To appear in the Proceedings of ECC’99
DISCRETE-TIME LIFTING VIA IMPLICIT
DESCRIPTOR SYSTEMS∗
Allan C. Kahane Leonid Mirkin† Zalman J. Palmor
Faculty of Mechanical Engineering, Technion — IIT
Haifa 32 000, Israel
†
Fax: +972-4-8324533; E-mail: [email protected]
Keywords: lifting technique, periodic systems, sampleddata systems, descriptor systems, Riccati equations.
Abstract
This paper introduces a new representation for the parameters of the discrete-time lifted plants. In this representation the lifted parameters are expressed as discrete-time
dynamic systems in the implicit descriptor form. We argue that the operations over these dynamic systems can
effectively be performed in the state-space, in terms of
the original plant parameters. Consequently, the computational efficiency of the discrete-time lifting technique is
improved, and the structures of the original problems can
easily be recovered from their lifted solutions.
The efficiency of the proposed approach is demonstrated
through the investigation of the discrete-time algebraic
Riccati equation associated with a discrete-lifted system.
1
Introduction
The discrete-time lifting technique is a powerful tool for
the analysis and design of discrete-time periodical systems, like those arising in sampled-data control (see [3]
and the references therein). The essence of lifting is the
conversion of a periodic system to an equivalent, in some
sense, time-invariant one. Lifting, however, increases the
dimensions of the input and output spaces. Consequently,
the parameters of the lifted systems might have considerably higher dimensions than those of the original ones.
Conventionally [4], the lifted parameters are dealt with
as unstructured matrices as if they were the parameters of
a plain discrete-time system. This approach suffers from
computational problems when high dimensional matrices
are involved. Moreover, the structure of the original problem is completely obscured using such an approach.
To overcome these difficulties, this paper introduces a
new representation of the parameters of the discrete-time
lifted systems. This new representation is based on expressing the lifted parameters as discrete-time dynamic
∗ This research was supported by the Israel Ministry of Science
under contract no. 8573-1-98.
systems operating over a finite time interval, thus exploiting the inherently Toeplitz structure of the lifted parameters. As a result, the algebraic manipulations over the
lifted parameters can be performed efficiently using rather
standard state-space techniques preserving the dimensions
and the structure of the original system.
This idea has already been exploited in the context of
the so-called continuous lifting [10]. Thus the central contribution of this paper is the extension of the approach in
[10] to the discrete-time lifting. Such an extension is not
trivial since the class of the discrete-time systems operating over a finite time interval (unlike its continuous-time
counterpart) is not closed under the adjoint and the inversion operations. In this respect, we define a broader class
of dynamic systems with two-point boundary conditions:
the discrete-time implicit descriptor systems (DIDS). It
is shown that this class is closed under all the operations
required for treating discrete-time lifted systems.
We demonstrate the potential capabilities of this new
representation by considering its application to the investigation of the discrete-time algebraic Riccati equation
(DARE) associated with a discrete-lifted plant. Using this
representation, a close connection between the solutions to
this DARE and those to the DARE associated with the
original plant is found. In particular, it is proven that the
stabilizing solutions to those two DARE’s coincide.
This paper is organized as follows. Section 2 briefly
reviews the discrete-time lifting technique and underlines
the difficulties arising from its application. In Section 3
it is argued that a possible remedy to those difficulties
might be a new representation of the discrete-lifted plant
parameters as dynamical systems. Moreover, the use of
DIDS for this purpose is motivated. Section 4 contains
both preliminaries and new results on discrete-time implicit descriptor systems with two-point boundary conditions. The new representation of the parameters of the
discrete-lifted systems is the subject matter of Subsection 5.1, while the mathematical tools required for using
this representation are developed in Subsection 5.2. An
application of those results to the investigation of DARE’s
associated with discrete-lifted plants is shown in Section 6.
Some concluding remarks are given in Section 7.
To appear in the Proceedings of ECC’99
The notation throughout the paper is as follows. M 0
means the transpose of a matrix M and O∗ — the adjoint of a Hilbert space operator O. Rn denotes the ndimensional Euclidean space and l2n [0, ν − 1] denotes the
Hilbert space of Rn valued sequences defined over a finite
time interval [0, ν−1]. Discrete-time signals and operators
in the time domain are highlighted by bars (like ζ̄), while
in the lifted domain, by vectors (like ~ζ). Transfer functions in the z domain are denoted
in terms of their state
A B
space realizations as C D . The lower linear fractional
transformation of K̄ over P̄ is denoted by Fl P̄, K̄ .
2
Discrete-time lifting
The notion of discrete-time lifting consists on establishing an one-to-one correspondence between a discrete-time
periodically shift-varying system and a shift-invariant one
(but with higher input and output dimensions). Thus it
enables the use of the well-established LTI tools for the
analysis and design of periodically shift-varying systems
arising in many multi-rate sampled-data control problems.
Define the discrete-time lifting operator W̄ν [3], which
transforms the Rn valued sequences to the Rnν valued
ones, as follows:

~ξ = W̄ν ξ̄
⇐⇒

ξ̄[νk]
 ξ̄[νk + 1] 

~ξ[k] = 

.
..


.
ξ̄[νk + ν − 1]
The usefulness of this operator follows from the fact that
.
for a ν-periodic system Ḡ its lifting ~G = W̄ν ḠW̄−1
ν is shiftinvariant. Also, since W̄ν is an isomorphism, the stability
properties are preserved under lifting, and since the restriction of W̄ν to `p is an isometry, induced norms of the
original system are equivalent to norms of the lifted one.
Lifting however, increases the input and output dimensions. For example, let Ḡ be a discrete-time LTI system
with the following transfer matrix:
. A B
Ḡ(z) =
,
(1)
C D
where A ∈ Rn×n , B ∈ Rn×m , C ∈ Rp×n , D ∈ Rp×m . Its
lifting, ~G, is also shift-invariant [3], and:
~ B
~
. A
~
G(z) = ~ ~
C D

=






Aν
Aν−1 B Aν−2 B
C
D
0
CA
CB
D
..
..
..
.
.
.
CAν−1 CAν−2 B CAν−3 B
...
...
...
..
.
...
B
0
0
..
.
D




.


(2)
In principle, (2) describes a standard discrete-time system.
Hence, dealing with ~G is conceptually not more complicated than with Ḡ. Moreover, the state dimensions in (1)
and (2) are equal. Yet the input and output dimensions
of ~G increase by the factor ν with respect to those of Ḡ.
~ C,
~ D.
~ ConseThat, in turn, “blows up” the matrices B,
quently, numerical difficulties associated with lifted solutions increase rapidly as ν grows. This fact reduces the
effectiveness of lifting, especially for large lifting frames ν.
3
Discrete-time lifting via implicit
descriptor systems.
As follows from the discussion at the end of the previous section, dealing with the lifted parameters as blockmatrices limits the efficiency of the discrete-time lifting.
Similar problems arise when the so-called continuoustime lifting [2, 3] is applied. In that case the parameters
of the lifted systems become infinite-dimensional operators. Thus, a direct manipulation over those parameters
is considerably more difficult than over the parameters in
(2). It was shown in [10], however, that those difficulties
can be overcome by representing the parameters of continuous lifted systems as continuous-time dynamic systems
with two-point boundary conditions (STPBC) of the form
ẋ(t) = Ax(t) + Bu(t), t ∈ [0, h],
y(t) = Cx(t) + Du(t),
Ωx(0) + Υx(h) = 0 ,
and by replacing the manipulations over the infinitedimensional operators with operations over STPBC. By
a subsequent extension of the results of [5], the latter manipulations can easily be performed in the state-space, in
terms of the matrix parameters of the original systems.
This fact suggests that the manipulations over the parameters of ~G in (2) can also be simplified by treating them
not as unstructured matrices, but rather as a representation of discrete dynamical systems. Yet the extension of
the results of [10] to the case of the discrete-time lifting is
not straightforward. The reason is that unlike the class of
the continuous-time STPBC, the class of the discrete-time
STPBC defined in [5], i.e.
x[k + 1] = Ax[k] + Bu[k],
y[k] = Cx[k] + Du[k],
Ωx[0] + Υx[ν] = 0 ,
k = 0, . . . , ν − 1
(3a)
(3b)
(3c)
is not closed under the adjoint (when det(A) = 0) and
inverse (when det(D) = 0) operations. At the same time,
singular A and D appear frequently in the sampled-data
control problems [3]. Hence, a broader class of discretetime STPBC should be used in order to cover all possible
singularities in the description of Ḡ.
The fact that the standard state space form is not closed
under the adjoint operation (conjugation) is actually well
known. For example, consider the discrete-time STPBC
x[k + 1] = u[k], k = 0, 1
y[k] = x[k], x[0] = 0,
To appear in the Proceedings of ECC’99
which describes the following input-output mapping:
y[0]
0 0 u[0]
=
.
y[1]
1 0 u[1]
y[0]
0 1 u[0]
=
,
y[1]
0 0 u[1]
does not belong to any STPBC of the form (3).
As a possible remedy, the descriptor representation (i.e.,
the representation with the state equation in the form
Ex[k + 1] = Ax[k] + Bu[k] for possibly singular E and A) is
considered, see, e.g., [13]. In this form, the adjoint system
from the example above can be represented as follows:
y[k] = 0 1 x[k],
Implicit descriptor systems
Let
The input-output mapping of the adjoint system, i.e.
00
1
x[k + 1] = x[k] −
u[k],
10
0
4
k = 0, 1
x[2] = 0 .
Clearly, this representation can not be reduced to the form
(3) since the coefficient matrix of x[k + 1] is singular.
The theory of the discrete-time descriptor STPBC is extensively developed in the literature [8, 9, 11]. The latter
class of systems, however, is not closed under the inversion
operation when det(D) = 0. On the other hand, unlike
the continuous-time systems, a discrete-time system can
be invertible (as an operator on `2 ) even when D = 0. For
instance, consider the system
x[k + 1] = 0.5x[k] + u[k], k = 0, 1
y[k] = 4x[k], 3x[0] + 4x[2] = 0,
which describes the input-output mapping
y[0]
−2 −4 u[0]
=
.
y[1]
3 −2 u[1]
The inverse of this system exists (since the input-output
mapping is not singular) and it is given by the following
set of equations:
x[k] = 0.25u[k], k = 0, 1
y[k] = x[k + 1] − 0.5x[k], 3x[0] + 4x[2] = 0.
This system, however, can not be represented as a
discrete-time descriptor STPBC, since the output y[k] is
determined by a dynamic equation1 .
This implies that a further generalization of the class of
descriptor systems is required. A possible way of closing
the descriptor model under inversion, suggested here, is
to represent the output equation in the descriptor form
as well. Such systems, called implicit, were studied in [1]
for the case of the infinite time interval. In the next section we define the class of discrete-time implicit descriptor
systems-with-two-point-boundary-conditions (DIDS) and
study its basic properties. The adjoint and the algebraic
operations over these systems are addressed in Section 5.
1 Actually, it can be proved that this system can be represented
as a discrete-time descriptor STPBC either by using time-varying
parameters or by increasing the descriptor-state dimension. Yet
dealing with these possibilities is considerably more difficult, and
complicates the solution.
Ex[k+1] = Fx[k] + Gy[k] + Hu[k], k = 0, . . . , ν− 1 (4a)
Ωx[0] + Υx[ν] = 0,
(4b)
describe a linear DIDS operating over a finite time interval. x[k] ∈ Rn , y[k] ∈ Rr , u[k] ∈ Rp are the descriptor
state, the system output and the system input at the time
instance k, respectively. The matrices in the so-called implicit descriptor equation (4a) are assumed to have the
following dimensions: E, F ∈ R(n+r)×n , G ∈ R(n+r)×r ,
H ∈ R(n+r)×p , while the matrices Ω and Υ in the socalled boundary condition (4b) are assumed to be square.
Definition 1. The DIDS (4) is said to be well-posed if its
unique solution, when u ≡ 0, is the trivial one, i.e., x ≡ 0
and y ≡ 0.
Definition 1 is an extension of the well-posed boundary
condition notion [5] to implicit descriptor systems and it
actually states that well-posed systems have a unique solution x, y for any given input u. In the remainder of this
section we present this solution and the necessary and sufficient conditions for the well-posedness of a DIDS.
It can be proven that a necessary condition
for well
posedness of (4) is that the matrix pair [E 0], [F G]
comprises a regular pencil, i.e.:
|[E 0]z − [F G]| 6≡ 0
(5)
Note that this condition is a generalization of the solvability condition of explicit descriptor equations [9] to the
implicit ones of the form (4a).
An important aspect of regular pencils is that they
can be transformed into forms that greatly simplify the
solution process of (4). Transformations like the forward/backward decomposition [8] or the reduction to the
standard form [11] are extensively used in the literature
to solve explicit descriptor systems. Usually, a matrix
pair {M1 , M2 } is said to be in the standard form if
αM1 + βM2 = I, for some real numbers α and β. Since,
in our case, the ‘M1 ’ matrix has a special form (that is
[E 0]), β can be omitted from the definition.
Definition 2. The DIDS (4) satisfying (5) is said to be
in the standard form if there exists α ∈ R for which:
α[E 0] + [F G] = I.
In order to transform a DIDS to the standard form, one
has to find any α for which |α[E 0]+[F G]| 6= 0 (its existence
is guaranteed by condition (5)) and to multiply (4a) from
−1
the left with the non singular matrix α[E 0] + [F G]
.
The standard form leads to the following simple wellposedness condition:
To appear in the Proceedings of ECC’99
Lemma 1. A DIDS (4) is well-posed if and only if:
(a) condition (5) is satisfied,
Theorem 1. For any LTI discrete-time system Ḡ, (1), its
lifting ~G, (2), is also LTI and its state-space realization has
the following representation:
(b) for (4) in standard form, the matrix
Ξ=
V i Mν
1
+
V f Mν
2
.
=
Ω0
[E
00
ν
0]
0
+ Υ
[F
0I
"
# ~ B
~
I0 0
I∗ν−1 0
A
Q
,
=
~ D
~
0 I
0 I
C
     


 I

A
00
AB
.
Q = 0, A, −I 0,  0 B , I, 0 .
 0

C
0I
CD
ν
G]
is invertible.
The next result shows how to compute the unique solution of a well-posed DIDS.
Lemma 2. Given any u, the unique solution of the wellposed DIDS (4) in standard form is:
ν−1
X
x[k]
=
G(k, l)Hu[l],
y[k]
Moreover:
"
#∗
∗ ~ C
~
A
I0 0 ∗ Iν−1 0
=
Q
,
~ D
~
0 I
0 I
B

      


 A0
I
00
A0 C0
Q∗ = A 0, 0,−I 0,  0 C 0 , 0, I .

 B0
0
0I
B0 D0
l=0
where
.
G(k, l) =
Mk2 M2 − M1ν−k Ξ−1 ΦMk1 Ml−k
Mν−l−1 Γ −1 ,
l≥k
1
l 2 k−l−1 −1
k −1
ν−k
ν−k
Γ , l<k
M1 M2
ωM1 − M2 Ξ ΦM2
M1
.
Φ = Vi M2 + ωVf M1 ,
.
Γ = ωMν+1
− M2ν+1 ,
1
and ω is any real number for which Γ is invertible.
In the sequel we will denote the DIDS (4) using the
following compact notation:
{E, F, G, H, Ω, Υ} .
The new representation
Having defined DIDS, we are now in the position to introduce the new representation of the parameters of the
discrete-lifted system. The purpose of this representation
is to reduce manipulations over the high dimensional parameters of the discrete-lifted system in (2) to simple algebraic manipulations over DIDS with the low dimensional
parameters of the original system (1).
5.1
Lifted parameters as DIDS
In this subsection the parameters of the discrete-time
lifted plant will be represented as DIDS. To this end, define the discrete impulse operator Iθ : Rn 7→ l2 [0, ν − 1],
θ = 0, . . . , ν − 1 as follows:
η, if k = θ
ζ = Iθ η ⇐⇒ ζ[k] =
0, else.
It can be shown that I∗θ : l2 [0, ν − 1] 7→ Rn , the adjoint
of the Hilbert space operator Iθ is given by:
η = I∗θ ζ[k] ⇐⇒ η = ζ[θ],
which in fact is the discrete-time sampling operator.
These operators, together with the DIDS defined in Section 4 lead to the main result of this paper:
(6b)
Proof. A straightforward application of Lemma 2.
Remark. It can be verified that for any A, B, C and D the
DIDS in (6) are well-posed.
Despite of the rather complicated form of (6), it will be
shown in the next section that: i) the manipulations over
DIDS can be easily performed in the descriptor state-space
using lower dimensional matrix computations, ii) the operators Iθ and I∗θ fit nicely into this framework.
5.2
5
(6a)
Manipulations over DIDS
As mentioned before, the solution process of various multirate sampled-data control problems using lifting techniques involves algebraic manipulations over the parameters of the lifted plant, such as addition, multiplication, inversion and lower linear fractional transformation. For ex~
~ 0 (D
~ 0 D)
~ −1 C
ample, computations of matrices of the form C
2
∞
arise in many multi-rate sampled-data H /H optimization problems. These computations, using formulae (2),
are cumbersome and require the inversion of highly dimensional matrices, even though the dimension of the results is
equal to that of the original plant state dimension n. This
subsection shows how these algebraic manipulations can
be performed in an effective manner using the representation in Theorem 1. It turns out that manipulations over
DIDS can be performed in terms of matrix parameters
of their representation, much like manipulations over the
standard LTI, causal systems given in the state-space representation. Moreover, it is shown that the impulse and
the ideal sampler operators can be “absorbed” into the
boundary conditions, thus enabling to extend these manipulations to operations over the lifted parameters (6).
Consider two linear operators P̄ and K̄ with an appropriately dimensioned partition of the form
P̄ =
P̄11 P̄12
,
P̄21 P̄22
K̄ =
K̄11 K̄12
,
K̄21 K̄22
To appear in the Proceedings of ECC’99
where K̄22 P̄22 is square. The operation
P̄ ? K̄ =
Fl (P̄, K̄22 )
P̄12 (I − K̄22 P̄22 )−1 K̄21
−1
K̄12 (I − P̄22 K̄22 ) P̄21
Fl (K̄, P̄22 )
is known as the Redheffer star product [12]. The following
lemma shows how to compute the star product of two DIDS.
One important property of DIDS is that the ideal sampler and impulse operators can be “absorbed” into the
boundary conditions.
Lemma 4. Let Ḡ, (1), be an LTI discrete-time system
~ B,
~ C
~ and D
~ be the parameters of its lifting ~G,
and let A,
(2). For any appropriately dimensioned matrices M, J:
Lemma 3. Let P̄1 and P̄2 be two well-posed DIDS, defined over the same time interval:
" #
" #
h
i
i ∗ ~∗
~∗ h
A
~ B
~ + C J C
~D
~ = I0 0 P̄ I0 0 ,
M
A
~∗
~∗
0 I
0 I
B
D
P̄i : {Ei , Fi , [Gi1 Gi2 ], [Hi1 Hi2 ], Ωi , Υi } , i = 1, 2,
and assume that the input/output partitions are compatible, i.e., xi ∈ Rni , ui1 ∈ Rpi , yi1 ∈ Rri , u12 , y22 ∈ Rm ,
u22 , y12 ∈ Rq . Then:
.
P̄? = P̄1 ? P̄2 = {E? , F? , G? , H? , Ω? , Υ? } ,
where:
E 0 00
,
E? = 1
0 E2 0 0
G
0
G? = 11
,
0 G21


Ω1 0 0 0
 0 Ω2 0 0

Ω? = 
 0 0 0 0,
0 0 00
F 0 G12 H12
F? = 1
,
0 F2 H22 G22
H
0
H? = 11
,
0 H21


Υ1 0 0 0
 0 Υ2 0 0

Υ? = 
 0 0 I 0.
0 0 0I
Operations like addition, multiplication, inversion and
lower linear fractional transformation are all particular
cases of the Redheffer star product. Simple formulae for
these operations can be reached by constraining the input
and output dimensions of P̄1 and P̄2 in Lemma 3. Owing
to space limitations, in most of the cases we present only
these constraints.
• addition: p1 = p2 , r1 = r2 and m = q = 0.
where P̄ is the well-posed DIDS:
 0   0  
A 0 I C JC 0
 0 I,0 A ,0
 00 0 I 00  I
B 0
• inverse: If P̄, (4), is a well-posed square DIDS then
its inverse exists if and only if:
|[E 0]z − [F H]| 6≡ 0,
Ω0
ν Υ 0 −1
ν −1
Ψ [F H] 6= 0,
Ψ [E 0] +
0 0
0 I
where Ψ = β[E 0] + [F H] and β is any real number
for which det(Ψ) 6= 0. If these conditions hold, then:
P̄−1 = {E, F, H, G, Ω, Υ} .
It is worth noting that these conditions are in fact the
well-posedness conditions for the DIDS P̄−1 .
• lower linear fractional transformation: the 11- and
22-blocks of the Redheffer star product.
0 D JC
Lemmas 3 and 4 make the representation proposed in
Theorem 1 a powerful tool for the analysis and design of
multi-rate sampled-data systems in the lifted domain.
6
DARE’s in the lifted domain
The results presented in the previous section were used
to investigate the discrete-time algebraic Riccati equation
(DARE) associated with the lifting ~G of the discrete-time
LTI plant Ḡ. Owing to space limitations, in this section
we only present the results of this analysis. Note that in
the sequel we neither require |A| 6= 0 nor |D| 6= 0. Thus,
we deal with the most general case.
Let J be an appropriately dimensioned square matrix
and associate with Ḡ, (1), the equation
A 0 XA − X + C 0 JC + (A 0 XB + C 0 JD)F = 0,
(7)
where F is the matrix gain associated with (7):
.
F = −(B 0 XB + D 0 JD)−1 (B 0 XA + D 0 JC).
• multiplication: p1 = r2 = q = 0.
It can easily be verified that both the sum and the product
of well-posed DIDS are also well-posed.



0 C 0 JC C 0 JD 

0
B 
, A
, 0 0 , IM .
0  0
0  0I 0 0
0 −I D 0 JC D 0 JD
(8)
Equation (7) is the well known DARE, and finding its
stabilizing solution is a crucial step in solving various
discrete-time
control problems, such as H2 (J = I) and H∞
(J =
I 0
0 −I
) optimizations [3, 15]. This equation is exten-
sively investigated in the literature [7] and the necessary
and sufficient conditions for the existence of its stabilizing solution are well understood. It is known [14], that
the solutions to the DARE (7) are closely related to the
following generalized eigenvalue problem:

A0

B

I

00
.
∆ − λΛ = −C 0 JC I −C 0 JD − λ0 A 0 0,
D 0 JC 0 D 0 JD
0 −B 0 0
and can be computed directly from the deflating subspaces
of the extended symplectic pencil (ESP) {Λ, ∆}.
In the same manner, associate with ~G the equation:
~ ∗ XA
~ −X+C
~ ∗ JC
~+ A
~ ∗ XB
~ +C
~ ∗ JD
~ ~F = 0,
A
(9)
To appear in the Proceedings of ECC’99
where
.
~F =
~ ∗ XB
~ +D
~ ∗ JD
~
− B
−1
~ ∗ JC
~ +B
~ ∗ XA
~ .
D
Equation (9) is also a DARE, which arises in many multirate sampled-data optimization problems, see, e.g., [4].
Even though (7) and (9) have solutions of the same
dimensions, the ESP associated with (9) might have much
larger dimensions than {Λ, ∆}. Hence, for (9), a higher
dimensional eigenvalue problem has to be solved. This
may make the computation of the stabilizing solution to
the DARE (9) sensitive to numerical errors.
Since ~G is equivalent to Ḡ from the input-output point
of view, it is likely that the DARE’s (7) and (9) are closely
connected. The internal structures of ~G and Ḡ, however,
are different: the state vector of ~G is the sampled version
of that of Ḡ. Hence, the relation between (7) and (9) might
not be obvious. The next theorem clarifies this relation.
Theorem 2. Any solution X to DARE (7) is also a solution to DARE (9). Moreover,
if the
generalized eigenvalν
ues λi of the extended pencil Λ, ∆ are such that λν
i 6= λj
whenever λi 6= λj , then the converse is also true.
The following Lemma is important for various multirate sampled-data optimization problems.
Lemma 5. A matrix X = X 0 is the stabilizing solution to
DARE (7) iff it is the stabilizing solution to DARE (9).
Moreover, if X is the stabilizing solution to those DARE’s,


then
~ +B
~ ~F = (A + BF)ν ,
A
F
 F(A + BF) 

~F = 

,
..


.
F(A + BF)ν−1
where F (8) is the matrix gain associated with DARE (7).
Thus, considering the stabilizing solution, the cumbersome DARE (9) is equivalent to the considerably simpler
and well understood DARE (7). Moreover, the computa~ +B
~ ~F is considerably simplified.
tion of ~F and A
7
Conclusions
In this paper we have introduced a new representation for
the parameters of discrete-lifted systems that considerably
simplifies the algebraic manipulations over those parameters. This new representation is based on the algebra of
discrete-time implicit descriptor systems with two-point
boundary conditions, operating over a finite time interval. We have shown that the algebraic manipulations over
those systems are simple and the results are explicitly expressed in terms of the original plant parameters.
The proposed approach has been used to analyze properties of DARE’s in the lifted domain. In particular, it has
been shown that although the lifted DARE might have
more solutions then the original one, their stabilizing solutions coincide.
It is believed that the tools developed in this paper
will help to derive simple solutions to various multi-rate
sampled-data control problems, both computationally and
conceptually. Those tools were used in [6] for the H2 optimal design of generalized sampling and hold functions
with waveform constraints.
References
[1] J. D. Aplevich, Implicit Linear Systems, Lecture Notes
in Control and Information Sciences, vol. 152, SpringerVerlag, NY, 1991.
[2] B. Bamieh, J. B. Pearson, B. A. Francis, and A. Tannenbaum, “A lifting technique for linear periodic systems
with applications to sampled-data control,” Systems &
Control Letters 17 (1991), pp. 79–88.
[3] T. Chen and B. A. Francis, Optimal Sampled-Data Control
Systems, Springer-Verlag, London, 1995.
[4] T. Chen and L. Qiu, “H∞ design of general multirate
sampled-data control systems,” Automatica 30 (1994),
no. 7, pp. 1139–1152.
[5] I. Gohberg and M. A. Kaashoek, “Time varying linear systems with boundary conditions and integral operators, I.
The transfer operator and its properties,” Integral Equations and Operator Theory 7 (1984), pp. 325–391.
[6] C. A. Kahane, L. B. Mirkin, and J. Z. Palmor, “H2 design
of generalized sampling and hold functions with waveform
constraints,” in Proceedings of the 5 th European Control
Conference (Karlsrue, Germany), July 1999.
[7] P. Lancaster and L. Rodman, Algebraic Riccati Equations,
Clarendon Press, Oxford, UK, 1995.
[8] F. L. Lewis, “Descriptor systems: Decomposition into forward and backward subsystems,” IEEE Transactions on
Automatic Control 29 (1984), no. 2, pp. 167–170.
[9] D. Luenberger, “Boundary recursion for descriptor variable systems,” IEEE Transactions on Automatic Control
34 (1989), no. 3, pp. 287–292.
[10] L. Mirkin and Z. J. Palmor, “A new representation of
parameters of lifted systems,” IEEE Transactions on Automatic Control 44 (1999), no. 4.
[11] R. Nikoukhah, A. S. Willsky, and B. C. Levy, “Boundaryvalue descriptor systems: Well-posedness, reachability
and observability,” International Journal of Control 46
(1987), no. 5, pp. 1751–1737.
[12] R. M. Redheffer, “On a certain linear fractional transformation,” Journal of Mathematics and Physics 39 (1960),
pp. 269–286.
[13] M. G. Safonov, D. J. N. Limebeer, and R. Y. Chiang,
“Simplifying the H∞ theory via loop shifting, matrix
pencil and descriptor concepts,” International Journal of
Control 50 (1990), pp. 2467–2488.
[14] P. Van Dooren, “A generalized eigenvalue approach for
solving Riccati equations,” SIAM Journal on Scientific
and Statistical Computing 2 (1981), no. 2, pp. 121–135.
[15] K. Zhou, J. C. Doyle, and K. Glover, Robust and Optimal
Control, Prentice-Hall, Englewood Cliffs, NJ, 1995.