Subspace technique for identification of hybrid complex systems. Lala H. Rajaoarisoa1 , Nacer K. M’sirdi1 , El-Kebir Boukas2 and Aziz Naamane1 1 Systems and Sciences Laboratory UMR6168, Domaine Universitaire de Saint-Jerˆ ome Avenue Escadrille Normandie-Niemen 13397 MARSEILLE CEDEX 20 [email protected], [email protected], [email protected] 2 Universit´ e de Montreal Canada [email protected]

Abstract— Switched and piecewise affine models are interesting when modeling a nonlinear and non stationary complex system. Identification of these models is a challenging problem that involves estimation of both the parameters of the submodels, and the partitioning of state and input space. In this paper, we introduce a subspace technique for identification of the class of multivariable hybrid complex system with unknown switching. Observability and Identifiability problems are discussed. The verification will be done for an experimental greenhouse. Index Terms— Hybrid Systems, Varying Complex Systems, Identification, Observability.

I. INTRODUCTION Hybrid systems are heterogeneous dynamical systems whose behavior is defined by interacting continuous and discrete dynamics. The continous dynamics is described by variables taking values from a continuous set, while the discrete dynamics is described by variables (or events) taking values from a discrete, typically finite, set. Such systems are common across a wide range of application areas. Examples include robotic assembly [1], manufacturing [2], power systems [3] and human control behavior [5]. Hybrid systems can be used to describe real phenomena that exhibit discontinuous behaviors. For instance, the trajectory of a mass-spring jumper results from alternating between flying phase and contact phase or free fall and elastic contact of the spring with the ground [25]. Moreover, hybrid models can be used to approximate continuous phenomena by concatening different simple models. In fact, we consider a hybrid systems operating in differents environments with changing features. Hence, different submodels are required [7], [6], [4]. For instance, a nonlinear dynamic system can be approximated by switching among various linear models. In this case, concerining the partitioning of state-space, two alternative approaches can be distinguished: either the partition is fixed a priori, or the partition is estimated along with the submodels. In the first case, data classification is very simple, and estimation of the submodels can be carried out by standart linear identification techniques as it shown in [8] and also argued in [9]. Effectively the identification of PieceWise Affine (PWA) and PieceWise Linear (PWL) systems for which the switching is known can be tackled using standard techniques. In the second case, the regions must be shaped to the clusters of data, and the strict relation among data classification, parameter estimation and regions for estimation must be verified.

In the literature we can find two subclasses of models to approximate continuous phenomena in a nonlinear dynamics, namely Hinging hyperplanes ARX (HHARX) [13], [12], [11], and PieceWise ARX (PWARX) models. These can be also solved via Mixed-Integer Linear Programming/ Mixed-Integer Quadratic Programming (MILP/MIQP) [14]. Unfortunately, these algorithm have too much computational complexity and are not suitable for multivariable systems framework. A. Paper contribution In the first part, this paper introduces a brief topic of models for hybrid systems by focusing in particular on the switched affine and PiecWise Affine models (PWA). PWA systems are a class of hybrid systems obtained by partitioning the stateinput domain into finite number of non-overlapping convex polyhedral regions, and by considering linear/affine sybsystems in each region [10]. In the second part, we try to use subspace technique for identification of multivariable hybrid complex systems with unknown switching. In order to do that, we use an interconnection between discrete event atomic model with a collection of state-space linear models. The atomic models is necessary to identify the operating modes of global system and the state-space model is used to describe the continuous dynamic of the process. Furthermore, statespace models are more suitable for dealing with multivariable inputs and outputs. B. Paper outline This paper is organized as follows. Firstly, section 2 introduces the classes of switched affine and piecewise affine models, and on the other hande describes the hybrid complex system models. Section 3 reports several formulations of the identification problem on the observability and identifiability of each submodel. Section 4 gives some experimental results for an identification of an experimental greenhouse. Finally, in section 5 we conclude and give some perspectives for a futur work. II. SWITCHED AFFINE AND PIECEWISE AFFINE MODELS

Switched affine models are defined as collections of linear/affine models, connected by switches that are indexed by a discrete-valued additional variable, called the discrete state. Models for which the discrete state is determined by a polyhedral partition of state-input domain, are called piecewise affine

models. Moreover, piecewise affine models are equivalent to several classes of hybrid models, and can therefore be used to describe systems exhibiting hybrid structures. In the literature we can find two categories: models in state space form and models in input-output form. A. Models in state space form A discrete-time switched affine model in state space form is described by the equations x(k + 1) = Aq(k) x(k) + Bq(k) u(k) + fq(k) + w(k) y(k) = Cq(k) x(k) + Dq(k) u(k) + gq(k) + v(k)

(1)

where x(k) ∈ Rn , u(k) ∈ Rm and y(k) ∈ R p are respectively, the continuous state, the input and the output of system at time k ∈ Z, and w(k) ∈ Rn and v(k) ∈ R p are noise/error terms. The discrete state q(k) describing in what affine dynamics the system is at time k, is assumed to take only finite number of values, i.e. q(k) ∈ {1, ..., s} where s is the number of affine submodels. In general, q(k) can be a function of (k, x(k), u(k)) or some other external input. The real matrices/vectors Ai , Bi ,Ci , Di , fi and gi , i = 1, ..., s, having appropriate dimensions, describe each affine dynamics. Hence, model 1 can be seen as a collection of affine models with continuous state x(k), connected by switches that are indexed by the discrete state q(k). B. Hybrid complex systems models Switched affine and piecewise affine models are interesting when modeling a nonlinear systems. Several simplifications are necessary in order to do that. These simplifications are based on the following assumptions: Assumption 2.1: we assume that: 1) The global dynamic behavior of the system can be stabilized around some operating modes. 2) The evolution of the continuous dynamics of the system can be approximated by linear models around each operating mode. 3) The system behaves like a statinary system around each operating mode. 4) Each operating mode can be associated with discrete event characterizing the global dynamic behavior. According to assumptions (2.1), to describe the global system’s behavior, we use an arbitrary interconnection between a piecewise affine models in state-space form with discrete events atomic models. The latter is used for relaxing fixation of partitioning of the state-space number studied in [8]. Moreover, these coupled models (figure 1) represent some advantages, for instance the use is simple and can be described by the following relations :   x(k + 1) = Ami x(k) + Bemi e(k) + Bvmi v(k) + w(k) y(k) = Cmi x(k) + Demi e(k) + Dvmi v(k) + µ (k) (2)  mi = hMode, I, O, δint , δext , λ ,ta i

where w(k) ∈ Rn and µ (k) ∈ R p are noise/error terms. v(k) is the input which define the external environment. The real matrices/vectors Ami , Bemi , Bvmi ,Cmi and Dmi , mi = 1, ..., s, with appropriate dimensions, describe each affine dynamics. Mode

Fig. 1.

Hybrid complex system representation

is the operating modes of the system (i.e. depending on the region of operation and the valid submodel). The discrete system is in Modemi when operation is in the region Ωmi , mi = 1, ..., s. I is the set of input event values (i.e. all the values that an input event can take). The events may be considered as defined by state x(k) of the system and any internal x(k)dependant variable. Then, events depend on input values of the mi associated to valid model noted I(k) [18]. As consequence the validity domain Ωmi depends also on this input. O is the set of output event values. λ is the output function which warrants the activity execution. δint is the internal transition function. It ensures mode switches when no exogenous events come out before elapsed time ti and time advance of this mode ta [17] : Mode j = δint (Modei ,ti + ta). δext is the external transit function. This function is used when exogenous events come out [17] : Mode j = δext (Modei ,ti , Ii ). III. HYBRID COMPLEX MODELS IDENTIFICATION A. Observability/Identifiability To ensure the exact reconstruction of the state infinite time we must ensure the observability properties of the system. Several researchers work in this framework [27], [28], [29], [31], [32]. In [30], Vidal et al. carried out the first systematic attempt to characterize observability concepts in switched linear systems. However, their work restricts the switching by imposing an unnecessary minimum separation time between consecutive switches and contains several flaws. Theorem 3.1 (Vidal [30]): If (C(k)−C(k′ ))A(k) is full rank for all k 6= k′ ∈ {1, ..., N} and the dwell time τk ≥ ε for all k ≥ 0, then the initial state the switching times are observable if and only if for all k 6= k′ ∈ {1, ..., N} we have rank([Oε (k) Oε (k′ )]) = 2n. Unfortunately Bemporad [26] show through counterexamples that observability properties cannot be easily deduced. Thus, an extension of this theorem is give in [28] to ensure the exact

Step 1

2 3

Operation a- Data Acquisition b- Data processing and scale adjustement c- Setting Input/Output variables of the supervisor a- Identification des modes opratoires c- Data Classification in each oprating mode a- Identification of parameters of each sub-models b- Validation of the structure

Variables Ymi ,Umi Ymi Norm ,Umi Norm Imi , Omi Mode mi © ª Ymi ,Umi ⊂ Ω(Modemi , Imi , Omi ) Aˆ mi , Bˆ mi , Cˆmi Aˆ mi , Bˆ mi , Cˆmi , Modemi , Imi , Omi

TABLE I I DENTIFICATION PROCEDURE FOR HYBRID COMPLEX SYSTEM

reconstruction of the state. This is based on the detectability and observability notion. Theorem 3.2 (De Santis [28]): A switching system is observable if the following conditions are satisfied: (i) S(qi ) is observable for any qi ∈ Q (ii) ∀p ∈ Ω(γ ), ∀qi , q j ∈ γ −1 (p), ∃k ∈ N ∪ {0} : Ci Aki Bi 6= C j Akj B j Moreover, if the maximum dwell time τM = ∞, then condition (i) and (ii) are necessary. Theorem 3.3: A switching system is detectable if the following conditions are satisfied: (iii) ∀p ∈ Ω(γ ), ∀qi , q j ∈ γ −1 (p), ∃k ∈ N ∪ {0} : Ci1 Aki11 Bi1 6= C j1 Akj11 B j1 µ ¶ ξ0 (iv) for all inital states ∈ Oi × Q, i ∈ J, and for any q0 ε > 0, there exist t > t0 such as kξ (t, j)k ≤ ξ for any t ≥ t. Leads to another important issue on the identifiability: under which conditions can we recover good model from data? In other words, whether the model itself can be inferred from data. B. Identification procedure This section adresses the identification of hybrid complex systems defined by the equation (2). Problem 3.1: Given a collection of N input-output pairs (y(k), u(k)), k = 1, ..., N, estimate the model order n, the number of submodels s, and the parameters (Ami , Bmi ,Cmi , Dmi ), mi = 1, ..., s. Moreover, estimate the operating modes Mode which define each validity domain Ωmi . In order to find all paramaters of equation (2), we propose the algorithm defined in the table I. The step 1 is acquisition of data and its transformation to ensure that all variables are in the same basis in different operating modes. This transformation is defined by Tr f (X) = 1 −

(MX − X) ; (MX − mX )

(3)

where Tr f (X) ∈ {0, 1} is the transformation applied for the variable X, MX and mX are respectively the maximum and minimum values of the input-output vector noted X. The step 2 consists on the one hand to the identification of all parameters of atomic model which define each operating modes {Imi , Omi } → Modemi and on the other hand ensures tha data classification for identification of submodels {Ymi ,Umi } ⊂ Ω(Modemi , Imi , Omi ).

The step 3 consists to the identification of all parameters of each submodel (Ami , Bmi ,Cmi , Dmi ). C. Discrete Events atomic model to identify operating modes Let us recall discrete event atomic model to represent sequences of events [15] with the condition that the state has a finite number of changes in any finite interval of time [15]: mi = hMode, I, O, δint , δext , λ ,ta i

(4)

So, an event is the representation of an instantaneous change in some part of a system that can be associated with one operating mode. An event can be characterized by a value and an ocurrence time also [17]. To identify each operating mode, let us consider the following definition and procedure: 1) Operating mode variables definition: The discrete state variables are: Mode = {phase, I,ta} (5) where phase variable is introduced in order to have a clean interpretation of the model behavior, I and ta is respectively the set of the input variables and the time advance for each operating mode. If we note k the current operating mode value, the possible phase can be represented in Table II. To obtain Phase Ph1 Ph2 . .. Phs

Mode value Mode = k, I = I1 Mode = k, I = I2 . .. Mode = k, I = Is

TABLE II TABLE OF PHASE

the different values (I1 , I2 , ..., Is ) of the input variables I, we adopt the following definition: 2) Input variables definition: The set of inputs, I, is: I = {ev1 , ev2 , ..., evs , I1 , I2 , ..., Is }

(6)

where (ev1 , ev2 , ..., evs ) is the set of event variables and (I1 , I2 , ..., Is ) are the set of different input values. These values are obtained from a discretization of continuous input variables to piecewise continuous segment following the next procedure (Figure 2): • define a threshold Ri , i = 1, ..., l, where l is a performance index for quantifying the input descrptive variable (table

the general concept in subspace identification that we consider as the most procedure to identify our system. Moreover, subspace identification is by now a well-accepted method for identification of multivariable linear system [19], [20], [22], [23]. The following input-output matrix equation [21], played a very important role in the development of subspace identification : Yi/i = Oi Xi/i + Hi Ui/i + Fi Vi/i + Wi/i

(8)

The reader can to refer at [35] for the definition of the different terms in this equation. Now, the basic idea of subspace identification is to try to recover the Oi Xi/i -term of the equation (8). This is a particularly interesting term since either the knowledge of Oi or Xi/i leads to the system parameters. How can an estimate of Oi Xi/i be extracted from above equation? For this we need to define the orthogonal projection. Definition 3.1: The orthogonal projection of the row space of M into the row space of N is denoted by MΠN as: Fig. 2.

•

MΠN = MN T (NN T )(−) N

Signal discretization

III), in order to transform the continuous segment in a piecewise constant segment (note that the discretization procedure is in the variable values regardlessness the time, i.e. this is not the output of classical sample and hold), a discrete event is then associated to each change of value in the piecewise constant segment. XX X Input

Values XX XX X I

I1

I2

I3

≤ R2

> R2 et ≤ R3

> R3

(9)

with .(−) is the Pseudo-inverse of Moore-Penrose. MΠN ⊥ is the projection of the row space of M into N ⊥ , the orthogonal complement of the row space of N, for which we have MΠN ⊥ = M − MΠN By projecting the row space of Yi/i into the orthogonal complement U⊥ i/i of the row space of Ui/i we find : Yi/i ΠU⊥ = Oi Xi/i ΠU⊥ + Fi Vi/i ΠU⊥ + Wi/i ΠU⊥ i/i

i/i

i/i

(10)

i/i

The following step consits in weighting this projection to the left and to the right with some matrices M∞ and M∈ can not be chosen arbitrarily but they should satisfy the following 3 conditions: 1) rank(M1 Oi ) = rank(Oi ) 2) rank(Xi/i ΠU⊥ M2 ) = rank(Xi/i )

TABLE III T HRESHOLD DEFINITION FOR INPUT VALUES OF ATOMIC MODEL

i/i

3) E[M1 (Fi Vi/i ΠU⊥ + Wi/i ΠU⊥ )M2 ] = 0 i/i

3) Output variables definition: An atomic model processes an input event trajectory and, according to that trajectory and its own initial conditions provokes an output event trajectory. So, when an input event arrives the mode changes instantaneously. The output variables O values can be associated with one selector that define the input-output (y(k), u(k)) pairs necessary to identify each submodel. So, we can write: O = {O1 , O2 , ..., Os } → {y(k), u(k)}/Mode

(7)

Therefore, if the switching sequence is known and the inputoutput (y(k), u(k)) pairs is defined, the matrices of each submodel can be tackled using standart techniques like subspace identification methods. D. Subspace techniques to identify each submodel Let us consider stable linear time-invariant discrete time systems Σmi of finite order n, valid in the operating zone Ωmi . One form of describing such system is in equation (1). To identify all parameters of this system we make reference on

i/i

The first two conditions guarantee that the rank-n property of Oi Xi/i is preserved after projection onto U⊥ i/i and weighting by M1 and M2 . The third condition expresses that M2 should be uncorrelated with noise sequences w(k) and v(k) to obtain unbiased estimations. 1) Determination of A and C: The matrices A and C can be determined from the extended observability matrix in different ways. All the methods, make use of the shift invariance property of the matrix Oi , which implies that: (−) Aˆ = O i O i , o Oi = Oi (1 : l(i − 1), :) et O i = Oi (l : li, :) (11)

and Cˆ = O i (1 : l, :)

(12)

2) Determination of B and D: After the determination of A and C, the system matrices B and D have to be computed. From the Input/Output equation (8), we define that: (−)

(−)

Oi⊥ Yi/i Ui/i = Oi⊥ Hi + Oi⊥ (Fi Vi/i + Wi/i )Ui/i

(13)

The noise sequences is uncorrelated with the Input, we have: (−)

E[Oi⊥ Yi/i Ui/i ] = Oi⊥ Hi

(14)

where Oi⊥ is a full row rank matrix satisfying Oi⊥ .Oi = 0. Here once again the noise is cancelled out due to the assumption that the input u(k) is correlated with the noise. Observe that with known matrices A,C, Oi⊥ , Ui/i and Yi/i , the equation (13) is linear in B and D. IV. EXPERIMENTAL RESULTS In this section we give some results in the identification of the greenhouse models. We consider one day of March (10 March) to identify all parameters [33], [34]. Therefore, we retain the variables Rg and V v like descriptive variables of the environment where our system evolves. The choice of these variables have been done in a previous work [7], [24]. Moreover, these variables are necessary to construct the transition table IV which used for the discretization of the descriptive variables. XXX Input

Values XX XX X Vv Rg

Lower

Middle

Upper

≤ R′1 ≤ R′1

− > R′1 and ≤ R′2

> R′1 > R′2

TABLE IV E NVIRONMENT SPECIFICATION DISCRETIZATION

Furthermore, the wind speed will take the following values V v = {V vL,V vU} which are respectively Lower and U pper values of V v while the global radiation of the sun will take Rg = {RgL, RgM, RgU} which are respectively Lower, Middle and U pper values of Rg. The combination of the descriptive variables values Rg et V v allows us to have 6 operating modes which are in the table V. Mode Mode1 Mode2 Mode3 Mode4 Mode5 Mode6

Phase ColdNight FreshNight ColdDaybreak FreshDaybreak ColdDay FreshDay

Input variable I1 = {RgL,V vU} I2 = {RgL,V vL} I3 = {RgM,V vU} I4 = {RgM,V vL} I5 = {RgU,V vL} I6 = {RgU,V vU}

TABLE V D ISCRETE MODE DESIGNATION OF THE SUPERVISION DEVICE

All operating modes are defined by: Mode = Mode1 , Mode2 , Mode3 , Mode4 , Mode5 , Mode6 . • Input variables defined by the set I = {I1 , I2 , I3 , I4 , I5 , I6 }, associated respectively by the input events set ev = {ev1 , ev2 , ev3 , ev4 , ev5 , ev6 }. • The set of corresponding output events is described by O = {O1 , O2 , O3 , O4 , O5 , O6 }. The submodels are in three categories : day, night, and daybreak. In each category they are two classes : Cold and •

Fig. 3. a) Atomic discrete events model. b) Switching result between the different operating modes for the identification procedure.

Fresh. This leads us six submodels associated with six operating modes. The atomic model associated with this system is illustrated in the figure (3a). The simulation done with the discrete events atomic model let us to have an idea on the detectability and observability of all operating modes of the system and the corresponding time range. This is illustrated in figure (3b). This result shows us a priori that our system is completely observable (resp. identifiable). On the other hand as for us, our system loses its observability due to fact that we have one very short total dwell time for several operating modes and they are hardly detectable. For instance, the modes 6, 3 and 1. However, a small-scale model can be defined to preserve the observability and the identifiability while eliminating modes with short dwell time. Thus, we can formulate the following proposition: Proposition 4.1: With a dwell time τk ≥ ε for all k ≥ 0 1) All sub-systems must be observable, 2) All operating modes must be detectable, mi 3) The total dwell time Tmi = ∑Nk=1 τk for one mode make it is possible to verify the conditions of persistently exciting input signal for identification of submodels. Therefore, the discreet events atomic model is ∗ composed of operating ª modes defined by Mode = © ∗ ∗ ∗ Mode1 , Mode ª 3 , input variables defined by the © 2 , Mode set I ∗ = I1∗ , I2∗© , I3∗ associated ª respectively by the input and the set of output events is events set ev∗ = ev©∗1 , ev∗2 , ev∗3 ª described by O∗ = O∗1 , O∗2 , O∗3 . This is illustrated in figure (4a). The switching results for the experimental greenhouse are illustrated in the figure (4b). Now, we can identify the continuous dynamical models parameters according to the

algorithm described above. Thus, we have the following models for each mode:

same way, we can show the effectiveness of this approach, in modeling case while considering only 3 modes where, as for us, our system is observable and identifiable. Commutation de modèles pour différents jours de simulation 6 Avec 3 Modes Observables/Identifiables Avec 6 Modes Non Observables/Non Identifiables

Modes Opératoires

5

4

3

2

1

0

8

16

33

41

50

58

66

75

83

Heures

Estimation des deux variables d état de la Serre

Température intérieure

35 Estimé Mesure 30

25

20

15

∗ for the mode µ : Mode2 ¶ µ ¶ −0.2694 −0.3000 1 0 AMode∗2 = ; CMode∗2 = ; 0 1 ¶ µ 0.3795 −0.4274 0 0 0 0 ; BuMode∗ = 2 µ0 0 0 0 ¶ 0.3946 0.3160 −0.0630 0.1335 v BMode∗ = ; 0.5616 0.6740 −0.1797 −0.9603 2 ∗ and for the ¶ ¶ µ µ mode : Mode3 1 0 −0.3235 −0.0513 ; ; CMode∗3 = AMode∗3 = 0 1 µ−0.2581 −1.1357 ¶ 0 −0.1304 −0.2003 −0.1020 BuMode∗ = ; 3 ¶ µ0 0.5993 −0.5851 0.0743 0.3186 0.1036 −0.0076 0.2572 v ; BMode∗ = 0.9125 0.9130 0.2010 −0.1596 3 In the experimentation, we have considered four days of March. Its the 10, 11, 12 and 15March. These choices were made owing to the fact that these days take into account even behaviors which all are not similar. The figure (5) gives us the result of simulation between the internal temperature and hygrometry of the estimated and mesured values. For the

8

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70 Hygrométrie intérieure

for the mode: Mode∗1 ¶ ¶ µ µ 1 0 −0.2128 0.0819 ∗ ∗ ; ; CMode1 = AMode1 = 0 1 µ 0.1259 −0.2162¶ 0.1469 0 0 0 BuMode∗ = ; −0.1480 0 0 0 1 ¶ µ 0.0848 −0.0679 −0.1323 0 v ; BMode∗ = 0.0109 0.1365 0.1780 0 1

0

Heures

Fig. 4. a) Atomic discrete events model with observability and identifiability constraints. b) Equivalent operating modes commutation for the identification procedure.

Estimé Mesure

60 50 40 30 20 10 0

0

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Fig. 5. 1) Operating modes commutation between 6 operating modes (solid line) and 3 operating modes (dashed line). 2) Simulation results between real (solid line) and estimated (dashed line) internal temperature. 3) Simulation result between real (solid line) and estimated (dashed line) internal humidity.

V. CONCLUSIONS AND FUTURE WORKS We propose of this work an approach to modeling a multivariable hybrid complex systems. The system is defined by an interconnection between discrete atomic model with a set of state-space linear models. The atomic model is used to identify the operating modes where our system evolves, and the statespace linear models to approximate continuous phenomena in a nonlinear and not stationary dynamical system. Each operating mode is identify from an interpretation of the process behavior and to identify all parameters of each state space model, we make reference on the general concept in subspace identification. It has shown with our motivated example some observability and identifiability problems. It has proposed to eliminate the operating modes which have a short dwell time

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