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IMA Journal of Mathematical Control and Information Page 1 of 22
doi:10.1093/imamci/dri000
Robust output feedback control for discrete-time systems with ellipsoidal
uncertainty
ARA SH SADEGHZADEH*
Faculty of Electrical Engineering, Shahid Beheshti University, Tehran, Iran
*Corresponding author: a.sadz@razi.ac.ir
HAMIDREZA MOMENI
Faculty of Electrical and Computer Engineering, Tarbiat Modares University, Tehran, Iran
[Received on October 2014]
This paper provides a new robust control synthesis method for linear uncertain multivariable discrete-
time systems subject to the ellipsoidal parametric uncertainty. This type of the parametric uncertainty
is delivered by classical prediction error identification methods in a full-order model structure. The un-
certainty is represented by linear fractional transformation (LFT). Fixed-order H2and H∞control design
problems are formulated in terms of solutions to a set of dilated linear matrix inequalities (LMIs). The
control designs build on quadratically parameter-dependent Lyapunov functions. Flexible structures for
the auxiliary variables are introduced to improve the performance and reduce the conservatism related
to the fixed-order controller synthesis. The employed structures may be updated by a proposed iterative
procedure. Numerical examples show the effectiveness of the proposed method.
Keywords: System identification; Robust control; Fixed-order controller; Ellipsoidal uncertainty; Dilated
linear matrix inequalities.
1. Introduction
System identification and robust control are two main fields of control systems research. This paper is a
step in an ongoing effort to connect these areas. The classical prediction error (PE) system identification
methods deliver models with ellipsoidal uncertainty, due to the measurement noise, in the case that the
set of parameterized models is flexible enough to capture the true system (Ljung, 1999). The uncertain
model obtained by the identification procedure is a parameterized model whose parameter vector θ∈Rm
is constrained to lie in the following ellipsoidal set
U={θ|(θ−ˆ
θ)TRe(θ−ˆ
θ)61},(1)
where the positive definite matrix Reand the identified vector ˆ
θare obtained by the identification pro-
cedure. This set contains the true system at some (user-chosen) probability level. In this paper, the
problems of robust H2and H∞output feedback control for systems with ellipsoidal uncertainty are
investigated. The problems of disturbance rejection and assessing the robustness with respect to un-
structured uncertainties can be formulated as the H∞problems (Zhou et al., 1995). H∞-optimization
minimizes the L2-induced gain from exogenous inputs to performance outputs. The standard H2prob-
lem is concerned with the minimization of the variance of the regulated performance output due to a
white noise input. H2-optimization removes the stochastics from LQG optimization with the minimiza-
tion of the 2-norm of the closed-loop system (Doyle et al., 1989) and allows the consideration of design
problems that the conventional LQG formulation and solution does not permit (Kwakernaak, 2002).
The author 2008. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.
2 of 22
The problem of robust performance analysis for systems with ellipsoidal uncertainty is addressed
in Biernacki et al. (1987); Guzzella et al. (1991); Iwasaki (1996); Braatz and Drisalle (1998); Bombois
et al. (2001); Chen and Seborg (2002); Barenthin et al. (2008); Bombois et al. (2010); Sadeghzadeh
(2014b).The problem of minimization of the maximum linear quadratic regulator (LQR) cost for all the
plants with parameters in an ellipsoidal set is considered in Lau et al. (1991a,b). The papers Mahon et al.
(1999); Larsson et al. (2011) are focused on the predictive control strategy for systems with ellipsoidal
uncertainty. The robust stabilization problem is considered in Crisalle et al. (1994); Crisalle and Balla-
mudi (1996); Peaucelle and Arzelier (1998); Raynaud et al. (2000); Henrion et al. (2001); Sadeghzadeh
and Momeni (2009) for ellipsoidally uncertain systems. The proposed method in Henrion et al. (2001)
is based on employing an ellipsoidal inner approximation of the stability domain in the polynomial coef-
ficient space to cope with the ellipsoidal uncertainty. The robust fragile controller design is investigated
in Sadeghzadeh and Momeni (2009).The design specification of the method proposed in Raynaud et al.
(2000) is that the closed-loop poles should be inside a stable disc region of a given radius. The paper
Barenthin and Hjalmarsson (2008) presents a method for the robust H∞state feedback control design
to cope with the ellipsoidal uncertainty thanks to the S-procedure. An iterative procedure for the robust
H2state feedback controller design is presented in Kanev et al. (2003). The problem of output feedback
control for uncertain discrete-time systems with input saturation is studied in Song and Wang (2013).
A delay partitioning approach is applied to solve the problem. In Rantzer and Megretski (1994), a full-
order robust controller synthesis method is presented that is based on infinite dimensional Youla-Kucera
parametrization. This approach cannot handle the reduced-order controller synthesis. A finite dimen-
sional convex optimization algorithm that solves the problem proposed in Rantzer and Megretski (1994)
is provided in Ghulchak and Rantzer (2002).
Robust fixed-order controller design is a challenging problem in control theory since full-order con-
trollers indeed impose restrictions on the scope of use in the practical applications (Keel and Bhat-
tacharyya, 1999). In the context of the robust fixed-order controller design, most of the existing ap-
proaches are limited to the uncertain polytopic systems. The results on the robust fixed-order H2control
can be found e.g. in Moreira et al. (2011); Sadeghzadeh and Karimi (2014); Dong and Yang (2007); Ger-
shon and Shaked (2006); Arzelier et al. (2003); Peaucelle and Arzelier (2001a); Sadeghzadeh (2014c).
The robust fixed-order H∞problem is investigated in Dong and Yang (2013); Agulhari et al. (2012);
Sadeghzadeh (2014a); Agulhari et al. (2010); Du and Yang (2008). Most of these approaches are based
on dilated or extended linear matrix inequalities (LMIs) which have been introduced for the robust anal-
ysis and synthesis problems by de Oliveira et al. and Peaucelle et al. (de Oliveira et al., 1999; Peaucelle
et al., 2000; de Oliveira et al., 2002). The main feature of these LMIs is that the Lyapunov variables have
no multiplication relation with the state space matrices by introducing auxiliary matrix variables. These
auxiliary variables provide extra free dimensions in the solution space for the H2and H∞optimization
problems and facilitate the use of parameter dependent Lyapunov functions. It is well known that fixed-
order controller design leads to either a non-convex rank constraint or bilinear matrix inequalities which
are computationally intractable. In the literature, to convexify the fixed-order control design problem
either a block triangular structure on the auxiliary variables is imposed (Dong and Yang, 2007; Du and
Yang, 2008) or auxiliary variables are pre-computed using an initial controller (Agulhari et al., 2012;
Moreira et al., 2011; Sadeghzadeh, 2014a; Sadeghzadeh and Karimi, 2014) that both of them suffer from
inherent conservatism.
To the best of our knowledge, the literature on the robust fixed-order controller synthesis to cope with
the ellipsoidal uncertainty is restricted to Single Input Single Output (SISO) systems. Fixed-order H∞
output feedback control synthesis for SISO systems based on the quadratic stability concept is developed
in Sadeghzadeh and Momeni (2011) using a polynomial method. As an extension to these methods,
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parameter dependent Lyapunov functions are used for fixed-order H∞control design problem for SISO
systems in Sadeghzadeh et al. (2011) thanks to the concept of strictly positive realness (SPRness) of
transfer functions. Combined pole placement with sensitivity function shaping in H2or H∞norm by
fixed-order controllers is considered in Sadeghzadeh (2012) for SISO ellipsoidally uncertain systems.
Based on these motivations, fixed-order control design problem for Multi Input Multi Output (MIMO)
systems with ellipsoidal parametric uncertainty is studied in this paper. The design of both static and
fixed-order dynamic H2and H∞controllers are addressed employing dilated or extended LMIs and using
quadratically parameter-dependent Lyaounov functions. In the proposed method, full structure auxil-
iary variables are utilized for the fixed-order controller design, as the main technical contribution of the
paper, whereas the Lyapunov variables and auxiliary matrices are all considered as decision variables
in the controller design step. It is not necessary to consider a special structure such as block diag-
onal structure or block triangular structure for the auxiliary variables in the presented method which
leads to effectively reducing the related conservatism. In addition, an iterative procedure is proposed to
improve the initially considered structures for the auxiliary variables. The iterations produce a mono-
tonically non-increasing sequence of the guaranteed cost by updating the auxiliary variable structures.
Note that the proposed design procedure does not need either a state feedback controller or an initial
output feedback controller for the design procedure initialization, contrary to the methods of Agulhari
et al. (2012); Moreira et al. (2011); Sadeghzadeh (2014a); Sadeghzadeh and Karimi (2014). The uncer-
tainty of the system is expressed by linear fractional transformation (LFT) in an uncertain part ∆(with
block-diagonal structure) only known to vary in the following set
U={∆=Ip⊗(θ−ˆ
θ)|θ∈U},
where ⊗denotes the Kronecker product (Zhou et al., 1995). Note that in contrast to the extensively
studied polytopic uncertainty which can be modeled as a convex hull of finite number of vertices, the el-
lipsoidal parametric uncertainty cannot be dealt with in this way and it makes the ellipsoidal uncertainty
treatment much harder. The proposed method in this paper is based on the quadratic separation concept
that has been previously employed for the robust stability and performance analysis (see, e.g., Peau-
celle and Arzelier (2001b); Iwasaki and Hara (1998)) as well as robust static output feedback control
design for continuous-time systems (Peaucelle and Arzelier, 2005). We utilize a general parametriza-
tion for the set of multipliers corresponding to the set U, introduced in Barenthin et al. (2008) for the
performance assessment, to formulate the robust controller synthesis in terms of solution to a tractable
LMI-based optimization problem. This way a new fixed-order robust H2and H∞control design method
for ellipsoidally uncertain multivariable systems via LMI formulation is provided in this paper.
The notation is fairly standard. Rn×mis the set of n×mreal matrices. Inis an n×nidentity matrix.
0n×mis an n×mzero matrix. The subscript for the dimension may be dropped if the sizes of matrices
are clear from the context. diag(·)is the block diagonal concatenation of input arguments. H e(A)is
a shorthand notation for A+AT.?AB denotes the Hermitian matrix BTAB. In a symmetric matrix, ?
denotes the transpose of an off-diagonal block. For a matrix Mgiven as follows
M=M11 M12
M21 M22 ,
the upper and lower LFTs with respect to ∆are defined as
Fu(M,∆) = M22 +M21∆(I−M11∆)−1M12,
Fl(M,∆) = M11 +M12∆(I−M22∆)−1M21,
respectively.
4 of 22
2. Problem formulation and preliminaries
Consider the following linear time-invariant uncertain discrete-time system
x(q+1)
z(q)
y(q)
=
A(∆)Bw(∆)Bu
Cz(∆)Dzw(∆)Dzu
Cy(∆)Dyw(∆)
x(q)
w(q)
u(q)
,(2)
where x∈Rnis the state vector, u∈Rnuis the vector of control inputs, w∈Rnwis the vector of
performance (exogenous) inputs, y∈Rnyis the vector of measured outputs and z∈Rnzis the vector
of performance outputs. The matrices are constant matrices of appropriate dimensions. The parametric
uncertainty of the system is modeled through an LFT structure. This representation corresponds to a
fictitious feedback v=∆ron the linear system
x(q+1)
r(q)
z(q)
y(q)
=
A BvBwBu
CrDrv Drw
CzDzv Dzw Dzu
CyDyv Dyw
x(q)
v(q)
w(q)
u(q)
,(3)
where ∆∈U. This way, we have
A(∆) = A+Bv∆aCr,Bw(∆) = Bw+Bv∆aDrw,
Cz(∆) = Cz+Dzv∆aCr,Dzw(∆) = Dzw +Dzv ∆aDrw ,
Cy(∆) = Cy+Dyv∆aCr,Dyw(∆) = Dyw +Dyv ∆aDrw ,
where, to be concise, we adopt the notation
∆a=∆(I−Drv∆)−1.(4)
In the sequel, the uncertain system is assumed to be well-posed, which is classically met. This fact
implies that ∆aalways exists and is unique for each ∆(i.e. I−Drv∆is considered to be invertible).
We start with the simpler static output feedback controller and then the modifications for the case of
dynamic controller are provided in the context of the paper. First, assume that
Dzu =0.
In the case that Dzu is not zero, one can add a low-pass filter with a sufficiently large bandwidth to
the performance outputs to obtain Dzu =0 for the augmented system. Let Tzw(∆)denote the uncertain
closed-loop transfer functions from wto zfor some static output feedback control law
u=Ky,
with the following state space realization
x(q+1)
z(q)=Acl (∆)Bcl (∆)
Ccl(∆)Dcl (∆) x(q)
w(q),(5)
where,
Acl (∆) = [A+Bv∆aCr] + BuK[Cy+Dyv∆aCr],
Bcl (∆) = [Bw+Bv∆aDrw] + BuK[Dyw +Dyv ∆aDrw ],(6)
Ccl(∆) = Cz+Dzv∆aCr,
Dcl (∆) = Dzw +Dzv∆aDrw .
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The problem addressed here is to design a stabilizing controller with H2or H∞performance for the
uncertain closed-loop system.
The following lemmas are required to proceed further.
Lemma 1 (Duan et al., 2006) Consider the transfer function Tzw with the state space realization (Acl ,Bcl ,Ccl ,Dcl ).
Then the following inequalities hold if, and only if, there exist auxiliary matrices F, G and a symmetric
Lyapunov matrix P =PT>0and a matrix Z satisfying
•for kTzwk2
2<γ(H2performance)
G+GT−P GAcl −FTGBcl
?P−He(FAcl )−F Bcl
? ? I
>0,(7)
Z Ccl Dcl
?P0
? ? I
>0,trace(Z)<γ,(8)
•for kTzwk2
∞<γ(H∞performance)
G+GT−P0GAcl −FTGBcl
?γI Ccl Dcl
? ? P−He(FAcl )−F Bcl
? ? ? I
>0.(9)
This lemma introduces dilated LMIs for evaluating the H2and H∞performances. The main feature of
these dilated LMIs is that the Lyapunov variables have no multiplication relation with the state space
matrices through introducing auxiliary variables Gand F. This feature facilitates the using of parameter
dependent Lyapunov functions in the robust synthesis. In what follows, a parametrization for the set of
multipliers corresponding to the uncertainty set Uis introduced.
Lemma 2 (Barenthin et al., 2008) Let Σbe the set of all matrices
Στ=Σ11 Σ12
ΣT
12 Σ22 ,(10)
which satisfy the following condition
I Iτ⊗∆TΣτI
Iτ⊗∆60,∀∆∈U.(11)
Then a subset of Σmay be defined, if we use a specific parametrization for Στ. That is Σ11 ∈R(τp)×(τp)
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an arbitrary negative definite matrix and Σ22 =−Σ11 ⊗Re+ϒ
with ϒ=
0ϒ12 ··· ϒ1(τp)
ϒT
12 0··· ϒ2(τp)
.
.
..
.
.....
.
.
ϒ1(τp)ϒ2(τp)··· 0
and Σ12 =
0Ψ12 ··· Ψ1(τp)
−Ψ12 0··· Ψ2(τp)
.
.
..
.
.....
.
.
−Ψ1(τp)−Ψ2(τp)··· 0
with arbitrary Ψij ∈R1×mand ϒi j =−ϒT
i j ∈Rm×mfor all i,j.
Proof 1 Pre- and post-multiplying of parameterized Στby I Iτ⊗∆Tand its transpose, respec-
tively, results that
I Iτ⊗∆TΣτI
Iτ⊗∆= (1−(θ−ˆ
θ)TRe(θ−ˆ
θ))Σ11.
For all θ∈U , we have (1−(θ−ˆ
θ)TRe(θ−ˆ
θ)) >0, this implies (11).
3. Static output feedback control
To derive the robust performance conditions, we apply the results of Lemma 1 to the uncertain closed-
loop system (5). Assume that Buis of full column rank, and let invertible matrix T, such that T Bu=
[I0]T.T∈Rn×ngenerally is not unique and can be obtained by
T=(BT
uBu)−1BT
u
B⊥
u,(12)
where B⊥
udenotes an orthogonal basis for the null space of Bu, i.e. B⊥
uBu=0.
We consider the following parameter-dependent Lyapunov and auxiliary matrices
P(∆) = I
∆aCrTP
1P
2
PT
2P
3 I
∆aCr,(13)
G(∆) = I
∆aCrTG
G∆T,(14)
F(∆) = I
∆aCrTF
F∆T,(15)
and to facilitate the presentation, we partition G,G∆,Fand F∆in the following blocked matrices:
G= [ G1G2],G∆= [ G∆1G∆2],F= [ F1F2],F∆= [ F∆1F∆2],(16)
where P
1∈Rn×n,P
2∈Rn×(pm),P
3∈R(pm)×(pm),G1,F1∈Rn×nu,G2,F2∈Rn×(n−nu),G∆1,F∆1∈
R(pm)×nu,G∆2,F∆2∈R(pm)×(n−nu).
The state-space characterizations of robust H2and H∞performances are presented in the next theo-
rem.
7 of 22
Theorem 1 For the closed-loop system (5), the following robust performances are guaranteed for all
∆∈U, if there exist symmetric matrices P
1, P
3, Z and matrices P
2and G, G∆, F, F∆given by (16) and
parameterized matrices Σ1,Σ2and Σ3given by (10) satisfying
•for kTzw(∆)k2
2<γ(Robust H2performance)
X>0,(17)
"Y?
CzDzw Dzv Dzv Z#>0,trace(Z)<γ,(18)
•for kTzw(∆)k2
∞<γ(Robust H∞performance)
"X?
0CzDzw 0Dzv Dzv γI#>0,(19)
P
1P
2
PT
2P
3+?Σ1CrDrv
0I>0,(20)
where
X=M1M2
MT
2M3+?Σ3diag(Cr,Cr,Drw)I3⊗Drv
0I,
Y=M4M5
MT
5M6+?Σ2diag(Cr,Drw)I2⊗Drv
0I,(21)
with
M1=
He(GT )−P
1GT A +ΦCy−TTFTGT Bw+ΦDyw
?P
1−He(F T A +ΨCy)−FT Bw−ΨDyw
? ? I
,(22)
M2=
−P
2+TTGT
∆GT Bv+ΦDyv −TTFT
∆GT Bv+ΦDyv
(G∆TA +Φ∆Cy)TP
2−F T Bv−ΨDyv −(F∆T A +Ψ∆Cy)T−F T Bv−ΨDyv
(G∆T Bw+Φ∆Dyw)T−(F∆TBw+Ψ∆Dyw)T0
,(23)
M3=
−P
3G∆T Bv+Φ∆Dyv G∆T Bv+Φ∆Dyv
?P
3−He(F∆T Bv+Ψ∆Dyv )−F∆T Bv−Ψ∆Dyv
? ? 0
,(24)
M4=P
10
0Inw,M5=P
20
0 0 ,M6=P
30
0 0(pm)×(pm).(25)
also consider the following change of variables
Φ=G1K,Ψ=F1K,Φ∆=G∆1K,Ψ∆=F∆1K.(26)
Proof 2 See Appendix A.
8 of 22
3.1 Controller design procedure
Since controller Khas multiplication relations with the auxiliary matrices, Theorem 1 cannot be used
directly for the controller synthesis (see eq. (26)). To overcome this non-convexity, one can employ
a bisection design procedure. First, (26) can be ignored to find initial values for G1,F1,G∆1and F∆1
(i.e., Φ,Ψ,Φ∆and Ψ∆are considered as free decision variables in the inequality conditions of Theorem
1). Then, employing the obtained values as the structures for the auxiliary variables, one can design a
robust controller. Consider the following three-step algorithm for the controller synthesis:
Step 1- For one of the following cases
–Case 1: Consider Φ,Φ∆,Ψand Ψ∆as free decision variables ((26) is ignored),
–Case 2: Consider Φ,Φ∆as free decision variables and impose the constraints F1=0, F∆1=
0, Ψ=0 and Ψ∆=0,
find a feasible solution for the LMI constraints (17)-(18) or (19)-(20) for the design of H2or H∞
controller, respectively, and then let ¯
G1=G1,¯
F1=F1,¯
G∆1=G∆1,¯
F∆1=F∆1.
Step 2- Consider
G1=¯
G1H,G∆1=¯
G∆1H,F1=¯
F1H,F∆1=¯
F∆1H,
Φ=¯
G1Q,Φ∆=¯
G∆1Q,Ψ=¯
F1Q,Ψ∆=¯
F∆1Q,
where, H∈Rnu×nuand Q∈Rnu×nyare free decision variables, then
min
γ,H,Q,G2,F2,G∆2,F∆2,P
1,P
2,P
3,Z,Σ1,Σ2,Σ3
γ(27)
s.t.(17)−(18)(for H2controller),
or
(19)−(20)(for H∞controller).
The required controller can be recovered by K=H−1Q.
Step 3- The obtained guaranteed cost γcan be improved if we update the values of ¯
G1,¯
G∆1,¯
F1
and ¯
F∆1in Step 2. Consider
Φ=G1K,Ψ=F1K,Φ∆=G∆1K,Ψ∆=F∆1K,
for the obtained controller Kfrom Step 2. Then
min
γ,G,F,G∆,F∆,P
1,P
2,P
3,Z,Σ1,Σ2,Σ3
γ(28)
s.t.(17)−(18)(for H2controller),
or
(19)−(20)(for H∞controller).
and let ¯
G1=G1,¯
F1=F1,¯
G∆1=G∆1,¯
F∆1=F∆1. Now, go to Step 2 again.
9 of 22
Similar methods have been used in the literature to solve the nonconvex matrix inequalities, see,
e.g. El Ghaoui et al. (1997); Rotea and Iwasaki (1994). However, the merit of the proposed algorithm
over the existing approaches is that there is no need to fix any decision variable, instead, we determine
the structure of the auxiliary variables with the unconstrained Lyapunov matrices. Imposing a special
structure on the auxiliary matrices is an innovative way to overcome the non-convexity related to the
robust controller or filter synthesis. See e.g. de Oliveira et al. (2002); Dong and Yang (2007) for the
robust controller design, or Duan et al. (2006) for the robust filter design. These methods all consider a
fixed predefined structure for the key auxiliary variables. In this note, we iteratively improve the chosen
structure.
Remark 1 Step 1 is just an initial step to choose a structure for the key auxiliary variables. Case 1 is a
necessary condition for the existence of the desired controller, using the proposed method. This means
that if it fails, we should consider a higher order controller (see Section 4). Case 2 is not too restrictive
in comparison with Case 1, since, F in Lemma 1 just provides extra free dimensions in the solution
space. However, even in this case F2and F∆2are free variables. Nothing can be said about which case
provides a better solution. It should be mentioned that, we do not minimize γin Step 1 but we just look
for a feasible solution. Note that we do not pretend that the obtained structure from Step 1 is the best
one, however, the restriction to have upper or lower triangular structure for the auxiliary variables is
removed, contrary to the presented methods in de Oliveira et al. (2002); Dong and Yang (2007); Du and
Yang (2008).
Remark 2 Step 3 is indeed a robust performance analysis procedure which results a guaranteed cost
γ0for a designed controller K0. Note that the optimization problem (27) using the obtained auxiliary
variables in Step 3 results a new controller K with an upper bound γnot higher than γ0, since in this
case for H =I and Q =K0(27) is equivalent to (28). Thus the improvement of the obtained controller
by the proposed iterations is guaranteed. In the context of the robust performance analysis, Step 3
can be considered as an alternative for the method of Sadeghzadeh (2014b) as well as a discrete-time
counterpart for Peaucelle and Arzelier (2001b).
Remark 3 The proposed iterative procedure can be initiated with an initial stabilizing controller (if
there exists) from Step 3. This initial controller can be a robust stabilizing controller designed for
the uncertain system using any existing methods. It can be either state feedback or output feedback
controller or one can try the designed controller for an arbitrary system in the model set.
Remark 4 Note that the algorithm defined above just provides sufficient conditions for the fixed-order
controller design. It means that no one can guarantee that the proposed iterative algorithm results in
a feasible solution even if there exists any. However, if an initial robust output feedback stabilizing
controller is available (see Remark 3) then the algorithm always results in a feasible solution with a
lower guaranteed cost γin comparison with that of the initial controller.
4. Dynamic output feedback control
Suppose that the problem is to design a dynamic controller of order nkwith the following state space
realization xk(q+1)
u(q)=Kxk(q)
y(q)with K=AkBk
CkDk,
for the system given by (2), where Ak∈Rnk×nkand Bk,CkDkhaving compatible dimensions. This
controller results the following closed-loop state space matrices
10 of 22
Acl (∆) = A(∆) + BuDkCy(∆)BuCk
BkCy(∆)Ak,Bcl(∆) = Bw(∆) + BuDkDyw (∆)
BkDyw(∆)
Ccl(∆) = Cz(∆)0,Dcl (∆) = Dzw (∆).
Following a well-developed methodology (see, e.g., Agulhari et al. (2012); El Ghaoui et al. (1997)),
it is easy to see that these matrices can be rewritten as
Acl (∆) = A0
0 0 +Bv
0∆aCr0+0Bu
I0K 0I
Cy0+0
Dyv ∆aCr0,
Bcl (∆) = Bw
0+Bv
0∆aDrw+0Bu
I0K 0
Dyw +0
Dyv ∆aDrw,
Ccl =Cz0+Dzv∆aCr0,
Dcl (∆) = Dzw +Dzv∆aDrw .
Now, due to comparison between these closed-loop state space matrices with those of (6), one can see
that the design of a dynamic controller can be regarded as the design of a static controller Kfor an
augmented system given by (3) with the following state space matrices
A BvBwBu
CrDrv Drw
CzDzv Dzw Dzu
CyDyv Dyw
=
A0BvBw0Bu
0 0nk0 0 I0
Cr0Drv Drw
Cz0Dzv Dzw 0Dzu
0I0 0
Cy0Dyv Dyw
.
5. Numerical illustrations
In this section two numerical examples will be given for illustrating the effectiveness of the proposed
method. The computer used is an Intel Core i7-3770 (3.40 GHz),8GB RAM, Windows 7 Ultimate SP1.
The LMIs are solved using YALMIP (L ¨
ofberg, 2004) under MATLAB 7.14. by using SeDuMi (Sturm,
1999) as the LMI solver.
5.1 Example 1
Let us consider a model of a mechanical system with two masses and two springs borrowed from Iwasaki
(1996). A discretized model of this system is presented in Agulhari et al. (2010). This model presents
four states (positions and speed of the masses m1and m2). k1and k2are the stiffness of the springs
and c0is the viscous friction coefficient. The parameters of the system are given by m1=2, m2=1,
k2=0.5 and consider the ellipsoidal uncertainty for the parameters k1and c0given by (1) with θ=
[k1c0]T,ˆ
θ= [ ˆ
k1ˆc0]T= [ 2.5 3.5]T,Re=diag(1/1.52,1/2.52)and ∆=I3⊗(θ−ˆ
θ). The
dynamic matrices of the system are given by
A=
1 0 0.1 0
0 1 0 0.1
−0.1(ˆ
k1+k2)
m1
0.1k2
m11−0.1 ˆc0
m10
0.1k2
m2−0.1k2
m20 1 −0.1 ˆc0
m2
,
11 of 22
1 2 3 4 5 6 7 8
9.5
10
10.5
11
11.5
12
12.5
13
13.5
Iteration
Upper−bound
FIG. 1. The upper bound on the H∞performance versus iterations for example 1.
123456
1
1.5
2
2.5
3
3.5
4
C0
k1
FIG. 2. Inner polytopic approximation with 8 vertices for the ellipsoidal set related to example 1.
Bv=
0 0 0 0 0 0
0 0 0 0 0 0
−0.1
m10 0 −0.1
m10 0
0 0 0 0 0 −0.1
m2
,Bw=
0
0.1
0.1
0
,Bu=
0
0
0.1
m1
0
Cr=
100
000
010
001
T
,Cz=
0
1
0
0
T
,Cy=
0 0
0 0
1 0
0 1
T
,
Drv Drw
Dzv Dzw Dzu
Dyv Dyw
=0.
The proposed method of Section 3.1 results the following static output feedback H∞controller
K∞=−17.7815 14.9196 ,with γ=9.55072
after 8 iterations of the auxiliary variable structure updates. The design procedure is initiated with the
following auxiliary variables obtained by Case 2 in Step 1.
¯
G1=6.5280 −0.4555 9.1207 −12.8018 T
¯
G∆1=−0.8688 0.0154 0.2293 −0.0005 0.1575 −0.719 T
The monotonic decreasing of the upper-bound for the H∞performance (√γ) versus iterations is shown
in Fig. 1. For the comparison purposes with the existing approaches, the ellipsoidal set is approximated
with an inner polytope with 8 vertices shown in Fig. 2.
Now, the presented methods in de Oliveira et al. (2002); Du and Yang (2008); Agulhari et al. (2010);
Chang and Yang (2014) are applied on the inner polytopic approximation. To evaluate the conservatism,
12 of 22
Table 1. The comparison of the H∞performance for example 1.
Methods √γw.c.p. NvNrTime(sec.)
proposed method 9.5507 9.5426 281 49 6.93
Chang and Yang (2014) 15.5849 15.7086 404 576 208.1
Agulhari et al. (2010) 11.3283 11.1445 172 1320 2.46
Du and Yang (2008) Fail - 207 80 -
de Oliveira et al. (2002) Fail - 95 84 -
Table 2. The comparison of the H2performance for example 1.
Methods √γw.c.p. NvNrTime(sec.)
proposed method 0.7540 0.7518 327 61 44.29
Moreira et al. (2011) 0.7976 0.7907 228 1328 3.21
Gershon and Shaked (2006) Fail - 202 265 -
Dong and Yang (2007) Fail - 2655 952 -
de Oliveira et al. (2002) Fail - 103 124 -
lower bounds for the worst case performance (w.c.p.) are calculated for the designed controllers by
griding the ellipsoidal set (with about 5100 points). Additionally, the numerical complexity associated
with the considered methods is analysed by the Nvnumber of scalar variables, Nrnumber of LMI rows
and the required computational time. None of the methods presented in de Oliveira et al. (2002); Du and
Yang (2008) provide feasible solution for the polytopic approximation, however, the presented method
in Agulhari et al. (2010) results a controller with the guaranteed cost γ=11.32832. By applying the
parameter dependent linear matrix inequalities approach of Oliveira and Peres (2007) on the presented
method in Chang and Yang (2014), one can obtain a robust static output feedback H∞controller with
the guaranteed cost γ=15.58492. This minimum value is resulted by an iterative search in the two-
dimensional space for two scalar parameters αand β, considering about 100 points where the optimal
values α=12.91 and β=0.01 are obtained. For this method, the guaranteed cost is smaller than the
related w.c.p., that is because of utilizing the inner polytopic approximation for the actual ellipsoidal
uncertainty set. All the results are summarized in Table 1.
Now, we consider the robust H2control problem. The proposed method results the following con-
troller
K2=−28.3482 17.3435 ,with γ=0.75402
after 40 iterations of the auxiliary variable structure updates. Now, the methods of de Oliveira et al.
(2002); Gershon and Shaked (2006); Dong and Yang (2007); Moreira et al. (2011) are considered for
the comparison, employing the mentioned inner polytopic approximation. The methods of de Oliveira
et al. (2002); Gershon and Shaked (2006) and Dong and Yang (2007) fail whereas the presented method
in Moreira et al. (2011) results a static controller with the guaranteed cost γ=0.79762. Note that the
first obtained controller by the proposed method (without iterations) yields γ=0.79672which is less
than that of Moreira et al. (2011). The results are summarized in Table 2.
13 of 22
5.2 Example 2
Consider the following system
G(q,θ) = 1
1+θ1q−1θ2q−1θ3q−1
θ4q−1θ5q−1,
borrowed from Sadeghzadeh (2014b) with the ellipsoidal uncertainty set, given by (1), resulted from
system identification procedure. Where θ= [ θ1θ2θ3θ4θ5]T,
Re=
721.636 −23.653 47.121 21.962 −22.171
−23.653 144.877 −2.923 −1.624 0.033
47.121 −2.923 147.813 0.033 −1.657
21.962 −1.624 0.033 156.969 −3.167
−22.171 0.033 −1.657 −3.167 160.149
,
and ˆ
θ= [ −0.7991 0.1952 −0.4465 −0.4090 0.3160 ]T.
The considered problem is to design a first order controller with the robust H∞performance
W(I+G(θ)K)−1
2
∞<γ,with W=I2⊗0.3(q−0.38)
q−0.97 ,
for the uncertain system.
To obtain the dynamic state space matrices, let Sbe the lower LFT of W(I+G(θ)K)−1with respect
to Kas follows (Zhou et al., 1995):
S=
W−G(θ)
W−G(θ)
.
Moreover, consider SGas the upper LFT of Swith respect to G(θ), given by
SG=
00I2
−I
−I W0
W0
Additionally, consider G∆as the upper LFT of G(θ)with respect to ∆=I2⊗(θ−ˆ
θ)as follows
(Sadeghzadeh, 2014b):
G∆=0 0
E21 E22 +I
E21∆0(I−E11 ∆0)−1E11 E12 ,
with ∆0=I2⊗ˆ
θ, where
E11 E12
E21 E22 =
−I2⊗(q−1eT
1)q−1I
eT
2eT
3
eT
4eT
502×2
,
is the upper LFT of G(θ)with respect to I2⊗θ, with eidefining the ith unit vector .
Now the state space matrices (3) is obtained by the Redheffer star-product (Zhou et al., 1995) of
the state space realizations of SGby G∆. These LFT representations are illustrated in Fig. 3 and the
obtained state space matrices are given in Appendix B.
14 of 22
K
K
S
S
G
G(θ)
K
S
G
G
∆
∆
G(θ)
S
z
z
w
w
w z
y
y
y
u
v
u
u
r
FIG. 3. The LFT representations for example 2.
REFERENCES 15 of 22
Now, the proposed method in Section 3.1 using the modifications provided in Section 4 results the
following controller
K=AkBk
CkDk=
0.9901 0.0001 0.0081
−43.5938 0.5156 −1.7468
−33.4174 −2.6287 −1.5508
,
with γ=0.83662. The procedure is initiated with Case 1 in Step 1. The obtained controller results the
worst case performance γw.c.p.=0.83562computed by the griding approach with about 12000 points
for the whole ellipsoidal set that shows a low level of conservatism for the obtained value γ.
Let ¯
G1=¯
F1= [ I0]Tand ¯
G∆1=¯
F∆1= [ I0]Tbe considered in Step 2 for the controller
synthesis. This assumption is equivalent to consider a triangular structure for the auxiliary variables
to convexify the problem similar to the methods of de Oliveira et al. (2002); Dong and Yang (2007);
Du and Yang (2008). But it cannot provide a feasible solution for this example. This reveals that the
obtained structure by Step 1 is more efficient than the triangular structure for the auxiliary variables at
least for this system.
Now, suppose that the synthesis procedure to be initiated from Step 3 by a given controller
K0=
0.97 1.202 −0.066
−0.172 −1.399 −1.393
−0.142 −0.705 2.299
that is the H∞controller designed by HIFOO (Popov et al., 2010) for the nominal system G(q,ˆ
θ)with
the performance γ=0.62762. The guaranteed cost γ=1.12772is obtained by the griding method for
the whole uncertain system for this initial controller. Step 3 results γ=1.12912for the initial controller
that shows a low level of conservatism related to the robust analysis condition (28). Some iterations
between Step 3 and Step 2 result a new robust controller with the guaranteed cost γ=0.83262.
6. conclusion
The problem of robust fixed-order output feedback control for time-invariant discrete-time systems with
ellipsoidal uncertainty has been considered. Recently developed dilated LMIs to assess the H2and
H∞performances are employed. Along with the parameter-dependent Lyapunov functions, a general
parametrization for the set of multipliers corresponding to the ellipsoidal uncertainty is employed. This
way, a tractable optimization problem is obtained for the controller synthesis. An iterative procedure has
been proposed for updating the structure of the auxiliary variables to obtain the monotonic decreasing of
the robust performance guaranteed cost. Numerical comparisons have shown that the proposed method
is indeed promising for achieving less conservative results. The proposed method of this paper can easily
be modified for the uncertain discrete-time polytopic systems employing an appropriate parametrization
for the set of multipliers.
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A. Proof of Theorem 1
To obtain the robust performance conditions, Lemma 1 is applied on the uncertain closed-loop system
(5). Firs, note that for the auxiliary matrices given in (15), we get
G(∆)Acl (∆) =GTA +ΦCy+ (GT Bv+ΦDyv )(∆aCr)+(∆aCr)T(G∆TA +Φ∆Cy)
+ (∆aCr)T(G∆T Bv+Φ∆Dyv)(∆aCr),
F(∆)Acl (∆) =FT A +ΨCy+ (FT Bv+ΨDyv )(∆aCr)+(∆aCr)T(F∆T A +Ψ∆Cy)
+ (∆aCr)T(F∆T Bv+Ψ∆Dyv)(∆aCr),
G(∆)Bcl (∆) =GT Bw+ΦDyw + (GT Bv+ΦDyv)(∆aDrw )+(∆aCr)T(G∆T Bw+Φ∆Dyw)
+ (∆aCr)T(G∆T Bv+Φ∆Dyv)(∆aDrw ),
F(∆)Bcl (∆) =FT Bw+ΨDyw + (F T Bv+ΨDyv )(∆aDrw)+(∆aCr)T(F∆T Bw+Ψ∆Dyw)
+ (∆aCr)T(F∆T Bv+Ψ∆Dyv)(∆aDrw),
20 of 22 REFERENCES
with Φ,Φ∆,Ψand Ψ∆are given by (26).
Proof of the robust H2performance:
Employing the parameter-dependent Lyapunov matrix given in (13) and taking into account that
diag(∆aCr,∆aCr,∆aDrw) = (I3⊗∆a)L1with L1=d iag(Cr,Cr,Drw),
condition (7) can be rewritten as
I
(I3⊗∆a)L1TM1M2
MT
2M3 I
(I3⊗∆a)L1>0,(29)
where M1,M2,M3are given in (22)-(24) which are constant matrices and do not depend on ∆.
Now, ∆can be written as
∆=∆(I−Drv∆) + ∆Dr v∆,
and post-multiplying by (I−Drv∆)−1that is invertible (based on the well-posedness of the system) and
using (4), we obtain ∆a=∆+∆Drv∆a. Then, we can immediately obtain the following constraint for
all ∆∈U
I3⊗∆−IL1I3⊗Drv
0I I
(I3⊗∆a)L1=0,(30)
which can be verified by a direct multiplication and taking into account (I3⊗∆)(I3⊗Drv)(I3⊗∆a) =
I3⊗(∆Drv∆a).
Now, Similar to the method of Peaucelle and Arzelier (2001b); Sadeghzadeh (2014b), application of
Finsler’s Lemma (de Oliveira and Skelton, 2001) to (29) and (30) implies the existence of a parameter-
dependent scalar α(∆)such that
M1M2
MT
2M3+?Σ(∆)L1I3⊗Drv
0I>0,(31)
where,
Σ(∆) = α(∆)I3⊗∆T
−II3⊗∆−I.(32)
Note, without loss of generality, we can assume that Σ(∆)>0 for all ∆∈U. Therefore, there exists a
constant Σsuch that
Σ(∆)>Σ,∀∆∈U.(33)
Thus a sufficient condition for the satisfaction of (31) can be given as follows:
X=M1M2
MT
2M3+?ΣL1I3⊗Drv
0I>0,(34)
subject to
I I3⊗∆TΣI
I3⊗∆60,∀∆∈U.(35)
This latter inequality is obtained by once more applying Finsler’s Lemma on (33). Since, (35) is not a
finite-dimensional inequality, one can restrict attention to a subset for all of the matrices Σsatisfy (35),
in expense of some conservatism. Lemma 2 using a special parametrization for Σ, introduces such a
REFERENCES 21 of 22
subset that is a general parametrization for the set of multipliers corresponding to set U. This way, the
inequality (17) is obtained.
Additionally, condition (8) for the closed-loop system (5), using the Schur complement formula
(Boyd et al., 1994) can be written as
Z>0,trace(Z)<γ,(36)
P(∆)0
0I−?Z−1Ccl(∆)Dcl (∆)>0.(37)
One can see that (37) can be rewritten as
? M4M5
MT
5M6−?Z−1CzDzw Dzv Dzv I
(I2⊗∆a)L2>0,(38)
where M4,M5and M6are given in (25). Following the same line as in the proof of (17), we obtain the
following sufficient condition
Y−?Z−1CzDzw Dzv Dzv >0,(39)
for the satisfaction of (8) for the closed-loop system (5). Yis given in (21).
Once again applying the Schur complement formula to (39) and taking into account (36) implies
(18). Note that P(∆)>0 is involved in (18) and this ends the proof.
Proof of the robust H∞performance:
The proof is the same as that of the H2performance and for the sake of brevity we just give a sketch
of that.
Using the Schur complement formula, it is easy to see that condition (9) is equivalent to
G+GT−P GAcl −FTGBcl
?P−He(FAcl )−F Bcl
? ? I
−?(γ−1I)0Ccl Dcl >0.
This condition for the uncertain closed-loop system (5) can be written as
? M1M2
MT
2M3−?(γ−1I)0CzDzw 0Dzv Dzv I
(I3⊗∆a)L1>0,(40)
and following the same line as in the proof of (17), one can obtain
X−?(γ−1I)0CzDzw 0Dzv Dzv >0,(41)
as a sufficient condition for the satisfaction of (9) for the closed-loop system (5). Then, by using the
Schur complement formula, it can be reformulated as (19).
Additionally, condition P(∆)>0 is equivalent to
P(∆) = I
∆aCrTP
1P
2
PT
2P
3 I
∆aCr>0,
that following the same line as in the proof of (17), one can obtain (20) as a sufficient condition for the
satisfaction of P(∆)>0. This concludes the proof.
22 of 22 REFERENCES
B. The state space matrices for Example 2
A BvBwBu
CrDrv Drw
CzDzv Dzw Dzu
CyDyv Dyw
=
0.8 0 0 0 0.7071 0 0 0 0 0 0 0 0 0 0 0 −0.7071 0
0 0.8 0 0 0 0 0 0 0 −0.7071 0 0 0 0 0 0 0 0.7071
0 0 0.97 0 0 0 0 0 0 0 0 0 0 0 0.5 0 0 0
0 0 0 0.97 0 0 0 0 0 0 0 0 0 0 0 0.5 0 0
−1.4142 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 1.4142 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0.2828 0.7071 0.354 0 0 −1 0 0 0 0 0 −1 0 0 0.3 0 0 0
−0.5657 −0.4243 0 0.354 0 0 0 −1 0 0 0 0 0 −1 0 0.3 0 0
0.2828 0.7071 0.354 0 0 −1 0 0 0 0 0 −1 0 0 0.3 0
−0.5657 −0.4243 0 0.354 0 0 0 −1 0 0 0 0 0 −1 0 0.3
