Convex Gradient Controller Design for Incrementally Passive Systems
with Quadratic Storage Functions
Takayuki Ishizaki1 , Asami Ueda1 , and Jun-ichi Imura1
Abstract— In this paper, we give an equivalence condition forincremental passivity in terms of convex gradients and perform
output regulator design for incrementally passive systems. To
derive the equivalence condition, we focus on the class ofincremental passive systems with quadratic storage functions,which can be transformed into a particular realization called aself-dual realization. On the basis of the self-dual realization, in
which the input matrix necessarily coincides with the transposeof the output matrix, we show that the convexity of potentialfunctions is necessary and sufficient for the incremental passivity of systems whose vector field is given as the gradient ofpotential functions. Furthermore, we show that the equilibriumof such convex gradient systems can be analyzed via the convex
conjugate defined by the Legendre-Fenchel transformation.
Combining these facts, we then develop a design method of
output regulators that have a potential to improve a degree ofstability while leaving the original equilibrium of integratorbased control systems invariant. The stability improvement isdemonstrated though an example of power systems control, in
which the resultant output regulator is shown to have a low-passproperty with the nonlinearity of input saturation.

I. I NTRODUCTION
In the real world, there can be found a number of systems
whose behavior is described as the gradient of physical (orperhaps virtual) energy functions. Examples include reactiondiffusion systems, Euler-Lagrange systems, port-Hamiltoniansystems, and so forth [1], [2]. As another example of such
gradient systems, dynamics stemming from the gradientof objective functions can be found in the optimization
theory [3]. In particular, a relation between optimizationbased gradient systems and power system control has also
been investigated recently [4]. Generally speaking, analyses
on fundamental system properties, such as stability andexistence of equilibria, would be a necessary first step toaim at systematically controlling them.
As for stability analyses, it is well known that passivity is
one of tractable properties of physical systems to discuss
the stability of interconnected systems. More specifically,
negative feedback interconnection of passive components
retains the passivity, thereby proving the stability of feedback
control systems. By making use of this fact, various typesof stability analyses based on passivity have been performedin the literature [5], [6]. For example, analysis and synthesis
methods for network synchronization and coordination havebeen developed in [7], [8] as an application of such passivitybased stability analyses.
1 Department of Mechanical and Environmental Informatics, Graduate
School of Information Science and Engineering, Tokyo Institute of Technology; 2-12-1, Meguro, Tokyo, Japan.
This research was supported by CREST, JST.
{ishizaki,ueda,imura}@cyb.mei.titech.ac.jp

For linear systems, the passivity can be characterizedby linear matrix inequalities, which are useful for boththeoretical and numerical analyses. On the other hand, forgeneral nonlinear systems, it is however difficult to find out
a simple characterization of passivity, except for the portHamiltonian systems known as a useful class of passive
nonlinear systems. In view of this, it is valuable to explore
a simple characterization towards systematic control systemdesign.
Against this background, this paper aims at giving a
tractable characterization of passive systems. In particular,
focusing on the class of incrementally passive systems withquadratic storage functions, which can be transformed into aparticular realization called a self-dual realization [9], weshow that the convexity of potential functions is necessary and sufficient for the incremental passivity of systems
whose vector field is given as the gradient of potentialfunctions. This clarification is based on the fact that the
input matrix necessarily coincides with the transpose of theoutput matrix in the self-dual realization. Furthermore, weshow that the equilibrium of such convex gradient systems
can be systematically analyzed via the conjugate of convex
functions, defined by the Legendre-Fenchel transformation[10]. Note that the incremental passivity [11]–[13], which is
generally stronger than the standard notion of passivity, is a
useful property to guarantee the stability of an equilibriumof interconnected nonlinear systems, especially in the casewhere the equilibrium is not at the origin of the state-space.
On the basis of the investigations above, we then develop a design method of output regulators that have apotential to improve a stability degree while leaving theoriginal equilibrium of integrator-based control systems invariant. The efficiency of the proposed method is numericallydemonstrated through an example of decentralized output
regulation of swing equations appearing in power systems,
where the resultant output regulator is shown to have a lowpass property with the nonlinearity of input saturation. Itshould be noted that input saturation generally causes theloss of passivity as shown in [14]. In contrast to this, weshow that, by giving an appropriate convex function, a lowpass controller subject to input saturation can be expressed asa convex gradient system possessing incremental passivity.
Furthermore, we demonstrate that a degree of system stability
can be improved as long as the time constant of controllers
is sufficiently small, i.e., as long as a singular perturbation
approximation of controllers is valid, even when the control
input is subject to saturation.
Finally, we give some references related to passivity-

based analyses of dynamical systems originating in convex
optimization. A passive controller design method for network
flow optimization is proposed in [15], where a dynamics
stemming from a source updating law is shown to havea passive property. It should be noted that the paper only
deals with a specific problem where the objective function is given as the sum of functions of scalar variable,
which leads to a network of one-dimensional subsystems.
Therefore, the result cannot be applied straightforwardly
to general optimization problems that involve multivariate
objective functions, yielding a network of multi-dimensional
subsystems. In a similar way, [16], [17] propose a passive
controller design method for market mechanisms in smart
grids. Furthermore, [18] shows that a class of passivity-based
cooperative control problems has a relation with network
flow optimization problems. In that paper, the convergence of
output agreement is discussed on the premise that subsystems
forming a cooperative network are assumed to be passive.
The remainder of this paper is structured as follows. In
Section II, summarizing the preliminary facts for incrementalpassivity and convex functions, we give an equivalence condition of incrementally passive systems in terms of convex
gradients. Furthermore, we show that the equilibrium of
convex gradient systems can be analyzed by conjugating
convex functions. Next, in Section III, we apply the resultsin Section II to output regulator design for incrementally
passive systems, where we show that a monotone property
of convex functions can be utilized to improve a stabilizing
performance of control systems. In Section IV, numericalsimulations are given to show the efficiency of the proposed
output regulator design method. Finally, concluding remarks
are provided in Section V.
II. A NALYSIS OF C ONVEX G RADIENT S YSTEMS
A. Preliminaries
1) Incrementally Passive Systems: Let us consider theclass of dynamical systems given by{ẋ = f (x) + BuΣ:(1)
y = Cxwhere x ∈ Rn denotes the state, u ∈ Rp denotes the
input signal, and y ∈ Rp denotes the output signal. In thefollowing, we denote the domains of x, u, and y by X , U,
and Y, respectively. Furthermore, let EΣ denote the set of
all triplets (x, u, y) ∈ X × U × Y that satisfy the dynamical
equation of Σ in (1). In this notation, we first introduce thefollowing notion of incremental passivity [11]–[13].
Definition 1: A dynamical system Σ in (1) is said tobe incrementally passive if there exist a positive definite
function S : Rn → R and a positive semidefinite function
δ : Rn × Rn → R such thatṠ(∆x) ≤ ∆uT ∆y − δ(x1 , x2 ),t≥0

for any (x1 , u1 , y1 ) ∈ EΣ and (x2 , u2 , y2 ) ∈ EΣ , where∆x := x1 − x2 ,∆u := u1 − u2 ,∆y := y1 − y2 .

(2)

In particular, it is said to be strictly incrementally passive if(2) is satisfied with δ being positive unless x1 = x2 .
One major advantage of the incremental passivity is that a
network system composed of the negative feedback interconnection of incrementally passive systems is shown to have astable nonzero equilibrium. This can be seen as follows. Let
Σ1 and Σ2 be incrementally passive systems with respect tothe storage functions S1 : Rn1 → R and S2 : Rn2 → R,
and suppose that at least one of them is strictly incrementally passive. Furthermore, let (x1 , u1 , y1 ) ∈ EΣ1 and(x2 , u2 , y2 ) ∈ EΣ2 be their feasible dynamical trajectories
whereas let (x∗1 , u∗1 , y1∗ ) ∈ EΣ1 and (x∗2 , u∗2 , y2∗ ) ∈ EΣ2 betheir stationary trajectories. In this notation, if Σ1 and Σ2
are interconnected with
u1 = y2 ,u2 = −y1 ,then for S(x1 , x2 ) := S1 (x1 ) + S2 (x2 ), which is positive
definite, we haveṠ(x1 − x∗1 , x2 − x∗2 ) = Ṡ1 (x1 − x∗1 ) + Ṡ2 (x2 − x∗2 )≤ (u1 − u∗1 )T (y1 − y1∗ ) + (u2 − u∗2 )T (y2 − y2∗ ) = 0,
where the strict inequality holds unless x1 = x∗1 and x2 =x∗2 . This implies that S serves as a Lyapunov function toprove the asymptotic stability of the equilibrium (x∗1 , x∗2 ).
2) Convex Functions: Next, we give some preliminary
facts for convex functions [3], [10]. A function F : Rn → R
is said to be convex if
F ((1 − λ)x + λx′ ) ≤ (1 − λ)F (x) + λF (x′ )(3)for all 0 < λ < 1 and for all pairs of x and x′ in the domainsuch that F is finite. Such a domain, denoted by XF , is called
the effective domain of F and it is known to be convex. In
particular, F is said to be strictly convex if (3) holds withthe strict inequality unless x = x′ .
For F being continuously differentiable, the gradient of F
∂F
is denoted by ∇F , whose ith element is given by ∂xwithixi denoting the ith element of x. By the gradient of F , theconvexity can be characterized by
F (x) − F (x′ ) ≥ ∇F T (x′ )(x − x′ )(4)

for all x, x′ ∈ XF . Furthermore, for F being twice continuously differentiable, the Hessian of F is denoted by HF ,2
F
whose (i, j)-element is given by ∂x∂i ∂x. The convexity ofj
F can also be characterized by the positive semidefinitenessof the Hessian as
HF (x) ⪰ 0(5)
for all x ∈ XF . In a similar manner, the strict notion of
convexity can be characterized by the strict inequalities of(4) and (5). In the rest of this paper, for simplicity, allconvex functions under consideration are assumed to be
twice continuously differentiable.
In many applications of convex analysis theory, such as
convex optimization, the following conjugate transformation

is often utilized for dual analyses. The conjugate of F isdefined by{}
F (z) := sup z T x − F (x) ,(6)the specific realization given as in (8). For linear systems,
i.e., f (x) = Ax, the specific realization satisfies thatA + AT ⪯ 0,

B = C T.

x∈X

where X is a convex domain. It is known that, as long as F
is convex, F is also convex and the conjugate of F coincides
with F , namely{}
F (x) = sup xT z − F (z) ,(7)z∈Z

where Z is the set of all z such that the supremum of (7) isfinite. Furthermore, the strict convexity of F is equivalent
to the smoothness of F . In convex analysis theory, thetransformation between (6) and (7) is called the LegendreFenchel transformation, and some of conjugate pairs can befound in closed forms.
B. An Equivalence Condition for Incremental Passivity
In this subsection, we derive an equivalence condition forincremental passivity. This derivation is based on a specificrealization having a symmetric nature, called a self-dual
realization [9] for the case of linear systems, as shown in
the following lemma.
Lemma 1: Let Σ in (1) be given and assume that it
is incrementally passive (strictly incremental passive) withrespect to S(x) = 12 xT V x, where V is positive definite.
Furthermore, let Vc be a Cholesky factor of V such that
V = VcT Vc . Then, the realization of{x̃˙ = Vc f (Vc−1 x̃) + Vc BuΣ̃ :(8)y = CVc−1 x̃
is incrementally passive (strictly incremental passive) withrespect to S̃(x̃) = 12 ∥x̃∥2 . In addition, Vc B = (CVc−1 )T .
Proof: To prove the claim for incremental passivity,
we can assume that δ = 0 without loss of generality. Then,for (x1 , u1 , y1 ) ∈ EΣ and (x2 , u2 , y2 ) ∈ EΣ , the incremental
inequality in (2) is represented byT{f (x1 ) − f (x2 )} V ∆x + (B∆u) V ∆x ≤ ∆u ∆y.
TT

By the coordinate transformation of x̃ = Vc x, we have{}Tf˜(x̃1 ) − f˜(x̃2 ) ∆x̃ + (B̃∆u)T ∆x̃ ≤ ∆uT ∆y(9)where ∆x̃ := Vc ∆x, f˜(x̃) := Vc f (Vc−1 x̃), and B̃ := Vc B.
This implies that Σ̃ in (8) is incrementally passive withrespect to S̃(x̃) = 21 ∥x̃∥2 . Note that (9) is satisfied for all
feasible trajectories of x1 , x2 ∈ X and u1 , u2 ∈ U. Thus, itfollows for C̃ := CVc−1 that{}Tf˜(x̃1 ) − f˜(x̃2 ) ∆x̃ ≤ 0, ∆uT (B̃ T − C̃)∆x̃ ≤ 0.
This leads to B̃ = C̃ T , because ∆x̃ and ∆u are arbitrary.
The same argument with the change of variables of δ proves
the claim for the strict notion of incremental passivity.
Lemma 1 shows that any incrementally passive systems
with a quadratic storage function can be transformed into

In the literature, this particular realization of passive systems
is known to have good compatibility with a passive controller
synthesis, model reduction, and so forth [9], [19]. The factsof B = C T and the storage function of S(x) = 12 ∥x∥2 in the
self-dual realization are essential in the following arguments.
Next, on the basis of the self-dual realization, we derive an
equivalence condition for incremental passivity in the context
of convex gradient. To this end, the following fact for convex
functions is useful [3].
Lemma 2: Let F : Rn → R and defineTδF (x1 , x2 ) := {∇F (x1 ) − ∇F (x2 )} (x1 − x2 ).

(10)Then, δF is nonnegative for all x1 , x2 ∈ XF if and only if
F is convex. In particular, δF is positive unless x1 = x2 if
F is strictly convex.
Lemma 2 shows that the positive semidefiniteness of δFin (10), appearing in the incremental passivity analysis, isequivalent to the convexity of F . This equivalence leads tothe following result.
Theorem 1: Let Σ in (1) be given and assume that there
exist a function F : Rn → R and a skew symmetric matrix
J ∈ Rn×n such thatf (x) = −∇F (x) + Jx.

(11)Then, Σ is incrementally passive with respect to S(x) =122 ∥x∥ if and only if F is convex. In particular, Σ is strictlyincrementally passive if F is strictly convex.
Proof: From Lemma 1, we see that B = C T holds if
Σ is incrementally passive with respect to S(x) = 12 ∥x∥2 .
In the notation of (2), we haveṠ(∆x) − ∆uT ∆y = −δF (x1 , x2 ),where δF is defined as in (10). Thus, by Lemma 2, it
is nonpositive for all x1 , x2 ∈ XF if and only if F isconvex. Similarly, the claim for strictly incremental passivity
is proven by the strict convexity of F , which implies that δF
is positive unless x1 = x2 .
Theorem 1 shows that, as long as the vector field is represented by the gradient of potential functions, the convexity ofpotential functions is necessary and sufficient for incrementalpassivity with respect to quadratic storage functions. On thebasis of this investigation, let us introduce the following
terminology.
Definition 2: A dynamical system Σ in (1) is said to bea convex gradient system if there exist a convex function
F : Rn → R and a skew symmetric matrix J ∈ Rn×n suchthat (11) holds. In particular, a convex gradient system Σ issaid to be self-dual if B = C T .
From the discussion above, it can be seen that a selfdual convex gradient system is incrementally passive withrespect to the unitary quadratic storage function. It should

be noted that any linear passive system is equivalent to aconvex gradient system, because its self-dual realization is
given by letting
1
F (x) = − xT (A + AT )x, J = A − AT ,2
which yields f (x) = Ax. From this viewpoint, we see thatthe class of convex gradient systems in Definition 2 involves
that of linear passive systems associated with quadratic
storage functions.
An interconnected system given by the negative feedbackof self-dual convex gradient systems again belongs to theclass of convex gradient systems. This can be seen as
[ ] [
] [
][ ]
−∇F (x)
ẋ
J
C TH
x
,
(12)
˙ξ = −∇G(ξ) + −H T C J ′ξwhere F and G are convex, J and J ′ are skew symmetric,
and C and H are matrices having compatible dimensions.
This type of negative feedback systems can appear in solving
convex optimization by gradient methods. To see this moreclearly, let us consider the convex-concave optimization{}
sup inf F (x) − (Hξ)T Cx − G(ξ) ,(13)ξ∈XG x∈XF

which can be viewed as a general form of saddle-point
problems arising in the Lagrange dual decomposition forconvex optimization [3], [10]. Denoting the objective function by L(x, ξ), we consider the dynamics stemming fromthe gradient descent and gradient ascent of (13) given asẋ = −∂L(x, ξ),∂x∂Lξ˙ =(x, ξ),∂ξ(14)

which corresponds to the Uzawa algorithm to solve convex
optimizations. It can be readily verified that (14) is identicalto (12) with J = 0 and J ′ = 0. Furthermore, the solutions x∗
and ξ ∗ of the convex-concave optimization in (13) coincidewith the stable equilibria of (12).
C. Equilibrium Analysis of Convex Gradient Systems
In this subsection, we give a method to analyze theequilibrium of convex gradient systems using the conjugate
of convex functions. To this end, we show the following
simple but useful fact; see, e.g., [10] for a proof.
Lemma 3: Let a convex gradient system Σ be given as in
Definition 2, and assume that F is strictly convex and J = 0.
Let (x∗ , u∗ ) be a feasible pair such that
∇F (x∗ ) = Bu∗ .

(15)

ξ = ∇G(−H T Cx).
Substituting this into the dynamical equation of x in (12), we
obtain the singular perturbation model as a convex gradientsystemΣ̂ : ẋ = −∇F̂ (x) + Jx
(17)
where
F̂ (x) := F (x) + G(−H T Cx).

(18)

Note that the stable equilibrium of Σ̂ is identical to that ofx in (12). In addition, one can say that a convergence rate to
the equilibrium is improved in view of the following stability
index.
Definition 3: Let a convex gradient system Σ be given as
in Definition 2, and assume that F is strictly convex. Letx∗ be a feasible equilibrium such that there exists some u∗
satisfying (16). Then, an incremental stability degree of Σwith respect to x∗ is defined byθΣ (x) := δF (x, x∗ ),(19)

where δF is defined as in (10).
The incremental stability degree θΣ corresponds to thedecreasing rate, i.e., time derivative, of the Lyapunov function S(x) = 21 ∥x − x∗ ∥2 , proving the stability of an
equilibrium x∗ , when injecting the corresponding constant
input u = u∗ such that (16). For linear systems with A
being negative definite, this function measures the degree ofnegative definiteness becauseθΣ (x) = −(x − x∗ )T A(x − x∗ ).
On the basis of this definition, we can see for Σ̂ in (17)that the incremental stability degree with respect to theequilibrium x∗ of Σ̂ becomes larger asθΣ (x) ≤ θΣ̂ (x)(20)

for all x ∈ XF ∩ XF̂ . The inequality in (20) is proven by themonotone property of convex functions such that any pair ofconvex functions F1 and F2 satisfies
δF1 (x1 , x2 ) ≤ δF1 +F2 (x1 , x2 )(21)

for all x1 , x2 ∈ XF2 ∩ XF2 . This result will be applied toconvex gradient regulator design in Section III.

Then, it follows thatx∗ = ∇F (Bu∗ )One application of this fact can be found in analyzing
a singular perturbation approximation of convex gradientsystems. To see this, let us consider applying the singular
perturbation approximation to the dynamics of ξ in (12) with
J ′ = 0. More specifically, replacing ξ˙ with 0, we have thetrajectory of ξ along x as

(16)

where F is the convex conjugate of F defined as in (6).
Lemma 3 shows that the conjugacy of convex functions
has a relation to the equilibrium analysis of convex gradientsystems. This is based on the fact that ∇F is the inverse
function of ∇F , and vice versa. For linear systems, it canbe verified that (16) is reduced to x∗ = −A−1 Bu∗ .

III. S YNTHESIS OF C ONVEX G RADIENT S YSTEMSIn this section, we consider designing an output regulator for incrementally passive systems, on the basis of theanalyses in Section II. In this output regulator design, we
explicitly use the equilibrium analysis based on Lemma 3 toleave the original equilibrium invariant, while guaranteeing
the closed-loop stability in a way similar to [13].

Let a convex gradient system be given as in Definition 2
and assume that it is self-dual. Our objective here is to steer
the output y towards a constant reference signal y ∗ . One
simple approach to do this is implementing the conventional
integrator as
] [
][ ]
[ ] [
−∇F (x)
ẋ
x
J
CT
=
+
.
(22)
λ
−C 0
λ̇
y∗
In fact, owing to the incremental passivity of the integrator,
the equilibrium (x∗ , λ∗ ) of (22) is proven to be asymptotically stable and y ∗ = Cx∗ is to be guaranteed, as long as F
is strictly convex.
However, the convergence rate of (22) may not be desirable because it relies only on the inherent system stability.
Towards improving it, one can consider implementing an
additional controller to the feedback system. As taking
advantage of incremental passivity, it would be reasonable
to suppose that the additional input signal u is injected as] [][ ] [ T ][ ] [−∇F (x)ẋ
J CT x
C


+
+
u
=
 λ̇
λ
−C 0
0
y∗
Σ:
[ ]
[
] x


 y = C 0.
λ(23)
To this system, we implement a convex gradient controller
with a time constant τ > 0 given as{τ ξ˙ = −∇G(ξ) + H T v
κ:(24)
w = Hξ,which is interconnected with Σ in (23) byv = −y,u = w.

Note that it can be transformed to√ a self-dual realization byscaling the state variable as ξ˜ = τ ξ.
In this formulation, as the integration of analyses in
Section II, we show the following result for designing output
regulating systems.
Theorem 2: Consider the feedback system composed ofa convex gradient system Σ in (23) and a convex gradientcontroller κ in (24), and let (x∗ , λ∗ ) denote the equilibriumof (22). If G : Rν → R is given as
G(ξ) = Γ (ξ) − (H T y ∗ )T ξ

(25)

where Γ : Rν → R is a strictly convex function such that∇Γ (0) = 0, thenlim (x, λ, ξ) = (x∗ , λ∗ , 0)

t→∞

(26)

for any initial values of (x, λ, ξ) and any time constant τ > 0.
In particular, in the limit of τ → 0, the resultant feedback
system approaches to a convex gradient system Σ̂ that isgiven as replacing F in (22) with F̂ in (18). Furthermore,
for the incremental stability degree with respect to (x∗ , λ∗ ),it follows thatθΣ (x, λ) ≤ θΣ̂ (x, λ)(27)
for all x ∈ XF ∩ XF̂ and λ ∈ R.

Proof: First, as shown in Lemma 3, we see that ∇Γ isthe inverse function of ∇Γ and vice versa. Thus, ∇Γ (0) = 0
is equivalent to ∇Γ (0) = 0. Furthermore, from the relation
among the affine transformation of conjugates [10], we seethat
G(µ) = Γ (µ + H T y ∗ )where G and Γ are the convex conjugates of G and Γ ,respectively. Then, using Theorem 3, we obtainξ ∗ = ∇G(−H T y ∗ ) = ∇Γ (0) = 0,
which implies u∗ = 0 at the equilibrium of the feedback
system. Hence, (26) is proven by the fact that the equilibriumis asymptotically stable owing to the incremental passivity.
Next, we show (27). In the limit of τ → 0, coinciding withthe singular perturbation approximation of κ, it approaches
to the static controller given as
w = H∇G(H T v).
This implies that the resultant feedback system is identicalto Σ̂, given as replacing F in (22) with F̂ in (18). Notethat G(−H T Cx) is convex. Thus, (27) follows from themonotonicity as shown in (20).
Theorem 2 shows that the additional implementation of
the convex gradient controller G in (25) does not change
the equilibrium (x∗ , λ∗ ) of Σ from (22). Furthermore, thestability degree of the equilibrium can be improved in viewof the stability index in Definition 3, as long as the time
constant τ is sufficiently small. As for a suitable choice ofτ , there can be a trade-off relation between the convergence
rate to equilibria and the sensitivity to measurement noise.
More specifically, a smaller value of τ would be desirable
to improve the convergence rate whereas a larger value ofτ would be desirable to suppress measurement noise in alarger frequency range.
IV. N UMERICAL E XAMPLES
A. System Description
Let us consider giving the system dynamics as{M ζ̈ + Dζ̇ + Kζ = Eu
y = E T ζ̇
where M ≻ 0 and D ≻ 0 denote mass and damper diagonal
matrices, K ≻ 0 denotes a spring stiffness matrix, and E isa matrix having a compatible dimension. This second-ordersystem is often used as a primary model of rotor dynamics
for power system stabilization [20], [21], called a swing
equation, and it is incrementally passive since the input andoutput are collocated [9].
With the state of x := [ζ T , ζ̇ T ]T , the system dynamics canbe represented as Σ in (1) with
[
]
[
]
0
I
0
A=
, B=
,
−M −1 K −M −1 D
M −1 E(28)[]
C = 0 ET ,where f (x) = Ax. For this system, we consider the situation where output regulation by the integrators in (22) isperformed in a decentralized fashion. To represent this, wegive E in (28) as
E = diag(1ni )i∈N ,N := {1, . . . , N }(29)where 1n ∈ Rn denotes the all-ones vector, N denotes thenumber of subsystems, and ni denotes the dimension of theith subsystem, satisfying n = n1 + · · · + nN .
B. Convex Gradient Controller with Input Saturation
Next, we consider implementing a set of decentralized
controllers that individually supervise clusters of several
subsystems, i.e., a disjoint set of subsystems. To represent
this, we define an aggregated output signal by∑ŷl := i∈Cl yi(30)
where yi denotes the ith element of y, and Cl ⊂ N denotes
a cluster defined as a consecutive index set such that∪max Cl < min Cl+1 , L := {1, . . . , L}l∈L Cl = N,
for L denoting the number of clusters.
To demonstrate the output regulator design based on
Theorem 2, we consider implementing a set of decentralized
controllers subject to input saturation. To this end, we givethe dynamics of a controller for the lth cluster as{τ ξ˙l = −∇Γl (ξl ) + kl (ŷl∗ + vl )
κl :
(31)
w l = k l ξl ,where kl > 0 denotes a controller gain, ŷl∗ denotes theaggregated reference signal defined similarly to (30), and∇Γl (ξl ) =αl ξl+
ln − ,2ξlΓl (ξl ) :=αl +(ξ ln ξl+ + ξl− ln ξl− )2 lfor ξl+ := αl + ξl and ξl− := αl − ξl . The set of controllers
is connected with Σ byvl = −ŷl ,ui = wl ,i ∈ Cl ,l ∈ L,

where ŷl is defined as in (30) and ui denotes the ith elementof u.
As shown in Theorem 2, the asymptotic stability of (26) isguaranteed because Γl is a strictly convex function satisfying∇Γl (0) = 0. Note that the controller state ξl is confined tothe effective domain XGl = (−αl , αl ). This implies that thecontrol input is subject to the saturation of|ui | ≤ kl αl ,∀i ∈ Cl .

(32)

In addition, we see that the linear approximation of κl in(31) becomesτ ẇl = −wl + kl2 (ŷl∗ + vl ),whose transfer function from ŷl∗ + vl to wl is kl2 /(1 + τ s).
Thus, κl can be considered as a one-dimensional low-passfilter subject to input saturation. Furthermore, in the limit ofτ → 0, κl approaches to the static controller()(33)
wl = kl ∇Γ l kl (ŷl∗ + vl )where the conjugate function is given byµlµl∇Γ l (µl ) = αl tanh , Γ l (µl ) := αl2 ln cosh .
αlαlFrom this expression, we see that the input saturation isrepresented as the hyperbolic tangent (sigmoid) function. As
shown in Theorem 2, the static controller in (33) has theability to improve the degree of stability in the sense of (27).
C. Simulation Results
We consider the network of 16 mass components, whichyields a 32-dimensional system Σ in (1) with (28). The
system network structure is depicted in Fig. 1, where thenodes represent the mass components and the edges represent
the interconnection among them. The parameter matrices are
given as M = 1.25 × I, D = 0.02 × I, and{−0.3, if nodes i and j are connected,
Ki,j =
0,otherwise,{∑16
0.3 − j=1,j̸=i Ki,j , i = 1,
Ki,i =∑16− j=1,j̸=i Ki,j ,otherwise,
where Ki,j denotes the (i, j)-element of K.
Let us regard the 32-dimensional system as the networkof four subsystems, each of which is composed of four masscomponents, as shown in the dotted circles in Fig. 1. This
corresponds to N = 4 and ni = 8 in (29). In this setting,
implementing the integrators as in (22), we plot the output
signals y by the dashed-dotted lines in the upper subfigureof Fig. 2, where the reference signal is set to y ∗ = 14 . From
this figure, we see that the amplitude of oscillations is notmade small in the time interval. This is due to the fact thatthe degree of inherent system stability is not very large.
To improve the convergence rate in the output regulation, we implement a set of dynamical controllers with thenonlinearity of input saturation, explained in Section IV-B.
For the controllers κl in (31), we construct the clusters as
C1 = {1, 2} and C2 = {3, 4}, which lead to the aggregatedoutput ŷl as in (30). The controller parameters are given
as kl = 0.8 and αl = 0.0625 for l = 1, 2, which resultin the input saturation of |ui | ≤ 0.05 as in (32). In thissetting, the resultant outputs y and inputs u are plotted in theupper and lower subfigures of Fig. 2, respectively, where thethin solid lines correspond to the case of the static controllerin (33) and the dashed lines correspond to the case of the
dynamical one with the time constant of τ = 0.16. Fromthese figures, we see that the static and dynamical controllers
result in the almost same output and input signals and bothcontrollers make the convergence rate higher, in spite of the
input saturation.
In addition, we show the results when giving the controller
time constants as τ = 1 and τ = 9. The resultant outputs andinputs are plotted in Fig. 3 where the solid and dashed lines
correspond to the cases of τ = 1 and τ = 9, respectively.
From this figure, we see that the convergence rates become
lower as increasing the value of time constants. As long as
the measurement output is not contaminated with noise, we2Outputs

1.5
1
0.5
0
0.1

Inputs

0.05
0

-0.05
-0.1

Pajek

Fig. 1.

System network structure and decentralized output regulators.

2

Outputs

1.5

0.5

4

6

8

10
Time

12

14

16

18

20

Fig. 3. Output and input signals in the cases of dynamical controllers with
τ = 1 and τ = 9.

R EFERENCES

0
0.1
0.05
Inputs2

strated the improvement of convergence rates, while showingthat a nonlinear low-pass filter subject to input saturation canbe regarded as a convex gradient system.

1

0

-0.05
-0.1

0

0

2

4

6

8

10
Time

12

14

16

18

20

Fig. 2. Output and input signals in the cases of simple integration, static
controller and dynamical controller with τ = 0.16.

can improve the convergence rate, i.e., the degree of closedloop stability, as giving a smaller time constant of controllers.
Explicit consideration of measurement noise, which can besuppressed by the dynamical action (low-pass property) ofcontrollers, is one of future works to pursue.
V. C ONCLUDING R EMARKSIn this paper, we have derived an equivalence condition
for incremental passivity in terms of convex gradients. In
particular, focusing on the class of incremental passive
systems with a quadratic storage function, which can betransformed into a self-dual realization, we show that theconvexity of potential functions is necessary and sufficientfor the incremental passivity of systems whose vector field isgiven as the gradient of potential functions. Furthermore, wehave analyzed the equilibrium of convex gradient systems
via the convex conjugate defined by the Legendre-Fencheltransformation.
On the basis of these theoretical investigations, we havedeveloped a design method of output regulators for incrementally passive systems. The effectiveness of the proposed
method has been shown though a numerical example ofdecentralized output regulation in the frequency control ofpower systems. In this numerical example, we have demon-[1] A. van der Schaft and D. Jeltsema, Port-Hamiltonian Systems Theory:
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