Aalborg Universitet
State Space Solutions to the H-infinity/LTR Design Problem
Stoustrup, Jakob; Niemann, H.H.
Published in:
International Journal of Robust and Nonlinear Control
DOI (link to publication from Publisher):
10.1002/rnc.4590030102
Publication date:
1993
Document Version
Tidlig version også kaldet pre-print
Link to publication from Aalborg University
Citation for published version (APA):
Stoustrup, J., & Niemann, H. H. (1993). State Space Solutions to the H-infinity/LTR Design Problem.
International Journal of Robust and Nonlinear Control, 1-45. https://doi.org/10.1002/rnc.4590030102
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INTERNATIONAL JOURNAL
OF
ROBUST AND NONLINEAR CONTROL, VOL.
3,1-45
(1993)
STATE-SPACE SOLUTIONS TO THE %/LTR
DESIGN
PROBLEM
JAKOB STOUSTRUP AND HANS HENRIK NIEMANN
Mathematical Institute, Technical University
of
Denmark, Building
303,
DK-2800
Lyngby, Denmark
SUMMARY
The LTR design problem using an
JC
optimality criterion is presented for two types of recovery errors,
the sensitivity recovery error and the input-output recovery error.
For
both errors two different
approaches are presented. First, following the classical LTR design philosophy, a Luenberger observer
based approach is proposed, where the
Z
part of the controller is appended to a standard full-order
observer. Second, allowing
for
general controllers,
an
JC
state-space problem is formulated directly from
the recovery errors. Both approaches lead to controller orders
of
at most
2n.
In the minimum phase case,
though, the order of the controllers can be reduced to
n
in all cases. The control problems corresponding
to the various controller types are given as four different singular state-space problems, and the
solutions are given in terms of the relevant equations and inequalities.
KEY
WORDS
Loop transfer recovery Singular
25%
theory Luenberger observer
Youla parameterization
1. INTRODUCTION
In the last decade the concept of loop transfer recovery (LTR) has emerged as an important
approach to the design of robust feedback controllers. The attractive theoretical properties
of
such controllers in combination with its conceptual and computational simplicity has
motivated its popularity and spread in the control community for continuous time
systems
L45,8,11.18.19,?.4,25
and for discrete time
system^.^"^.'^
The LTR philosophy establishes
a systematic two-step method for the design of dynamical measurement feedback controllers.
The first step
is
to design
a
static state feedback which performs according to the specifications.
The second step
is
to design
a
dynamical measurement feedback controller, which ‘behaves
almost’ like the static state feedback. In the two steps entirely different design methodologies
can be applied, which make LTR an attractive alternative to ‘one-shot’ methods. The objective
of this paper is
to
provide a complete
%
design method for the second step.
Design methods as LQGILTR, ES/LTR (i.e. eigenstructure assignment LTR) and singular
perturbation LTR are
all
based on sufficient conditions for obtaining recovery. Further,
practically only controllers based on full-order
or
minimal-order observers have been used for
recovery design.
As
a
result
of
these two drawbacks no guarantee can be given that the best
controller type
is
selected for a given problem.
This
work
is
supported in part
by
the Danish Technical Research Council, under grant no.
16-4885-1.
This paper was recommended
for
publication by editor
M,
J.
Grimble
1O49-8923/93/01ooO1-45$27.50
0
1993 by John Wiley
&
Sons, Ltd.
Received
6
November 1990
Revised 20 February 1992
2
J.
STOUSTRUP
AND
H.
H.
NIEMANN
A first step towards
a
more systematic description of conditions for obtaining recovery has
been done by Goodman' by the introduction of the open loop recovery error for full order
observer based controllers. In Reference
14
the recovery error concept has been extended to
both open and closed loop recovery errors and more general controllers have been investigated:
the Luenberger observed based controller and the output feedback controller (the unknown
input observer based controller). In this work, it was shown that
a
certain matrix-valued
function, the so-called recovery matrix, plays
a
crucial role in the LTR problem. In the
asymptotic recovery case, it turns out that both recovery error types tend to zero if and only
if the recovery matrix does. Accordingly, both necessary and sufficient conditions for achieving
asymptotic recovery were obtained by the study
of
the recovery matrix.
Methods as LQG/LTR, ES/LTR and singular perturbation LTR are mainly ad hoc, in the
sense that they try to reduce
a
more or less unspecified part of the system. In References
21
and
23
an Z/LTR problem is formulated in order to reduce the norm of the recovery matrix.
We shall refer to this method as an
indirect
design method. One significant problem caused
by using indirect design methods is that no guarantee
exists
that the norm
of
the recovery
errors decreases when the norm of the recovery matrix decreases (except in the asymptotic
recovery case-see above). In fact, in the course of this paper we shall study an example, where
the recovery errors even increase when the norm
of
the recovery matrix decreases.
Moreover, for non-minimum phase systems, most contributions
so
far deal only with
analysis.
17~1929
Hence, there is
a
need for systematic design procedures.
A
more advantageous
approach to LTR controller design is therefore to use the recovery errors directly in the
recovery design problem formulation, which we shall call the direct LTR design method. Till
this point, only one such method has been investigated.
l2
This recovery method imposes an
Z
constraint on the sensitivity recovery error. The method is based on coprime factorizations
of
the sensitivity recovery error for
a
system where the direct feedthrough term is assumed to
have full rank. This method has two drawbacks: first, the order of the final observer based
controllers is
2n
for square system and
3n
-
1
otherwise in the minimum phase case. However,
it is always possible to reduce the
Z
norm of the recovery errors by nth order controllers in
the minimum phase case.
21s23
Second, for non-minimum phase systems, only the minimum
phase part is considered in Reference
12
and no norm bounds are guaranteed for the overall
system. But, as
a
matter of fact, the main importance of direct design methods, are their
application to non-minimum phase systems, as will appear in the course of this paper.
The key contribution of this paper is to formulate the recovery design problem as a direct
Z
optimization problem of the recovery errors and to derive the associated Z/LTR
controller in state-space form. The basis of this contribution
is
the general recovery description
given in Reference
14
which is summarized in Section
2.
Concurrently, two different controller
structures are considered for both the sensitivity recovery problem and the input-output
recovery problem. First, the so-called Q-observer is considered which is a structure, where the
Z
part
of
the controller is appended (in
a
block-diagram sense) to a standard full order
observer based controller. Second,
a
description is given, where the
Z
standard problem
emerges directly from the recovery errors.
22
Effectively, five different
3%
problem
formulations are given at the end
of
Section
2.
The five
Z
problem formulated in Section
2,
are all singular, i.e., they do not fulfil the
usual assumptions about the rank of the direct feedthrough term. This problem is often
overcome by approximation techniques, but
a
complete generalization of the
Z
problem to
include singular
D
matrices have been given by Stoorvogel.
2o
The results needed in this
presentation are cited in Appendix A, along with some easy corollaries. In Appendix
B
the
relevant algorithms for the singular
Z
approach are given.
.Xk./LTR
DESIGN
3
In
the following two sections (Sections
3
and
4)
the solutions to the four direct
,%
problems
are given. State-space formulations
of
the
4%
problems are given; the controllers are given in
state-space form and are expressed in terms of certain quadratic matrix inequalities which are
generalizations
of
the matrix Riccati equations known from Reference
3
(and solved by similar
techniques).
The results from Sections
3
and
4
are briefly summarized in Section
5.
In Section 6 an
exhaustive discussion of a non-minimum phase example is given. Finally, some concluding
remarks are given in Section
7.
2. THE LTR PROBLEM FORMULATION
In this section we shall shortly describe the significance of LTR and give
a
brief introduction
of
the Luenberger observer. Further, Q-parameterized controllers will be introduced, both as
Luenberger observer based controllers and as general controllers.
2.1.
The principle
of
recovery design
Loop transfer recovery (LTR) is a tool applied in robust multivariable control. LTR design
is
the last step in a two-step design procedure for constructing dynamic compensators. The first
step in the procedure is a specification of the desired properties for the final feedback control
system and the design of a target loop, using
a
state feedback for which the specifications are
satisfied. Then the LTR step follows, where the target loop is ‘recovered’ by an admissible
measurement based controller.
Suppose the design specifications are given as bounds on the sensitivity transfer function
S(
-
)
and the complementary sensitivity transfer function
T(
-
).
where
\I
-
11-
is the
4%
norm. The performance specifications (e.g. asymptotic tracking,
bandwidth) are expressed by the weight function
W1(
)
on the sensitivity function.’ The
weight
WZ(
-
)
on
T(
)
reflects system uncertainties such as disturbances, noise and modelling
errors. In the sequel, the specifications will always be reflected to the input node. Independent
of
the selected dynamic controller type, see Sections 2.2-2.6, the systematic LTR design
procedure can be applied to the design problem. First a state feedback design, the target
design, which satisfies
(l),
is designed,
‘.19
resulting in the target loop transfer function.
Second, the LTR step is performed, where the target design is recovered systematically for each
frequency by using a dynamic controller
C(s).
Often the system is assumed to be minimum
phase, which has been shown to be a sufficient condition
l4
for achieving asymptotic recovery,
i.e. recovering each frequency arbitrarily well. The minimum phase condition is not necessary
for asymptotic recovery. Necessary and sufficient conditions are known but rather
complicated. The LTR design originated as an approach to design of full order observer based
controller^,^'^
but it is possible to design other controller types by the LTR principle.
l4
At this
point we would like to stress, that the target design can be performed completely independent
of the specific LTR procedure chosen. For non-minimum phase systems, though, it might in
some cases be beneficial to design a state feedback which facilitates asymptotic recovery.
l7
4
J.
STOUSTRUP
AND
H.
H.
NIEMANN
2.2.
Recovery errors
represented by a state-space realization
(A,
B,
C,
0):
Let us consider a finite-dimensional, linear, time-invariant (FDLTI) plant model,
=r=h+Bu
z=cx
with transfer function:
where
xE
IR",
UE
IRm,
G(S)
=
C(SI
-
A)-'B
(3)
z
E
R',
with
m
>
r
and
A,
B
and
C
are matrices of appropriate
dimensions. The system is assumed to be stabilizable, detectable and left invertible. Moreover,
we shall make the technical assumption, that
A(A)
n
6
=
0.
Note, however, that this can
always be achieved by applying
a
preliminary static output feedback. Furthermore, this
preliminary static output feedback can be chosen arbitrarily small.
The associated sensitivity and input-output transfer functions for the target design and the
full loop design are given by:
(4)
(5)
&FL(S)
=
(I
-
F(s1-
A)-'B)-'
SI(S)
=
(I
-
C(s)G(s))-'
where
F
is the target static state feedback, and
C(s)
is the controller to be designed. Using
these transfer functions, two possible types of recovery errors can be defined.
Dejnition
2.1
The sensitivity recovery error
ES
and the input-output recovery error
EIO
are defined by
Es
(s)
=
STFL(S)
-
SI
(s)
EIO(S)
=
GFL(S)
-
GIO(S)
(8)
(9)
The objective in the rest
of
this paper is to describe how the norm of these two recovery errors
can be made small when applying different kinds
of
controllers, using methods. The
various controller types will be introduced in the following.
2.3.
The Luenberger observer based controller
Suppose that we wish to control the plant by a control law of the form:
u
=
F%+
r
=
w
+
r
(10)
where
P
is an estimate of the plant state. In the Luenberger observer
w
=
F%
is given by the
following equations:
(1 1)
=
DZ
+
GU
+
Ey
w
=
Pz
+
vy