Motivations and Historical Perspective

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1 Motivations and Historical Perspective Giovanni Marro DEIS, Universit of Bologna, Ital MTNS - Jl 5-9, 2010 G. Marro (Bologna, Ital) MTNS - Jl 5-9, / 32

2 Geometric Control Theor for Linear Sstems Block 1: Fondations [10: ]: Talk 1: Motivation and historical perspective, G. Marro [10:30-11:00] Talk 2: Invariant sbspaces, L. Ntogramatzidis [11:00-11:30] Talk 3: Controlled invariance and invariant zeros, D. Prattichizzo [11:30-12:00] Talk 4: Conditioned invariance and state observation, F. Morbidi [12:00-12:30] Block 2: Problems and applications [15: ]: Talk 5: Stabilization and self-bonded sbspaces, L. Ntogramatzidis [15:30-16:00] Talk 6: Distrbance decopling problems, L. Ntogramatzidis [16:00-16:30] Talk 7: LQR and H 2 control problems, D. Prattichizzo [16:30-17:00] Talk 8: Spectral factorization and H 2 -model following, F. Morbidi [17:00-17:30] G. Marro (Bologna, Ital) MTNS - Jl 5-9, / 32

3 What the geometric approach is The geometric approach is a collection of tools, properties and algorithms for the analsis and control of dnamic linear and nonlinear sstems in a coordinate-free context. Statements in terms of sbspaces (or, more generall, sets) instead of matrices are compact and clean, and insight into their meaning is highl facilitated. This mini-corse is an attempt to present in simple terms the origin, motivation and growth of the geometric approach throgh almost for decades. G. Marro (Bologna, Ital) MTNS - Jl 5-9, / 32

4 Otline The beginning The main tools The most important solved problems G. Marro (Bologna, Ital) MTNS - Jl 5-9, / 32

5 The beginning Or first papers in English G. Marro (Bologna, Ital) MTNS - Jl 5-9, / 32

6 The beginning Or first papers in English G. Marro (Bologna, Ital) MTNS - Jl 5-9, / 32

7 The beginning Independentl The following words can often be fond in the literatre: the geometric approach was introdced independentl b Basile and Marro and Wonham and Morse. This is tre. The following papers are sall cited: G. Basile and G. Marro, Controlled and conditioned invariant sbspaces in linear sstem theor, J. Optim. Theor Appl., vol. 3, no. 5, pp , W. M. Wonham and A. S. Morse, Decopling and pole assignment in linear mltivariable sstems: a geometric approach, SIAM J. Contr., vol. 8, no. 1, pp. 1 18, Let s look at two interesting docments concerning that period. G. Marro (Bologna, Ital) MTNS - Jl 5-9, / 32

8 The beginning The end of the period of independence (Janar 29, 1970) G. Marro (Bologna, Ital) MTNS - Jl 5-9, / 32

9 The beginning Another significant letter received in that period G. Marro (Bologna, Ital) MTNS - Jl 5-9, / 32

The beginning The three books on the geometric approach Wonham (1974,

neglected in Wonham s book. Software for Matlab is inclded.

10 The beginning The three books on the geometric approach Wonham (1974, 1979, 1985) has the great merit of having widel pblicized the geometric approach. Basile and Marro (1992) emphasize dalit and dal problems, that were neglected in Wonham s book. Software for Matlab is inclded. Trentelman, Stoorvogel and Hats (2001) provide a bridge towards achieving minimal H2 and H norm soltions. G. Marro (Bologna, Ital) MTNS - Jl 5-9, / 32

11 Otline The beginning The main tools The most important solved problems G. Marro (Bologna, Ital) MTNS - Jl 5-9, / 32

12 The main tools The concept of trajector in the state space (inpt) x (state) (otpt) X ẋ(t) x(t) Continos-time: ẋ(t) = Ax(t)+B (t) (t) = C x(t) ẋ(t) = Ax(t)+B (t) (t) = C x(t)+d(t) the term D(t) is called feedthrogh. G. Marro (Bologna, Ital) MTNS - Jl 5-9, / 32

13 The main tools The concept of trajector in the state space. X..... x(k) x(k +1) Discrete-time: x(k +1) = Ax(k)+B (k) (k) = C x(k) x(k +1) = Ax(k)+B (k) (k) = C x(k)+d(k) Ver often in practice the discrete-time sstems are prodced b sampling continos-time sstems, for instance with ZOH (zero order hold) eqivalence. In these case a positive sampling time is associated to the conversion process. G. Marro (Bologna, Ital) MTNS - Jl 5-9, / 32

14 The main tools The Matlab Representations of LTI Sstems Prel dnamic continos-time sstems >> ss=ss(a,b,c,0); Continos-time sstems with feedthrogh >> ss=ss(a,b,c,d); Prel dnamic discrete-time sstems >> ss1=ss(a,b,c,0,tc); (Tc = 1 if a sampling time is not specified) Discrete-time sstems with feedthrogh >> ss1=ss(a,b,c,d,tc); Conversion from continos to discrete time with the ZOH eqivalence >> ss1=c2d(ss,tc); (Tc > 0 - in this case a sampling time is specified) G. Marro (Bologna, Ital) MTNS - Jl 5-9, / 32

15 The main tools Some properties of sstems expressed in geometric terms x >> V=vstar(ss); >> S=sstar(ss); >> r=reldeg(ss); >> z=gazero(ss); Left invertibilit V S ={0} Right invertibilit V +S =X Relative degree minimal ρ sch that V +S ρ =X Minimalit of phase Z() C g where C g is the set of stable modes in the complex plane. G. Marro (Bologna, Ital) MTNS - Jl 5-9, / 32

16 The main tools Sstems with mltiple sets of inpts and otpts h e h : distrbance e : controlled otpt : control inpt : measred otpt The sstem eqation are ẋ(t) = Ax(t)+Hh(t)+B (t) e(t) = E x(t) +D 1 (t) (t) = C x(t)+d 2 h(t) ẋ(t) e(t) (t) or = A H B E 0 D 1 C D 2 0 x(t) h(t) (t) or else, b sing a poplar notation that points ot the existence of feedthroghs terms, we can denote the sstem with A H B = E 0 D 1 C D 2 0 G. Marro (Bologna, Ital) MTNS - Jl 5-9, / 32

17 Dalit The main tools The dal of sstem : (A,B,C,D) is defined as T : (A T,C T,B T,D T ) ū T 3 T 1 T ȳ Consider the interconnection of sstems shown above: the overall dal sstem is obtained b reversing the order of seriall connected sstems and interchanging branching points with smming jnctions and vice versa: = A B 1 0 A 2 0 B 2 B 3,1 C 1 B 3,2 C 2 A C 3 0, T = A T 1 0 C T 1 BT 3,1 0 0 A T 2 C T 2 BT 3, A T 3 C T 3 B T 1 B T G. Marro (Bologna, Ital) MTNS - Jl 5-9, / 32

18 Otline The beginning The main tools The most important solved problems G. Marro (Bologna, Ital) MTNS - Jl 5-9, / 32

19 The most important solved problems The distrbance decopling problem ( ) h F x Eqations ẋ(t) = Ax(t)+B (t)+hh(t) (t) = C x(t) Let B=imB, H=imH, C=kerC. The distrbance decopling problem (1969) It will be shown that the soltion in geometric terms depends on A, B, H, C. W. M. Wonham and A. S. Morse, Decopling and pole assignment in linear mltivariable sstems: a geometric approach, SIAM J. Contr., vol. 8, no. 1, pp. 1 18, G. Basile and G. Marro, L invarianza rispetto ai distrbi stdiata nello spazio degli stati, in Rendiconti della LXX Rinione Annale AEI, paper , Rimini, Ital, 1969, G. Marro (Bologna, Ital) MTNS - Jl 5-9, / 32

20 The most important solved problems The distrbance decopling problem ( ) h F x The distrbance decopling problem The soltion of a snthesis problem with geometric techniqes as a rle incldes A strctral condition A stabilizabilit condition G. Marro (Bologna, Ital) MTNS - Jl 5-9, / 32

21 The most important solved problems Other basic problems (primal and dal) h c T e T c + + ẽ η Measrable signal decopling Unknown-inpt observation S. P. Bhattachara, Distrbance rejection in linear sstems, Int. J. Sstems Science, vol. 5, no. 7, pp , R. Laschi and G. Marro, Alcne considerazioni sll osservabilità dei sistemi dinamici con ingressi inaccessibili in Rendiconti della LXX Rinione Annale AEI, paper , Rimini, Ital, 1969, G. Marro (Bologna, Ital) MTNS - Jl 5-9, / 32

22 The most important solved problems A review of exact feedforward model following h c m m + η=0 _ Exact model following as a measrable distrbance decopling problem with stabilit. NOTE: If m consists of p independent SISO sstems connected in parallel, we achieve exact row-b-row decopling at no cost. G. Marro (Bologna, Ital) MTNS - Jl 5-9, / 32

23 The most important solved problems From exact feedforward to exact feedback model following r + h _ c + η=0 r + h _ c + η=0 m m _ m m _ Exact feedback model following. A strctrall eqivalent connection. h m c r m + + m Another strctrall eqivalent connection. + η=0 _ r + h _ c m m + η=0 Exact feedback model following, possibl with row-b-row decopling and mltiple internal models. _ G. Marro (Bologna, Ital) MTNS - Jl 5-9, / 32

24 The most important solved problems Other basic problems (primal and dal) α 1 c,1 α 2 c, T T c,1 T c,2 α 1 α 2 The noninteracting control problem The falt detection and isolation problem A. S. Morse and W. M. Wonham, Decopling and pole assignment b dnamic compensation, SIAM J. Contr., vol. 8, no. 3, pp , G. Basile and G. Marro, A state space approach to noninteracting controls, Ricerche di Atomatica, vol. 1, pp , M. Massomnia, G. C. Verghese, and A. S. Willsk, Failre detection and identification, IEEE Trans. At. Contr., vol. 34, pp , G. Marro (Bologna, Ital) MTNS - Jl 5-9, / 32

25 The most important solved problems Other basic problems d e c Distrbance Decoplig Problem with dnamic otpt feedback J. M. Schmacher, Compensator snthesis sing (C,A,B)-pairs, IEEE Trans. At. Contr., vol. AC-25, pp , J. C. Willems and C. Commalt, Distrbance decopling with measrement feedback with stabilit or pole placement, SIAM J. Contr. Optim., vol. 19, pp , G. Basile, G. Marro, and A. Piazzi, Stabilit withot eigenspaces in the geometric approach: Some new reslts, in Freqenc Domain and State Space Methods for Linear Sstems, C. A. Brnes and A. Lindqist, Eds., pp North-Holland (Elsevier), Amsterdam, G. Marro (Bologna, Ital) MTNS - Jl 5-9, / 32

26 The most important solved problems From exact to H 2 -optimal and H -optimal d F x e d c e The H 2 -optimal distrbance decopling problem The main criticism to which the geometric approach is sbjected to is that exact soltions to the above problems cannot be implemented in practice becase of ncertainties in parameters. In this approach standard optimal control and filtering problems can be treated in a minimal H 2 or H norm context b simpl sbstitting sstem with the corresponding Hamiltonian sstem. Ths the geometric approach provides insight and tools for the treatment of singlar and cheap cases of optimal control. Interesting reslts in this area are de to Stoorvogel, Chen, Saberi and Sannti ( ). G. Marro (Bologna, Ital) MTNS - Jl 5-9, / 32

27 The most important solved problems Spectral factorization and H 2 -optimal model following, 1 1 ẋ(t) = Ax(t)+B (t) (t) = C x(t)+d(t) The Stoorvogel problem 1 (t) = C 1 x(t)+d 1 (t) A. Stoorvogel, A. Saberi, and B. M. Chen, Fll and redced observer-based controller design for H 2 optimization, Int.. J. Contr., vol. 58, pp , A. Stoorvogel, The singlar H 2 control problem, Atomatica, vol. 28, pp , >> ss=ss(a,b,c,d); >> [C1,D1]=stoor(A,B,C,D,[Tc]); G. Marro (Bologna, Ital) MTNS - Jl 5-9, / 32

28 The most important solved problems Spectral factorization and H 2 -optimal model following Φ Φ ȳ ȳ >> ss=ss(a,b,c,d); >> adjss=ss(-a,-c,b,d ); (adjoint sstem) >> ss1=ss(a,b,c1,d1); >> adjss1=ss(-a,-c1,b,d1 ); The spectral factorization problem (the spectrm Φ is the same) G. Marro, F. Morbidi and D. Prattichizzo, A geometric soltion to the cheap spectral factorization problem, in Proc. ECC2009, Bdapest, G. Marro (Bologna, Ital) MTNS - Jl 5-9, / 32

29 The most important solved problems The mltivariable reglator problem e r e + _ c d The mltivariable reglator problem with internal model ( ) B. A. Francis, The linear mltivariable reglator problem, SIAM J. Contr. Optim., vol. 15, pp , G. Basile, G. Marro and A Piazzi, Revisiting the reglator problem in the geometric approach. Part II: Asmptotic tracking and reglation in the presence of distrbances, J. Optim. Theor Appl., 53, 23 26, G. Marro (Bologna, Ital) MTNS - Jl 5-9, / 32

The most important solved problems The mltivariable reglator problem r + h _ c m d m + _ e r + h _ c m 1 m d + _ e o Exact feedback model following (2002) H 2 -optimal feedb. model following (2007) G.

30 The most important solved problems The mltivariable reglator problem r + h _ c m d m + _ e r + h _ c m 1 m d + _ e o Exact feedback model following (2002) H 2 -optimal feedb. model following (2007) G. Marro, F. Morbidi and D. Prattichizzo, H 2 psedo-optimal model following: a geometric approach, Proc. 3rd IFAC Smposim on Sstem Strctre and Control, Foz de Igaç, Brazil, October 17-19, G. Marro (Bologna, Ital) MTNS - Jl 5-9, / 32

31 Conclsion The geometric approach has been developed for abot fort ears gradall, as new linear and nonlinear problems arose and were analzed. It provides a significant insight into sstems and control, based on ver few elementar tools, on which the overall theor is based. Althogh the geometric tools are ver simple and spported b exhastive comptational machiner, it is rather difficlt to get a complete panorama of them, since their presentation in the literatre b several athors is not niform in stle and has ver often been covered in nnecessaril heav mathematics. G. Marro (Bologna, Ital) MTNS - Jl 5-9, / 32

32 A great friend To conclde, let me remember a great friend, Giseppe Basile, who worked with me on the above research and is no longer with s. I recall, in particlar, his intition for new ideas and his livel enthsiasm in discssion for developing them. Thank o! G. Marro (Bologna, Ital) MTNS - Jl 5-9, / 32

Education in Linear System Theory with the Geometric Approach Tools

Education in Linear System Theory with the Geometric Approach Tools Giovanni Marro DEIS, University of Bologna, Italy GAS Workshop - Sept 24, 2010 Politecnico di Milano G. Marro (University of Bologna)