Decentralized and distributed control

Transcription

1 Decentralized and distributed control Models of large-scale systems M. Farina 1 G. Ferrari Trecate 2 1 Dipartimento di Elettronica e Informazione (DEI) Politecnico di Milano, Italy farina@elet.polimi.it 2 Dipartimento di Informatica e Sistemistica (DIS) Università degli Studi di Pavia, Italy giancarlo.ferrari@unipv.it EECI-HYCON2 Graduate School on Control 2012 Supélec, France Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

2 Outline 1 Information on the lecture 2 Introduction 3 Unstructured and interaction-oriented models 4 Decomposition of input-output pairs RGA matrix Grammians 5 Non-overlapping decompositions Preliminaries Decomposition in weakly-interacting systems Decomposition in cascaded systems 6 Overlapping decompositions Motivating example Inclusion principle Completely overlapping decompositions 7 Conclusions Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

3 Schedule of the course Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

4 Suggested readings Books 1. J. Lunze, Feedback Control of Large Scale Systems, Prentice Hall, D. D. Siljak, Decentralized Control of Complex Systems, New York, Academic Press, S. Skogestad and I. Postlethwaite, Multivariable Feedback Control: Analysis and Design, Wiley, J. B. Rawlings and D. Q. Mayne. Model Predictive Control: Theory and Design. Nob Hill Publishing, Madison, WI, Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

5 Suggested readings Papers and reports 5. R. Scattolini, Report on literature survey and preliminary definition of the selected methods for the definition of system decomposition and hierarchical control architectures. HD-MPC deliverable D2.1, http: // 6. M.E. Salgado and A. Conley. MIMO interaction measure and controller structure selection. International Journal of Control, 77(4):367383, Šiljac, D. D. and Vukčević, M. B. Decentralizetion, stabilization, and estimation in large-scale systems. IEEE Trans. on Automatic Control, 21: , Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

6 Suggested readings Papers and reports 8. Pichai, V. and Sezer, M. E. and Šiljac, D. D. A graph-theoretic algorithm for hierarchical decomposition of dynamic systems with application to estimation and control. IEEE Trans. on Systems, Man, and Cybernetics, 13(3): , Ikeda, M. and Šiljac, D. D. and White, D. E. An inclusion principle for dynamic systems. IEEE Trans. on Automatic Control, 29(3): , K. E. Häggblom. Partial relative gain: a new tool for control structure selection. In AIChE Annual Meeting, Los Angeles, CA, USA, Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

7 Outline 1 Information on the lecture 2 Introduction 3 Unstructured and interaction-oriented models 4 Decomposition of input-output pairs RGA matrix Grammians 5 Non-overlapping decompositions Preliminaries Decomposition in weakly-interacting systems Decomposition in cascaded systems 6 Overlapping decompositions Motivating example Inclusion principle Completely overlapping decompositions 7 Conclusions Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

8 Introduction Large-scale systems ẋ o = A o x o + B o u y = C o x o Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

9 Introduction Centralized control: approach all the measured data y are conveyed to the central control station; a high order centralized state estimator is implemented; the control action u is computed; the control signals are communicated to all the actuators. Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

10 Introduction Centralized control: problems high communication burden: transducers control station; control station actuators; high computational load: observer implementation; control action computation; high memory required: to store model information (Ao,B o,c o ); to store variables (xo,u). Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

11 Introduction Centralized control: consequences high communication burden: transmission delays; high transmission failure rate (e.g., packet losses); high computational load: computational delays; limitations on the size of the problems; information storage: low robustness with respect to possible uncertainties on the model; change in a part requires the re-design of the overall control system (costly). Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

12 Introduction Top down approach Decompose the synthesis problem into a number of independent (or almost independent, if possible) small-scale problems. Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

13 Introduction Top-down approach For each sub-system the measured data y [i] are conveyed to a local control station; a low order local state estimator is implemented; the local control action u [i] is computed; the control signal is directly transmitted to the actuator. Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

14 Introduction Top-down approach Solutions: reduced communication burden (cable): local transducer local control station; local control station local actuator; reduced computational load: observer implementation; control action computation; reduced memory required: to store model information (local subsystem); to store local variables. Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

15 Introduction Top-down approach Consequences: reduced communication burden: no transmission delays; no transmission failures; reduced computational load: fast computation; scalability (regardless of the size of the large-scale problem, the size of the reduced problems is limited); information storage: robustness with respect to possible uncertainties on the model (e.g., on the mutual interactions); change in a part does not require the re-design of the overall control system. Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

16 Introduction Top-down approach The key point is to decompose the model in small scale interacting sub-models, i.e., to partition the large scale system model into a number of interconnected small-scale models. Model decompositions can be carried out based on physical/structural decomposability features of the plant into sub-plants, the mathematical-algebraic properties of the available mathematical model. Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

17 Outline 1 Information on the lecture 2 Introduction 3 Unstructured and interaction-oriented models 4 Decomposition of input-output pairs RGA matrix Grammians 5 Non-overlapping decompositions Preliminaries Decomposition in weakly-interacting systems Decomposition in cascaded systems 6 Overlapping decompositions Motivating example Inclusion principle Completely overlapping decompositions 7 Conclusions Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

18 Unstructured and interaction-oriented models Unstructured model The evolution of the large-scale system can be described by the equations ẋ o = A o x o + B o u y = C o x o where x o R n, u R m, and y R p. Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

19 Unstructured and interaction-oriented models Interaction-oriented model Assume that the large scale system results as an interaction of a number M of subsystems S i, i = 1,...,M, where the coupling terms may consist in energy, material, and information flow. Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

20 Unstructured and interaction-oriented models Assumption Each subsystem S i is endowed with its own input and output variables u i and y i, respectively. Therefore (up to a state permutation) u = u 1. u M, y = with u i R m i, y i R p i with M i=1 m i = m and M i=1 p i = p. y 1. y M Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

21 Unstructured and interaction-oriented models Interactions Additional input variables s i and output variables z i are used to describe the interconnection terms with the neighboring subsystems. Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

22 Unstructured and interaction-oriented models Subsystem S i model ẋ i = A ii x i + B ii u i + E i s i y i = C ii x i + F i s i z i = C zi x i + D zi u i Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

23 Unstructured and interaction-oriented models Subsystem S i model ẋ i = A ii x i + B ii u i + E i s i y i = C ii x i + F i s i z i = C zi x i + D zi u i Interaction model s i = M j=1 L ij z j L ij are interconnection matrices. We also define the interconnection gain l ij = L ij such that s i M j=1 l ij z j Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

24 Unstructured and interaction-oriented models Non-overlapping decomposition Unstructured model ẋ o y = A o x o + B o u = C o x o Let x i R n i be a non-overlapping partition of x o (possibly under a suitable reordering of the state variables), i.e. x 1 x o =. with M i=1 n i = n, and where x i defines the state of subsystem S i. x M Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

25 Unstructured and interaction-oriented models Non-overlapping decomposition The unstructured model results in: ẋ 1 A A 1M x 1 B B 1M u 1. = ẋ M A M1... A MM x M B M1... B MM u M y 1. y M = C C 1M x C M1... C MM x M The equation for subsystem S i results to be: ẋ i = A ii x i + B ii u i + i j (A ij x j + B ij u j ) y i = C ii x i + i j C ij x j Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

26 Unstructured and interaction-oriented models Non-overlapping decomposition Interaction-oriented model ẋ i = A ii x i + B ii u i + E i s i y i = C ii x i + F i s i z i = C zi x i + D zi u i Non-overlapping decomposition ẋ i = A ii x i + B ii u i + i j (A ij x j + B ij u j ) y i = C ii x i + i j C ij x j Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

27 Unstructured and interaction-oriented models Non-overlapping decomposition Interaction-oriented model ẋ i = A ii x i + B ii u i + E i s i y i = C ii x i + F i s i z i = C zi x i + D zi u i Non-overlapping decomposition ẋ i = A ii x i + B ii u i + i j (A ij x j + B ij u j ) y i = C ii x i + i j C ij x j [ ] [ ] [ ] Ini 0ni m If C zi =, D 0 zi = i xi then z mi n i I i =. mi u i [ ] Aij B Setting L ij = ij, then s C ij 0 i = pi m j i j C ij x j [ i j (A ij x j + B ij u j ) If E i = [ I ni 0 ni p i ], Fi = [ 0 pi n i I pi ], the equivalence is proved. ]. Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

28 Unstructured and interaction-oriented models Influence graphs Definition If l ij = L ij = 0 we say that subsystem S j is a neighbor of subsystem S i, i.e., there is a direct influence of S j on subsystem S i. Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

29 Unstructured and interaction-oriented models Influence graphs Consider a generic interaction-oriented system. We define the corresponding direct graph G = (V,E ) as follows: for each subsystem S i we define a node i (set: V ); an edge (set: E ) of the graph between nodes j and i exists iff l ij 0 (i.e., j is i s neighbor). Example Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

30 Unstructured and interaction-oriented models Influence graphs Definitions S j and S i are weakly coupled if l ij is small (in a relative sense); S j interacts with S i if there exists a direct path from j to i (it can be a direct interaction or not); S j and S i are strongly coupled if both S j interacts with S i and S i interacts with S j. weak and strong coupling are not opposite concepts! given a interaction-oriented representation there exists a unique interaction graph; for a generic unstructured system there exist several different interaction-oriented representations, and so several different interaction graphs. Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

31 Unstructured and interaction-oriented models Issues Some questions: Given an unstructured model, how to decompose the input-output pair (u,y) in a partition of M input-output pairs (u i,y i )? Chose overlapping or non-overlapping state partitions? In both cases, how to perform a suitable state decomposition? Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

32 Outline 1 Information on the lecture 2 Introduction 3 Unstructured and interaction-oriented models 4 Decomposition of input-output pairs RGA matrix Grammians 5 Non-overlapping decompositions Preliminaries Decomposition in weakly-interacting systems Decomposition in cascaded systems 6 Overlapping decompositions Motivating example Inclusion principle Completely overlapping decompositions 7 Conclusions Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

33 Decomposition of input-output pairs From an unstructured model ẋ o y = A o x o + B o u = C o x o we aim to find a suitable (possibly the most suitable) decomposition of the input-output pair (u,y) into M input-output sub-pairs (u i,y i ) such that the coupling between u i and y i is maximized, for all i = 1,...,M, the coupling between u j and y i is minimized, i,j = 1,...,M, j i. Two methods are discussed: methods based on the relative gain array (RGA) matrix, methods based on Grammians. Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

34 Outline 1 Information on the lecture 2 Introduction 3 Unstructured and interaction-oriented models 4 Decomposition of input-output pairs RGA matrix Grammians 5 Non-overlapping decompositions Preliminaries Decomposition in weakly-interacting systems Decomposition in cascaded systems 6 Overlapping decompositions Motivating example Inclusion principle Completely overlapping decompositions 7 Conclusions Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

35 Decomposition of input-output pairs Relative Gain Array Main assumptions A o is asymptotically stable, the unstructured system has no invariant zeros at the origin i.e., ([ ]) Ao B rank o = n C o 0 m = p, i.e., the number of input variables is equal to the number of output variables. The system transfer function is with steady-state gain G(s) = C o (si A o ) 1 B o G 0 = G(0) whose individual elements are denoted by g ij, i,j = 1,2,...,m. Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

36 Decomposition of input-output pairs Relative Gain Array Definition The system Relative Gain Array (RGA) matrix is defined as Λ = G 0 (G 1 0 )T where denotes element-wise product. The entries of Λ are equal to: λ ij = ( y i/ u j ) ul j constant ( y i / u j ) yk i constant Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

37 Decomposition of input-output pairs Relative Gain Array For example, consider the case m = p = 2: and consider, for example [ ] G11 (s) G G(s) = 12 (s) G 21 (s) G 22 (s) λ 11 = ( y 1/ u 1 ) u2 constant ( y 1 / u 1 ) y2 constant Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

38 Decomposition of input-output pairs Relative Gain Array λ 11 = ( y 1/ u 1 ) u2 constant ( y 1 / u 1 ) y2 constant Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

39 Decomposition of input-output pairs Relative Gain Array λ 11 = ( y 1/ u 1 ) u2 constant ( y 1 / u 1 ) y2 constant ( y 1 / u 1 ) u2 constant = G 11 (0) is the open-loop gain. It is the gain from input 1 to output 1 when input 2 is not controlled (i.e, it is kept constant). Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

40 Decomposition of input-output pairs Relative Gain Array λ 11 = ( y 1/ u 1 ) u2 constant ( y 1 / u 1 ) y2 constant ( y 1 / u 1 ) u2 constant = G 11 (0) is the open-loop gain. It is the gain from input 1 to output 1 when input 2 is not controlled (i.e, it is kept constant). ( y 1 / u 1 ) u2 constant = G 11 (0) is the gain from input 1 to output 1 when the other loop is closed, and y 2 is controlled (with integral action) using u 2. Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

41 Decomposition of input-output pairs Relative Gain Array In general λ ij = ( y i/ u j ) ul j constant ( y i / u j ) yk i constant represent the ratio between the process gain for the pairing y i u j in an isolated loop and the process gain when the rest of the system is under integral feedback control. Selection of the suitable pairings In the selection of input and output pairings for the design of SISO decentralized controllers select those pairs that maintain roughly the same gain in open-loop and closed-loop configurations, i.e. λ ij 1 the pairings for which a change of sign of these gains occurs, i.e. λ ij < 0, must be avoided. Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

42 Decomposition of input-output pairs Relative Gain Array Properties of RGA 1. its elements are independent of the adopted units; 2. the sum of the elements of any row is equal to 1; 3. the sum of the elements of any column is equal to 1; 4. it is equal to the identity if G(s) is a diagonal or a triangular matrix. the result of RGA is independent of the state-space model; if a pair has λ ij 1, then it is generally the best pairing for both the input u j and the output y i ; for simple structures (cascade structures on lack of interactions) the result of RGA is the trivial one. Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

43 Decomposition of input-output pairs Relative Gain Array So far, RGA was used to decompose (u,y) in m SISO pairs (u j,y i ). This can be extended for decomposition in M < m MIMO subsystems. Consider a generic system, 1. compute the steady state gain G 0 and the RGA Λ; 2. decompose G 0 and Λ, e.g., where G 22 is non-singular; 3. the matrix Λ 11 results to be equal to [ ] [ ] G11 G G 0 = 12 Λ11 Λ,Λ = 12 G 21 G 22 Λ 21 Λ 22 Λ 11 = G 11 (G 1 11 ) T where G 11 = G 11 G 12 G 1 22 G 21 is the Schur complement of G 22. It is also the gain (matrix) of the system from the vector u 1 and the vector y 1 when the rest of the system is closed under feedback integral control. Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

44 Decomposition of input-output pairs Relative Gain Array In general given a generic system, whose RGA is Λ, let G sub be a (square) sub-matrix of G 0 ; let Λ sub be a (square) sub-matrix of Λ; Λ sub results Λ sub = G sub (G 1 m ) T where G m is the gain of the subsystem when the rest is closed under feedback integral control. In this way G m is defined; define the block relative gain (BRG) Λ B sub = G sub(g 1 m ) T which is used as a generalization of RGA for blocks. Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

45 Decomposition of input-output pairs Relative Gain Array From the analysis of the RGA it is often not possible to find a unique, dominating solution. In these cases, it is convenient to use the Niederlinski index (NI). Niederlinski Index The NI N G0 (G 0 ) is defined as N G0 (G 0 ) = det(g 0) det( G 0 ) where G 0 is obtained by setting to zero all the elements of G 0 that do not correspond to an input-output pairing in the (block-) decentralized control structure. Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

46 Decomposition of input-output pairs Relative Gain Array Criterion for use of NI Among the possible sets of pairings selected by looking at the RGA, choose the pairings with a positive NI. In fact, G(s) has the potentiality to be decentralized integral controllable only if N G0 (G 0 ) > 0 Decentralized integral controllability (DIC) A system is DIC if there exists a decentralized controller such that the closed loop system is stable and such that each individual controller may be detuned independently by a factor ε i, ε i [0,1], without introducing instability. DIC is a desirable property since it allows individual controllers to be arbitrarily detuned. Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

47 Decomposition of input-output pairs Relative Gain Array RGA does not properly capture the effect of closing one or more feedback loops. Then, a recursive procedure and the partial relative gain criterion is suggested: from the analysis of the RGA choose an input-output pairing (i.e. u j y i ), reorder the matrix G 0 so as to obtain [ G11 G G 0 = 12 G 21 G 22 ] [ Λ11 Λ,Λ = 12 Λ 21 Λ 22 where G 22 corresponds to the selected input-output pair, recompute the RGA for the other subsystem (G 11 ), i.e., the partial relative gain (PRG), given that G 22 is in an integral feedback loop, i.e. Λ P = G 11 (G 1 11 ) T from the analysis of Λ P choose a new pairing, repeat the procedure until all the u y pairs have been chosen. ] Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

48 Decomposition of input-output pairs Relative Gain Array Note that: the PRG is different from the term Λ 11. In fact Λ 11 = G 11 (G 1 11 ) T the PRG describes exactly the closed-loop behaviour (it is denoted closed-loop RGA method) of the system. Other selection criteria are possible e.g., decomposed relative interaction array; decomposed relative gain array; Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

49 Outline 1 Information on the lecture 2 Introduction 3 Unstructured and interaction-oriented models 4 Decomposition of input-output pairs RGA matrix Grammians 5 Non-overlapping decompositions Preliminaries Decomposition in weakly-interacting systems Decomposition in cascaded systems 6 Overlapping decompositions Motivating example Inclusion principle Completely overlapping decompositions 7 Conclusions Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

50 Decomposition of input-output pairs Grammians Consider a stable system Controllability Grammian ẋ o y = A o x o + B o u = C o x o The controllability Grammian P is a symmetric non negative definite matrix which satisfies the Lyapunov equation A o P + PA T o + B o B T o = 0 Observability Grammian The observability Grammian Q is a symmetric non negative definite matrix which satisfies the Lyapunov equation A T o Q + QA o + C T o C o = 0 Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

51 Decomposition of input-output pairs Grammians The matrices P and Q can be expressed as P = 0 e A ot B o B T o e AT o t dt Q = 0 e AT o t C T o C o e A ot dt Let B j,o be the j th column of matrix B o and C i,o the i th row of C o. Define by P j and Q i the controllability and observability Grammians for the elementary system (A o,b j,o,c i,o ). Properties The original system controllability and observability Grammians P and Q can be written as P = Σ m j=1 P j Q = Σ m i=1 Q i and PQ = Σ m j=1 Σm i=1 P jq i Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

52 Decomposition of input-output pairs Grammians Partecipation matrix Define the partecipation matrix Φ as Φ = {φ ij φ ij = trace(p jq i ) trace(pq) } 0 < φ ij < 1 Properties: φ ij s are state realization independent, 0 φ ij 1, for all i,j, Σ m i=1 Σm j=1 φ ij = 1. Selection criterion Select the pairings (u j,y i ) with greater values of φ ij. Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

53 Decomposition of input-output pairs Grammians Alternatively, the following algorithm can be used: define the Henkel norm of the subsystems: G ij (s) H = λ MAX (P j Q i ) = σij H where λ MAX denotes the maximum eigenvalue of a matrix, define the Henkel interaction array Σ H as the matrix collecting the terms σ H ij, normalize the matrix Σ H. There are different ways e.g., similarly to RGA. Σ H ( Σ T H ) 1 chose the pairings corresponding to the greater values of Σ H (or a normalized version of Σ H ). Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

54 Outline 1 Information on the lecture 2 Introduction 3 Unstructured and interaction-oriented models 4 Decomposition of input-output pairs RGA matrix Grammians 5 Non-overlapping decompositions Preliminaries Decomposition in weakly-interacting systems Decomposition in cascaded systems 6 Overlapping decompositions Motivating example Inclusion principle Completely overlapping decompositions 7 Conclusions Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

55 Outline 1 Information on the lecture 2 Introduction 3 Unstructured and interaction-oriented models 4 Decomposition of input-output pairs RGA matrix Grammians 5 Non-overlapping decompositions Preliminaries Decomposition in weakly-interacting systems Decomposition in cascaded systems 6 Overlapping decompositions Motivating example Inclusion principle Completely overlapping decompositions 7 Conclusions Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

56 Non-overlapping decompositions Preliminaries Digraph Consider the unstructured system: ẋ o y = A o x o + B o u = C o x o For all the input variables, state variables, and output variables we introduce a node of a digraph (directed-graph). Example: Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

57 Non-overlapping decompositions Preliminaries Interconnection matrix Consider the unstructured system: ẋ o y = A o x o + B o u = C o x o we define the system matrix as follows: A o B o 0 n p S = 0 m n 0 m m 0 m p C o 0 p m 0 p p The structural system matrix [S] is defined by setting all the non-zero entries of S to 1, and is denoted interconnection matrix. Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

58 Non-overlapping decompositions Preliminaries Example: the system A o = 0 2 3,B o = C o = [ 1 0 ] corresponds to the following interconnection matrix: S = [S] = Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

59 Non-overlapping decompositions Preliminaries The interconnection matrix defines the edges (direct dependencies) of the digraph associated to the system: Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

60 Non-overlapping decompositions Preliminaries The interconnection matrix defines the edges (direct dependencies) of the digraph associated to the system: Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

61 Non-overlapping decompositions Preliminaries The interconnection matrix defines the edges (direct dependencies) of the digraph associated to the system: Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

62 Non-overlapping decompositions Preliminaries The interconnection matrix defines the edges (direct dependencies) of the digraph associated to the system: Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

63 Non-overlapping decompositions Preliminaries The interconnection matrix defines the edges (direct dependencies) of the digraph associated to the system: Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

64 Non-overlapping decompositions Preliminaries The interconnection matrix defines the edges (direct dependencies) of the digraph associated to the system: Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

65 Non-overlapping decompositions Preliminaries Therefore, there is equivalence between the two representations: digraph and interconnection matrix They represent only the direct interconnections from inputs to states (from B o ); from states to states (from A o ); from states to outputs (from C o ). On the other hand, to represent influences (both direct and indirect interconnections) between nodes, we derive the reachability matrix. Step 1: s-steps reachability matrix R s = [S] [S] 2 [S] s where defines the element-wise boolean OR operator (1 1 = 1, 1 0 = 0 1 = 1, and 0 0 = 0). Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

66 Non-overlapping decompositions Preliminaries The ij-th entry of the s-steps reachability matrix is 1 iff there is a path of at most s steps from the j-th node and the i-th node of the digraph. E.g., the 2-steps reachability matrix of the example is: [S] = R 2 = Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

67 Non-overlapping decompositions Preliminaries Reachability matrix There exists s such that R s = R s for all s s. R s corresponds with the reachability matrix R. The ij-th entry of the s-steps reachability matrix is 1 iff there is a path of any length from the j-th node and the i-th node of the digraph. For example: [S] = R = Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

68 Non-overlapping decompositions Preliminaries Example: R = Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

69 Non-overlapping decompositions Preliminaries Decomposition of the reachability matrix The reachability matrix R is decomposed in matrices F, G, H, θ, such that F G 0 n p R = 0 m n 0 m m 0 m p H θ 0 p p Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

70 Non-overlapping decompositions Preliminaries Decomposition of the reachability matrix The reachability matrix R is decomposed in matrices F, G, H, θ, such that F G 0 n p R = 0 m n 0 m m 0 m p H θ 0 p p F represents how state variables influence state variables, G represents how input variables influence state variables, H represents how state variables influence output variables, θ represents how input variables influence output variables. Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

71 Non-overlapping decompositions Preliminaries In the example: F = 0 1 1, G = [ ] [ ] H =, θ = Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

72 Non-overlapping decomposition Preliminaries Input reachability A state variable x i is input-reachable if there exists a path from at least an input variable to x i, the system is input-reachable if each state variable is input-reachable. Output reachability A state variable x i is output-reachable if there exists a path from x i to at least an output variable, the system is output-reachable if each state variable is output-reachable. Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

73 Non-overlapping decomposition Preliminaries Main result A system is input-reachable if and only if the binary matrix G has no zero rows, a system is output-reachable if and only if the binary matrix H has no zero columns. Remarks STRUCTURAL PROPERTY NUMERICAL PROPERTY Input-reachability Controllability Output-reachability Observabiility Structural properties require graph-theoretical tools, and are computationally easier to verify. Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

74 Non-overlapping decomposition Preliminaries Example: 1 1 G = 1 1 the system is input-reachable 1 0 H = [ 1 0 ] the system is output-reachable Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

75 Outline 1 Information on the lecture 2 Introduction 3 Unstructured and interaction-oriented models 4 Decomposition of input-output pairs RGA matrix Grammians 5 Non-overlapping decompositions Preliminaries Decomposition in weakly-interacting systems Decomposition in cascaded systems 6 Overlapping decompositions Motivating example Inclusion principle Completely overlapping decompositions 7 Conclusions Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

76 Non-overlapping decomposition Decomposition in weakly-interacting systems From an unstructured model ẋ o y = A o x o + B o u = C o x o we aim to find the most appropriate non-overlapping state/input/output partition x 1 x M u 1 u M y 1. = x (perm) o,. = u (perm),. y M = y (perm) where x o (perm), u (perm), and y (perm) are the permutated state vector x o, input vector u, and output vector y, respectively. That is x (perm) o = P x,perm x o, u (perm) = P u,perm u, and y (perm) = P y,perm y where P x,perm, P u,perm, and P y,perm are permutation matrices. Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

77 Non-overlapping decomposition Decomposition in weakly-interacting systems We aim to obtain the interaction-oriented model ẋ i = A ii x i + B ii u i + E i s i y i = C ii x i + F i s i z i = C zi x i + D zi u i where (see previous slides): [ ] [ ] Ini 0ni m C zi =, D 0 zi = i mi n i I mi and E i = [ I ni 0 ni p i ], Fi = [ 0 pi n i I pi ] Note that the entries of these matrices are equal either to 0 or to 1. The interaction terms must be sufficiently weak (the level of interaction is pre-assigned and equal to ε). The interactions among subsystems are indeed given by s i = L ij z j Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

78 Non-overlapping decomposition Decomposition in weakly-interacting systems L ij, i,j = 1,...,M, are the interconnection matrices [ ] Aij B L ij = ij C ij 0 pi m j Weak interactions The aim is to obtain a state partition such that L ij is sufficiently small. Specifically, all entries of L ij must be ε in absolute value. This implies that the entries of A ij, B ij, C ij must be in absolute values. ε Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

79 Non-overlapping decomposition Decomposition in weakly-interacting systems We define, as detailed before, a digraph G o. Recall that the strength of the edge between two generic states x j and x i is given by the absolute value of the (i,j)-th entry of matrix A o, the input u j and the state x i is given by the absolute value of the (i,j)-th entry of matrix B o, the state x j and the output y i is given by the absolute value of the (i,j)-th entry of matrix C o. Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

80 Non-overlapping decomposition Decomposition in weakly-interacting systems The algorithm: main steps I) define a graph G o ε = (V,E ε ), where V is the same set of nodes as the one of G o, and E ε is obtained by eliminating from G o all the edges whose strength is smaller than ε; II) cluster together all the nodes which are connected together by Gε o. The number of resulting sub-networks G i,ε = (V i,ε,e i,ε ) is actually M, i.e., the number of obtained partitions. Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

81 Non-overlapping decomposition Decomposition in weakly-interacting systems Example: maximum level of interaction ε = 0.5. Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

82 Non-overlapping decomposition Decomposition in weakly-interacting systems Example: maximum level of interaction ε = B o = Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

83 Non-overlapping decomposition Decomposition in weakly-interacting systems Example: maximum level of interaction ε = A o = Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

84 Non-overlapping decomposition Decomposition in weakly-interacting systems Example: maximum level of interaction ε = 0.5. C o = [ 1 0 ] Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

85 Non-overlapping decomposition Decomposition in weakly-interacting systems Example: maximum level of interaction ε = 0.5. The system is partitioned in M = 2 subsystems. Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

86 Non-overlapping decomposition Decomposition in weakly-interacting systems Conditions to be verified (a posteriori) all the subsystems must contain input, output, and state vertices: otherwise the obtained partition results to be useless, at least as far as control applications are concerned, for each sub-graph, it is advisable -but not necessary- that the input-output reachability property be verified. In the case the obtained partition does not satisfy the requirements, the subgraphs should be combined together and/or the threshold parameter ε must be chosen more appropriately. Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

87 Non-overlapping decomposition Decomposition in weakly-interacting systems ε-nested decomposition The presented method for system decomposition in weakly-interacting systems is denoted ε-nested decomposition. the threshold ε is a design parameter; the partition obtained in this way can be further decomposed in subsystems with a bigger threshold ε > ε: the decomposition obtained with ε is nested within the one obtained with ε. Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

88 Non-overlapping decomposition Decomposition in weakly-interacting systems Example: consider the problem of controlling the temperature of the building Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

89 Non-overlapping decomposition Decomposition in weakly-interacting systems Model in the example cρv ṪA = s r u 2 (T B T A ) + s r u 1 (T C T A ) + s e U(T E T A ) + q A cρv ṪB = s r u 2 (T A T B ) + s r u 1 (T D T B ) + s e U(T E T B ) + q B cρv ṪC = s r u 1 (T A T C ) + s r u 2 (T D T C ) + s e U(T E T C ) + q C cρv ṪD = s r u 1 (T B T D ) + s r u 2 (T C T D ) + s e U(T E T D ) + q D c: specific heat of the air; V : volume of the rooms; T i, i = A,B,C,D: temperatures of the rooms; s r : surface of the walls between two rooms; s e : surface of the walls between a room and the environment; u 1,u 2,U: transmittances; T E : external temperature; q i, i = A,B,C,D: inputs (heating power). Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

90 Non-overlapping decomposition Decomposition in weakly-interacting systems Equilibrium condition: T E = 0 C, T A = T B = T C = T D = T = 20 C, q A = q B = q C = q D = q = s e U T. We define: δt A = T A T, δt B = T B T, δt C = T C T, δt D = T D T, δq A = (q A q)/cρv, δq B = (q B q)/cρv, δq C = (q C q))/cρv and δq D = (q D q)/cρv. Around the given equilibrium condition: δt A γ ol γ 2 γ 1 0 δt A δq A δt B δt C = γ 2 γ ol 0 γ 1 δt B γ 1 0 γ ol γ 2 δt C + δq B δq C δt D 0 γ 1 γ 2 γ ol δt D δq D where γ 1 = s r u 1 cρv and γ 2 = s r u 2 cρv A, γ e = s eu cρv, γ ol = γ 1 + γ 2 + γ e. Importantly, γ ol > γ 1 > γ 2. Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

91 Non-overlapping decomposition Decomposition in weakly-interacting systems Model: δt A γ ol γ 2 γ 1 0 δt A δq A δt B δt C = γ 2 γ ol 0 γ 1 δt B γ 1 0 γ ol γ 2 δt C + δq B δq C δt D 0 γ 1 γ 2 γ ol δt D δq D Digraph: Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

92 Non-overlapping decomposition Decomposition in weakly-interacting systems Model: δt A γ ol γ 2 γ 1 0 δt A δq A δt B δt C = γ 2 γ ol 0 γ 1 δt B γ 1 0 γ ol γ 2 δt C + δq B δq C δt D 0 γ 1 γ 2 γ ol δt D δq D Threshold ε = γ 2 Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

93 Non-overlapping decomposition Decomposition in weakly-interacting systems Model: δt A γ ol γ 2 γ 1 0 δt A δq A δt B δt C = γ 2 γ ol 0 γ 1 δt B γ 1 0 γ ol γ 2 δt C + δq B δq C δt D 0 γ 1 γ 2 γ ol δt D δq D Threshold ε = γ 1 Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

94 Non-overlapping decomposition Decomposition in weakly-interacting systems For ε = γ 2, the obtained partitioned model is [ ] [ ][ ] δta (γ1 + γ = 2 + γ e ) γ 1 δta + δt C γ 1 (γ 1 + γ 2 + γ e ) δt C [ ] δtb δt D = [ (γ1 + γ 2 + γ e ) γ 1 γ 1 (γ 1 + γ 2 + γ e ) ][ δtb δt D ] + [ δqa δq C [ δqb δq D ] [ ] δtb + γ 2 δt D ] [ ] δta + γ 2 δt C Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

95 Outline 1 Information on the lecture 2 Introduction 3 Unstructured and interaction-oriented models 4 Decomposition of input-output pairs RGA matrix Grammians 5 Non-overlapping decompositions Preliminaries Decomposition in weakly-interacting systems Decomposition in cascaded systems 6 Overlapping decompositions Motivating example Inclusion principle Completely overlapping decompositions 7 Conclusions Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

96 Non-overlapping decomposition Decomposition in cascaded systems Motivation The scope of this method is to decompose the unstructured system in M hierarchically structured interconnected subsystems: { ẋi = A ii x i + B ii u i + i 1 j=1 (A ijx j + B ij u j ) y i = C ii x i + i 1 j=1 C ijx j where the subsystems are connected in a cascaded scheme where system i depends on system j only if j < i. Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

97 Non-overlapping decomposition Decomposition in cascaded systems This decomposition exists provided that there exists a permutation such that the submatrices of the resulting reachability matrix F G 0 n p R = 0 m n 0 m m 0 m p H θ 0 p p are lower-block-triangular (LBT). For this reason, this decomposition is denoted LBT decomposition. For example, if M = 2: [ ] [ ] F11 0 n1 n 2 G11 0 n1 m 2 0 F 21 F 22 G 21 G n p 22 R = [H11 0 m n 0 m m 0 m p ] [ ] 0 p1 n 2 θ11 0 p1 m 2 0 H 21 H 22 θ 21 θ p p 22 Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

98 Non-overlapping decomposition Decomposition in cascaded systems Aim The aim is to obtain an input/output partition of the digraph G o such that (i) the reachability matrix R has an input-output LBT form; (ii) each subgraph G i is input-output reachable. This decomposition is important since a decentralized control scheme designed by controlling the tuples (A ii,b ii,c ii ) by closing M independent loops (this is guaranteed by (ii)) conserves the triangular structure of the system and makes the overall scheme asymptotically stable! Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

99 Non-overlapping decomposition Decomposition in cascaded systems Remarks: a necessary condition for input/output partition is that the original graph G o be input-output reachable, to obtain the maximum benefit from the present decomposition it is important that a method for input-output decompositions guarantee that the subgraphs result to be irreducible (i.e., not further decomposable). This motivates the development of a recursive algorithm. Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

100 Non-overlapping decomposition Decomposition in cascaded systems Sketch of the algorithm 1. Start. 2. Compute the reachability matrix R. 3. If G o is not reducible, then output G o and end. Otherwise proceed. 4. Decompose G o into an acyclic i/o partition G 1 G 2, where G 1 is irreducible. 5. Output G Rename G o = G 2 and go to step 2. Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

101 Non-overlapping decomposition Decomposition in cascaded systems - Step 3.: reducibility tests Recalling that the reducibility matrix is structured as follows F G 0 n p R = 0 m n 0 m m 0 m p H θ 0 p p the reducibility tests focus on the matrix θ. Recall also that: { θij = 1 the output y i depends on the input u j θ ij = 0 otherwise. Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

102 Non-overlapping decomposition Decomposition in cascaded systems - Step 3.: reducibility tests Definition: dominating rows of θ The h-th row dominates the k-th row if θ hj θ kj for all j = 1,...,m and if there exists at least a value j such that θ h j > θ k j (recall that the entries of θ can take values 0 or 1). In other words, the h-th row dominates the k-th row if the output y h depends by the same input variables that affect y k, and at least one more. For example: Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

103 Non-overlapping decomposition Decomposition in cascaded systems - Step 3.: reducibility tests Conditions for reducibility Necessary condition: a graph G o is reducible only if θ contains at least one zero entry. Necessary and sufficient condition: a graph G o is irreducible if only if θ does not contain any dominating rows. Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

104 Non-overlapping decomposition Decomposition in cascaded systems - Step 4.: reachability tests After defining the two digraphs G 1 and G 2 it is necessary to test their reachability properties. A simple test is provided. I) Rearranging the inputs G = [ G11 0 n1 m 2 G 21 G 22 Define the columns ^G 11, ^G 21, and ^G 22 by applying the boolean OR operator to the columns of G 11, G 21, and G 22, respectively, i.e., (col. 1) ^G ij = G ij G (col. m j ) ij, i,j = 1,2 ] Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

105 Non-overlapping decomposition Decomposition in cascaded systems II) Rearranging the outputs H = [ H11 0 p1 n 2 H 21 H 22 Define the rows ^H 11, ^H 21, and ^H 22 by applying the boolean OR operator to the rows of H 11, H 21, and H 22, respectively, i.e., (row 1) ^H ij = H ij H (row m i ) ij, i,j = 1,2 III) Rearranging both inputs and outputs θ = [ θ11 0 p1 m 2 θ 21 θ 22 Define the scalars ^θ 11, ^θ 21, and ^θ 22 by applying the boolean OR operator to all the elements of θ 11, θ 21, and θ 22, respectively. ] ] Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

106 Non-overlapping decomposition Decomposition in cascaded systems IV) Define [ ^G11 0 ^G = n1 1 ^G 21 ^G22 ] [, ^H ^H11 0 = 1 n2 ^H 12 ^H22 ] ] [^θ, ^θ = 11 0 ^θ 21 ^θ 22 V) Compute ^S = ^G ^H T R n 2 (element-wise boolean AND operator: 1 1 = 1, 1 0 = 0 1 = 0, 0 0 = 0). Main result The sub-graphs G 1 and G 2 are input-output reachable if and only if both the following conditions are verified: i) ^θ 11 = ^θ 22 = 1; ii) the matrix ^S contains exactly a single 1 on each row. Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

107 Non-overlapping decomposition Decomposition in cascaded systems Example: consider the system illustrated in the following Figure, consisting in a cascade interconnection of three tanks. Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

108 Non-overlapping decomposition Decomposition in cascaded systems Model in the example Aẋ 1 = k 3 x3 k 1 x1 Aẋ 2 = k 2 x2 + u 2 Aẋ 3 = k 2 x2 k 3 x3 + u 1 y 1 = x 2 y 2 = x 1 A = 1 m 2 : section of each tank; x i, i = 1,2,3: water level; u 2 : input water volume flow, k 1 = k 2 = k 3 = 2 m 5 2 /s. Equilibrium condition: x 1 = x 2 = x 3 = 1 m, ū 1 = 0 m 3 /s, and ū 2 = 2 m 3 /s. We define: δx i = x i x i, for i = 1,2,3, and δu i = u i ū i, δy i = y i ȳ i, for i = 1,2. Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

109 Non-overlapping decomposition Decomposition in cascaded systems Linearized system matrices: A o = 0 1 0, B o = 0 1, C o = x o = (δx 1,δx 2,δx 3 ), u = (δu 1,δu 2 ), and y = (δy 1,δy 2 ). Interconnection matrix: [S] = [ 0 1 ] Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

110 Non-overlapping decomposition Decomposition in cascaded systems Reachability matrix: R = Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

111 Non-overlapping decomposition Decomposition in cascaded systems If the subgraphs G 1 and G 2 are such that V u,1 = {u 2 }, V x,1 = {x 2 }, V y,1 = {y 1 }, V u,2 = {u 1 }, V x,2 = {x 1,x 3 }, and V y,2 = {y 2 }, the permutated reachability matrix is R = Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

112 Non-overlapping decomposition Decomposition in cascaded systems Following steps I) IV ), we obtain that [ ] 1 0 ^θ =, 1 1 ^G = , = 1 1 [ 1 0 ] In this way (i) is verified, and ^S = ^G ^H T = = verifying condition (ii). It is therefore verified that the two subsystems, in which the overall system are partitioned, are input-output reachable. Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

113 Non-overlapping decomposition Decomposition in cascaded systems The two obtained submodels are: δx 2 = δx 2 + δu 2 δy 1 = δx 2 [ ] δx1 δx 3 = [ ][ ] 1 1 δx δx 3 ] δy 2 = [ 1 0 ][ δx 1 δx 3 [ ] 0 δu [ ] 0 δx 1 2 Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

114 Outline 1 Information on the lecture 2 Introduction 3 Unstructured and interaction-oriented models 4 Decomposition of input-output pairs RGA matrix Grammians 5 Non-overlapping decompositions Preliminaries Decomposition in weakly-interacting systems Decomposition in cascaded systems 6 Overlapping decompositions Motivating example Inclusion principle Completely overlapping decompositions 7 Conclusions Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

115 Outline 1 Information on the lecture 2 Introduction 3 Unstructured and interaction-oriented models 4 Decomposition of input-output pairs RGA matrix Grammians 5 Non-overlapping decompositions Preliminaries Decomposition in weakly-interacting systems Decomposition in cascaded systems 6 Overlapping decompositions Motivating example Inclusion principle Completely overlapping decompositions 7 Conclusions Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

116 Overlapping decomposition Introduction There exist many cases when the coupling strength between subsystems is relevant, the decomposition into disjoint partitions (non-overlapping decomposition) is uneffective (for control purposes). Main idea of overlapping decompositions In those cases it may be advisable to decompose the system into models that have one or more equations and states in common. Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

117 Overlapping decomposition Motivating example Consider the problem of controlling the temperature of the building: Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

118 Overlapping decomposition Motivating example Model cρv A Ṫ A = s r u(t B T A ) + s r u(t C T A ) + s A U(T E T A ) + q A cρv B,C Ṫ B = s r u(t A T B ) + s B,C U(T E T B ) + q B cρv B,C Ṫ C = s r u(t A T C ) + s B,C U(T E T C ) + q C c: specific heat of the air; V i, i = A,B,C: Volume of room i; s r : wall surface between A and B (and C); s A : wall surface between A and the environment; s B,C : wall surface between B (and C) and the environment; u,u: transmittances; q i, i = A,B,C: inputs (heating power). Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

119 Overlapping decomposition Motivating example Equilibrium condition: T E = 0 C, T A = T B = T C = T = 20, q A = q A = s A U T W, q B = q C = q B = q C = s B,C u T W. We define: δt A = T A T, δt B = T B T, δt C = T C T, δq A = q A q A, δq B = q B q B, and δq C = q C q C. Around the given equilibrium condition: δt B δt A = (Γ + γ r ) Γ 0 γ (2γ + γ A ) γ δt 1 B cρv 0 0 B,C δt A cρv 0 δq B A δq A δt C 0 Γ (Γ + γ r ) δt C δq cρv C B,C γ = s r u cρv, Γ = s r u A cρv, γ B,C A = s AU cρv, and γ r = s B,C U A cρv. B,C Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

120 Overlapping decomposition Motivating example Model: δt B δt A = (Γ + γ r ) Γ 0 γ (2γ + γ A ) γ δt 1 B cρv 0 0 B,C δt A cρv 0 δq B A δq A δt C 0 Γ (Γ + γ r ) δt C δq cρv C B,C There are different possible non-overlapping decompositions: I non-overlapping decomposition δt B = (Γ + γ r )δt B + cρv 1 δq B,C B + ΓδT A [ ] [ ][ ] [ 1 ] [δqa ] [ δta (2γ + γa ) γ δta cρv 0 = + A γ δt C Γ (Γ + γ r ) δt 1 + δt C 0 δq cρv B,C C 0] B Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

121 Overlapping decomposition Motivating example Model: δt B δt A = (Γ + γ r ) Γ 0 γ (2γ + γ A ) γ δt 1 B cρv 0 0 B,C δt A cρv 0 δq B A δq A δt C 0 Γ (Γ + γ r ) δt C δq cρv C B,C There are different possible non-overlapping decompositions: II non-overlapping decomposition δt C = (Γ + γ r )δt C + cρv 1 δq B,C C + ΓδT A [ ] [ ][ ] [ δtb (Γ + γr ) Γ δtb = + δt A γ (2γ + γ A ) δt A 1 cρv 0 B,C 1 0 cρv A ] [δqb ] [ 0 + δt δq A γ] C Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

122 Overlapping decomposition Motivating example Model: δt B δt A = (Γ + γ r ) Γ 0 γ (2γ + γ A ) γ δt 1 B cρv 0 0 B,C δt A cρv 0 δq B A δq A δt C 0 Γ (Γ + γ r ) δt C δq cρv C B,C There are different possible non-overlapping decompositions: III non-overlapping decomposition δt A = (2γ + γ A )δt A + cρv 1 δq A A + γ(δt B + δt C ) [ ] [ ][ δtb (Γ + γr ) 0 δtb = δt C 0 (Γ + γ r ) ] [ ] + δt cρv 1 δqb + C B,C δq C [ Γ Γ] δt A Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

123 Overlapping decomposition Motivating example Model: δt B δt A = (Γ + γ r ) Γ 0 γ (2γ + γ A ) γ δt 1 B cρv 0 0 B,C δt A cρv 0 δq B A δq A δt C 0 Γ (Γ + γ r ) δt C δq cρv C B,C There are different possible non-overlapping decompositions: IV non-overlapping decomposition δt B = (Γ + γ r )δt B + 1 cρv B,C δq B + ΓδT A δt A = (2γ + γ A )δt A + 1 cρv A δq A + γ(δt B + δt C ) δt C = (Γ + γ r )δt C + 1 cρv B,C δq C + ΓδT A Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

124 Overlapping decomposition Motivating example Model: δt B δt A = (Γ + γ r ) Γ 0 γ (2γ + γ A ) γ δt 1 B cρv 0 0 B,C δt A cρv 0 δq B A δq A δt C 0 Γ (Γ + γ r ) δt C δq cρv C B,C A solution is the overlapping decomposition: Overlapping decomposition [ ] δtb δt (1) A [ ] (2) δt A δt C = = [ ][ ] (Γ + γr ) Γ δtb γ (2γ + γ A ) δt (1) + A [ (2γ + γa ) γ Γ (Γ + γ r ) ][ (2) δt A δt C ] + [ 1 cρv 0 B,C 1 0 cρv A ] [δqb δq A ] + [ 1 ] [δqa ] cρv 0 A δq cρv B,C C [ γ ][ δt (2) A δt C ] [ ][ ] γ 0 δtb 0 0 δt (1) A Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

125 Overlapping decomposition Motivating example In this way a model expansion has been carried out (equations and states have been duplicated), i.e. δt B δt A = (Γ + γ r ) Γ 0 γ (2γ + γ A ) γ δt 1 B cρv 0 0 B,C δt A cρv 0 δq B A δq A δt C 0 Γ (Γ + γ r ) δt C δq cρv C B,C δt 1 B δt (1) (Γ + γ r ) Γ 0 0 δt cρv 0 0 B A δt (2) = γ (2γ + γ A ) 0 γ δt (1) B,C γ 0 (2γ + γ A ) γ A δt (2) cρv 0 A 0 1 δq B δq A 0 0 Γ (Γ + γ δt r ) A cρv 0 A A δq C C δt C cρv B,C Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

126 Outline 1 Information on the lecture 2 Introduction 3 Unstructured and interaction-oriented models 4 Decomposition of input-output pairs RGA matrix Grammians 5 Non-overlapping decompositions Preliminaries Decomposition in weakly-interacting systems Decomposition in cascaded systems 6 Overlapping decompositions Motivating example Inclusion principle Completely overlapping decompositions 7 Conclusions Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

127 Overlapping decomposition Inclusion principle More formally: Contracted model (C) ẋ o y = A o x o + B o u = C o x o Expanded model (E) x = Ā x + Bu y = C x Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

128 Overlapping decomposition Inclusion principle Definition The model (E) includes (C) if there exists a pair of matrices (T,T ) such that x o = T x and T T = I n T : expansion map, T : contraction map. Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

129 Overlapping decomposition Inclusion principle Expansion: main result System expansions can be obtained by setting: Ā B C = TA o T + M = TB o + N = C o T + L where matrices M, N, and L must satisfy: for all i = 1,...,dim( x). T M i T = 0 T M i 1 N = 0 LM i 1 T = 0 LM i 1 N = 0 Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

130 Overlapping decomposition Inclusion principle Expansion: main result The proof of the result relies on the fact that, provided the conditions on M, N, and L are verified, then the free and forced motions of x o (t) and T x(t) are equivalent (the same must hold for the outputs). Namely, for example: x o (t) = x FREE o x FREE x FORCED o (t) + x FORCED o (t), where o (t) = e A o(t t 0 ) x o (0) (t) = t t 0 e Ao(t ν) B o u(ν)dν, T x(t) = T x FREE (t) + T x FORCED (t), where x FREE (t) = eā(t t 0) x(0) x FORCED (t) = t t eā(t ν) 0 Bu(ν)dν. Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

131 Overlapping decomposition Inclusion principle For example, consider the free motion of the state T x FREE (t) = T eā(t t 0) Tx o (0) Recalling that Ā = TA ot + M, and that, for a generic matrix Q we obtain that e Q(t t 0 ) 1 = i! ( Q(t t 0 )) i i=0 T x FREE 1 (t) = ( ) i i! T (TA o T + M)(t t 0 ) Txo (0) i=0 Since T T = I n and from the fact that T M i T = 0 for all i = 1,...,dim( x) (which implies that this holds for all i = 1,...,, see e.g., the Cayley-Hamilton Theorem), then ( ) i T (TA o T + M)(t t 0 ) T = (Ao (t t 0 )) i and therefore T x FREE 1 (t) = i! (A o(t t 0 )) i x o (0) = e A o(t t 0 ) x o (0) = x FREE o (t) i=0 Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

132 Overlapping decomposition Inclusion principle Control procedure decompose the system with an overlapping decomposition, prove that the overlapping decomposition satisfies the contraction/expansion conditions, develop control strategies for the subsystems in such a way that the expanded system is stabilized. Main question Is it enough to guarantee that the original system (contraction) is stabilized? The answer is yes: this issue is solved using the inclusion principle. Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

133 Overlapping decomposition Inclusion principle Inclusion principle Given the contraction (C) ẋ o y = A o x o + B o u = C o x o and the expansion (E) x = Ā x + Bu y = C x then the asymptotic stability of (E) implies the asymptotic stability of (C). Remark: in the literature, this issue has been extensively studied e.g., extensions of these concepts and results to the case of non linear systems are available. Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

134 Outline 1 Information on the lecture 2 Introduction 3 Unstructured and interaction-oriented models 4 Decomposition of input-output pairs RGA matrix Grammians 5 Non-overlapping decompositions Preliminaries Decomposition in weakly-interacting systems Decomposition in cascaded systems 6 Overlapping decompositions Motivating example Inclusion principle Completely overlapping decompositions 7 Conclusions Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

135 Overlapping decomposition Completely overlapping decomposition Subsystems in dynamically decoupled form A number of distributed control methods developed in the literature (e.g., based on MPC, see next lectures) require the interaction-oriented model of a system to be dynamically decoupled: { ẋi = A ii x i + M j=1 B iju j y i = C i x i Note that the dynamics of each subsystem do not depend on the state of the other subsystems, in general the dynamics of each subsystem depend by the inputs of all the subsystems, Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

136 Overlapping decomposition Completely overlapping decomposition Subsystems in dynamically decoupled form { ẋi = A ii x i + M j=1 B iju j y i = C i x i It is rarely possible to obtain a representation of this type by simply applying a non-overlapping decomposition. To obtain this representation it is generally required to apply a completely overlapping decomposition. Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

137 Overlapping decomposition Completely overlapping decomposition Consider the unstructured model ẋ o y = A o x o + B o u = C o x o Step 1. Decompose the input-output pair (u,y) into M input-output sub-pairs (u i,y i ) such that the coupling between u i and y i is maximized, for all i = 1,...,M, the coupling between u j and y i is minimized, i,j = 1,...,M, j i. This can be done: based on physical considerations (e.g., for geographically/structurally distributed systems endowed with local actuators and local transducers), based on mathematical considerations (e.g., RGA matrix, Grammians). Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

138 Overlapping decomposition Completely overlapping decomposition Step 2. Decompose matrices B o and C o according to the partition obtained on u and y, respectively. Specifically (possibly under permutation) B o = [ ] B 1,o... B M,o, Co = C 1,o. C M,o in such a way that for all i. B o u = M i=1 B i,ou i y i = C i,o x o Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

139 Overlapping decomposition Completely overlapping decomposition Step 3. Note that, in general, from all inputs u j, j = 1,...,M to all outputs y i, i = 1,...,M there is a non-zero transfer function ( direct and/or indirect interaction). For each input/output pair (u j,y i ), using the Kalman canonical form, we obtain the minimal (both reachable and observable) representation of the tuple (A o,b j,o,c i,o ), i.e., (A ii,j,b ij,c i,j) (and, correspondingly, a state vector x i,j, i,j = 1,...,M). The dependence of each output y i upon u j solely is given by: { ẋi,j = A ii,j x i,j + B ij u j y i = C i,j x i,j Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

140 Overlapping decomposition Completely overlapping decomposition Step 4. In view of the superposition principle, the dynamics of y i is described by ẋ i,1 = A ii,1 x i,1 + Bi1 u 1... ẋ i,m = A ii,m x i,m + BiM u M y i = [ ] C i,1..., C i,m x i,1. x i,m Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

141 Overlapping decomposition Completely overlapping decomposition Step 5. Define x i,1 x i =.,A ii = diag(a ii,1,...,a ii,m ) x i,m. C i = [ ] 0 C i,1..., C i,m,bij = Bij } j-th position 0. The dynamics of the i-th subsystem is described by the system in the dynamically-decoupled form, as required: { ẋi = A ii x i + M j=1 B ij u j y i = C i x i Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

142 Overlapping decomposition Completely overlapping decomposition Remarks: it is always possible to apply the sketched decomposition method, the obtained representation is not minimal (the state dimension, for each subsystem, is generally greater than n), but the input and output vector dimensions (for each subsystem) are reduced, in view of the Kalman canonical form reduction in Step 3., the state variables of the subsystem do not generally correspond to state variables of the original (unstructured) system. Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136

143 Overlapping decomposition Completely overlapping decomposition - Example Consider the motivating example: Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 136