to guarantee safety:

min

u∈Rmku−r(x)k2

s.t.Lfh(x) + LGh(x)u≥ −αh(h(x)).

(5)

III. PROBLEM STATEMENT

In agreement with the results in [4]–[6], [10], we note that

a control law synthesized by the CBF QP in (5) introduces

undesired local equilibria on the boundary of the safe set C

for the system in (2). Hence, our objective is to design a new

control synthesis method, which simultaneously guarantees

safety and prevents undesired local equilibria for the closed-

loop system.

Suppose the assumption below holds throughout the paper.

Assumption 1. The safety requirements for system (2) are

speciﬁed by a set C={x∈Rn|h(x)≥0} ⊂ D such that h

is a CBF of Cwith h(0)>0and LGh(x)6=0,∀x∈ D.

Assumption 1 requires that the safe set Cis deﬁned by a

CBF with relative degree 1and the origin is in the interior of

C. Under this assumption, the CBF-QP control law in (5) can

be obtained in closed-form using the KKT conditions.

Proposition 1 ([10, Thm. 1]).Consider system (2) with CLF

Von Dand CBF hthat satisﬁes Assumption 1. Then, the

CBF QP in (5) has a unique closed-form solution:

u∗(x) = (r(x),∇h(x)>e(x) + αh(h(x)) ≥0,

u(x),∇h(x)>e(x) + αh(h(x)) <0,(6)

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ControlSynthesisforStabilityandSafetybyDifferentialComplementarityProblemYinzhuangYi,ShumonKoga,BogdanGavrea,andNikolayAtanasovAbstractThispaperdevelopsanovelcontrolsynthesismethodforsafestabilizationofcontrol-afnesystemsasaDifferentialComplementarityProblem(DCP).OurdesignusesacontrolLyapunovfunct...

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Control Synthesis for Stability and Safety by Differential Complementarity Problem Yinzhuang Yi Shumon Koga Bogdan Gavrea and Nikolay Atanasov

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