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Distributed control of multi-agent systems with pulse-width-modulated controllers
Automatica 119 (2020) 109020
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Distributed control of multi-agent systems with pulse-width-modulated controllers✩ ∗
Tengfei Liu a , , Zhong-Ping Jiang b a b
State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang, 110004, China Department of Electrical and Computer Engineering, New York University, 370 Jay Street, Brooklyn, NY 11201, USA
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Article history: Received 21 December 2018 Received in revised form 4 December 2019 Accepted 19 April 2020 Available online xxxx
a b s t r a c t This paper studies the distributed control of multi-agent systems with pulse-width-modulated (PWM) control inputs. A necessary and sufficient condition on the PWM control laws is given for uniform asymptotic state agreement with directed and switched information exchange topologies. The methodology is validated by a computer-based simulation. © 2020 Elsevier Ltd. All rights reserved.
Keywords: Pulse-width modulation Multi-agent systems Switched topologies State agreement
1. Introduction In the past two decades, considerable attention has been devoted to the distributed control of groups of agents for coordinated movement. In addition to practical motivations, one of the technical reasons for this is that conventional control theory cannot be readily used to address group behaviors. Increasingly challenging control problems have been addressed for multiagent systems, in terms of growing complexity of system dynamics (from integrators to general linear and nonlinear dynamic models) and enhanced flexibility of interconnection topologies (from fixed and undirected to switched and directed). Physical constraints on the states and control inputs, and communication constraints on information exchange amongst the agents have also been taken into consideration. See Bertsekas and Tsitsiklis (1989) and Jadbabaie, Lin, and Morse (2003) for the early works on distributed computing and control, and the books (Bai, Arcak, & Wen, 2011; Bullo, Cortés, & Martínez, 2009; Lewis, Zhang, Hengster-Movric, & Das, 2014; Ren & Beard, 2008) and review articles (Cao, Yu, Ren, & Chen, 2013; Chebotarev & Agaev, 2009; ✩ This work was supported in part by the National Natural Science Foundation of China under Grants 1633007, and 61533007, in part by the U.S. National Science Foundation under Grant EPCN-1903781, and in part by State Key Laboratory of Intelligent Control and Decision of Complex Systems at BIT. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Wei Ren under the direction of Editor Christos G. Cassandras. ∗ Corresponding author. E-mail addresses: [email protected] (T. Liu), [email protected] (Z.-P. Jiang). https://doi.org/10.1016/j.automatica.2020.109020 0005-1098/© 2020 Elsevier Ltd. All rights reserved.
Martínez, Cortés, & Bullo, 2007; Olfati-Saber, Fax, & Murray, 2007) as well as many references therein for the results. Actuation is an important part of the control systems. In this paper, we focus on pulse-width modulation (PWM), which is a common actuation technique widely used in industry. PWM generates rectangular pulse waveforms with different average values by adjusting the pulse width, and is able to process large signals with high efficiency and low sensitivity to noise. PWM modules are available with many popular microcontrollers, and PWM signals are commonly used to approximate analog signals. As an example, high-frequency PWM of on/off power supplies has been extensively used for (almost) continuous control of electrical drives with digital controllers; see, for instance, Gelig and Churilov (1998), Holmes and Lipo (2003) and Skoog and Blankenship (1970). The study of PWM control theory can be traced back to the early 1960s. A variety of techniques including linearization, Lyapunov method, Popov criterion, LaSalle’s invariance principle, and input–output stability were used to address the stability problem in the early results (Balestrino, Eisinberg, & Sciavicco, 1974; Datta, 1972; Delfeld & Murphy, 1961; Gelig, 1968; Kadota & Bourne, 1961; Kuntsevich & Chekhovoi, 1971; Min, Slivinsky, & Hoft, 1977; Murphy & Wu, 1964; Pavlidis & Jury, 1965; Polak, 1961; Skoog & Blankenship, 1970). More recently, techniques from the fields of nonlinear control, singular system and hybrid system theories are adopted to the study of PWM control; see, e.g., Hou and Michel (2001), Komaee (2019), Sira-Ramirez (1989) and Taylor (1992). With respect to multi-agent systems, PWM control is capable of reducing the amount of communication and simplifying the actuators as the duty ratios are only updated on discrete
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time instants and the control signals only take three values. This motivates the study in this paper on state agreement of a class of multi-agent systems with PWM control inputs. Our interest is also motivated by the substantial challenges caused by the coexistence of the hybrid PWM control inputs and the complex system topology. In this research direction, Meng, Meng, Chen, Dimarogonas, and Johansson (2016) studied the distributed PWM control of multi-agent systems with fixed interconnection topologies. In that paper, each agent is controlled by a PWM signal with unit amplitude, and the duty ratio of each PWM control input is designed as a saturated linear function of the diffusionlike interconnection. The spanning tree condition on the fixed interconnection topology is proved to be necessary and sufficient for asymptotic state agreement. Notice that such connectivity condition has been assumed in the prior work of others without considering PWM actuation. In view of the research trend of addressing more general topologies, it is expected to develop a PWM counterpart for multi-agent systems with time-dependent topologies. Moreover, besides the mostly studied connectivity condition on topologies, a designer may also be interested in obtaining some necessary and sufficient condition on the PWM control laws for group coordination. This paper focuses on the state agreement objective, which aims to steer the states of all the agents to a common value, and is known as one of the fundamental group behaviors. The contribution of the paper lies in the proposed necessary and sufficient condition on the PWM control laws for the state agreement of multi-agent systems with directed and switched interconnection topologies, thereby extending some of the results, e.g., Lin, Francis, and Maggiore (2007) and Shi and Hong (2009), from non-PWM case to PWM case, and extending (Meng et al., 2016) from fixed topologies to flexible topologies. As shown in this paper, a trajectory of a PWM-controlled agent with larger initial state may not remain larger, and the range of the states of all the PWM-controlled agents, considered as a Lyapunov function candidate in some existing results, may not be decreasing on the time-line. Furthermore, different PWM control signals are allowed to have different periods and amplitudes, which causes additional complexity for the discussions. In this paper, the coordination problem is formulated as a stabilization problem, and the main Theorem 1 is proved by finding two signals which are upper and lower bounds of the states of the agents, and converge in sense of uniform asymptotic convergence. The paper is organized as follows. Section 2 introduces some basic concepts on the multi-agent systems and PWM control, and presents the problem formulation of distributed PWM control. Section 3 gives the necessary and sufficient condition for uniform asymptotic state agreement of a class of PWM-controlled multiagent systems. The proof of the main result is given in Section 4. Section 5 employs numerical simulations to verify the theoretical results. Section 6 gives some concluding remarks. Notation and terminology Throughout the paper, sgn(r) is the sign of r: sgn(r) = 1 if r > 0; sgn(r) = 0 if r = 0; sgn(r) = −1 if r < 0. Id is the identity function defined on R+ , that is, Id(s) = s for all s ∈ R+ . For all s ∈ R+ , ⌈s⌉ represents the smallest integer that is not less than s, and ⌊s⌋ represents the largest integer that is not larger than s. |x| denotes the Euclidean norm of x ∈ Rn . ∅ represents the empty set. A function α : R+ → R+ is positive definite if α (s) > 0 for all s > 0 and α (0) = 0. α : R+ → R+ is a class K function if it is continuous, strictly increasing and α (0) = 0; it is a class K∞ function if it is a class K function, and also satisfies α (s) → ∞ as s → ∞. A function β : R+ × R+ → R+ is a class KL function if, for each fixed t ∈ R+ , function β (·, t) is a class K function
and, for each fixed s ∈ R+ , function β (s, ·) is decreasing and limt →∞ β (s, t) = 0. A digraph G0 is a pair (N0 , E0 ) where N0 is a nonempty, finite set, and E0 is a subset of N0 × N0 with (i, i) ∈ / E0 for all i ∈ N0 . Elements of N0 are referred to as nodes, and an element (i, j) of E0 is referred to as an edge from i to j. A digraph G0 = (N0 , E0 ) is quasi-strongly connected (QSC) if there exists some c ∈ N0 such that there is a directed path from c to i for each i ∈ N0 \{c }; the node c is called the center of G0 . Clearly, if digraph G0 is QSC, then it has a spanning tree with c as the root. For a switching digraph G (t) = (N , E (t)), we denote the ⋃ union digraph over time interval [t1 , t2 ) as G ([t1 , t2 )) = (N , t ∈[t ,t ) E (t)). A switching 1 2 digraph G (t) is called uniformly quasi-strongly connected (UQSC) with time constant T > 0 if G ([t , t + T )) is QSC for all t ≥ 0. A switching digraph G (t) has a positive edge dwell time ∆E > 0 if for any t ∈ [0, ∞) and any directed edge (i1 , i2 ) ∈ E (t), there exists a te ≥ 0 depending on t and (i1 , i2 ) such that t ∈ [te , te +∆E ) and (i1 , i2 ) ∈ E (τ ) for all τ ∈ [te , te + ∆E ). 2. Preliminaries and problem formulation Consider a multi-agent system defined by x˙ i (t) = ui (t)
(1)
where xi ∈ R is the state and ui ∈ R is the control input, for all i ∈ N = {1, . . . , N }. Normally, the PWM control input is in the form of
{ ui (t) =
mi si (k), 0,
f
for t ∈ [k∆i + tir (k), k∆i + ti (k)), k ∈ Z+ otherwise
(2)
where mi and ∆i are prescribed positive constants representing the amplitude and the period of the PWM signal, respectively, si (k) ∈ {−1, 0, 1} represents the sign of the pulse during the kth f period, and tir (k) and ti (k) represent the time instants of the rising edge and the falling edge during the kth period, respectively. To make the PWM signal well-defined, it is required that 0 ≤ tir (k) ≤ f f ti (k) ≤ ∆i . For the kth period of the PWM signal, (ti (k) − tir (k)) is usually known as the pulse width, and the popular notion of duty ratio (also called duty cycle) is defined as f
di (k) =
(ti (k) − tir (k))
∆i
.
(3)
Clearly, 0 ≤ di (k) ≤ 1. Normally, the duty ratio di and the sign si are considered as the control variables. We consider diffusion-like interconnections between the agents. By using zi to represent the error information used for the control of agent i, we have
{∑
j∈Ni (t) aij (xi (t)−xj (t))
zi (t) =
∑
0,
j∈Ni (t) aij
, if Ni (t) ̸= ∅,
(4)
otherwise,
where aij are positive constants for any pair i, j ∈ N and i ̸ = j, and Ni represents the neighbors of agent i, that is, j ∈ Ni (t) if xj (t) is available to agent i. As usual, we employ a switching digraph G (t) = (N , E (t)) with N = {1, . . . , N } and E (t) = {(j, i) : i ∈ N , j ∈ Ni (t)} to represent the switched information exchange topology amongst the agents. Throughout the paper, the following assumption is made on the connectivity of the interconnection digraph and the periods of the PWM control signals. Assumption 1. The switching digraph G (t) for all t ≥ 0 is UQSC with time constant T , and has an edge dwell-time ∆E ≥ ∆i for all i = 1, . . . , N.
T. Liu and Z.-P. Jiang / Automatica 119 (2020) 109020
With di and si as the control inputs and zi as the measured information, this paper aims to develop a general class of distributed PWM control laws for uniform asymptotic state agreement: there exist a β ∈ KL and a constant c ≥ 0 such that for any xi0 ∈ R with i ∈ N and any t0 ≥ 0, if xi (t0 ) = xi0 for all i ∈ N , then max xi (t) − min xi (t) ≤ β i∈N
i∈N
(
max xi0 − min xi0 + c , t − t0 i∈N
)
i∈N
(5)
for all t ≥ t0 . For practical implementations, it is required that the calculation of control signals di (k) and si (k) only uses the sampled data zi (k∆i ) for k ∈ Z+ . Remark 1. UQSC has been recognized to be the basic connectivity condition for uniform agreement of multi-agent systems. Edge dwell-time basically represents a lower bound of the time durations of all the edges ever appearing. Assumption 1 means that if edge (j, i) appears, then there is at least one period of the PWM control input ui that is able to make use of (xi − xj ). Remark 2. In practice, the position of each pulse with respect to the corresponding PWM period is predetermined, and different types of pulse positions are available for the flexibility of physical implementations: Type A: the pulse center is fixed in the center of the time window of the corresponding period, f i.e., (ti (k) + tir (k))/2 = (k + 1/2)∆i ; Type B: the rising edge occurs at the beginning of the corresponding period, i.e., tir (k) = k∆i ; Type C: the falling edge occurs at the end of the corresponding f period, i.e., ti (k) = (k + 1)∆i . By considering the signs si and the duty ratios di as the control inputs, this paper aims to develop a class of distributed PWM control laws, which is valid for different types of PWMs in terms of the pulse positions. Remark 3. The constant c in (5) is used to characterize the influence of the discontinuous PWM control input during the period before t0 . This term is considered as an off-set with the initial state in the literature of nonlinear systems theory, and generally it does not influence the convergence and robustness properties (Bao, Liu, & Jiang, 2019; Jiang, Teel, & Praly, 1994). 3. State agreement of distributed PWM-controlled multiagent systems 3.1. Properties of single PWM-controlled agents This paper considers a multi-agent system as an interconnection of individual agents. Before designing the distributed PWM controller for the multi-agent system, this subsection studies the properties of a class of PWM-controlled individual agents:
ξ˙ (t) = µ(t) { ms(k), µ(t) = 0,
(6) for t ∈ [k∆ + t r (k), k∆ + t f (k)), k ∈ Z+ otherwise
(7)
where ξ ∈ R is the state, µ ∈ R is the PWM signal, m and ∆ are positive constants, s ∈ {−1, 0, 1} represents the sign of the pulse. The sign s(k) and the duty ratio d(k) = (t f (k) − t r (k))/∆ are usually considered as the control inputs, and the most fundamental control law would be in the form of s(k) = − sgn(ξ (k∆) − ω(k∆))
(8)
d(k) = ∆φ (|ξ (k∆) − ω(k∆)|)
(9)
where φ : R+ → [0, 1] is a continuous function, and ω : R+ → R is a bounded and piece-wise continuous function. Here, ω is considered as an external input for the closed-loop system composed of (6)–(9). In the case of ω ≡ 0, it can be directly
3
recognized that the PWM control law is negative feedback, that is, the sign of the PWM signal is opposite to the sign of ξ . For distributed control design in the following subsection, ω will be used to represent the interconnection between the agents, and the boundedness of ω will be guaranteed by first proving the boundedness of the states of all the agents by using Lemma 1. Lemma 1. that
Consider the PWM-controlled system (6)–(9). It holds
ξ ((k + 1)∆) = ξ (k∆) − m∆ sgn(ξ (k∆) − ω(k∆))φ (|ξ (k∆) − ω(k∆)|) (10) for all k ∈ Z+ , and moreover,
ξ (t) ∈ [ξ (k∆), ξ ((k + 1)∆)]
(11)
for all t ∈ [k∆, (k + 1)∆) with k ∈ Z+ . Lemma 1 can be proved by directly taking the integration of both sides of (6) and using (7)–(9); see, e.g., Gelig and Churilov (1998). Lemma 1 means that the evolution of the PWM-controlled state on discrete time-line with period ∆ depends only on the pulse width, and is not influenced by the positions of the pulses with respect to the corresponding periods, though different positions of the pulse lead to different state trajectories during each period. Lemma 2. that
Consider the PWM-controlled system (6)–(9). Suppose
ω(t) ∈ Ω
(12)
for all 0 ≤ t ≤ T , with Ω ⊂ R being a compact interval and T ≥ ∆. ˘ = Ω u − Ω l . If there Denote Ω u = max Ω , Ω l = min Ω , and Ω exist continuous and positive definite functions δ1 , δ2 < Id such that φ : R+ → [0, 1] satisfies
δ1 (s) ≤ φ (s) ≤
(Id −δ2 )(s) m∆
(13)
for all s ∈ R+ , then for the specific T , there exist an α(Ω˘ )1 (which ˘ , and for any specific Ω ˘ , is continuous, continuously depends on Ω positive definite and less than Id), and an α2 (which is continuous, positive definite and less than Id), such that
• if ξ (0) > Ω u , then for all t ∈ [∆, T ], ξ (t) ≤ Ω u + α(Ω˘ )1 (ξ (0) − Ω u );
(14)
• if ξ (0) ≤ Ω u , then for all t ∈ [∆, T ], ξ (t) ≤ Ω u − α2 (Ω u − ξ (0));
(15)
• if ξ (0) < Ω l , then for all t ∈ [∆, T ], ξ (t) ≥ Ω l − α(Ω˘ )1 (Ω l − ξ (0));
(16)
• if ξ (0) ≥ Ω l , then for all t ∈ [∆, T ], ξ (t) ≥ Ω l + α2 (ξ (0) − Ω l ).
(17)
Proof. We only give the proof for (14) and (15). Properties (16) and (17) can be proved in the same way. We study the behavior of ξ (t) in the cases of ξ (k∆) ≤ Ω u and ξ (k∆) > Ω u separately.
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T. Liu and Z.-P. Jiang / Automatica 119 (2020) 109020
• Case 1: ξ (k∆) > Ω u . In this case, sgn(ξ (k∆) − ω(k∆)) = 1. Then, by using Lemma 1, we have
ξ ((k + 1)∆) = ξ (k∆) − m∆φ (ξ (k∆) − ω(k∆)) ≤ ξ (k∆) − m∆ min φ (ξ (k∆) − ω)
Lemmas 3 and 4 give some direct consequences of dissatisfaction of condition (13), and are used for the proof of the necessity part of the main result.
= ξ (k∆) − m∆ min φ (ξ (k∆) − Ω + ζ )
Lemma 3. Consider the PWM-controlled system (6)–(9). If φ does not satisfy the first inequality in (13), then there exists a λ > 0 such that
≤ ξ (k∆) − m∆ min δ1 (ξ (k∆) − Ω u + ζ ) (18)
|ξ (k∆) − ω(k∆)| = λ ⇒ ξ ((k + 1)∆) = ξ (k∆).
ω∈Ω
u
˘ 0≤ζ ≤Ω
(28)
˘ 0≤ζ ≤Ω
where condition (13) is used for the last inequality. Define
α(′Ω˘ ) (s) = m∆ min δ1 (s + ζ )
(19)
˘ 0≤ζ ≤Ω
(29)
Choose λ = s . In the case of ξ (k∆) − ω(k∆) = λ, (6)–(9) imply
ξ ((k + 1)∆) ≤ ξ (k∆) − α(Ω˘ ) (ξ (k∆) − Ω ). u
′
(20)
A direct consequence of condition (13) is that m∆δ1 (s) < s for all s ∈ R+ . From its definition, α ′ ˘ (·) continuously (Ω )
˘ , and for any specific Ω ˘ , it is continuous, depends on Ω positive definite and less than Id. • Case 2: ξ (k∆) ≤ Ω u . In this case, if ξ (k∆) ≤ ω(k∆), then
ξ ((k + 1)∆) = ξ (k∆) + m∆φ (ω(k∆) − ξ (k∆)) ≤ ξ (k∆) + m∆ max u φ (ω − ξ (k∆)) ω∈[ξ (k∆),Ω ]
≤ ξ (k∆) +
φ (s∗ ) = 0. ∗
for all s ∈ R+ . Then, we have
= ξ (k∆) + m∆
Proof. If the first inequality in (13) does not hold, then due to the continuity of φ , there exists an s∗ > 0 such that
max
ζ ∈[0,Ω u −ξ (k∆)]
max
ζ ∈[0,Ω u −ξ (k∆)]
(21)
where condition (13) is used for the last inequality. Define
α (s) = max (Id −δ2 )(δ ) ′′
δ∈[0,s]
(22)
for all s ∈ R+ . Thus,
ξ ((k + 1)∆) ≤ ξ (k∆) + α ′′ (Ω u − ξ (k∆)).
(23)
Condition (13) means that δ2 is less than Id. By also using the continuity and positive definiteness of δ2 , one can prove that α ′′ is continuous, positive definite and less than Id. In the same case, if ξ (k∆) > ω(k∆), then
ξ ((k + 1)∆) = ξ (k∆) − m∆φ (ξ (k∆) − ω(k∆)) ≤ ξ (k∆).
= ξ (k∆).
(30)
The case for ω(k∆) − ξ (k∆) = λ can be proved in the same way. This ends the proof of Lemma 3. □ Lemma 4. Consider the PWM-controlled system (6)–(9). If φ does not satisfy the second inequality in (13), then there exists a λ > 0 such that
|ω(k∆) − ξ (k∆)| = λ ⇒ ξ ((k + 1)∆) = ω(k∆).
φ (ζ )
(Id −δ2 )(ζ )
ξ ((k + 1)∆) = ξ (k∆) − m∆φ (ξ (k∆) − ω(k∆)) = ξ (k∆) − m∆φ (s∗ )
(24)
(31)
Proof. If the second inequality in (13) is not satisfied, then due to the continuity of φ , there exists an s∗ > 0 such that
φ (s∗ ) =
s∗ m∆
.
(32)
Choose λ = s∗ . If ξ (k∆) − ω(k∆) = −λ, then (6)–(9) imply
ξ ((k + 1)∆) = ξ (k∆) + m∆φ (ω(k∆) − ξ (k∆)) = ξ (k∆) + m∆φ (s∗ ) = ξ (k∆) + s∗ = ω(k∆).
(33)
The case for ξ (k∆) − ω(k∆) = λ can be proved in the same way. This ends the proof of Lemma 4. □ 3.2. Main result on the state agreement of the PWM-controlled agents
From both (23) and (24), we have
ξ ((k + 1)∆) ≤ ξ (k∆) + α ′′ (Ω u − ξ (k∆)).
(25)
Recall that α ′′ is less than Id. This means that, if ξ (k∆) ≤ Ω u , then ξ ((k + 1)∆) ≤ Ω u . Denote m(T ) = ⌈T /∆⌉. Based on the discussions above, we have
• if ξ (0) > Ω u , then (20) implies ξ (k∆) ≤ Ω u + (Id −α(′Ω˘ ) )(ξ (0) − Ω u )
di (k) = φi (|zi (k∆i )|)
(34)
si (k) = − sgn(zi (k∆i ))
(35)
where φi : R+ → [0, 1] is a continuous function. By defining
{∑
(26)
wi (t) =
for all 1 ≤ k ≤ m(T ); • if ξ (0) ≤ Ω u , then (25) implies
ξ (k∆) ≤ Ω u − (Id −α ′′ )m(T ) (Ω u − ξ (0))
For the PWM controlled multi-agent system (1)–(2), motivated by the standard distributed control law, the idea of ‘‘negative feedback’’ is used in the design of the PWM control law:
(27)
for all 1 ≤ k ≤ m(T ), where (Id −α ′′ )m(T ) represents the composition of m(T ) number of (Id −α ′′ ). Then, properties (14) and (15) can be proved by defining
α(Ω˘ )1 (s) = (Id −α(′Ω˘ ) )(s) and α2 (s) = (Id −α ′′ )m(T ) (s) for all s ∈ R+ , and using the monotonicity of ξ (t) during each interval [k∆, (k + 1)∆) for all k ∈ Z+ . This ends the proof of Lemma 2. □
j∈Ni (t) ∑
aij xj (t)
j∈Ni (t) aij
xi (t),
, if Ni (t) ̸= ∅, otherwise,
(36)
to represent the influence of other agents, zi (t) given by (4) can be rewritten as zi (t) = xi (t) − wi (t),
(37)
and the proposed distributed PWM controller (34)–(35) can be equivalently written in the form of (8)–(9). Theorem 1 gives the necessary and sufficient condition on φi for asymptotic state agreement of PWM-controlled multi-agent systems.
T. Liu and Z.-P. Jiang / Automatica 119 (2020) 109020
5
Theorem 1. Consider the PWM-controlled multi-agent system described by (1), (2), (3), (4), (34) and (35) with N ≥ 3. Uniform asymptotic state agreement (5) is achievable for any G satisfying Assumption 1, if and only if for each i ∈ N , there exist continuous and positive definite functions δi1 , δi2 such that φi : R+ → [0, 1] satisfies
δi1 (s) ≤ φi (s) ≤
(Id −δi2 )(s) m i ∆i
(38)
for all s ∈ R+ . The proof of Theorem 1 is given in Section 4. Remark 4. With the first inequality in (38) satisfied, the PWM control law is in accordance with the standard diffusion-like distributed control laws in the sense that negative feedback of the interconnection is used. A special requirement caused by the switching topology is the upper-bound condition given by the second inequality. Please see Section 4.1 for the proof of the necessity of the upper-bound condition of φi . Remark 5. The closed-loop system can be described by a specific class of retarded functional differential equations, for which the existence and uniqueness of the solution have been well studied in the literature of dynamic systems and mathematical control theory; see, e.g., Hale and Lunel (1993) and Karafyllis and Jiang (2011). For the specific class of systems studied in this paper, the analytical solution during each PWM period can be calculated by taking the integration of both sides of the differential equation. Also, according to the definition of PWM signals, there is only one pulse during each predetermined, fixed period, which means that no chattering happens in the closed-loop system.
Fig. 1. An example for the necessity of the second inequality in (38): x1 (•) and x3 (◦) keep constant, and x2 (⋄) oscillates between x1 and x3 .
• Set φ2 as one of the control laws that do not satisfy the second inequality in (38). By using Lemma 4, there exists a λ > 0 such that |ω2 (k∆2 ) − x2 (k∆2 )| = λ ⇒ x2 ((k + 1)∆2 ) = ω2 (k∆2 ). Choose E (t) such that E ([0, ∞)) = {(1, 2), (3, 2), (2, 3)} ∪ ⋃ k̸ =1,2,3 {(1, k), (2, k), (3, k)}, and in particular, (1, 2) exists for all t ∈ [3k∆E , (3k + 1)∆E ), (3, 2) exists for all t ∈ [(3k + 1)∆E , (3k + 2)∆E ), and (2, 3) exists for all t ∈ [(3k + 2)∆E , 3(k + 1)∆E ) with ∆E > max{∆1 , ∆2 , ∆3 }. In this case, if x1 (0) − x2 (0) = λ and x2 (0) = x3 (0), then x1 (t) = x1 (0) and x3 (t) = x3 (0) for all t ≥ 0, and x2 (t) oscillates between x1 (0) and x3 (0). The evolution of the states on the discrete time instants k∆E (k ∈ Z+ ) is shown in Fig. 1. The necessity part is proved. 4.2. Sufficiency
Remark 6. The proposed result covers the situation of fixed topologies having spanning trees and the situation of saturated linear control laws. In particular, φi (s) = min{κi s, 1} for all s ∈ R+ satisfies condition (38) as long as 0 < κi < 1/mi ∆i . Also, to the best of our knowledge, the necessary and sufficient condition for uniform asymptotic state agreement of PWM-controlled multiagent systems reported by this paper is for the first time in the literature.
The following discussion is for any specific initial time t0 ≥ 0. The forward completeness of the PWM-controlled multi-agent system can be proved by considering the ‘‘worst case’’: for each i ∈ N , −mi ≤ x˙ i ≤ mi implies −mi t ≤ xi (t) − xi (t0 ) ≤ mi t for all t ≥ t0 . The forward completeness is guaranteed. For convenience of discussions, denote x(t) = maxj∈N xj (t), x(t) = minj∈N xj (t), and x˘ (t) = x(t) − x(t). To prove the asymptotic convergence property, define
4. Proof of the main result
T o = T + 2∆E ,
We divide the proof of Theorem 1 into two parts. Section 4.1 focuses on the necessity, and Section 4.2 is devoted to the proof of the sufficiency part.
to represent different time scales. In the following proof, we find an asymptotically converging upper bound of x˘ (t). Claims 1 and 2 give boundedness and convergence results, respectively.
4.1. Necessity
Claim 1. min
The necessity part can be proved if we can find a PWMcontrolled multi-agent system with specific interconnection digraph G satisfying Assumption 1, such that if at least one of the PWM controllers does not satisfy condition (38), then asymptotic convergence cannot be achieved. Consider the following two cases:
• Set φ2 as one of the control laws that do not satisfy the first inequality in (13). By using Lemma 3, there exists a λ > 0 such that x2 (k∆2 )−ω2 (k∆2 ) = λ ⇒ x2 ((k+1)∆2 ) = x2⋃ (k∆2 ). Choose E (t) such that E ([0, ∞)) = {(1, 2)} ∪ j̸ =1,2 {(1, j), (2, j)}, and in particular, (1, 2) exists for all t ≥ 0. In this case, if x2 (0) − x1 (0) = λ, then x1 (t) = x1 (0) and x2 (t) = x2 (0) for all t ≥ 0. Convergence cannot be achieved.
T ∗ = NT o + ∆E
(39)
For any t ∗ ≥ 0,
max{0,t ∗ −∆E }≤τ ≤t ∗
x(τ ) ≤ x(t) ≤ x(t) ≤
max
max{0,t ∗ −∆E }≤τ ≤t ∗
x(τ )
(40)
for all t ≥ t ∗ , where ∆E is the edge dwell-time of G given by Assumption 1. Proof of Claim 1. Property (40) is proved by contradiction. Recognizing that the first and the last inequalities in (40) can be proved in the same way, we only give the proof for the last inequality. For a specific t ∗ > 0, suppose that the last inequality in (40) does not hold for some t ≥ t ∗ . Then, there exist a tc > t ∗ and an i∗ ∈ N such that x(t) ≤ maxmax{0,t ∗ −∆E }≤τ ≤t ∗ x(τ ) for all t ∈ [t ∗ , tc ] and x(t) > maxmax{0,t ∗ −∆E }≤τ ≤t ∗ x(τ ) for all t ∈ (tc , tc + tϵ ) with some tϵ > 0. Define kc ∈ Z+ such that tc ∈ [kc ∆i , (kc + 1)∆i ). Then, we have kc ∆i ≤ tc and kc ∆i ≥ max{0, tc − ∆i } ≥ t ∗ − ∆E . Condition ∆E ≥ ∆i given by Assumption 1 is used here. By also
6
T. Liu and Z.-P. Jiang / Automatica 119 (2020) 109020
for all t ∈ [kr T ∗ , (kr + 1)T ∗ ). By using Lemma 2, we have
using the definition of ωi in (36), we have
ωi (kc ∆i ) ≤ x(kc ∆i ) ≤ =
max
max{0,t ∗ −∆E }≤τ ≤t ∗
max
max{0,t ∗ −∆E }≤τ ≤t ∗
x(τ )
xi1 (t) ≥ η(kr ),
x(τ )
(41)
for all 0 ≤ t ≤ tc . In this case, it holds that
≤ η(kr ) −
xi ((kc + 1)∆i ) = xi (kc ∆i )
− mi ∆i sgn(xi (kc ∆i ) − ωi (kc ∆i ))· φi (|xi (kc ∆i ) − ωi (kc ∆i )|) = xi (kc ∆i ) + mi ∆i φi (ωi (kc ∆i ) − xi (kc ∆i )) ≤ xi (kc ∆i ) + (Id −δi2 )(ωi (kc ∆i ) − xi (kc ∆i )) ≤ ωi (kc ∆i ) ≤
max
max{0,t ∗ −∆E }≤τ ≤t ∗
x(τ ),
(42)
where condition (38) is used in the first inequality; • if xi (kc ∆i ) ≥ ωi (kc ∆i ), then
max
max{0,t ∗ −∆E }≤τ ≤t ∗
x(τ ),
(43)
where condition (38) is used in the first inequality. Recall the monotonicity of xi (t) for all t ∈ [kc ∆i , (kc + 1)∆i ); see Lemma 1. Thus, there does not exist a tϵ > 0 such that x(t) > maxmax{0,t ∗ −∆E }≤τ ≤t ∗ x(τ ) for all t ∈ (tc , tc + tϵ ), which leads to a contradiction. Finally, property (40) is proved. □ Claim 2. There exists α : R+ → R+ which is continuous, positive definite and less than Id, such that for any k∗ ∈ Z+ , one can find η, η : {k∗ , k∗ + 1, . . . , ∞} → R satisfying
η(k) ≥ η(k)
(44)
η(k + 1) ≤ η(k)
(45)
η(k + 1) ≥ η(k)
(46)
η(k + 1) − η(k + 1) ≤ α (η(k) − η(k))
(47)
for all k ∈ {k , k + 1, . . . , ∞}, and ∗
η(k) ≤ x(t) ≤ x(t) ≤ η(k)
o
∗
η(k ) =
xij (t) ≤ η(k ) − α(η˘ (kr ))ij (η˘ (k ))
(49)
min
x(τ ).
(50)
η(k ) + η(k ) r
2
.
(51)
For convenience of discussions, denote η˘ = η − η. Initial Step: From the definition of ωi1 , we have
η(kr ) ≤ ωi1 (t) ≤ η(kr )
(56)
for all t ∈ [k T + jT , (k + 1)T ), where α(η˘ (kr ))ij continuously depends on η˘ (kr ), and is continuous, positive definite and less than Id, for all j = 1, . . . , l − 1. By using the definitions of il and edge dwell-time, there exist a til ∈ [kr T ∗ + (l − 1)T o , kr T ∗ + (l − 1)T o + ∆E + T ) and an l′ ∈ {1, . . . , l − 1} such that (il′ , il ) ∈ E (t) for all t ∈ [til , til + ∆E ). Thus, we have
ωil (t) ≥ η(kr ) ∑ ωil (t) ≤
j∈Ni
o
∗
r
∗
(57) l
r r r ˘ (kr ))) (t)\{i ′ } ail jη (k ) + ail il′ (η (k ) − α(η˘ (k ))il′ (η l
∑
j∈Ni (t) l
ail j
ail il′ α(η˘ (kr ))il′ (η˘ (k )) ≤ η(kr ) − ∑ r
j∈Ni ([0,∞)) l
=: η(k ) − α r
0 (η˘ (kr ))il (
ail j
η˘ (k )) r
(58)
for all t ∈ [til , til + ∆E ). Clearly, α(0η˘ (kr ))i continuously depends on l η˘ (kr ), and is continuous, positive definite and less than Id. Then, by using Lemma 2, we have xil (til + ∆E ) ≥ η(kr )
(59)
+
η˘ (k ))
0 (η˘ (kr ))il (
r
max
α(1η˘ (kr ))il (ζ )
=: η(kr ) − α(2η˘ (kr ))il (η˘ (kr ))
Claim 1 guarantees that property (48) holds for k = k∗ . Suppose that property (48) holds for a specific kr ≥ k∗ . We study the motion of the PWM-controlled agents during the interval [kr T ∗ , (kr + 1)T ∗ ). For all t ∈ [kr T ∗ , (kr + 1)T ∗ ), define r ∗ o r ∗ i1 , . . . , iN ∈ N such that i⋃ 1 = c and E ([k T + (l − 1)T + ∆E , k T + (l − 1)T o + ∆E + T )) ∩ l′ =1,...,l−1 (il′ , il ) ̸ = ∅. The existence of such i1 , . . . , iN is guaranteed by the UQSC property of G with time constant T ; see Assumption 1. We consider the case of xi1 (kr T ∗ ) ≤
r
0≤ζ ≤α 0 r (η˘ (kr )) (η˘ (k ))il
x(τ ),
r
(55)
r
xil (til + ∆E ) ≤ η(k ) − α
max
max{0,k∗ T ∗ −∆E }≤τ ≤k∗ T ∗
∗
xij (t) ≥ η(kr )
(48)
max{0,k∗ T ∗ −∆E }≤τ ≤k∗ T ∗
(54)
r
r
Proof of Claim 2. Choose
∗
α(′η˘ (kr ))i1 (ζ )
for all k T + T ≤ t < (k + 1)T . Recursive Step: Suppose that r
for all t ∈ [kT ∗ , (k + 1)T ∗ ) with k ∈ {k∗ , k∗ + 1, . . . , ∞}.
η(k∗ ) =
min
η˘ (kr )/2≤ζ ≤η˘ (kr )
=: η(kr ) − α(η˘ (kr ))i1 (η˘ (kr ))
r
xi ((kc + 1)∆i ) = xi (kc ∆i ) − mi ∆i φi (xi (kc ∆i ) − ωi (kc ∆i ))
∗
for all kr T ∗ + T o ≤ t < (kr + 1)T ∗ , and there exists a continuous and positive definite α(η˘ (kr ))i1 which continuously depends on η˘ (kr ) and is less than Id such that xi1 (t) ≤ η(kr ) − α(η˘ (kr ))i1 (η(kr ) − xi1 (kr T ∗ ))
• if xi (kc ∆i ) < ωi (kc ∆i ), then
≤ xi (kc ∆i ) ≤
(53)
(52)
(60)
It can be directly checked that α(2η˘ (kr ))i also continuously depends l on η˘ (kr ), and is continuous, positive definite and less than Id. r ∗ For any t ∈ [til + ∆E , (k + 1)T ), we have
η(kr ) ≤ ωil (t) ≤ η(kr ).
(61)
Thus, by using Lemma 2, we have xil (t) ≥ η(kr )
(62)
xil (t) ≤ η(k ) − α r
≤ η(k ) −
η(k ) − xil (til + ∆E ))
3 (η˘ (kr ))il (
r
r
min
α(2η˘ (kr ))i (η˘ (kr ))≤ζ ≤η˘ (kr )
α(3η˘ (kr ))il (ζ )
l
=: η(kr ) − α(η˘ (kr ))il (η˘ (kr ))
(63)
for all t ∈ [til + ∆E , (kr + 1)T ∗ ). Since til ∈ [kr T ∗ + (l − 1)T o , kr T ∗ + (l − 1)T o + ∆E + T ), we have [kr T ∗ + lT o , (kr + 1)T ∗ ) ⊆ [til + ∆E , (kr + 1)T ∗ ). Thus, from the discussions above, if (55)–(56) hold for all t ∈ [kr T ∗ + (l − 1)T o , (kr + 1)T ∗ ), then (62)–(63) hold for all t ∈ [kr T ∗ + lT o , (kr + 1)T ∗ ). This ends the recursive step.
T. Liu and Z.-P. Jiang / Automatica 119 (2020) 109020
Define
α (s) = min {α(s)ij (s)}
(64)
j=1,...,N
for all s ∈ R+ . Then, α is continuous, positive definite and less than Id. Define
η(kr + 1) = η(kr ) − α (η˘ (kr )),
(65)
η(kr + 1) = η(kr ).
(66)
From the discussions above, we have
η(kr + 1) ≤ x(t) ≤ x(t) ≤ η(kr + 1)
(67)
holds for all t ∈ [k T + NT , (k + 1)T ) = [(k + 1)T ∗ − ∆E , (kr + 1)T ∗ ). Then, Claim 1 implies that (48) holds for all t ∈ [(kr + 1)T ∗ , (kr + 2)T ∗ ). The discussions above assume (51). If r
o
∗
r
r
∗
η(kr ) + η(kr )
xi1 (kr T ∗ ) >
(68)
2
holds instead, then the definition of η(kr + 1) and η(kr + 1) would be in the form of
η(kr + 1) = η(kr ),
(69)
η(k + 1) = η(k ) + α (η˘ (k )). r
r
r
(70)
This ends the proof of Claim 2. □ With (47), there exists a βη˘ ∈ KL such that 0 ≤ x˘ (t) ≤ η˘ (k) ≤ βη˘ (η˘ (k∗ ), k − k∗ )
(71)
with k = ⌊t /T ⌋, for all t ≥ k T . For any xi0 ∈ R with i ∈ N and any t0 ≥ 0, if xi (t0 ) = xi0 for all i ∈ N , then we have ∗
∗ ∗
x0 − m(T ∗ + ∆E ) ≤ x(t) ≤ x(t) ≤ x0 + m(T ∗ + ∆E )
(72)
for all max{0, k0 T − ∆E } ≤ t ≤ k0 T , where k0 = ⌊t0 /T ⌋, x0 = maxi∈N xi0 , and x0 = mini∈N xi0 . For all t ≥ t0 , by also using (71), we have ∗
(
x˘ (t) ≤ βη˘
η˘ (k0 ),
⌊
∗
⌋
t T∗
(
⌊ −
(
⌋)
t0 T∗
≤ βη˘ x˘ 0 + 2m(T + ∆E ), ∗
⌊
t
⌋
⌊ −
T∗
≤ βη˘ x˘ 0 + 2m(T + ∆E ), max ∗
∗
{
T∗
t − t0 T∗
⌋)
t0
β (s, t) =
− 2, 0
(1 + ι − ιt /2T )βη˘ (s, 0),
for t ≤ 2T ,
βη˘ (s, t /T − 2),
for t > 2T ∗ ,
∗
∗
with respect to t is impossible. Basically, the uniformness of the main result is guaranteed by the periodicity of both topology connectivity and PWM control signals. Remark 8. The condition that aij are positive constants is used in deriving (58). The proposed result is still valid when aij are time-varying and there exist constants 0 < aij ≤ aij such that aij ≤ aij (t) ≤ aij for all t ≥ 0. Remark 9. The main theorem focuses on the case of N ≥ 3. When the system is reduced to one composed of only two agents, the condition required for state agreement could be relaxed. Indeed, one specific relaxation is allowing δ12 or δ22 in (38) to be zero. Since there are only two agents, the convergence of one agent to the other means the agreement of the states of all the agents. The validity of the relaxed condition can still be proved similarly as for the proof for Theorem 1. However, on the contrary, such relaxed condition cannot be readily proved to be necessary for asymptotic state agreement. Due to space limitation, we leave this issue to further research. 5. A numerical example In this section, we validate the proposed theoretical result by means of a numerical simulation. Consider a group of six PWM-controlled agents (N = {1, . . . , 6}), with the PWM parameters chosen as: mi = 0.5i and ∆i = 0.05i. Regarding the pulse positions, agents 1,2, agents 3,4, and agents 5,6 are of Type A, Type B, and Type C, respectively. See Remark 2 on the pulse positions. For each i ∈ N , the PWM controller is designed as
φi (s) = min
{
(Id −δ i2 )(s) mi ∆i
,1
}
eij (t) = t − sup te ≤ t : cij (te ) = 0
.
(73)
∗
(74)
for all s, t ∈ R+ , with ι being a positive constant. Then, β ∈ KL. Property (5) can be proved by choosing c = 2m(T ∗ + ∆E ). This ends the proof of Theorem 1. Remark 7. Given a multi-agent system subject to timedependent topologies, designers are concerned about what controller works. However, some doubt might still be cast upon the necessity of UQSC of G for uniform asymptotic state agreement. For this problem, one may consider a contraposition-based reasoning as given by Lin et al. (2007), Moreau (2005) and Shi and Hong (2009). Precisely, one may assume that G does not have a finite time constant T for UQSC (i.e., T → ∞ as t → ∞), then the functions α(·) defined in the proof above should depend on the initial time, and constructing a function β that is uniform
(75)
with δ i2 (s) = 0.1(1 + 0.1 sin(is))s for all s ∈ R+ . It can be directly checked that δ i2 is continuous, positive definite and less than Id, and thus φi satisfies (38) for all i ∈ N . We use a time dependent (0, 1)-matrix C (t) = [cij (t)]6×6 to represent the time-varying topology, that is, cij (t) = 1 if (i, j) ∈ E (t) with E (t) representing the set of edges. A time dependent matrix E(t) = [eij (t)]6×6 ∈ R6×6 is used to represent the hold times of the edges. Without loss of generality, assume that C and E are right-continuous. In particular,
{
})
Define
{
7
}
(76)
for all t ≥ 0. Choose cij (t) as:
• For all (i, j) ∈ E0 , if maxt −T +δT ≤τ ≤t cij (τ ) = 0, then set cij (t) = 1; otherwise, if eij (t) ≥ ∆E , then set cij (t) = random1 , and if eij (t) = 0, then set cij (t) = random2 . • For all (i, j) ∈ / E0 , if eij (t) ≥ ∆E or eij (t) = 0, then set cij (t) = random3 . Here, randomk with k = 1, 2, 3 randomly takes a value from {0, 1}, and δT can be any positive constant that is smaller than T . By choosing E0 such that G0 = (N , E0 ) is QSC, we can guarantee that G = (N , E ) is UQSC with time constant T and edge dwelltime ∆E . For the simulation, we set E0 = {(1, 2), (2, 3), (3, 4), (4, 5), (5, 6)}, T = 1, ∆E = 0.1, δT = 0.02, and randomi = round((oi − rand)/2), with ‘‘rand’’ generating values from the uniform distribution on the interval [0, 1], ‘‘round’’ finding the nearest integer, and o1 = 1.8 and o2 = o3 = 1.01. Also, we set aij = 1 for all i ̸ = j. Fig. 2 shows the state trajectories of the PWM-controlled multi-agent system with the initial states x(0) = [x1 (0), x2 (0), x3 (0), x4 (0), x5 (0), x6 (0)]T = [0, 1, 0.5, 0, −0.5, −1]T .
8
T. Liu and Z.-P. Jiang / Automatica 119 (2020) 109020
Fig. 2. State agreement of a PWM-controlled multi-agent system.
6. Conclusions This paper has shown that uniform asymptotic state agreement is achievable by PWM-controlled multi-agent systems with directed and switched topologies. A necessary and sufficient condition on the distributed PWM control laws is proposed. Following this line of research, the next step is to examine the distributed control problem with other types of pulse modulation, e.g., amplitude modulation, density modulation, position modulation, and their combinations. Instead of considering continuous feedback in this paper, a more general case is that some or all of the design functions φi in the proposed control law are allowed to be discontinuous, to tackle the logical control designs. It is also of theoretical importance and practical relevance to study the distributed PWM control problem for multi-agent systems with moving leaders. Acknowledgments The authors would like to thank the anonymous AE and reviewers for their invaluable comments for the improvement of the paper. References Bai, H., Arcak, M., & Wen, J. (2011). Cooperative control design: A systematic, passivity-based approach. Springer. Balestrino, A., Eisinberg, A., & Sciavicco, L. (1974). A generalized approach to the stability analysis of PWM feedback control systems. Journal of Franklin Institute, 298, 45–58. Bao, A., Liu, T., & Jiang, Z. P. (2019). An IOS small-gain theorem for large-scale hybrid systems. IEEE Transactions on Automatic Control, 64, 1295–1300. Bertsekas, D., & Tsitsiklis, J. (1989). Parallel and distributed computation: Numerical methods. Prentice-Hall. Bullo, F., Cortés, J., & Martínez, S. (2009). Distributed control of robotic networks. Princeton. Cao, Y., Yu, W., Ren, W., & Chen, G. (2013). An overview of recent progress in the study of distributed multi-agent coordination. IEEE Transactions on Industrial Informatics, 9, 427–438. Chebotarev, P. Y., & Agaev, R. P. (2009). Coordination in multiagent systems and Laplacian spectra of digraphs. Automation and Remote Control, 70, 128–142. Datta, K. (1972). Stability of pulse-width-modulated feedback systems. International Journal of Control, 16, 977–983. Delfeld, F. R., & Murphy, G. J. (1961). Analysis of pulse-width modulated control systems. IEEE Transactions on Automatic Control, 6, 283–292. Gelig, A. K. (1968). Absolute stability of nolinear sampled-data systems with pulse-width and pulse-time modulation. Automation and Remote Control, 29, 1043–1054. Gelig, A. K., & Churilov, A. N. (1998). Stability and oscillations of nonlinear pulse-modulated systems. Boston: Birkhauser. Hale, J. K., & Lunel, S. M. V. (1993). Introduction to functional differential equations. NY: Springer. Holmes, D. G., & Lipo, T. A. (2003). Pulse width modulation for power converters: Principle and practice. Wiley-IEEE Press. Hou, L., & Michel, A. N. (2001). Stability analysis of pulse-width-modulated feedback systems. Automatica, 37, 1335–1349. Jadbabaie, A., Lin, J., & Morse, A. (2003). Coordination of groups of mobile autonomous agents using nearest neighbor rules. IEEE Transactions on Automatic Control, 48, 988–1001.
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Tengfei Liu received the B.E. and the M.E. degrees from South China University of Technology and the Ph.D. degree from RSISE, the Australian National University. He was a postdoc with faculty fellowship at Polytechnic Institute of New York University. He is a faculty member at Northeastern University, China. His research interests include mathematical control theory of interconnected nonlinear systems.
Zhong-Ping Jiang received the M.Sc. degree in statistics from the University of Paris XI, France, in 1989, and the Ph.D. degree in automatic control and mathematics from the Ecole des Mines de Paris (now, called ParisTech-Mines), France, in 1993, under the direction of Prof. Laurent Praly. Currently, he is a Professor of Electrical and Computer Engineering at the Tandon School of Engineering, New York University. His main research interests include stability theory, robust/adaptive/distributed nonlinear control, robust adaptive dynamic programming, learning-based control and their applications to information, mechanical and biological systems. In these fields, he has written five books and is author/co-author of over 450 peerreviewed journal and conference papers. Dr. Jiang has served as Deputy Editor-in-Chief, Senior Editor and Associate Editor for numerous journals. Prof. Jiang is a Fellow of the IEEE, a Fellow of the IFAC, a Fellow of the CAA and is among the Clarivate Analytics Highly Cited Researchers.
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Brief paper
Distributed control of multi-agent systems with pulse-width-modulated controllers✩ ∗
Tengfei Liu a , , Zhong-Ping Jiang b a b
State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang, 110004, China Department of Electrical and Computer Engineering, New York University, 370 Jay Street, Brooklyn, NY 11201, USA
article
info
Article history: Received 21 December 2018 Received in revised form 4 December 2019 Accepted 19 April 2020 Available online xxxx
a b s t r a c t This paper studies the distributed control of multi-agent systems with pulse-width-modulated (PWM) control inputs. A necessary and sufficient condition on the PWM control laws is given for uniform asymptotic state agreement with directed and switched information exchange topologies. The methodology is validated by a computer-based simulation. © 2020 Elsevier Ltd. All rights reserved.
Keywords: Pulse-width modulation Multi-agent systems Switched topologies State agreement
1. Introduction In the past two decades, considerable attention has been devoted to the distributed control of groups of agents for coordinated movement. In addition to practical motivations, one of the technical reasons for this is that conventional control theory cannot be readily used to address group behaviors. Increasingly challenging control problems have been addressed for multiagent systems, in terms of growing complexity of system dynamics (from integrators to general linear and nonlinear dynamic models) and enhanced flexibility of interconnection topologies (from fixed and undirected to switched and directed). Physical constraints on the states and control inputs, and communication constraints on information exchange amongst the agents have also been taken into consideration. See Bertsekas and Tsitsiklis (1989) and Jadbabaie, Lin, and Morse (2003) for the early works on distributed computing and control, and the books (Bai, Arcak, & Wen, 2011; Bullo, Cortés, & Martínez, 2009; Lewis, Zhang, Hengster-Movric, & Das, 2014; Ren & Beard, 2008) and review articles (Cao, Yu, Ren, & Chen, 2013; Chebotarev & Agaev, 2009; ✩ This work was supported in part by the National Natural Science Foundation of China under Grants 1633007, and 61533007, in part by the U.S. National Science Foundation under Grant EPCN-1903781, and in part by State Key Laboratory of Intelligent Control and Decision of Complex Systems at BIT. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Wei Ren under the direction of Editor Christos G. Cassandras. ∗ Corresponding author. E-mail addresses: [email protected] (T. Liu), [email protected] (Z.-P. Jiang). https://doi.org/10.1016/j.automatica.2020.109020 0005-1098/© 2020 Elsevier Ltd. All rights reserved.
Martínez, Cortés, & Bullo, 2007; Olfati-Saber, Fax, & Murray, 2007) as well as many references therein for the results. Actuation is an important part of the control systems. In this paper, we focus on pulse-width modulation (PWM), which is a common actuation technique widely used in industry. PWM generates rectangular pulse waveforms with different average values by adjusting the pulse width, and is able to process large signals with high efficiency and low sensitivity to noise. PWM modules are available with many popular microcontrollers, and PWM signals are commonly used to approximate analog signals. As an example, high-frequency PWM of on/off power supplies has been extensively used for (almost) continuous control of electrical drives with digital controllers; see, for instance, Gelig and Churilov (1998), Holmes and Lipo (2003) and Skoog and Blankenship (1970). The study of PWM control theory can be traced back to the early 1960s. A variety of techniques including linearization, Lyapunov method, Popov criterion, LaSalle’s invariance principle, and input–output stability were used to address the stability problem in the early results (Balestrino, Eisinberg, & Sciavicco, 1974; Datta, 1972; Delfeld & Murphy, 1961; Gelig, 1968; Kadota & Bourne, 1961; Kuntsevich & Chekhovoi, 1971; Min, Slivinsky, & Hoft, 1977; Murphy & Wu, 1964; Pavlidis & Jury, 1965; Polak, 1961; Skoog & Blankenship, 1970). More recently, techniques from the fields of nonlinear control, singular system and hybrid system theories are adopted to the study of PWM control; see, e.g., Hou and Michel (2001), Komaee (2019), Sira-Ramirez (1989) and Taylor (1992). With respect to multi-agent systems, PWM control is capable of reducing the amount of communication and simplifying the actuators as the duty ratios are only updated on discrete
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T. Liu and Z.-P. Jiang / Automatica 119 (2020) 109020
time instants and the control signals only take three values. This motivates the study in this paper on state agreement of a class of multi-agent systems with PWM control inputs. Our interest is also motivated by the substantial challenges caused by the coexistence of the hybrid PWM control inputs and the complex system topology. In this research direction, Meng, Meng, Chen, Dimarogonas, and Johansson (2016) studied the distributed PWM control of multi-agent systems with fixed interconnection topologies. In that paper, each agent is controlled by a PWM signal with unit amplitude, and the duty ratio of each PWM control input is designed as a saturated linear function of the diffusionlike interconnection. The spanning tree condition on the fixed interconnection topology is proved to be necessary and sufficient for asymptotic state agreement. Notice that such connectivity condition has been assumed in the prior work of others without considering PWM actuation. In view of the research trend of addressing more general topologies, it is expected to develop a PWM counterpart for multi-agent systems with time-dependent topologies. Moreover, besides the mostly studied connectivity condition on topologies, a designer may also be interested in obtaining some necessary and sufficient condition on the PWM control laws for group coordination. This paper focuses on the state agreement objective, which aims to steer the states of all the agents to a common value, and is known as one of the fundamental group behaviors. The contribution of the paper lies in the proposed necessary and sufficient condition on the PWM control laws for the state agreement of multi-agent systems with directed and switched interconnection topologies, thereby extending some of the results, e.g., Lin, Francis, and Maggiore (2007) and Shi and Hong (2009), from non-PWM case to PWM case, and extending (Meng et al., 2016) from fixed topologies to flexible topologies. As shown in this paper, a trajectory of a PWM-controlled agent with larger initial state may not remain larger, and the range of the states of all the PWM-controlled agents, considered as a Lyapunov function candidate in some existing results, may not be decreasing on the time-line. Furthermore, different PWM control signals are allowed to have different periods and amplitudes, which causes additional complexity for the discussions. In this paper, the coordination problem is formulated as a stabilization problem, and the main Theorem 1 is proved by finding two signals which are upper and lower bounds of the states of the agents, and converge in sense of uniform asymptotic convergence. The paper is organized as follows. Section 2 introduces some basic concepts on the multi-agent systems and PWM control, and presents the problem formulation of distributed PWM control. Section 3 gives the necessary and sufficient condition for uniform asymptotic state agreement of a class of PWM-controlled multiagent systems. The proof of the main result is given in Section 4. Section 5 employs numerical simulations to verify the theoretical results. Section 6 gives some concluding remarks. Notation and terminology Throughout the paper, sgn(r) is the sign of r: sgn(r) = 1 if r > 0; sgn(r) = 0 if r = 0; sgn(r) = −1 if r < 0. Id is the identity function defined on R+ , that is, Id(s) = s for all s ∈ R+ . For all s ∈ R+ , ⌈s⌉ represents the smallest integer that is not less than s, and ⌊s⌋ represents the largest integer that is not larger than s. |x| denotes the Euclidean norm of x ∈ Rn . ∅ represents the empty set. A function α : R+ → R+ is positive definite if α (s) > 0 for all s > 0 and α (0) = 0. α : R+ → R+ is a class K function if it is continuous, strictly increasing and α (0) = 0; it is a class K∞ function if it is a class K function, and also satisfies α (s) → ∞ as s → ∞. A function β : R+ × R+ → R+ is a class KL function if, for each fixed t ∈ R+ , function β (·, t) is a class K function
and, for each fixed s ∈ R+ , function β (s, ·) is decreasing and limt →∞ β (s, t) = 0. A digraph G0 is a pair (N0 , E0 ) where N0 is a nonempty, finite set, and E0 is a subset of N0 × N0 with (i, i) ∈ / E0 for all i ∈ N0 . Elements of N0 are referred to as nodes, and an element (i, j) of E0 is referred to as an edge from i to j. A digraph G0 = (N0 , E0 ) is quasi-strongly connected (QSC) if there exists some c ∈ N0 such that there is a directed path from c to i for each i ∈ N0 \{c }; the node c is called the center of G0 . Clearly, if digraph G0 is QSC, then it has a spanning tree with c as the root. For a switching digraph G (t) = (N , E (t)), we denote the ⋃ union digraph over time interval [t1 , t2 ) as G ([t1 , t2 )) = (N , t ∈[t ,t ) E (t)). A switching 1 2 digraph G (t) is called uniformly quasi-strongly connected (UQSC) with time constant T > 0 if G ([t , t + T )) is QSC for all t ≥ 0. A switching digraph G (t) has a positive edge dwell time ∆E > 0 if for any t ∈ [0, ∞) and any directed edge (i1 , i2 ) ∈ E (t), there exists a te ≥ 0 depending on t and (i1 , i2 ) such that t ∈ [te , te +∆E ) and (i1 , i2 ) ∈ E (τ ) for all τ ∈ [te , te + ∆E ). 2. Preliminaries and problem formulation Consider a multi-agent system defined by x˙ i (t) = ui (t)
(1)
where xi ∈ R is the state and ui ∈ R is the control input, for all i ∈ N = {1, . . . , N }. Normally, the PWM control input is in the form of
{ ui (t) =
mi si (k), 0,
f
for t ∈ [k∆i + tir (k), k∆i + ti (k)), k ∈ Z+ otherwise
(2)
where mi and ∆i are prescribed positive constants representing the amplitude and the period of the PWM signal, respectively, si (k) ∈ {−1, 0, 1} represents the sign of the pulse during the kth f period, and tir (k) and ti (k) represent the time instants of the rising edge and the falling edge during the kth period, respectively. To make the PWM signal well-defined, it is required that 0 ≤ tir (k) ≤ f f ti (k) ≤ ∆i . For the kth period of the PWM signal, (ti (k) − tir (k)) is usually known as the pulse width, and the popular notion of duty ratio (also called duty cycle) is defined as f
di (k) =
(ti (k) − tir (k))
∆i
.
(3)
Clearly, 0 ≤ di (k) ≤ 1. Normally, the duty ratio di and the sign si are considered as the control variables. We consider diffusion-like interconnections between the agents. By using zi to represent the error information used for the control of agent i, we have
{∑
j∈Ni (t) aij (xi (t)−xj (t))
zi (t) =
∑
0,
j∈Ni (t) aij
, if Ni (t) ̸= ∅,
(4)
otherwise,
where aij are positive constants for any pair i, j ∈ N and i ̸ = j, and Ni represents the neighbors of agent i, that is, j ∈ Ni (t) if xj (t) is available to agent i. As usual, we employ a switching digraph G (t) = (N , E (t)) with N = {1, . . . , N } and E (t) = {(j, i) : i ∈ N , j ∈ Ni (t)} to represent the switched information exchange topology amongst the agents. Throughout the paper, the following assumption is made on the connectivity of the interconnection digraph and the periods of the PWM control signals. Assumption 1. The switching digraph G (t) for all t ≥ 0 is UQSC with time constant T , and has an edge dwell-time ∆E ≥ ∆i for all i = 1, . . . , N.
T. Liu and Z.-P. Jiang / Automatica 119 (2020) 109020
With di and si as the control inputs and zi as the measured information, this paper aims to develop a general class of distributed PWM control laws for uniform asymptotic state agreement: there exist a β ∈ KL and a constant c ≥ 0 such that for any xi0 ∈ R with i ∈ N and any t0 ≥ 0, if xi (t0 ) = xi0 for all i ∈ N , then max xi (t) − min xi (t) ≤ β i∈N
i∈N
(
max xi0 − min xi0 + c , t − t0 i∈N
)
i∈N
(5)
for all t ≥ t0 . For practical implementations, it is required that the calculation of control signals di (k) and si (k) only uses the sampled data zi (k∆i ) for k ∈ Z+ . Remark 1. UQSC has been recognized to be the basic connectivity condition for uniform agreement of multi-agent systems. Edge dwell-time basically represents a lower bound of the time durations of all the edges ever appearing. Assumption 1 means that if edge (j, i) appears, then there is at least one period of the PWM control input ui that is able to make use of (xi − xj ). Remark 2. In practice, the position of each pulse with respect to the corresponding PWM period is predetermined, and different types of pulse positions are available for the flexibility of physical implementations: Type A: the pulse center is fixed in the center of the time window of the corresponding period, f i.e., (ti (k) + tir (k))/2 = (k + 1/2)∆i ; Type B: the rising edge occurs at the beginning of the corresponding period, i.e., tir (k) = k∆i ; Type C: the falling edge occurs at the end of the corresponding f period, i.e., ti (k) = (k + 1)∆i . By considering the signs si and the duty ratios di as the control inputs, this paper aims to develop a class of distributed PWM control laws, which is valid for different types of PWMs in terms of the pulse positions. Remark 3. The constant c in (5) is used to characterize the influence of the discontinuous PWM control input during the period before t0 . This term is considered as an off-set with the initial state in the literature of nonlinear systems theory, and generally it does not influence the convergence and robustness properties (Bao, Liu, & Jiang, 2019; Jiang, Teel, & Praly, 1994). 3. State agreement of distributed PWM-controlled multiagent systems 3.1. Properties of single PWM-controlled agents This paper considers a multi-agent system as an interconnection of individual agents. Before designing the distributed PWM controller for the multi-agent system, this subsection studies the properties of a class of PWM-controlled individual agents:
ξ˙ (t) = µ(t) { ms(k), µ(t) = 0,
(6) for t ∈ [k∆ + t r (k), k∆ + t f (k)), k ∈ Z+ otherwise
(7)
where ξ ∈ R is the state, µ ∈ R is the PWM signal, m and ∆ are positive constants, s ∈ {−1, 0, 1} represents the sign of the pulse. The sign s(k) and the duty ratio d(k) = (t f (k) − t r (k))/∆ are usually considered as the control inputs, and the most fundamental control law would be in the form of s(k) = − sgn(ξ (k∆) − ω(k∆))
(8)
d(k) = ∆φ (|ξ (k∆) − ω(k∆)|)
(9)
where φ : R+ → [0, 1] is a continuous function, and ω : R+ → R is a bounded and piece-wise continuous function. Here, ω is considered as an external input for the closed-loop system composed of (6)–(9). In the case of ω ≡ 0, it can be directly
3
recognized that the PWM control law is negative feedback, that is, the sign of the PWM signal is opposite to the sign of ξ . For distributed control design in the following subsection, ω will be used to represent the interconnection between the agents, and the boundedness of ω will be guaranteed by first proving the boundedness of the states of all the agents by using Lemma 1. Lemma 1. that
Consider the PWM-controlled system (6)–(9). It holds
ξ ((k + 1)∆) = ξ (k∆) − m∆ sgn(ξ (k∆) − ω(k∆))φ (|ξ (k∆) − ω(k∆)|) (10) for all k ∈ Z+ , and moreover,
ξ (t) ∈ [ξ (k∆), ξ ((k + 1)∆)]
(11)
for all t ∈ [k∆, (k + 1)∆) with k ∈ Z+ . Lemma 1 can be proved by directly taking the integration of both sides of (6) and using (7)–(9); see, e.g., Gelig and Churilov (1998). Lemma 1 means that the evolution of the PWM-controlled state on discrete time-line with period ∆ depends only on the pulse width, and is not influenced by the positions of the pulses with respect to the corresponding periods, though different positions of the pulse lead to different state trajectories during each period. Lemma 2. that
Consider the PWM-controlled system (6)–(9). Suppose
ω(t) ∈ Ω
(12)
for all 0 ≤ t ≤ T , with Ω ⊂ R being a compact interval and T ≥ ∆. ˘ = Ω u − Ω l . If there Denote Ω u = max Ω , Ω l = min Ω , and Ω exist continuous and positive definite functions δ1 , δ2 < Id such that φ : R+ → [0, 1] satisfies
δ1 (s) ≤ φ (s) ≤
(Id −δ2 )(s) m∆
(13)
for all s ∈ R+ , then for the specific T , there exist an α(Ω˘ )1 (which ˘ , and for any specific Ω ˘ , is continuous, continuously depends on Ω positive definite and less than Id), and an α2 (which is continuous, positive definite and less than Id), such that
• if ξ (0) > Ω u , then for all t ∈ [∆, T ], ξ (t) ≤ Ω u + α(Ω˘ )1 (ξ (0) − Ω u );
(14)
• if ξ (0) ≤ Ω u , then for all t ∈ [∆, T ], ξ (t) ≤ Ω u − α2 (Ω u − ξ (0));
(15)
• if ξ (0) < Ω l , then for all t ∈ [∆, T ], ξ (t) ≥ Ω l − α(Ω˘ )1 (Ω l − ξ (0));
(16)
• if ξ (0) ≥ Ω l , then for all t ∈ [∆, T ], ξ (t) ≥ Ω l + α2 (ξ (0) − Ω l ).
(17)
Proof. We only give the proof for (14) and (15). Properties (16) and (17) can be proved in the same way. We study the behavior of ξ (t) in the cases of ξ (k∆) ≤ Ω u and ξ (k∆) > Ω u separately.
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T. Liu and Z.-P. Jiang / Automatica 119 (2020) 109020
• Case 1: ξ (k∆) > Ω u . In this case, sgn(ξ (k∆) − ω(k∆)) = 1. Then, by using Lemma 1, we have
ξ ((k + 1)∆) = ξ (k∆) − m∆φ (ξ (k∆) − ω(k∆)) ≤ ξ (k∆) − m∆ min φ (ξ (k∆) − ω)
Lemmas 3 and 4 give some direct consequences of dissatisfaction of condition (13), and are used for the proof of the necessity part of the main result.
= ξ (k∆) − m∆ min φ (ξ (k∆) − Ω + ζ )
Lemma 3. Consider the PWM-controlled system (6)–(9). If φ does not satisfy the first inequality in (13), then there exists a λ > 0 such that
≤ ξ (k∆) − m∆ min δ1 (ξ (k∆) − Ω u + ζ ) (18)
|ξ (k∆) − ω(k∆)| = λ ⇒ ξ ((k + 1)∆) = ξ (k∆).
ω∈Ω
u
˘ 0≤ζ ≤Ω
(28)
˘ 0≤ζ ≤Ω
where condition (13) is used for the last inequality. Define
α(′Ω˘ ) (s) = m∆ min δ1 (s + ζ )
(19)
˘ 0≤ζ ≤Ω
(29)
Choose λ = s . In the case of ξ (k∆) − ω(k∆) = λ, (6)–(9) imply
ξ ((k + 1)∆) ≤ ξ (k∆) − α(Ω˘ ) (ξ (k∆) − Ω ). u
′
(20)
A direct consequence of condition (13) is that m∆δ1 (s) < s for all s ∈ R+ . From its definition, α ′ ˘ (·) continuously (Ω )
˘ , and for any specific Ω ˘ , it is continuous, depends on Ω positive definite and less than Id. • Case 2: ξ (k∆) ≤ Ω u . In this case, if ξ (k∆) ≤ ω(k∆), then
ξ ((k + 1)∆) = ξ (k∆) + m∆φ (ω(k∆) − ξ (k∆)) ≤ ξ (k∆) + m∆ max u φ (ω − ξ (k∆)) ω∈[ξ (k∆),Ω ]
≤ ξ (k∆) +
φ (s∗ ) = 0. ∗
for all s ∈ R+ . Then, we have
= ξ (k∆) + m∆
Proof. If the first inequality in (13) does not hold, then due to the continuity of φ , there exists an s∗ > 0 such that
max
ζ ∈[0,Ω u −ξ (k∆)]
max
ζ ∈[0,Ω u −ξ (k∆)]
(21)
where condition (13) is used for the last inequality. Define
α (s) = max (Id −δ2 )(δ ) ′′
δ∈[0,s]
(22)
for all s ∈ R+ . Thus,
ξ ((k + 1)∆) ≤ ξ (k∆) + α ′′ (Ω u − ξ (k∆)).
(23)
Condition (13) means that δ2 is less than Id. By also using the continuity and positive definiteness of δ2 , one can prove that α ′′ is continuous, positive definite and less than Id. In the same case, if ξ (k∆) > ω(k∆), then
ξ ((k + 1)∆) = ξ (k∆) − m∆φ (ξ (k∆) − ω(k∆)) ≤ ξ (k∆).
= ξ (k∆).
(30)
The case for ω(k∆) − ξ (k∆) = λ can be proved in the same way. This ends the proof of Lemma 3. □ Lemma 4. Consider the PWM-controlled system (6)–(9). If φ does not satisfy the second inequality in (13), then there exists a λ > 0 such that
|ω(k∆) − ξ (k∆)| = λ ⇒ ξ ((k + 1)∆) = ω(k∆).
φ (ζ )
(Id −δ2 )(ζ )
ξ ((k + 1)∆) = ξ (k∆) − m∆φ (ξ (k∆) − ω(k∆)) = ξ (k∆) − m∆φ (s∗ )
(24)
(31)
Proof. If the second inequality in (13) is not satisfied, then due to the continuity of φ , there exists an s∗ > 0 such that
φ (s∗ ) =
s∗ m∆
.
(32)
Choose λ = s∗ . If ξ (k∆) − ω(k∆) = −λ, then (6)–(9) imply
ξ ((k + 1)∆) = ξ (k∆) + m∆φ (ω(k∆) − ξ (k∆)) = ξ (k∆) + m∆φ (s∗ ) = ξ (k∆) + s∗ = ω(k∆).
(33)
The case for ξ (k∆) − ω(k∆) = λ can be proved in the same way. This ends the proof of Lemma 4. □ 3.2. Main result on the state agreement of the PWM-controlled agents
From both (23) and (24), we have
ξ ((k + 1)∆) ≤ ξ (k∆) + α ′′ (Ω u − ξ (k∆)).
(25)
Recall that α ′′ is less than Id. This means that, if ξ (k∆) ≤ Ω u , then ξ ((k + 1)∆) ≤ Ω u . Denote m(T ) = ⌈T /∆⌉. Based on the discussions above, we have
• if ξ (0) > Ω u , then (20) implies ξ (k∆) ≤ Ω u + (Id −α(′Ω˘ ) )(ξ (0) − Ω u )
di (k) = φi (|zi (k∆i )|)
(34)
si (k) = − sgn(zi (k∆i ))
(35)
where φi : R+ → [0, 1] is a continuous function. By defining
{∑
(26)
wi (t) =
for all 1 ≤ k ≤ m(T ); • if ξ (0) ≤ Ω u , then (25) implies
ξ (k∆) ≤ Ω u − (Id −α ′′ )m(T ) (Ω u − ξ (0))
For the PWM controlled multi-agent system (1)–(2), motivated by the standard distributed control law, the idea of ‘‘negative feedback’’ is used in the design of the PWM control law:
(27)
for all 1 ≤ k ≤ m(T ), where (Id −α ′′ )m(T ) represents the composition of m(T ) number of (Id −α ′′ ). Then, properties (14) and (15) can be proved by defining
α(Ω˘ )1 (s) = (Id −α(′Ω˘ ) )(s) and α2 (s) = (Id −α ′′ )m(T ) (s) for all s ∈ R+ , and using the monotonicity of ξ (t) during each interval [k∆, (k + 1)∆) for all k ∈ Z+ . This ends the proof of Lemma 2. □
j∈Ni (t) ∑
aij xj (t)
j∈Ni (t) aij
xi (t),
, if Ni (t) ̸= ∅, otherwise,
(36)
to represent the influence of other agents, zi (t) given by (4) can be rewritten as zi (t) = xi (t) − wi (t),
(37)
and the proposed distributed PWM controller (34)–(35) can be equivalently written in the form of (8)–(9). Theorem 1 gives the necessary and sufficient condition on φi for asymptotic state agreement of PWM-controlled multi-agent systems.
T. Liu and Z.-P. Jiang / Automatica 119 (2020) 109020
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Theorem 1. Consider the PWM-controlled multi-agent system described by (1), (2), (3), (4), (34) and (35) with N ≥ 3. Uniform asymptotic state agreement (5) is achievable for any G satisfying Assumption 1, if and only if for each i ∈ N , there exist continuous and positive definite functions δi1 , δi2 such that φi : R+ → [0, 1] satisfies
δi1 (s) ≤ φi (s) ≤
(Id −δi2 )(s) m i ∆i
(38)
for all s ∈ R+ . The proof of Theorem 1 is given in Section 4. Remark 4. With the first inequality in (38) satisfied, the PWM control law is in accordance with the standard diffusion-like distributed control laws in the sense that negative feedback of the interconnection is used. A special requirement caused by the switching topology is the upper-bound condition given by the second inequality. Please see Section 4.1 for the proof of the necessity of the upper-bound condition of φi . Remark 5. The closed-loop system can be described by a specific class of retarded functional differential equations, for which the existence and uniqueness of the solution have been well studied in the literature of dynamic systems and mathematical control theory; see, e.g., Hale and Lunel (1993) and Karafyllis and Jiang (2011). For the specific class of systems studied in this paper, the analytical solution during each PWM period can be calculated by taking the integration of both sides of the differential equation. Also, according to the definition of PWM signals, there is only one pulse during each predetermined, fixed period, which means that no chattering happens in the closed-loop system.
Fig. 1. An example for the necessity of the second inequality in (38): x1 (•) and x3 (◦) keep constant, and x2 (⋄) oscillates between x1 and x3 .
• Set φ2 as one of the control laws that do not satisfy the second inequality in (38). By using Lemma 4, there exists a λ > 0 such that |ω2 (k∆2 ) − x2 (k∆2 )| = λ ⇒ x2 ((k + 1)∆2 ) = ω2 (k∆2 ). Choose E (t) such that E ([0, ∞)) = {(1, 2), (3, 2), (2, 3)} ∪ ⋃ k̸ =1,2,3 {(1, k), (2, k), (3, k)}, and in particular, (1, 2) exists for all t ∈ [3k∆E , (3k + 1)∆E ), (3, 2) exists for all t ∈ [(3k + 1)∆E , (3k + 2)∆E ), and (2, 3) exists for all t ∈ [(3k + 2)∆E , 3(k + 1)∆E ) with ∆E > max{∆1 , ∆2 , ∆3 }. In this case, if x1 (0) − x2 (0) = λ and x2 (0) = x3 (0), then x1 (t) = x1 (0) and x3 (t) = x3 (0) for all t ≥ 0, and x2 (t) oscillates between x1 (0) and x3 (0). The evolution of the states on the discrete time instants k∆E (k ∈ Z+ ) is shown in Fig. 1. The necessity part is proved. 4.2. Sufficiency
Remark 6. The proposed result covers the situation of fixed topologies having spanning trees and the situation of saturated linear control laws. In particular, φi (s) = min{κi s, 1} for all s ∈ R+ satisfies condition (38) as long as 0 < κi < 1/mi ∆i . Also, to the best of our knowledge, the necessary and sufficient condition for uniform asymptotic state agreement of PWM-controlled multiagent systems reported by this paper is for the first time in the literature.
The following discussion is for any specific initial time t0 ≥ 0. The forward completeness of the PWM-controlled multi-agent system can be proved by considering the ‘‘worst case’’: for each i ∈ N , −mi ≤ x˙ i ≤ mi implies −mi t ≤ xi (t) − xi (t0 ) ≤ mi t for all t ≥ t0 . The forward completeness is guaranteed. For convenience of discussions, denote x(t) = maxj∈N xj (t), x(t) = minj∈N xj (t), and x˘ (t) = x(t) − x(t). To prove the asymptotic convergence property, define
4. Proof of the main result
T o = T + 2∆E ,
We divide the proof of Theorem 1 into two parts. Section 4.1 focuses on the necessity, and Section 4.2 is devoted to the proof of the sufficiency part.
to represent different time scales. In the following proof, we find an asymptotically converging upper bound of x˘ (t). Claims 1 and 2 give boundedness and convergence results, respectively.
4.1. Necessity
Claim 1. min
The necessity part can be proved if we can find a PWMcontrolled multi-agent system with specific interconnection digraph G satisfying Assumption 1, such that if at least one of the PWM controllers does not satisfy condition (38), then asymptotic convergence cannot be achieved. Consider the following two cases:
• Set φ2 as one of the control laws that do not satisfy the first inequality in (13). By using Lemma 3, there exists a λ > 0 such that x2 (k∆2 )−ω2 (k∆2 ) = λ ⇒ x2 ((k+1)∆2 ) = x2⋃ (k∆2 ). Choose E (t) such that E ([0, ∞)) = {(1, 2)} ∪ j̸ =1,2 {(1, j), (2, j)}, and in particular, (1, 2) exists for all t ≥ 0. In this case, if x2 (0) − x1 (0) = λ, then x1 (t) = x1 (0) and x2 (t) = x2 (0) for all t ≥ 0. Convergence cannot be achieved.
T ∗ = NT o + ∆E
(39)
For any t ∗ ≥ 0,
max{0,t ∗ −∆E }≤τ ≤t ∗
x(τ ) ≤ x(t) ≤ x(t) ≤
max
max{0,t ∗ −∆E }≤τ ≤t ∗
x(τ )
(40)
for all t ≥ t ∗ , where ∆E is the edge dwell-time of G given by Assumption 1. Proof of Claim 1. Property (40) is proved by contradiction. Recognizing that the first and the last inequalities in (40) can be proved in the same way, we only give the proof for the last inequality. For a specific t ∗ > 0, suppose that the last inequality in (40) does not hold for some t ≥ t ∗ . Then, there exist a tc > t ∗ and an i∗ ∈ N such that x(t) ≤ maxmax{0,t ∗ −∆E }≤τ ≤t ∗ x(τ ) for all t ∈ [t ∗ , tc ] and x(t) > maxmax{0,t ∗ −∆E }≤τ ≤t ∗ x(τ ) for all t ∈ (tc , tc + tϵ ) with some tϵ > 0. Define kc ∈ Z+ such that tc ∈ [kc ∆i , (kc + 1)∆i ). Then, we have kc ∆i ≤ tc and kc ∆i ≥ max{0, tc − ∆i } ≥ t ∗ − ∆E . Condition ∆E ≥ ∆i given by Assumption 1 is used here. By also
6
T. Liu and Z.-P. Jiang / Automatica 119 (2020) 109020
for all t ∈ [kr T ∗ , (kr + 1)T ∗ ). By using Lemma 2, we have
using the definition of ωi in (36), we have
ωi (kc ∆i ) ≤ x(kc ∆i ) ≤ =
max
max{0,t ∗ −∆E }≤τ ≤t ∗
max
max{0,t ∗ −∆E }≤τ ≤t ∗
x(τ )
xi1 (t) ≥ η(kr ),
x(τ )
(41)
for all 0 ≤ t ≤ tc . In this case, it holds that
≤ η(kr ) −
xi ((kc + 1)∆i ) = xi (kc ∆i )
− mi ∆i sgn(xi (kc ∆i ) − ωi (kc ∆i ))· φi (|xi (kc ∆i ) − ωi (kc ∆i )|) = xi (kc ∆i ) + mi ∆i φi (ωi (kc ∆i ) − xi (kc ∆i )) ≤ xi (kc ∆i ) + (Id −δi2 )(ωi (kc ∆i ) − xi (kc ∆i )) ≤ ωi (kc ∆i ) ≤
max
max{0,t ∗ −∆E }≤τ ≤t ∗
x(τ ),
(42)
where condition (38) is used in the first inequality; • if xi (kc ∆i ) ≥ ωi (kc ∆i ), then
max
max{0,t ∗ −∆E }≤τ ≤t ∗
x(τ ),
(43)
where condition (38) is used in the first inequality. Recall the monotonicity of xi (t) for all t ∈ [kc ∆i , (kc + 1)∆i ); see Lemma 1. Thus, there does not exist a tϵ > 0 such that x(t) > maxmax{0,t ∗ −∆E }≤τ ≤t ∗ x(τ ) for all t ∈ (tc , tc + tϵ ), which leads to a contradiction. Finally, property (40) is proved. □ Claim 2. There exists α : R+ → R+ which is continuous, positive definite and less than Id, such that for any k∗ ∈ Z+ , one can find η, η : {k∗ , k∗ + 1, . . . , ∞} → R satisfying
η(k) ≥ η(k)
(44)
η(k + 1) ≤ η(k)
(45)
η(k + 1) ≥ η(k)
(46)
η(k + 1) − η(k + 1) ≤ α (η(k) − η(k))
(47)
for all k ∈ {k , k + 1, . . . , ∞}, and ∗
η(k) ≤ x(t) ≤ x(t) ≤ η(k)
o
∗
η(k ) =
xij (t) ≤ η(k ) − α(η˘ (kr ))ij (η˘ (k ))
(49)
min
x(τ ).
(50)
η(k ) + η(k ) r
2
.
(51)
For convenience of discussions, denote η˘ = η − η. Initial Step: From the definition of ωi1 , we have
η(kr ) ≤ ωi1 (t) ≤ η(kr )
(56)
for all t ∈ [k T + jT , (k + 1)T ), where α(η˘ (kr ))ij continuously depends on η˘ (kr ), and is continuous, positive definite and less than Id, for all j = 1, . . . , l − 1. By using the definitions of il and edge dwell-time, there exist a til ∈ [kr T ∗ + (l − 1)T o , kr T ∗ + (l − 1)T o + ∆E + T ) and an l′ ∈ {1, . . . , l − 1} such that (il′ , il ) ∈ E (t) for all t ∈ [til , til + ∆E ). Thus, we have
ωil (t) ≥ η(kr ) ∑ ωil (t) ≤
j∈Ni
o
∗
r
∗
(57) l
r r r ˘ (kr ))) (t)\{i ′ } ail jη (k ) + ail il′ (η (k ) − α(η˘ (k ))il′ (η l
∑
j∈Ni (t) l
ail j
ail il′ α(η˘ (kr ))il′ (η˘ (k )) ≤ η(kr ) − ∑ r
j∈Ni ([0,∞)) l
=: η(k ) − α r
0 (η˘ (kr ))il (
ail j
η˘ (k )) r
(58)
for all t ∈ [til , til + ∆E ). Clearly, α(0η˘ (kr ))i continuously depends on l η˘ (kr ), and is continuous, positive definite and less than Id. Then, by using Lemma 2, we have xil (til + ∆E ) ≥ η(kr )
(59)
+
η˘ (k ))
0 (η˘ (kr ))il (
r
max
α(1η˘ (kr ))il (ζ )
=: η(kr ) − α(2η˘ (kr ))il (η˘ (kr ))
Claim 1 guarantees that property (48) holds for k = k∗ . Suppose that property (48) holds for a specific kr ≥ k∗ . We study the motion of the PWM-controlled agents during the interval [kr T ∗ , (kr + 1)T ∗ ). For all t ∈ [kr T ∗ , (kr + 1)T ∗ ), define r ∗ o r ∗ i1 , . . . , iN ∈ N such that i⋃ 1 = c and E ([k T + (l − 1)T + ∆E , k T + (l − 1)T o + ∆E + T )) ∩ l′ =1,...,l−1 (il′ , il ) ̸ = ∅. The existence of such i1 , . . . , iN is guaranteed by the UQSC property of G with time constant T ; see Assumption 1. We consider the case of xi1 (kr T ∗ ) ≤
r
0≤ζ ≤α 0 r (η˘ (kr )) (η˘ (k ))il
x(τ ),
r
(55)
r
xil (til + ∆E ) ≤ η(k ) − α
max
max{0,k∗ T ∗ −∆E }≤τ ≤k∗ T ∗
∗
xij (t) ≥ η(kr )
(48)
max{0,k∗ T ∗ −∆E }≤τ ≤k∗ T ∗
(54)
r
r
Proof of Claim 2. Choose
∗
α(′η˘ (kr ))i1 (ζ )
for all k T + T ≤ t < (k + 1)T . Recursive Step: Suppose that r
for all t ∈ [kT ∗ , (k + 1)T ∗ ) with k ∈ {k∗ , k∗ + 1, . . . , ∞}.
η(k∗ ) =
min
η˘ (kr )/2≤ζ ≤η˘ (kr )
=: η(kr ) − α(η˘ (kr ))i1 (η˘ (kr ))
r
xi ((kc + 1)∆i ) = xi (kc ∆i ) − mi ∆i φi (xi (kc ∆i ) − ωi (kc ∆i ))
∗
for all kr T ∗ + T o ≤ t < (kr + 1)T ∗ , and there exists a continuous and positive definite α(η˘ (kr ))i1 which continuously depends on η˘ (kr ) and is less than Id such that xi1 (t) ≤ η(kr ) − α(η˘ (kr ))i1 (η(kr ) − xi1 (kr T ∗ ))
• if xi (kc ∆i ) < ωi (kc ∆i ), then
≤ xi (kc ∆i ) ≤
(53)
(52)
(60)
It can be directly checked that α(2η˘ (kr ))i also continuously depends l on η˘ (kr ), and is continuous, positive definite and less than Id. r ∗ For any t ∈ [til + ∆E , (k + 1)T ), we have
η(kr ) ≤ ωil (t) ≤ η(kr ).
(61)
Thus, by using Lemma 2, we have xil (t) ≥ η(kr )
(62)
xil (t) ≤ η(k ) − α r
≤ η(k ) −
η(k ) − xil (til + ∆E ))
3 (η˘ (kr ))il (
r
r
min
α(2η˘ (kr ))i (η˘ (kr ))≤ζ ≤η˘ (kr )
α(3η˘ (kr ))il (ζ )
l
=: η(kr ) − α(η˘ (kr ))il (η˘ (kr ))
(63)
for all t ∈ [til + ∆E , (kr + 1)T ∗ ). Since til ∈ [kr T ∗ + (l − 1)T o , kr T ∗ + (l − 1)T o + ∆E + T ), we have [kr T ∗ + lT o , (kr + 1)T ∗ ) ⊆ [til + ∆E , (kr + 1)T ∗ ). Thus, from the discussions above, if (55)–(56) hold for all t ∈ [kr T ∗ + (l − 1)T o , (kr + 1)T ∗ ), then (62)–(63) hold for all t ∈ [kr T ∗ + lT o , (kr + 1)T ∗ ). This ends the recursive step.
T. Liu and Z.-P. Jiang / Automatica 119 (2020) 109020
Define
α (s) = min {α(s)ij (s)}
(64)
j=1,...,N
for all s ∈ R+ . Then, α is continuous, positive definite and less than Id. Define
η(kr + 1) = η(kr ) − α (η˘ (kr )),
(65)
η(kr + 1) = η(kr ).
(66)
From the discussions above, we have
η(kr + 1) ≤ x(t) ≤ x(t) ≤ η(kr + 1)
(67)
holds for all t ∈ [k T + NT , (k + 1)T ) = [(k + 1)T ∗ − ∆E , (kr + 1)T ∗ ). Then, Claim 1 implies that (48) holds for all t ∈ [(kr + 1)T ∗ , (kr + 2)T ∗ ). The discussions above assume (51). If r
o
∗
r
r
∗
η(kr ) + η(kr )
xi1 (kr T ∗ ) >
(68)
2
holds instead, then the definition of η(kr + 1) and η(kr + 1) would be in the form of
η(kr + 1) = η(kr ),
(69)
η(k + 1) = η(k ) + α (η˘ (k )). r
r
r
(70)
This ends the proof of Claim 2. □ With (47), there exists a βη˘ ∈ KL such that 0 ≤ x˘ (t) ≤ η˘ (k) ≤ βη˘ (η˘ (k∗ ), k − k∗ )
(71)
with k = ⌊t /T ⌋, for all t ≥ k T . For any xi0 ∈ R with i ∈ N and any t0 ≥ 0, if xi (t0 ) = xi0 for all i ∈ N , then we have ∗
∗ ∗
x0 − m(T ∗ + ∆E ) ≤ x(t) ≤ x(t) ≤ x0 + m(T ∗ + ∆E )
(72)
for all max{0, k0 T − ∆E } ≤ t ≤ k0 T , where k0 = ⌊t0 /T ⌋, x0 = maxi∈N xi0 , and x0 = mini∈N xi0 . For all t ≥ t0 , by also using (71), we have ∗
(
x˘ (t) ≤ βη˘
η˘ (k0 ),
⌊
∗
⌋
t T∗
(
⌊ −
(
⌋)
t0 T∗
≤ βη˘ x˘ 0 + 2m(T + ∆E ), ∗
⌊
t
⌋
⌊ −
T∗
≤ βη˘ x˘ 0 + 2m(T + ∆E ), max ∗
∗
{
T∗
t − t0 T∗
⌋)
t0
β (s, t) =
− 2, 0
(1 + ι − ιt /2T )βη˘ (s, 0),
for t ≤ 2T ,
βη˘ (s, t /T − 2),
for t > 2T ∗ ,
∗
∗
with respect to t is impossible. Basically, the uniformness of the main result is guaranteed by the periodicity of both topology connectivity and PWM control signals. Remark 8. The condition that aij are positive constants is used in deriving (58). The proposed result is still valid when aij are time-varying and there exist constants 0 < aij ≤ aij such that aij ≤ aij (t) ≤ aij for all t ≥ 0. Remark 9. The main theorem focuses on the case of N ≥ 3. When the system is reduced to one composed of only two agents, the condition required for state agreement could be relaxed. Indeed, one specific relaxation is allowing δ12 or δ22 in (38) to be zero. Since there are only two agents, the convergence of one agent to the other means the agreement of the states of all the agents. The validity of the relaxed condition can still be proved similarly as for the proof for Theorem 1. However, on the contrary, such relaxed condition cannot be readily proved to be necessary for asymptotic state agreement. Due to space limitation, we leave this issue to further research. 5. A numerical example In this section, we validate the proposed theoretical result by means of a numerical simulation. Consider a group of six PWM-controlled agents (N = {1, . . . , 6}), with the PWM parameters chosen as: mi = 0.5i and ∆i = 0.05i. Regarding the pulse positions, agents 1,2, agents 3,4, and agents 5,6 are of Type A, Type B, and Type C, respectively. See Remark 2 on the pulse positions. For each i ∈ N , the PWM controller is designed as
φi (s) = min
{
(Id −δ i2 )(s) mi ∆i
,1
}
eij (t) = t − sup te ≤ t : cij (te ) = 0
.
(73)
∗
(74)
for all s, t ∈ R+ , with ι being a positive constant. Then, β ∈ KL. Property (5) can be proved by choosing c = 2m(T ∗ + ∆E ). This ends the proof of Theorem 1. Remark 7. Given a multi-agent system subject to timedependent topologies, designers are concerned about what controller works. However, some doubt might still be cast upon the necessity of UQSC of G for uniform asymptotic state agreement. For this problem, one may consider a contraposition-based reasoning as given by Lin et al. (2007), Moreau (2005) and Shi and Hong (2009). Precisely, one may assume that G does not have a finite time constant T for UQSC (i.e., T → ∞ as t → ∞), then the functions α(·) defined in the proof above should depend on the initial time, and constructing a function β that is uniform
(75)
with δ i2 (s) = 0.1(1 + 0.1 sin(is))s for all s ∈ R+ . It can be directly checked that δ i2 is continuous, positive definite and less than Id, and thus φi satisfies (38) for all i ∈ N . We use a time dependent (0, 1)-matrix C (t) = [cij (t)]6×6 to represent the time-varying topology, that is, cij (t) = 1 if (i, j) ∈ E (t) with E (t) representing the set of edges. A time dependent matrix E(t) = [eij (t)]6×6 ∈ R6×6 is used to represent the hold times of the edges. Without loss of generality, assume that C and E are right-continuous. In particular,
{
})
Define
{
7
}
(76)
for all t ≥ 0. Choose cij (t) as:
• For all (i, j) ∈ E0 , if maxt −T +δT ≤τ ≤t cij (τ ) = 0, then set cij (t) = 1; otherwise, if eij (t) ≥ ∆E , then set cij (t) = random1 , and if eij (t) = 0, then set cij (t) = random2 . • For all (i, j) ∈ / E0 , if eij (t) ≥ ∆E or eij (t) = 0, then set cij (t) = random3 . Here, randomk with k = 1, 2, 3 randomly takes a value from {0, 1}, and δT can be any positive constant that is smaller than T . By choosing E0 such that G0 = (N , E0 ) is QSC, we can guarantee that G = (N , E ) is UQSC with time constant T and edge dwelltime ∆E . For the simulation, we set E0 = {(1, 2), (2, 3), (3, 4), (4, 5), (5, 6)}, T = 1, ∆E = 0.1, δT = 0.02, and randomi = round((oi − rand)/2), with ‘‘rand’’ generating values from the uniform distribution on the interval [0, 1], ‘‘round’’ finding the nearest integer, and o1 = 1.8 and o2 = o3 = 1.01. Also, we set aij = 1 for all i ̸ = j. Fig. 2 shows the state trajectories of the PWM-controlled multi-agent system with the initial states x(0) = [x1 (0), x2 (0), x3 (0), x4 (0), x5 (0), x6 (0)]T = [0, 1, 0.5, 0, −0.5, −1]T .
8
T. Liu and Z.-P. Jiang / Automatica 119 (2020) 109020
Fig. 2. State agreement of a PWM-controlled multi-agent system.
6. Conclusions This paper has shown that uniform asymptotic state agreement is achievable by PWM-controlled multi-agent systems with directed and switched topologies. A necessary and sufficient condition on the distributed PWM control laws is proposed. Following this line of research, the next step is to examine the distributed control problem with other types of pulse modulation, e.g., amplitude modulation, density modulation, position modulation, and their combinations. Instead of considering continuous feedback in this paper, a more general case is that some or all of the design functions φi in the proposed control law are allowed to be discontinuous, to tackle the logical control designs. It is also of theoretical importance and practical relevance to study the distributed PWM control problem for multi-agent systems with moving leaders. Acknowledgments The authors would like to thank the anonymous AE and reviewers for their invaluable comments for the improvement of the paper. References Bai, H., Arcak, M., & Wen, J. (2011). Cooperative control design: A systematic, passivity-based approach. Springer. Balestrino, A., Eisinberg, A., & Sciavicco, L. (1974). A generalized approach to the stability analysis of PWM feedback control systems. Journal of Franklin Institute, 298, 45–58. Bao, A., Liu, T., & Jiang, Z. P. (2019). An IOS small-gain theorem for large-scale hybrid systems. IEEE Transactions on Automatic Control, 64, 1295–1300. Bertsekas, D., & Tsitsiklis, J. (1989). Parallel and distributed computation: Numerical methods. Prentice-Hall. Bullo, F., Cortés, J., & Martínez, S. (2009). Distributed control of robotic networks. Princeton. Cao, Y., Yu, W., Ren, W., & Chen, G. (2013). An overview of recent progress in the study of distributed multi-agent coordination. IEEE Transactions on Industrial Informatics, 9, 427–438. Chebotarev, P. Y., & Agaev, R. P. (2009). Coordination in multiagent systems and Laplacian spectra of digraphs. Automation and Remote Control, 70, 128–142. Datta, K. (1972). Stability of pulse-width-modulated feedback systems. International Journal of Control, 16, 977–983. Delfeld, F. R., & Murphy, G. J. (1961). Analysis of pulse-width modulated control systems. IEEE Transactions on Automatic Control, 6, 283–292. Gelig, A. K. (1968). Absolute stability of nolinear sampled-data systems with pulse-width and pulse-time modulation. Automation and Remote Control, 29, 1043–1054. Gelig, A. K., & Churilov, A. N. (1998). Stability and oscillations of nonlinear pulse-modulated systems. Boston: Birkhauser. Hale, J. K., & Lunel, S. M. V. (1993). Introduction to functional differential equations. NY: Springer. Holmes, D. G., & Lipo, T. A. (2003). Pulse width modulation for power converters: Principle and practice. Wiley-IEEE Press. Hou, L., & Michel, A. N. (2001). Stability analysis of pulse-width-modulated feedback systems. Automatica, 37, 1335–1349. Jadbabaie, A., Lin, J., & Morse, A. (2003). Coordination of groups of mobile autonomous agents using nearest neighbor rules. IEEE Transactions on Automatic Control, 48, 988–1001.
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Tengfei Liu received the B.E. and the M.E. degrees from South China University of Technology and the Ph.D. degree from RSISE, the Australian National University. He was a postdoc with faculty fellowship at Polytechnic Institute of New York University. He is a faculty member at Northeastern University, China. His research interests include mathematical control theory of interconnected nonlinear systems.
Zhong-Ping Jiang received the M.Sc. degree in statistics from the University of Paris XI, France, in 1989, and the Ph.D. degree in automatic control and mathematics from the Ecole des Mines de Paris (now, called ParisTech-Mines), France, in 1993, under the direction of Prof. Laurent Praly. Currently, he is a Professor of Electrical and Computer Engineering at the Tandon School of Engineering, New York University. His main research interests include stability theory, robust/adaptive/distributed nonlinear control, robust adaptive dynamic programming, learning-based control and their applications to information, mechanical and biological systems. In these fields, he has written five books and is author/co-author of over 450 peerreviewed journal and conference papers. Dr. Jiang has served as Deputy Editor-in-Chief, Senior Editor and Associate Editor for numerous journals. Prof. Jiang is a Fellow of the IEEE, a Fellow of the IFAC, a Fellow of the CAA and is among the Clarivate Analytics Highly Cited Researchers.