1 Gain-Scheduling Controller Synthesis for Nested Systems with Full Block Scalings

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1
Gain-Scheduling Controller Synthesis for Nested
Systems with Full Block Scalings
Christian A. R ¨
osinger and Carsten W. Scherer, Fellow, IEEE
Abstract—This work presents a framework to synthe-
size structured gain-scheduled controllers for structured
plants whose dynamics change according to time-varying
scheduling parameters. Both the system and the controller
are assumed to admit descriptions in terms of a linear
time-invariant system in feedback with so-called schedul-
ing blocks, which collect all scheduling parameters into a
static system. We show that such linear fractional repre-
sentations permit to exploit a so-called lifting technique in
order to handle several structured gain-scheduling design
problems. These could arise from a nested inner and outer
loop control configuration with partial or full dependence
on the scheduling variables. Our design conditions are
formulated in terms of convex linear matrix inequalities and
permit to handle multiple performance objectives.
Index Terms—Control system synthesis, linear matrix
inequalities, decentralized control, optimal scheduling.
I. INTRODUCTION
IN this work, we consider gain-scheduled synthesis based
on linear fractional representations (LFRs) for the standard
configuration in Fig. 1, as motivated by the early works [1],
[2]. Here, G(∆) is a linear parametrically-varying (LPV) sys-
tem affected by some matrix-valued time-varying uncertainty
∆whose current value can change arbitrarily fast and is
measured online. For instance, in a concrete application, ∆can
represent the rotor speed of the generators of a wind turbine
[3], or the variation of the longitudinal speed in a car [4].
The philosophy of gain-scheduling synthesis [1], [2], [5]–[8]
is based on the idea to design a ∆-dependent controller K(∆)
in Fig. 1 which achieves better performance if compared to a
robust controller that does not depend on ∆.
We present a flexible synthesis framework encompassing
gain-scheduled problems for different nested interconnections
of LPV systems. As inspired by [9], one specific configuration
covered by our approach is shown in Fig. 2, to which we refer
as partial gain-scheduling in the sequel. This configuration
involves an outer loop with a linear time invariant (LTI)
Funded by Deutsche Forschungsgemeinschaft (DFG, German Re-
search Foundation) under Germany’s Excellence Strategy - EXC 2075
- 390740016. We acknowledge the support by the Stuttgart Center for
Simulation Science (SimTech).
The authors are with the Institute of Mathematical Methods in
the Engineering Sciences, Numerical Analysis and Geometrical Mod-
eling, Department of Mathematics, University of Stuttgart, 70569
Stuttgart, Germany (e-mail: christian.roesinger@imng.uni-stuttgart.de,
carsten.scherer@imng.uni-stuttgart.de). (Corresponding author: Chris-
tian A. R¨
osinger.)
G(∆)
K(∆)
zpwp
u
y
Fig. 1. Gain-scheduling configuration
plant P2and a controller C2, interconnected with a gain-
scheduled inner loop consisting of an LPV system P1(∆)
and a scheduled controller C1(∆). Note that the outer loop
with P2and C2is affected by P1(∆) and C1(∆) by one-
sided communication links ξand η, respectively. Such nested
configurations are of practical interest, e.g., in the control of
induction motors [10], where the physical constraints impose
a fast ∆-dependent inner loop with ∆being the rotor speed
of the motor, and a slow outer mechanical loop. Further, our
framework permits us to handle the case that P2=P2(∆) and
C2=C2(∆) are also ∆-dependent in Fig. 2, which is called
triangular gain-scheduling for reasons to be seen later. Such
structures emerge, for instance, in a wind park if wind turbines
are interacting in a nested fashion and where ∆depends on
the wind speed and some torque coefficients [11].
Since we are interested in embedding these problems into a
unifying framework for analysis and synthesis, we show how
to translate these nested configurations into Fig. 1 by making
use of the flexibility of LFRs. This leads to controller design
problems for particularly structured G(∆) and K(∆). One of
the main contributions of this work is to show, for the first
time, that these structured design problems can be solved by
convex optimization techniques. This is achieved through a
general design framework for nested gain-scheduled control
problems based on linear matrix inequalities (LMIs).
Moreover, a central aspect of our design framework lies in
the flexibility for handling a combination of different criteria
in one shot, such as, e.g., stability, H∞-and H2-performance
objectives. Since H2-control requires to guarantee finiteness
of the closed-loop norm, we also show how to incorporate the
recent approaches [12], [13] based on D-, positive real and
full block scalings into our framework. These works focus on
a certain structured H2-design problem to render the direct
feedthrough term of wp→zpzero in Fig. 1, which in turn
guarantees finiteness of the closed-loop H2-norm by design.
On the one hand, if P1(∆) and C1(∆) are ∆-independent
LTI systems in Fig. 2, LMI solutions are given for nominal,
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reuse of any copyrighted component of this work in other works. DOI: 10.1109/TAC.2023.3329851
arXiv:2210.03712v3 [math.OC] 29 Nov 2023
2
P1(∆)
P2
C1(∆)
C2
ξ
y1u1
η
y2u2
Fig. 2. Nested gain-scheduling loop
nested H∞- and H2-design in [14], [15], while [16] uses
coupled Riccati equations to solve the H2-case. On the other
hand, without the outer loop in Fig. 2, a gain-scheduled H∞-
solution with full block scalings is given, e.g., in [17] to
design a suitable ∆-dependent controller C1(∆) for some
parameter-dependent plant P1(∆). In order to handle the H2-
analogue, we have recently shown in [13] how to use the so-
called lifting technique in the context of gain-scheduling. This
lifting technique embeds the original gain-scheduled problem
into some new design framework such that synthesis can be
performed using LMIs. As our main technical contribution,
we show that lifting is the key enabling technique to also
handle nested gain-scheduling. This includes Fig. 2 (partial
gain-scheduling) and triangular gain-scheduling, which turns
out to be the most challenging case since it involves coupled
∆-structures between the inner and the outer loop. To the best
knowledge of the authors, no other methods exist to solve the
partial/triangular gain-scheduling problem in this generality.
The paper is organized as follows. After introducing some
notation, we illustrate the main design steps for a special H2-
gain-scheduling problem without nested structures in Sec. II.
The exposition is tailored to the seamless extension to nested
gain-scheduling in Sec. III, with the corresponding analysis
and synthesis conditions presented for multiple objectives in
Secs. IV and V, respectively. We conclude this work by giving
an illustrative numerical example in Sec. VI.
Notation. We give the basic notations here, and particular
ones for structured matrices and inequalities in Secs. III
and IV, respectively. If N0is the set of nonnegative integers,
Np
0denotes the set of p-tuples a= (a1, . . . , ap)∈Np
0of length
|a|:= a1+· · ·+ap. If m, n ∈Np
0, we denote by Rm×nthe set
of real |m| × |n|matrices that carry a row/column-partition as
induced by the entries of the tuples m/n, while Sn⊂Rn×nis
the associated subset of partitioned real symmetric matrices.
Further, Iis the identity matrix, an n-partition of which being
specified as In:= diag(In1, . . . , Inp). We use •for irrelevant
matrix entries and col(M1, . . . , Mk) := (MT
1, . . . , MT
k)Tfor
vectors or matrices M1, . . . , Mk. If P∈Rm×m,M∈Rm×n,
let tr(P)be the trace of P,eig(P)be the set of eigenvalues of
P, and we write (•)TP M := MTP M and He(P) := P+PT.
II. A SPECIAL CASE:H2-GAIN-SCHEDULING
After giving a brief introduction of the gain-scheduled
synthesis problem, we motivate the cornerstones of our design
procedure, including the lifting technique, for the situation of
H2-gain-scheduling without nested constraints.
Let us consider the standard loop for gain-scheduling in
Fig. 1. For some given value set V=Co{∆1,...,∆N},
the convex hull of finitely many matrices ∆i∈Rr∆×s∆, let
0∈Vand assume that ∆lies in the set ∆:= C([0,∞),V)
of matrix-valued, arbitrarily fast time-varying (continuous)
uncertainties. Moreover, G(∆) is assumed to admit the LFR
˙x
ˆz
zp
y
=
ˆ
A11 ˆ
A12 ˆ
Bp
1ˆ
B1
ˆ
A21 ˆ
A22 ˆ
Bp
2ˆ
B2
ˆ
Cp
1ˆ
Cp
2ˆ
Dpˆ
E
ˆ
C1ˆ
C2ˆ
Fˆ
D
x
ˆw
wp
u
,ˆw=ˆ
∆(∆)ˆz(1)
where ˆ
∆ : V→Rr×sis an affine map and ∆is contained in
∆; note that the dimension of ˆ
∆(∆) might differ from that of
∆. In this representation, ˆw→ˆzis the uncertainty channel,
wp→zpserves to impose performance specifications while
u→yis the channel to interconnect LPV controllers. Recall
that this encompasses standard LFRs with ˆ
∆(∆) admitting
a block-diagonal structure with several repeated time-varying
parametric uncertainties on the diagonal [1], [2], [18]; see [18]
for a general introduction to LFRs.
Example 1: If G(∆) in Fig. 1 is given as the uncertain
system ˙x
y=∆23
4 0 (x
u)with ∆∈∆where V= [−1,1],
we sequentially define ˆwi:= ∆ˆzifor i= 1,2with ˆz1:= x,
ˆz2:= ˆw1to obtain the LFR
˙x
ˆz1
ˆz2
y
=
0 0 1 3
1 0 0 0
0 1 0 0
4 0 0 0
x
ˆw1
ˆw2
u
,ˆw1
ˆw2=∆ 0
0 ∆ ˆz1
ˆz2;
this indeed matches (1) with block-diagonal ˆ
∆(∆) := ( ∆ 0
0 ∆ ).
Analogously to the plant, we describe the gain-scheduled
controller K(∆) in Fig. 1 as an LFR
˙xc
ˆzc
u
=
ˆ
Ac
11 ˆ
Ac
12 ˆ
Bc
1
ˆ
Ac
21 ˆ
Ac
22 ˆ
Bc
2
ˆ
Cc
1ˆ
Cc
2ˆ
Dc
xc
ˆwc
y
,ˆwc=ˆ
∆c(∆)ˆzc(2)
with ∆∈∆and a so-called scheduling function
ˆ
∆c:V→Rrc×sc,∆7→ ˆ
∆c(∆).
The interconnection of (1) and (2) (through the shared
signals uand y) then admits the LFR
˙xe
ˆze
zp
=
ˆ
A11 ˆ
A12 ˆ
B1
ˆ
A21 ˆ
A22 ˆ
B2
ˆ
C1ˆ
C2ˆ
D
xe
ˆwe
wp
,ˆwe=ˆ
∆e(∆)ˆze(3)
with the closed-loop signals xe:= col(x, xc),ˆze:= col(ˆz, ˆzc),
ˆwe:= col( ˆw, ˆwc)and the extended scheduling block
ˆ
∆e(∆) := diag( ˆ
∆(∆),ˆ
∆c(∆)) (4)
of dimension (r, rc)×(s, sc). Recall that the closed-loop matri-
ces ˆ
Aij ,ˆ
Bi,ˆ
Cj,ˆ
Dcan be obtained by standard computations,
as shown in Sec. II-C for a different scenario.
Gain-scheduling synthesis then means to design matrices
ˆ
Ac
ij ,ˆ
Bc
i,ˆ
Cc
j,ˆ
Dcand a possibly nonlinear scheduling function
ˆ
∆c(.)such that the controlled system (3) satisfies a desired
3
performance specification for all ∆∈∆, as made precise in
the sequel. We emphasize that the controller (2) is indeed gain-
scheduled, in the sense that it requires knowledge of the value
of ˆ
∆c(∆(t)) at each time instant t≥0for its implementation;
explicit bounds on the to-be-designed controller order and the
size of ˆ
∆c(.)are given for each (nested) synthesis result. For
logical similarities, we denote the system matrices associated
to the integrator and the scheduling parameter in a similar
fashion by using different indices, as e.g., ˆ
Aij ,ˆ
Bi,ˆ
Cjin (1).
A. Problem formulation
Let us now formulate the H2-gain scheduling problem. We
assume that the direct feedthrough term of u→yin (1)
vanishes, i.e. ˆ
D= 0. This assumption ensures well-posedness
of the controlled interconnection (3), and, as essential for
synthesis in Sec. II-D, renders the closed-loop matrices affinely
dependent on the controller matrices (2). As widely spread
over the existing gain-scheduling literature, we consider the
problem without any other structural constraints in the LFRs
(1), (2), i.e., the scheduling function and the describing ma-
trices are unstructured with
ˆ
A11 ˆ
A12 ˆ
Bp
1ˆ
B1
ˆ
A21 ˆ
A22 ˆ
Bp
2ˆ
B2
ˆ
Cp
1ˆ
Cp
2ˆ
Dpˆ
E
ˆ
C1ˆ
C2ˆ
Fˆ
D
∈
Rn×nRn×rRn×mpRn×m
Rs×nRs×rRs×mpRs×m
Rkp×nRkp×rRkp×mpRkp×m
Rk×nRk×rRk×mp0k×m
,
ˆ
∆(∆) = ∆ being of dimension r×s=r∆×s∆,
and
ˆ
Ac
11 ˆ
Ac
12 ˆ
Bc
1
ˆ
Ac
21 ˆ
Ac
22 ˆ
Bc
2
ˆ
Cc
1ˆ
Cc
2ˆ
Dc
∈
Rnc×ncRnc×rcRnc×k
Rsc×ncRsc×rcRsc×k
Rm×ncRm×rcRm×k
.
This is addressed as the unstructured gain-scheduling problem
and briefly expressed by
G(∆) ∈ G1and K(∆) ∈ K1.
Recall that the controlled LFR (3) is called well-posed if I−
ˆ
∆e(∆) ˆ
A22 is non-singular for all ∆∈V. For brevity, we call
(3) stable if the system is exponentially stable, i.e., there exist
constants α > 0and c≥0such that every solution of (3) for
wp= 0 and for any xe(0) ∈Rn,∆∈∆satisfies ∥xe(t)∥ ≤
ce−αt∥xe(0)∥for all t≥0. Since our scaling parameter ∆∈
∆is time-varying in (3), we use the definition of the H2-norm
for linear time-varying systems in the stochastic context [19].
Problem 1: For a given plant G(∆) ∈ G1and γ > 0, find
a controller K(∆) ∈ K1such that the closed-loop (3) is well-
posed, stable, and such that the squared H2-norm of wp→zp
is smaller than γfor xe(0) = 0 and for all ∆∈∆.
B. Closed-loop analysis for original systems
For H2-performance, we suppose that the plant and con-
troller LFR (1), (2) are built such that, after interconnecting
ˆ
∆e(∆) in (3), the direct feedthrough term of wp→zp
vanishes. In Sec. III-C, we show that our general design
procedure comes with the strong advantage that we can
enforce this condition with tailored LFRs for (1), (2). Under
this hypothesis, let us recall a well-known analysis result based
on the full block S-procedure. By using the class of multipliers
ˆ
P:= nˆ
P ∈ S(r,rc,s,sc)(•)Tˆ
Pˆ
∆e(∆)
I(s,sc)≻0∀∆∈Vo(5)
for the extended scheduling block (4), to which we also refer
as full block scalings, we can characterize the requirements in
Problem 1 by the feasibility of two standard matrix inequalities
as follows [17].
Theorem 1: Problem 1 is solved for G(∆) ∈ G1,K(∆) ∈
K1if there exist X1≻0and ˆ
P ∈ ˆ
P,Z≻0with tr(Z)<1
such that the closed-loop system (3) fulfills
(•)T
0X10 0
X10 0 0
0 0 ˆ
P0
0 0 0 −γI
I(n,nc)0 0
ˆ
A11 ˆ
A12 ˆ
B1
0I(r,rc)0
ˆ
A21 ˆ
A22 ˆ
B2
0 0 I
≺0,
(•)T
−X10 0
0ˆ
P0
0 0 Z−1
I(n,nc)0
0I(r,rc)
ˆ
A21 ˆ
A22
ˆ
C1ˆ
C2
≺0.
(6)
Theorem 1 forms the basis for a convex characterization of
the existence of a gain-scheduled controller (2) such that the
formulated analysis conditions are satisfied for some full block
scaling ˆ
P ∈ ˆ
P. Technically, all existing scaling approaches to
obtain such gain-scheduled synthesis results are based on the
elimination of the ingredients defining the controller, as seen
for special plants in [17]. However, it is well-known that, even
for nominal synthesis, such an elimination step is infeasible
for the full H2-conditions in (6), since two inequalities are
involved which are coupled through the matrices ˆ
A21,ˆ
A22. It
is among the key contributions of this paper to overcome this
deficiency through what we call lifting for gain-scheduling.
C. Closed-loop analysis for lifted systems
The motivation for lifting is as follows. If ˆ
P ∈ ˆ
Pis fixed,
the anti-diagonal structure of 0X1
X10in (6) is essential to
obtain convex conditions for synthesizing nominal controllers
with a suitable transformation of ˆ
Ac
ij ,ˆ
Bc
i,ˆ
Cc
jand ˆ
Dcin (2), as
addressed in detail in [20], [21]. In gain-scheduling synthesis,
ˆ
Pis an unstructured variable and it remains fully unclear
how to modify the transformation from [20], [21] to convexify
(6). It is a decisive innovation of this paper to overcome this
trouble for (even nested) gain-scheduling synthesis by lifting
the descriptions of the system and the controller such that,
in the resulting analysis conditions, ˆ
Pis replaced by an anti-
diagonally structured block (0P
P0)with a suitable new scaling
matrix P. In our notation, we use a hat for the initial system
components (1)-(4) and the scalings (5) to distinguish them
from the lifted descriptions denoted without a hat in the sequel.
Lifting amounts to building an augmented LFR of G(∆)
in (1) as follows. The uncertainty equation ˆw=ˆ
∆(∆)ˆzin
(1) can be expressed as ˆw=−ˆw+ 2 ˆ
∆(∆)ˆzwhich leads to
w= ∆l(∆)zfor ∆∈∆with
w:= z:= ˆw
ˆz,∆l(∆) := −Ir2ˆ
∆(∆)
0Is.(7)
4
By employing analogous steps for the LTI system in (1), we
arrive at an equivalent reformulation of the overall system as
˙x
z
zp
y
=
A11 A12 Bp
1B1
A21 A22 Bp
2B2
Cp
1Cp
2DpE
C1C2F0
x
w
wp
u
=
ˆ
A11 ˆ
A12 0ˆ
Bp
1ˆ
B1
0Ir0 0 0
2ˆ
A21 2ˆ
A22 −Is2ˆ
Bp
22ˆ
B2
ˆ
Cp
1ˆ
Cp
20ˆ
Dpˆ
E
ˆ
C1ˆ
C20ˆ
F0
x
w
wp
u
, w = ∆l(∆)z.
(8)
Note that the augmented uncertainty channel w→zis square
and of size |l|×|l|for the partition l:= (r, s).
Definition 1: We abbreviate (8) by Gl(∆) and call it lifted
LFR, while we refer to ∆l(∆) from (7) as the lifted block.
Further, we say that Gl(∆) ∈ Gl
1if G(∆) ∈ G1.
By defining wc:= zc:= col( ˆwc,ˆzc), the controller (2) is
lifted accordingly to
˙xc
zc
u
=
ˆ
Ac
11 ˆ
Ac
12 0ˆ
Bc
1
0Irc0 0
2ˆ
Ac
21 2ˆ
Ac
22 −Isc2ˆ
Bc
2
ˆ
Cc
1ˆ
Cc
20ˆ
Dc
xc
wc
y
,
wc= ∆c(∆)zc:= −Irc2ˆ
∆c(∆)
0Isczc
(9)
with a square lifted scheduling block ∆c(.)of dimension |lc|×
|lc|for the partition lc:= (rc, sc). Interconnecting (8) and (9)
gives rise to the closed-loop LFR
˙xe
ze
zp
=
A11 A12 B1
A21 A22 B2
C1C2D
xe
we
wp
(10)
with ze:= col(z, zc),we:= col(w, wc)and
we= ∆e(∆)ze:= diag(∆l(∆),∆c(∆))ze.(11)
This leads to a key link between the initial and lifted setting.
Lemma 1 (Lifting Lemma): There exists a permutation ma-
trix Πof size |l|+|lc|such that X1,Z,ˆ
P ∈ ˆ
Pfulfill (6),
(•)Tˆ
PI(r,rc)
0≺0,(•)Tˆ
P0
I(s,sc)≻0(12)
for the initial closed-loop interconnection (1)-(3) if and only
if X1,Z,P:= 1
2ΠTˆ
PΠsatisfy
(•)T
0X10 0 0
X10 0 0 0
0 0 0 P0
0 0 P0 0
0 0 0 0 −γI
I(n,nc)0 0
A11 A12 B1
0I(l,lc)0
A21 A22 B2
0 0 I
≺0,
(•)T
−X10 0 0
0 0 P0
0P0 0
0 0 0 Z−1
I(n,nc)0
0I(l,lc)
A21 A22
C1C2
≺0
(13)
and
(•)T0P
P0∆e(∆)
I≻0for all ∆∈V(14)
for the lifted closed-loop interconnection (8)-(10).
Proof: Let the first inequality in (12) be true for ˆ
P ∈ ˆ
P.
If ∆∈V, we infer by the definition of ˆ
Pthe inequality
He"−I(r,rc)ˆ
∆e(∆)
0I(s,sc)T
(1
2ˆ
P)I(r,rc)ˆ
∆e(∆)
0I(s,sc)#=
=
−(•)Tˆ
PI(r,rc)
00
0 (•)Tˆ
Pˆ
∆e(∆)
I(s,sc)
≻0.
By a congruence transformation with −I(r,rc)ˆ
∆e(∆)
0I(s,sc)and if
inserting ˆ
∆e(∆) from (4), the latter inequality is equivalent to
0≺He"(1
2ˆ
P) −I(r,rc)2ˆ
∆(∆) 0
0 2 ˆ
∆c(∆)
0I(s,sc)!#.
If recalling (11), this can be equivalently expressed as
0≺Heh(1
2ˆ
P)Π∆e(∆)ΠTiwith Π :=
Ir0 0 0
0 0 Irc0
0Is0 0
0 0 0 Isc
(15)
being a permutation matrix. By a congruence transformation
with Π, the inequality in (15) transforms into (14) for the
multiplier P=1
2ΠTˆ
PΠ. The converse is shown by reversing
the steps. If P=1
2ΠTˆ
PΠholds, an analogous computation
shows that (6) and the second inequality in (12) are equivalent
to (13), which is omitted for reasons of space.
The lifting lemma can be interpreted as follows. For the
subclass of scalings ˆ
P ∈ ˆ
Pwith (12), we can equivalently
reformulate the analysis conditions for the original inter-
connection (1)-(3) to the ones based on (13) for the lifted
interconnection (8)-(10) and multipliers satisfying (14). At the
level of the lifted systems, however, we are confronted with
a highly structured controller (9), and it is unknown how to
formulate convex conditions for synthesizing such controllers.
As a further central step, we drop the structural constraint and
synthesize, instead, an unstructured LPV controller - still with
a square scheduling block - for the lifted system. This is the
key toward convexification as exposed in detail next.
To this end, let us assume that the LPV controller is again
described as in (2), but now with rc=sc=: lcas motivated
above. The interconnection of (2) with (8) leads, again, to (10)
with ze:= col(z, ˆzc),we:= col(w, ˆwc)and scheduled as
we= ∆lc(∆)ze:= diag(∆l(∆),ˆ
∆c(∆))ze(16)
of dimension (r, s, lc)×(r, s, lc). Let us now compactly
express the closed-loop matrices as
Aij Bi
CjD=
=
Aij 0Bp
i
0 0 0
Cp
j0Dp
+
0Bi
I0
0E
ˆ
Ac
ij ˆ
Bc
i
ˆ
Cc
jˆ
Dc!0I0
Cj0F.(17)
As motivated by Lemma 1, we introduce the scaling class
P:= nP ∈ S(l,lc)(•)T(0P
P0)∆lc(∆)
I≻0∀∆∈Vo
(18)
摘要：

1Gain-SchedulingControllerSynthesisforNestedSystemswithFullBlockScalingsChristianA.R¨osingerandCarstenW.Scherer,Fellow,IEEEAbstract—Thisworkpresentsaframeworktosynthe-sizestructuredgain-scheduledcontrollersforstructuredplantswhosedynamicschangeaccordingtotime-varyingschedulingparameters.Boththesyste...

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分类：图书资源 价格：10玖币 属性：16 页 大小：733.23KB 格式：PDF 时间：2025-04-28

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