LPV hard constraints IJCAS resubmit3 · nored, such as space telescope pointing [21] and machine...
Transcript of LPV hard constraints IJCAS resubmit3 · nored, such as space telescope pointing [21] and machine...
Submission to International Journal of Control, Automation, and Systems 1
Optimal LPV control with hard constraints
Andrew White, Guoming Zhu, and Jongeun Choi
Abstract: This paper considers the optimal control of polytopic, discrete-time linear parameter
varying (LPV) systems with a guaranteed ℓ2 to ℓ∞ gain. Additionally, to guarantee robust stability
of the closed-loop system under parameter variations, H∞ performance criterion is also consid-
ered as well. Controllers with a guaranteed ℓ2 to ℓ∞ gain and a guaranteed H∞ performance
(ℓ2 to ℓ2 gain) are a special family of mixed H2/H∞ controllers. Normally, H2 controllers are
obtained by considering a quadratic cost function that balances the output performance with the
control input needed to achieve that performance. However, to obtain an optimal controller with a
guaranteed ℓ2 to ℓ∞ gain (closely related to the physical performance constraint), the cost function
used in the H2 control synthesis minimizes the control input subject to maximal singular-value
performance constraints on the output. This problem can be efficiently solved by a convex opti-
mization with linear matrix inequality (LMI) constraints. The main contribution of this paper is
the characterization of the control synthesis LMIs used to obtain an LPV controller with a guar-
anteed ℓ2 to ℓ∞ gain and H∞ performance. A numerical example is presented to demonstrate the
effectiveness of the convex optimization.
Keywords: Linear parameter varying (LPV) systems, hard constraints, ℓ2 to ℓ∞ gain, linear matrix
inequality (LMI), LPV control.
1. INTRODUCTION
The design of multi-objective, mixed H2/H∞ controllers
has been a topic of interest for sometime [1–9]. The goal
of using both H2 and H∞ performance criteria is to de-
sign a controller which can meet multiple performance
objectives. In [1] and [2] mixed H2/H∞ control was
introduced by minimizing the H2 norm of a closed-loop
transfer function subject to an H∞ norm constraint of an-
other closed-loop transfer function. In [3], mixed H2/H∞
state-feedback and output-feedback controllers were de-
signed for continuous-time systems by using a convex op-
timization approach to solve the coupled nonlinear matrix
Riccati equations and in [4] a similar approach is used for
discrete-time systems. The state-feedback H2/H∞ design
with regional pole placement was addressed by [5] using
the linear matrix inequality (LMI) approach. In [6] and
[7], the LMI approach for multi-objective control synthe-
sis for output-feedback controllers is presented. In [10], an
extra instrumental variable was added to the LMI stability
conditions to build a parameter dependent Lyapunov func-
tion capable of proving the stability of uncertain linear-
time-invariant (LTI) systems. The new extended LMI con-
ditions in [10] were used in [8] to develop H2 and H∞
LMI conditions for linear state-feedback and output-feedback
Andrew White, Guoming Zhu, and Jongeun Choi are withthe Department of Mechanical Engineering, Michigan StateUniversity, East Lansing, MI 48824 (e-mail: {whitea23,zhug, jchoi}@egr.msu.edu)
Guoming Zhu, and Jongeun Choi are also with the Depart-ment of Electrical Engineering, Michigan State University, EastLansing, MI 48824
controllers for uncertain LTI systems. The extended LMI
conditions provided by [8] were utilized in [11] and [12]
to develop linear parameter varying (LPV) static output
feedback controllers that meet H2 [11] and H∞ [12] per-
formance bounds for linear time-varying (LTV) systems
with polytopic uncertainty. The results presented in [11]
and [12] were extended in [9] to cover the synthesis of
multi-objective H2/H∞ gain-scheduled output feedback
controllers. These LPV control techniques have been suc-
cessfully applied to engine system applications [13–16].
Gain scheduling controllers designed using the LPV method
have traditionally included H∞ performance constraints.
This is largely due to the fact that H∞ controllers can pro-
vide robust stability margins that H2 controllers cannot
provide [17]. However, since the H∞ norm is defined as
the root-mean-square gain, or ℓ2 to ℓ2 gain, from the ex-
ogenous input to the regulated output, controllers designed
with only H∞ performance constraints are not suitable for
applications when hard constraints on responses or actua-
tor signals must be met.
When hard constraints on responses or actuator signals
must be met, a controller with a guaranteed ℓ2 to ℓ∞ gain
is required, which is a special type of H2 controller [18].
The ℓ2-ℓ∞ filtering has been a recent topic of interest for
various nonlinear and stochastic systems [19, 20]. A con-
troller with a guaranteed ℓ2 to ℓ∞ gain provides hard bounds
on the regulated output while minimizing the control in-
put as much as possible. This problem was solved for LTI
systems in [18], where it is referred to as the output co-
variance constraint (OCC) problem. The OCC problem
2 Submission to International Journal of Control, Automation, and Systems
defined in [18] is to find a controller for a given system to
minimize the weighted control input cost subject to a set of
output constraints. The OCC problem has two interesting
interpretations: stochastic and deterministic. The stochas-
tic interpretation is obtained by first assuming that the H2
exogenous inputs are uncorrelated zero-mean white noises
with a given intensity. Then the OCC problem minimizes
the weighted control input covariance subject to the output
covariance constraints, such that the constraints are inter-
preted as constraints on the variance of the performance
variables. The deterministic interpretation is obtained by
assuming that the H2 exogenous inputs are unknown but
belong to a bounded ℓ2 energy set. Then the OCC prob-
lem minimizes the weighted control input while ensuring
that the maximum singular values, or ℓ∞ response, of the
regulated outputs are less than the corresponding output
constraints. In other words, the OCC problem is the prob-
lem of minimizing the weighted sum of worst-case peak
values on the control signal subject to the constraints on
the worst-case peak values of the performance variables.
This interpretation is important in applications where hard
constraints on responses or actuator signals cannot be ig-
nored, such as space telescope pointing [21] and machine
tool control. For both interpretations, a solution to the
OCC control problem results in a controller with a guar-
anteed ℓ2 to ℓ∞ gain.
The idea of an LPV controller design with a guaranteed
ℓ2 to ℓ∞ gain is not an entirely new concept and has been
previously studied before. In [22], the ℓ2 to ℓ2 and ℓ2 to
ℓ∞ gains of LPV sampled-data systems was investigated.
LMI conditions for the synthesis of output-feedback con-
trollers that provide the closed loop sampled-data system
with the desired gain were provided as well. However,
all of the LMI conditions provided in [22] are infinite di-
mensional, such that gridding of the parameter space is
required to obtain a finite dimensional set of LMI condi-
tions. In contrast, the LMI conditions presented in this pa-
per are finite dimensional such that parameter space grid-
ding is not required.
The main contributions of this paper are the guaranteed
ℓ2 to ℓ∞ gain controller synthesis LMIs for gain-scheduled
state-feedback and dynamic output-feedback control for
discrete-time polytopic LPV systems in Section 4.. When
these LMIs are satisfied, the optimal state-feedback or dy-
namic output-feedback LPV controller obtained guaran-
tees that for a finite disturbance energy, hard constraints
on the regulated output are met. The guaranteed ℓ2 to ℓ∞
gain is achieved by modifying H2 control synthesis LMIs
provided by [9] to minimize the weighted control input
cost while ensuring the output covariances meet the per-
formance constraints. A preliminary version of this paper
has been included as a chapter in [16].
The paper is organized as follows. The H∞ and guar-
anteed ℓ2 to ℓ∞ gain performances are provided in Sec-
tion 2. Then, in Section 3, the modeling of the uncer-
tainty domain where the time-varying barycentric coordi-
nates takes their values is presented. The controller syn-
thesis conditions for both state-feedback and dynamic out-
put feedback are presented in Section 4. In section 5, a nu-
merical example is presented for both state-feedback and
output-feedback control to illustrate the performance of
the control synthesis LMIs. Conclusions of this work are
given in Section 6.
2. PERFORMANCE OF DISCRETE-TIME
SYSTEMS WITH HARD CONSTRAINTS
Consider the closed-loop, asymptotically stable, discrete-
time LPV system H with the following finite-dimensional
state space realization:
H :=
{
x(k+ 1) = A (αk)x(k)+B(αk)w(k)
z(k) = C (αk)x(k)+D(αk)w(k)(1)
where x is the state, w is the exogenous input, and z is the
system output. The matrices A (α(k)), B(α(k)), C (α(k)),and D(α(k)) belong to the polytope
D=
{
(A ,B,C ,D)(α(k)) : (A ,B,C ,D)
=N
∑i=1
αi(k)(A ,B,C ,D)i,α(k) ∈ ΛN
}
,
(2)
where Ai, Bi, Ci, and Di are the vertices of the polytope
and α(k) ∈ RN is the vector of time-varying barycentric
coordinates lying in the unit simplex ΛN , given by
ΛN =
{
ζ ∈ RN :
N
∑i=1
ζi = 1,ζi ≥ 0, i = 1, · · · ,N
}
. (3)
2.1. H∞ Performance
The H∞ performance of system H, given by (1), is de-
fined as
‖H‖∞ = sup‖w(k)‖2 6=0
‖z(k)‖2
‖w(k)‖2
(4)
with w(k) ∈ ℓr2 and z(k) ∈ ℓp
2 . Based on the bounded real
lemma, an upper bound for the H∞ performance of system
H can be computed using an extended LMI characteriza-
tion, as shown in the following lemma given by [9, 23].
Lemma 1: Consider the system H given by (1). If
there exist a bounded matrix G(αk)∈Rn×n and a bounded
symmetric positive-definite matrix P(αk) ∈ Rn×n, for all
αk ∈ΛN , such that the LMI (5) is satisfied, then the system
H is exponentially stable and
‖H‖∞ ≤ infP(αk),G (αk),η
η . (6)
Submission to International Journal of Control, Automation, and Systems 3
P(αk+1) A (αk)G (αk) B(αk) 0
G (αk)T A (αk)
T G (αk)+G (αk)T −P(αk) 0 G (αk)
T C (αk)T
B(αk)T 0 ηI D(αk)
0 C (αk)G (αk) D(αk) ηI
> 0, (5)
2.2. Minimum Energy Control with Guaranteed ℓ2 to ℓ∞
Gain
To define the minimum energy control with guaranteed
ℓ2 to ℓ∞ gain problem for the system H, the following as-
sumptions are first made:
1. The system output z(k) is partitioned into
z(k) = [zp(k)T , zu(k)
T ]T ,
2. and the feed-through matrix D(αk) = 0
such that the system output z(k) is given by
z(k) :=
[zp(k)zu(k)
]
=
[Cp(αk)Cu(αk)
]
x(k) (7)
where the vector zp(k)∈Rc contains all the variables whose
dynamic responses are of interest and the vector zu(k) con-
tains the weighted control variables to be minimized. De-
fine the ℓ2 and ℓ∞ norms as
‖zp‖2∞ := sup
k≥0
zTp (k)zp(k), (8)
‖w‖22 :=
∞
∑ℓ=0
wT (ℓ)w(ℓ). (9)
Then the ℓ2 to ℓ∞ gain of (1) is less than or equal to the
square root of the maximum singular value of the matrix
constraint Zp if
supw∈ℓ2−{0}
‖zp‖∞
‖w‖2
≤√
σ(Z p) (10)
where σ(·) denotes the maximum singular value operator.
Corollary 2: Consider the asymptotically stable sys-
tem (1) with the performance output defined by (7). The
ℓ2 to ℓ∞ gain of (1) is less than or equal to the square root
of σ(Zp
), where Z p = Z
Tp > 0 is a given matrix constraint,
if there exists a bounded an continuous matrix function
P(αk) = P(αk)T > 0 such that
A (αk)P(αk)A (αk)T −P(αk+1)+B(αk)B(αk)
T < 0
(11)
Cp(αk)P(αk)Cp(αk)T −Zp < 0
(12)
The proof for Corollary 2, which follows from Theo-
rem 4 of [22], is provided in the following text.
Proof: Using the Schur complement formula, condi-
tion (11) is equivalent to[
P(αk)−1 0
0 I
]
−
[A (αk)
T
B(αk)T
]
P(αk+1)−1
[A (αk) B(αk)
]> 0.
(13)
Let w be any signal having finite energy, and x be the so-
lution of (1). Then, multiply (13) by [xT (k) wT (k)] and its
transpose from left and right, to obtain
xT (k)P(αk)−1x(k)+wT (k)w(k)
− xT (k+ 1)P(αk+1)−1x(k+ 1)< 0
Taking the summation from k = 0 to k = n− 1, we have
xT (n)P(αn)−1x(n)<
n−1
∑k=0
‖w(k)‖22.
Using the Schur complement formula again, this is equiv-
alent to[
∑n−1k=0 ‖w(k)‖2
2 xT (n)x(n) P(αn)
]
> 0 (14)
which implies[
1 00 Cp(αn)
][
∑n−1k=0 ‖w(k)‖2
2 xT (n)x(n) P(αn)
][1 0
0 C Tp (αn)
]
+
[
wT (n)0
]
[ w(n) 0 ]≥ 0
(15)
or equivalently[
∑n−1k=0 ‖w(k)‖2
2 +‖w(n)‖22 zT
p (n)
zp(n) Cp(αn)P(αn)Cp(αn)T
]
≥ 0.
(16)
Using the Schur complement formula again, we obtain
n
∑k=0
‖w(k)‖22−zT
p (n)[Cp(αn)P(αn)Cp(αn)
T]−1
zp(n)≥ 0.
(17)
After re-arranging and noting that (12) implies
Zp > Cp(αn)P(αn)Cp(αn)T
σ(Zp
)I > Cp(αn)P(αn)Cp(αn)
T(18)
we have that
zTp (n)
[Cp(αn)P(αn)Cp(αn)
T]−1
zp(n)≤n
∑k=0
‖w(k)‖22
zTp (n)
[σ(Zp
)]−1zp(n)≤
n
∑k=0
‖w(k)‖22
4 Submission to International Journal of Control, Automation, and Systems
such that
zTp (n)zp(n)≤ σ
(Z p
) n
∑k=0
‖w(k)‖22.
This implies that
‖zp‖∞ ≤√
σ(Zp
)‖w‖2 (19)
such that the ℓ2 to ℓ∞ gain is less than or equal to the square
root of σ(Zp
). �
Suppose that some a priori information about the con-
straints on the performance of zp are known such that an
output covariance bound Zp can be constructed. Then
it would be desirable to design a minimum energy con-
troller such that the closed-loop system H has the follow-
ing property:
‖zp‖2∞ ≤ σ
(Zp
)‖w‖2
2, (20)
where
Zp(αk) = Cp(αk)P(αk)Cp(αk)T ≤ Zp, (21)
and P(αk) is the solution to the time-varying Lyapunov
equation
P(αk+1) = A (αk)P(αk)A (αk)T +B(αk)B(αk)
T
(22)
This problem, which we call the minimum energy control
with guaranteed ℓ2 to ℓ∞ gain problem, is defined as fol-
lows: find a state feedback or full-order dynamic output
feedback controller to minimize the control energy
Zu(αk) = trace{Cu(αk)P(αk)Cu(αk)
T}, (23)
of the closed-loop system H, subject to the hard constraint
Z p.
Theorem 3: Consider the system H, given by (1) with
the performance output given by (7). Given the output
covariance Zp, if there exist parameter-dependent matrices
G (αk), P(αk) = P(αk)T > 0, and W (αk) = W (αk)
T >0, for all αk ∈ ΛN , such that
P(αk+1) A (αk)G (αk) B(αk)⋆ G (αk)+G (αk)
T −P(αk) 0
⋆ ⋆ I
> 0,
(24)[
W (αk) Cu(αk)G (αk)⋆ G (αk)+G (αk)
T −P(αk)
]
> 0,
(25)
Zp −Cp(αk)P(αk)Cp(αk)T > 0,(26)
where ⋆ represents entries that follow from symmetry, then
the closed-loop system (1) is exponentially stable with a
guaranteed ℓ2 to ℓ∞ performance given by
supw∈ℓ2
‖zp‖2∞
‖w‖22
≤ σ(Z p
), (27)
and a control energy bounded by
Zu = infP(αk),G (αk),W (αk)
supαk∈ΛN
trace{W (αk)} ,
≥ trace{Cu(αk)P(αk)Cu(αk)
T}≥ Zu(αk).
(28)
Proof: The LMIs (24) and (25) ensure the stability of
the closed-loop system H with the control energy bounded
by (28) [9]. However, the guaranteed ℓ2 to ℓ∞ gain perfor-
mance (27) is a result of (26). Since (24) implies that
P(αk+1)> A (αk)P(αk)A (αk)T +B(αk)B(αk)
T ,(29)
there exist matrices M(αk) = M(αk)T > 0 such that
P(αk+1)=A (αk)P(αk)A (αk)T +B(αk)B(αk)
T +M(αk).(30)
Consequently, P(αk)> P(αk) ∀k≥ 0, which shows that
Z p ≥ Cp(αk)P(αk)Cp(αk)T
≥ Cp(αk)P(αk)Cp(αk)T = Zp(k).
(31)
Thus, it follows that the guaranteed ℓ2 to ℓ∞ gain (27) is
satisfied. The Schur complement of the LMI (25) is
W (αk)−Cu(αk)G (αk)(G (αk)+G (αk)
T −P(αk))−1
×G (αk)TCu(αk)
T > 0.
(32)
In order for the LMI (25) to be feasible, it follows that
G (αk)+G (αk)T > P(αk)> 0, such that
W (αk)> Cu(αk)G (αk)(G (αk)+G (αk)
T −P(αk))−1
×G (αk)TCu(αk)
T ,
> Cu(αk)P(αk)Cu(αk)T ,
> Cu(αk)P(αk)Cu(αk)T ,
(33)
such that (28) holds. �
3. MODELING THE UNCERTAINTY DOMAIN
In this section, the modeling of the uncertainty domain,
which is covered in full detail in [9], is briefly presented.
For all k ∈ Z≥0, the rate of variation of the parameters
∆αi(k) = αi(k+ 1)−αi(k), i = 1, · · · ,N, (34)
is assumed to be limited by an a priori known bound b ∈R
such that
−b ≤ ∆αi(k)≤ b, i = 1, . . . ,N, (35)
Submission to International Journal of Control, Automation, and Systems 5
with b ∈ [0,1]. Since α(k) ∈ ΛN , it is clear from (34) that
N
∑i=1
∆αi(k) = 0. (36)
The uncertainty domain, where the vector (α(k),∆α(k))T ∈R
2N takes values, can be modeled by the compact set
Γb =
{
δ ∈ R2N : δ ∈ co{g1, . . . ,gM},
g j =
(f j
h j
)
, f j ∈ RN , h j ∈R
N ,
N
∑i=1
fj
i = 1 with fj
i ≥ 0, i = 1, . . . ,N,
N
∑i=1
hji = 0, j = 1, . . . ,M
}
(37)
defined as the convex combination of the vectors g j, for
j = 1, . . . ,M, given a priori. This definition of Γb ensures
that α(k) ∈ ΛN and that (36) holds for all k ≥ 0. For a
given bound b, the columns of Γb can be generated as
shown in [9].
4. CONTROLLER SYNTHESIS
This section considers the design of minimum energy
gain-scheduled controllers that provide guaranteed hard
constraints for the closed-loop system, while also satis-
fying some other H∞ performance criteria for robustness.
Thus, in this section we consider the following discrete-
time polytopic time-varying systems: the system Hh with
hard constraints that must be satisfied given by
Hh :=
xp(k+ 1) = A(αk)xp(k)+Bh(αk)wh(k)
+Bu(αk)u(k)
zp(k) =Cp(αk)xp(k)
zu(k) = Dhu(αk)u(k)
y(k) =Cy(αk)xp(k)+Dyh(αk)wh(k)
(38)
and the H∞ weighted system H∞ given by
H∞ :=
xp(k+ 1) = A(αk)xp(k)+B∞(αk)w∞(k)
+Bu(αk)u(k)
z∞(k) =C∞(αk)xp(k)+D∞(αk)w∞(k)
+D∞u(αk)u(k)
y(k) =Cy(αk)xp(k)+Dy∞(αk)w∞(k)(39)
where xp(k) ∈ Rn is the state, wh(k) ∈ R
rh and w∞ ∈ Rr∞
are the exogenous inputs, and u(k) ∈ Rm is the control
input. The outputs zu(k) ∈ Rph and z∞(k) ∈ R
p∞ are the
weighted system performance outputs for the mixed con-
trol synthesis, while the output zp(k) ∈ Rc contains all
variables whose dynamic responses have hard constraints
that must be met. The output vector y(k) ∈Rq is the mea-
surement to be used for control. The goal is to provide
a finite-dimensional set of LMIs for the synthesis of both
gain-scheduled state feedback controllers of the form
u(k) = K(αk)x(k), (40)
and gain-scheduled, strictly proper, output feedback con-
trollers of the form
xc(k+ 1) = Ac(αk)xc(k)+Bc(αk)y(k),
u(k) =Cc(αk)xc(k),(41)
such that the closed-loop systems given by
Hhcl :=
x(k+ 1) = A (αk)x(k)+Bh(αk)wh(k),
zp(k) = Cp(αk)x(k),
zu(k) = Cu(αk)x(k),
(42)
and
H∞cl :=
{
x(k+ 1) = A (αk)x(k)+B∞(αk)w∞(k),
z∞ = C∞(αk)x(k)+D∞(αk)w∞(k),(43)
are exponentially stable and satisfy hard constraints on de-
sired performance outputs for all possible trajectories of
the parameter αk ∈ ΛN , while minimizing the control en-
ergy for Hhcl and satisfying a robustness criteria defined as
an H∞ performance bound for H∞cl .
4.1. Gain-Scheduled State Feedback Control Synthesis
In this section, it is assumed that the state vector, xp(k),is available for feedback without corruption from the ex-
ogenous inputs wh(k) or w∞(k). This is a standard as-
sumption, and as in [8] can be enforced on the measure-
ment equation in (38) and (39) by assigning to the matri-
ces Cy(αk), Dyh(αk), and Dy∞(αk) the values Cy(αk) = I,
Dyh(αk) = 0, and Dy∞(αk) = 0. The feedback structure
provided by the gain-scheduled state-feedback controller
(40), produces the closed-loop systems Hhcl and H∞
cl in (42)
and (43) where x(k) = xp(k) and the closed-loop system
matrices are given by
A (αk) = A(αk)+Bu(αk)K(αk),
Bh(αk) = Bh(αk),
Cp(αk) =Cp(αk),
Cu(αk) = Dhu(αk)K(αk),
B∞(αk) = B∞(αk),
C∞(αk) =C∞(αk)+D∞u(αk)K(αk),
D∞(αk) = D∞(αk).
(44)
The parameter-dependent, full-state feedback controller
is solved for by performing a convex optimization over a
set of linear matrix inequalities. The LMIs in this sec-
tion are an extension of the work presented in [9]. To ob-
tain a finite-dimensional set of LMI conditions, an affine
6 Submission to International Journal of Control, Automation, and Systems
parameter-dependent structure is imposed on the Lyapunov
matrix P(αk) such that
P(αk) =N
∑i=1
αi(k)Pi, αk ∈ ΛN . (45)
With the uncertainty set Γb, each αi(k) and ∆αi(k) for i =1,2, . . . ,N are given by
αi(k) =M
∑j=1
fj
i γ j(k) and ∆αi(k) =M
∑j=1
hji γ j(k) (46)
such that
P(αk) = P(γ(k)) =M
∑j=1
γ j(k)Pj (47)
with Pj =∑Ni=1 f
ji Pi as shown in [9]. Using the same struc-
ture for αk, the system matrices in H∞ and Hh are also
converted to the new representation in terms of γ(k)∈ΛM ,
such that
A(αk) = A(γ(k)) =M
∑j=1
γ j(k)A j (48)
with A j = ∑Ni=1 f
ji Ai. All other matrices in H∞ and Hh are
converted the same way. Also, by combining (46) with the
fact that αk+1 = αk +∆αk,
P(αk+1) = P(γ(k)) =M
∑j=1
γ j(k)Pj (49)
with Pj = ∑Ni=1
(
fj
i + hji
)
Pi. Using these parameteriza-
tions, the finite-dimensional LMIs in the following theo-
rem can be solved to obtain a full-state feedback controller
(40) such that the closed-loop systems for Hh and H∞ have
a guaranteed ℓ2 to ℓ∞ and H∞ gain, respectively.
Theorem 4: Consider the system Hh, given by (38).
Assume that the vectors f j and h j of Γb are given. Given
Z p, if there exists, for i = 1,2, . . . ,N, matrices, Gi ∈ Rn×n
and Zi ∈ Rm×n, and symmetric positive-definite matrices
Ph,i ∈ Rn×n and Wi ∈ R
ph×ph such that
Ph, j ⋆ ⋆GT
j ATj + ZT
j BTu, j G j + GT
j − Ph, j ⋆
BTh, j 0 I
= Φ j > 0
(50)
for j = 1,2, . . . ,M, and
Ph, j + Ph,ℓ ⋆ ⋆Φ21, jℓ Φ22, jℓ ⋆
BTh, j + BT
h,ℓ 0 2I
= Φ jℓ > 0 (51)
with
Φ21, jℓ = GTj AT
ℓ + GTℓ AT
j + ZTj BT
u,ℓ+ ZTℓ BT
u, j
Φ22, jℓ = G j + GTj + Gℓ+ GT
ℓ − Ph, j − Ph,ℓ
for j = 1, . . . ,M− 1 and ℓ= j+ 1, . . . ,M, and
[Wj ⋆
ZTj DT
hu, j G j + GTj − Ph, j
]
= Ψ j > 0 (52)
for j = 1,2, . . . ,M and
[Wj +Wℓ ⋆
ZTj DT
hu,ℓ+ ZTℓ DT
hu, j Ψ22, jℓ
]
= Ψ jℓ > 0 (53)
with
Ψ22, jℓ = G j + GTj + Gℓ+ GT
ℓ − Ph, j − Ph,ℓ
for j = 1, . . . ,M− 1 and ℓ= j+ 1, . . . ,M and
Zp −Cp,iPh,iCTp,i ≥ 0, i = 1,2, . . . ,N, (54)
with
Ph, j =N
∑i=1
(
fj
i + hji
)
Ph,i, Ph, j =N
∑i=1
fj
i Ph,i,
G j =N
∑i=1
fj
i Gi, Z j =N
∑i=1
fj
i Zi, and Wj =N
∑i=1
fj
i Wi.
then the parameter-dependent full state feedback gain
K(αk) = Z(αk)G(αk)−1 (55)
with
Z(αk) =N
∑i=1
αi(k)Zi and G(αk) =N
∑i=1
αi(k)Gi (56)
stabilizes the the system Hh with a guaranteed (weighted)
control energy bounded by Zu given by
Zu = minP∞,i,Pσ ,i,Gi,Zi ,Wi
maxi
trace{Wi}
≥ trace{Dhu(α)K(α)Ph(α)K(α)T Dhu(α)T }= Zu(αk)
(57)
while also ensuring that the hard constraint Zp is satisfied.
The equality for Zu(αk) is obtained by substituting the
value for Cu(αk) from (44) into the expression (23) and
by noting that the Lyapunov variable is represented here
by Ph(α). Additionally, consider the system H∞, given by
(39). If there exist, for i = 1,2, . . . ,N, symmetric positive-
definite matrices P∞,i ∈ Rn×n such that
P∞, j ⋆ ⋆ ⋆GT
j ATj + ZT
j BTu, j G j + GT
j − P∞, j ⋆ ⋆
BT∞, j 0 ηI ⋆
0 C∞, jG j + D∞u, jZ j D∞, j ηI
= Θ j > 0
(58)
Submission to International Journal of Control, Automation, and Systems 7
for j = 1,2, . . . ,M and
P∞, j + P∞,ℓ ⋆ ⋆ ⋆Θ21, jℓ Θ22, jℓ ⋆ ⋆
BT∞, j + BT
∞,ℓ 0 2ηI ⋆
0 Θ42, jℓ D∞, j + D∞,ℓ 2ηI
= Θ jℓ > 0
(59)
with
Θ21, jℓ = GTj AT
ℓ + GTℓ AT
j + ZTj BT
u,ℓ+ ZTℓ BT
u, j
Θ22, jℓ = G j + GTj + Gℓ+ GT
ℓ − P∞, j − P∞,ℓ
Θ42, jℓ = C∞, jGℓ+ C∞,ℓG j + D∞u, jZℓ+ D∞u,ℓZ j
for j = 1,2, . . . ,M− 1 and ℓ= j+ 1, . . . ,M, where
P∞, j =N
∑i=1
(
fj
i + hji
)
P∞,i, P∞, j =N
∑i=1
fj
i P∞,i,
then the parameter-dependent full-state feedback gain K(αk)given by (55) also stabilizes the system H∞ with a guaran-
teed H∞ performance bounded by η .
Proof: The proof is organized as follows. The follow-
ing properties are a consequence of applying Theorems 8
and 9 of [9]:
• The system H∞ is stabilized with a guaranteed H∞ per-
formance bounded by η when the LMIs (58) and (59) are
satisfied.
• The system Hh is stabilized with a guaranteed (weighted)
control energy bounded by Zu (57) when the LMIs (50),
(51), (52), and (53) are satisfied.
However, the fact that the output constraint (20) is satisfied
for i = 1,2, . . . ,N follows from the LMI constraint (54)
Z p −Cp,iPh,iCTp,i ≥ 0, i = 1,2, . . . ,N.
Since the LMIs (50),(51),(52), and (53) are all required to
be positive-definite, from [9] it can be shown that
Ph(αk) =N
∑i=1
αi(k)Ph,i > P(αk), ∀k ≥ 0,
where P(αk) is the controllability Gramian satisfying (22).
Thus, it is also true that
Z p −Cp(αk)P(αk)Cp(αk)T ≥ 0
such that
Zp(αk) =Cp(αk)P(αk)Cp(αk)T ≤ Zp.
�
4.2. Output Feedback LMIs
The feedback structure provided by the gain-scheduled
dynamic output-feedback controller (41), produces the closed-
loop systems Hhcl and H∞
cl in (42) and (43) where x(k) =[xT
p (k) xTc (k)]
T with the closed-loop system matrices given
by
A (αk) =
[A(αk) Bu(αk)Cc(αk)
Bc(αk)Cy(αk) Ac(αk)
]
,
Bh(αk) =
[Bh(αk)
Bc(αk)Dyh(αk)
]
,
Cp(αk) =[
Cp(αk) 0],
Cu(αk) =[
0 Dhu(αk)K(αk)],
B∞(αk) =
[B∞(αk)
Bc(αk)Dy∞(αk)
]
,
C∞(αk) =[
C∞(αk) D∞u(αk)Cc(αk)],
D∞(αk) = D∞(αk).
(60)
As for the full-state feedback problem, the parameter-
dependent, strictly proper output feedback controller is
also solved for by performing a convex optimization over
a set of linear matrix inequalities. It is clear that when sub-
stituting the closed-loop matrices (60) into the matrix in-
equalities of the performance conditions in Lemma 1 and
Theorem 3 nonlinear matrix inequalities result, due to the
multiplication between the unknown controller matrices
and the slack variable G (αk). Also, to obtain a set of LMI
conditions to synthesize an LPV controller that will rely
on measurements of only the current time-varying param-
eter αk, it is necessary to enforce the slack variable G (αk)to be independent of the scheduling parameter αk, such
that G (αk) = G (see [23]). Then the slack variable G , its
inverse G −1, the Lyapunov matrix P(αk), and the con-
troller matrices K (αk) are partitioned as
G :=
[X Z1
U Z2
]
, G−1 :=
[Y T Z3
V T Z4
]
,
P(αk) :=
[P(αk) P2(αk)
P2(αk)T P3(αk)
]
,
K (αk) :=
[Ac(αk) Bc(αk)Cc(αk) 0
]
.
From the definition of G and G −1, it is clear that the fol-
lowing relationship must hold:
G G−1 =
[X Z1
U Z2
][Y T Z3
V T Z4
]
=
[XY T +Z1V T XZ3 +Z1Z4
UY T +Z2V T UZ3 +Z2Z4
]
=
[I 0
0 I
]
,
such that XY T + Z1V T = I and UY T + Z2V T = 0. Now,
the parameter-independent transformation matrix
T :=
[I Y T
0 V T
]
(61)
8 Submission to International Journal of Control, Automation, and Systems
used in [6–8,23] is introduced and the following nonlinear
parameter-dependent change of variables are defined:
[Q(αk) F(αk)L(αk) 0
]
:=
[Y
0
]
A(αk)[
X 0]
+
[V Y Bu(αk)0 I
]
K (αk)
[U 0
Cy(αk)X I
]
,
(62)
[P(αk) J(αk)J(αk)
T H(αk)
]
:= TTPT , (63)
S := Y X +VU. (64)
The nonlinear matrix inequalities that result from sub-
stituting the closed-loop matrices into the matrix inequal-
ities in Theorem 3 can be transformed into the LMIs (65),
(66), and (67) by using the congruence transformations
T1 = diag(T ,T , I) and T2 = diag(I,T )
on the first and second nonlinear matrix inequalities, re-
spectively. While the matrix inequalities in (65), (66), and
(67) are linear, they are still infinite dimensional as they
must be evaluated for all values of time-varying parameter
αk. To obtain a finite-dimensional set of LMIs, the param-
eter dependent structure imposed on the Lyapunov matrix
in (45) is imposed on the Lyapunov matrix here as well.
In the following, the same parameterizations used for the
state-feedback case are utilized for the gain-scheduled dy-
namic output-feedback control problem.
Theorem 5: Consider the system Hh, given by (38).
Assume that the vectors f j and h j of Γb are given. Given
Z p, if there exists matrices X ∈ Rn×n, Y ∈ R
n×n, and for
i = 1,2, . . . ,N, matrices, Jh,i ∈ Rn×n, Li ∈ R
m×n, and Fi ∈R
n×q, and symmetric positive-definite matrices Ph,i ∈Rn×n,
Hh,i ∈Rn×n, and Wi ∈ R
ph×ph such that
Ph, j Jh, j Ψ13, j A j Bh, j
⋆ Hh, j Q j Ψ24, j Ψ25, j
⋆ ⋆ Ψ33, j Ψ34, j 0
⋆ ⋆ ⋆ Ψ44, j 0
⋆ ⋆ ⋆ ⋆ I
= Ψ j > 0 (68)
with
Ψ13, j = A jX + Bu, jL j
Ψ24, j = YA j + FjCy, j
Ψ25, j = YBh, j + FjDyh, j
Ψ33, j = X +XT − Ph, j
Ψ34, j = I+ ST − Jh, j
Ψ44, j = Y +YT − Hh, j
for j = 1, . . . ,M, and
Ψ11, jℓ Ψ12, jℓ Ψ13, jℓ A j + Aℓ Bh, j + Bh,ℓ
⋆ Ψ22, jℓ Q j + Qℓ Ψ24, jℓ Ψ25, jℓ
⋆ ⋆ Ψ33, jℓ Ψ34, jℓ 0
⋆ ⋆ ⋆ Ψ44, jℓ 0
⋆ ⋆ ⋆ ⋆ 2I
= Ψ jℓ > 0
(69)
with
Ψ11, jℓ = Ph, j + Ph,ℓ
Ψ12, jℓ = J j + Jℓ
Ψ13, jℓ = A jX + AℓX + Bu, jLℓ+ Bu,ℓL j
Ψ22, jℓ = Hh, j + Hh,ℓ
Ψ24, jℓ = YA j +YAℓ+ FjCy,ℓ+ FℓCy, j
Ψ25, jℓ = YBh, j +YBh,ℓ+ FjDyh,ℓ+ FℓDyh, j
Ψ33, jℓ = 2X + 2XT − Ph, j − Ph,ℓ
Ψ34, jℓ = 2I+ 2ST − Jh, j − Jh,ℓ
Ψ44, jℓ = 2Y + 2YT − Hh, j − Hh,ℓ
for j = 1, . . . ,M− 1 and ℓ= j+ 1, . . . ,M, and
Wj Dhu, jL j 0
⋆ X +XT − Ph, j I + ST − Jh, j
⋆ ⋆ Y +Y T − Hh, j
= Φ j > 0
(70)
for j = 1, . . . ,M, and
Wj +Wℓ Φ12, jℓ 0
⋆ Φ22, jℓ Φ23, jℓ
⋆ ⋆ Φ33, jℓ
= Φ jℓ > 0 (71)
with
Φ12, jℓ = Dhu, jLℓ+ Dhu,ℓL j
Φ22, jℓ = 2X + 2XT − Ph, j − Ph,ℓ
Φ23, jℓ = 2I+ 2ST − Jh, j − Jh,ℓ
Φ33, jℓ = 2Y + 2YT − Hh, j − Hh,ℓ
for j = 1, . . . ,M− 1 and ℓ= j+ 1, . . . ,M, and
Zp −Cp,iPh,iCTp,i ≥ 0, i = 1,2, . . . ,N, (72)
where
Ph, j =N
∑i=1
( fj
i + hji )Ph,i,
Jh, j =N
∑i=1
( fj
i + hji )Jh,i,
Hh, j =N
∑i=1
( fj
i + hji )Hh,i,
(73)
Submission to International Journal of Control, Automation, and Systems 9
P(αk) J(αk) A(αk)X +Bu(αk)L(αk) A(αk) Bw(αk)
⋆ H(αk) Q(αk) YA(αk)+F(αk)Cy(αk) Y Bw(αk)⋆ ⋆ X +XT −P(αk) I + ST − J(αk) 0
⋆ ⋆ ⋆ Y +Y T −H(αk) 0
⋆ ⋆ ⋆ ⋆ I
> 0 (65)
W (αk) Dzu(αk)L(αk) 0
⋆ X +XT −P(αk) I+ ST − J(αk)⋆ ⋆ Y +YT −H(αk)
> 0 (66)
Zp −Cp(αk)P(αk)Cp(αk)T > 0 (67)
and
Ph, j =N
∑i=1
fj
i Ph,i, Jh, j =N
∑i=1
fj
i Jh,i,
Hh, j =N
∑i=1
fj
i Hh,i, Q j =N
∑i=1
fj
i Qi,
L j =N
∑i=1
fj
i Li, Fj =N
∑i=1
fj
i Fi,
A j =N
∑i=1
fj
i Ai, Bu, j =N
∑i=1
fj
i Bu,i,
Bw, j =N
∑i=1
fj
i Bw,i, Cz, j =N
∑i=1
fj
i Cz,i,
Dzu, j =N
∑i=1
fj
i Dzu,i, Dzw, j =N
∑i=1
fj
i Dzw,i,
Cy, j =N
∑i=1
fj
i Cy,i, Dyw, j =N
∑i=1
fj
i Dyw,i,
(74)
then the parameter-dependent, strictly proper output feed-
back controller
xc(k+ 1) = Ac(αk)xc(k)+Bc(αk)y(k),
u(k) =Cc(αk)xc(k),(75)
with matrices computed by
Ac(αk) =V−1
(
Q(αk)−YA(αk)X
−YBu(αk)L(αk)−F(αk)Cy(αk)X
)
U−1,
Bc(αk) =V−1F(αk),
Cc(αk) = L(αk)U−1,
(76)
stabilizes the system Hh with a guaranteed (weighted) con-
trol energy bounded by Zu while also ensuring that the
hard constraint Zp is satisfied. Additionally, consider the
system H∞, given by (39). If there exist, for i= 1,2, . . . ,N,
matrices J∞,i ∈ Rn×n and symmetric positive-definite ma-
trices P∞,i ∈Rn×n and H∞,i ∈ R
n×n such that
P∞, j J∞, j Θ13, j A j B∞, j 0
⋆ H∞, j Q j Θ24, j Θ25, j 0
⋆ ⋆ Θ33, j Θ34, j 0 Θ36, j
⋆ ⋆ ⋆ Θ44, j 0 CT∞, j
⋆ ⋆ ⋆ ⋆ ηI DT∞, j
⋆ ⋆ ⋆ ⋆ ⋆ ηI
= Θ j > 0
(77)
with
Θ13, j = A jX + Bu, jL j
Θ24, j = Y A j + FjCy, j
Θ25, j = Y B∞, j + FjDy∞, j
Θ33, j = X +XT − P∞, j
Θ34, j = I+ ST − J∞, j
Θ36, j = XTCT∞, j + LT
j DT∞u, j
Θ44, j = Y +Y T − H∞, j
for j = 1, . . . ,M, and
Θ11, jℓ Θ12, jℓ Θ13, jℓ A j + Aℓ Θ15, jℓ 0
⋆ Θ22, jℓ Q j + Qℓ Θ24, jℓ Θ25, jℓ 0
⋆ ⋆ Θ33, jℓ Θ34, jℓ 0 Θ36, jℓ
⋆ ⋆ ⋆ Θ44, jℓ 0 Θ46, jℓ
⋆ ⋆ ⋆ ⋆ 2ηI Θ56, jℓ
⋆ ⋆ ⋆ ⋆ ⋆ 2ηI
= Θ j > 0
(78)
10 Submission to International Journal of Control, Automation, and Systems
with
Θ11, jℓ = P∞, j + P∞,ℓ
Θ12, jℓ = J∞, j + J∞,ℓ
Θ13, jℓ = A jX + AℓX + Bu, jLℓ+ Bu,ℓL j
Θ15, jℓ = B∞, j + B∞,ℓ
Θ22, jℓ = H∞, j + H∞,ℓ
Θ24, jℓ = Y A j +YAℓ+ FjCy,ℓ+ FℓCy, j
Θ25, jℓ = Y B∞, j +YB∞,ℓ+ FjDy∞,ℓ+ FℓDy∞, j
Θ33, jℓ = 2X + 2XT − P∞, j − P∞,ℓ
Θ34, jℓ = 2I+ 2ST − J∞, j − J∞,ℓ
Θ36, jℓ = XTCT∞, j +XTCT
∞,ℓ+ LTj DT
∞u,ℓ+ LTℓ DT
∞u, j
Θ44, jℓ = 2Y + 2YT − H∞, j − H∞,ℓ
Θ46, jℓ = CT∞, j + CT
∞,ℓ
Θ56, jℓ = DT∞, j + DT
∞,ℓ
for j = 1, . . . ,M − 1 and ℓ= j+ 1, . . . ,M with
P∞, j =N
∑i=1
( fj
i + hji )P∞,i, P∞, j =
N
∑i=1
fj
i P∞,i,
J∞, j =N
∑i=1
( fj
i + hji )J∞,i, J∞, j =
N
∑i=1
fj
i J∞,i,
H∞, j =N
∑i=1
( fj
i + hji )H∞,i, H∞, j =
N
∑i=1
fj
i H∞,i.
(79)
then the parameter-dependent, strictly proper output feed-
back controller also stabilizes the system H∞ with a guar-
anteed H∞ performance bounded by η .
Proof: The proof, which is similar to the proof for The-
orem 4, is organized as follows:
• The system H∞ is stabilized with a guaranteed H∞ per-
formance bounded by η when the LMIs (77) and (78) are
satisfied.
• The system Hh is stabilized with a guaranteed (weighted)
control energy bounded by Zu when the LMIs (68), (69),
(70), and (71) are satisfied.
The output constraint (20) is satisfied for i = 1,2, . . . ,N as
a result of the LMI constraint (72)
Z p −Cp,iPh,iCTp,i ≥ 0, i = 1,2, . . . ,N.
Since the LMIs (68), (69), (70), and (71) are all required
to be positive definite, it can be shown that
[Ph(αk) Jh(αk)Jh(αk)
T Hh(αk)
]
=N
∑i=1
αi(k)
[Ph,i Jh,i
JTh,i Hh,i
]
> P(αk), ∀k ≥ 0,
where P(αk) is the controllability Gramian satisfying (22).
Thus, it is also true that
Zp −Cp(αk)P(αk)Cp(αk)T ≥ 0
such that
Zp(αk) = Cp(αk)P(αk)Cp(αk)T ≤ Zp.
�
5. NUMERICAL EXAMPLE
The approach proposed in this paper is demonstrated
with a numerical example. Consider the discrete-time LPV
system (originally used in [24], and later used in [25] and
[8])
xp(k+ 1) =
2+ δ1 0 1
1 0.5 0
0 1 −0.5
︸ ︷︷ ︸
A(δ1(k))
xp(k)
+
1+ δ2
0
0
︸ ︷︷ ︸
Bu(δ2(k))
u(k)+
0
1
0
︸ ︷︷ ︸
Bh
wp(k)
zp(k) =
1 0 0
0 1 0
0 0 1
︸ ︷︷ ︸
Cp
xp(k)
zu(k) = u(k)
(80)
where δi, i = 1,2 are the time-varying parameters. In this
section, two design examples with different performance
constraints are considered for both state-feedback and dy-
namic output-feedback control.
5.1. State Feedback Control
For the state-feedback controller design, the time-varying
parameters were assumed to have the following parameter
variation bounds:
δ1 ∈ [−1, 1], and δ2 ∈ [−0.5, 0.5]. (81)
The discrete-time LPV system (80) is converted to the
discrete-time polytopic LPV system (38) by solving A(δ1)and Bu(δ2) at the vertices of the parameter space polytope
of δ1 and δ2. The exogenous ℓ2 disturbance wp is a scalar
and the performance variable zp has three components.
In the following, we consider two different ℓ2 to ℓ∞ gain
designs. The designs differ in the grouping of the perfor-
mance variables inside of zp used to define the constraints
(20). The constraints for each design are given as follows:
Design 1: Zp ≤ 1.85× I3, (82)
Design 2: Zp,1 ≤ 1.85, Zp,2 ≤ 1.85× I2, (83)
Submission to International Journal of Control, Automation, and Systems 11
where for design 1, Zp denotes the (3 × 3) output co-
variance matrix corresponding to the all performance out-
puts in zp grouped together. In design 2, Zp,1 denotes the
(1× 1) output variance corresponding to the first perfor-
mance output of zp and Zp,2 denotes the (2× 2) output
covariance matrix corresponding to the second and third
performance outputs grouped together.
For each design, to enhance the robustness of the closed-
loop system with the controller K(αk) with respect to un-
certainty in the measurements of the time-varying param-
eters δ1 and δ2, the closed-loop H∞ norms of the trans-
fer functions of some appropriately defined extra inputs
and outputs that ‘pull out’ [8, 24] the uncertain parame-
ters are bounded. Specifically, the following H∞ system is
defined:
H∞ =
x(k+ 1) = A(δ1(k))x(k)+Bu(δ2(k))u(k)
+
1
0
0
w∞,1(k)+
1
0
0
w∞,2(k)
z∞,1(k) =[
1 0 0]
x(k)
z∞,2(k) = u(k)(84)
so that the robustness requirement is given by
‖Hz∞,iw∞,i(α)‖∞ < η = 100, i = 1,2, (85)
where η defines the robustness level. Note that the nota-
tion used here, specifically w∞,1(k) and w∞,2(k) with the
same input matrix, was selected to match what is found in
the literature [8, 24].
For each of the ℓ2-ℓ∞ designs (82)-(83), the LMIs in
Theorem 4 are programmed into MATLAB using the LMI
parser YALMIP [26] and solved with SeDuMi [27] to min-
imize the cost function (57). As shown in Fig. 1 and
Fig. 3A, each design is feasible and the achieved covari-
ance bound is tight with the design bound in at least one
dimension. The constraint in design 1 ensures that the
covariance bound ellipsoid of Zp remains inside of the
sphere displayed in Fig. 1A. Side views of the covariance
bound Zp are displayed in Fig. 1B, Fig. 1C, and Fig. 1D.
As displayed in Fig. 1C, the output covariance Zp is tight
with the bound in the zp,2-zp,3 plane.
For design 2, the constraints ensure that the variance of
the first output of zp will be below 1.85 and the covariance
bound of second and third outputs of zp will remain inside
of the circle in Fig. 3A. The dashed ellipses in Fig. 3A are
the obtained output covariances at each of the vertices for
i = 1, . . . ,4, and as shown they are tight with the bound.
To test the performance of each design, we simulate
each of the controllers with a positive impulse (I1), an
ℓ2 excitation, followed by a negative impulse (I2) as dis-
played in Fig. 4A. To see the effect of the time-varying
parameters, the parameters δ1 and δ2 are varied as dis-
played in Fig. 4B. The values used to compute the con-
troller at each time step k are the noisy measurements dis-
played with a gray dashed line. The response to the ℓ2
zp,1
zp,1zp,1
zp,2
z p,2
zp,2
z p,3
z p,3
z p,3
A B
C D 2
2
2
2
2
2
2
22
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0 0
-1
-1
-1
-1
-1
-1
-1
-2-2
-2-2
-2-2
-2
-2 -2
Fig 1: Design 1: The covariance bound Zp achieved com-
pared to the constraint (82).
disturbance wp(k) for design 1 is displayed in Fig. 2. The
response in Fig. 2 is plotted inside of the ℓ∞ norm con-
straint (the square root of the covariance bound) sphere
and the achieved ℓ∞ norm bound ellipsoid. In Fig. 3B, the
response of design 2 is plotted inside of the ℓ∞ norm con-
straint circle and the achieved ℓ∞ norm bound ellipse. The
path of the response, with respect to each of the impulses
(I1) and (I2), is also displayed in Fig. 3B. As shown in
Fig. 2 and Fig. 3B, the response for each design stays in-
side of the ℓ∞ bound.
5.2. Dynamic Output-Feedback Control
For the dynamic output-feedback controller design, the
measurement equation in [8, 24, 25] is given as
y(k) =[
0 1+ δ3(k) 0]
x(k), (86)
where δ3 is an additional time-varying parameter. As one
might guess, this is much more restrictive than the state
feedback case. The good news is that the system is still
observable (assuming δ3 6=−1). However, the bad news is
that in order to obtain a feasible controller with the LMIs
provided by Theorem 5, some modifications need to be
made to the problem. The modified discrete-time LPV is
12 Submission to International Journal of Control, Automation, and Systems
zp,1
zp,1zp,1
zp,2
z p,2
zp,2
z p,3
z p,3
z p,3
A B
C D 1.51.5
1.51.5
11
11
1
0
0
0
0
0
0
0
0 0
-1-1
-1-1 -1
-1.5-1.5
-1.5
-1.5
Fig 2: Design 1: The output response of zp,1, zp,2, and
zp,3 plotted against each other for design 1 simu-
lated with a positive (I1) and negative (I2) impulse
function and compared with the ℓ∞ norm bound.
zp,2zp,2
z p,3
z p,3 I1I2
A B2
2
1.5
1.5
1
1
0
0
0
0
-1
-1-1.5
-1.5-2-2
Fig 3: Design 2: A. The covariance bound Zp,2 achieved
compared to the second constraint in (83). B. The
output response of zp,2 plotted against zp,3 for de-
sign 2 simulated with a positive and negative im-
pulse function and compared with the ℓ∞ norm
bound.
δ1(k)
δ2(k)
k
wp(k)
A
B
I1
I2
1
1
0.5
0
0
0
0-0.5
-1
-1
100 200 300
-0.8 -0.6 -0.4 -0.2 0.80.60.40.2
True
Measured
Fig 4: The ℓ2 disturbance (A) and the parameter variation
(B) used to simulate each controller design.
given by
xp(k+ 1) =
2+ δ (k) 0 1
1 0.5 0
0 1 −0.5
︸ ︷︷ ︸
A(δ (k))
xp(k)
+
1+ δ (k)0
0
︸ ︷︷ ︸
Bu(δ (k))
u(k)+
0
0.10
︸ ︷︷ ︸
Bh
wp(k),
zp(k) =
1 0 0
0 1 0
0 0 1
︸ ︷︷ ︸
Cp
xp(k),
zu(k) = u(k),
y(k) =[
0 1+ δ (k) 0]
︸ ︷︷ ︸
Cy(δ (k))
x(k)+ 0.01v(k).
(87)
Notice that in the modified system, each of the time-varying
parameters δi, i = 1,2,3 have been set equal to each other
such that δ = δ1 = δ2 = δ3, which was originally done
by the authors in [25]. Also, as in [25], the time-varying
parameter δ is assumed to have the following parameter
variation bound:
δ ∈ [−0.2, 0.2]. (88)
As for the state-feedback design case, the discrete-time
LPV system (87) is converted to the discrete-time poly-
topic LPV system (38) by solving A(δ ), Bu(δ ), and Cy(δ )at the vertices of the parameter space polytope of δ . For
the dynamic output-feedback design, the exogenous ℓ2 dis-
turbance is given by the process disturbance wp(k) and the
measurement disturbance v(k), such that w(k)= [wp(k), v(k)]T .
The performance variable zp, again has three components.
As we did for the state-feedback control example, we
also consider two different ℓ2 to ℓ∞ gain designs for the
dynamic output-feedback control example. As before, the
designs differ in the grouping of the performance variables
inside of zp used to define the constraints (20). The con-
straints for each design are given as follows:
Design 1: Zp ≤ 5× I3, (89)
Design 2: Zp,1 ≤ 5, Zp,2 ≤ 5× I2, (90)
where for design 1, Zp denotes the (3 × 3) output co-
variance matrix corresponding to the all performance out-
puts in zp grouped together. In design 2, Zp,1 denotes the
(1× 1) output variance corresponding to the first perfor-
mance output of zp and Zp,2 denotes the (2× 2) output
covariance matrix corresponding to the second and third
performance outputs grouped together.
As in the state-feedback control example, H∞ perfor-
mance criteria is used to enhance the robustness of the
Submission to International Journal of Control, Automation, and Systems 13
zp,1
zp,1zp,1
zp,2
z p,2
zp,2
z p,3
z p,3
z p,3
A B
C D 5
5
5
5
5
5
5
55
0
0
0
0
0
0
0
0 0
-5-5
-5-5
-5-5
-5
-5 -5
Fig 5: Design 1: The covariance bound Zp achieved com-
pared to the constraint (82).
closed-loop system using the dynamic output-feedback con-
troller with respect to uncertainty in the measurements of
the time-varying parameter δ . The system H∞ used for
the dynamic output-feedback design is the same as given
in (84), with the following additions:
y(k) =Cy(δ (k))x(k)+w∞,3(k)
z∞,3(k) =[
0 1 0]
x(k)(91)
such that the robustness requirement is now given by
‖Hz∞,iw∞,i(α)‖∞ < η = 100, i = 1,2,3, (92)
where η defines the robustness level.
For each of the ℓ2 to ℓ∞ designs (89)-(90), the LMIs
in Theorem 5 are programmed into MATLAB and solved
with LMI Lab [28] to minimize the control energy Zu.
As shown in Fig. 5 and Fig. 7A, each design is feasible
and the achieved covariance bound is tight with the de-
sign bound in at least one dimension. The constraint in
design 1 ensures that the covariance bound ellipsoid of
Zp remains inside of the sphere displayed in Fig. 1A. Side
views of the covariance bound Zp are displayed in Fig. 5B,
Fig. 5C, and Fig. 5D. As displayed in Fig. 5C, the output
covariance Zp is tight with the bound in the zp,2-zp,3 plane.
For design 2, the constraints ensure that the variance of
the first output of zp will be below 5 and the covariance
bound of second and third outputs of zp will remain inside
of the circle in Fig. 7A. The dashed ellipses in Fig. 7A are
the obtained output covariances at each of the vertices for
i = 1,2, and as shown they are tight with the bound.
To test the performance of each design, we again sim-
ulate each of the controllers with a positive impulse (I1)
followed by a negative impulse (I2) as displayed in Fig. 8A.
To see the effect of the time-varying parameter, the param-
eter δ was varied as displayed in Fig. 8B. As before, the
zp,1
zp,1zp,1
zp,2
z p,2
zp,2
z p,3
z p,3
z p,3
A B
C D2
2
2
2
2
2
2
22
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0 0
-1
-1
-1
-1
-1
-1
-1
-2
-2
-2
-2
-2
-2
-2
-2 -2
Fig 6: Design 1: The output response of zp,1, zp,2, and
zp,3 plotted against each other for design 1 simu-
lated with a positive (I1) and negative (I2) impulse
function and compared with the ℓ∞ norm bound.
zp,2zp,2
z p,3
z p,3
I1
I2
A B5
5
2
2
1
1
0
0
0
0
-1
-1
-2
-2-5
-5
Fig 7: Design 2: A. The covariance bound Zp,2 achieved
compared to the second constraint in (83). B. The
output response of zp,2 plotted against zp,3 for de-
sign 2 simulated with a positive and negative im-
pulse function and compared with the ℓ∞ norm
bound.
k
δ(k)
k
wp(k)
A
B
I1
I2
1
0.5
0
0
0
0
-0.5
-1
0.2
0.1
-0.1
-0.2
100
100
200
200
300
300
True
Measured
Fig 8: The ℓ2 disturbance (A) and the parameter variation
(B) used to simulate each controller design.
14 Submission to International Journal of Control, Automation, and Systems
values used to compute the controller at each time step k
are the noisy measurements displayed with a gray dashed
line. The response to the ℓ2 disturbance wp(k) for design 1
is displayed in Fig. 6. The response in Fig. 6 is plotted
inside of the ℓ∞ norm constraint (the square root of the co-
variance bound) sphere and the achieved ℓ∞ norm bound
ellipsoid. In Fig. 7B, the response of design 2 is plotted
inside of the ℓ∞ norm constraint circle and the achieved ℓ∞
norm bound ellipse. The path of the response, with respect
to each of the impulses (I1) and (I2), is also displayed in
Fig. 7B. As shown in Fig. 6 and Fig. 7B, the response for
each design stays inside of the ℓ∞ bound. It is interesting
to note that the response to the positive impulse (I1) is
larger than the response to the negative impulse (I2). This
is caused by the different time-varying parameter δ (k) val-
ues at the time of each impulse, as is displayed in Fig. 8.
6. CONCLUSION
In this paper, discussion motivating the necessity of the
ℓ2 to ℓ∞ gain performance criteria was provided. Then, the
ℓ2 to ℓ∞ gain performance criteria was introduced to allow
for the specification of hard constraints when designing
gain-scheduling controllers. Controller synthesis LMIs
are provided for the synthesis of minimum energy state-
feedback and dynamic output-feedback controllers with
guaranteed ℓ2 to ℓ∞ gain and H∞ performance. To demon-
strate the effectiveness of the controller synthesis LMIs
provided in this paper, gain-scheduled state-feedback and
dynamic output-feedback controllers are designed for a
numerical example. One of the future work will be to
apply our approach to a practical application such as en-
gine control problems [13–15] with hard constraints on
responses or actuators.
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Andrew White received his Ph.D., M.S.
and B.S. degrees in Mechanical Engi-
neering from Michigan State University
in 2012, 2008 and 2006 respectively. Cur-
rently, he is working in the powertrain
virtual analysis department at Chrysler.
His research interests include adaptive
and robust control, and parameter esti-
mation, with applications to engine con-
trol, robotics, and non-destructive testing and evaluation. He
is a member of IEEE and ASME.
Dr. Guoming (George) Zhu is a profes-
sor of mechanical engineering and elec-
trical/computer engineering at Michigan
State University. Prior to joining the ME
and ECE departments, he was a techni-
cal fellow in advanced powertrain sys-
tems at the Visteon Corporation. He also
worked for Cummins Engine Co. as a
technical advisor. Dr. Zhu earned his
PhD (1992) in aerospace engineering at Purdue University.
His BS and MS degrees (1982 and 1984 respectively) were
from Beijing University of Aeronautics and Astronautics in
China. His current research interests include closed-loop com-
bustion control, adaptive control, closed-loop system identi-
fication, LPV control of automotive systems, hybrid power-
train control and optimization, and thermoelectric generator
management system. Dr. Zhu has over 30 years of experience
related to control theory and applications. He has authored or
co-authored more than 140 refereed technical papers and re-
ceived 40 US patents. He was an associate editor for ASME
Journal of Dynamic Systems, Measurement and Control and
a member of editorial board of International Journal of Pow-
ertrain. Dr. Zhu is a Fellow of SAE and ASME.
Jongeun Choi received his Ph.D. and
M.S. degrees in Mechanical Engineer-
ing from the University of California at
Berkeley in 2006 and 2002 respectively.
He also received a B.S. degree in Me-
chanical Design and Production Engineer-
ing from Yonsei University at Seoul, Re-
public of Korea in 1998. He is currently
an Associate Professor with the Depart-
ments of Mechanical Engineering and Electrical and Com-
puter Engineering at the Michigan State University. His cur-
rent research interests include systems and control, LPV and
robust control, system identification, and Bayesian methods,
with applications to mobile robotic sensors, environmental
adaptive sampling, engine control, human motor control sys-
tems, and biomedical problems. He is an Associate Editor for
the Journal of Dynamic Systems, Measurement and Control
(JDSMC) and a Co-Editor for a Special Issue on Stochas-
tic Models, Control, and Algorithms in Robotics in JDSMC.
His papers were finalists for the Best Student Paper Award
at the 24th American Control Conference (ACC) 2005 and
the Dynamic System and Control Conference (DSCC) 2011
and 2012. He was a recipient of an NSF CAREER Award in
2009. Dr. Choi is a member of ASME and IEEE.