- Email: [email protected]
Distributed Systems - A brief review of theory and practice
14th IFAC Conference on Programmable Devices and Embedded 14th Systems 14th IFAC IFAC Conference Conference on on Programmable Programmable Devices Devices and and Embedded Embedded 14th IFAC Conference on Programmable Devices and Embedded Systems Available online at www.sciencedirect.com October 5-7, 2016. Brno, Czech Republic Systems Systems October 5-7, 2016. Brno, Czech Republic October October 5-7, 5-7, 2016. 2016. Brno, Brno, Czech Czech Republic Republic
ScienceDirect
IFAC-PapersOnLine 49-25 (2016) 318–323
Distributed Systems -- A brief Distributed Systems A Distributed Systems -practice A brief brief theory and theory and practice theory and practice
review review review
of of of
Stefan Misik ∗∗ Arben Cela ∗∗ Zdenek Bradac ∗∗∗ ∗ Arben Cela ∗∗ ∗∗ Zdenek Bradac ∗∗∗ ∗∗∗ Stefan Misik ∗ ∗∗ Stefan Misik Arben Cela Stefan Misik Arben Cela Zdenek Zdenek Bradac Bradac ∗∗∗ ∗ ∗ Department of Control and Instrumentation, Faculty of Electrical ∗ Department of Control and Instrumentation, Faculty of Electrical ∗Engineering Department of and Faculty of Communication, Brno University of Technology, Department and of Control Control and Instrumentation, Instrumentation, Faculty of Electrical Electrical Engineering and Communication, Brno University of Technology, Engineering and Communication, Brno Brno, Czech Republic (e-mail: [email protected]) Engineering and Communication, Brno University University of of Technology, Technology, Brno, Czech (e-mail: [email protected]) ∗∗ Brno, Czech Republic Republic (e-mail:Science [email protected]) of Computer and Telecommunication, Brno, Czech Republic (e-mail: [email protected]) ∗∗ Department ∗∗ Department of Computer Science and Telecommunication, ∗∗ Department of Computer Science and Universit´ e Paris-Est, ESIEE Paris, Noisy Grand, France (e-mail: Department of Computer Science andle Telecommunication, Telecommunication, Universit´ ee Paris-Est, ESIEE Paris, Noisy le Grand, France (e-mail: Universit´ Paris-Est, ESIEE Paris, Noisy le [email protected]) Universit´e Paris-Est, ESIEE Paris, Noisy le Grand, Grand, France France (e-mail: (e-mail: [email protected]) ∗∗∗ [email protected]) Department of Control and Instrumentation, Faculty of Electrical [email protected]) ∗∗∗ ∗∗∗ Department of Control and Instrumentation, Faculty of Electrical ∗∗∗ Departmentand of Control Control and Instrumentation, Instrumentation, Faculty of Electrical Electrical Engineering Communication, Brno University of Technology, Department of and Faculty of Engineering Communication, Brno University of Technology, Engineering and Communication, Brno University Brno, and Czech Republic (e-mail: [email protected]) Engineering and Communication, Brno University of of Technology, Technology, Brno, Czech Republic (e-mail: [email protected]) Brno, Brno, Czech Czech Republic Republic (e-mail: (e-mail: [email protected]) [email protected]) Abstract: Due to technological and economical reasons plants, manufacturing systems and Abstract: Due to technological and economical reasons plants, manufacturing Abstract: to and economical reasons manufacturing systems and networks developed with an ever complexity. Such high complexitysystems systems and are Abstract:areDue Due to technological technological andincreasing economical reasons plants, plants, manufacturing systems and networks are developed with an ever increasing complexity. Such high complexity systems are networks are developed with an an ever ever increasing complexity. Such high acomplexity complexity systems are also calledare Large Scale Systems. The increasing aim of thiscomplexity. paper is toSuch present brief review of theory networks developed with high systems are also called Large Scale Systems. The aim of this paper is to present a brief review of theory also called Large Scale Systems. The aim of this paper is to present a brief review of theory used to model specific of distributed scale systems. second this paper also also called Large Scaleclass Systems. The aim large of this paper is to In present a part brief of review of theory to model specific of distributed large scale systems. In second part of this paper also used to specific class of large scale In second part of aused practical example of class modelling process and simulation of such presented. used to model model specific class of distributed distributed large scale systems. systems. In system second is part of this this paper paper also also a practical example of modelling process and simulation of such system is presented. a practical example of modelling process and simulation of such system is presented. a© practical example of modelling process and simulation of such system is presented. 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Large Scale Systems, Distributed Computer Control Systems, Modelling, Keywords: Large Scale Systems, Systems, Distributed Computer Computer Control Systems, Systems, Modelling, Keywords: Scale Distributed Discretization, Distributed Models Keywords: Large Large Scale Systems, Distributed Computer Control Control Systems, Modelling, Modelling, Discretization, Distributed Models Discretization, Discretization, Distributed Distributed Models Models 1. INTRODUCTION models are usually vectors of first-order differential or 1. INTRODUCTION INTRODUCTION models are are usually usually vectors vectors of first-order first-order differential or 1. models of or difference for continuous-time discrete-time 1. INTRODUCTION models areequations usually vectors of first-orderordifferential differential or difference equations for continuous-time or discrete-time difference equations systems differencerespectively. equations for for continuous-time continuous-time or or discrete-time discrete-time systems respectively. respectively. Dynamic systems divided into many component subsys- systems Dynamic systems divided divided into many component component subsys- systems respectively. Dynamic systems into many subsystems by sparsity of their inter-component connections Dynamic systems divided into many component subsystemscalled by sparsity sparsity of their their inter-component connections tems of inter-component µC and are distributed systems. As described connections in Shamma tems by by sparsity of their inter-component connections PlantM A-D µC and are called distributed systems. As described in Shamma µC and are called distributed systems. As described Shamma (2007), Scattolini (2009), and Riverso (2013), in apart from D-A Plant µC and A-D are called distributed systems. As described in Shamma PlantM A-D M Plant A-D (2007), Scattolini (2009), and Riverso (2013), apart from D-A M (2007), Scattolini and (2013), apart from D-A the multitude of (2009), their components limited (2007), Scattolini (2009), and Riverso Riverso and (2013), apart interfrom D-A the multitude of their components and limited interthe of limited component connections, they are alsoand characterized by the multitude multitude of their their components components and limited interintercomponent connections, connections, they are also also characterized by component are by information being sensed they only locally at characterized each component. component connections, they are also characterized by information being sensed only locally at each component. .. .. information being sensed only locally at each component. This leads tobeing distribution information, whencomponent. in contrast information sensed of only locally at each . . µC and This leads leads to distribution distribution of information, information, when inaccess contrast .. Plant A-D This to of when in contrast 1 ... µC and with centralized approaches, no component has to This leads to distribution of information, when in contrast .. µC and D-A Plant11 . µC and A-D with centralized approaches, no component has access to Plant A-D with centralized approaches, no component Plant1 the by all A-D D-A withinformation centralized gathered approaches, no components. component has has access access to to D-A D-A the information gathered by all components. the gathered by all the information information all components. components. Examples of suchgathered systems by include mobile sensor networks, Examples of such systems include mobile sensor networks, Examples of such systems include mobile networks, network congestion control and routing, transportation Examples of such systems include mobile sensor sensor networks, network congestion control and routing, transportation network congestion and transportation systems and Kotsialos (2002); Bellemans Global network Papageorgiou congestion control control and routing, routing, transportation systems Papageorgiou and Kotsialos Kotsialos (2002); distributed Bellemans Bus Bus Global systems Papageorgiou and (2002); et al. (2006), autonomous vehicle systems, Global Control Algorithm systems Papageorgiou and Kotsialos (2002); Bellemans Bellemans Global Bus Bus Bus Bus et al. al. (2006), (2006), power autonomous vehicle systems, distributed Control Algorithm Bus Bus et autonomous vehicle systems, distributed computation, systems Saadat (1998); Riverso and Control Algorithm et al. (2006), autonomous vehicle systems, distributed Control Algorithm computation, power power systems SaadatEtemadi (1998); etRiverso Riverso and computation, systems Saadat and Ferrari-Trecate (2012), micro-grids al. (2012); computation, power systems Saadat (1998); (1998); Riverso and Computer Systems Ferrari-Trecate (2012), micro-grids Etemadi et al. (2012); (2012); Ferrari-Trecate micro-grids al. Computer Systems Bolognani and (2012), Zampieri (2013), Etemadi building et temperature Ferrari-Trecate (2012), micro-grids Etemadi et al. (2012); Computer Computer Systems Systems Bolognani and Zampieri (2013), building building temperature Bolognani (2013), temperature models Ma and et al.Zampieri (2012); Oldewurtel et al. (2012), etc. Bolognani and Zampieri (2013), building temperature Fig. 1. Distributed computer control models Ma et al. (2012); Oldewurtel et al. (2012), etc. models Ma et al. (2012); Oldewurtel et al. (2012), etc. Fig. 1. Distributed computer control models Ma et al. (2012); Oldewurtel et al. (2012), etc. Fig. 1. Distributed computer control Due to technological and economical reasons plants, man- Fig. 1. Distributed computer control The main aim of this paper is to present a review of the Due to technological and economical reasons plants, manDue and economical plants, manufacturing systems and arereasons developed with an The main aim of this paper is present review the Due to to technological technological andnetworks economical reasons plants, manThe main main concerning aim of of this this distributed paper is is to to systems present aaaand review ofscale the literature largeof ufacturing systems and networks are developed with an The aim paper to present review of the ufacturing systems and are ever increasing complexity, such system is shown inwith Fig.an 1. literature concerning distributed systems and large scale ufacturing systems and networks networks are developed developed with an literature concerning distributed systems and large scale systems modelling and conjunction of these subjects. In ever increasing complexity, such system is shown in Fig. 1. literature concerning distributed systems and large scale ever system is in 1. These systems, complexity, also known such as Large Scale Systems, often modelling and conjunction subjects. In ever increasing increasing complexity, such system is shown shown in Fig. Fig. 1. systems systems modelling and conjunction of these subjects. In the experimental part, an example ofof a these large scale system These systems, also known as Large Scale Systems, often systems modelling and conjunction of these subjects. In These systems, also known as Large Scale Systems, often encompass manyalso interacting can be often diffi- the experimental part, an example of a large scale system These systems, known assubsystems Large Scaleand Systems, the experimental experimental part, an an example example of of aa large large scale scale system system model design is presented. encompass many interacting subsystems and can be diffithe part, encompass many interacting subsystems and can be difficult to control using classic control approaches (Scattolini encompass many interacting subsystems and can be diffi- model model design design is is presented. presented. cult to to control using classic control approaches approaches (Scattolini model design is presented. cult classic control (Scattolini (2009)), which using are mostly concerned only by input/output cult to control control using classic control approaches (Scattolini 2. THEORETICAL PART (2009)), which are mostly concerned only by input/output (2009)), are mostly only input/output description of the systems Lewis (1992). 2. THEORETICAL PART (2009)), which which are dynamic mostly concerned concerned only by by input/output 2. description of of the the dynamic dynamic systems systems Lewis Lewis (1992). (1992). 2. THEORETICAL THEORETICAL PART PART description description of the dynamic systems Lewis (1992). More modern approach in modeling and control is there- As mentioned in the introduction, to carry out design mentioned in the introduction, to out Moresuitable modern for approach indefined modeling andofcontrol control is therethereAs the mentioned in the thediscrete-time introduction,domain, to carry carrya state-space out design design More and is of control in fore abovein class systems. This As As mentioned in the introduction, to carry out design More modern modern approach approach in modeling modeling and control is thereof the control in the discrete-time domain, a state-space fore suitable for above defined class of systems. This of the control in the discrete-time domain, a state-space fore suitable for above defined class of systems. This mathematical (1) compriseddomain, of first aorder vector approach dealsfor with time-domain state-space description the control model in the discrete-time state-space fore suitable above defined class of systems. This of model (1) comprised first order vector approach deals with with time-domain state-space description mathematical model (1) comprised of approach deals time-domain state-space description difference equations used. of (Lewis (1992)). Mathematical tool for expressing such mathematical mathematical modelare (1)often comprised of first first order order vector vector approach deals with time-domain state-space description difference equations are often used. (Lewis (1992)). Mathematical tool for expressing such (Lewis difference equations equations are are often often used. used. (Lewis (1992)). (1992)). Mathematical Mathematical tool tool for for expressing expressing such such difference Copyright © 2016, 2016 IFAC 318Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright 2016 IFAC 318 Copyright © 2016 IFAC 318 Peer review© of International Federation of Automatic Copyright ©under 2016 responsibility IFAC 318Control. 10.1016/j.ifacol.2016.12.057
2016 IFAC PDES October 5-7, 2016. Brno, Czech Republic
Stefan Misik et al. / IFAC-PapersOnLine 49-25 (2016) 318–323
x(k + 1) = Ax(k) + Bu(k) y(k) = Cx(k) + Du(k)
(1)
where k ∈ Z+ is number of the sample and corresponds to the time kTS , TS is a sampling period, x(k) ∈ Rn is a vector of discrete-time internal states, u(k) ∈ Rm is a vector of discrete-time control inputs, y(k) ∈ Rp is a vector of discrete-time measured outputs, and A ∈ Rn×n , B ∈ Rn×m , C ∈ Rp×n , D ∈ Rp×m are matrices whose elements specify the static and dynamic properties and behavior of the discrete-time plant model. Discrete-time dynamic systems describe either (I) inherently discrete systems, e.g. bank saving account balance at the k-th month, (II) Transformed continuous-time systems, while the latter is much more common case. 2.1 Large Scale Systems
y3
x2
u2
y2
x2
Sybsystem 1 x1
u3
u1
y1
Systems denoted by the name Large Scale Systems are often described as a result of many subsystems interacting through the coupling of physical variables or the transmission of information over a communication network. It is common to represent a LSS as coupling graph (Fig. 2), i.e. a directed graph where nodes represent subsystems and edges are couplings Riverso (2013). Couplings from subsystem i to subsystem j, in Fig. 2 represented by an arrow with label xi , denote that, the dynamics of the j-th subsystem depend on the signal xi . Also systems i and j are said to be in parent-child relation.
Sybsystem 3 x3
x1
Sybsystem 2 x2
x1
x3
x5
x2
319
xi (k + 1) = Aii xi (k) + Bi ui (k) +
j∈Ni
yi (k) = Ci xi (k) + Di ui (k)
Aij xj (3)
where k ∈ Z+ is number of the sample and corresponds to the time kTS , TS is a sampling period, i ∈ M and j ∈ M are subsystem indices, xi (k) ∈ Rni is a vector of discretetime internal states of the subsystem i, ui (k) ∈ Rmi is a vector of discrete-time control inputs of the subsystem i, yi (k) ∈ Rpi is a vector of discrete-time measured outputs of the subsystem i, and Aij ∈ Rni ×nj , Bi ∈ Rni ×mi , Ci ∈ Rpi ×ni , Di ∈ Rpi ×mi are matrices whose elements specify the dynamic properties and behavior of the subsystem i. By comparing model of a subsystem (3) to a general model (1), it can be seen that in former a term with sum is added. This part of the (3) equation describes the coupling between subsystems. From the (3) is clear that all the subsystem are input and output decoupled, as the B, C, D matrices of the overall system model (2) are block diagonal matrices, which is shown in (4).
A11 · · · A1M .. A = ... . . . . AM 1 · · · AM M B1 · · · 0 B = ... . . . ... 0 · · · BM C1 · · · 0 C = ... . . . ... 0 · · · CM D1 · · · 0 D = ... . . . ... 0 · · · DM
(4)
2.2 Mixed Euler-ZOH Discretization Sybsystem 4 x4
Sybsystem 5 x5 u5
y5
u4
y4
Fig. 2. LSS coupling graph Interaction Modeling A considered LSS is comprised of M subsystems which together form whole system, which can be modeled by (2). System (2) is then partitioned into M low order interconnected non-overlapping systems Farina and Scattolini (2012) whose indices are contained in the set M, i.e. M = {i ∈ Z+ ; 1 ≤ i ≤ M }. x(k + 1) = Ax(k) + Bu(k) y(k) = Cx(k) + Du(k)
(2)
Each decentralized node i ∈ M of the system (Fig. 2) is represented by the model of the subsystem Σi and its neighborhood Ni , which is set of subsystem indices Ni = {j ∈ M : Aij = 0, i = j}. 319
It is common that the models of LSS are developed in continuous time (i. e. models expressed as differential equations (5)), while in most cases control synthesis methods are designed for discrete-time systems. Especially for LSS, and in general sparse systems, the most recent synthesis methods based on methods such as Model Predictive Control are developed in discrete-time Scattolini (2009). x˙ c (t) = Ac x(t) + Bc uc (t) yc (t) = Cxc (t) + Duc (t)
(5)
In (5) and (6) the matrices, vectors, sets and indices have similar meaning as in (2) and (3) respectively, the added subscript ”c ” denotes that they are associated with the continuous-time system. Similarly as a discrete-time LSS, at the beginning of the section, a continuous-time LSS (5) results from the interaction and/or coordination of a large number of interconnected subsystems (6). It is desirable that the discretetime model, used to design controller(s), has the same
Stefan Misik et al. / IFAC-PapersOnLine 49-25 (2016) 318–323
x˙ c,i (t) = Ac,ii xc,i (t) + Bc,i uc,i (t) +
Ac,ij xc,j (6)
j∈Nc,i
yc,i (t) = Cc,i xc,i (t) + Dc,i uc,i (t)
From the properties described above follows that the Nc,i = Ni , ∀i ∈ M and Ac,ij = 0 =⇒ Aij = 0, ∀i ∈ M, j ∈ M, i = j, which express preservation of the topology of a LSS.
described, using theory outlined in the previous section, as a set of subsystems Σi , i ∈ M. 3.1 Model Each mass constitutes one subsystem with state variables xi = (xi1 , xi2 , xi3 , xi4 ) [m, m s−1 , m, m s−1 ] and inputs ui = (ui1 , ui2 ) [hN, hN], where xi1 and xi3 are vertical and horizontal displacement from the equilibrium respectively, xi2 and xi4 are the vertical and horizontal velocities respectively, ui1 and ui2 are the vertical and horizontal forces respectively applied to the mass. Each subsystem Σi has an associated set of neighbor subsystems Σj , j ∈ Ni .
mj1
Af E (TS ) = I + TS Ac
(7)
B f E (TS ) = TS Bc
This method combines forward Euler discretization (7), which preserves sparsity but easily discards stability of the discretized system, and zero-order-hold discretization, which preserves stability for larger range of sampling periods TS . Combination of these methods, as presented in Farina et al. (2013), is shown in (8).
ki,j4
ki,j2
mj4
m j2
mi bi,j4
bi,j2 bi,j3
One such method is called Mixed Euler zero-order-hold discretization and was proposed in Colaneri et al. (2012); Farina et al. (2013).
bi,j1
It is well known that these properties are lost by implementing exact zero-order-hold discretization or generaltype Tustin discretization Farina et al. (2013), therefore it is crucial to find another methods to transform continuoustime LSS into discrete-time equivalent.
ki,j1
interconnection topology as the continuous-time model. In other words, it is beneficial for the continuous-time model (5) to be transformed into discrete-time model (2) in the way its continuous-time subsystems (6) are mapped into discrete-time subsystems (3) respectively for all i ∈ M.
ki,j3
2016 IFAC PDES 320 October 5-7, 2016. Brno, Czech Republic
mj3 Fig. 3. Mechanical distributed system
A(TS ) = I + G(TS )Ac B(TS ) = G(TS )Bc TS G = diag eAc,11 t dt, . . . , 0
TS
e
Ac,M M t
0
dt
(8)
Equations (9) are obtained by expanding (8) into separate matrix expressions for each subsystem. By comparing (9) with a ZOH discretization, it is apparent that Mixed EulerZOH discretization can be alternatively perceived as a ZOH discretization applied on separate subsystems of a LSS, while the coupling is treated as another exogenous input. Aii = eAc,ii TS TS Bi = Bc,i eAc,ii τ dτ 0 TS Aij = Ac,ij eAc,ii τ dτ 0
∀i ∈ M ∀i ∈ M
(9)
∀i ∈ M, j ∈ M, i = j
3. EXPERIMENTAL PART In this section a distributed LSS is considered which consists of masses connected by the combination of springs and dampers, as shown in Fig. 3. This system can be 320
From the Fig. 3 is evident that ki,j and bi,j , where i ∈ M; j ∈ M; i = j, are spring and damper constants respectively. Also can be observed that the order of the constant indices does not make any change. k = kj,i ∀i ∈ M; ∀j ∈ M; i = j i,j bi,j = bj,i Each subsystem from Fig. 3 can be mathematically described by set of differential equations (10). x˙ c,i1 =xc,i1 1 x˙ c,i2 = − ki,j xc,i1 − xc,j1 − mi j∈Ni uc,i1 bi,j xc,i2 − xc,j2 + mi x˙ c,i3 =xc,i4 1 − ki,j xc,i3 − xc,j3 − x˙ c,i4 = mi j∈Ni uc,i2 bi,j xc,i4 − xc,j4 + mi
(10)
In (10) mi is veight of i-th mass, xc,i1 , xc,i2 , xc,i3 , and xc,i4 are continuous-time state variables of subsystem i and uc,i1 , uc,i2 are continuous-time inputs of the subsystem i.
2016 IFAC PDES October 5-7, 2016. Brno, Czech Republic
Stefan Misik et al. / IFAC-PapersOnLine 49-25 (2016) 318–323
Differential equations (10) can be transformed into a LSS of the form (3), introduced in the previous section.
0
1
−Σki,j Ac,ii = mi 0 0
0 0 0
0 0 1
−Σki,j mi
−Σbi,j mi
−Σbi,j mi
0 0
0
0
kmi,j i
Ac,ij = 0 0
bi,j mi
0 0
0
1 mi Bc,i = 0 0 In (11a) Σki,j stands for j∈Ni bi,j where i ∈ M.
j∈Ni
0 0 0 ki,j mi
0 0 0
0 0 0
321
Results The initial and final positions of the masses associated with their respective subsystems are shown in Fig. 4 and Fig. 5 respectively.
(11a)
(11b)
bi,j mi
(11c)
1 mi
ki,j and Σbi,j stands for
Fig. 4. Initial positions of masses in the system
Model used in this paper‘s experimental part consists of an array of masses each connected in the 4-neighborhood by combination of spring and dampers, in similar fashion as shown in Fig. 3. Mass of each subsystem is selected randomly from the interval [5, 10] kg. The spring and damper constants are for all subsystems identical and are equal to 0.5 N m−1 and 0.5 N s m−1 respectively. 3.2 Discretization To obtain discrete time system model with preserved separation of matrices Aii , Aij and Bi the Mixed EulerZOH discretization, described by (9), is applied to the continuous model matrices (11). To chose the sampling period TS a frequency bandwidth of the system is to be considered. The sampling theorem says that for the sinusoidal signal to be recoverable from its values in discrete points, it is necessary to sample the signal with sampling period equal to the half of the signal’s period ˚ Astr¨ om and Wittenmark (1997). Using this knowledge and by empirical examination, sampling period TS = 0.2 s was chosen for the described system. 3.3 Simulation Theory reviewed in this paper is here applied and created model of a Large Scale System is here presented by a simulation of such system. Model Model used in the simulation consists of 3 × 4 array of the masses each connected in the fashion described above, as shown in Fig. 4 and Fig. 5. These figures are for illustrative purposes only and do not have any physical dimensions in any scale. Simulated values accompanied with physical units can be read from Fig. 6 and Fig. 7 for displacement and velocity state variables respectively. Initially the state of the model is set to be disturbed from its equilibrium, which can be seen in the Fig. 4. 321
Fig. 5. Final positions of masses in the system The evolution of the state variables of the systems during the simulation are shown in Fig. 6 and Fig. 7, which display model‘s displacement and velocity progression respectively. As mentioned earlier, initially, the state of the overall system is randomly disturbed, which is shown in Fig. 4. This can also be observed in Fig. 6 at the time 0 s. Initial velocities of the masses in the system are, however, set to 0 m s−1 , which is reflected in Fig. 7 at the time 0 s. State evolutions shown in figures Fig. 6 and Fig. 7 are further discussed in the discussion section of this paper. 4. DISCUSSION Initially the system of masses is disturbed into state shown in Fig. 4, where the weights of masses are illustrated by the radii of the disks. Similarly, the final positions of the masses are presented in Fig. 5. Note that, for simplicity‘s sake, the connections among masses are not illustrated by the figures Fig. 4 and Fig. 5, but they are present in the model.
2016 IFAC PDES 322 October 5-7, 2016. Brno, Czech Republic
Stefan Misik et al. / IFAC-PapersOnLine 49-25 (2016) 318–323
Purpose behind this paper was to present the theory needed to design models of real-word Large Scale Systems and to describe the whole design process of such model, including description of the physical system using differential equations, discretization of the continuous-time model, and verification of the model using simulation.
1.5
Displacement [m]
1
0.5
From the discussion at the end of this paper it follows, that the theory reviewed in this paper is suitable to create models of distributed dynamic systems, which can be further used to design discrete-time control approaches on such systems.
0
−0.5
−1
ACKNOWLEDGEMENTS −1.5
0
10
20
30
40
50 60 Time [s]
70
80
90
100
This paper was made possible by grant No. FEKT-S-142429 - ”The research of new control methods, measurement procedures and intelligent instruments in automation”, which was funded by the Internal Grant Agency of Brno University of Technology.
Fig. 6. Displacement state variables of the system model 0.6
REFERENCES
0.4
Velocity [m s−1 ]
0.2
0
−0.2
−0.4
−0.6
0
10
20
30
40
50 60 Time [s]
70
80
90
100
Fig. 7. Velocity state variables of the system model From the final positions of masses (Fig. 5) and the evolution of velocity variables (Fig. 7) it is obvious, that system converges towards some equilibrium. This equilibrium, however, is an arbitrary one with offsets added to displacement state variables, which is visible in Fig. 6. The displacement state variables are in Fig. 6 divided into two groups, each of them converging into different value. This two groups are accounting for horizontal and vertical displacement state variables and are converging to some arbitrary offset in which the system reaches its newly obtained equilibrium. This vertical and horizontal offsets depend on the weights of the masses, spring constants, and the initial conditions of the system. In conclusion, this behavior is very much similar to the expected behavior of a real-world system, which consists of an array of masses connected by combination of springs and dampers floating in two-dimensional void space without any external force acting upon it. 5. CONCLUSION This paper contains two parts, first part reviews the theory needed to model distributed large scale systems. In the second part, the prototype model of a large scale system is designed, and the simulation results are presented. 322
˚ Astr¨om, K.J. and Wittenmark, B. (1997). Computercontrolled systems. Prentice Hall, Upper Saddle River, New Jersey 07458, 3rd edition. Bellemans, T., Schutter, B.D., and Moor, B.D. (2006). Model predictive control for ramp metering of motorway traffic: A case study. Control Engineering Practice, 14(7), 757 – 767. doi: http://dx.doi.org/10.1016/j.conengprac.2005.03.010. URL http://www.sciencedirect.com/science/ article/pii/S0967066105000900. Bolognani, S. and Zampieri, S. (2013). A distributed control strategy for reactive power compensation in smart microgrids. IEEE Transactions on Automatic Control, 58(11), 2818–2833. doi:10.1109/TAC.2013.2270317. Colaneri, P., Farina, M., Scattolini, R., and Shorten, R. (2012). A note on discretization of sparce linear systems. In J.D. Bendtsen (ed.), Robust Control Design, volume 7, 97–102. Aalborg Congress & Culture Centre, Denmark. doi:10.3182/20120620-3-DK-2025. 00123. URL http://www.ifac-papersonline.net/ Detailed/53261.html. Etemadi, A.H., Davison, E.J., and Iravani, R. (2012). A decentralized robust control strategy for multi-der microgrids; part i: Fundamental concepts. IEEE Transactions on Power Delivery, 27(4), 1843–1853. doi:10.1109/ TPWRD.2012.2202920. Farina, M., Colaneri, P., and Scattolini, R. (2013). Block-wise discretization accounting for structural constraints. Automatica, 49(11), 3411 – 3417. doi: http://dx.doi.org/10.1016/j.automatica.2013.08.013. URL http://www.sciencedirect.com/science/ article/pii/S0005109813004147. Farina, M. and Scattolini, R. (2012). Distributed predictive control: A non-cooperative algorithm with neighbor-to-neighbor communication for linear systems. Automatica, 48(6), 1088 – 1096. doi: http://dx.doi.org/10.1016/j.automatica.2012.03.020. URL http://www.sciencedirect.com/science/ article/pii/S000510981200129X. Lewis, F.L. (1992). Applied optimal control. Prentice Hall, Englewood Cliffs, N.J.
2016 IFAC PDES October 5-7, 2016. Brno, Czech Republic
Stefan Misik et al. / IFAC-PapersOnLine 49-25 (2016) 318–323
Ma, Y., Borrelli, F., Hencey, B., Coffey, B., Bengea, S., and Haves, P. (2012). Model predictive control for the operation of building cooling systems. IEEE Transactions on Control Systems Technology, 20(3), 796–803. doi:10.1109/TCST.2011.2124461. Oldewurtel, F., Parisio, A., Jones, C.N., Gyalistras, D., Gwerder, M., Stauch, V., Lehmann, B., and Morari, M. (2012). Use of model predictive control and weather forecasts for energy efficient building climate control. Energy and Buildings, 45, 15 – 27. doi:http://dx.doi.org/10.1016/j.enbuild.2011.09. 022. URL http://www.sciencedirect.com/science/ article/pii/S0378778811004105. Papageorgiou, M. and Kotsialos, A. (2002). Freeway ramp metering: an overview. IEEE Transactions on Intelligent Transportation Systems, 3(4), 271–281. doi: 10.1109/TITS.2002.806803. Riverso, S. (2013). Distributed and Plug-and-Play control for constrained systems. Ph.D. thesis, Universit`a degli Studi di Pavia.
323
323
Riverso, S. and Ferrari-Trecate, G. (2012). Hycon2 benchmark: Power network system. Technical report, Universit`a degli Studi di Pavia. URL http://arxiv.org/abs/ 1207.2000. Saadat, H. (1998). Power System Analysis. McgrawHill College. URL http://www.amazon.com/PowerSystem-Analysis-Hadi-Saadat/dp/0075616343% 3FSubscriptionId%3D0JYN1NVW651KCA56C102% 26tag%3Dtechkie-20%26linkCode%3Dxm2%26camp% 3D2025%26creative%3D165953%26creativeASIN% 3D0075616343. Scattolini, R. (2009). Architectures for distributed and hierarchical model predictive control – a review. Journal of Process Control, 19(5), 723 – 731. doi:http://dx.doi.org/10.1016/j.jprocont.2009.02. 003. URL http://www.sciencedirect.com/science/ article/pii/S0959152409000353. Shamma, J.S. (2007). Cooperative control of distributed multi-agent systems. John Wiley, Hoboken, NJ.
ScienceDirect
IFAC-PapersOnLine 49-25 (2016) 318–323
Distributed Systems -- A brief Distributed Systems A Distributed Systems -practice A brief brief theory and theory and practice theory and practice
review review review
of of of
Stefan Misik ∗∗ Arben Cela ∗∗ Zdenek Bradac ∗∗∗ ∗ Arben Cela ∗∗ ∗∗ Zdenek Bradac ∗∗∗ ∗∗∗ Stefan Misik ∗ ∗∗ Stefan Misik Arben Cela Stefan Misik Arben Cela Zdenek Zdenek Bradac Bradac ∗∗∗ ∗ ∗ Department of Control and Instrumentation, Faculty of Electrical ∗ Department of Control and Instrumentation, Faculty of Electrical ∗Engineering Department of and Faculty of Communication, Brno University of Technology, Department and of Control Control and Instrumentation, Instrumentation, Faculty of Electrical Electrical Engineering and Communication, Brno University of Technology, Engineering and Communication, Brno Brno, Czech Republic (e-mail: [email protected]) Engineering and Communication, Brno University University of of Technology, Technology, Brno, Czech (e-mail: [email protected]) ∗∗ Brno, Czech Republic Republic (e-mail:Science [email protected]) of Computer and Telecommunication, Brno, Czech Republic (e-mail: [email protected]) ∗∗ Department ∗∗ Department of Computer Science and Telecommunication, ∗∗ Department of Computer Science and Universit´ e Paris-Est, ESIEE Paris, Noisy Grand, France (e-mail: Department of Computer Science andle Telecommunication, Telecommunication, Universit´ ee Paris-Est, ESIEE Paris, Noisy le Grand, France (e-mail: Universit´ Paris-Est, ESIEE Paris, Noisy le [email protected]) Universit´e Paris-Est, ESIEE Paris, Noisy le Grand, Grand, France France (e-mail: (e-mail: [email protected]) ∗∗∗ [email protected]) Department of Control and Instrumentation, Faculty of Electrical [email protected]) ∗∗∗ ∗∗∗ Department of Control and Instrumentation, Faculty of Electrical ∗∗∗ Departmentand of Control Control and Instrumentation, Instrumentation, Faculty of Electrical Electrical Engineering Communication, Brno University of Technology, Department of and Faculty of Engineering Communication, Brno University of Technology, Engineering and Communication, Brno University Brno, and Czech Republic (e-mail: [email protected]) Engineering and Communication, Brno University of of Technology, Technology, Brno, Czech Republic (e-mail: [email protected]) Brno, Brno, Czech Czech Republic Republic (e-mail: (e-mail: [email protected]) [email protected]) Abstract: Due to technological and economical reasons plants, manufacturing systems and Abstract: Due to technological and economical reasons plants, manufacturing Abstract: to and economical reasons manufacturing systems and networks developed with an ever complexity. Such high complexitysystems systems and are Abstract:areDue Due to technological technological andincreasing economical reasons plants, plants, manufacturing systems and networks are developed with an ever increasing complexity. Such high complexity systems are networks are developed with an an ever ever increasing complexity. Such high acomplexity complexity systems are also calledare Large Scale Systems. The increasing aim of thiscomplexity. paper is toSuch present brief review of theory networks developed with high systems are also called Large Scale Systems. The aim of this paper is to present a brief review of theory also called Large Scale Systems. The aim of this paper is to present a brief review of theory used to model specific of distributed scale systems. second this paper also also called Large Scaleclass Systems. The aim large of this paper is to In present a part brief of review of theory to model specific of distributed large scale systems. In second part of this paper also used to specific class of large scale In second part of aused practical example of class modelling process and simulation of such presented. used to model model specific class of distributed distributed large scale systems. systems. In system second is part of this this paper paper also also a practical example of modelling process and simulation of such system is presented. a practical example of modelling process and simulation of such system is presented. a© practical example of modelling process and simulation of such system is presented. 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Large Scale Systems, Distributed Computer Control Systems, Modelling, Keywords: Large Scale Systems, Systems, Distributed Computer Computer Control Systems, Systems, Modelling, Keywords: Scale Distributed Discretization, Distributed Models Keywords: Large Large Scale Systems, Distributed Computer Control Control Systems, Modelling, Modelling, Discretization, Distributed Models Discretization, Discretization, Distributed Distributed Models Models 1. INTRODUCTION models are usually vectors of first-order differential or 1. INTRODUCTION INTRODUCTION models are are usually usually vectors vectors of first-order first-order differential or 1. models of or difference for continuous-time discrete-time 1. INTRODUCTION models areequations usually vectors of first-orderordifferential differential or difference equations for continuous-time or discrete-time difference equations systems differencerespectively. equations for for continuous-time continuous-time or or discrete-time discrete-time systems respectively. respectively. Dynamic systems divided into many component subsys- systems Dynamic systems divided divided into many component component subsys- systems respectively. Dynamic systems into many subsystems by sparsity of their inter-component connections Dynamic systems divided into many component subsystemscalled by sparsity sparsity of their their inter-component connections tems of inter-component µC and are distributed systems. As described connections in Shamma tems by by sparsity of their inter-component connections PlantM A-D µC and are called distributed systems. As described in Shamma µC and are called distributed systems. As described Shamma (2007), Scattolini (2009), and Riverso (2013), in apart from D-A Plant µC and A-D are called distributed systems. As described in Shamma PlantM A-D M Plant A-D (2007), Scattolini (2009), and Riverso (2013), apart from D-A M (2007), Scattolini and (2013), apart from D-A the multitude of (2009), their components limited (2007), Scattolini (2009), and Riverso Riverso and (2013), apart interfrom D-A the multitude of their components and limited interthe of limited component connections, they are alsoand characterized by the multitude multitude of their their components components and limited interintercomponent connections, connections, they are also also characterized by component are by information being sensed they only locally at characterized each component. component connections, they are also characterized by information being sensed only locally at each component. .. .. information being sensed only locally at each component. This leads tobeing distribution information, whencomponent. in contrast information sensed of only locally at each . . µC and This leads leads to distribution distribution of information, information, when inaccess contrast .. Plant A-D This to of when in contrast 1 ... µC and with centralized approaches, no component has to This leads to distribution of information, when in contrast .. µC and D-A Plant11 . µC and A-D with centralized approaches, no component has access to Plant A-D with centralized approaches, no component Plant1 the by all A-D D-A withinformation centralized gathered approaches, no components. component has has access access to to D-A D-A the information gathered by all components. the gathered by all the information information all components. components. Examples of suchgathered systems by include mobile sensor networks, Examples of such systems include mobile sensor networks, Examples of such systems include mobile networks, network congestion control and routing, transportation Examples of such systems include mobile sensor sensor networks, network congestion control and routing, transportation network congestion and transportation systems and Kotsialos (2002); Bellemans Global network Papageorgiou congestion control control and routing, routing, transportation systems Papageorgiou and Kotsialos Kotsialos (2002); distributed Bellemans Bus Bus Global systems Papageorgiou and (2002); et al. (2006), autonomous vehicle systems, Global Control Algorithm systems Papageorgiou and Kotsialos (2002); Bellemans Bellemans Global Bus Bus Bus Bus et al. al. (2006), (2006), power autonomous vehicle systems, distributed Control Algorithm Bus Bus et autonomous vehicle systems, distributed computation, systems Saadat (1998); Riverso and Control Algorithm et al. (2006), autonomous vehicle systems, distributed Control Algorithm computation, power power systems SaadatEtemadi (1998); etRiverso Riverso and computation, systems Saadat and Ferrari-Trecate (2012), micro-grids al. (2012); computation, power systems Saadat (1998); (1998); Riverso and Computer Systems Ferrari-Trecate (2012), micro-grids Etemadi et al. (2012); (2012); Ferrari-Trecate micro-grids al. Computer Systems Bolognani and (2012), Zampieri (2013), Etemadi building et temperature Ferrari-Trecate (2012), micro-grids Etemadi et al. (2012); Computer Computer Systems Systems Bolognani and Zampieri (2013), building building temperature Bolognani (2013), temperature models Ma and et al.Zampieri (2012); Oldewurtel et al. (2012), etc. Bolognani and Zampieri (2013), building temperature Fig. 1. Distributed computer control models Ma et al. (2012); Oldewurtel et al. (2012), etc. models Ma et al. (2012); Oldewurtel et al. (2012), etc. Fig. 1. Distributed computer control models Ma et al. (2012); Oldewurtel et al. (2012), etc. Fig. 1. Distributed computer control Due to technological and economical reasons plants, man- Fig. 1. Distributed computer control The main aim of this paper is to present a review of the Due to technological and economical reasons plants, manDue and economical plants, manufacturing systems and arereasons developed with an The main aim of this paper is present review the Due to to technological technological andnetworks economical reasons plants, manThe main main concerning aim of of this this distributed paper is is to to systems present aaaand review ofscale the literature largeof ufacturing systems and networks are developed with an The aim paper to present review of the ufacturing systems and are ever increasing complexity, such system is shown inwith Fig.an 1. literature concerning distributed systems and large scale ufacturing systems and networks networks are developed developed with an literature concerning distributed systems and large scale systems modelling and conjunction of these subjects. In ever increasing complexity, such system is shown in Fig. 1. literature concerning distributed systems and large scale ever system is in 1. These systems, complexity, also known such as Large Scale Systems, often modelling and conjunction subjects. In ever increasing increasing complexity, such system is shown shown in Fig. Fig. 1. systems systems modelling and conjunction of these subjects. In the experimental part, an example ofof a these large scale system These systems, also known as Large Scale Systems, often systems modelling and conjunction of these subjects. In These systems, also known as Large Scale Systems, often encompass manyalso interacting can be often diffi- the experimental part, an example of a large scale system These systems, known assubsystems Large Scaleand Systems, the experimental experimental part, an an example example of of aa large large scale scale system system model design is presented. encompass many interacting subsystems and can be diffithe part, encompass many interacting subsystems and can be difficult to control using classic control approaches (Scattolini encompass many interacting subsystems and can be diffi- model model design design is is presented. presented. cult to to control using classic control approaches approaches (Scattolini model design is presented. cult classic control (Scattolini (2009)), which using are mostly concerned only by input/output cult to control control using classic control approaches (Scattolini 2. THEORETICAL PART (2009)), which are mostly concerned only by input/output (2009)), are mostly only input/output description of the systems Lewis (1992). 2. THEORETICAL PART (2009)), which which are dynamic mostly concerned concerned only by by input/output 2. description of of the the dynamic dynamic systems systems Lewis Lewis (1992). (1992). 2. THEORETICAL THEORETICAL PART PART description description of the dynamic systems Lewis (1992). More modern approach in modeling and control is there- As mentioned in the introduction, to carry out design mentioned in the introduction, to out Moresuitable modern for approach indefined modeling andofcontrol control is therethereAs the mentioned in the thediscrete-time introduction,domain, to carry carrya state-space out design design More and is of control in fore abovein class systems. This As As mentioned in the introduction, to carry out design More modern modern approach approach in modeling modeling and control is thereof the control in the discrete-time domain, a state-space fore suitable for above defined class of systems. This of the control in the discrete-time domain, a state-space fore suitable for above defined class of systems. This mathematical (1) compriseddomain, of first aorder vector approach dealsfor with time-domain state-space description the control model in the discrete-time state-space fore suitable above defined class of systems. This of model (1) comprised first order vector approach deals with with time-domain state-space description mathematical model (1) comprised of approach deals time-domain state-space description difference equations used. of (Lewis (1992)). Mathematical tool for expressing such mathematical mathematical modelare (1)often comprised of first first order order vector vector approach deals with time-domain state-space description difference equations are often used. (Lewis (1992)). Mathematical tool for expressing such (Lewis difference equations equations are are often often used. used. (Lewis (1992)). (1992)). Mathematical Mathematical tool tool for for expressing expressing such such difference Copyright © 2016, 2016 IFAC 318Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright 2016 IFAC 318 Copyright © 2016 IFAC 318 Peer review© of International Federation of Automatic Copyright ©under 2016 responsibility IFAC 318Control. 10.1016/j.ifacol.2016.12.057
2016 IFAC PDES October 5-7, 2016. Brno, Czech Republic
Stefan Misik et al. / IFAC-PapersOnLine 49-25 (2016) 318–323
x(k + 1) = Ax(k) + Bu(k) y(k) = Cx(k) + Du(k)
(1)
where k ∈ Z+ is number of the sample and corresponds to the time kTS , TS is a sampling period, x(k) ∈ Rn is a vector of discrete-time internal states, u(k) ∈ Rm is a vector of discrete-time control inputs, y(k) ∈ Rp is a vector of discrete-time measured outputs, and A ∈ Rn×n , B ∈ Rn×m , C ∈ Rp×n , D ∈ Rp×m are matrices whose elements specify the static and dynamic properties and behavior of the discrete-time plant model. Discrete-time dynamic systems describe either (I) inherently discrete systems, e.g. bank saving account balance at the k-th month, (II) Transformed continuous-time systems, while the latter is much more common case. 2.1 Large Scale Systems
y3
x2
u2
y2
x2
Sybsystem 1 x1
u3
u1
y1
Systems denoted by the name Large Scale Systems are often described as a result of many subsystems interacting through the coupling of physical variables or the transmission of information over a communication network. It is common to represent a LSS as coupling graph (Fig. 2), i.e. a directed graph where nodes represent subsystems and edges are couplings Riverso (2013). Couplings from subsystem i to subsystem j, in Fig. 2 represented by an arrow with label xi , denote that, the dynamics of the j-th subsystem depend on the signal xi . Also systems i and j are said to be in parent-child relation.
Sybsystem 3 x3
x1
Sybsystem 2 x2
x1
x3
x5
x2
319
xi (k + 1) = Aii xi (k) + Bi ui (k) +
j∈Ni
yi (k) = Ci xi (k) + Di ui (k)
Aij xj (3)
where k ∈ Z+ is number of the sample and corresponds to the time kTS , TS is a sampling period, i ∈ M and j ∈ M are subsystem indices, xi (k) ∈ Rni is a vector of discretetime internal states of the subsystem i, ui (k) ∈ Rmi is a vector of discrete-time control inputs of the subsystem i, yi (k) ∈ Rpi is a vector of discrete-time measured outputs of the subsystem i, and Aij ∈ Rni ×nj , Bi ∈ Rni ×mi , Ci ∈ Rpi ×ni , Di ∈ Rpi ×mi are matrices whose elements specify the dynamic properties and behavior of the subsystem i. By comparing model of a subsystem (3) to a general model (1), it can be seen that in former a term with sum is added. This part of the (3) equation describes the coupling between subsystems. From the (3) is clear that all the subsystem are input and output decoupled, as the B, C, D matrices of the overall system model (2) are block diagonal matrices, which is shown in (4).
A11 · · · A1M .. A = ... . . . . AM 1 · · · AM M B1 · · · 0 B = ... . . . ... 0 · · · BM C1 · · · 0 C = ... . . . ... 0 · · · CM D1 · · · 0 D = ... . . . ... 0 · · · DM
(4)
2.2 Mixed Euler-ZOH Discretization Sybsystem 4 x4
Sybsystem 5 x5 u5
y5
u4
y4
Fig. 2. LSS coupling graph Interaction Modeling A considered LSS is comprised of M subsystems which together form whole system, which can be modeled by (2). System (2) is then partitioned into M low order interconnected non-overlapping systems Farina and Scattolini (2012) whose indices are contained in the set M, i.e. M = {i ∈ Z+ ; 1 ≤ i ≤ M }. x(k + 1) = Ax(k) + Bu(k) y(k) = Cx(k) + Du(k)
(2)
Each decentralized node i ∈ M of the system (Fig. 2) is represented by the model of the subsystem Σi and its neighborhood Ni , which is set of subsystem indices Ni = {j ∈ M : Aij = 0, i = j}. 319
It is common that the models of LSS are developed in continuous time (i. e. models expressed as differential equations (5)), while in most cases control synthesis methods are designed for discrete-time systems. Especially for LSS, and in general sparse systems, the most recent synthesis methods based on methods such as Model Predictive Control are developed in discrete-time Scattolini (2009). x˙ c (t) = Ac x(t) + Bc uc (t) yc (t) = Cxc (t) + Duc (t)
(5)
In (5) and (6) the matrices, vectors, sets and indices have similar meaning as in (2) and (3) respectively, the added subscript ”c ” denotes that they are associated with the continuous-time system. Similarly as a discrete-time LSS, at the beginning of the section, a continuous-time LSS (5) results from the interaction and/or coordination of a large number of interconnected subsystems (6). It is desirable that the discretetime model, used to design controller(s), has the same
Stefan Misik et al. / IFAC-PapersOnLine 49-25 (2016) 318–323
x˙ c,i (t) = Ac,ii xc,i (t) + Bc,i uc,i (t) +
Ac,ij xc,j (6)
j∈Nc,i
yc,i (t) = Cc,i xc,i (t) + Dc,i uc,i (t)
From the properties described above follows that the Nc,i = Ni , ∀i ∈ M and Ac,ij = 0 =⇒ Aij = 0, ∀i ∈ M, j ∈ M, i = j, which express preservation of the topology of a LSS.
described, using theory outlined in the previous section, as a set of subsystems Σi , i ∈ M. 3.1 Model Each mass constitutes one subsystem with state variables xi = (xi1 , xi2 , xi3 , xi4 ) [m, m s−1 , m, m s−1 ] and inputs ui = (ui1 , ui2 ) [hN, hN], where xi1 and xi3 are vertical and horizontal displacement from the equilibrium respectively, xi2 and xi4 are the vertical and horizontal velocities respectively, ui1 and ui2 are the vertical and horizontal forces respectively applied to the mass. Each subsystem Σi has an associated set of neighbor subsystems Σj , j ∈ Ni .
mj1
Af E (TS ) = I + TS Ac
(7)
B f E (TS ) = TS Bc
This method combines forward Euler discretization (7), which preserves sparsity but easily discards stability of the discretized system, and zero-order-hold discretization, which preserves stability for larger range of sampling periods TS . Combination of these methods, as presented in Farina et al. (2013), is shown in (8).
ki,j4
ki,j2
mj4
m j2
mi bi,j4
bi,j2 bi,j3
One such method is called Mixed Euler zero-order-hold discretization and was proposed in Colaneri et al. (2012); Farina et al. (2013).
bi,j1
It is well known that these properties are lost by implementing exact zero-order-hold discretization or generaltype Tustin discretization Farina et al. (2013), therefore it is crucial to find another methods to transform continuoustime LSS into discrete-time equivalent.
ki,j1
interconnection topology as the continuous-time model. In other words, it is beneficial for the continuous-time model (5) to be transformed into discrete-time model (2) in the way its continuous-time subsystems (6) are mapped into discrete-time subsystems (3) respectively for all i ∈ M.
ki,j3
2016 IFAC PDES 320 October 5-7, 2016. Brno, Czech Republic
mj3 Fig. 3. Mechanical distributed system
A(TS ) = I + G(TS )Ac B(TS ) = G(TS )Bc TS G = diag eAc,11 t dt, . . . , 0
TS
e
Ac,M M t
0
dt
(8)
Equations (9) are obtained by expanding (8) into separate matrix expressions for each subsystem. By comparing (9) with a ZOH discretization, it is apparent that Mixed EulerZOH discretization can be alternatively perceived as a ZOH discretization applied on separate subsystems of a LSS, while the coupling is treated as another exogenous input. Aii = eAc,ii TS TS Bi = Bc,i eAc,ii τ dτ 0 TS Aij = Ac,ij eAc,ii τ dτ 0
∀i ∈ M ∀i ∈ M
(9)
∀i ∈ M, j ∈ M, i = j
3. EXPERIMENTAL PART In this section a distributed LSS is considered which consists of masses connected by the combination of springs and dampers, as shown in Fig. 3. This system can be 320
From the Fig. 3 is evident that ki,j and bi,j , where i ∈ M; j ∈ M; i = j, are spring and damper constants respectively. Also can be observed that the order of the constant indices does not make any change. k = kj,i ∀i ∈ M; ∀j ∈ M; i = j i,j bi,j = bj,i Each subsystem from Fig. 3 can be mathematically described by set of differential equations (10). x˙ c,i1 =xc,i1 1 x˙ c,i2 = − ki,j xc,i1 − xc,j1 − mi j∈Ni uc,i1 bi,j xc,i2 − xc,j2 + mi x˙ c,i3 =xc,i4 1 − ki,j xc,i3 − xc,j3 − x˙ c,i4 = mi j∈Ni uc,i2 bi,j xc,i4 − xc,j4 + mi
(10)
In (10) mi is veight of i-th mass, xc,i1 , xc,i2 , xc,i3 , and xc,i4 are continuous-time state variables of subsystem i and uc,i1 , uc,i2 are continuous-time inputs of the subsystem i.
2016 IFAC PDES October 5-7, 2016. Brno, Czech Republic
Stefan Misik et al. / IFAC-PapersOnLine 49-25 (2016) 318–323
Differential equations (10) can be transformed into a LSS of the form (3), introduced in the previous section.
0
1
−Σki,j Ac,ii = mi 0 0
0 0 0
0 0 1
−Σki,j mi
−Σbi,j mi
−Σbi,j mi
0 0
0
0
kmi,j i
Ac,ij = 0 0
bi,j mi
0 0
0
1 mi Bc,i = 0 0 In (11a) Σki,j stands for j∈Ni bi,j where i ∈ M.
j∈Ni
0 0 0 ki,j mi
0 0 0
0 0 0
321
Results The initial and final positions of the masses associated with their respective subsystems are shown in Fig. 4 and Fig. 5 respectively.
(11a)
(11b)
bi,j mi
(11c)
1 mi
ki,j and Σbi,j stands for
Fig. 4. Initial positions of masses in the system
Model used in this paper‘s experimental part consists of an array of masses each connected in the 4-neighborhood by combination of spring and dampers, in similar fashion as shown in Fig. 3. Mass of each subsystem is selected randomly from the interval [5, 10] kg. The spring and damper constants are for all subsystems identical and are equal to 0.5 N m−1 and 0.5 N s m−1 respectively. 3.2 Discretization To obtain discrete time system model with preserved separation of matrices Aii , Aij and Bi the Mixed EulerZOH discretization, described by (9), is applied to the continuous model matrices (11). To chose the sampling period TS a frequency bandwidth of the system is to be considered. The sampling theorem says that for the sinusoidal signal to be recoverable from its values in discrete points, it is necessary to sample the signal with sampling period equal to the half of the signal’s period ˚ Astr¨ om and Wittenmark (1997). Using this knowledge and by empirical examination, sampling period TS = 0.2 s was chosen for the described system. 3.3 Simulation Theory reviewed in this paper is here applied and created model of a Large Scale System is here presented by a simulation of such system. Model Model used in the simulation consists of 3 × 4 array of the masses each connected in the fashion described above, as shown in Fig. 4 and Fig. 5. These figures are for illustrative purposes only and do not have any physical dimensions in any scale. Simulated values accompanied with physical units can be read from Fig. 6 and Fig. 7 for displacement and velocity state variables respectively. Initially the state of the model is set to be disturbed from its equilibrium, which can be seen in the Fig. 4. 321
Fig. 5. Final positions of masses in the system The evolution of the state variables of the systems during the simulation are shown in Fig. 6 and Fig. 7, which display model‘s displacement and velocity progression respectively. As mentioned earlier, initially, the state of the overall system is randomly disturbed, which is shown in Fig. 4. This can also be observed in Fig. 6 at the time 0 s. Initial velocities of the masses in the system are, however, set to 0 m s−1 , which is reflected in Fig. 7 at the time 0 s. State evolutions shown in figures Fig. 6 and Fig. 7 are further discussed in the discussion section of this paper. 4. DISCUSSION Initially the system of masses is disturbed into state shown in Fig. 4, where the weights of masses are illustrated by the radii of the disks. Similarly, the final positions of the masses are presented in Fig. 5. Note that, for simplicity‘s sake, the connections among masses are not illustrated by the figures Fig. 4 and Fig. 5, but they are present in the model.
2016 IFAC PDES 322 October 5-7, 2016. Brno, Czech Republic
Stefan Misik et al. / IFAC-PapersOnLine 49-25 (2016) 318–323
Purpose behind this paper was to present the theory needed to design models of real-word Large Scale Systems and to describe the whole design process of such model, including description of the physical system using differential equations, discretization of the continuous-time model, and verification of the model using simulation.
1.5
Displacement [m]
1
0.5
From the discussion at the end of this paper it follows, that the theory reviewed in this paper is suitable to create models of distributed dynamic systems, which can be further used to design discrete-time control approaches on such systems.
0
−0.5
−1
ACKNOWLEDGEMENTS −1.5
0
10
20
30
40
50 60 Time [s]
70
80
90
100
This paper was made possible by grant No. FEKT-S-142429 - ”The research of new control methods, measurement procedures and intelligent instruments in automation”, which was funded by the Internal Grant Agency of Brno University of Technology.
Fig. 6. Displacement state variables of the system model 0.6
REFERENCES
0.4
Velocity [m s−1 ]
0.2
0
−0.2
−0.4
−0.6
0
10
20
30
40
50 60 Time [s]
70
80
90
100
Fig. 7. Velocity state variables of the system model From the final positions of masses (Fig. 5) and the evolution of velocity variables (Fig. 7) it is obvious, that system converges towards some equilibrium. This equilibrium, however, is an arbitrary one with offsets added to displacement state variables, which is visible in Fig. 6. The displacement state variables are in Fig. 6 divided into two groups, each of them converging into different value. This two groups are accounting for horizontal and vertical displacement state variables and are converging to some arbitrary offset in which the system reaches its newly obtained equilibrium. This vertical and horizontal offsets depend on the weights of the masses, spring constants, and the initial conditions of the system. In conclusion, this behavior is very much similar to the expected behavior of a real-world system, which consists of an array of masses connected by combination of springs and dampers floating in two-dimensional void space without any external force acting upon it. 5. CONCLUSION This paper contains two parts, first part reviews the theory needed to model distributed large scale systems. In the second part, the prototype model of a large scale system is designed, and the simulation results are presented. 322
˚ Astr¨om, K.J. and Wittenmark, B. (1997). Computercontrolled systems. Prentice Hall, Upper Saddle River, New Jersey 07458, 3rd edition. Bellemans, T., Schutter, B.D., and Moor, B.D. (2006). Model predictive control for ramp metering of motorway traffic: A case study. Control Engineering Practice, 14(7), 757 – 767. doi: http://dx.doi.org/10.1016/j.conengprac.2005.03.010. URL http://www.sciencedirect.com/science/ article/pii/S0967066105000900. Bolognani, S. and Zampieri, S. (2013). A distributed control strategy for reactive power compensation in smart microgrids. IEEE Transactions on Automatic Control, 58(11), 2818–2833. doi:10.1109/TAC.2013.2270317. Colaneri, P., Farina, M., Scattolini, R., and Shorten, R. (2012). A note on discretization of sparce linear systems. In J.D. Bendtsen (ed.), Robust Control Design, volume 7, 97–102. Aalborg Congress & Culture Centre, Denmark. doi:10.3182/20120620-3-DK-2025. 00123. URL http://www.ifac-papersonline.net/ Detailed/53261.html. Etemadi, A.H., Davison, E.J., and Iravani, R. (2012). A decentralized robust control strategy for multi-der microgrids; part i: Fundamental concepts. IEEE Transactions on Power Delivery, 27(4), 1843–1853. doi:10.1109/ TPWRD.2012.2202920. Farina, M., Colaneri, P., and Scattolini, R. (2013). Block-wise discretization accounting for structural constraints. Automatica, 49(11), 3411 – 3417. doi: http://dx.doi.org/10.1016/j.automatica.2013.08.013. URL http://www.sciencedirect.com/science/ article/pii/S0005109813004147. Farina, M. and Scattolini, R. (2012). Distributed predictive control: A non-cooperative algorithm with neighbor-to-neighbor communication for linear systems. Automatica, 48(6), 1088 – 1096. doi: http://dx.doi.org/10.1016/j.automatica.2012.03.020. URL http://www.sciencedirect.com/science/ article/pii/S000510981200129X. Lewis, F.L. (1992). Applied optimal control. Prentice Hall, Englewood Cliffs, N.J.
2016 IFAC PDES October 5-7, 2016. Brno, Czech Republic
Stefan Misik et al. / IFAC-PapersOnLine 49-25 (2016) 318–323
Ma, Y., Borrelli, F., Hencey, B., Coffey, B., Bengea, S., and Haves, P. (2012). Model predictive control for the operation of building cooling systems. IEEE Transactions on Control Systems Technology, 20(3), 796–803. doi:10.1109/TCST.2011.2124461. Oldewurtel, F., Parisio, A., Jones, C.N., Gyalistras, D., Gwerder, M., Stauch, V., Lehmann, B., and Morari, M. (2012). Use of model predictive control and weather forecasts for energy efficient building climate control. Energy and Buildings, 45, 15 – 27. doi:http://dx.doi.org/10.1016/j.enbuild.2011.09. 022. URL http://www.sciencedirect.com/science/ article/pii/S0378778811004105. Papageorgiou, M. and Kotsialos, A. (2002). Freeway ramp metering: an overview. IEEE Transactions on Intelligent Transportation Systems, 3(4), 271–281. doi: 10.1109/TITS.2002.806803. Riverso, S. (2013). Distributed and Plug-and-Play control for constrained systems. Ph.D. thesis, Universit`a degli Studi di Pavia.
323
323
Riverso, S. and Ferrari-Trecate, G. (2012). Hycon2 benchmark: Power network system. Technical report, Universit`a degli Studi di Pavia. URL http://arxiv.org/abs/ 1207.2000. Saadat, H. (1998). Power System Analysis. McgrawHill College. URL http://www.amazon.com/PowerSystem-Analysis-Hadi-Saadat/dp/0075616343% 3FSubscriptionId%3D0JYN1NVW651KCA56C102% 26tag%3Dtechkie-20%26linkCode%3Dxm2%26camp% 3D2025%26creative%3D165953%26creativeASIN% 3D0075616343. Scattolini, R. (2009). Architectures for distributed and hierarchical model predictive control – a review. Journal of Process Control, 19(5), 723 – 731. doi:http://dx.doi.org/10.1016/j.jprocont.2009.02. 003. URL http://www.sciencedirect.com/science/ article/pii/S0959152409000353. Shamma, J.S. (2007). Cooperative control of distributed multi-agent systems. John Wiley, Hoboken, NJ.