Control refinement for discrete-time
descriptor systems: a behavioural approach
via simulation relations
F. Chen ∗ , S. Haesaert ∗ , A. Abate ∗∗ , and S. Weiland ∗
∗

arXiv:1704.01672v1 [] 6 Apr 2017

Department of Electrical Engineering
Eindhoven University of Technology, Eindhoven, The Netherlands
∗∗
Department of Computer Science
University of Oxford, Oxford, United Kingdom
Abstract: The analysis of industrial processes, modelled as descriptor systems, is often
computationally hard due to the presence of both algebraic couplings and difference equations
of high order. In this paper, we introduce a control refinement notion for these descriptor
systems that enables analysis and control design over related reduced-order systems. Utilising
the behavioural framework, we extend upon the standard hierarchical control refinement for
ordinary systems and allow for algebraic couplings inherent to descriptor systems.
Keywords: Descriptor systems, simulation relations, control refinement, behavioural theory.
1. INTRODUCTION
Complex industrial processes generally contain algebraic
couplings in addition to differential (or difference) equations of high order. These systems, referred to as descriptor
systems (Kunkel and Mehrmann, 2006; Dai, 1989), are
commonly used in the modelling of mechanical systems.
The presence of algebraic equations, or couplings, together
with large state dimensions renders numerical simulation
and controller design challenging. Instead model reduction
methods (Antoulas, 2005) can be applied to replace the
systems with reduced order ones. Even though most methods have been developed for systems with only ordinary
difference equations, recent research also targets descriptor
systems (Cao et al., 2015).
In this paper, we newly target the use of descriptor systems
of reduced order for the verifiable design of controllers.
A rich body of literature on verification and formal controller synthesis exists for systems solely composed of
difference equations. This includes the algorithmic design
of certifiable (hybrid) controllers and the verification of
pre-specified requirements (Tabuada, 2009; Kloetzer and
Belta, 2008). Usually, these methods first reduce the original, concrete systems to abstract systems with finite or
smaller dimensional state spaces over which the verification or controller synthesis can be run. A such controller
obtained for the abstract system can be refined over the
concrete system leveraging the existence of a similarity relation, e.g., an (approximate) simulation relation, between
the two systems (Tabuada, 2009; Girard and Pappas,
2011). For the application of these relations in control
problems, a hierarchical control framework is presented
by (Girard and Pappas, 2009). Currently, the control synthesis over descriptor systems cannot be dealt with in this
fashion due to the presence of algebraic equations.
The presence of similarity relations between descriptor systems has also been a topic under investigation

in (Megawati and Van der Schaft, 2015). This work on
similarity relations deals with continuous-time descriptor
systems that are unconstrained and non-deterministic, and
focuses on the conditions for bisimilarity and on the construction of similarity relations. Instead in this work, we
specifically consider the control refinement problem for
discrete-time descriptor systems via simulation relations
within a behavioural framework, such that properties verified over the future behaviour of the abstract system are
also verified over the concrete controlled system. Within
the behavioural theory (Willems and Polderman, 2013),
a formal distinction is made between a system (its behaviour) and its representations, enabling us to investigate descriptor systems and refinement control problems
without having to directly deal with their inherent anticausality.
In the next section, we define the notion of dynamical
systems and control within a behavioural framework and
use it to formalise the control refinement problem. Subsequently, Section 3 is dedicated to the exact control refinement for descriptor systems and contains the main results
of the paper. The last section closes with the conclusions.
2. THE BEHAVIOURAL FRAMEWORK
2.1 Discrete-time descriptor systems
As introduced by (Willems and Polderman, 2013), we
define dynamical systems as follows.
Definition 1. A dynamical system Σ is defined as a triple
Σ = (T, W, B)
with the time axis T, the signal space W, and the behaviour
B ⊂ WT . ✷
In this definition, WT denotes the collection of all timedependent functions w : T → W. The set of trajectories
or time-dependent functions given by B represents the

trajectories that are compatible with the system. This set
is referred to as the behaviour of the system (Willems
and Polderman, 2013). Generally, the representation of the
behaviour of a dynamical system by equations, such as a
set of ordinary differential equations, state space equations
and transfer functions, is non-unique. Hence we distinguish
a dynamical system (its behaviour) from the mathematical
equations used to represent its governing laws.
We consider dynamical systems evolving over discrete-time
(T := N = {0, 1, 2, . . .}) that can be represented by a
combination of linear difference and algebraic equations.
The dynamics of such a linear discrete-time descriptor
system (DS) are defined by the tuple (E, A, B, C) as
Ex(t + 1) = Ax(t) + Bu(t),
(1)
y(t) = Cx(t),
with the state x(t) ∈ X = Rn , the input u(t) ∈ U = Rp ,
and the output y(t) ∈ Y = Rk and t ∈ N. Further, E, A ∈
Rn×n , B ∈ Rn×p and C ∈ Rk×n are constant matrices and
we presume that rank(B) = p and rank(C) = k.
We say that a trajectory w = (u, x, y), with w : N → (U ×
X × Y), satisfies (1) if for all t ∈ N the equations in
(1) evaluated at u(t), x(t), x(t + 1), y(t) hold. Then the
collection of all trajectories w defines the full behaviour,
or equivalently the input-state-output behaviour as
Bi/s/o := {(u, x, y) ∈ (U × X × Y)N | (1) is satisfied}. (2)

Σ := (T, W, B) = (N, U × X × Y, Binit
i/s/o )
with
• the time axis T := N = {0, 1, 2, . . .},
• the full signal space W := U × X × Y, and
• the initialised behaviour 1
N
Binit
i/s/o = {w ∈ W |w = (u, x, y) s.t. (1)

and s.t. x(0) = x0 ∈ X0 }.
2.2 Control of descriptor systems
Controller synthesis amounts to synthesising a system Σc ,
called a controller, which, after interconnection with Σ,
restricts the behaviour B of Σ to desirable (or controlled)
trajectories. Thus, in the behavioural framework, control
is defined through interconnections (or via variable sharing
as specified next), rather than based on the causal transmission of signals or information, as in classical system theory. Let Σ1 = (T, C1 ×W, B1 ) and Σ2 = (T, C2 ×W, B2 ) be
two dynamical systems. Then, as depicted in Fig. 1a and
defined in (Willems and Polderman, 2013), the interconnection of Σ1 and Σ2 over W, denoted by Σ = Σ1 ×w Σ2
with the shared variable w ∈ W, yields the dynamical
system Σ = (T, C1 × C2 × W, B) with B = {(c1 , c2 , w) :
T → C1 × C2 × W | (c1 , w) ∈ B1 , (c2 , w) ∈ B2 }.

The variable x is considered as a latent variable, therefore
the manifest, or equivalently the input-output behaviour
associated with (1) is defined by
Bi/o:= {(u, y)|∃x ∈ XN s.t. (u, x, y) ∈ Bi/s/o }.
If E is non-singular, we refer to the corresponding dynamical system as a non-singular DS. In that case, we can
transform (1) into standard state space equations, as
x(t + 1) = Ãx(t) + B̃u(t),
y(t) = Cx(t),
with Ã = E

−1

A, B̃ = E

−1

(3)

B. Further Bi/s/o as in (2) is

{(u, x, y) ∈ (U × X × Y)N | (u, x, y) s.t. (3) holds}.
Similarly, if E is non-singular, Bi/o can be defined by (3).
The tuple with dynamics (1) defines a dynamical system
Σ evolving over the combined signal space W = U × X × Y
with behaviour B := Bi/s/o given in (2). Similarly, for W
restricted to input-output space, the tuple (N, U×Y, Bi/o )
defines the manifest or induced dynamical system.
We are specifically interested in the behaviour initialised
at t = 0 with a given set of initial states X0 ⊂ X. For this,
we say that a trajectory w : N → (U × X × Y) is initialised
with X0 if (1) holds and x(0) = x0 ∈ X0 . Such a trajectory,
initialised with x0 ∈ X0 , is also called the continuation
of x0 . We refer to the collection of initialised trajectories
related to X0 as the initialised behaviour Binit
i/s/o . This
allows us to formalise our definition of the descriptor
system evolving over N.
Definition 2. (Discrete-time descriptor systems (DS)). A
(discrete-time) descriptor system is defined as a dynamical
system Σ initialised with X0 , whose behaviour can be
represented by the combination of algebraic equations and
difference equations given in (1), that is

(4)

c1

c2

w
Σ1

Σ2

(a) The interconnected system Σ
obtained via the shared variables
w in W between dynamical systems Σ1 and Σ2 with signal spaces
C1 × W and C2 × W.

BΣ BΣ×Σc

BΣ c

(b) The controlled behaviour
BΣ×Σc = BΣ ∩ BΣc is given as
the intersection of the behaviours
of the dynamical system Σ and
its controller Σc .

Fig. 1. The left figure (a) portrays the general interconnection of two dynamical systems. In figure (b), the more
specific case of behavioural intersection for a system
and its controller is depicted.
Observe that w ∈ WT contains the signals shared by
both Σ1 and Σ2 , while c1 ∈ CT1 only belongs to Σ1 and
c2 ∈ CT2 only belongs to Σ2 . So, in the interconnected
system, the shared variable w satisfies the laws of both
B1 and B2 . Note that it is always possible to trivially
extend the signal spaces of Σ1 and Σ2 (and the associated
behaviour) such that a full interconnection structure is
obtained, that is, such that both C1 and C2 are empty and
the behaviour of the interconnected system is B = B1 ∩
B2 . Hence, a full interconnection of Σ = (T, W, BΣ ) and
Σc = (T, W, BΣc ) is simply Σ ×w Σc = (T, W, BΣ ∩
BΣc ), with the intersection of the behaviours, denoted by
BΣ×Σc , as portrayed in Fig. 1b. That is, interconnection
and intersection are equivalent in full interconnections.
Further, we define a well-posed controller Σc for Σ as
follows.
Definition 3. Consider a dynamical system Σ = (T, W, B),
with initialised behaviour as defined in (4). We say that a
system Σc = (T, W, Bc ) is a well-posed controller for Σ if
the following conditions are satisfied:
1

In the sequel the indexes init and i/s/o will be dropped.

(1) BΣ×Σc := BΣ ∩ BΣc 6= {∅};
(2) For every initial state x0 ∈ X0 , there exists a unique
continuation in BΣ×Σc .
Denote with C(Σ) the collection of all well-posed controllers for Σ.
We want a controller that accepts any initial state of
the system. This is formalised in the second condition
by requiring that for any initial state of Σ, there exists a
unique continuation in BΣ×Σc . We elucidate the properties
of a well-posed linear controller as follows.
Example 1. For a system Σ as in (1), consider a controller
Σc , which is a DS, and has dynamics given as
Ec x(t + 1) = Ac x(t) + Bc u(t),
(5)
with Ec , Ac ∈ Rnc ×n and Bc ∈ Rnc ×p . Suppose that the
controller shares the variables u and x with the system Σ.
That is, w = (u, x). The interconnected system Σ ×w Σc
yields the state evolutions of the combined system as
 
 
 
B
A
E
u(t),
(6)
x(t) +
x(t + 1) =
Bc
Ac
Ec
and can be rewritten
  

to
A
E −B x(t + 1)
=
x(t).
Ac
Ec −Bc
u(t)

(7)

If for any x(t) ∈ X, there exists a pair (x(t + 1), u(t)) such
that (7) holds, then this implies that for any initial state
x0 ∈ X0 of Σ there exists a continuation in the controlled
behaviour. In addition, if the pair (x(t + 1), u(t)) is unique
for any x(t) ∈ X, then this continuation is unique and we
say that Σc ∈ C(Σ). This existence and uniqueness of the
pairs (x(t+ 1), u(t)) depends on the solutions of the matrix
equality (7). We use the classical results on the solutions
of matrix equalities (cf. (Abadir and Magnus, 2005)) to
conclude that the first well-posedness condition is satisfied
if and only if




E B
E B A
rank
= rank
.
(8)
Ec Bc
Ec Bc Ac
If in addition,
rank



E B A
Ec Bc Ac



= n + p,

(9)

then the second condition is also satisfied and Σc ∈ C(Σ).
Of interest is the design of well-posed controllers subject
to specifications over the future output behaviour of the
controlled system. We thus consider specifications defined
over the output space. In order to analyse the output
behaviour, we introduce a projection map. For B ⊂ (W1 ×
W2 )T we denote with ΠW2 a projection given as
ΠW2 (B) := {w2 ∈

WT2 |∃w1

∈

WT1

(E, A, B, C) as in (1) and initialised with X0 . A well-posed
controller for Σ is referred to Σc ∈ C(Σ). The controlled
concrete system is the interconnected system Σ×w Σc with
the shared variables w = (u, x).
Now, we consider a simpler DS Σa , related to the concrete
DS Σ, with dynamics given as (Ea , Aa , Ba , Ca ) and initialised with Xa0 . We assume that the synthesis of a wellposed controller Σca for Σa is substantially easier than for
Σ. We refer to this simpler system Σa as the abstract DS,
and we note that its signals take values ua (t), xa (t), ya (t)
with xa (t) ∈ Xa = Rm , ua (t) ∈ Ua = Rq , ya (t) ∈ Ya =
Y = Rk and t ∈ N. With respect to the concrete system,
the abstract DS is generally a reduced-order system. The
controlled abstract system Σa ×wa Σca is the interconnected system with the shared variables wa = (ua , xa ).
If we assume that we can compute a well-posed controller
for the abstract system, then the control synthesis problem
reduces to a control refinement problem.
Definition 4. (Exact control refinement). Let Σa and Σ be
the abstract and concrete DS, respectively. We say that
controller Σc ∈ C(Σ) refines the controller Σca ∈ C(Σa ) if
ΠY (BΣ×Σc ) ⊆ ΠY (BΣa ×Σca ).
Then we formalise the exact control refinement problem.
Problem 1. Let Σa and Σ be the abstract and concrete
DS, respectively. For any Σca ∈ C(Σa ), refine Σca to Σc ,
s.t. Σc ∈ C(Σ) and ΠY (BΣ×Σc ) ⊆ ΠY (BΣa ×Σca ).
In the next section, we will show that the existence of
a solution to this problem hinges on certain conditions
involving similarity relations between the concrete and
abstract DS. For this, we will first introduce simulation
relations to formally characterise this similarity.
3. EXACT CONTROL REFINEMENT
3.1 Similarity relations between DS
We give the notion of simulation relation as defined in
(Tabuada, 2009) for transition systems and applied to
pairs of DS Σ1 and Σ2 that share the same output space
Y1 = Y2 = Y.
Definition 5. Let Σ1 and Σ2 be two DS with respective
dynamics (E1 , A1 , B1 , C1 ) and (E2 , A2 , B2 , C2 ) over state
spaces X1 and X2 . A relation R ⊆ X1 × X2 is called a
simulation relation from Σ1 to Σ2 , if ∀(x1 , x2 ) ∈ R,
(1) for all (u1 , x+
1 ) ∈ U1 × X1 subject to
E1 x+
1 = A1 x1 + B1 u1
there exists (u2 , x+
2 ) ∈ U2 × X2 subject to

s.t. (w1 , w2 ) ∈ B}.

We focus here on finding a controller Σc for a given dynamical system Σ such that the output behaviour ΠY (BΣ×Σc )
of the interconnected system satisfies some specifications.
2.3 Exact control refinement & problem statement
Let us refer to the original DS that represents the real
physical system as the concrete DS. It is for this system
that we would like to develop a well-posed controller.
Recall that the DS is a dynamical system Σ with dynamics

such that

E2 x+
2 = A2 x2
+
+
(x1 , x2 ) ∈ R, and

+ B2 u 2

(2) we have C1 x1 = C2 x2 .
We say that Σ1 is simulated by Σ2 , denoted by Σ1  Σ2 , if
there exists a simulation relation R from Σ1 to Σ2 and if in
addition ∀x10 ∈ X10 , ∃x20 ∈ X20 such that (x10 , x20 ) ∈ R.
We call R ⊆ X1 × X2 a bisimulation relation between Σ1
and Σ2 , if R is a simulation relation from Σ1 to Σ2 and
its inverse R−1 ⊆ X2 × X1 is a simulation relation from

Σ2 to Σ1 . We say that Σ1 and Σ2 are bisimilar, denoted
by Σ1 ∼
= Σ2 , if Σ1  Σ2 w.r.t. R and Σ2  Σ1 w.r.t. R−1 .
Simulation relations as defined above are transitive. Let
R12 and R23 be simulation relations respectively, from Σ1
to Σ2 and from Σ2 to Σ3 . Then a simulation relation from
Σ1 to Σ3 is given as a composition of R12 and R23 , namely
R12 ◦R23 = {(x1 , x3 ) | ∃x2 : (x1 , x2 ) ∈ R12 ∧(x2 , x3 ) ∈ R23 }.
We also have that Σ1  Σ2 and Σ2  Σ3 implies Σ1  Σ3
and, in addition, Σ1 ∼
= Σ3 .
= Σ3 implies Σ1 ∼
= Σ2 and Σ2 ∼
Simulation relations have also implications on the properties of the output behaviours of the two systems. More
precisely, if a system is simulated by another system then
this implies output behaviour inclusion. This follows from
Proposition 4.9 in (Tabuada, 2009) and is formalised next.
Proposition 6. Let Σ1 and Σ2 be two DS with simulation
relations as defined in Definition 5. Then,
Σ1  Σ2 =⇒ ΠY (BΣ1 ) ⊆ ΠY (BΣ2 ),
Σ1 ∼
= Σ2 =⇒ ΠY (BΣ1 ) = ΠY (BΣ2 ).
Simulation relations can also be used for the controller
design for deterministic systems such as nonsingular DS
(Tabuada, 2009; Fainekos et al., 2007; Girard and Pappas,
2009). This will be used in the next subsection, where we
consider the exact control refinement for non-singular DS.
After that, we introduce a transformation of a singular
DS to an auxiliary nonsingular DS representation, referred
to as a driving variable (DV) system. The exact control
refinement problem is then solved based on the introduced
notions.
3.2 Control refinement for non-singular DS
Let us consider the simple case where the concrete and
abstract systems of interest are given with non-singular
dynamics. For these systems, the existence of a simulation
relation also implies the existence of an interface function
(Girard and Pappas, 2009), which is formulated as follows.
Definition 7. (Interface). Let Σ1 and Σ2 be two nonsingular DS defined over the same output space Y with a
simulation relation R from Σ1 to Σ2 . A mapping F : U1 ×
X1 × X2 7→ U2 is an interface related to R, if ∀(x1 , x2 ) ∈ R
and for all u1 ∈ U1 , u2 := F (u1 , x1 , x2 ) ∈ U2 is such that
+
(x+
1 , x2 ) ∈ R with
+
x+
1 = A1 x1 + B1 u1 and x2 = A2 x2 + B2 u2 .
It follows from Definition 5 that there exists at least one
interface related to R if two deterministic, or non-singular
systems are in a simulation relation. As such we can solve
the exact refinement problem as follows.
Theorem 8. Let Σ1 and Σ2 be two non-singular DS defined over the same output space Y with dynamics
(I, A1 , B1 , C1 ) and (I, A2 , B2 , C2 ), which are initialised
with X10 and X20 , respectively. If there exists a relation
R ⊆ X1 × X2 such that

(1) R is a simulation relation from Σ1 to Σ2 , and
(2) ∀x20 ∈ X20 , ∃x10 ∈ X10 s.t. (x10 , x20 ) ∈ R,
then for any controller Σc1 ∈ C(Σ1 ), there exists a
controller Σc2 ∈ C(Σ2 ) that is an exact control refinement
for Σc1 and thus achieves with
ΠY (BΣ2 ×Σc2 ) ⊆ ΠY (BΣ1 ×Σc1 ).

Proof. Since R is a simulation relation from Σ1 to Σ2 ,
there exists an interface function F : U1 × X1 × X2 → U2
as given in Definition 7, cf (Tabuada, 2009; Girard and
Pappas, 2009). Additionally, due to (2) there exists a map,
F0 : X20 → X10 such that for all x20 ∈ X20 it holds that
(F0 (x20 ), x20 ) ∈ R.
Next, we construct the controller Σc2 that achieves exact
control refinement for Σc1 as
Σc2 := (Σ1 ×w1 Σc1 ) ×w1 ΣF ,
where w1 = (u1 , x1 ) and where ΣF := (N, W, BF ) is a
dynamical system taking values in the combined signal
space with
BF := {(x1 , u1 , x2 , u2 ) ∈ W|x10 = F0 (x20 ) and
u2 = F (x1 , u1 , x2 )}.
The dynamical system Σc2 is a well-posed controller for
Σ2 with Σ2 ×w2 Σc2 sharing w2 = (u2 , x2 ). Denote with
BΣ2 ×Σc2 the behaviour of the controlled system, then due
to the construction of ΣF it follows that BΣ2 ×Σc2 is nonempty and ∀x20 ∈ X20 , ∃x10 ∈ X10 such that (x10 , x20 ) has
a unique continuation in BΣ2 ×Σc2 . Furthermore it holds
that ΠY (BΣ2 ×Σc2 ) ⊆ ΠY (BΣ1 ×Σc1 ). ✷
The design of the controller Σc2 that achieves exact control
refinement for Σc1 is similar to that in (Tabuada, 2009),
which also holds in the behavioural framework.
3.3 Driving variable systems
Since it is difficult to control and analyse a DS directly, we
develop a transformation to a system representation that
is in non-singular DS form and is driven by an auxiliary
input. We refer to this non-singular DS as the driving
variable (DV) system (Weiland, 1991). We investigate
whether the DS and the obtained DV system are bisimilar
and behaviourally equivalent. Let us first introduce with
a simple example the apparent non-determinism or anticausality in the DS. Later-on, we show the connections
between a DS and its related DV system.
Example 2. Consider the DS with dynamics (E, A, B, C)
defined as
h1i
h 0 iT
h1 0 0i
h −1 0 0 i
E = 0 0 1 , A = 0 1 0 , B = 1 , C = 0.2 , (10)
0 0 0

1

0 0 1

0.5

T

and x(t) = [x1 (t) x2 (t) x3 (t)] . In this case, the input
u(t) = −x3 (t) is constrained by the third state component.
Now the state trajectories of (10) can be found as follows:
• for a given input sequence u : N → U, we have
x2 (t) = −u(t)−u(t+1), and thus we can use this anticausal relation of the DS to find the corresponding
state trajectories;
• alternatively, we can allow the next state x2 (t + 1)
to be freely chosen, and for arbitrary state x2 (t),
the equations (10) impose constraints on the input
sequence that is, therefore, no longer free as u(t) =
−x3 (t).
We embrace the latter, non-deterministic interpretation of
the DS.
This non-determinism can be characterised by introducing
an auxiliary driving input of a so-called DV system. We

reorganise the state evolution of (1). For simplicity we omit
the time index in x(t) and u(t) and denote x(t + 1) as x+
 +
x
= Ax,
(11)
M
u
where M = [E −B]. For any x, we notice that the pairs
(u, x+ ) are non-unique due to the non-determinism related
to x+ . If M has full row rank, then it has a right inverse.
This always holds when the DS is reachable (cf. Definition
2-1.1 (Dai, 1989)). In that case we can characterise the
non-determinism as follows. Let M + be a right inverse of
M such that M M + = I and N be a matrix such that
im N = ker M and N T N = I. Then all pairs (u, x+ ) that
are compatible with state x in (11) are parametrised as
 +
x
= M + Ax + N s,
(12)
u
where s is a free variable. We now claim that all transitions
(x, u, x+ ) in (12) for some variable s satisfy (11). To see
this, multiply M on both sides of (12) to regain (11). Now
assume that there exists a tuple (x, u, x+ ) satisfying (11)
that does not satisfy (12). Then there exists an s and a
vector z 6= 0 that is not an element of the kernel of M and
such that the right side of (12) becomes M + Ax + N s +
z. Multiplying again with M , we infer that there is an
additional non-zero term M z and that (11) cannot hold.
In conclusion any transition of (11) is also a transition of
(12) and vice versa.
Example 3. [Example 2: cont’d] For the DS of Example 2,
the related DV system ΣDV is developed as
h −1 0 −1 i
h 0 i
x(t + 1) = 0 0 0 x(t) + −1 s(t)
0 1 −1
0
(13)
u(t) = [ 0 0 −1 ]x(t)
y(t) = [ 0 0.2 0.5 ]x(t).
As indicated by (13), the input u(t) is a function of
the state trajectory. The non-determinism of x2 (t + 1) is
characterised by −s(t) for which the auxiliary input s can
be freely selected.
Let us now formalise the notion of a driving variable
representation. We associate a driving variable representation with any given DS (1) by defining a tuple
(Ad , Bd , Cu , Du , C) and setting
 
 
Ad
Bd
= M + A,
= N,
(14)
Cu
Du
where N ∈ R(n+p)×p has orthonormal columns, that is
N T N = I. For any given DS, this tuple defines the driving
variable system ΣDV = (N, W, BΣDV ), which maintains
the same set of initial states X0 and has dynamics
x(t + 1) = Ad x(t) + Bd s(t)
u(t) = Cu x(t) + Du s(t)
(15)
y(t) = Cx(t),
thereby yielding the initialised behaviour
BΣDV := {w ∈ WN |w =(u, x, y), ∃s ∈ SN
s.t. (15) and x0 ∈ X0 }.
Next, we propose the following assumption for DS, which
will be used in the sequel to develop our main results.
Assumption 1. The given DS Σ is a dynamical system
with dynamics (E, A, B, C) such that M = [E −B] has
full row rank.

The relationship between a DS and its related DV system
is characterised as follows.
Theorem 9. Let the DS Σ be given as in (1) satisfying
Assumption 1 and let ΣDV = (N, W, BΣDV ) be defined as
in (15). Then
(1) Σ and ΣDV are bisimilar, that is, Σ ∼
= ΣDV ,
(2) Σ and ΣDV have equal behaviour, i.e., BΣDV = BΣ ,
(3) Σ and ΣDV have equal output behaviour, that is,
ΠY (BΣ ) = ΠY (BΣDV ).
Proof. For the first statement (1), we define the diagonal
relation as I := {(x, x) | x ∈ X}. Then I is a bisimulation
relation between Σ and ΣDV , because by construction their
state evolutions can be matched, hence stay in I; and they
share the same output map. In addition, since they have
the same set of initial states it follows that Σ ∼
= ΣDV .
The second part (2) follows immediately from the derivation of ΣDV , because by construction all the transitions
in Σ can be matched by those of ΣDV and vice versa, in
addition, they have the same output map. Hence, they
share the same signal space (U × X × Y) and we can
conclude that Σ and ΣDV have equal behaviour.
Additionally, we have that (2) implies (3); via Proposition
6 also (1) implies (3). ✷
3.4 Main result: exact control refinement for DS
Based on the results developed in the previous subsections,
we now derive the solution to the exact control refinement
problem in Problem 1. More precisely, subject to the
assumption that there exists a simulation relation R from
Σa to Σ, for which in addition holds that ∀x0 ∈ X0 , ∃xa0 ∈
Xa0 s.t. (xa0 , x0 ) ∈ R, we show that for any Σca ∈ C(Σa ),
there exists a controller Σc for Σ that refines Σca such that
Σc ∈ C(Σ) and ΠY (BΣ×Σc ) ⊆ ΠY (BΣa ×Σca ).
In the case of Assumption 1, we construct DV systems
ΣDV and ΣDVa for the respective DS systems Σ and Σa
as a first step. For these systems, we develop the following
results on exact control refinement:
i) The exact control refinement for the DV systems:
∀ΣcDVa ∈ C(ΣDVa ), ∃ΣcDV ∈ C(ΣDV ), s.t.


ΠY BΣDV ×Σc
;
⊆ ΠY BΣDVa ×Σc
DV

DVa

ii) The exact control refinement from Σa to ΣDVa :
∀Σca ∈ C(Σa ), ∃ΣcDVa ∈ C(ΣDVa ), s.t.


;
ΠY BΣa ×Σca = ΠY BΣDVa ×Σc
DVa

iii) The exact control refinement from ΣDV to Σ:
∀ΣcDV ∈ C(ΣDV ), ∃Σc ∈ C(Σ), s.t.


= ΠY (BΣ×Σc ) .
ΠY BΣDV ×Σc
DV

It will be shown that the combination of the elements
i)–iii) also implies the construction of the exact control
refinement for the concrete and abstract DS.
i) Exact control refinement for the DV systems. From
Theorem 9, we know that Σ ∼
= ΣDVa
= ΣDV and Σa ∼
with respective diagonal relations I := {(x, x)|x ∈ X} and
Ia := {(xa , xa )|xa ∈ Xa }. Hence as depicted in Fig. 2 and
based on the transitivity of simulation relations, we also
derive that R is a simulation relation from ΣDVa to ΣDV .

Σ∼
= ΣDV , w.r.t. I

Σ

R = Ia ◦ R ◦ I
(∃xa0 , ∀x0 ) ∈ R

R
(∃xa0 , ∀x0 ) ∈ R

Σa

ΣDV

DVa

ΣDVa

Σa ∼
= ΣDVa , w.r.t. Ia

Fig. 2. Connection between DS and DV systems for the
exact control refinement.
Since the DV systems ΣDV and ΣDVa share the same
initial states as the respective DS Σ and Σa , it also holds
that ∀x0 ∈ X0 , ∃xa0 ∈ Xa0 s.t. (xa0 , x0 ) ∈ R. According
to Theorem 8, we know that we can do exact control
refinement, that is, we have shown
∀ΣcDVa ∈ C(ΣDVa ), ∃ΣcDV ∈ C(ΣDV ), s.t.


⊆ ΠY BΣDVa ×Σc
.
ΠY BΣDV ×Σc
DVa

DV

ii) Exact control refinement from Σa to ΣDVa . Denote
with ΣDVa the abstract DV system related to Σa , with dynamics (Ada , Bda , Cua , Dua , Ca ) and initialised with Xa0 .
We first derive the static function Sa mapping transitions
of Σa to the auxiliary input sa of ΣDVa . From the definition
of DV systems, we can also derive the transitions of ΣDVa
indexed with a, which is similar to the derivation of (12).
 +
xa
= Ma+ Aa xa + Na sa .
(16)
ua
Multiplying NaT on both sides of (16), Sa is derived as
 +
T xa
−NaT Ma+ Aa xa . (17)
Sa : sa = Sa (x+
,
u
,
x
)
=
N
a
a
a
a
ua
Sa maps the state evolutions of Σa ×wa Σca to the auxiliary
input sa for ΣDVa , where wa = (ua , xa ). Now, we consider
the exact control refinement from the abstract DS to the
abstract DV system.
Theorem 10. Let Σa be the abstract DS with dynamics
(Ea , Aa , Ba , Ca ) satisfying the condition of Assumption 1
and let ΣDVa be its related DV system with dynamics
(Ada , Bda , Cua , Dua , Ca ) such that both systems are initialised with Xa0 . Then, for any Σca ∈ C(Σa ), there exists
a controller ΣcDVa ∈ C(ΣDVa ) that is an exact control
refinement for Σca as defined in Definition 4 with


.
ΠY BΣa ×Σca = ΠY BΣDVa ×Σc
DVa

Proof. Denote with xa and xda the state variables of Σa
and ΣDVa , respectively. Next, we construct the controller
ΣcDVa that achieves exact control refinement for Σca as
ΣcDVa := (Σa ×wa Σca ) ×wa ΣSa ,
where wa = (ua , xa ) and where ΣSa := (N, W, BSa ) is a
dynamical system with
BSa := {(xa , ua , xda , sa ) ∈ W|xa0 = xda0 and
sa = Sa (x+
a , ua , xa )}.
c
The dynamical system ΣDVa is a well-posed controller for
ΣDVa with ΣDVa ×wad ΣcDVa sharing wad = (sa , xda ). Denote
the behaviour of the controlled system.
with BΣDVa ×Σc
DVa

By construction, we know that the set of the behaviour
is non-empty and there is a unique continuation for any
xda0 ∈ Xa0 . Further based on the construction of ΣSa , the
behaviour is such that xda (t) = xa (t), ∀t ∈ N. Additionally,
since Σa and ΣDVa share the same set of initial states
 Xa0 ,

it holds that ΠY BΣa ×Σca = ΠY BΣDVa ×Σc
. ✷
The proof is actually constructive in the design of the
controller ΣcDVa that achieves exact control refinement for
Σc a .
iii) Exact control refinement from ΣDV to Σ. Now, we
consider the exact control refinement from ΣDV to Σ.
Suppose we are given a well-posed controller ΣcDV for ΣDV ,
which shares the free variable s and the state variable x
with ΣDV . We want to design a well-posed controller for
Σ over w = (u, x), for which we consider the dynamical
system ΣC = (N, W, B) over the signal space W = U ×
X × S, the behaviour of which can be defined by
BdT x(t + 1) = BdT Ad x(t) + BdT Bd s(t)
(18)
u(t) = Cu x(t) + Du s(t).
Then the dynamics of the interconnected system Σ ×w ΣC
as a function
of x and
as 



 s is derived
BDu
A + BCu
E
s(t). (19)
x(t)
+
x(t
+
1)
=
BdT Bd
BdT
BdT Ad
Note that A + BCu = EAd and BDu = EBd by
multiplying M = [E −B] on the left-hand side of the two
equations in (14). Therefore, (19) is simplified to
 
 
 
E
E
E
Bd s(t).
(20)
Ad x(t) +
x(t + 1) =
BdT
BdT
BdT
T

Furthermore E T Bd has full column rank because the
 T T
is square and has full rank. Hence
matrix M N
 T
T
E Bd has a left inverse and the dynamics of Σ ×w ΣC
in (20) can be simplified as
x(t + 1) = Ad x(t) + Bd s(t),
which is exactly the same as the state evolutions of ΣDV
as shown in (15). Next we construct Σc := ΣC ×wd ΣcDV
with wd = (s, xd ) and it is a well-posed controller for Σ.
This allows us to state the following theorem regarding the
control refinement from ΣDV to Σ.
Theorem 11. Let Σ be the concrete DS with dynamics
(E, A, B, C) satisfying Assumption 1 and let ΣDV be its
related DV system with dynamics (Ad , Bd , Cu , Du , C) such
that both systems are initialised with X0 . Then, for any
ΣcDV ∈ C(ΣDV ), there exists a controller Σc ∈ C(Σ)
that is an exact control refinement for ΣcDV as defined in
Definition 4 with


ΠY BΣDV ×Σc

DV

= ΠY (BΣ×Σc ) .

Proof. Denote with x and xd the state variables of the
Σ and ΣDV , respectively. Next, we construct the controller
Σc that achieves exact control refinement for ΣcDV as
Σc := ΣC ×wd ΣcDV ,
where wd = (s, xd ) and the dynamics of ΣC is defined as
(18). Then, we can show that the dynamical system Σc is a
well-posed controller for Σ. Based on the analysis of (20),
it is shown that Σ ×w ΣC = ΣDV with w = (u, x), then we

can derive Σ ×w Σc = ΣDV ×wd ΣcDV . Therefore,
we can

= ΠY BΣ×Σc
conclude Σc ∈ C(Σ) with ΠY BΣDV ×Σc
DV
immediately follows from ΣcDV ∈ C(ΣDV ). ✷
Exact control refinement for descriptor systems. We can
now argue that there exists exact control refinement from
Σa to Σ, as stated in the following result.
Theorem 12. Consider two DS Σa (abstract, initialised
with Xa0 ) and Σ (concrete, initialised with X0 ) satisfying
Assumption 1 and let R be a simulation relation from Σa
to Σ, for which in addition holds that ∀x0 ∈ X0 , ∃xa0 ∈
Xa0 s.t. (xa0 , x0 ) ∈ R. Then, for any Σca ∈ C(Σa ), there
exists a controller Σc ∈ C(Σ) such that

ΠY (BΣ×Σc ) ⊆ ΠY BΣa ×Σca .

Proof. Based on Assumption 1, we first construct ΣDV
and ΣDVa . Then to prove this we need to construct the
exact control refinement. This can be done based on
the subsequent control refinements given in Theorem 10,
Theorem 8 and Theorem 11. ✷
Theorem 12 claims the existence of such controller Σc that
achieves exact control refinement for Σca . More precisely,
we have shown in the proof that the refined controller Σc
is constructive, which provides the solution to Problem 1.
To elucidate how such an exact control refinement is
constructed, we consider the following example.
Example 4. [Example 2,3: cont’d] Consider the DS of
Example 2 and its related DV system (cf. Example 3)
such that both systems are initialised with X0 = {x0 |
x0 ∈ [−1, 1]3 ⊂ R3 }. According to Silverman-Ho algorithm (Dai, 1989), we can select an abstract DS Σa =
(Ea , Aa , Ba , Ca ) that is the minimal realisation of Σ and
is initialised with Xa0 = R2 , in addition
[ 01 00 ], Aa

[ 10 01 ], Ba

[ 10 ], Ca

T
[ 0.7
0.2 ] .

Ea =
=
=
=
Similarly, the related DV system ΣDVa of Σa is given as
 0 
xa (t + 1) = [ 00 10 ]xa (t) + −1
sa (t)
(21)
ua (t) = [ −1 0 ]xa (t)
ya (t) = [ 0.7 0.2 ]xa (t).
Subsequently,
R := {(xa , x) | xa = Hx, xa ∈ Xa , x ∈ X}
is a simulation relation from Σa to Σ with


1
H = 00 01 −1
.
This can be proved through verifying the two properties of
Definition 5. In addition, the condition ∀x0 ∈ X0 , ∃xa0 ∈
Xa0 s.t. (xa0 , x0 ) ∈ R holds. According to Theorem 12,
we can refine any Σca ∈ C(Σa ) to attain a well-posed
controller Σc for Σ that solves Problem 1 as follows: Define
Σca ∈ C(Σa ) with dynamics as
[ 1 1 ]xa (t + 1) = [ 0.5 0.5 ]xa (t) + ua (t).
The controlled system Σa ×wa Σca is derived as
 0

1
xa (t + 1) = −0.5
−0.5 xa (t)
ya (t) = [ 0.7 0.2 ]xa (t),
with wa = (ua , xa ) and ua (t) = [ −1 0 ]xa (t). Then Σa ×wa
Σca is stable. According to Theorem 10, we derive the map
Sa for ΣDVa as sa (t) = [ 0 −1 ]xa (t + 1) = [ 0.5 0.5 ]xa (t).
Next, the related interface from ΣDVa to ΣDV is developed

as s(t) = sa (t) − [ 0 1 −1 ]x(t). According to Theorem 11,
we derive the well-posed controller Σc as
[ 0 −1 0 ]x(t + 1) = [ 0 −1 1 ]x(t) + [ 0.5 0.5 ]xa (t)
u(t) = [ 0 0 −1 ]x(t),
and the interconnected system Σ ×w Σc with w = (u, x), is
derived as
i
h1 0 1 i
h 0
0
x(t + 1) = 0 1 −1 x(t) + −0.5 −0.5 xa (t)
0 1 −1

0

0

y(t) = [ 0 0.2 0.5 ]x(t).
Since (xa , x) ∈ R, that is xa = Hx, Σ ×w Σc can be
simplified by replacing xa (t):
h1 0 1 i
x(t + 1) = 0 0.5 −1 x(t)
0 1 −1

y(t) = [ 0 0.2 0.5 ]x(t).


Finally, Σc ∈ C(Σ) and ΠY (BΣ×Σc ) ⊆ ΠY BΣa ×Σca are
achieved.
4. CONCLUSION
In this paper, we have developed a control refinement
procedure for discrete-time descriptor systems that is
largely based on the behavioural theory of dynamical
systems and the theory of simulation relations among
dynamical systems. Our main results provide complete
solutions of the control refinement problem for this class
of discrete-time systems.
The exact control refinement that has been developed
in this work also opens the possibilities for approximate
control refinement notions, to be coupled with approximate similarity relations: these promise to leverage general
model reduction techniques and to provide more freedom
for the analysis and control of descriptor systems.
The future research includes a comparison of the control
refinement approach for descriptor systems to results in
perturbation theory, as well as control refinement for
nonlinear descriptor systems.
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