Online Feedback Equilibrium Seeking Giuseppe Belgioioso Dominic Liao-McPherson Mathias Hudoba de Badyn Saverio Bolognani Roy S. Smith John Lygeros and Florian D orfler

2025-05-02 0 0 1.88MB 16 页 10玖币

侵权投诉

Online Feedback Equilibrium Seeking

Giuseppe Belgioioso*, Dominic Liao-McPherson*, Mathias Hudoba de Badyn,

Saverio Bolognani, Roy S. Smith, John Lygeros, and Florian D¨

orﬂer

Abstract—This paper proposes a unifying design framework

for dynamic feedback controllers that track solution trajectories

of time-varying generalized equations, such as local minimizers

of nonlinear programs or competitive equilibria (e.g., Nash) of

non-cooperative games. Inspired by the feedback optimization

paradigm, the core idea of the proposed approach is to re-purpose

classic iterative algorithms for solving generalized equations

(e.g., Josephy–Newton, forward-backward splitting) as dynamic

feedback controllers by integrating online measurements of

the continuous-time nonlinear plant. Sufﬁcient conditions for

closed-loop stability and robustness of the algorithm-plant cyber-

physical interconnection are derived in a sampled-data setting

by combining and tailoring results from (monotone) operator,

ﬁxed-point, and nonlinear systems theory. Numerical simulations

on smart building automation and competitive supply-chain

management are presented to support the theoretical ﬁndings.

I. INTRODUCTION

Online feedback optimization (FO) [1] is an emerging con-

trol paradigm for optimal steady-state operation of complex

systems based on their direct closed-loop interconnection with

optimization algorithms. FO controllers can handle control

objectives beyond set-point regulation, typically tracking (a-

priori unknown) solution trajectories of time-varying con-

strained optimization problems. In recent years, FO controllers

have been proposed for a wide variety of problem settings [1]–

[7]. These can be categorized by the type of control objective

(e.g., convex or nonconvex) and constraints (e.g., hard or

soft), the dynamics of the plant (e.g., nonlinear, linear, or

algebraic), the type of algorithm (discrete or continuous-time),

and the stability analysis (e.g., continuous-time, discrete-time,

or hybrid), see [1] for a comprehensive list. FO has found

widespread application in various domains, including power

systems (e.g., for optimal power reserve dispatch [2], [3], or

frequency regulation in AC grids [4], [5]), communication

networks (e.g., for network congestion control [6]), and trans-

portation systems (e.g., for ramp metering control [7]).

These large-scale engineering infrastructures comprise mul-

tiple subsystems with local decision authority and preferences,

commonly known as agents. These agents are typically self-

interested, hence, a more general notion of “efﬁciency” is

needed to model desirable (i.e., safe, locally optimal, and

strategically stable) operating points for such systems, motivat-

ing the use of game-theoretic equilibria (e.g., Nash, Wardrop)

[8]. A timely example are modern supply-chain systems,

*These authors contributed equally to this work. G. Belgioioso, S. Bolog-

nani, R. Smith, J. Lygeros, and F. D¨

orﬂer are with the ETH Z¨

urich

Automatic Control Laboratory, {gbelgioioso, bsaverio, rsmith,

jlygeros, dorfler}@ethz.ch. D. Liao-McPherson is with the Uni-

versity of British Columbia, dliaomcp@mech.ubc.ca. M. Hudoba de

Badyn is with the University of Oslo, mathias.hudoba@its.uio.no.

This work is supported by the SNSF via NCCR Automation (Grant 180545).

whereas suppliers, manufacturers, and retailers compete over

the available resources (e.g., raw materials, market demand)

to maximize their local proﬁts [9], [10].

There are several classes of control approaches for driving

non-cooperative multi-agent systems to steady-state conﬁgura-

tions given by game-theoretic equilibria. One class uses ﬁrst-

order algorithms and builds on operator-theoretic and passivity

arguments [11]–[13]. Passivity is leveraged in [11] to design

a distributed feedback control law to drive agents with single-

integrator dynamics to a steady-state operating point given by

a Nash equilibrium (NE). This approach is extended to multi-

integrator agents affected by partially-known disturbances in

[12]. The case of games with coupling constraints and agents

with mixed-order integrator dynamics is considered for the ﬁrst

time in [7]. All the these works consider agents coupled via

their objective functions (and constraints) but with decoupled

dynamics and use continuous-time ﬂows.

A second class of methods are sampled-data approaches

based on the model-free extremum seeking (ES) framework,

for ﬁnding optima [14], game-theoretic equilibria [15], [16],

and solutions to dynamic inclusions (which subsume the two

previous cases) [17], [18]. In [15], an ES controller is designed

for NE seeking in games in which the agents have nonlinear

decoupled dynamics. The extension to games with coupling

constraints is presented in [16]. In both cases, practical stabil-

ity is proven in the disturbance-free case. In [17], a general

framework is developed for the design and analysis of a class

ES controllers applicable to optimization as well as non-

cooperative games. Practical stability is established by relying

on the mathematical framework of hybrid dynamic inclusions.

To accelerate convergence, [18] extends the previous frame-

work from periodic to event-triggered sampled-data control.

These methods require nominal stability of the plant and,

typically, time-scale separation assumptions.

A third class of methods are extensions of economic model

predictive control to non-cooperative systems, including multi-

objective MPC [19] and various ﬂavours of game-theoretic

MPC [20], [21]. Unlike the ﬁrst-order or ES methods, these

consider transient operation and can handle unstable systems.

In exchange, they require dynamic models of the plant and

are typically computationally heavier due to the need to ﬁnd

accurate equilibria of trajectory games at each sampling time.

In this paper, we propose feedback equilibrium seeking

(FES), an extension of FO that seeks to drive pre-stabilized

dynamical systems to “efﬁcient” operating points encoded by

time-varying generalized equations (GEs). GEs contain con-

strained optimization as a special case and can model a broad

range of equilibrium problems (e.g., Nash, Wardrop). FES con-

trollers are most similar to the ﬁrst class of control approaches

discussed above [11]–[13]. They are typically based on ﬁrst-

arXiv:2210.12088v3 [math.OC] 14 Feb 2024

order methods and require an (approximate) steady-state input-

output sensitivity of the plant, which is assumed to stable. As a

result, they are faster than model-free ES methods; on the other

hand, they are slower but computationally lighter than MPC-

based methods, and require only static models. Therefore, FES

controllers occupy a unique middle ground between ES and

MPC. Their greatest advantage emerges in situations where

steady-state input-output sensitivity information is available,

but detailed dynamic models are lacking. Such conditions are

commonly found in sectors like power systems and process

optimization [2]–[5]. Unlike [11]–[13], our framework en-

compasses a broad class of algorithms and systems beyond

continuous-time ﬂows and multi-integrators. Moreover, we

use discrete-time algorithms which is especially relevant for

embedded distributed implementations, as communication in

multi-agent engineering networks is a discrete process.

Some other recent works [22], [23] study the problem of

tracking time-varying generalized NEs (GNEs) with discrete-

time algorithms. A distributed algorithm for tracking the solu-

tion trajectory of an exogenously varying strongly monotone

game is developed in [22] and extended to monotone games in

[23]. In [24], a GNE-seeking algorithm is adapted for online

operation by integrating measurements from a physical system,

however, the plant is treated as an algebraic map. All the

above works neglect dynamic interactions between the plant

and the algorithm (either the variation is exogenous or the

plant is algebraic) while we consider closed-loop stability in

the sampled-data case. This is signiﬁcant as digital control of

continuous-time plants is currently the dominant paradigm.

Our contribution to this area of research is threefold:

(i) We propose a general framework for designing FES

controllers for continuous-time pre-stabilized nonlinear

systems by tapping into a broad class of ﬁrst- and second-

order discrete-time algorithms for generalized equations.

This class is broad, and include many optimization

and game-theoretic algorithms (e.g., sequential convex

programming (SCP), proximal-gradient) as special cases.

(ii) We derive sufﬁcient conditions for stability and robust-

ness of the sampled-data algorithm-plant interconnection.

Speciﬁcally, we prove practical input-to-state stability

(Def. 1) of the closed loop with respect to exogenous dis-

turbances, under three fundamental conditions: 1. robust

stability of the plant, 2. strong regularity of the control

objective, and 3. robust convergence of the iterative algo-

rithm. Moreover, we prove a sharper notion of input-to-

state stability if an additional small-gain condition holds.

As by-product of our analysis, we also demonstrate that

some popular classes of algorithms are not suitable for

constructing sampled-data FES (or FO) controllers.

(iii) We showcase the utility of our framework by design-

ing two novel controllers based on a SCP algorithm

for nonconvex nonsmooth optimization, and a forward-

backward splitting controller for distributed Nash equi-

librium seeking in dynamically-coupled games. Further,

we illustrate their effectiveness through numerical simu-

lations on smart building and supply chain examples.

We build on our preliminary work [25] which addresses the

Fig. 1. In feedback equilibrium seeking, measurements from a countinuous-

time dynamical system are incorporated into a discrete-time equilibrium

seeking algorithm resulting in a coupled sampled-data cyber-physical system.

special case of strongly monotone GEs, globally linearly con-

vergent algorithms, and exponentially stable plants. Our results

here provide a signiﬁcant extension that allows non-monotone

GEs, locally-stable nonlinear plants and non-linearly locally

convergent algorithms. This algorithmic extension is partic-

ularly important since it captures many practically relevant

cases, such as algorithms for nonconvex optimization, which

are not globally convergent, and primal-dual and distributed al-

gorithms which typically exhibit sub-linear convergence only.

II. PRELIMINARIES

Basic notation: Denote by Iand id the identity matrix and

operator, respectively; by lim and sup the limit and essential

suprema, respectively. Given P=P⊤≻0and x∈Rn,

|x|P:= √x⊤P x with the convention |x|=|x|I.

Systems Theory: Continuous-time signals are denoted by

Ln={f:R≥0→Rn}and sequences by ℓn={f:

Z≥0→Rn}. Given x∈Lnand y∈ℓn, we consider

the signal/sequence norms ∥x∥= supt≥0|x(t)|and ∥y∥=

supk≥0|yk|. A function γ:R≥0→R≥0is of class Kif

it is continuous, strictly increasing, and satisﬁes γ(0) = 0.

If it is also unbounded, then γ∈ K∞. Similarly, a function

σ:R≥0→R≥0is said to be of class Lif it is continuous,

strictly decreasing, and satisﬁes σ(s)→0as s→ ∞. A

function β:R≥0×R≥0→R≥0is of class KL if β(·, s)∈ K

for each ﬁxed s≥0and β(r, ·)∈ L for ﬁxed r. Class K

functions obey a weak triangle inequality [26, Lemma 10]:

∀a, b ≥0and α∈ K, α(a+b)≤α(2a) + α(2b).(1)

For γ1, γ2:R→R,γ1◦γ2denotes the composition. Given

ξ∈Rn,u∈Lmand f:Rn×Rm→Rn,x∈Lnis a

solution of the initial value problem

˙x(t) = f(x(t), u(t)), x(0) = ξ. (2)

If it is uniformly continuous, obeys the initial condition and

satisﬁes the differential equation almost everywhere.

Deﬁnition 1 (LISpS [27]).Consider a system

x′(t) = f(x(t), u(t)), x(0) = ξ(3)

with solution x(t, ξ, u)(where either x′(t) = ˙x(t)and t∈R≥0

or x′(t) = x(t+ 1) and t∈Z≥0) and let ¯xbe a reference

signal. The system (3) is said to be Locally Input-to-State

Practically Stable (LISpS) about ¯xwith respect to uif there

exists ϵ > 0,β∈ KL,γ∈ K, and b > 0such that,

∀t≥0,|x(t, ξ, u)−¯x(t)| ≤ β(|ξ−¯x(0)|, t) + γ(∥u∥) + b,

provided |ξ| ≤ ϵand ∥u∥ ≤ ϵ. If b= 0 then (3) is Locally

Input-to-State Stable (LISS) about ¯xwith respect to u.

Operator Theory [28]: Given a closed convex set Ω⊆Rn,

ιΩ:Rn→ {0,∞} denotes its indicator function, NΩ: Ω ⇒

Rnis its normal cone operator, and projΩ:Rn→Ωis the

Euclidean projection onto Ω. A set-valued mapping F:Rn⇒

Rnis µ-strongly monotone, if (u−v)⊤(x−y)≥µ∥x−y∥2

for all x̸=y∈Rn,u∈ F(x),v∈ F(y), and monotone if

µ= 0;ﬁx(F) = {x∈Rn|x∈ F(x)}and zer(F) = {x∈

Rn|0∈ F(x)}denote the set of ﬁxed points and of zeros of

F. For a convex function f:Rn→R,∂f :Rn⇒Rndenotes

the subdifferential mapping in the sense of convex analysis.

Deﬁnition 2 (Strong Regularity [29]).A set-valued mapping

Ψ : Rn⇒Rnis said to be strongly regular at (x, y)if and

only if y∈Ψ(x)and there exist neighborhoods Uof xand

Vof ysuch that the truncated inverse mapping ˜

Ψ−1:V7→

Ψ−1(V)∩Uis a Lipschitz continuous function on V.

III. PROBLEM SETTING

We consider the problem of efﬁciently operating a physical

plant described by the following nonlinear state-space system

˙x(t) = f(x(t), u(t), w(t)),(4a)

y(t) = g(x(t), w(t)) (4b)

where x∈Lnxis the state, y∈Lnyis the output, u∈Lnuis

the control input, with u(t)∈ U for all t∈R≥0, and w∈Lnw

is a disturbance satisfying w(t)∈ W for all t∈R≥0.

We adopt the “stabilize-then-optimize” paradigm, and as-

sume that (4) is stable1and has a steady-state map p:

U × W → Rnxsatisfying f(p(u, w), u, w)=0for all

u∈ U, w ∈ W and a steady-state input-output map

h(u, w) = g(p(u, w), w).(5)

Formally, we assume the system (4) satisﬁes the following

properties that ensure its solution trajectories and steady-state

mappings are well deﬁned.

Assumption 1 (Robust Stability of the Plant (4) with respect

to parameter variations).(i) fis locally Lipschitz; (ii) gis

globally Lg-Lipschitz; (iii) wis continuously differentiable,

and ˙w∈Lnwsatisﬁes ∥˙w∥<∞; (iv) Wand Uare

compact and convex; (v) (4) is LISS [27], namely, there exists

1If the physical plant (4) is not stable, it can be pre-stabilized by replacing

f(x, u, w)with f(x, k(x, v), w), where vis the new input (for example, a

reference command) and kis a stabilizing controller.

ϵx, ϵw, α3, α4, α5>0, a continuously differentiable function

V, and σ1∈ K such that for any constant u∈ U

α3|x−p(u, w)|2≤V(x, u, w)≤α4|x−p(u, w)|2,

V(x(t), u, w(t)) ≤ −α5V(x(t), u, w(t)) + σ1(|˙w(t)|),

if V(x(t), u, w(t)) ≤ϵxand |˙w(t)| ≤ ϵw.□

Our control objective is to design an output feedback con-

troller that drives (4) and maintain it near efﬁcient operating

conditions. We encode “efﬁciency” using the following struc-

tured generalized equation (GE), parameterized by w∈ W:

0∈G(z, s) + A(z)(efﬁciency objective) (6a)

s=h(u, w),(steady-state map) (6b)

u=q(z),(ctrl state to input map)(6c)

where z∈Rnzis an auxiliary variable, sis the steady-state

output of (4), G:Rnz×Rny→Rnzis a single-valued

mapping2,A:Rnz⇒Rnzis a set-valued mapping, and

q:Rnz→ U is the output mapping. The notion of efﬁciency

encoded in the GE (6) is ﬂexible and encompasses a wide

variety of useful objectives, notably constrained optimization

(e.g., minimizing energy consumption as described in VII-A)

and generalized Nash equilibria. The auxiliary variable zis the

internal state of the controller, and it also allows the modelling

of optimality and equilibrium conditions that involve dual

variables, such as KKT critical points. Two concrete examples

are provided at the end of this section.

Our control objective is to maintain the system (4) near

solutions of (6); we must impose some regularity conditions

to ensure that this is a well-deﬁned problem. Substituting (6b)

and (6c) into (6a) yields the following compact GE:

0∈G(z, w) + A(z),(7)

where G(z, w) = G(z, h(q(z), w)). The parameter-to-solution

mapping S:W → Rnzis deﬁned as

S(w) = {z∈Rnz|0∈G(z, w) + A(z)}.(8)

For a given w∈Lnw, the set of solution trajectories is

S(w) = {z∈Lnz| ∀t≥0, z(t)∈S(w(t))}.(9)

The following assumption ensures that tracking solution tra-

jectories in Sis a well-posed problem.

Assumption 2 (Strong Regularity of the GE).(i) qis globally

Lq-Lipschitz continuous, (ii) Gis continuously differentiable,

and (iii) the mapping G(·, w) + A(·)in (7) is strongly regular

at all points satisfying z∈S(w), for all w∈ W.□

Conditions for strong regularity are context-dependent, as

detailed later in this section with two examples. In optimiza-

tion contexts, it reduces to having a strong local minimizer

and unique dual variables, which follows by second-order

sufﬁcient conditions and suitable constraint qualiﬁcations.

In general, the set S(w)can be complex and multi-valued,

thankfully Assumption 2 imposes some structure on S(w).

2For notational simplicity Gdoes not depend on w, this is without loss of

generality since wcan be included in s.

Theorem 1. [30, Theorem 3.2] Under Assumption 2, for

any w∈Lnw, there exist mLipschitz continuous mappings

¯zi∈Lnz,i∈ {1, . . . , m}, such that S(w) = {¯z1,...,¯zm}.□

Remark 1. Theorem 1 demonstrates that the solution map

consists of misolated w-parameterized trajectories called

“branches”. These branches generalize the notion of a isolated

local solution of a GE (in the context of optimization, a strict

local minimizer) to a parameterized setting. Our analysis will

eventually establish tracking error bounds and the existence

of a region of attraction around each branch.

We can now formally state our control problem.

Problem. Design an output feedback controller so that the

system (4) tracks the input and output trajectories3u∗=q(z∗)

and y∗=h(u∗, w), for some z∗∈ S(w)and w∈Lnw.

The GE (6) allows us to naturally express complex control

objectives, e.g., constraints can be readily encoded into A

using normal cone operators (see Example 1.A). Similarly

to FO [1], we assume that the dynamic model of the plant

(4) is unknown and that the exogenous disturbance wis not

measurable. However, the output ycan be measured and the

input-output steady-state sensitivity ∇uh(u, w), or an approxi-

mation of it, is available. Robustness of FO controllers to static

input-output modelling error is analytically studied in [31] and

experimentally in [32]. The inability to measure wis common

in practical applications. For example, in power systems, w

represents variable micro-generation and uncontrollable loads

caused by consumers drawing power from the grid.

Example 1.A. Nonlinear Programming (NLP)

Consider the w-parameterized nonlinear program (NLP)

min

ξ,u ϕ(ξ, u) + φ(ξ)(10a)

s.t. ξ=h(u, w), u ∈ U (10b)

where ϕ:Rny×Rnu→Rand hare twice continuously

differentiable, and φ:Rny→R∪ {∞} is a proper lower

semicontinuous convex function [28, Def. 9.12], e.g., an ℓ1-

norm. We assume that (10) is always feasible, namely,

{(ξ, u)|ξ∈Dom φ, u ∈ U, ξ =h(u, w)} ̸=∅,∀w∈ W.

The associated partial Lagrangian function is

L(ξ, u, λ, w) = ϕ(ξ, u) + λ⊤(h(u, w)−ξ),(11)

where λ∈Rny

≥0is a dual variable, and the KKTs for (10) are











0∈ ∇ξL(ξ, u, λ, w) + ∂φ(ξ)

0∈ ∇uL(ξ, u, λ, w) + NU(u)

0 = ∇λL(ξ, u, λ, w) = h(u, w)−ξ

(12)

This is a special case of the GE in (6), with z= (ξ, u, λ),

0∈

∇ξL(ξ, u, λ, w)

∇uL(ξ, u, λ, w)

s−ξ



| {z }

G(z,s)

+



∂φ(ξ)

NU(u)

0



| {z }

A(z)

,(13)

3With an abuse of notation, we extend the mappings hand qto take signals

as arguments, in the sense of u∗(t) = q(z∗(t)) for all t∈R≥0.

s=h(u, w)and q(z) = [0 I0]z. The following statement

provides sufﬁcient conditions for strong regularity of (13).

Lemma 1. Consider any w∈ W, a KKT point ¯z= (¯

ξ, ¯u, ¯

λ)∈

S(w), and let ζ= (ξ, u). Then, the condensed GE (7) associ-

ated with (13) is strongly regular at ¯zif ζ⊤∇2

ζL(¯z, w)ζ > 0

for all ζ= (ξ, u)satisfying ξ=∇uh(¯u, w)u.

Proof. The proof is identical to that of [33, Theorem 7].

The condition in Lemma 1 is known as a second order

sufﬁcient condition and can be checked numerically4if ¯zis

known. Conditions of this type are standard in the analysis of

Newton’s method and sequential quadratic programming, see

e.g., [34, Theorem 1.13, Theorem 4.14]. ▲

Example 2.A. Generalized Nash Equilibrium Problems

Consider a set of Nagents labelled by i∈ I:={1, . . . , N }.

Each agent i∈ I chooses a control action uifrom a polyhedral

set Ui:= {ξ∈Rni|Biξ≤bi}, and has a cost function

Ji(ui, s)that depends on its own action, ui, and the actions

of other agents, u−i= (u1, . . . , ui−1, ui+1, . . . , uN), through

the steady state of the physical system s=h(u, w), where u=

(u1, . . . , uN). Moreover, the actions of all agents are linked

via afﬁne coupling constraints of the form Au ≤b, modelling

limited availability of shared resources [8]. The agents are

self-interested and want to optimize their own local cost. The

resulting collection of the Nparameterized inter-dependent

optimization problems constitutes a parametrized game:

∀i∈ I :(min

Ji(ui, h(u, w)) =: Ji(ui, u−i, w)

s.t. Au ≤b, ui∈ Ui

(14)

From a game-theoretic perspective, a relevant solution concept

for (14) is the generalized Nash equilibrium, where no agent

can unilaterally reduce its cost [35].

Deﬁnition 3. A feasible action proﬁle ¯u= (¯u1,...,¯uN)is a

generalized Nash equilibrium (GNE) of the game (14) if

Ji(¯ui,¯u−i, w)≤min

ui∈Ui{Ji(ui,¯u−i, w)|A(ui,¯u−i)≤b},

holds for all agents i∈ I.□

If each Jiis continuously differentiable and convex with

respect to ui, a GNE can be found by solving a system of

coupled KKT conditions of the problems (14) [35, Th. 4.8]:

(0∈ ∇uiJi(ui, u−i, w) + A⊤

iλ+NUi(ui),∀i∈ I

0∈b−Au +NRm

≥0(λ),(15)

where λ∈Rmis a dual variable associated with the coupling

constraints Au−b≤0. The set of solutions to (15) is a special

subclass of GNEs, known as variational GNEs (v-GNEs) [35].

To write (15) in a compact form, we deﬁne the pseudo-

gradient F(u, w) = F(u, h(u, w)) = [∇uiJi(ui, h(u, w))]i∈I

obtained by stacking up the partial gradients of the local cost

functions Ji, i.e., ∇uiJi(ui, h(u, w)) =: Fi(u, h(u, w)). The

4Speciﬁcally, if Zis a basis for the nullspace of [I− ∇uh(¯u, w)] then

the regularity condition is equivalent to ZT∇2

ζL(¯z, w)Z≻0which can be

checked using e.g., a Cholesky factorization.

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

OnlineFeedbackEquilibriumSeekingGiuseppeBelgioioso*,DominicLiao-McPherson*,MathiasHudobadeBadyn,SaverioBolognani,RoyS.Smith,JohnLygeros,andFlorianD¨orflerAbstract—Thispaperproposesaunifyingdesignframeworkfordynamicfeedbackcontrollersthattracksolutiontrajectoriesoftime-varyinggeneralizedequations,suc...

展开>> 收起<<

Online Feedback Equilibrium Seeking Giuseppe Belgioioso Dominic Liao-McPherson Mathias Hudoba de Badyn Saverio Bolognani Roy S. Smith John Lygeros and Florian D orfler.pdf

共16页,预览4页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Online Feedback Equilibrium Seeking Giuseppe Belgioioso Dominic Liao-McPherson Mathias Hudoba de Badyn Saverio Bolognani Roy S. Smith John Lygeros and Florian D orfler

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: