Multiresolved Control of Discrete-Time Linear Systems Based on
Redundant Realization via Wedderburn Rank Reduction
Takayuki Ishizaki, Masakazu Koike, Takuro Kato, Jun-ichi Imura1,2
Abstract— In this paper, we propose a design method ofmultiresolved control for discrete-time linear systems. In the
proposed control system, we implement a transitory compensator specialized in controlling a short-term system behavior
into a standard controller that is designed for controlling along-term behavior. To establish such control architecture, weconstruct a low-rank model having the same reachable andobservable subspaces as those of the original system in therange of its rank. Then, we derive a redundant state-spacerealization associated with the low-rank model. A cascaded
structure of the redundant realization enables to systematicallydesign a transitory compensator that stabilizes the short-term
system behavior while cooperating with a standard controller.
The efficiency of the multiresolved control is shown through an
illustrative example of frequency control in power networks.

I. I NTRODUCTION
In the real world, there can be found a number of systems
whose behavior is captured as an interaction among subsystems having different spatiotemporal scales. For example,
biological systems [1] are composed of molecules, proteins,
cells, and organs that hierarchically interact on multiple
spatiotemporal scales. To deal with such nonuniformly distributed systems in the spatiotemporal point of view, it is
crucial to figure out and take advantage of essential systemproperties depending on objectives to be accomplished.
Towards the development of a systematic framework for
spatiotemporally multiresolved control [2], this paper proposes a design method of multiple time scale control, in
which we explicitly consider the short-term to long-term
behavior of systems to be stabilized. To this end, we first
construct a low-rank model that can properly capture a shortterm system behavior, which can be represented by lowdimensional reachable and observable subspaces. Next, wederive a redundant state-space realization associated with the
low-rank model, and then we develop a design method ofmultiresolved control. By virtue of a cascaded structure of
the redundant realization, we can systematically design atransitory compensator that stabilizes the states in possible
contingencies, cooperating with a standard controller for
normal circumstances.
As the demonstration for the effectiveness of this multiresolved control, we perform a numerical simulation on thestabilization of frequency variations in a power network. In
this simulation, we show that a low-dimensional transitory
1 Department of Mechanical and Environmental Informatics, Graduate
School of Information Science and Engineering, Tokyo Institute of Technology; 2-12-1, Meguro, Tokyo, Japan.
2 Japan Science and Technology Agency, CREST; 4-1-8 Honcho,
Kawaguchi, Saitama, 332-0012, Japan.
{ishizaki,koike,kato,imura}@cyb.mei.titech.ac.jp

compensator, attached to the average feedback controller, has
good ability to stabilize contingency frequency variations
arising in a local area.
To clarify our contribution, we give some references as follows. In [3], a hierarchical control architecture is considered
where a low-dimensional approximant is used to construct anadditional input signal such that the error between the outputsof the approximant and its original system converges to zero
asymptotically. However, the hierarchical control system ispractically difficult to implement because it is based on thepremise of the possibilities of an exact model reduction, i.e.,
the low-dimensional approximant can exactly reconstruct theoriginal system behavior, and of the state feedback of theoriginal system.
From the viewpoint of time scale separation, the proposed
multiresolved control has a similarity to a control synthesis method based on singular perturbation theory [4]. In
this approach, an asymptotic expansion is generally used
to analyze the degradation of control performance due tosingular perturbation approximation. By contrast, our approach has an advantage that, on the basis of the redundantrealization having a tractable cascaded structure, we cananalytically manage an approximation error of the lowdimensional model, which corresponds to a long-term systembehavior. This redundant realization is different from thoseused in [5], [6] in the sense that we use state-space expansion
to derive a multiresolved state-space representation in a
temporal viewpoint, whereas the existing works use it toapproximately decouple interconnected systems in a spatial
viewpoint.
This paper is organized as follows. In Section II, we derivea low-rank model that can capture the short-term behaviorof systems on the basis of matrix decomposition, called
Wedderburn rank reduction [7]. Then, in Section III, we
propose a synthesis method of multiresolved control basedon the redundant realization associated with the low-rank
model. Section IV shows an illustrative example for thestabilization of frequency variations in a power network.
Finally, concluding remarks are provided in Section V.
Notation: We denote the set of real numbers by R, the ndimensional identity matrix by In , the ith column of In byeni , the rank of a matrix M by rank(M ), the image of amatrix M by im M , and the kernel of a matrix M by ker M .
A matrix A ∈ Rn×n is said to be Schur if
lim xt = 0,t→∞∀x0 ∈ Rn(1)

for the associated recurrence formula xt+1 = Axt . Further-more, a switching matrix At ∈ {A(1) , A(2) } with a fixedswitching mode sequence is said to be transitionally Schurif (1) holds for xt+1 = At xt .
II. S YSTEM R EDUCTION BASED ON W EDDERBURNR ANK R EDUCTION
Let A ∈ Rn×n be given and denote ν := rank(A). For
some sequences of vectors ri , oi ∈ Rn , we consider thefollowing biconjugation process associated with A:
ui := ri −riT Avjuj ,uT Avjj=1 j

vi := oi −

i−1
∑

uT
j Aoi

j=1

uT
j Avj

vj , (2)where u1 := r1 and v1 := o1 . It has been shown in [7] thatviT Auj = 0 for all i ̸= j, or equivalently, forΩk := diag(ω1 , . . . , ωk ),ωi := viT Aui ,(3)

it follows thatVkT AUk = Ωk ,k ∈ {1, . . . , ν}(4)where Uk := [u1 , . . . , uk ] and Vk := [v1 , . . . , vk ]. The
biconjugation process in (2) can be regarded as a functionthat produces a biorthogonal pair satisfying (4). This process
is to be denoted by(Uk , Vk ) = WA (Rk , Ok )(5)

where Rk := [r1 , . . . , rk ] and Ok := [o1 , . . . , ok ]. Note thatthey satisfy
im Uk = im Rk ,im Vk = im Ok(6)for all k ∈ {1, . . . , ν}.
In this notation, it is also shown that A is decomposed
into the sum of rank-one matrices as{ν∑
ϕi := Aui−1TA=
ωi ϕi ψi ,(7)
ψi := AT vi
i=1

for any Rν and Oν , as long as uTi Avi ̸= 0 for all i ∈{1, . . . , ν}. On the basis of this rank-one decomposition, we
can define a matrix having rank k as
Ak :=

k
∑

Aik B = Ai B,

Let (A, B, C) denote the system matrices of{
xt+1 = Axt + But
yt = Cxt .

k ∈ {1, . . . , ν},

(8)

i=1

which is called a Wedderburn matrix [7]. This low-rank
reduction can be represented in a matrix form as
T
Ak = Φk Ω−1
k Ψk ,

Φk := AUk ,TΨk := A Vk .

Then, the following lemma gives a link between the Wedderburn rank reduction and linear systems theory:
Lemma 1: Let A ∈ Rn×n , B ∈ Rn , and C ∈ R1×n .
Consider the biconjugation process in (5) with
Rk = [B, AB, . . . , Ak−1 B]
Ok = [C T , (CA)T , . . . , (CAk−1 )T ]

(10)

(9)

(11)

Lemma 1 shows that, by giving the original bases Rk andOk as in (9), the biconjugation process in (5) produces asystem matrix Ak such that the low-rank model (Ak , B, C)
has the same (k + 1)-dimensional reachable and observable
subspaces as those of (A, B, C).
B. Relation to the Krylov Projection
Let us consider the following biconjugation process in (5)associated with In :(Pk , Qk ) = WIn (Rk , Ok )(12)

where Rk and Ok are defined as in (9). From the relationsin (4) and (6), we see thatim Pk = im Rk ,QTk P k = Λk

im Qk = im Ok ,where Λk is a diagonal matrix. In model reduction theory, thebiorthogonal projection of linear systems onto their reachable
and observable subspaces is called the Krylov projection.
This model reduction technique can produce an approximate
model that preserves the first 2k Markov parameters of theoriginal system [8], similarly to (10). The Krylov projection
model of (A, B, C) is given as (Pk† APk , Pk† B, CPk ) whereT
Pk† := Λ−1
k Qk ,(13)

which satisfies Pk† Pk = Ik .
To show a relation between the Krylov projection and the
low-rank model in Section II-A, we use the following facts[7]. There exists an upper bidiagonal matrix Mk ∈ Rk×k ,whose diagonal elements are all one and superdiagonal
elements are all nonzero, such that
Pk = Uk Mk ,

ωi−1 ϕi ψiT ,CAik = CAi

for all i ∈ {1, . . . , k}.

A. Wedderburn Rank Reductioni−1∑

and assume that Ωk in (3) is nonsingular. Then, for Ak in(8), it follows that

Qk = Vk Mk .

(14)

In addition, for Ωk in (3), it follows thatT
QT
k Ak Pk = Mk Ωk Mk ,(15)

which is a symmetric tridiagonal matrix. In this notation, thefollowing fact is proven:
Lemma 2: Let A ∈ Rn×n , B ∈ Rn , and C ∈ R1×n
be such that (A, B) is reachable and (A, C) is observable.
Consider the biconjugation processes in (5) and (12) with
Rk and Ok in (9). Then, (16) holds for Ak in (8), where
β0 := CB and βi denotes the ith superdiagonal element ofMk such that (14). Furthermore, defineÂk := Pk† Ak−1 Pk ,B̂k := Pk† B,

Ĉk := CPk

(17)



ω1
ω1 β1
 ω1 β1 ω2 + ω1 β12 ω2 β2


..
..
T
.
.
Qk Ak−1 Pk = 


..

.


..
..

.
ωk−1 βk−1

where Pk† is defined as in (13). Then, for any R ∈ Rn×mand S ∈ Rp×n , it follows that
S(zIn − Ak−1 )−1 B = Ŝ(zIk − Âk )−1 B̂k
C(zIn − Ak−1 )−1 R = Ĉk (zIk − Âk )−1 R̂(18)

for all k ∈ {2, . . . , ν}, where R̂ := Pk† R and Ŝ := SPk .
In particular, if (Ak−1 , R) and (Ak−1 , S) are reachable andobservable, respectively, then (Âk , R̂, Ĉk ) and (Âk , B̂k , Ŝ)
are minimal realizations.
Lemma 2 shows that the Krylov projection of the lowrank model (Ak−1 , B, C) onto the k-dimensional reachable
and observable subspaces of (A, B, C) leads to the exact
dimension reduction. This can be regarded as the extractionof a minimal realization, i.e., a reachable and observablerealization, from the low-rank model.
III. M ULTIRESOLVED C ONTROL
A. Observation and Stabilization via Redundant Realization
Consider a discrete-time linear system xt+1 = Axt + But + Rwt
y t = CxtΣ:z t = Sxt

(19)

where A ∈ Rn×n , B ∈ Rn , C ∈ R1×n , R ∈ Rn×m ,S ∈ Rp×n , and x0 ∈ Rn . In the rest of this paper, we assume
that (A, B) is reachable, (A, R) is stabilizable, (A, C) isobservable, and (A, S) is detectable. For convenience ofarguments, we express measurable (available) quantities by
the bold face, e.g., ut and y t in (19).
In the following, we design a compensator that aims at stabilizing the transitory behavior of systems while cooperatingwith a standard controller. More specifically, we consider
a situation where a transitory compensator begins to workat t = 0 for stabilizing an unknown disturbance, described
as an unknown initial state. Furthermore, we suppose that astandard controller has observed the system state for t < 0
to accomplish a control objective in normal circumstances.
We first introduce the following redundant state-spacerealization, similar to one used in [9]:Lemma 3: Consider Σ in (19). Let Σ̃ be given by (23)where x̂0 is a constant, Ak is defined as in (8) and
Γk :=

ν
∑

ωi−1 ϕi ψiT .

(21)

i=k

Then, it follows that
(1)

(2)

xt = xt + xt ,∀t ∈ {0, 1, . . .}

.

(22)

and




,


ωk−1 βk−1 2ωk−1 βk−1
{




β0
 0 


 .. TT

Qk B = (CPk ) =  . 
 . 
 .. 
0

(1)

(16)

(2)

y t = yt + yt
(1)
(2)
z t = zt + zt ,∀t ∈ {0, 1, . . .}(23)

for any x0 ∈ Rn and any sequences of ut ∈ R and wt ∈ Rm .
The expanded system Σ̃ in (23) can be regarded as aredundant state-space realization of Σ in (19). The constantx̂0 is to be used as a guess of the initial state x0 available
from past observation, i.e., the state observation by a standard
controller for t < 0. In other words, x0 − x̂0 in (23)corresponds to an unknown disturbance arising at t = 0.
The cascaded structure of Σ̃ provides a possibility to use(1)(2)
the input signals ut and wt for controlling xt and xt in an(1)
individual manner. Note that xt has an ability to capture theshort-term behavior of Σ owing to the property of the low(2)rank model shown in Lemma 1. Accordingly, xt represents
the remaining behavior in the sense of the superposition asin (22).
As shown in the relation of (23), even though y t and z t(i)are available as the measurement signals, the outputs yt(i)and zt of the redundant realization Σ̃ cannot be measured
as individual signals from the system Σ. To manage thisdifficulty, we use the following fact, which stems from thematching of the observable subspace shown in Lemma 1:
Lemma 4: Consider Σ̃ in (23) and let Σ̃obs be given by(1)(2)(27), where the sequences of ht and ht denote external(2)
inputs. If ht = 0 for all t ∈ {0, . . . , k − 1}, then
(2)

yt

(2)

= ŷ t ,∀t ∈ {0, . . . , k − 1}(25)

for any x0 ∈ Rn and any sequences of ut ∈ R, wt ∈ Rm ,(1)
and ht ∈ Rn .
(2)

Lemma 4 shows that, as long as ht = 0, the virtual(2)
output yt of the redundant realization Σ̃ can be exactly(2)
constructed as the output ŷ t of its Luenberger-type observer Σ̃obs during the limited interval of t ∈ {0, . . . , k − 1}.
(1)
From this fact in conjunction with (23), we see that yt is(2)measurable as y t − ŷ t during the limited interval. This idea
leads to the following result:
Theorem 1: Consider Σ̃ in (23) and Σ̃obs in (27). Let(1)(2)
ht , ht ∈ Rn in (27) be given by
(
)
(1)
(1)
(2)
ht = δ t H (1) y t − ŷ t − ŷ t
(
)
(26)
(1)
(2)
(2)
,
ht = δt H (2) z t − ẑ t − ẑ twhere y t ∈ R and z t ∈ Rp associated with Σ in (19) satisfy(23), and{0, t ∈ {0, . . . , k − 1}δt :=δ t := 1 − δt . (27)1, t ∈ {k, k + 1 . . . , },

 [
]
[
] [ (2) ] [
]
(2)

A
Γ
x
xt
Rwt

k
t+1

=
+
,

(1)
(1)

0 Ak−1
But

xt+1
xt



[
]
[
]


(2)
(2)
yt
Cxt
Σ̃ :
=
(1)
(1)

Cxt

 [ yt ]

[
]


(2)
(2)

zt
Sxt


=

(1)
(1)

zt
Sxt

[

(2)

x0
(1)
x0

(1)

(1)

(1)

(2)

(2)

(2)

for any sequences of ut ∈ R and wt ∈ Rm . Furthermore, if
Ak−1 − δ t H (1) C is transitionally Schur and A − H (2) S is
Schur, then(1)

lim ϵt

t→∞

= 0,

(2)

lim ϵtt→∞=0

for any x0 ∈ R .
nTheorem 1 shows that, by switching the output feedback
as in (26), the observer Σ̃obs can estimate the states of
the redundant realization Σ̃ in an individual manner. The
feedback of estimated states leads to a dynamical stabilizing
controller as follows:
Theorem 2: In the same notation as that in Theorem 1,
assume that Ak−1 −δ t H (1) C is transitionally Schur and A−
H (2) S is Schur. Let(1)

ut = F (1) x̂t ,(2)wt = F (2) x̂t .

(29)

For Σ̃ in (23) satisfying (22) with respect to Σ in (19), if
Ak−1 + BF (1) and A + RF (2) are Schur, then(1)

lim xt

t→∞

= 0,

(2)

lim xt

t→∞

=0

(30)

for any x0 ∈ Rn .
The switching control system in Theorem 2, which we calla multiresolved control system, consists of the cascaded connection of a low-rank component and a full-rank component(i.e., low and high resolution components), each of which(1)(2)
stabilizes the states xt and xt of the redundant realization
Σ̃. Clearly, (30) implies the stability of the original xt in theclosed-loop system.

[
=

x̂0
x0 − x̂0

]

(23)

 [
]
[
] [ (2) ] [] [ (2) ]
(2)

x̂t
ht
x̂t+1
A
Γk
Rwt


=
+
+
,

(1)
(1)
(1)

0
A
Bu

k−1
t
x̂
ht
x̂

t
t+1


]
]
[
[


(2)
(2)
C x̂t
ŷ t
Σ̃obs :
=
(1)
(1)

C x̂t
ŷ t



[
]
[
]


(2)
(2)

ẑ
S
x̂

t
t

=

(1)
(1)

ẑ t
S x̂t

Then, ϵt := xt − x̂t and ϵt := xt − x̂t obey
{
(2)
(2)
(1)
ϵt+1 = (A − δt H (2) S)ϵt + (Γk − δt H (2) S)ϵt
(28)
(1)
(1)
ϵt+1 = (Ak−1 − δ t H (1) C)ϵt

]

[

(2)

x̂0
(1)
x̂0

]

[
=

x̂0
0

]

(27)

B. Implementation in Minimal Dimension
In this subsection, we consider implementing the low-rankcomponent in the multiresolved control system by a lowerdimensional realization.
Lemma 5: Consider A ∈ Rn×n , B ∈ Rn , and C ∈ R1×n .
For Ak−1 defined as in (8), there exist Fk and Hk such that
Ak−1 +BFk is Schur, Ak−1 −δ t Hk C is transitionally Schur,andim FkT ⊂ im Ok , im Hk ⊂ im Rk ,(31)
where Rk and Ok are defined as in (9).
On the basis of this fact in conjunction with the extractionof the minimal realization in Lemma 2, we obtain a tractablerealization of the multiresolved control system as follows:
Theorem 3: Consider Σ in (19). Let ξ t+1 = AK ξ t + δt H(z t − ẑ t ) + ϕk γ t , ξ0 = x̂0wt = F ξt
K:ŷ t = Cξ t
(32)
where δt is defined as in (27) andAK := A + RF − δt HS.
Furthermore, with Âk , B̂k , and Ĉk in (17), letη= Âπ η t+1 + δ t Ĥk (y t − ŷ t ), η 0 = 0


 t+1
ut = F̂k η t+1
π:
γ t = (ekk )T η t+1ẑ t = Ŝη t+1(33)

where δ t is defined as in (27), Ŝ := SPk , andÂπ := Âk + B̂k F̂k − δ t Ĥk Ĉk .
For Σ̃ in (23) satisfying (22) with respect to Σ, if A + RF ,A − HS, and Âk + B̂k F̂k are Schur and Âk − δ t Ĥk Ĉk istransitionally Schur, then (30) holds for any x0 ∈ Rn .
Theorem 3 shows that the multiresolved control system
can be implemented as the conventional observer-based state

1Angular velocity

Cooperation

Area 2

Area3

Area 1

Area 4

0.5

0

-0.5

Area5
-1
-10

Fig. 1.

Interconnection structure of power generators.

IV. I LLUSTRATIVE E XAMPLE
A. Frequency Variation Model for Power Networks
In this section, we perform a numerical simulation on thestabilization of frequency variations. Let us consider a power
network consisting of five areas shown in Fig. 1, where thenodes represent power generators and the edges represent
the interconnection among the generators. In this network,
the generators in each area are densely connected whereas
they are sparsely connected among the five areas. Thus, thestates of generators belonging to the same area tend to besynchronized each other.
In the following, we model the dynamics of the ith
generator as [10]∑mi θ̈i + di θ̇i +ki,j (θi − θj ) = ui + wi
(34)
i̸=j

where mi denotes the inertial constant, di denotes the damping constant, ki,j denotes the coupling strength coefficient
between generators, θi denotes the angle deviation withrespect to a basis generator, and ui denotes the input torque.
The constants are given as follows. As for the couplingstrength coefficient between interconnected nodes, we give
ki,j = −1 if the ith and jth nodes belong to an identicalarea, and ki,j = −0.1 if they belong to different areas.
Furthermore∑
1−ki,j , i ∈ {1, 25}


i̸
=
j
∑
ki,i =

ki,j , i ∈ {1, . . . , 25}\{1, 25}.

 −
i̸=j

The damping constant is given as di = 0.02 and the inertial
constant is given randomly as mi ∈ [0.35, 0.65] for each i.
For the control input wi , the same signal is supposed tobe applied to all the generators as(35)

On the other hand, the control input ui is applied to thegenerators belonging to the first area asu := u1 = · · · = u5 ,

u6 = · · · = u25 = 0.

10

20

30

40

50

Time [s]feedback controller K in (32) to which the low-dimensional
compensator π in (33) is attached. The low-dimensional partcan be regarded as a transitory compensator that stabilizes theshort-term system behavior captured by the low-rank modelin Section II.

w := w1 = · · · = w25 .

0

(36)

Fig. 2.

Average feedback control for synchronous frequency variation.

Furthermore, the measurement output for constructing the
input signal wi is supposed to be the frequency variation
averaged over all generators, denoted byz := θ̇1 + · · · + θ̇25 .

(37)

Similarly, the measurement output for the input signal ui isavailable as the frequency variation averaged over the firstarea generators, denoted by
y := θ̇1 + · · · + θ̇5 .

(38)

In this setting, we obtain Σ in (19) by applying the temporal
discretization based on the zero-order hold with the sampling
period of one second.
B. Simulation Result
1) The Case of Average Feedback Control: First, for thefrequency variation model in Section IV-A, we design an
average feedback controller that uses wt and z t , which are
the temporally discretized versions of w in (35) and z in (37),
as its input and output signals. This controller corresponds
to K in (32), for which, on the basis of the LQR design
technique, we find the feedback gains F and H such thatA + RF and A − HS are Schur.
From the setting of wt and z t , we can expect that thisaverage feedback controller works well for frequency variations synchronized in all generators. Supposing that suchsynchronous frequency variations arise as normal circumstances, we calculate the response of the control system witha frequency variation that is almost uniformly distributed
over all generators. Furthermore, we set the state of K being
identical to that of Σ, which can stem from past observation.
The trajectories of the angular velocity (frequency) deviations are shown in Fig. 2, where the color of each set of lines
corresponds to that of each area in Fig. 1. From this figure,
we see that the frequency variation is properly stabilized by
the average feedback controller K as time goes on.
Next, in the same controller setting, we calculate theresponse of the control system where a local frequency
variation is additionally injected into only the first area at
t = 0. More specifically, we consider the situation that theguess of the initial state satisfiesx̂0 = Ax̂−1 + Rw−1 ,w−1 = F x̂−1 ,where x̂−1 = x−1 holds, and the initial state of Σ satisfies
x0 = x̂0 + ζ where ζ denotes the local frequency variation,

Angular velocity

1

Local variation

0.5

0

-0.5

-1
-10

0

10

20

30

40

50

Time [s]Fig. 3.

Average feedback control for asynchronous frequency variation.

Angular velocity

1

Local variation

0.5

0

-0.5
Only

-1
-10

&

0

V. C ONCLUSION
10

20

30

40

50

Time [s]Fig. 4.

controllers possibly occurs if they are implemented at thesame time.
In this example, the frequency deviation averaged over the
first area generators is fedback to the transitory compensator
only for t ∈ {0, . . . , 7}. After t = 8, the average deviation
over all generators is fedback to the average feedback controller. This switching is represented by δt and δ t in (32)and (33). Note that the transitory compensation fades out astime goes on, and then the multiresolved control coincides
with the average feedback control if the state of the transitorycompensator converges to zero. On the basis of this control
architecture, by designing multiple transitory compensatorsthat are specialized for individual areas in advance, we canrealize multiresolved control that can deal with unexpecteddisturbances arising in a specific area. Such a control strategy
is reasonable because contingent disturbances can be dealt
with several transitory compensators depending on situations,while a conventional controller can focus on accomplishinga control objective in normal circumstances.

Multiresolved control for asynchronous frequency variation.

demonstrating a contingency circumstance. From the result
shown in Fig. 3, we see that the settling time becomes
longer than that in Fig. 2. This result implies that the
average feedback controller does not work well for frequency
variations arising in a specific area whereas it is effective foruniform variations.
2) The Case of Multiresolved Control: In the following,
we show that local frequency variations can be effectively
stabilized by implementing a low-dimensional transitorycompensator π in (33), which uses ut and y t , denoting thediscretized versions of u in (36) and y in (38), as its input
and output signals. Note that, because these signals containthe information on the generators in the first area, they
are expected to be suitable for stabilizing local frequency
variations arising in the first area.
When k = 7, it turns out that Ak−1 defined as in (8) is
Schur. Using this Wedderburn matrix, we construct a sevendimensional model (Âk , B̂k , Ĉk ), for which, on the basis of
the LQR design technique, we find the feedback gains F̂kand Ĥk such that Âk + B̂k F̂k and Âk − Ĥk Ĉk are Schur.
In Fig. 4, we show the system response in the same
frequency variation settings as those in Fig. 3 when implementing the seven-dimensional transitory compensator into
the average feedback controller in Section IV-B.1. From thisfigure, where the transitory compensator begins to work at
t = 0, we see that the stabilization of the frequency variation
is well performed in comparison with Fig. 3. Note that thisstabilization is achieved by the suitable cooperation of K
and π in (32) and (33), demonstrating the efficiency of the
proposed multiresolved control. Such cooperation work ofdifferent controllers cannot be realized by a naive designof each controller because negative interference between

In this paper, we have proposed a design method ofmultiresolved control for discrete-time linear systems. In
the multiresolved control, we implement a transitory compensator specialized in stabilizing the short-term systembehavior into a conventional controller that is designed forstabilizing the long-term behavior. This control architecture
enables efficient handling of possible contingencies, suchas unexpected disturbances, besides accomplishing a control
objective in normal circumstances. The efficiency of themultiresolved control has been shown through an illustrative
example of frequency control in power networks.
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