THE LQG control problem [1] concerns linear systems

Transcription

1 Orhogonal Funcions Approach o LQG Conrol B M Mohan Senior Member, IEEE, and Sanjeeb Kumar Kar Absrac In his paper a unified approach via block-pulse funcions (BPFs) or shifed Legendre polynomials (SLPs) is presened o solve he linear-quadraic-gaussian (LQG) conrol problem Also a recursive algorihm is proposed o solve he above problem via BPFs By using he elegan operaional properies of orhogonal funcions (BPFs or SLPs) hese compuaionally aracive algorihms are developed To demonsrae he validiy of he proposed approaches a numerical example is included Keywords Linear quadraic Gaussian conrol; Linear quadraic esimaor; Linear quadraic regulaor; Time-invarian sysems; Orhogonal funcions; Block-pulse funcions; Shifed Legendre polynomials I INTRODUCTION THE LQG conrol problem concerns linear sysems disurbed by addiive whie Gaussian noise, incomplee sae informaion and quadraic coss The LQG conroller is simply he combinaion of a linear-quadraic esimaor (LQE), ie Kalman filer wih a linear quadraic regulaor (LQR) The separaion principle guaranees ha hese can be designed and compued independenly Orhogonal funcions approach 8, 9 has been recognized as an efficien and useful approach compuaionally o solve variey of problems in sysems and conrol In 6 he soluion of he LQG conrol design problem was obained by employing general orhogonal polynomials In he auhors considered he problem of LQG conrol sysem and showed is applicaion as an informaion ransmission problem The discree ime LQG problem was considered in and showed is applicaions over lossy daa neworks Very recenly, applicaions of orhogonal funcion approach is exended o differen ype of sysems, ie sysems described by inegro-differenial equaions, muli-delay sysems, 7, disribued parameer sysems 4, delay sysems wih reverse ime funcions, singular sysems 6 and o nonlinear sysems 8 In his paper, we consider linear ime-invarian sysems and propose a unified approach, based on using BPFs or SLPs, o solve LQG conrol problem of such sysems We call his approach unified approach because i can be used via SLPs or BPFs In addiion o he unified approach a recursive algorihm is proposed using BPFs I is very imporan o noe ha he LQG conrol problem is no ye sudied via BPFs The paper is organized as follows : The nex secion deals wih BPFs and SLPs, and heir properies The LQG conrol B M Mohan is wih he Deparmen of Elecrical Engineering, IIT Kharagpur, India (mohan@eeiikgpernein) Sanjeeb Kumar Kar is wih he Deparmen of Elecrical Engineering, Insiue of Technical Educaion and Research, SOA Universiy, Bhubaneswar- 7, INDIA (skkarier@rediffmailcom) Manuscrip received March ; revised design problem is discussed in Secion The mehod of obaining soluion of he LQG conrol design problem is presened in Secion 4 A numerical example is considered in Secion The las secion concludes he paper II ORTHOGONAL FUNCTIONS AND THEIR PROPERTIES We consider wo classes of orhogonal funcions, namely BPFs and SLPs, and discuss heir properies A BPFs and heir properies 4, 7 A se of m BPFs, orhogonal over, f ), is defined as, + it < B i () = +(i +)T (), oherwise for,,,,m, T = f, he block-pulse widh () m A square inegrable funcion f () on f can be approximaely represened in erms of BPFs as f() f i B i () =f T B() () f = T f, f,, f (4) is an m - dimensional block-pulse specrum of f (), and B() = B (), B (),, B () T () an m - dimensional BPF vecor f i in Eq () is given by +(i+)t f i = f()d (6) T +it which is he average value of f () over + it + (i +)T The produc of wo BPFs B i () and B j () can be expressed as if i j B i ()B j () = (7) B i () if i = j Operaional marix of forward inegraion 4 : Inegraing B() from o and expressing he resul in m-se of BPFs, we have B(τ)dτ P f B() (8) P f = T (9) 7

2 is called he operaional marix of forward inegraion of BPFs and i is an m m upper riangular marix Operaional marix of backward inegraion 7 : We inegrae B() from f o and express he resul in m-se of BPFs o obain B(τ)dτ P b B() () f P b = T = Pf T () We call P b he operaional marix of backward inegraion of BPFs I is an m m lower riangular marix B SLPs and heir properies 8 SLPs saisfy he recurrence relaion L i+ () = for i =,,, wih (i +) (i +) ϕ() L i i() (i +) L i () () ϕ() = ( ) ( f ) () L () =, and L () = ϕ() (4) A funcion f () ha is square inegrable on, f can be represened in erms of SLPs as f() f i L i () =f T L() () Here f is called Legendre specrum of f (), given in Eq (4) and L() = L (), L (),, L () T (6) is called SLP vecor f i in Eq () is given by f i = (i +) ( f ) f f()l i ()d (7) The produc of wo SLPs L i () and L j () can be expressed as L i ()L j () ψ ijk = k = ψ ijk L k () (8) (k +) ( f ) π ijk (9) π ijk = π ikj = π jik = π jki = π kji = π kij () π ijk = al a j l a i j+l a i+l ( f ) (i+l)+ if k = i j +l if k i j +l () for i j a =, a l+ = for l =,,, (l +) (l +) a l () Operaional marix of forward inegraion 8 : Inegraing L() from o, and expressing he resul in erms of he same se of SLPs, we have L(τ)dτ P f L() () P f = ( f ) (4) m which is called he operaional marix of forward inegraion of SLPs Operaional marix of backward inegraion : If we inegrae L() from f o and express he resul in erms of he same se of SLPs, we have L(τ)dτ P b L() () f P b = ( f ) m (6) which is called he operaional marix of backward inegraion of SLPs III THE LQG CONTROL PROBLEM Consider he linear dynamic sysem ẋ() = Ax()+Bu()+v() (7) z() = Cx()+w() (8) x() is n dimensional sae vecor, u() p dimensional conrol vecor, and z() q dimensional oupu vecor, and v() and w() he addiive zero-mean whie Gaussian sysem noise and measuremen noise, respecively, ie E v()v T (τ) } = Q δ( τ) (9) E w()w T (τ) } = R δ( τ) () Q is posiive semi-definie and R is posiive definie symmeric marices Also v() is uncorrelaed wih w(), ie E v()w T (τ) } = () Assume ha he iniial condiion x( ) is Gaussian wih mean x( ) and covariance marix P = P ( ) = E x( } ) x( ) x( ) x( ) T 8

3 which is symmeric posiive semi-definie, and E v() x T } = E w() x T } = for () Given his sysem, he objecive is o find he conrol inpu u() which a every ime may depend only on he pas measuremens z( ), <such ha he cos funcion J = E xt ( f )Sx( f ) + f x T ()Q x()+u T ()R u() } d () is minimized, he marix R is posiive definie symmeric, and S and Q are symmeric posiive semi-definie marices The LQG conroller ha solves he LQG conrol problem is specified by he equaions ˆx() = Aˆx()+Bu()+K ()z() Cˆx() (4) ˆx( ) = E x( ) = x u() = K ()ˆx() () The marix K () is called he Kalman gain of he associaed Kalman filer represened by Eq (4) A each ime his filer generaes esimaes ˆx() of he sae x() using he pas measuremens and inpus The Kalman gain is deermined hrough he associaed marix Riccai differenial equaion P () = AP ()+P ()A T P ()C T R CP ()+Q (6) P ( ) = P Given he soluion P (), f he Kalman gain equals K () = P ()C T R (7) The marix K () is called he feedback gain marix which is deermined hrough he associaed marix Riccai differenial equaion P () = A T P ()+P ()A P ()BR BT P ()+Q (8) P ( f ) = S Given he soluion P (), f he feedback gain equals K () = R BT P () (9) Observe he similariy of he wo marix Riccai differenial Equaions (6) and (8); he firs one running forward in ime, and he second one running backward in ime The firs one solves he LQE problem and he second one solves LQR problem So he LQG problem separaes ino LQE and LQR problems ha can be solved independenly The block diagram of he LQG problem is presened in Figure IV ORTHOGONAL FUNCTIONS APPROACH Inegraing he Riccai equaion (8) backward in ime from f o, we obain P () S = A T P (τ)+p (τ)a f P (τ)fp (τ)+q dτ (4) F = BR BT Expressing P (), P ()FP (), Q and S in erms of orhogonal funcions φ i ()}, which may be BPFs B i ()} or SLPs L i ()}, wehave Then and P () P i φ i () = P (φ() I n ) (4) P = P, P,, P, (4) A T P () P (φ() I n ) (4) P ()A ˆP (φ() I n ) (44) P = A T P, A T P,, A T P, (4) ˆP = P A, P A,, P, A (46) P ()FP () P i FP i φ i () (47) if BPFs are used P i FP j ψ ijk φ k ()(48) j =k = if SLPs are used F (φ() I n ) (49) F = P FP, P FP,,, P, FP, if φ()isb() = ψ ij P i FP j, j =, if φ()is L() j = ψ ij, P i FP j () () Q = Q (φ() I n ) () Q = Q, Q,, Q if φ()isb() () = Q,,, if φ()isl() (4) S = S (φ() I n ) () 9

4 v() w() B + x () y( ) + z() + + xˆ( ) C K () + s s + + u() A - K () C A Fig Opimum linear combined esimaion and conrol S = S, S,, S if φ()isb() (6) = S,,, if φ()isl() (7) and is he Kronecker produc Subsiuing Eqs (4), (4), (44), (49), () and () ino Eq (4) and making use of he backward inegraion operaional propery in Eq () or (), we have P + S = P + ˆP F + Q (P b I n ) P + P + ˆP F (P b I n )= S Q (P b I n ) (8) which is o be solved for he specrum of P () Similarly, he specrum of P () can also be found from he Riccai equaion (6), and is given by P P + ˆP G (P f I n )= P + Q (P f I n ) (9) P = AP, AP,, AP, (6) ˆP = P A T, P A T,, P, A T (6) and G = C T R C Noice ha boh he Riccai equaions are hus reduced o he non-linear algebraic equaions (8) and (9), which can easily be solved using Newon-Raphson mehod A Recursive algorihm via BPFs For a scalar sysem i is possible o obain a recursive algorihm if BPFs are used This poin is discussed here Subsiuing he operaional marix of backward inegraion P b in Eq () ino Eq (8) and simplifying, we obain he following recursive algorihm : V NUMERICAL EXAMPLE Consider he linear sysem, 6 wih he measuremen ẋ() = x()+u()+v() x() = z() = x()+w() and he cos funcion J = E x ( f )S + f x ()+u () } d E v()v(τ)} = δ( τ) E w()w(τ)} = δ( τ) E x() x() } = If S =and f =, he exac soluions of P () and P () are given by P () = + anh ( ) and P () = + anh ( +an / )} So wih m =4and 4 he above recursive algorihm via BPFs and wih m =4he nonrecursive approach in Secion 4 via SLPs are applied, and P () and P () are compued as shown in Figs and The exac soluions are also shown in he same figures for comparison sake The resuls are quie saisfacory even wih four SLPs VI CONCLUSION A unique mehod o deermine he filer gain and he regulaor gain in LQG conrol problem is proposed I is shown ha he applicaion of orhogonal funcions (BPFs and SLPs) reduces differenial calculus o algebra A BPF based recursive algorihm is presened o solve he LQG conrol problem of linear ime-invarian scalar sysems An illusraive example is

5 Recursive algorihm : ( ) F T A + Q F P, = ( ) F T A + P,j = ( ) F T A + ( ) F T A + Q F + F (6) ( ) T + A P,j+ P,j+ (6) for j = m,m,,, Similarly, subsiuing he operaional marix of forward inegraion P f in Eq (9) ino Eq (9) we have P, = ( ) ( ) G T A + G T A + Q (64) G P,j = ( ) G T A ( ) + G T A + Q G + ( ) G T + A P,j P,j (6) for j =,,,m Such a recursive algorihm is no possible wih SLPs Exac SLPs wih m = 4 BPFs wih m = 4 BPFs wih m = P () P () 6 4 P () P () ime Exac SLPs wih m = 4 BPFs wih m = 4 BPFs wih m = ime Fig Exac, SLP and BPF soluions of P () Fig Exac, SLP and BPF soluions of P () included o demonsrae he usefulness of he unified approach via SLPs and he recursive algorihm via BPFs As can be seen from Figs and, only four SLPs are good enough o obain he resul which is almos following he exac soluion One has o consider a large number of BPFs o improve upon he accuracy This is because we are using piecewise consan funcions (BPFs) o represen he smooh funcions P () and P () in he presen conex Every approach (SLP or BPF) has is own advanage and disadvanage SLP mehod does no require large number of polynomials in series expansion o represen smooh funcions, bu compuaionally i is no as much aracive as BPF mehod because SLPs are o be compued and used for signal represenaion while i is no so in BPF mehod as BPFs are all uniy and disjoin REFERENCES Ahans, M, The role and use of he sochasic linear-quadraic- Gaussian problem in conrol sysem design, IEEE Trans Auomaic Conrol, vol 6, no 6, pp: 9-, 97 Sage, A P and Whie, C C, Opimum Sysems Conrol, Prenice-Hall, Inc, Englewood Cliffs, New Jersey, 977 Brewer, J W, Kronecker producs and marix calculus in sysem heory, IEEE Trans Circuis and Sysems, vol, no 9, pp: 77-78, Rao, G P, Piecewise Consan Orhogonal Funcions and Their Applicaion o Sysems and Conrol, LNCIS, Springer, Berlin, 98

6 Hwang, C and Chen, M Y, Analysis and opimal conrol of imevarying linear sysems via shifed Legendre polynomials, In J Conrol, vol 4, no, pp: 7-, 98 6 Chang, Y F and Lee, T T, General orhogonal polynomials approximaions of he linear-quadraic-gaussian conrol design, In J Conrol, vol 4, no 6, pp: , Jiang, Z H and Schaufelberger, W, Block-Pulse Funcions and Their Applicaions in Conrol Sysems, LNCIS 79, Spinger, Berlin, 99 8 Daa, K B and Mohan, B M, Orhogonal Funcions in Sysems and Conrol, Advanced Series in Elecrical and Compuer Engineering, vol 9, World Scienific, Singapore, 99 9 Para, A and Rao, G P, General Hybrid Orhogonal Funcions and Their Applicaions in Sysems and Conrol, LNCIS, Springer, London, 996 Gupa, V, Hassibi, B and Murray, R M, Opimal LQG conrol across packe-dropping links, Sysems & Conrol Leers, vol 6, no 6, pp: , 7 Sinopoli, B, Schenao, L, Franceschei, M, Poolla, K and Sasry, S, Opimal linear LQG conrol over lossy neworks wihou packe acknowledgmen, Asian J Conrol, vol, no, pp: -, 8 Kar, S K, Orhogonal funcions approach o opimal conrol of linear ime-invarian sysems described by inegro-differenial equaions, KLEKTRIKA, vol, no, pp: -8, 9 Mohan, B M and Kar, S K, Opimal Conrol of Muli-Delay Sysems via Orhogonal Funcions, In J Advanced Research in Engineering and Technology, vol, no, pp: -4, 4 Kar, S K, Opimal conrol of a linear disribued parameer sysem via shifed Legendre polynomials, In J Elecrical and Compuer Engineering (WASET), vol, no, pp: 9-97, Mohan, B M and Kar, S K, Orhogonal funcions approach o opimal conrol of delay sysems wih reverse ime erms, J The Franklin Insiue, vol 47, no 9, pp: 7-79, 6 Mohan, B M and Kar, S K, Opimal Conrol of Singular Sysems via Orhogonal Funcions, In J Conrol, Auomaion and Sysems, vol 9, no, pp: 4-, 7 Mohan, B M and Kar, S K, Opimal conrol of muli-delay sysems via shifed Legendre polynomials, In Conf on Energy, Auomaion and Signals (ICEAS), Bhubaneswar, INDIA, December 8-, 8 Mohan, B M and Kar, S K, Opimal conrol of nonlinear sysems via orhogonal funcions, In Conf on Energy, Auomaion and Signals (ICEAS), Bhubaneswar, INDIA, December 8-, Sanjeeb Kumar Kar was born in Bankura, India in 97 He received his BTech in Elecrical Engineering from College of Engineering & Technology, OUAT, Bhubaneswar in 99, M Tech in Conrol Sysems from Indian Insiue of Technology, Kharagpur in 4 and PhD from Indian Insiue of Technology, Kharagpur in From March 4 o February he worked in Odisha Sae Elecriciy Board as Apprenice Engineer, from April 99 o March 997 worked in Pal Texile, Balasore as Assisan Manager From Augus 997 he joined CET, OUAT, Bhubaneswar as a faculy in Elecrical Engineering and coninued upo Ocober 8 Since November 8 he has been on he faculy of Elecrical Engineering Deparmen, Insiue of Technical Educaion & Research (ITER), Bhubaneswar A presen he is working as Associae Professor and Head of he Deparmen of Elecrical Engineering, ITER, SOA Universiy, Bhubaneswar He auhored/co-auhored several papers in inernaional journals and conferences He is a life member of Sysems Sociey of India His research ineress include Esimaion, Analysis, and Conrol of Dynamical Sysems using Orhogonal Funcions B M Mohan was born in Tapeswaram, India in 96 He received he Bachelors degree in Elecrical Engineering from Osmania Universiy in 98, he Masers degree in Elecrical Engineering (wih Conrol Sysems specializaion) from Andhra Universiy in 98, and he Docoral degree in Engineering from Indian Insiue of Technology, Kharagpur, 989 From July 989 o Ocober 99 he was on he faculy of Elecrical and Elecronics Engineering Deparmen, Regional Engineering College (now called Naional Insiue of Technology), Tiruchirapalli Since November 99 he has been on he faculy of Elecrical Engineering Deparmen, Indian Insiue of Technology, Kharagpur he is currenly a Professor He was a Visiing Professor a he Deparmen of Elecrical Engineering - Sysems, Universiy of Souhern California, and Los Angeles, California in 6 His research ineress include Idenificaion, Analysis, and Conrol of Dynamical Sysems using Fuzzy Logic and Orhogonal Funcions He coauhored he research monograph Orhogonal Funcions in Sysems and Conrol (World Scienific, Singapore, 99), and several papers in inernaional journals and conferences He is a member of Asian Conrol Associaion, life member of Sysems Sociey of India, senior member of he IEEE, and life fellow of Insiuion of Engineers (India) He is he Associae Edior of In J Auomaion & Conrol (IJAAC), he Ediorial Advisory Board member of Auo sof J Inelligen Auomaion & Sof Compuing, and he Ediorial Board member of In J Mahemaics and Engineering wih Compuers (IJMAEC) He is he reviewer for Compuing Reviews He was he Chairman of Conrol Sysems Chaper, IEEE Kharagpur Secion He was he chair of Conrol, Roboics & Moion Conrol rack, ICIIS 8 and Engineering wih Compuers