Apr 6, 2017 - Eindhoven University of Technology, Eindhoven, The Netherlands ..... controller shares the variables u and x with the system Î£. That is...

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arXiv:1704.01672v1 [cs.SY] 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) = Ax(t) + Bu(t), (3) y(t) = Cx(t), ˜ = E −1 B. Further B with A˜ = E −1 A, B as in (2) is i/s/o

N

{(u, x, y) ∈ (U × X × Y) | (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. REFERENCES Abadir, K.M. and Magnus, J.R. (2005). Matrix algebra. Cambridge University Press. Antoulas, A.C. (2005). Approximation of large-scale dynamical systems. SIAM. Cao, X., Saltik, M., and Weiland, S. (2015). Hankel model reduction for descriptor systems. In 2015 54th IEEE CDC, 4668–4673. Dai, L. (1989). Singular control systems. Springer-Verlag New York, Inc. Fainekos, G.E., Girard, A., and Pappas, G.J. (2007). Hierarchical synthesis of hybrid controllers from temporal logic specifications. In International Workshop on HSCC, 203–216. Girard, A. and Pappas, G.J. (2009). Hierarchical control system design using approximate simulation. Automatica, 45(2), 566–571. Girard, A. and Pappas, G.J. (2011). Approximate bisimulation: A bridge between computer science and control theory. European Journal of Control, 17(5), 568–578. Kloetzer, M. and Belta, C. (2008). A fully automated framework for control of linear systems from temporal logic specifications. IEEE Transactions on Automatic Control, 53(1), 287–297.

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