arXiv:1705.01332v1  [cs.SY]  3 May 2017
1
LiDAR-based Control of Autonomous Rotorcraft
for the Inspection of Pier-like Structures: Proofs
Bruno J. Guerreiro, Carlos Silvestre, Rita Cunha, David Cabecinhas
Abstract—This is a complementary document to the paper
presented in [1], where more detailed proofs are provided for
some results. The main paper addresses the problem of trajectory
tracking control of autonomous rotorcraft in operation scenarios
where only relative position measurements obtained from LiDAR
sensors are possible. The proposed approach defines an alterna-
tive kinematic model, directly based on LiDAR measurements,
and uses a trajectory-dependent error space to express the
dynamic model of the vehicle. An LPV representation with
piecewise affine dependence on the parameters is adopted to
describe the error dynamics over a set of predefined operating
regions, and a continuous-time H2 control problem is solved
using LMIs and implemented within the scope of gain-scheduling
control theory. In this document, Section A presents the stability
analysis of the attitude inner-loop presented in [1, Section II.B],
whereas Section B presents a more detailed version of the stability
and performance guarantees for the LPV system, extending the
results presented in [1, Section IV.A].
APPENDIX A
INNER-LOOP DYNAMICS
This appendix presents a possible inner-loop stabilization
method, within the framework of feedback linearization and
Lyapunov stability methods, that results in a second-order
linear model for the pitch and roll angular motion, as well
as a first-order linear model for the yaw angular velocity.
Theorem 1 (Inner-loop Stability). Consider the control law
given by
next =S(ωωω) JB ωωω + JB Q
−1(λλλ)[ ˙Q(λλλ)ωωω−K2 ( ˙λλλ −e3 eT
3 uIL)
−ΠΠΠT
e3 K1 ΠΠΠe3 (λλλ −uIL)]
(1)
where K1 ∈R2 and K2 ∈R3 are positive definite diagonal
matrices and the inner-loop input vector is denoted as uIL =
uφ
uθ
u ˙ψ
T , accounting for the desired roll angle, pitch
This work was partially funded by the Macau Science and Technology De-
velopment Fund (FDCT), grant FDCT/048/2014/A1, by project MYRG2015-
00127-FST of the Univ. of Macau, by the European Union’s Horizon
2020 research and innovation programme under grant agreement No 731667
(MULTIDRONE), and by the Portuguese Foundation for Science and Tech-
nology (FCT), under projects LARSyS UID/EEA/50009/2013 and LOTUS
PTDC/EEI-AUT/5048/2014. The work of B. Guerreiro and R. Cunha were
respectively supported by the FCT Post-doc Grant SFRH/BPD/110416/2015
and FCT Investigator Programme IF/00921/2013. This publication reflects the
authors’ views only. The European Commission and other funding institutions
are not responsible for any use that may be made of the information it contains.
B.
Guerreiro
and
R.
Cunha
are
with
the
Institute
for
Systems
and
Robotics
(ISR-Lisbon),
LARSyS,
Instituto
Superior
T´ecnico
(IST),
Universidade
de
Lisboa,
Av.
Rovisco
Pais
1,
1049
Lisboa,
Portugal.
(bguerreiro@isr.tecnico.ulisboa.pt
and
rita@isr.tecnico.ulisboa.pt)
C. Silvestre and D. Cabecinhas are with the Department of Electrical and
Computer Engineering, Faculty of Science and Technology, University of
Macau, Taipa, Macau, China, and C. Silvestre is also on leave from IST,
Portugal. (csilvestre@umac.mo and dcabecinhas@umac.mo)
angle, and yaw angular rate, respectively. Then, the resulting
attitude dynamics is given by
¨λλλ = −K2 ( ˙λλλ −e3eT
3 uIL) −ΠΠΠT
e3 K1 ΠΠΠe3 (λλλ −uIL) .
(2)
which, for constant inputs, guarantees that the equilibrium
point (ΠΠΠe3λλλ, ˙λλλ) = (ΠΠΠe3uIL, e3eT
3 uIL) is exponentially stable.
Proof. Recalling the rigid body dynamics, the angular motion
equations can be written as
(
˙ωωω = J−1
B [−S(ωωω) JBωωω + next]
˙λλλ = Q(λλλ)ωωω
(3a)
(3b)
Noting that the angular velocity can be defined as ωωω =
Q−1(λλλ) ˙λλλ and taking the time derivative of angular kinematics,
the angular motion dynamics can be expressed as
¨λλλ = ˙Q(λλλ)ωωω + Q(λλλ) J−1
B [−S(ωωω) JB ωωω + next]
(4)
Consider the desired roll angle, pitch angle, and yaw rate to
be denoted as uφ ∈R, uθ ∈(−π/2, π/2), and u ˙ψ ∈R,
respectively, indicating that they will be considered as inputs
in the overall system. Let also the input vector of this inner-
loop system be defined as uIL =
uφ
uθ
u ˙ψ
T .
Thus, a feedback linearizing control law can be designed as
next =S(ωωω) JB ωωω + JB Q
−1(λλλ)[ ˙Q(λλλ)ωωω−K2 ( ˙λλλ −e3 eT
3 uIL)
−ΠΠΠT
e3 K1 ΠΠΠe3 (λλλ −uIL)]
(5)
where K1 ∈R2×2, K2 ∈R3×3, e3 =

0
0
1
T , and as
such e3eT
3 uIL =
0
0
u ˙ψ
T , while ΠΠΠe3 =
I2
02×1

such that ΠΠΠe3uIL =
uφ
uθ
T . Replacing the control law
(5) in the angular dynamics (4), it can be seen that the closed-
loop dynamics results in
¨λλλ = −K2 ( ˙λλλ −e3 eT
3 uIL) −ΠΠΠT
e3 K1 ΠΠΠe3 (λλλ −uIL) .
(6)
Considering that uIL ∈UIL ⊂R3 and define a state vector
as xIL =

ΠΠΠe3λλλ
˙λλλ
T ∈XIL, the resulting system is a linear
time-invariant (LTI) system of the form
˙xIL = AIL xIL + BIL uIL ,
where the system matrices are given by
AIL =

02×2
ΠΠΠe3
−ΠΠΠT
e3 K1
−K2

,
BIL =

02×3
ΠΠΠT
e3 K1 ΠΠΠe3 + K2 e3 eT
3

.
It can be seen that matrix AIL is Hurwitz for any positive
definite matrices K1 and K2, and, therefore, the system is
locally input-to-state stable, noting that the local part of the
2
result is a direct consequence of the domain of the Euler angles
not being R3.
Considering the matrices K1 = diag(kφ, kθ) and K2 =
diag(k ˙φ, k ˙θ, k ˙ψ) the system described in (6) can also be
written as







¨φ = −k ˙φ ˙φ −kφ (φ −uφ)
¨θ = −k ˙θ ˙θ −kθ (θ −uθ) .
¨ψ = −k ˙ψ ( ˙ψ −u ˙ψ)
(7a)
(7b)
(7c)
It can be seen that, with constant inputs and the change of
variables ˜φ = φ −uφ, ˜θ = θ −uθ, and ˙˜ψ = ˙ψ −u ˙ψ, the
resulting autonomous system is exponentially stable. Thus,
it can be concluded that the state variables φ, θ, and
˙ψ
converge exponentially to the constant inputs uφ, uθ, and u ˙ψ,
respectively.
APPENDIX B
CONTROLLER SYNTHESIS
In this section an LMI approach is used to tackle the
continuous-time state feedback H2 synthesis problem for
polytopic LPV systems. Consider a general LPV system of
the form
 ˙x = A(ξξξ)x + Bw(ξξξ)w + B(ξξξ)u
z = C(ξξξ)x + D(ξξξ)w + E(ξξξ)u
(8a)
(8b)
where x is the state, u is the control input, z denotes the error
signal to be controlled, and w denotes the exogenous input
signal. The system is parameterized by ξξξ, which is a possibly
time-varying parameter vector and belongs to the convex set
Ej = co(Ej
0). Here, the operator co(.) denotes the convex
hull of the elements of the argument set, Ej
0 = {ξξξ1, . . . ,ξξξnj},
where ξξξ1 to ξξξnj are the vertices of a polytope. It is also
noted that the controller synthesis presented in the following
subsection will only be valid for a specific operating region,
here represented by Ej ⊂E.
Applying the static state feedback law given by u = K x
to (8) results in the closed-loop system given by
Tzw(ξξξ) :=
 ˙x = Ac(ξξξ) x + Bc(ξξξ) w
z = Cc(ξξξ) x + Dc(ξξξ) w
(9a)
(9b)
where Tzw(ξξξ) denotes the resulting closed-loop operator from
the disturbance input w to the performance output z, and the
system matrices are defined as Ac(ξξξ) = A(ξξξ) + B(ξξξ) K,
Bc(ξξξ) = Bw(ξξξ), Cc(ξξξ) = C(ξξξ)+E(ξξξ) K and Dc(ξξξ) = D(ξξξ).
The closed-loop system can be characterized in terms of
quadratic stability using the following definition, where X ≻0
denotes that matrix X is positive definite.
Definition 1 (Quadratic stability, [2]). The system is said to
be quadratically stable if there exists a matrix X ≻0 such
that AT
c (ξξξ) X + X Ac(ξξξ) ≺0 is satisfied for all ξξξ ∈Ej.
It can be seen that testing for stability or solving the synthesis
problem without any further result, involves an infinite number
of LMIs. Thus, several different structures for LPV systems
have been proposed which reduce the problem to that of
solving a finite number of LMIs.
In this section, an affine polytopic description is adopted,
which can also be used to model a wide spectrum of systems
and, as shown in the results presented in [?], is an adequate
choice for the system at hand.
Definition 2 (Affine polytopic LPV system). The system (8)
is said to be a polytopic LPV system if the system matrix
P(ξξξ) =
A(ξξξ)
Bw(ξξξ)
B(ξξξ)
C(ξξξ)
D(ξξξ)
E(ξξξ)

(10)
verifies P(ξξξ) ∈co (P1, . . . , Pnr) for all ξξξ ∈Ej, where
Pi =

Ai
Bwi
Bi
Ci
Di
Ei

for all i = 1, . . . , nj. Moreover, if Ej is a polytopic set, such as
Ej = co(Ej
0), Ej
0 = {ξξξ1, . . . ,ξξξnj}, and P(ξξξ) depends affinely
on ξξξ, then Pi = P(ξξξi) for all i = 1, . . . , nj, i.e., the vertices of
the parameter set can be uniquely identified with the vertices
of the system.
This polytopic structure used with the following lemma,
enables the use of a powerful set of results.
Proposition 2 ( [3, Proposition 1.19]). Let f : Ej →R be a
convex function defined on the convex set Ej = co(Ej
0). Then,
for some γ ∈R, f(ξξξ) ≤γ for all ξξξ ∈Ej if and only if
f(ξξξ) ≤γ for all ξξξ ∈Ej
0.
Thus, the quadratic stability of an affine polytopic LPV system
can be easily established if there exists a matrix X ≻0 such
that AT
c (ξξξ) X + X Ac(ξξξ) ≺0 is satisfied for all ξξξ ∈Ej
0.
The H2 synthesis problem can be described as that of
finding a control matrix K that stabilizes the closed-loop
system and minimizes the H2-norm of Tzw(ξξξ), denoted by
∥Tzw(ξξξ)∥H2. It is assumed that matrix D(ξξξ) = 0 in order to
guarantee that ∥Tzw(ξξξ)∥H2 is finite for every internally sta-
bilizing and strictly proper controller. The following theorem
is used for controller design and relies on results available
in [4] and [3], after being rewritten for the case of polytopic
LPV systems. In the following, tr (.) denotes the trace of the
argument matrix.
Theorem 3 (Polytopic stability). If there are real matrices
X = XT ≻0, Y ≻0, and W such that

A(ξξξ)X+XAT(ξξξ)+B(ξξξ)W+WTBT(ξξξ) Bw(ξξξ)
BT
w(ξξξ)
−I

≺0 (11a)

Y
C(ξξξ) X + E(ξξξ) W
X CT (ξξξ) + WT ET (ξξξ)
X

≻0 (11b)
tr (Y)<γ2 (11c)
for all ξξξ ∈Ej
0, where the static feedback controller is defined
as K = W X−1, then, the closed-loop system is quadratically
stable and there exists an upper-bound γ for the continuous-
time H2-norm of the closed-loop operator Tzw(ξξξ) for all ξξξ ∈
Ej, i.e.,
∥Tzw(ξξξ)∥H2 < γ , ∀ξξξ ∈Ej .
(12)
Proof. Using Proposition 2 and assuming an affine polytopic
LPV system, it can be seen that satisfying the LMI system for
all ξξξ ∈Ej
0 is equivalent to satisfying the same system for all
3
ξξξ ∈Ej. The proof that the LMI system (11) implies (12) can
be obtained from the definition of H2-norm of Tzw(ξξξ). Let the
transfer function matrix of the closed-loop operator Tzw(ξξξ) be
denoted as Tizw(s) for some parameter vector ξξξi ∈Ej
0, and
defined as
Tizw(s) =
= Cc(ξξξi) [s I −Ac(ξξξi)]−1 Bc(ξξξi) + Dc(ξξξi)
= [C(ξξξi) + E(ξξξi) K] [s I −A(ξξξi) −B(ξξξi) K]−1 Bw(ξξξi)
+ D(ξξξi) .
Then, the H2-norm of Tizw(s) is defined as
∥Tizw∥2
H2 = 1
2 πtr
Z +∞
−∞
TH
izw(jω) Tizw(jω) dω

which using Parseval’s theorem can be rewritten as
∥Tizw∥2
H2 = tr
Z +∞
0
HT
i (t) Hi(t) dt

where Hi(t) is the impulse response matrix of Tizw(s).
Noting that the impulse response can be defined as
Hi(t) = Cc(ξξξi) eAc(ξξξi) t Bc(ξξξi) ,
it is possible to see that, after some algebraic manipulation,
∥Tizw∥2
H2 = tr
Cc(ξξξi) Wctr(ξξξi) CT
c (ξξξi)


,
where
Wctr(ξξξi) =
Z +∞
0
eAc(ξξξi) t Bc(ξξξi) BT
c (ξξξi) eAT
c (ξξξi) t dt
stands for the controllability Grammian, given by the symmet-
ric positive definite solution of the Lyapunov equation
Ac(ξξξi) Wctr(ξξξi) + Wctr(ξξξi) AT
c (ξξξi) + Bc(ξξξi) BT
c (ξξξi) = 0 .
It can be seen that, for each parameter vector ξξξi ∈Ej
0, γ is
an upper bound for the H2-norm of the closed-loop operator
Tizw if and only if there exists X ≻0, 0 ≺Wctr(ξξξi) ≺X
such that
Ac(ξξξi) X + X Ac(ξξξi)T + Bc(ξξξi) BT
c (ξξξi) ≺0
(13)
and
tr
Cc(ξξξi) X CT
c (ξξξi)


< γ2.
(14)
Equation (13) can be rewritten as

Ac(ξξξi) X + X AT
c (ξξξi) + Bc(ξξξi) BT
c (ξξξi)
0
0
−I

≺0 ,
that, using Schur complements, becomes
Ac(ξξξi) X + X AT
c (ξξξi)
X Bc(ξξξi)
BT
c (ξξξi) X
−I

≺0 ,
or equivalently, introducing the matrix W = K X yields

A(ξξξi)X+XAT(ξξξi)+B(ξξξi)W+WT BT(ξξξi) B(ξξξi)
BT (ξξξi)
−I

≺0. (15)
Under the conditions of the theorem, it can be seen that (15)
is satisfied for all ξξξi ∈Ej
0 and that (11b) can be written as

Y
Cc(ξξξi)
CT
c (ξξξi)
X−1

≻0 ,
which, applying once again Schur complements, implies that

Y −Cc(ξξξi) X CT
c (ξξξi)
0
0
X−1

≻0
and consequently
tr
Cc(ξξξi) X Cc(ξξξi)T 

< tr (Y) < γ2 ,
also, for all ξξξi ∈Ej
0. Thus, (14) is implied and the bound on the
H2-norm is established. As (13) is implied by the conditions
of the theorem, it can be noted that
Ac(ξξξi) X + X Ac(ξξξi)T ≺−Bc(ξξξi) BT
c (ξξξi) ⪯0 ,
for all ξξξi ∈Ej
0, implying the quadratic stability of the closed-
loop and, thus, concluding the proof.
With this result, the optimal solution for the continuous-time
H2 control problem is approximated through the minimization
of γ subject to the LMIs of Theorem 3.
REFERENCES
[1] B. J. Guerreiro, C. Silvestre, R. Cunha, and D. Cabecinhas, “Lidar-based
control of autonomous rotorcraft for the inspection of pier-like structures,”
IEEE Transactions in Control Systems Technology, 2017.
[2] S. Boyd, L. Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix
Inequality in System and Control Theory, ser. SIAM studies in applied
mathematics.
Philadelphia, PA: Society for Industrial and Applied
Mathematics, SIAM, 1994, vol. 15.
[3] C. Scherer and S. Weiland, “Lecture notes on linear matrix inequality
methods in control,” Dutch Institute of Systems and Control, 2000.
[4] L. Ghaoui and S.-I. Niculescu, Advances in Linear Matrix Inequality
Methods in Control. Philadelphia, PA: Society for Industrial and Applied
Mathematics, SIAM, 1999.
arXiv:1705.01332v1  [cs.SY]  3 May 2017
1
LiDAR-based Control of Autonomous Rotorcraft
for the Inspection of Pier-like Structures: Proofs
Bruno J. Guerreiro, Carlos Silvestre, Rita Cunha, David Cabecinhas
Abstract—This is a complementary document to the paper
presented in [1], where more detailed proofs are provided for
some results. The main paper addresses the problem of trajectory
tracking control of autonomous rotorcraft in operation scenarios
where only relative position measurements obtained from LiDAR
sensors are possible. The proposed approach defines an alterna-
tive kinematic model, directly based on LiDAR measurements,
and uses a trajectory-dependent error space to express the
dynamic model of the vehicle. An LPV representation with
piecewise affine dependence on the parameters is adopted to
describe the error dynamics over a set of predefined operating
regions, and a continuous-time H2 control problem is solved
using LMIs and implemented within the scope of gain-scheduling
control theory. In this document, Section A presents the stability
analysis of the attitude inner-loop presented in [1, Section II.B],
whereas Section B presents a more detailed version of the stability
and performance guarantees for the LPV system, extending the
results presented in [1, Section IV.A].
APPENDIX A
INNER-LOOP DYNAMICS
This appendix presents a possible inner-loop stabilization
method, within the framework of feedback linearization and
Lyapunov stability methods, that results in a second-order
linear model for the pitch and roll angular motion, as well
as a first-order linear model for the yaw angular velocity.
Theorem 1 (Inner-loop Stability). Consider the control law
given by
next =S(ωωω) JB ωωω + JB Q
−1(λλλ)[ ˙Q(λλλ)ωωω−K2 ( ˙λλλ −e3 eT
3 uIL)
−ΠΠΠT
e3 K1 ΠΠΠe3 (λλλ −uIL)]
(1)
where K1 ∈R2 and K2 ∈R3 are positive definite diagonal
matrices and the inner-loop input vector is denoted as uIL =
uφ
uθ
u ˙ψ
T , accounting for the desired roll angle, pitch
This work was partially funded by the Macau Science and Technology De-
velopment Fund (FDCT), grant FDCT/048/2014/A1, by project MYRG2015-
00127-FST of the Univ. of Macau, by the European Union’s Horizon
2020 research and innovation programme under grant agreement No 731667
(MULTIDRONE), and by the Portuguese Foundation for Science and Tech-
nology (FCT), under projects LARSyS UID/EEA/50009/2013 and LOTUS
PTDC/EEI-AUT/5048/2014. The work of B. Guerreiro and R. Cunha were
respectively supported by the FCT Post-doc Grant SFRH/BPD/110416/2015
and FCT Investigator Programme IF/00921/2013. This publication reflects the
authors’ views only. The European Commission and other funding institutions
are not responsible for any use that may be made of the information it contains.
B.
Guerreiro
and
R.
Cunha
are
with
the
Institute
for
Systems
and
Robotics
(ISR-Lisbon),
LARSyS,
Instituto
Superior
T´ecnico
(IST),
Universidade
de
Lisboa,
Av.
Rovisco
Pais
1,
1049
Lisboa,
Portugal.
(bguerreiro@isr.tecnico.ulisboa.pt
and
rita@isr.tecnico.ulisboa.pt)
C. Silvestre and D. Cabecinhas are with the Department of Electrical and
Computer Engineering, Faculty of Science and Technology, University of
Macau, Taipa, Macau, China, and C. Silvestre is also on leave from IST,
Portugal. (csilvestre@umac.mo and dcabecinhas@umac.mo)
angle, and yaw angular rate, respectively. Then, the resulting
attitude dynamics is given by
¨λλλ = −K2 ( ˙λλλ −e3eT
3 uIL) −ΠΠΠT
e3 K1 ΠΠΠe3 (λλλ −uIL) .
(2)
which, for constant inputs, guarantees that the equilibrium
point (ΠΠΠe3λλλ, ˙λλλ) = (ΠΠΠe3uIL, e3eT
3 uIL) is exponentially stable.
Proof. Recalling the rigid body dynamics, the angular motion
equations can be written as
(
˙ωωω = J−1
B [−S(ωωω) JBωωω + next]
˙λλλ = Q(λλλ)ωωω
(3a)
(3b)
Noting that the angular velocity can be defined as ωωω =
Q−1(λλλ) ˙λλλ and taking the time derivative of angular kinematics,
the angular motion dynamics can be expressed as
¨λλλ = ˙Q(λλλ)ωωω + Q(λλλ) J−1
B [−S(ωωω) JB ωωω + next]
(4)
Consider the desired roll angle, pitch angle, and yaw rate to
be denoted as uφ ∈R, uθ ∈(−π/2, π/2), and u ˙ψ ∈R,
respectively, indicating that they will be considered as inputs
in the overall system. Let also the input vector of this inner-
loop system be defined as uIL =
uφ
uθ
u ˙ψ
T .
Thus, a feedback linearizing control law can be designed as
next =S(ωωω) JB ωωω + JB Q
−1(λλλ)[ ˙Q(λλλ)ωωω−K2 ( ˙λλλ −e3 eT
3 uIL)
−ΠΠΠT
e3 K1 ΠΠΠe3 (λλλ −uIL)]
(5)
where K1 ∈R2×2, K2 ∈R3×3, e3 =

0
0
1
T , and as
such e3eT
3 uIL =
0
0
u ˙ψ
T , while ΠΠΠe3 =
I2
02×1

such that ΠΠΠe3uIL =
uφ
uθ
T . Replacing the control law
(5) in the angular dynamics (4), it can be seen that the closed-
loop dynamics results in
¨λλλ = −K2 ( ˙λλλ −e3 eT
3 uIL) −ΠΠΠT
e3 K1 ΠΠΠe3 (λλλ −uIL) .
(6)
Considering that uIL ∈UIL ⊂R3 and define a state vector
as xIL =

ΠΠΠe3λλλ
˙λλλ
T ∈XIL, the resulting system is a linear
time-invariant (LTI) system of the form
˙xIL = AIL xIL + BIL uIL ,
where the system matrices are given by
AIL =

02×2
ΠΠΠe3
−ΠΠΠT
e3 K1
−K2

,
BIL =

02×3
ΠΠΠT
e3 K1 ΠΠΠe3 + K2 e3 eT
3

.
It can be seen that matrix AIL is Hurwitz for any positive
definite matrices K1 and K2, and, therefore, the system is
locally input-to-state stable, noting that the local part of the
2
result is a direct consequence of the domain of the Euler angles
not being R3.
Considering the matrices K1 = diag(kφ, kθ) and K2 =
diag(k ˙φ, k ˙θ, k ˙ψ) the system described in (6) can also be
written as







¨φ = −k ˙φ ˙φ −kφ (φ −uφ)
¨θ = −k ˙θ ˙θ −kθ (θ −uθ) .
¨ψ = −k ˙ψ ( ˙ψ −u ˙ψ)
(7a)
(7b)
(7c)
It can be seen that, with constant inputs and the change of
variables ˜φ = φ −uφ, ˜θ = θ −uθ, and ˙˜ψ = ˙ψ −u ˙ψ, the
resulting autonomous system is exponentially stable. Thus,
it can be concluded that the state variables φ, θ, and
˙ψ
converge exponentially to the constant inputs uφ, uθ, and u ˙ψ,
respectively.
APPENDIX B
CONTROLLER SYNTHESIS
In this section an LMI approach is used to tackle the
continuous-time state feedback H2 synthesis problem for
polytopic LPV systems. Consider a general LPV system of
the form
 ˙x = A(ξξξ)x + Bw(ξξξ)w + B(ξξξ)u
z = C(ξξξ)x + D(ξξξ)w + E(ξξξ)u
(8a)
(8b)
where x is the state, u is the control input, z denotes the error
signal to be controlled, and w denotes the exogenous input
signal. The system is parameterized by ξξξ, which is a possibly
time-varying parameter vector and belongs to the convex set
Ej = co(Ej
0). Here, the operator co(.) denotes the convex
hull of the elements of the argument set, Ej
0 = {ξξξ1, . . . ,ξξξnj},
where ξξξ1 to ξξξnj are the vertices of a polytope. It is also
noted that the controller synthesis presented in the following
subsection will only be valid for a specific operating region,
here represented by Ej ⊂E.
Applying the static state feedback law given by u = K x
to (8) results in the closed-loop system given by
Tzw(ξξξ) :=
 ˙x = Ac(ξξξ) x + Bc(ξξξ) w
z = Cc(ξξξ) x + Dc(ξξξ) w
(9a)
(9b)
where Tzw(ξξξ) denotes the resulting closed-loop operator from
the disturbance input w to the performance output z, and the
system matrices are defined as Ac(ξξξ) = A(ξξξ) + B(ξξξ) K,
Bc(ξξξ) = Bw(ξξξ), Cc(ξξξ) = C(ξξξ)+E(ξξξ) K and Dc(ξξξ) = D(ξξξ).
The closed-loop system can be characterized in terms of
quadratic stability using the following definition, where X ≻0
denotes that matrix X is positive definite.
Definition 1 (Quadratic stability, [2]). The system is said to
be quadratically stable if there exists a matrix X ≻0 such
that AT
c (ξξξ) X + X Ac(ξξξ) ≺0 is satisfied for all ξξξ ∈Ej.
It can be seen that testing for stability or solving the synthesis
problem without any further result, involves an infinite number
of LMIs. Thus, several different structures for LPV systems
have been proposed which reduce the problem to that of
solving a finite number of LMIs.
In this section, an affine polytopic description is adopted,
which can also be used to model a wide spectrum of systems
and, as shown in the results presented in [?], is an adequate
choice for the system at hand.
Definition 2 (Affine polytopic LPV system). The system (8)
is said to be a polytopic LPV system if the system matrix
P(ξξξ) =
A(ξξξ)
Bw(ξξξ)
B(ξξξ)
C(ξξξ)
D(ξξξ)
E(ξξξ)

(10)
verifies P(ξξξ) ∈co (P1, . . . , Pnr) for all ξξξ ∈Ej, where
Pi =

Ai
Bwi
Bi
Ci
Di
Ei

for all i = 1, . . . , nj. Moreover, if Ej is a polytopic set, such as
Ej = co(Ej
0), Ej
0 = {ξξξ1, . . . ,ξξξnj}, and P(ξξξ) depends affinely
on ξξξ, then Pi = P(ξξξi) for all i = 1, . . . , nj, i.e., the vertices of
the parameter set can be uniquely identified with the vertices
of the system.
This polytopic structure used with the following lemma,
enables the use of a powerful set of results.
Proposition 2 ( [3, Proposition 1.19]). Let f : Ej →R be a
convex function defined on the convex set Ej = co(Ej
0). Then,
for some γ ∈R, f(ξξξ) ≤γ for all ξξξ ∈Ej if and only if
f(ξξξ) ≤γ for all ξξξ ∈Ej
0.
Thus, the quadratic stability of an affine polytopic LPV system
can be easily established if there exists a matrix X ≻0 such
that AT
c (ξξξ) X + X Ac(ξξξ) ≺0 is satisfied for all ξξξ ∈Ej
0.
The H2 synthesis problem can be described as that of
finding a control matrix K that stabilizes the closed-loop
system and minimizes the H2-norm of Tzw(ξξξ), denoted by
∥Tzw(ξξξ)∥H2. It is assumed that matrix D(ξξξ) = 0 in order to
guarantee that ∥Tzw(ξξξ)∥H2 is finite for every internally sta-
bilizing and strictly proper controller. The following theorem
is used for controller design and relies on results available
in [4] and [3], after being rewritten for the case of polytopic
LPV systems. In the following, tr (.) denotes the trace of the
argument matrix.
Theorem 3 (Polytopic stability). If there are real matrices
X = XT ≻0, Y ≻0, and W such that

A(ξξξ)X+XAT(ξξξ)+B(ξξξ)W+WTBT(ξξξ) Bw(ξξξ)
BT
w(ξξξ)
−I

≺0 (11a)

Y
C(ξξξ) X + E(ξξξ) W
X CT (ξξξ) + WT ET (ξξξ)
X

≻0 (11b)
tr (Y)<γ2 (11c)
for all ξξξ ∈Ej
0, where the static feedback controller is defined
as K = W X−1, then, the closed-loop system is quadratically
stable and there exists an upper-bound γ for the continuous-
time H2-norm of the closed-loop operator Tzw(ξξξ) for all ξξξ ∈
Ej, i.e.,
∥Tzw(ξξξ)∥H2 < γ , ∀ξξξ ∈Ej .
(12)
Proof. Using Proposition 2 and assuming an affine polytopic
LPV system, it can be seen that satisfying the LMI system for
all ξξξ ∈Ej
0 is equivalent to satisfying the same system for all
3
ξξξ ∈Ej. The proof that the LMI system (11) implies (12) can
be obtained from the definition of H2-norm of Tzw(ξξξ). Let the
transfer function matrix of the closed-loop operator Tzw(ξξξ) be
denoted as Tizw(s) for some parameter vector ξξξi ∈Ej
0, and
defined as
Tizw(s) =
= Cc(ξξξi) [s I −Ac(ξξξi)]−1 Bc(ξξξi) + Dc(ξξξi)
= [C(ξξξi) + E(ξξξi) K] [s I −A(ξξξi) −B(ξξξi) K]−1 Bw(ξξξi)
+ D(ξξξi) .
Then, the H2-norm of Tizw(s) is defined as
∥Tizw∥2
H2 = 1
2 πtr
Z +∞
−∞
TH
izw(jω) Tizw(jω) dω

which using Parseval’s theorem can be rewritten as
∥Tizw∥2
H2 = tr
Z +∞
0
HT
i (t) Hi(t) dt

where Hi(t) is the impulse response matrix of Tizw(s).
Noting that the impulse response can be defined as
Hi(t) = Cc(ξξξi) eAc(ξξξi) t Bc(ξξξi) ,
it is possible to see that, after some algebraic manipulation,
∥Tizw∥2
H2 = tr
Cc(ξξξi) Wctr(ξξξi) CT
c (ξξξi)


,
where
Wctr(ξξξi) =
Z +∞
0
eAc(ξξξi) t Bc(ξξξi) BT
c (ξξξi) eAT
c (ξξξi) t dt
stands for the controllability Grammian, given by the symmet-
ric positive definite solution of the Lyapunov equation
Ac(ξξξi) Wctr(ξξξi) + Wctr(ξξξi) AT
c (ξξξi) + Bc(ξξξi) BT
c (ξξξi) = 0 .
It can be seen that, for each parameter vector ξξξi ∈Ej
0, γ is
an upper bound for the H2-norm of the closed-loop operator
Tizw if and only if there exists X ≻0, 0 ≺Wctr(ξξξi) ≺X
such that
Ac(ξξξi) X + X Ac(ξξξi)T + Bc(ξξξi) BT
c (ξξξi) ≺0
(13)
and
tr
Cc(ξξξi) X CT
c (ξξξi)


< γ2.
(14)
Equation (13) can be rewritten as

Ac(ξξξi) X + X AT
c (ξξξi) + Bc(ξξξi) BT
c (ξξξi)
0
0
−I

≺0 ,
that, using Schur complements, becomes
Ac(ξξξi) X + X AT
c (ξξξi)
X Bc(ξξξi)
BT
c (ξξξi) X
−I

≺0 ,
or equivalently, introducing the matrix W = K X yields

A(ξξξi)X+XAT(ξξξi)+B(ξξξi)W+WT BT(ξξξi) B(ξξξi)
BT (ξξξi)
−I

≺0. (15)
Under the conditions of the theorem, it can be seen that (15)
is satisfied for all ξξξi ∈Ej
0 and that (11b) can be written as

Y
Cc(ξξξi)
CT
c (ξξξi)
X−1

≻0 ,
which, applying once again Schur complements, implies that

Y −Cc(ξξξi) X CT
c (ξξξi)
0
0
X−1

≻0
and consequently
tr
Cc(ξξξi) X Cc(ξξξi)T 

< tr (Y) < γ2 ,
also, for all ξξξi ∈Ej
0. Thus, (14) is implied and the bound on the
H2-norm is established. As (13) is implied by the conditions
of the theorem, it can be noted that
Ac(ξξξi) X + X Ac(ξξξi)T ≺−Bc(ξξξi) BT
c (ξξξi) ⪯0 ,
for all ξξξi ∈Ej
0, implying the quadratic stability of the closed-
loop and, thus, concluding the proof.
With this result, the optimal solution for the continuous-time
H2 control problem is approximated through the minimization
of γ subject to the LMIs of Theorem 3.
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