IET Control Theory & Applications
Brief Paper
Robust consensus of linear systems on
directed graph with non-uniform delay
ISSN 1751-8644
Received on 19th July 2016
Accepted on 2nd September 2016
E-First on 21st October 2016
doi: 10.1049/iet-cta.2016.0970
www.ietdl.org
Dongjun Lee1
1Department
of Mechanical & Aerospace Engineering and IAMD, Seoul National University, Seoul 151-744, Republic of Korea
E-mail: [email protected]
Abstract: The authors propose a consensus control framework for multiple heterogeneous general single-input single-output
linear systems with no zeros at s = 0 on a directed information graph with constant, yet, non-uniform and unknown delays. The
proposed consensus control is easy to design via loop-shaping like graphical approach, robust against plant uncertainty and
constant delay, and completely decentralisable (i.e. locally synthesisable without consulting other agents). Consensus proof
using multi-input multi-output Nyquist theorem and algebraic graph theory is given, with a numerical example to illustrate the
theory.
1 Introduction
One of fundamental problems in multiagent cooperative control is
consensus, i.e. among � agents, | | ��(�) − ��(�) | | → 0,
∀�, � ∈ {1, 2, …, �}, where ��(�) is a certain variable of interest of
the agent � (e.g. position, orientation, opinion etc.). In this paper,
we propose a robust consensus framework for multiple
heterogeneous SISO (single-input single-output) linear systems,
which can assume any form of real rational transfer function (e.g.
arbitrary system order, non-minimum phase dynamics, unstable
poles/zeros, relative degree etc.) as long as it does not has zeros at
� = 0 and evolves on a directed fixed information graph with
constant, yet, non-uniform and unknown inter-agent delays.
For this, we propose a two-degree-of-freedom control
architecture: each agent's dynamics is first stabilised by a local
feedback control, and then, if necessary, the stabilised dynamics is
further shaped by a pre-filter so that the closed-loop dynamics of
each agent exhibits unit gain at zero frequency (i.e. unit dc-gain)
and gain strictly less than unity elsewhere. Then, applying MIMO
(multi-input multi-output) Nyquist theorem [1] and algebraic graph
theory, we show that, with this condition ensured by each agent,
the consensus can be achieved if and only if the information graph
has a globally-reachable node [2]. Thanks to the flexibility
provided by the pre-filter, our proposed consensus framework is
easy to design even to be robust against plant uncertainty (e.g.
loop-shaping like graphical method possible), delay independent,
and completely decentralisable (i.e. each agent can design their
own control without consulting others).
The majority of consensus results with delay is limited either to
identical agents (e.g. all single integrators [3–5], all double
integrators [6–8], or possibly high-order, yet, still identical agent
dynamics [9, 10]); or to agents with specific properties and only on
undirected/balanced graph (e.g. convexity or passivity [11, 12]
excluding agents with non-minimum phase dynamics or relative
degree > 1). Consensus results for general heterogeneous systems
with delay are much rarer, and most closely-related to our result in
this paper are [13–19]. The results of [13, 14] are, respectively,
restricted to identical agents, and non-identical, yet, minimumphase agents, all only on undirected graph. The results of [15, 16]
consider general linear agents on a directed graph with delay, yet,
focuses only on the issue of (strict) stabilisation and the
(marginally-stable) consensus problem is not explored therein. The
work [17], a sequel of [15, 16], studies the consensus problem of
linear systems, yet, limited only to undirected graph. The result of
[18]
attains
high-order
consensus
(i.e.
| | (d���(�))/d�� − (d�� �(�))/d�� | | → 0) of linear agents on
directed graph with delay, yet, zero dynamics is not allowed (when
� = 0) and stability assumed there instead of being constructively
established. A condition for this consensus stability, which is
similar to our consensus condition as stated above and also
originally proposed in [20], is proved in [19]. This proof however
is incomplete similar to that in [20] with some important details not
fully taken into account (e.g. closed-loop poles at � = 0 not
properly addressed without indenting Nyquist plot around the
origin; instability with Nyquist plot passing through the origin not
considered). One of the major contributions of this paper is in fact
to spell out a complete proof of this consensus condition as
compared with those in [19, 20]. To our knowledge, consensus of
heterogeneous non-minimum phase agents on directed graph with
delay is achieved in this paper for the first time. The usage of
MIMO Nyquist theorem and loop-shaping like graphical approach
for robust synthesis as adopted/presented in this paper are, we
believe, also novel in the field of consensus.
The rest of this paper is organised as follows. In Section 2,
some notions and results of graph theory are introduced. Our
proposed consensus framework is presented and detailed in Section
3, and illustrated with a numerical example in Section 4 including
robust consensus synthesis. Some concluding remarks are given in
Section 5.
This paper stems from its conference version paper [20], with
the complete consensus proof with all technical details now
explicitly spelled out, many of which were only vaguely alluded or
even overlooked in [20]. The pre-filter is also newly adopted in this
paper, which turns out to significantly improve practicality and
robustness of our proposed consensus framework. Simulation with
a high-order non-minimum phase agent is also newly performed in
this paper.
2 Information graph
Consider �-agents with their information exchange specified by an
information graph
� := {�, ℰ, �, �}
where (i) � := {�1, …, ��} is the set of nodes (i.e. agents); (ii)
ℰ ⊆ � × � the set of edges among the nodes, with ��� ∉ ℰ,
∀� ∈ {1, …, �} (i.e. self-joining edges excluded) and the information
neighbours of �� denoted by
�� := {� | ��� = (��, ��) ∈ ℰ}
IET Control Theory Appl., 2016, Vol. 10 Iss. 18, pp. 2574-2579
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Fig. 1 Examples of information graphs: only (�, ℰ) depicted
i.e. the set of all the tails of ��; (iii) �: ℰ → ℝ+ ∪ {0} the delay map
defined s.t., for ��� ∈ ℰ, �(���) = ���, where ��� ≥ 0 is a constant
delay; and (iv) �: ℰ → ℝ+ the weight map defined s.t., for ��� ∈ ℰ,
�(���) = ���, where ��� > 0 is a weight (e.g. reliability of
information via ���) with ∑� ∈ � ��� = 1. In this paper, we allow
�
the graph � be directed with non-uniform delays ��� and weights
���. See Fig. 1.
We say that a node �� ∈ � is a globally reachable node of �, if
there exists a directed path from it to all the other nodes of � [2,
21]. We say an information graph � globally-reachable if it
possesses a globally-reachable node. The adjacency matrix
�(�) ∈ ℝ� × � of � is defined by
���(�) =
��� if ��� = (��, ��) ∈ ℰ
0
otherwise
(1)
with ���(�) = 0, ∀� = 1, 2, …, � (with ��� ∉ ℰ). The graph
Laplacian matrix �(�) ∈ ℝ� × � of � is defined by
���(�) =
�
∑
�=1
���
−���
if � = �
with the following properties: (i) all the eigenvalues ��(�) have
non-negative real part; (ii) 0 is always its eigenvalue with
� := [1, 1, . . , 1]T ∈ ℝ� being the corresponding right eigenvector
(from ∑� ∈ � ��� = 1); and (iii) this 0 is a simple eigenvalue of
�
�(�) if and only if � is globally-reachable [2, 21]. We also define
the in-degree of �� to be ���(�) := ���(�). Note that ���(�) ∈ {0, 1},
and, further, if ���(�) = 0, ���(�) = 0, ∀� = 1, 2, …, � (with
��� > 0).
3 Consensus control design
Now, suppose that each agent has its own variable of interest
��(�) ∈ ℝ. What we want to achieve here is consensus, i.e.
lim | ��(�) − ��(�) | = 0, with ��(�), ��(�) → �
(3)
∀�, � ∈ {1, 2, …, �}, where � ∈ ℝ is generally a non-zero consensus
value depending on initial condition. For this, we propose the twodegree-of-freedom consensus architecture as shown in Fig. 2,
where the open-loop dynamics of each agent ��(�) is first stabilised
by a local feedback control ��(�) and then shaped by a pre-filter
��(�) so as to attain the consensus condition as specified in
Theorem 1. Here, all ��(�), ��(�), ��(�) are real rational transfer
functions. This architecture turns out to allow for robust consensus
synthesis for general linear agent dynamics ��(�), as long as it does
not possess zeros at � = 0 – see Lemma 1.
The closed-loop dynamics of the agent � is then given by:
��(�) = ��(�)��(�) ��(�)
∑
� ∈ ��
− ����
����
��(�) − ��(�) +
and ��� are the delay and weighting factor of the link ���, with
��� = 0 if ��� ∉ ℰ.
If in�(�) ≠ 0 (i.e., �� ≠ ∅), the closed-loop dynamics of the
agent � can be rewritten as:
where
��(�) = ��(�)
��(�) :=
��(�)
where ��(�) ∈ ℂ is the Laplace transform of ��(�), Δ��(�) ∈ ℂ is the
polynomial of � related to the initial conditions of ��, ��, and ���
∑
� ∈ ��
− ����
����
��(�) + ��(�) ⋅ Δ��(�)
��(�)��(�)
� (�),
1 + ��(�)��(�) �
��(�) :=
(4)
1
1 + ��(�)��(�)
or, if ���(�) = 0 (i.e., agent � does not receive information from
any other agents), it simply becomes
��(�) = ��(�) ⋅ Δ��(�)
(2)
otherwise
�→∞
Fig. 2 Consensus architecture: ��(�), ��(�) and ��(�) are the plant, the
local feedback control, and the pre-filter of the agent �; and ��� and ��� are
the delay and the weight of the edge ��� = (��, � �), with ��� = 0, if ��� ∉ ℰ
(5)
implying ��(�) → 0 if ��(�) is stable. Here, notice that
��(�)
and
��(�) :=
��(�)��(�)
��(�)
=
1 + ��(�)��(�)
��(�)
are respectively the sensitivity and the complementary sensitivity
functions [22].
Theorem 1: Suppose that ��(�) in (4) is strictly stable with no
+
cancelation of poles and zeros in ℂ̄ := {� + �� | � ≥ 0} during its
construction and also designed to possess the following properties:
lim ��(�) = 1, lim ��(�) = 0
�→0
�→∞
(6)
|��(��) | < 1, ∀� > 0
(7)
�(�) = �(�)�(�) + �(�)Δ�(�)
(8)
∀� ∈ {1, 2, …, �}. Assume also that ���(�) ≠ 0 ∀� ∈ {1, 2, …, �}.
Then, consensus (3) is achieved if and only if the information
graph � has a globally reachable node.
Proof: Stacking up (4) for each agent, we can obtain the closedloop dynamics of the total group s.t.,
where
�(�) := [�1(�); �2(�); …; ��(�)]
Δ�(�) := [Δ1�(�); Δ2�(�);
��
�2(�), …, ��(�)] ∈ ℂ
…; Δ�� (�)]
, and �(�) ∈ ℂ
���(�) := ����
0
∈ ℂ�,
��
∈ ℂ�,
�(�) := diag[�1(�);
is defined by
− ����
��(�) if ��� = (��, ��) ∈ ℰ
otherwise
(9)
with ���(�) ∈ ℂ being the ��th component of �(�). Note that all
the diagonal elements of �(�) are zeros, since ��� ∉ ℰ. Note also
that lim� → 0 �(�) = �(�).
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From (8), we can then compute �(�) s.t.,
�(�) = [� − �(�)]−1�(�)Δ�(�)
(10)
where with the assumptions made above, all the poles of �(�)Δ�(�)
are in ℂ− := {� + �� | � < 0}. Thus, the stability of the group
consensus dynamics (10) hinges upon [� − �(�)]−1 or the
characteristic roots of det (� − �(�)) = 0, which is equivalent to
the stability problem of the self-joining loop in Fig. 3.
For this, we utilise generalised MIMO Nyquist theorem [1].
More precisely, let us define the region � and the contour
∂� := ∪�3 = � ∂�� as shown in Fig. 3 with the counter-clockwise
+
indentation ∂�� around � = 0. Note that � = ℂ̄ ∖ {(0, 0)}. Since
all the ���(�) are strictly stable, all the open-loop poles are strictly
outside �. This then means that the closed-loop dynamics (10) will
not have poles in �, if the Nyquist plot det (� − �(�))� ∈ ∂� does
not make encirclements of the origin (0, 0) or pass through (0, 0).
This is in fact true under the assumptions of Theorem 1 as shown
below.
Notice first from the assumption (6) that
lim det (� − �(�)) = 1
�→∞
This then implies that, if the Nyquist plot det (� − �(�))� ∈ ∂�
makes the encirclement of, or passes through, the origin (0, 0),
there should exist � ∈ (0, 1] and �′ ∈ ∂� ∖ ∂�2 s.t.,
det (� − ��(�′)) = 0. This, however, cannot happen under the
assumptions of Theorem 1:
• For the segment of ∂�1,
det (� − ��(��)) =
�
∏
�=1
1 − ���(�(��)) ≠ 0
∀� ∈ (0, + ∞), where ��(�(��)) is the �th eigenvalue of
�(��) with |��(�(��)) | < 1, ∀� > 0 due to the Gershgorin's
theorem [23] and the structure of �(�) in (9) with the
assumption (6) and (7);
• For the segment ∂��, we have, with � ≃ 0 and � > 0,
|��(�) |� = �� �� ≃
�¯ 1� �� �� + 1
(11)
�
�¯ 1�� �� + 1
�
where � ∈ [ − �/2, + �/2], �¯ �1 := ��1 /��� and �¯ 1 := ��1 /��� are,
respectively, the coefficients �1� , ��1 of �1 of the numerator and the
denominator of ��(�) normalised by the coefficient of ��, with
��� = ��� from ��(0) = 1 in (6). Here, we argue that
¯�
¯�
�1 > | �1 | > 0, since: (i) all the coefficients of the denominator
of ��(�) should be of the same sign, as ��(�) is strictly stable;
and (ii) if � = �� with � ≃ 0 and � > 0, similar to (11), we
must have
|��(��) | ≃
�¯ 1� �� + 1
�
�¯ 1 �� + 1
=
(�¯ 1� �)2 + 1
�
(�¯ 1�)2 + 1
<1
due to the assumption (7). Then, with �� �� = �(cos � + �sin �),
� ∈ [ − �/2, + �/2], from (11), we have
|��(�� ��) | ≃
(1 + �¯ 1� �cos �) + ��sin �
�
(1 + �¯ 1�cos �) + ��sin �
<1
i.e. |��(�) |� ∈ ∂� = | ��(�� ��) | < 1. Further, applying again
�
Gershgorin's theorem [23] to �(�) with the structure of �(�) in
2576
Fig. 3 Positive unit feedback loop of the total group consensus dynamics
(10); and the region � and the contour ∂� := ∪3� = � ∂�� for generalised
Nyquist stability analysis
(9) and |��(�) |� ∈ ∂� < 1, we have |��(�(�)) | < 1, and,
consequently
�
det (� − ��(�))� ∈ ∂� =
�
≠0
�
∏ [1 − ���(�(�))�
�=1
∈ ∂��
]
• The above argument for the segment ∂�3 also holds for the
segment ∂�2 due to the fact that |�(�) | = | �(�¯ )|, where �¯ is
the complex conjugate of � ∈ ℂ.
We have just shown that the closed-loop consensus dynamics (10),
under the assumptions of Theorem 1, does not have any poles in �
of Fig. 3. In other words, all the poles of the closed-loop dynamics
(10) are either strictly within ℂ− (i.e. LHP) or at � = 0. The
characteristic quasi-polynomial of (10) is given by
det (� − �(�)) =
�
∏ ��(� − �(�)) = 0
(12)
�=1
which indeed has a root at � = 0 if ���(�) ≠ 0 ∀� ∈ {1, 2, …, �},
since, in this case, det (� − �(0)) = det (� − �) = det (�) = 0
(see Lemma 1 for the case where a globally-reachable node has
zero in-degree). The consensus dynamics (10) will then be stable if
this pole at � = 0 is simple, or unstable otherwise.
Now, suppose that the information graph � has a globallyreachable node. It is obvious that a globally-reachable node is
necessary for consensus, thus, the necessity proof of Theorem 1 is
omitted here. Then, from (12), there should be at least one
��(� − �(�)), � = {1, 2, …, �} s.t., ��(� − �(�)) = 0 when � = 0.
Moreover, if � has a globally-reachable node, only one of the
eigenvalues satisfies ��(� − �(�)) = 0 when � = 0, since, if not,
there exists � ≠ � s.t.,
��(� − �(�)) |� = 0 = ��(� − �(0)) = 0
which cannot happen with a globally reachable node, since, in this
case, � = � − �(0) must have a simple eigenvalue at 0.
This then means that det (� − �(�)) has roots at � = 0, which
is given by the only one of the eigenvalues, ��(� − �(�)) = 0. The
consensus dynamics (10) will be unstable if this pole at � = 0 is
repeated, or, for instance
��(� − �(�)) = �2�(�)
(13)
where �(�) is a smooth function with bounded �(0), from
smoothness of all the elements of �(�) composed of rational
− ��
transfer functions ��(�) and quasi-polynomial � ��. Here, we
assume double poles at � = 0 only for simplicity: our conclusion
below is equally applicable when high-order poles at � = 0 are
assumed.
��(� − �(0)) = 1 − ��(�(0)) = 0,
From
we
have
��(�(0)) = 1. Moreover, using Gershgorin's theorem [23] with the
IET Control Theory Appl., 2016, Vol. 10 Iss. 18, pp. 2574-2579
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assumption (7), it is also easy to show that |��(�(��)) | < 1 if
� ≠ 0. These together then implies that
∂ | ��(�(��))|
∂�
�=0
<0
(14)
which we will show is not possible if the pole at � = 0 is repeated.
For this, from (13), similar as before, we have
��(�(��)) = �2�(��) + 1 = �2[�(�) + ��(�)] + 1
where �(�) = Re [�(��)] and �(�) = Im [�(��)], both of which
are smooth and bounded at � = 0 due to the same reasons as
above. We then have
|��(�(��)) | = (1 + �2�(�))2 + (�2�(2))2
and further we can obtain (see equation below), implying that, to
uphold the condition (14), the pole at � = 0 should be simple.
So far, we have established that, if ���(�) ≠ 0 ∀� ∈ {1, 2, …, �}
and the graph � is globally-reachable, under the assumptions of
Theorem 1, the consensus dynamics (10) is stable, with all the
poles strictly within ℂ− (i.e. LHP) and one simple pole at � = 0. It
is this simple pole at � = 0, that produces a steady-state constant
value �¯ = lim� → ∞ [�1(�); �2(�); …; ��(�)] = lim� → 0 ��(�) in
(10). To compute this �¯ , let us apply the final value theorem to (8),
i.e.
�¯ = lim ��(�) = lim ��(�)�(�) + lim ��(�)Δ�(�)
�→0
�→0
�→0
(15)
= �(0)�¯
where lim� → 0 ��(�)Δ�(�) = 0 since all the characteristic roots of
�(�)Δ�(�) are strictly within ℂ−. We then have
[� − �(0)]�¯ = ��¯ = 0
implying that �¯ = �1, i.e. consensus (3) is achieved. Note also
from (10) that this consensus value �1 is linearly dependent on the
initial condition Δ�(�) and is a result of the sustained dccomponent of (10) stemming from the simple pole of [� − �(�)]−1
at � = 0. □
In Theorem 1, we assume that all the agents receive information
from others, i.e. ���(�) ≠ 0 ∀� ∈ {1, 2, …, �}. Now, suppose that
���(�) = 0 for some agents. For a globally-reachable graph �,
there can exist at most one such an agent with ���(�) = 0 and that
agent is the only globally-reachable node of �. This ‘root’ agent
will then stabilise by itself to zero according to (5) and this
information will propagate throughout � to drive �� → 0,
∀� ∈ {1, 2, …, �}, as formalised in the following Lemma 1.
Lemma 1: Suppose that ��(�) in (4) is designed to satisfy all the
assumptions of Theorem 1. Suppose further that the information
graph � has a globally-reachable node with zero in-degree. Then,
consensus (3) is achieved with � = 0.
Proof: As stated above, for a globally-reachable graph �, there
can exist only one node with zero in-degree and that node is the
only globally-reachable node of �. Without loss of generality,
denote this ‘root’ agent by agent 1. Then, the closed-loop dynamics
of the total group still has the structure of (8) with �1�(�) = 0,
∀� = 1, 2, …, �, and, following the same reasoning of the proof of
Theorem 1 using MIMO Nyquist theorem (with � shown in Fig.
3), all the characteristic roots of (10) can be either strictly within
ℂ− or at � = 0, with the latter, that comes from
∂ | ��(�(��))|
∂�
�=0
=
det (� − �(�)) = 0, dictating stability of the closed-loop
consensus dynamics (10).
Yet, with the root node, the characteristic roots of (10) cannot
happen at � = 0. To show this, notice first that, with �1�(�) = 0,
� = 1, 2, …, �,
� − �(0) =
1
01 × (� − 1)
�(� − 1) × �
= diag[1, 0, …, 0] + �
where �(� − 1) × � is the bottom portion of the Laplacian matrix
� ∈ ℜ� × �, with �1� = 0, ∀� = 1, …, �. This Laplacian matrix � has
1D null-space � = [1, 1, …, 1]T ∈ ℜ� associated with its simple
eigenvalue 0 [24, Lemma 2.4]. This null-space � = [1, 1, …, 1]T of
�, however, is not shared by diag[1, 0, …, 0], whose null-space is
given by [�, 0, …, 0], � ∈ ℜ. This then implies that � − �(0) does
not possess any non-trivial null-space, thereby, excluding the
possibility of the closed-loop consensus dynamics (10) having
characteristic roots at � = 0 (since det (� − �(0)) ≠ 0).
Now that the closed-loop dynamics (10) can have characteristic
roots only strictly within ℂ−, we can conclude that
�(�) = [�1(�); �2(�); . . . ��(�)] → 0 ∀�(0) ∈ ℜ�. □
The pre-filter ��(�) in Fig. 2 substantially facilitates the
synthesis of ��(�) satisfying the consensus conditions (6) and (7):
we can first stabilise ��(�) by using ��(�) and then shape the
stabilised dynamics by using ��(�) in a separate/sequential manner.
Synthesising ��(�), which is both stabilising and achieving the
gain condition (7) at the same time is typically much more
involved, particularly when ��(�) is non-minimum phase, as the
closed-loop dynamics is in general under some constraints (e.g.
complementary sensitivity integral condition [22]). In fact, this prefilter ��(�) and its two-degree-of-freedom architecture in Fig. 2
allow us to grant the consensus conditions (6) and (7) as long as
��(�) has no zeros at � = 0.
Lemma 2: The consensus conditions (6) and (7) in Theorem 1
can be granted by using the local feedback ��(�) and the pre-filter
��(�), if and only if the agent dynamics ��(�) does not have any
zero at � = 0.
Proof (Necessity): Suppose ��(�) has zeros at � = 0. Since
open-loop zeros are not affected by ��(�), to attain (6), ��(�)
should contain the same number of poles at � = 0, resulting in
pole-zero cancellation at � = 0, which is not allowed in Theorem
1.
(Sufficiency): Suppose ��(�) has no zeros at � = 0. Then, using
��(�) = (1/�)�′(�)
with high-enough order ��′(�) and ��′(0) ≠ 0
�
(or ��(�) = ��′(�) if ��(�) itself contains integrators), we can not
only stabilise any ��(�), but also attain
��(0) =
(1/�)��(�)��′(�)
1 + (1/�)��(�)�′(�)
�
�=0
= 1,
where ��(�) is the complementary sensitive function defined after
(4). We can further factorise ��(�) s.t.,
��(�) = ��(�) ⋅ �¯ �(�)
where ��(�) ∈ ℂ is the stable all-pass filter with all RHP zeros of
��(�) embedded therein, and �¯ �(�) ∈ ℂ is a stable minimum-phase
transfer function with |�¯ �(��) | = | ��(��)| for all � ≥ 0. Then, if
we choose any ��(�) s.t., |��(��) | ≤ (1/|�¯ �(��)|, ∀� ≥ 0, with
[1 + �2�(�)] ⋅ [2��(�) + �2((∂�(�))/∂�] + �2�(�) ⋅ [2��(�) + �2((∂�(�))/∂�]
(1 + �2�(�))2 + (�2�(2))2
�=0
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=0
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��(0) = 1, the consensus conditions (6) and (7) can be granted.
One such an example of a stable proper ��(�) is
��(�) =
1
�¯ (�)
�
⋅
rel (��)
∏
�=1
��
� + ��
(16)
where �� > 0 is some suitable cut-off frequency and rel(��) is the
relative degree of ��(�).□
As stated in the proof above, if ��(�) has zeros at � = 0, to
achieve (6), those zeros should be cancelled out by the poles of
��(�) at � = 0. This pole-zero cancellation, when the pole/zero at
� = 0 is simple, may appear benign, as it renders the closed-loop
system still marginally stable. Even this simple pole-zero
cancelation at � = 0, yet, will lead into instability for the consensus
with ���(�) ≠ 0 ∀� ∈ {1, 2, …, �}, since it will resonate with the dc
consensus value � = lim� → ∞ ��(�). Also, although we provide a
closed-form solution of ��(�) in (16), we found that a simpler
graphical method to shape |��(��)| by designing ��(�) to be a
bank of low-pass (and/or band-rejection) filters and lead/lag
compensators to satisfy (6) and (7) is typically more convenient,
leading into a lower-order ��(�) and often better performance than
(16). The following Corollary 1 on robust consensus is a direct
consequence of Theorem 1 and Lemma 2.
���(�) ≠ 0
Corollary
1:
Consider
�-agents
with
∀� ∈ {1, 2, …, �}. Suppose that each agent's dynamics ��(�) is
strictly proper and uncertain s.t.,
~
��(�) ∈ � � := {(1 + ��� (�)�(�))��� (�)}
where ��� (�), �(�), ��� (�) ∈ ℂ are respectively the uncertainty
weight, unit-gain bound (with |�(��) | < 1), and nominal
dynamics transfer functions, all stable and proper. Suppose further
~
that, ∀��(�) ∈ � �(�), ��(0) ≠ 0, the number of RHP-poles of ��(�)
is the same as that of ��� (�), and there exist proper robust stability
~
control ��(�) for � �. Then, if we choose a stable proper ��(�) s.t.,
|��(��) | ≤
inf
~
��(�) ∈ � �
1
, ∀� > 0, with ��(0) = 1
|�¯ �(��)|
the consensus (3) is achieved if and only if the information graph �
has a globally reachable node.
See Section 4 for an illustration of this robust consensus
synthesis of Corollary 1. Lastly, note that the consensus control
proposed in this paper is robust against any constant/unknown
time-delay, and its synthesis is also completely decentralisable (i.e.
each agent can design its own control without consulting others).
4 Illustrative example
We consider four heterogeneous agents on the first information
graph as shown in Fig. 1. We impose non-uniform constant delays:
�12 = 0.1 s, �23 = 0.3 s, �31 = 1.5 s, and �43 = 0.15 s.
Agent 3 is a second-order system with the following dynamics:
���¨ �(�) + ���˙ � = ��(�), where ��, �� > 0 are uncertain mass and
damping parameters. We then design the consensus control
��(�) := − ��(��(�) − ∑� ∈ � �����(� − ���)) so that, for Fig. 2,
�
��(�) = 1/(���2 + ���), ��(�) = ��, ��(�) = 1 and
��(�) =
2
��
��� + ��� + ��
where, if we set �� ≤ (�2� /4��) (i.e. critically-damped) for all
possible ��, ��, the consensus conditions (6) and (7) of Theorem 1
can be robustly ensured.
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Fig. 4 Bode plots of ��(�) = (��(�)��(�))/(1 + ��(�)��(�)) (top) and
��(�) = ��(�)��(�) (bottom) of agent 1 after applying ��(�) to enforce the
consensus conditions (6) and (7)
Agents 2 and 4 are first-order systems with the following
dynamics: �˙ �(�) = ��(�), with the consensus control ��(�) designed
by ��(�) := − ��(��(�) − ∑� ∈ � �����(� − ���)), �� > 0. Then, for
�
Fig. 2, we have ��(�) = 1/�, ��(�) = ��, and ��(�) = 1, with
��(�) =
��
� + ��
which satisfies the consensus conditions (6) and (7) in Theorem 1.
Agent 1 is a non-minimum phase fourth-order system with
relative-degree 3:
~
��(�) ∈ � � :=
� − ��
(� + 1)(�2 + ��� + ��)
(17)
where �� ∈ [2, 18] and �� ∈ [150, 250] with �� = �� /50 (i.e. dcgain ��(0) = − 1/50). First, we design a robust stable local
feedback control ��(�) s.t.,
��(�) := 200
�−1
�(� + 10)
with which ��(�) = (��(�)��(�))/(1 + ��(�)��(�)) is stable
~
∀��(�) ∈ � � with ��(0) = 1. Yet, as shown in Fig. 4, |��(��) | > 1
for some frequencies, violating the condition (7). To shape this
|��(��)|, we design the pre-filter ��(�) to be
��(�) := 15
� + 0.6
1
⋅
� + 9 100� + 1
(18)
where the last term is a low-pass roll-off filter, while the first term
a lead compensator, which turns out important to speed up the
consensus. With this ��(�), as shown in Fig. 4, ��(�) robustly
satisfies the consensus conditions (6) and (7). Instead of (18), we
may also use (16), which however turns out to be slower than (18)
with its order ( ≥ 5) also higher than that of (18). This also clearly
evidences the substantial flexibility and advantage of adopting the
pre-filter ��(�) for consensus control synthesis.
Simulation result is presented in Fig. 5, where consensus is
achieved, even though the agents dynamics are heterogeneous (e.g.
inverse response of third-order non-minimum phase agent 1; sharp
turns of first-order agents 2 and 4 at the first data reception after
delays); and the information graph � is only directed with nonuniform/unknown delays.
5 Conclusion
We propose a novel consensus control framework for multiple
heterogeneous general SISO linear systems, which can assume any
form of real rational transfer function ��(�) (e.g. arbitrary system
order, zero dynamics, relative degree etc.) as long as it does not
possess zeros at � = 0, on a directed information graph with
constant, yet, non-uniform and unknown delays. The proposed
IET Control Theory Appl., 2016, Vol. 10 Iss. 18, pp. 2574-2579
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[4]
[5]
[6]
[7]
[8]
Fig. 5 Consensus of heterogeneous agents on directed graph with nonuniform delay: agent 1 is third-order non-minimum phase system (with
�� = 2, �� = 250), agent 3 second-order system, and agents 2 and 4 firstorder systems
consensus framework, with the adoption of a certain pre-filter, is
also easy to use (similar to loop-shaping), robust against plant
uncertainty, and completely decentralisable (i.e. locally
synthesisable without consulting other agents). Some future
research topics include extensions to MIMO systems and switching
graphs.
6 Acknowledgments
Research supported by the Basic Science Research Program
(2015R1A2A1A15055616), the Global Frontier R&D Program
(2013M3A6A3079227) and the Convergence Technology
Development Program (2015M3C1B2052817) all through the
National Research Foundation (NRF) funded by MSIP, Korea; and
by the Industrial Strategic Technology Development Program
(10060070) and the Industrial Convergence Core Technology
Development Program (10063172) both funded by MOTIE, Korea.
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
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