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Mixed H2/H∞ Control via Nonsmooth Optimization
P. Apkarian, Dominikus Noll, Aude Rondepierre
To cite this version:
P. Apkarian, Dominikus Noll, Aude Rondepierre. Mixed H2/H∞ Control via Nonsmooth Optimiza-
tion. SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics,
2008, 47 (3), pp.1516-1546. �10.1137/070685026�. �hal-00634512�
MIXED H
2
/H
∞
CONTROL VIA NONSMOOTH OPTIMIZATION
P. APKARIAN
∗
, D. NOLL
†
, AND A. RONDEPIERRE
† ‡
Abstract. We present a new approach to mixed H
2
/H
∞
output feedback control synthesis.
Our method uses non-smooth mathematical programming techniques to compute locally optimal
H
2
/H
∞
-controllers, which may have a pre-defined structure. We prove global convergence of our
method and present numerical tests to validate it numerically.
Key words. Mixed H
2
/H
∞
output feedback control, multi-objective control, robustness and
performance, non-smooth optimization, trust region technique.
AMS subject classifications. 93B36, 93B50, 90C29, 49J52, 90C26, 90C34, 49J35
1. Introduction. Mixed H
2
/H
∞
output feedback control is a prominent exam-
ple of a multi-objective design problem, where the feedback controller has to respond
favorably to several performance specifications. Typically in H
2
/H
∞
synthesis, the
H
∞
channel is used to enhance the robustness of the design, whereas the H
2
channel
guarantees good performance of the system. Due to its importance in practice, mixed
H
2
/H
∞
control has been addressed in various ways over the years, and we briefly
review the main trends.
The interest in H
2
/H
∞
synthesis was originally risen by three publications [22,
23, 27] in the late 1980s and early 1990s. The numerical methods proposed by these
authors are based on coupled Riccati equations in tandem with homotopy metho ds,
but the numerical success of these strategies remains to be established. With the
rise of L MIs in the later 1990s, different strategies which convexify the problem be-
came increasingly popular. The price to pay for convexifying is either a considerable
conservatism, or that controllers have large state dimension [29, 25].
In [45, 47, 48] Scherer developed LMI formations for H
2
/H
∞
synthesis for full-
order controllers [48], and reduced the problem to solving LMIs in tandem with non-
linear algebraic equalities [48, 45]. In this form, H
2
/H
∞
problems could in principle
be solved via nonlinear semidefinite programming techniques like specSDP [24, 39, 49]
or Pennon [31, 32, 36], if only these techniques were suited for medium or large size
problems. Alas, one of the disappointing lessons learned in recent years from investi-
gating BMI and LMI problems is that this is just not the case. Due to the presence of
Lyapunov variables, whose number grows quadratically with the sys tem size, [13, p.
20ff], BMI and LMI programs quickly lead to problem sizes where existing numerical
methods fail.
Following [3, 4, 5, 6, 7], we address H
2
/H
∞
-synthesis by a new strategy which
avoids the use of Lyapunov variables. This leads to a non-smooth and semi-infinite
optimization program, which we solve with a spectral bundle method, inspired by
the non-convex spectral bundle method of [37, 38] and [3, 5]. Important forerunners
[19, 40, 28] are based on convexity and optimize functions of the form λ
1
◦A with affine
A. We have developed our method further to deal with typical control applications like
multi-disk [7] and multi frequency band synthesis [6], design under integral quadratic
constraints (IQCs) [4, 9, 8], and to loop-shaping techniques [2, 1].
∗
CERT-ONERA, 2, avenue Edouard Belin, 31055 Toulouse, France
†
Universit´e Paul Sabatier, Institut de Math´ematiques, 118 route de Narbonne, 31062 Toulouse,
France
‡
Corresponding author
1
2 P. Apkarian, D. Noll and A. Rondepierre
The structure of the paper is as follows. The problem setting is given in Section 2.
Computing the H
2
and H
∞
norm is briefly r ecalled in Sections 3 and 4. The algorithm
and its rationale are presented in Section 5. Global convergence is established in
Section 6. The implementation is discussed in Section 7, and numerical test examples
are discussed in Section 8.
2. Problem setting. We consider a plant in state space form
P :
˙x
z
∞
z
2
y
=
A
B
∞
B
2
B
C
∞
D
∞
0 D
∞u
C
2
0 0 D
2u
C
D
y ∞
D
y 2
0
x
w
∞
w
2
u
(2.1)
where x ∈ R
n
x
is the state, u ∈ R
n
u
the control, y ∈ R
n
y
the output, and where
w
∞
→ z
∞
is the H
∞
, w
2
→ z
2
the H
2
performance channel. We seek an output
feedback controller
K :
˙x
K
u
=
A
K
B
K
C
K
D
K
x
K
y
(2.2)
where x
K
∈ R
n
K
is the state of the controller, such that the closed-loop system,
obtained by s ubstituting (2.2) into (2.1), satisfies the following properties:
1. Internal stability. K stabilizes P exponentially in closed-loop.
2. Fixed H
∞
performance. The H
∞
performance channel has a pre-specified
performance level kT
w
∞
→z
∞
(K)k
∞
≤ γ
∞
.
3. Optimal H
2
performance. The H
2
performance kT
w
2
→z
2
(K)k
2
is mini-
mized among all K satisfying 1. and 2.
We will solve the H
2
/H
∞
synthesis problem by way of the following mathematical
program
minimize f(K) := kT
w
2
→z
2
(K)k
2
2
subject to g(K) := kT
w
∞
→z
∞
(K)k
2
∞
≤ γ
2
∞
(2.3)
where T
w
2
→z
2
(K, s) denotes the transfer function of the H
2
closed-loop performance
channel, while T
w
∞
→z
∞
(K, s) stands for the H
∞
robustness channel. Notice that
f(K) is a smooth function, whereas g(K) is not, being an infinite maximum of max-
imum eigenvalue functions. The unknown K is in the space R
(n
K
+n
u
)×(n
K
+n
y
)
, so
the dimension n = (n
K
+ n
y
)(n
K
+ n
u
) of (2.3) is usually small, which is particularly
attractive when small or medium size controllers for large systems are s ought. Notice
that as a BMI or LMI problem, H
2
/H
∞
synthesis (2.3) would feature n
2
x
additional
Lyapunov variables, which would arise through the use of the bounded real lemma.
See e.g. [46, 13].
Remark. Naturally, the approach chosen in (2.3) to fix the H
∞
performance and
optimize H
2
performance is just one among many other strategies in multi-objective
optimization. One could just as well optimize the H
∞
norm subject to a H
2
-norm
constraint, or minimize a weighted sum or even the maximum of both criteria. Other
ideas have b een considered, and even game theoretic approaches exist [35].
3. The H
2
norm. In program (2.3) we minimize composite functions f = k·k
2
2
◦
T
w
2
→z
2
, where k · k
2
denotes the H
2
-norm. Let us for brevity write T
2
:= T
w
2
→z
2
for
the H
2
transfer channel in (2.1). The corresponding plant P
2
is obtained by deleting
the w
∞
column and the z
∞
line in P . The objective function can be written as
f(K) = kT
2
(K, ·)k
2
2
=
1
2π
Z
+∞
−∞
Tr(T
2
(K, jω)
H
T
2
(K, jω))dω.
Mixed H
2
/H
∞
control via nonsmooth optimization 3
Algorithmically it is convenient to compute function values using a state space real-
ization of P
2
:
P
2
(s) =
0 D
2u
D
y 2
0
+
C
2
C
(sI − A)
−1
[ B
2
B ].
Introducing the closed-loop state space data:
A(K) =
A + BD
K
C BC
K
B
K
C A
K
, B
2
(K) =
B
2
+ BD
K
D
y 2
B
K
D
y 2
,
C
2
(K) = [ C
2
+ D
2u
D
K
C D
2u
C
K
], D
2
(K) = D
2u
D
K
D
y 2
= 0,
we either assume D
2u
= 0 or D
y 2
= 0, or that the controller K is strictly proper,
to ensure finiteness of the H
2
norm. Then a realization of the closed-loop transfer
function T
2
is given as:
T
2
(K, s) = C
2
(K)(sI − A(K))
−1
B
2
(K)
and (see e.g. [21]) the objective function f may be re-written as
f(K) = Tr(B
2
(K)
T
X(K) B
2
(K)) = Tr(C
2
(K) Y (K) C
2
(K)
T
),
where X(K) and Y (K) are the solutions of two Lyapunov equations:
A(K)
T
X(K) + X(K)A(K) + C
2
(K)
T
C
2
(K) = 0,
A(K)Y (K) + Y (K)A(K)
T
+ B
2
(K)B
2
(K)
T
= 0.
(3.1)
As observed in [42, Section 3], one proves differentiability of the objective f over
the set D of closed-loop stabilizing controllers K. In order to write the derivative
f
′
(K)dK in a gradient form, we introduce the gradient ∇f(K) of f at K defined by:
f
′
(K)dK = Tr[∇f(K)
T
dK],
meaning that ∇f(K) is now an element of the same matrix space as K. These results
lead to the following lemma which is an extension of [42, Theorem 3.2.]:
Lemma 3.1. The objective function f is differentiable on the open set D of
closed-loop stabilizing gains. For K ∈ D, the gradient of f at K is:
∇f(K) = 2
B
T
X(K) + D
T
2u
C
2
(K)
Y (K)C
T
+ 2B
T
X(K)B
2
(K)D
T
y 2
,
where X(K) and Y (K) solve (3.1).
4. The H
∞
-norm. The next element required in (2.3) is the constraint function
g = k · k
2
∞
◦ T
w
∞
→z
∞
, a composite function of the H
∞
-norm. To compute it we will
use a frequency domain repres entation of the H
∞
norm. Let us for brevity write
T
∞
:= T
w
∞
→z
∞
. The corresponding plant is P
∞
, obtained by deleting the w
2
column
and the z
2
line in P . The constraint function g may be written as
g(K) = max
ω ∈[ 0, ∞]
σ (T
∞
(K, jω))
2
= max
ω ∈[ 0, ∞]
λ
1
T
∞
(K, jω)
H
T
∞
(K, jω)
,
where
σ is the maximum singular value of a matrix, λ
1
the maximum eigenvalue of a
Hermitian matrix. We re-write this as
g(K) = max
ω ∈[ 0, ∞]
g(K, ω), g(K, ω) = λ
1
T
∞
(K, jω)
H
T
∞
(K, jω)
.
4 P. Apkarian, D. Noll and A. Rondepierre
Then it is clear that g(K) is nonsmooth with two possible sources of non-smoothness,
the infinite maximum, and the maximum eigenvalue function, which is convex but
nonsmooth. We present two basic results, which allow to exploit the structure of g
algorithmically. The following can be found in several places, e.g. [12, 11]:
Lemma 4.1. Let K be closed-loop stabilizing. Then g(K) = kT
∞
(K)k
2
∞
< ∞, and
the set of active frequencies at K, defined as Ω(K) = {ω ∈ [0, ∞] : g(K) = g(K, ω)}
is either finite, or Ω(K) = [0, ∞].
The case Ω(K) = [0, ∞] is when the closed-loop system is all-pass. It may very
well arise in practice, for instance, full order (n
x
= n
K
) optimal H
∞
controllers are
all-pass; see [26]. A similar result holds for full order H
2
/H
∞
-control; see [20]. But
we never observed it in cases where the order of the controller n
K
< n
x
is way smaller
than the order of the system.
The following result was already used in [5, 7]. It allows to compute Clarke
subgradients of the H
∞
norm and its comp osite function g. To represent it, we find
it convenient to introduce the notation
T
∞
(K, s) G
∞
12
(K, s)
G
∞
21
(K, s) ⋆
=
C
∞
(K)
C
(sI − A(K))
−1
[ B
∞
(K) B ]
+
D
∞
(K) D
∞u
D
y ∞
⋆
where the closed-loop state-space data (A(K), B
∞
(K), C
∞
(K), D
∞
(K)) are given by:
A(K) =
A + BD
K
C BC
K
B
K
C A
K
, B
∞
(K) =
B
∞
+ BD
K
D
y ∞
B
K
D
y ∞
,
C
∞
(K) =
C
∞
+ D
∞u
D
K
C D
∞u
C
K
, D
∞
(K) = D
∞
+ D
∞u
D
K
D
y ∞
.
Lemma 4.2. (See [5, Section IV], [14, p. 304]). Suppose K is closed-loo p
stabilizing and Ω(K) is finite. Then the Clarke subdifferential of g at K is the set
∂g(K) =
Φ
Y
: Y = (Y
ω
)
ω ∈Ω( K)
, Y
ω
0,
X
ω ∈Ω( K)
Tr(Y
ω
) = 1, Y
ω
∈ S
r
ω
,
where r
ω
is the multiplicity of λ
1
T
∞
(K, jω)
H
T
∞
(K, jω)
, and where
Φ
Y
=
X
ω ∈Ω( K)
2Re
G
∞
21
(K, jω)T
∞
(K, jω)
H
Q
ω
Y
ω
Q
H
ω
G
∞
12
(K, jω)
T
.
Here the columns of the m×r
ω
matrix Q
ω
form an orthonormal basis of the eigenspace
of T
∞
(K, jω)
H
T
∞
(K, jω) ∈ S
m
associated with its maximum eigenvalue.
Remark. Notice that the result extends to the all-pass case by replacing convex
combinations over a finite set Ω(K) by Radon probability measures on [0, ∞]. This
may still be exploited algorithmically, should the case of an all-pass system ever arise
in practice. Since this never occurred in our tests, this line is not investigated here.
5. Nonsmooth algorithm. In this central Section we present our main result,
a nonsmooth and nonconvex optimization method for program (2.3). In subsection
5.1 we will have a look at the necessary optimality conditions for program (2.3). The
![Fig. 7.1. H2/H∞ optimal static controllers K(γ∞) = (K1(γ∞),K2(γ∞)) ∈ R2 for the vehicular suspension control problem. [‖T∞(K∞)‖∞, ‖T∞(K2)‖∞] ∋ γ∞ 7→ K(γ∞) continuously transforms the H∞ optimal gain K∞ into the H2 optimal gain K2.](/figures/fig-7-1-h2-h-optimal-static-controllers-k-g-k1-g-k2-g-r2-for-19dpemm3.webp)