Combined design of robust controller and disturbance observer in a fixed-order H∞ control framework

Abstract

The disturbance observer (DOB) and H∞ based robust control methods are widely used in feedback control applications due to their disturbance rejection capability and robustness improvement. However, the simultaneous synthesizing of DOB and H∞ controller considering the real uncertain parameters and dynamic uncertainties of the models is an open problem. This paper presents a new fixed-order μ-synthesis algorithm based optimization methodology for the combined design of DOB and robust controller in the presence of mixed uncertainty. The fixed-order robust H∞ control framework is adopted to optimize structured plant inverse, so-called Q filter and feedback controller. In order to guarantee the robust stability of the main feedback loop and inner DOB loop, the overall closed-loop system is considered with uncertainties during the design procedure. The proposed approach renders the need for an explicit plant inverse unnecessary. Thanks to this advantage and generality of μ-synthesis techniques, the proposed method can be directly applied to both the non-minimum phase and multiple-input and multiple-output (MIMO) uncertain systems without any approximation. The theoretical design approach is experimentally verified on an electromechanical control actuation system of an air vehicle and numerically on a non-minimum phase system.

Full text

Combined Design of Robust Controller and Disturbance Observer in a Fixed-Order
H∞Control Framework
Ersin Da¸s
Abstract
The disturbance observer (DOB) and H∞based robust control methods are widely used in feedback control applications
due to their disturbance rejection capability and robustness improvement. However, the simultaneous synthesizing of
DOB and H∞controller considering the real uncertain parameters and dynamic uncertainties of the models is an open
problem. This paper presents a new fixed-order µ-synthesis algorithm based optimization methodology for the combined
design of DOB and robust controller in the presence of mixed uncertainty. The fixed-order robust H∞control framework
is adopted to optimize structured plant inverse, so-called Qfilter and feedback controller. In order to guarantee the robust
stability of the main feedback loop and inner DOB loop, the overall closed-loop system is considered with uncertainties
during the design procedure. The proposed approach renders the need for an explicit plant inverse unnecessary. Thanks
to this advantage and generality of µ-synthesis techniques, the proposed method can be directly applied to both the
non-minimum phase and multiple-input and multiple-output (MIMO) uncertain systems without any approximation.
The theoretical design approach is experimentally verified on an electromechanical control actuation system of an air
vehicle and numerically on a non-minimum phase system.
Keywords: robust control, fixed-order H∞, disturbance observer, µ-synthesis, mixed uncertainty, electromechanical
system
1. Introduction
Model uncertainties resulting from several parameters
such as unmodeled system dynamics, modelling errors,
process nonlinearities, real parameter variations and op-
erating point changes are almost unavoidable in practi-
cal control applications. Therefore, they might cause con-
troller performance degradation, and in some cases, even
lead to system instability. In addition to these uncertain-
ties, external disturbances such as external load, sensor
noise, friction may deteriorate the efficiency of today’s
complex dynamical control systems [1]. Therefore, the de-
sign of feedback controllers, which can compensate for the
effects of uncertainties and external disturbances, is one
of the active problems in control engineering [2]. Among
them, robust control and disturbance observer (DOB) tech-
niques have received particular attraction over the past
forty years due to their ability to improve the stability
and disturbance rejection performance of control systems
in practice [3, 4, 5, 6, 7, 8].
The H∞control theory was introduced to deal with
the dynamic uncertainties, exogenous inputs, performance
outputs, and user-defined weighting functions in controller
synthesis processes [9, 10]. Classical sub-optimal H∞con-
trol problem becomes convex when the model is nominal
Email address: ersindas@caltech.edu (Ersin Da¸s)
1E. Da¸s is with the Mechatronics Department, T ¨
UB˙
ITAK SAGE
and the controller is full-order, i.e., unstructured [11]. The
well-known convex optimization based solution techniques
of this full-order H∞control problem are state-space [12]
and linear matrix inequalities (LMI) [13] algorithms. In
these approaches, as the order of the nominal model is
increased to satisfy high accuracy system modelling, the
order of the resulting controller also increases. Moreover,
the total order of user-defined weighting functions also in-
creases the order of the controller. This disadvantage is
one of the reasons that limit the practical implementation
and readjustment of the unstructured H∞controller syn-
thesis methods [14]. On the other hand, fixed-order con-
trollers with a specific structure, such as Proportional In-
tegrator Derivative (PID) type controller, are preferred in
industrial control system applications due to their simplic-
ity and implementation advantages. To address the above
mentioned insufficiencies, the practicality of fixed-order
control methods and robustness utility of the H∞control
technique has been combined within different types of op-
timization framework in recent years [15]. However, these
optimization problems become an inherently NP-hard non-
convex when a low order controller structure is considered
instead of a full-order controller [16]. The hinfstruct and
systune functions of MATLAB [17] and H2−H∞Fixed
Order Optimization (HIFOO) MATLAB package [18] are
well-known algorithms to solve these non-convex fixed-
order H∞control design problems. The standard solution

of the H∞control problem [12] is limited to dealing with
unstructured uncertainty models, such as multiplicative,
additive, or coprime factor uncertainty. However, real-
time control systems generally include mixed uncertainty,
i.e., real parameter variations and unstructured dynam-
ical uncertainties. The transfer function based classical
uncertainty modelling approach produces a considerably
larger uncertainty magnitude than the parametric uncer-
tainty model. Therefore, the unstructured uncertainty
model may cause controller performance degradation due
to the conservative weighting function. When an overall
uncertainty model is described by a block diagonal form,
it is called structured uncertainty. The structured singu-
lar value (SSV), denoted simply as µ, based robust con-
troller synthesis framework, i.e., µ-synthesis approach, was
proposed to satisfy the robust stability and robust perfor-
mance conditions of a closed-loop system with structured
uncertainty [19].
The DOB based two degrees-of-freedom (2-DOF) con-
trol scheme is a robust control methodology that attenu-
ates external disturbance and compensates uncertainties.
In this control tool, first, the outer loop is designed sepa-
rately to satisfy nominal performance and stability, then
the inner DOB loop, which includes the inverse of the nom-
inal plant model and a low-pass filter Q, is synthesized to
estimate and reject disturbances and uncertainties [2]. Re-
cently, the standard H∞controller design techniques have
been adopted to the synthesis and analysis of DOB based
2-DOF control systems to improve the robustness and per-
formance [20, 21, 22]. The optimal Qsynthesizing algo-
rithm is formulated within the H∞control framework in
these approaches. However, these methods generate a full-
order (high order) Qfilter due to the employed state-space
and LMI based solution techniques; therefore, these solu-
tion techniques need an additional model reduction step.
In [23], a disturbance and uncertainty estimator based ro-
bust H∞control design method for non-minimum phase
(NMP) systems has been introduced. However, only an
unstructured uncertainty model is considered, and the to-
tal order of the main feedback controller and observer is
relatively high. An H∞DOB design method, which is
based on the minimization of the H∞norm of the weighted
transfer function from input disturbance to estimated dis-
turbance, is presented in [24]. Although the design proce-
dure does not require an explicit plant inverse, this method
only deals with the DOB synthesis for the uncertainty-free
system. More recently, [25] proposed a heuristic two-stage
algorithm for robustly stable fixed-order DOB design with
closed-loop consideration. This method needs a mathe-
matical inverse model of the controlled plant.
This study introduces a novel combined design approach
for robust feedback controller and DOB in the H∞con-
trol framework. To address the limitations of the available
methods mentioned earlier, we aim to combine the robust-
ness of DOB and the practicality of the structured con-
trollers within the potentially non-convex fixed-order H∞
control problem via simultaneous synthesizing of DOB and
the main controller. In order to perform this goal, three
main contributions are presented in this study:
•The fixed-order µ-synthesis algorithm based control
theory is adopted for DOB based 2-DOF robust con-
trol structure, which consists of the main feedback
loop and a DOB loop. The proposed method opti-
mizes the fixed-order (structured) controller, Qfilter
and inverse of the nominal model together; therefore,
the main feedback control loop and DOB loop are de-
signed under the same closed-loop constraints with
a robust performance objective.
•In pursuit of bridging the gap between the experi-
mental system data and the accuracy of an uncertain
model, this study considers the mixed type uncer-
tainty. This algorithm reduces the conservatism and
improves the robust performance of the proposed
method.
•Thanks to the stably model inversion approach, the
proposed method renders the need for an explicit
plant inverse unnecessary. This advantage and gen-
erality of µ-synthesis techniques make the proposed
method directly applicable to the NMP and MIMO
uncertain systems without any approximation.
Finally, the proposed approach is experimentally verified
on an electromechanical control actuation system of an air
vehicle. For the sake of completeness, numerical compar-
isons are made between the classical full-order H∞based
DOB design method and the proposed fixed-order 2-DOF
control approach on an NMP system model of a vertical
take-off and landing (VTOL) platform.
The remainder of this paper is organized as follows:
The preliminaries and DOB based control structure are
introduced in Section II. We will present the proposed de-
sign method of DOB and robust controller in Section III.
Application results are presented in Section IV, and Sec-
tion V concludes the paper.
2. Disturbance Observer
2.1. Preliminaries
The notation used in this study is fairly standard. N,
Rrepresent the set of natural and real numbers, respec-
tively. Rn×mand Cn×mdenote n×mreal and com-
plex matrices, respectively. The notations G(s), G(z) and
G(jω) represent the transfer function of the plant Gin the
Laplace domain, z-domain and frequency domain, respec-
tively. RH∞shows the stable rational transfer function
matrices. The H∞norm of a n×mtransfer function ma-
trix Tm×ncan be written as
kTm×nk∞:= sup
<(s)>0Tm×n(s)= sup
w∈RTm×n(jω)(1)
which represents the peak of the maximum singular value
of the Tm×n(jω) [11]. Finally, diag(·) stands for the block
diagonal matrices.
2

K(s)
+eur
−
+ + G(s)
+G−1
n(s)+
−
Q(s)
ym
v
−
d
de
ut
yn
DOB loop
DOB
Figure 1: Classical feedback control system with conventional DOB
framework.
We consider a single-input single-output (SISO) lin-
ear time invariant (LTI) controllable system written in the
state-space form:
˙x(t) = Ax(t) + Bu(t) + d(t)
y(t) = Cx(t)
G(s) = C(sI −A)−1B
(2)
where x(t)∈Rn×1is the state vector, u(t)∈Ris the con-
trol input, d(t)∈Ris the external disturbance, y(t)∈R
is the measured output and A∈Rn×n, B ∈Rn×1, C ∈
R1×nare the state-space matrices of the controllable and
observable system. Without loss of generality, through-
out the text, we assume that the d(t) is an additive vary-
ing disturbance input, i.e., d(t) is in the matched distur-
bance form, and there exists a constant dm∈R+
0such that
supt≥0||d(t)|| ≤ dm.
2.2. Conventional DOB-Based Control System
The main idea of the classical DOB structure, also
known as frequency-domain DOB, is to estimate the ex-
ternal disturbance by filtering the difference between the
calculated input signal using the inverse of the nominal
model and the control signal without using any sensors [5].
The block diagram representation of classical DOB based
robust feedback control system is shown in Fig. 1, where
r, e, u, ut, d, de, ym, yn, v ∈Rare the reference input,
error signal, controller output, control signal, external dis-
turbance input, estimated disturbance, measured system
output, feedback signal and noise input, respectively. In
this figure, G(s) and G−1
n(s) denote the uncertain plant
model and the inverse of the nominal model, respectively.
K(s) is the baseline feedback controller, and Q(s) is a sta-
ble low-pass filter. Note that the dependence in complex
frequency swill be omitted for simplicity throughout the
rest of the paper, and it will be used only if necessary.
The sensitivity function SDOB , which represents the
transfer function from output disturbance, i.e., G·dto ym,
and the complementary sensitivity function TDOB, which
represents the transfer function from dto ut, of the inner
DOB-loop can be directly derived from Fig. 1 as:
SDOB =Gn(1 −Q)
Gn+ (G−Gn)Q(3)
TDOB =GQ
Gn+ (G−Gn)Q.(4)
Q(s)
K(s)+G(s)
++ e
−−
r
d
+
v
G−1
n(s)
+
−
z1
z2
ue
uut
yn
de
P
Figure 2: Construction of the generalized plant Pfor H∞-based Q
filter synthesis problem.
Similarly, the sensitivity and complementary sensitivity
transfer functions, which are the transfer functions from
G·dand rto ymrespectively, of the outer-loop system are
given as:
SCL =Gn(1 −Q) + GQ
Gn+ (G−Gn)Q+GGnK(5)
TCL =GGnK
Gn+ (G−Gn)Q+GGnK.(6)
In order to investigate the disturbance estimation achieve-
ment of DOB-loop, a transfer function from dto decan be
defined as follows [26]:
Td=G(1 + GK)Q
Gn+GGnK+Q(G−Gn).(7)
Since Qis generally a low-pass filter and |Q(jω)| ≈ 1,
|G(jω)G−1
n(jω)| ≈ 1 at low frequencies, the above trans-
fer functions become |SDOB (j ω)| ≈ 0, TDOB(jω)| ≈ 1,
SCL (jω)| ≈ 0, TCL (jω)| ≈ 1 and |Td(jω)| ≈ 1, which
mean that, assuming the multivariable system is internally
stable, the actual external disturbance input is entirely es-
timated and rejected by using DOB, therefore, inner-loop
improves the robustness of the overall closed-loop control
system. On the other hand, when |Q(j ω)| ≈ 0 at high fre-
quencies, sensitivity functions of DOB-loop are derived as
|SDOB (jω)| ≈ 1 and |TDOB (jω)| ≈ 0. This indicates that
the DOB-loop is ineffective for the disturbance rejection
requirement in the high-frequency range.
The conventional DOB structure in Fig. 1 includes a
non-proper inverse model G−1
n. Furthermore, this direct
model inversion approach produces unstable poles in DOB
transfer function in case of non-minimum phase system
model Gnwith right half plane zeros [27]. In order to
obtain a realizable and stable DOB transfer function, the
relative degree of Qshould be no less than that of Gn, and
Q, G−1
nshould be stable.
2.3. H∞Based Q Filter Synthesis Framework
This subsection presents a full-order H∞control theory-
based Qfilter optimization methodology.
The general linear fractional transformation (LFT) model
of the 2-DOF closed-loop system for DOB synthesis and
analysis is given in Fig. 2. The LFT plant Pdenotes the
input-output relationships between inputs [r d v de]Tand
3

outputs [z1z2ue]T. This model is obtained by separat-
ing the Qfilter from the conventional DOB framework in
Fig. 1. Hence, the Pcan be partitioned in matrix form as:
P:z
de="P11 P12
P21 P22 #w
ue
P11 =1
X−GK G GK
−K−GK K 
P12 =1
XG
1
P21 =1
X1 + KGn
Gn
G(1 + KGn)
Gn
(G−Gn)K
Gn
P22 =1
XG−Gn
Gn
(8)
where X,1 + KG,z= [z1z2]T∈R2is a vector of
performance outputs, w= [r d v]T∈R3is a vector of
exogenous inputs, ue∈Ris the difference between control
signal ut∈Rand output of the inverse model, respectively.
The basic idea of the H∞based DOB synthesis frame-
work is to optimize the Qfilter that minimizes the effects
of the exogenous inputs [r d v] on selected performance
outputs [z1z2]. The transfer function Tw z, which includes
transfer functions from exogenous inputs to performance
outputs, can be calculated by the LFT as:
Twz :z=Fl(P, Q)
Twz =P11 +P12Q(I−P22 Q)−1P21
(9)
where Fl() denotes the lower LFT. Hence, the components
of matrix Twz are computed as:
1
Y−G(Q+GnK)−(Q−1)GGnGGnK
(Q−1)GnK(Q−1)GGnK(QG −QGn+Gn)K
(10)
where Y,Q(G−Gn) + Gn(1 + GK).
The H∞based Qfilter synthesis problem is: find all
admissible Q, i.e., it internally stabilizes the closed-loop
system, such that
kTwz k∞:= sup
<(s)>0
[Twz ] = sup
ω∈R
[Twz (jw)] (11)
is minimized. However, the solution of this H∞based
global optimal DOB filter Qoptimization problem is not
simple [11]. In order to overcome this difficulty, a sub-
optimal control problem can be considered as: find an ad-
missible Qfilter such that
kTwz k∞< γ (12)
where γ > 0. This full-order H∞control problem can be
solved iteratively by using LMI [28] based algorithm.
It is clear that in the H∞controller synthesis theory
based Qfilter design problem given in (12), both a causal
K
+eur
−
+ +
+G−1
n+
−
Q
ym
wn
−
d
de
ut
yn
Wr
Wd
Wp
wr
+
−
wd
∆
G11 G12
G21 G22 
Wn
v
We
zezpy∆
u∆
Figure 3: Augmented robust H∞interconnection for the combined
synthesis of DOB and feedback controller.
inverse model G−1
nand a stabilizing main feedback con-
troller Kare assumed to be known in advance, and both
are necessary for the solution. The DOB controller com-
posed with the resulting Qfilter and known inverse model
G−1
nis an add-on to the closed-loop system whose stabil-
ity is guaranteed before. Since the transfer functions, SC L
and TCL given in (3) and (4) are also dependent on the Q
filter model, the stability and performance of the overall
2-DOF closed-loop system including DOB should be re-
analysed after Qfilter synthesis process. Moreover, model
uncertainties that may cause instability of the closed-loop
system are ignored in the H∞controller synthesis theory
based Qfilter design approach. An alternative way to
design a robust Qfilter for the uncertain system model,
which includes model uncertainties, is to use the well-
known DK-iteration, i.e., µ-synthesis, method [29, 19].
However, the degree of the synthesized Qwith this method
doubles in each iteration; therefore, its real-time applica-
tion becomes more and more difficult with each iteration.
3. Design of Robust Controller and DOB in Fixed-
Order H∞Control Framework
In this section, the simultaneous optimization prob-
lem of DOB and robust main feedback controller in the
presence of mixed uncertainty is transferred into the fixed-
order µ-synthesis framework. First, we define a fixed-order
controller design problem for the uncertain 2-DOF closed-
loop system, using a structured controller K, a filter Qand
a model inverse G−1
n. After that, the recently presented
structured robust controller design method against mixed
uncertainty [16] is adopted for the solution of this opti-
mization problem to the combined design of K, Q, G−1
n
parameters, which guarantee the robust stability and per-
formance for DOB-based 2-DOF closed-loop system.
The interconnection of the proposed DOB based 2-
DOF feedback control system with an uncertainty block
∆ and user-defined weighting functions is given in Fig. 3,
where Wris the reference weighting function, Wdis the
disturbance input weighting function and Wnsensor noise
weighting function, respectively. Weand Wpblocks shown
in these figures are the frequency-dependent weighting func-
tions representing the performance requirements defined
4

u
de
∆
y∆
u∆
K0 0
0−Q QG−1
n
P


e
ut
yn




wr
wd
wn

ze
zp
Figure 4: Generalized control system configuration with LFT struc-
ture for the DOB based 2-DOF feedback control system design and
analysis.
on the tracking error and disturbance estimation error sig-
nal outputs, respectively. A general representation of un-
certain plant model Gby LFT is used in Fig. 3 to take
into account unmodeled system dynamics in the controller
synthesis process. Here,
G=Fu G11 G12
G21 G22 ,∆
G=G22 +G21∆(I−G11 ∆)−1G12
(13)
where G22 =Gn, i.e., nominal model and Fu() denotes the
upper LFT. Note that, Fl() and Fu() LFTs can be defined
utilizing Redheffer star product [10], which is denoted by ?
symbol, of corresponding partitioned system matrices with
appropriate dimensions as
Fl(G, K):=G?K
Fu(G, ∆) := ∆ ? G.
(14)
The control system scheme is shown in Fig. 3 can be
transformed into a corresponding robust controller synthe-
sis framework given by Fig. 4. This representation is ob-
tained by pulling out the unknown parts, i.e., K, G−1
n, Q,
and ∆ from the given parts of the closed-loop system in
Fig. 3. The augmented plant Pis partitioned in matrix
form with uncertainty channel u∆7→ z∆as:
y∆zezpe utynT=Pu∆wrwdwnu deT(15)
where [wrwdwn]T7→ [zezp]Tis the performance channel
and [u de]T7→ [e utyn]Tis the control channel. The
block-structured uncertainty ∆ is assumed as
∆ = ∆p0
0 ∆d(16)
where the set of real parametric uncertainty ∆pand LTI
normalized dynamic uncertainty ∆d, which is a square ma-
trix, have the following form:
∆p:=ndiag δ1Ir1, . . . , δSpIrSp:δi∈R,|δi| ≤ 1o(17)
where Spis the number of real parameters δ1, . . . , δSpand
r1, . . . , rSpare the number of their iterations,
∆d:=ndiag ∆1,...,∆Sdo(18)
with ∆i∈Cpi×qi,k∆ik∞≤1, i = 1,...Sd. We consider
that the parametric uncertainty is normalized. A bounded
real parametric uncertainty kpmin ≤kp0≤kpmax can be
represented in its normalized form δkp∈[−1,1] as [30]:
kp=kp0+kp0kpvar δkp, δkp∈[−1,1]
kp0= (kpmin +kpmax )/2
kpvar = (kpmin −kpmax )/(kpmin +kpmax ).
(19)
To define the decision variable QG−1
nin a compact
transfer function form, the new variable L=QG−1
nis
introduced. Then, the SISO controllers are parameterized
as general fixed-order transfer functions:
K=knK
mK−1smK−1+. . . +knK
0
smK+kdK
mK−1smK−1+. . . +kdK
0
Q=knQ
mQ−1smQ−1+. . . +knQ
0
smQ+kdQ
mQ−1smQ−1+. . . +kdQ
0
L=knL
mL−1smL−1+. . . +knL
0
smL+kdL
mL−1smL−1+. . . +kdL
0
(20)
where k=knK
mK−1. . . kdL
0∈R(2(mK+mQ+mL)) are tun-
able parameters of controller Λ(k) which is shown in Fig. 4
as:
Λ(k) = K0 0
0−Q L.(21)
Note that in knK
mK−1representation, the superscripts knK, kdK
show the numerator (nK) and denominator (dK) parame-
ters of the controller (21).
By using the generalized feedback control system con-
figuration with LFT structure for the DOB based 2-DOF
controller design shown in Fig. 4, the fixed-order synthesis
problem can be defined in a standard robust H∞form:
min
kmax
∆kFu(∆,Fl(P, Λ))k∞(22)
where Twz (∆, k):=Fu(∆,Fl(P, Λ)) is the corresponding
uncertain closed-loop model of the control system. This
optimization problem involves an infinite number of ∆ sce-
narios; therefore, it is semi-infinite programming. In order
to transform this problem into a tractable one, we con-
sider the outer relaxation approach based on the concept
of well-posedness [31]. Motivated by the recently pub-
lished structured robust control design approach [16, 32],
we introduce the steps of the DOB based 2-DOF control
algorithm in the sequel.
On the basis of aforementioned dynamic uncertainty
matrix ∆d, we can now define a block structured D-scaling
matrix D(s) that commutes with ∆d, i.e., ∆dD=D∆d,
as
D:=diaghd1(s)Ipi, . . . , dSd(s)IpSdi(23)
where di(s), di(s)−1are bi-proper stable minimum phase
transfer functions. Similarly, consider a block-structure
5

dynamic multiplier Φ(s) associated with the ∆psuch that
∆pΦ = Φ∆p:
Φ:=diaghΦ1(s),...,ΦNp(s)i(24)
where φi(s) is stable and kΦi(s)k∞≤1. In addition, we
define a matrix Γ(Φ, D) given by
Γ(Φ, D):=



−Φ 0 1 + Φ 0
000D
1−Φ 0 Φ 0
0D−10 0




(25)
such that the closed-loop model ∆ ?Γ(Φ, D) is internally
stable and k∆?Γ(Φ, D)k∞≤1. The inverse of Γ(Φ, D)
regarding to ?operation Γ(Φ, D)−?is given by
Γ(Φ, D)−?:=



Φ 0 1 −Φ 0
000D−1
1 + Φ 0 −Φ 0
0D0 0




(26)
which satisfies Γ(Φ, D)?Γ(Φ, D)−?principle [16]. Then,
in order to include the robust performance channel in µ-
synthesis based 2-DOF DOB design framework, we con-
sider the scaled version of the generalized plan Pin (15)
as
Pγ=0γ−1γ−10 0 0P(27)
where the performance channel [wrwdwn]T7→ [zezp]Tof
Pγis scaled with parameter 1/γ.
The uncertain closed-loop interconnection T(∆, Pγ,Λ),
which is shown in Fig. 4, is given by
Twz (∆, Pγ,Λ) :=Fu∆,Fl(P, Λ)= ∆ ? Pγ?Λ (28)
and can be converted into the following form using the
Γ(Φ, D)?Γ(Φ, D)−?=Iproperty of Redheffer star prod-
uct
Twz (∆, Pγ,Λ) = ∆ ?Γ(Φ, D)?Γ(Φ, D)−?? Pγ?Λ.(29)
By associativity of ?operation, kTwz (∆, Pγ,Λ)k∞<1
constraint is equivalent to
k∆?Γ(Φ, D)k∞kΓ(Φ, D)−?? Pγ?Λk∞<1 (30)
where k∆?Γ(Φ, D)k∞<1 due to the definition of Γ(Φ, D)
matrix. Therefore, Twz (∆, Pγ,Λ) is internally stable and
satisfies bounded H∞-norm condition kTwz (∆, Pγ,Λ)k∞<
1 if
kΓ(Φ, D)−?? Pγ?Λk∞<1.(31)
This means that if we can find elements of Γ and Λ sat-
isfying the above constraint, then we can guarantee that
Twz (∆, Pγ,Λ) is stable over defined uncertainty set ∆. It
also satisfies robust performance requirements. Note that
(31) is an outer relaxation of (30); therefore, an optimal
solution to the (30) is a feasible solution to the (31). Fig. 5
∆p
−ΦI+Φ
I−Φ Φ 
K0 0
0−Q L
Pγ
ΦI−Φ
I+Φ−Φ
D−1∆dD
D
D−1


wr
wd
wn


ze
zp
Γ(Φ, D)−⋆⋆Pγ
∆⋆Γ(Φ, D)
Figure 5: Interconnection of the outer relaxation method for robust
controller synthesis framework.
depicts the interconnection of the introduced outer relax-
ation method corresponding to LFT representation of un-
certain scaled plant with structured controller matrix.
Finally, after embedding relaxation matrix, Γ and un-
certain model Gin an augmented plant Pγwith robust-
ness and performance channels, the fixed-order µ-synthesis
based 2-DOF controller design problem becomes
min
γ, k, D , Φγ
s.t. kΓ(Φ, D)−?? Pγ?Λk∞<1
kΦ(s)k∞≤1
γ∈R+, k ∈R
(32)
which can be solved using numerically available non-smooth
techniques and Matlab Robust Control Toolbox solvers.
4. Synthesis Examples
We present two case studies in this section: (i) exper-
imental position control of a stable minimum phase elec-
tromechanical system and (ii) attitude control of a non-
minimum phase system model of a vertical VTOL plat-
form, respectively.
4.1. Experimental Implementation
This subsection applies the proposed DOB-based con-
trol design methodology to an electromechanical control
actuation system (CAS) of an air vehicle using the experi-
mental test bench. This system is shown in Fig. 6, where a
test side (actuation system) and a loading side (electrical
load simulator-ELS) are illustrated.
The CAS consists of a brushless DC (BLDC) electric
motor with an encoder, a ballscrew and bearing elements
with associated mechanical parts. This CAS is operated
6

Figure 6: The experimental setup containing CAS, torsion spring,
reaction type torque sensor and ELS.
by a 120A10 servo driver, a commercial product of AD-
VANCED Motion Controls (AMC). The actuation system
controls the air vehicle by changing the direction of the
aerodynamic control surface, which is exposed to distur-
bance hinge moment.
The equation of the motion of the CAS in terms of
equivalent moments of inertia Je, equivalent viscous damp-
ing Be, equivalent Coulomb friction Fe, load torque (dis-
turbance) TLand CAS torque constant Ktis given by
Je¨
θ+Be˙
θ+Ksθ+Fesign(˙
θ) + TL=Tm=Ktim(33)
where imis the motor phase current, Ksis the stiffness
constant, Tmis the output CAS torque, and θis the de-
flection angle of CAS. The input of the system is u=im
and the output is ym=θ.
A closed-loop torque control-based ELS is used to test
CAS under aerodynamic loading conditions on the load-
ing side. This load simulator is actuated by a permanent
magnet synchronous motor (PMSM) with a gearhead to
increase motor torque and a custom-made torsion spring
package to improve the accuracy of applied torque. The
actuation system and loading system are coupled through
this spring package. Therefore, the output of the loading
system treats as an external disturbance to the position
control loop of the actuation system. On the other hand,
the output angle of the actuation system is an external dis-
turbance to the load torque control loop. A reaction type
torque sensor with a signal conditioner is used to measure
the loading torque for the torque control loop. Hence, we
can easily measure the applied disturbance input to the
CAS and compare it with the output of the disturbance
observer.
We use an NI 6221 data acquisition (DAQ) board,
and an NI SCB-68 shielded input/output (I/O) connector
block, which has a signal conditioning capability for filter-
ing the signals with 68 screws terminals to receive the mea-
surement data and to send the control signals. The DOB
based closed-loop control system is implemented using the
xPC Target toolbox of Matlab software. This toolbox in-
cludes a discrete-time controller matrix, communication
protocols and signal type converters. The transfer function
of the obtained controller is digitalized using the bilinear
transformation method. The digital closed-loop position
Kt
d+ut
+
1
Jes2+Bes+Ks
ym
y∆p∆pδKt∆dWA
+
u∆py∆du∆d
Tm
Figure 7: The uncertain CAS model block diagram.
and torque control loops of experimental CAS and ELS
systems are operated at 2 kHz frequency. A host com-
puter is used for offline programming of the closed-loop
control algorithm.
The expression of the uncertain electromechanical CAS
model, Gin Fig. 3, can be constructed based on the follow-
ing relationships according to the block diagram illustrated
in Fig. 7:
G11 G12
G21 G22 :


y∆p
y∆d
ym


=




0 0 1
δKt0Kt
δKt
Jes2+Bes+Ks
WA
Kt
Jes2+Bes+Ks






u∆p
u∆d
d+ut



(34)
G:ym=FuG11 G12
G21 G22 ,∆p0
0 ∆dd+ut
Kt+δKt∆p(Jes2+Bes+Ks)WA∆d+ 1
Jes2+Bes+Ks
(35)
where WAis the additive type dynamic uncertainty weight-
ing function and δKtis the parametric uncertainty describ-
ing the level of torque constant variation.
The nominal plant model was estimated with Box-
Jenkins (BJ) system identification method results in
Gn:= θ(s)
Im(s)=352300
s2+ 69.59s+ 0.00487 (36)
which has a 93.5% model fit percentage. The paramet-
ric uncertainty of Ktwas computed as 10% according to
the result of numerous identification processes for vary-
ing conditions. Similarly, a 3rd-order dynamic uncertainty
transfer function was obtained as:
WA=3236000s+ 886200
s3+ 83.28s2+ 953s+ 0.0066 (37)
which led to a good frequency response fitting of relative
errors between the nominal model and the other identified
models.
The disturbance estimation capability requirements for
the CAS was described by weighting:
Wp=s+ 1125
1.5s+ 1125 (38)
7

which penalizes the differences between disturbance input
and estimated disturbance as illustrated in Fig. 3. The
performance weight Weis chosen to reflect tracking error
requirements such that the angle of the control surface
should track the reference commands with a settling time
of 0.04 seconds. Therefore, the weighting function was
defined as
We=0.015s+ 0.045
s+ 0.5.(39)
In our CAS operating conditions, the external distur-
bance generally affects the system at the low-frequency
range. Similarly, the reference command input is typically
concentrated on a low-frequency range with a 12-degree
control surface deflection magnitude. On the other hand,
sensor noise usually has high-frequency dynamics. Hence,
the disturbance input weighting Wd, the reference input
weighting function Wrand the sensor noise weighting func-
tion Wnwere chosen as:
Wd=0.0026s+ 30.16
s+ 2.513
Wr=0.0048s+ 7.54
s+ 0.6283
Wn=0.4395s+ 1.104
s+ 25.13 .
(40)
Next, the K,Qand Lcontrollers were structured as gen-
eral 2nd, 1st and 3rd -order transfer functions with respect
to the (20), respectively. Finally, the optimization problem
(32) was solved using Matlab software.
The transfer functions of the obtained fixed-order ro-
bust controller are given by
K=0.9592s2+ 81.01s+ 124.9
s2+ 1637s+ 1591
Q=388.4
s+ 388.4
L=2.63s3+ 183s2+ 76.32s+ 9.726
s3+ 2727s2+ 949100s+ 149900.
(41)
The resulting controller system satisfies the robust perfor-
mance condition such that kΓ(Φ, D)−??Pγ?Λk∞= 0.24 <
1.
Real-time hardware in the loop tests were carried out
to verify the performance of the synthesized DOB based
2-DOF controller. This controller was applied to the ex-
perimental system in the real-time hardware in the loop
test. In the first experimental study, a filtered step in-
put signal with 10-degree control surface deflection am-
plitude was applied as a reference command to the sys-
tem to determine the reference tracking the performance
of the controlled CAS and disturbance estimation and re-
jection achievements of the robust system DOB loop. In
this experiment, we added the torque disturbance, which
includes step and 1 Hz sinusoidal parts, to the CAS using
ELS, and we tested the closed-loop control system with
and without DOB loop cases. The first and second parts
Figure 8: Reference command tracking performance of the designed
control system with/without DOB loop and disturbance estimation
achievement of the proposed DOB scheme.
of Fig. 8 respectively show the reference command tracking
accomplishment of the CAS with and without DOB loop
and disturbance estimation performance of the presented
DOB design scheme.
As can be seen from this figure, the maximum tracking
error is around 0.4 degrees, and the applied disturbance
input causes a 0.4-degree steady-state tracking error with-
out the DOB loop. However, the DOB loop reduces the
maximum error to 0.1 degrees and disturbance rejection
time to 0.2 seconds in the 2-3 second time range. There-
fore, the reference command tracking performance of the
CAS is improved using the proposed 2-DOF DOB based
robust control algorithm. Furthermore, it is observed from
the second part of Fig. 8 that the DOB can effectively esti-
mate actual disturbance within the estimation bandwidth.
The difference between the measured ELS output torque
and the estimated disturbances by the DOB is caused by
the unmodeled (ignored) system dynamics such as param-
eter uncertainty, friction, free-play, sensor dynamics, servo
driver dynamics and coupling effect of the ELS.
4.2. Simulation Example
A simulation example is presented to demonstrate the
applicability of the described methodology on the NMP
systems. This example was taken from [24] in which a
full-order H∞based DOB controller design problem is de-
fined for the control of an NMP vertical VTOL platform.
In this work, a PID controller is designed for the feed-
back path. First, to make a fair comparison between this
study and our fixed-order 2-DOF method, the same PID
controller, which is taken from [33], and weighting func-
tions with nominal system model were preferred. There-
fore, only the 1st-order Qfilter and 3r d-order L=QG−1
n
8

were optimized (PID-DOB). Second, we applied the pro-
posed method to obtain K,Qand Lcontrollers as 2nd, 1st,
3rd-order transfer functions, respectively (H∞−DOB).
The transfer function of the controlled nominal plant
is given by
Gn(s) = (s−9.58)(s+ 1.32)(s+ 7.68)(s2+ 25)
s(s+ 2.08)(s+ 0.18)(s2+ 26.48) ,(42)
which is a non-minimum phase model. The disturbance
and noise weighting functions designed in [24] are given
by
Wd=0.01s+ 9.42
s+ 0.1
Wn=100s+ 14.43
s+ 942.6,
(43)
respectively. Then, the weighting functions Wr,Wewere
chosen for H∞−DOB control design problem as
Wr=0.002s+ 31.42
s+ 3.142
We=0.003s+ 0.09
s+ 0.0001 .
(44)
The implementation of the proposed DOB design algo-
rithm leads to PID-DOB controller:
Q=8.316
s+ 8.316
L=0.0085s3+ 0.16s2+ 2934s+ 12310
s3+ 468.7s2+ 22850s+ 410000
(45)
with kΓ(Φ, D)−??Pγ?Λk∞= 0.99 <1 guaranteed nominal
performance level. Next, the H∞−DOB controller was
obtained as
K=0.476s2+ 2.976s+ 15.08
s2+ 36.56s
Q=9.596
s+ 9.596
L=0.009s3+ 4.806s2
s3+ 52.12s2+ 443.8s+ 499.3,
(46)
such that kΓ(Φ, D)−?? Pγ?Λk∞= 0.96 <1.
In order to illustrate the performance and practical-
ity of the fixed-order DOB, the Bode magnitude plot of
the Tdis given in Fig. 9 with the results obtained from
H∞−DOB, DOB-PID, and full-order DOB method [24].
It can be observed from Fig. 9 that the obtained distur-
bance estimation performance with the proposed method
is similar to those obtained with the classical H∞ap-
proach. However, the resulting full-order H∞based DOB
controller has the same order as the generalized plant that
is 48 for this simulation example. Therefore, this unstruc-
tured high order DOB complicates the closed-loop control
system implementation and readjustment of the DOB ma-
trix. Time-domain step disturbance input responses of the
Figure 9: Comparison of the Bode magnitude plot of the Tdobtained
from the proposed, full-order H∞[24], and PID-DOB method.
Figure 10: Time-domain comparison between the disturbance re-
jection capabilities and disturbance estimation achievements of the
control methods.
system are given in Fig. 10. In this numerical simulation, a
step disturbance input was applied to the system. The re-
sponses of the feedback control only (without DOB) cases
to the same input are also given in the same figure for
comparison purposes. Fig. 10 also shows the comparison
of the disturbance estimation achievements for both con-
trollers. The simulation results show that the synthesized
H∞−DOB and DOB-PID controllers improve the precise
positioning capability of the closed-loop system and ensure
the required disturbance rejection objective.
5. Conclusion
This paper proposes a synthesis approach for the com-
bined robust DOB loop and main feedback controller in
the presence of mixed uncertainty based on a fixed-order
µ-synthesis algorithm. This algorithm has been adopted to
simultaneously optimize structured plant inverse, Qfilter,
and feedback controller. Therefore, the proposed approach
renders the need for an explicit plant inverse unnecessary
and, thanks to this advantage, can be easily applied to
the NMP systems. The robust stability and performance
conditions of the control system with DOB based 2-DOF
controller structure have been explicitly derived and in-
cluded as a constraint in an optimization problem. Since
the mixed type model uncertainty has been considered in
9

the µ-synthesis based robust controller synthesis process,
the need for robust stability analysis has been eliminated
for the designed closed-loop system. Experimental results
reveal that the proposed fixed-order 2-DOF DOB based
controller design approach improves the disturbance rejec-
tion performance compared to the closed-loop system with
a 1-DOF feedback controller without a DOB loop. Finally,
the simulation case study on an NMP vertical VTOL plat-
form verifies the usefulness of the proposed method for
both NMP systems and robust control problems in which
the main loop controller is known, but an additional DOB
loop is desired to be synthesized.
Future research is directed towards applying the pre-
sented approach to multiple input MIMO systems with
uncertainty. Additionally, the presented methods can be
extended to data-driven fixed-order DOB synthesis theory
to make the DOB design approach more practical.
Acknowledgment
The authors would like to thank the Scientific and
Technological Research Council of Turkey Defense Indus-
tries Research and Development Institute-T ¨
UB˙
ITAK SAGE
for their support in this study.
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