Frequency Theorem for discrete time stochastic system with multiplicative noise

F requency Theorem for discrete time sto c hastic system w ith m ultiplicative noise P eter Situm b ek o Nalitolela ∗ Departmen t of M a thematics, Univ ersit y of Dar Es Salaam Dar Es Salaam, T anzania Nik olai Dokuc haev † Departmen t of Mathematics & Statistics, Curtin Univ ersit y P erth, Australia Abstract In this paper we consider the problem o f minimizing a quadratic functiona l for a discrete- time linear sto chastic system with m ultiplicative noise, on a standard probability space, in infinite time horizon. W e show that the necessary and sufficient conditions for the existence of the optimal con tro l can be formulated as matrix inequalities in freq uency do ma in. F urther - more, we s how that if the optimal control exists, then certain Lyapunov e q uations m ust hav e a so lution. The optimal control is obtained b y solv ing a deterministic linea r-quadr atic opti- mal control pro blem whose functional de p ends on the solution to the Lyapunov e q uations. Moreov er, we show that under certain conditions, solv ability o f the Lyapuno v equa tions is guaranteed. W e a lso show tha t, if the frequency inequalities are strict, then the solution is unique up to equiv ale nc e . Mathematics Sub ject Classification : 93 E 20, 49N10 Key w ords : Sto c hastic Optimal Con trol, F requency T h eorem, Kalman–Y akub o vich Lemma, Kalman–Szego Lemm a, Ly apu no v equations. ∗ p eter.nalitolela@gma il.com † N.Dokuchaev@curtin.edu.au 1 1 In tro d uction Kalman–Y a ku b ovi c h Lemm a (KY Lemma) was a groun dbreaking result th at pav ed w a y for a solutions to lots of problems in con trol theory , includ ing op timal con trol. The fi r st v ariant of the Lemma wa s derived by Y akub o vich in 1962 (see [22]). Th e follo w in g y ear, the discrete-time v ersion of that resu lt was derive d b y Szeg˝ o and Kalman (see [20]). It is called sometimes the Kalman–Szeg˝ o Lemm a (KS Lemma); see [5, 13, 17] for a comprehensive review of v arious r esults in con trol theory deriv ed from the KY Lemma. V arious w orks suc h as [1 –4 ] consider ed problems with qu ad r atic fun ctionals whereas Y akub o vic h (see [25, 26]) deriv ed the KY L emma for the case in wh ic h b oth th e con trol and state v ectors are b oth Hilb ert sp aces. Dokuc haev [6] consider ed a con tinuous time sto c hastic linear-quadratic optimal con trol prob- lem, with the state ev olution d escrib ed by Itˆ o equations, with state dep en den t coefficien ts; a generalizat ion of the F requency Theorem w as obtained. W e consider a discrete-t ime analogy of th e p roblem studied in [6]. W e sho w that the n ecessary and sufficien t conditions for the existence of the optimal control can b e f orm ulated as matrix inequalities in frequency domain. F urth er m ore, we s ho w that if the optimal con trol exists, then certain Lyapuno v equations must ha ve a s olution. The optimal control is obta ined by solving a deterministic linear-quadratic optimal con trol problem w hose functional dep ends on the solution to the Lyapuno v equations. Moreo v er, we sho w that un der certain conditions, solv abilit y of the Ly apu no v equations is guar- an teed. W e also sho w th at, if the frequency inequ alities are strict, then the solution is uniqu e up to equiv alence. 2 Problem Statemen t W e consider th e follo wing optimization pr ob lem on a s tand ard p robabilit y space, (Ω , F , P ). Φ ( u . ) = + ∞ X t =0 Minimize E [ x ∗ t Gx t + 2 Re x ∗ t γ u t + u ∗ t Γ u t ] (2.1) o v er the set U = ( u t ∈ R m : + ∞ X t =0 | u t | 2 < + ∞ ) (2.2) sub ject to x t +1 = Ax t + bu t + C x t ξ t +1 , t = 0 , 1 , 2 , . . . (2.3) x 0 = a. (2.4) 2 Here x t is a rand om n -vect or o f states, u t is an m -vec tor of con tr ols and U is the set of admissible con trols. Matrices A ∈ R n × n , b ∈ R n × m , C ∈ R n × n , G = G ⊤ ∈ R n × n , γ ∈ R n × n , and Γ = Γ ⊤ ∈ m × m are constant. The scalar ξ t ∈ R is the d iscrete-time white n oise adapted to a flow of n on-decreasing σ -algebras F t ⊂ F suc h that E ξ t = 0, V ar ( ξ t ) = 1. The ve ctor a is random, m easurable with r esp ect to F 0 , in dep end en t of { ξ t } + ∞ t =0 and is such that E | a | 2 < + ∞ and E   aa ⊤   2 < + ∞ ; w e denote by | . | the E u clidean norm f or v ectors and F rob enius n orm for matrices. W e assum e all the matrices in (2.2) and (2.3) are real and w e restrict our considerations to the case when all eigen v alues λ ( A ) of A lie inside the u n it disk on th e complex plane (that is, the sp ectral r adius of A is ρ ( A ) < 1). Moreo v er, w e assume that the system is stable in mean-square sense for u t ≡ 0. V arious sufficien t conditions of this stabilit y can b e foun d in [7–12, 15, 17, 18, 21 ] and other wo r ks. F or random x t , y t ∈ C n w e denote the inn er p ro du ct ( x . , y . ) b y ( x . , y . ) = + ∞ P t =0 E x ⊤ t y t and the norm by k x . k = p ( x . , x . ). F u rthermore, we write k x . k 1 = + ∞ P t =0 E | x t | 1 where | x | 1 is the l 1 -norm | x | 1 = P i | x i | of a vect or x or an en tryw ise l 1 -norm | x | 1 = P ij | x ij | of a matrix x . 3 Main Results Condition 3.1. Ther e exist symmetric matric es H and Θ in C n × n satisfying A ⊤ H A − H + Θ = 0 , (3.5) Θ − C ⊤ H C − G = 0 . (3.6) Let Θ b e the matrix satisfying Condition 3.1. Consider the hermitian form F : C n × C m 7→ R giv en b y F ( x, u ) = x ∗ Θ x + 2 Re x ∗ γ u + u ∗ Γ u. (3.7) Let g : C 7→ C n × n b e the matrix-v alued function g ( z ) = ( z I − A ) − 1 , (3.8) W e denote the unit circle by ζ = { z ∈ C : | z | = 1 } . The f ollo wing Theorem establishes necessary and sufficien t conditions for the existence of optimal u o for the problem (2.1 )-(2.4 ) Theorem 3.1. If ther e exists exists a u o ∈ U such that Φ ( u o ) ≤ Φ ( u ) , for al l u ∈ U then 3 i) it is ne c essary that F ( g ( z ) bu, u ) ≥ 0 , ( ∀ z ∈ ζ , ∀ u ∈ C m ) . (3.9) ii) F urthermor e, if ther e exists a δ > 0 such that F ( g ( z ) bu, u ) ≥ δ | u | 2 2 , ( ∀ z ∈ ζ , ∀ u ∈ C m ) , (3.10) then u o is unique (up to e quivalenc e). Theorem 3.1 ab o ve is an analog of KS Lemma for discrete-time optimal sto c h astic control problem (2.1)-(2.4). This is a discrete time ve r sion of a cont inuous-time result obtained in [6] for the case wh en γ = 0 and in Chapter 5 of [16]) for the general γ . 3.1 Pro of of Theorem 3.1 Lemma 3.1. If u t ∈ U , then sup t ≥ 0 E | x t | 2 < + ∞ for the solution of system (2.3) - (2.4) . Pr o of. Let µ t = E x t , (3.11) M t = E x t x ⊤ t (3.12) F rom (2.3)-(2.4) and (3.11), w e ha ve µ t +1 = Aµ t + bu t t = 0 , 1 , 2 , . . . , (3.13) µ 0 = E a. (3.14) Note that | E a | 2 = n P i =1 ( E a i ) 2 ≤ n P i =1 E a 2 i = E | a | 2 < + ∞ . Thus, usin g the f act that u t ∈ U and ρ ( A ) < 1, it follo ws fr om (3.13)-(3.14) that k µ t k < + ∞ . F rom (2.3)-(2.4) and (3.12), w e ha ve M t +1 = AM t A ⊤ + Aµ t u ⊤ t b ⊤ + bu t µ ⊤ t A ⊤ + bu t u ⊤ t b ⊤ + C M t C ⊤ , (3.15) M 0 = E aa ⊤ . (3.16) Let Q t = Aµ t u ⊤ t b ⊤ + bu t µ ⊤ t A ⊤ + bu t u ⊤ t b ⊤ . L et us d enote the j -th colum of a matrix D by D ( j ) . 4 W e define the v ectors q t , m t ∈ C n 2 as q t =                 Q (1) t Q (2) t . . . Q ( n ) t                 , m t =                 M (1) t M (2) t . . . M ( n ) t                 . (3.17) The v ectors q t and m t are formed by stac king up the columns of the matrices Q t and M t , resp ectiv ely . Set A = A ⊗ A + C ⊗ C (w h ere ⊗ d enotes the Kronec ker pro duct). W e can th en rewrite (3.1 5 ) as m t +1 = A m t + q t . (3.18) Note that the system in (3.18) is of dimension n 2 , how ev er, due symmetry , it can b e reduced to a sys tem of dimension n 2 + n 2 . The assumption that the sy s tem (2.3 )-(2.4) is stable in the mean-square sense for u t = 0, is equiv alen t to m t b eing stable for q t = 0, which is true if and only if the sp ectral radius of A is ρ ( A ) < 1. F rom the solution of (3.18), we can s h o w, using H¨ older’s inequalit y and Y oung’s theorem, that k m t k 1 < + ∞ , therefore sup t ≥ 0 E | x t | 2 < + ∞ . This comp oletes the pro of of Lemma 3.1. It follo ws from Lemma 3.1 that the Z -transform , ˆ x ( z ), of x t , exists, and it’s radius of con ve r gence con tains the unit circle, ζ . If w e set x t = 0, u t = 0 for all t < 0 we can then tak e the Z -transform of the s y s tem (2.3) -(2.4) and obtain ˆ x ( z ) = z g ( z ) a + g ( z ) b ˆ u ( z ) + g ( z ) C ∞ X t = −∞ ξ t +1 x t z t . (3.19) Let D b e an n × n r eal symmetric matrix and let T 7→ R n × n × R n × n b e defined by T ( D ) = 1 2 π i I ζ C ⊤ g ( z ) ⊤ D g ( z ) C 1 z dz . (3.20) Lemma 3.2. Condition 3.1 is satisfie d if and only if Θ satisfies G = Θ − T (Θ) . (3.21) Pr o of. Supp ose there exists a Θ suc h that (3.21) holds. Let H = 1 2 π i I ζ g ( z ) ⊤ Θ g ( z ) 1 z dz . (3.22) 5 It follo ws f r om P arsev al’s identit y that H = + ∞ P t =0  A ⊤  t Θ A t = Θ + A ⊤ H A . It therefore follo ws from (3.21) that G = Θ − C ⊤ H C . Hence Condition 3.1 is satisfied. Con versely , sup p ose (3.5) h olds, then + ∞ P t =0  A ⊤  t H A t − + ∞ P t =0  A ⊤  t +1 H A t +1 = + ∞ P t =0  A ⊤  t Θ A t . It follo ws from Parsev al’s Iden tit y that H = 1 2 π i I ζ g ( z ) ⊤ Θ g ( z ) 1 z dz and it follo w s f r om (3.6) and (3.20) that G = Θ − T (Θ). T hus (3.21) holds. T his completes the pr o of of Lemma 3.2. Lemma 3.3. If the system (2.3 ) - (2.4) is stable in the me an-squar e sense for u t ≡ 0 then Condition 3.1 holds. Pr o of. Let us d enote the i -th column of a matrix D by D .,i , let the matrices H and G b e as in Condition 3.1 and let h =  H ⊤ ., 1 , . . . , H ⊤ .,n  , θ =  Θ ⊤ ., 1 , . . . , Θ ., n ⊤  and g =  G ⊤ ., 1 , . . . , G ⊤ .,n  . Let A 1 = A ⊤ ⊗ A ⊤ − I n 2 and A 2 = − C ⊤ ⊗ C ⊤ , where I n 2 is the n 2 × n 2 iden tity matrix. W e can rewrite (3.5)-(3 .6 ) as       A 1 I n 2 A 2 I n 2             h θ       =       0 g       . (3.23) Please n otice that the sys tem in (3.23) w ould b e degenerate if and only if A 1 = A 2 ; ho wev er, this would require that A = A ⊗ A + C ⊗ C = I n 2 whic h w ould violate the assumption that the matrix A from (3.18) satisfies ρ ( A ) < 1 (which is equiv alen t to the requiremen t that the system (2.3 )-(2.4) b e s table in the mean-squ are sense for u t = 0). Therefore, if th e mean-squ are stabilit y is satisfied, we can assume that the s y s tem in (3.23) alw a ys has a solution,  h ⊤ , g ⊤  ⊤ . Hence matrices H and G exist (that is, Condition 3.1 holds). This pro v es Lemma 3.3. It follo ws f r om Lemma 3.2 that G = Θ − T (Θ). Therefore, if w e set x t = 0 and u t = 0 for t < 0, we can rewrite (2.1) as Φ ( u . ) = + ∞ X t = −∞ E F ( x t , u t ) − + ∞ X t = −∞ E x ∗ t T (Θ) x t (3.24) Let the matrix-v alued fun ction Π : C 7→ C m × m b e defined by Π ( z ) = b ⊤ g ( z ) ⊤ Θ g ( z ) b + b ⊤ g ( z ) ⊤ γ + γ ⊤ g ( z ) b + Γ , (3.25) 6 and let ( ˆ u ( z ) , R ˆ u ( z )) = 1 2 π i I ζ ˆ u ( z ) ∗ Π ( z ) ˆ u ( z ) 1 z dz , (3 .26) ( r , ˆ u ( z )) = 1 2 π i I ζ E z a ⊤ g ( z ) ⊤ [ Gg ( z ) b + γ ] ˆ u ( z ) 1 z dz , (3.27) ρ = 1 2 π i I ζ E a ⊤ g ( z ) ⊤ Gg ( z ) a 1 z dz . (3.28) It follo ws f rom (3.26)-(3.28) and Parsev al’s iden tity that w e can rewr ite (3.24) as Φ ( u . ) = ( ˆ u ( z ) , R ˆ u ( z )) + ( r , ˆ u ( z )) + ρ. (3.29) Th us. Φ ( u . ) is a quadratic form in ˆ u ( . ). Consider the deterministic con trol problem b elo w. Minimize Φ 1 ( u . ) = + ∞ X t =0 [ y ∗ t Θ x t + 2 Re y ∗ t γ u t + u ∗ t Γ u t ] (3.30) o v er the set U = ( u t ∈ R m : + ∞ X t =0 | u t | 2 < + ∞ ) (3.31) sub ject to y t +1 = Ay t + bu t , t = 0 , 1 , 2 , . . . (3.32) y 0 = E a. (3.33) Here y t is an n -v ector of states and u t is an m -vec tor of controls. Let matrices G , γ , Γ, A , and b and the vec tor a ha ve the same p rop erties as in the sto chastic optimizatio n p roblem (2.1)-(2.4) ab o ve, and let the matrix Θ b e su c h that (3.21) is satisfied. Using Parsev al’s iden tity , w e can rewrite (3.3 0 ) as Φ 1 ( u ( . )) = ( ˆ u ( . ) , R 1 ˆ u ( . )) + ( r 1 , ˆ u ( . )) + ρ 1 , where ( ˆ u ( z ) , R 1 ˆ u ( z ) ) = 1 2 π i I ζ ˆ u ( z ) ∗ Π ( z ) ˆ u ( z ) 1 z dz , (3.34) ( r 1 , ˆ u ( z ) ) = 1 2 π i I ζ E z a ⊤ g ( z ) ⊤ [ Gg ( z ) b + γ ] ˆ u ( z ) 1 z dz , (3.35) ρ 1 = 1 2 π i I ζ E a ⊤ g ( z ) ⊤ Gg ( z ) a 1 z dz . (3.36) Theorem 3.2. A n optimal c ontr ol u o t for the sto chastic optimization pr oblem (2.1) - (2.4) exists if and only if an optima l c ontr ol for the deterministic optimization pr oblem (3.30) - (3.33) exists. F u rthermor e, if (3.10) holds then the optimal c ontr ols in opt imization pr oblems (2.1) - (2.4) and (3.30) - (3.33) ar e identic al and unique to within e quiv alenc e. 7 Pr o of. Note that, the necessary and sufficien t conditions for the existence of optimal u o that minimizes th e qu adratic form ( u, R u ) + ( r , u ) + ρ dep end on R and r . Moreo ve r , the optimal u o , when it exists, is giv en by the solution to Ru o + r = 0. W e can see from (3.2 6 )-(3.28) and (3.34)-(3.36) that R 1 = R and r 1 = r for the fun ctionals Φ and Φ 1 . I t therefore follo w s that the solution to (2.1)-(2.4) exists if and only if the solution to (3.30)-(3.3 3 ) exists. F ur th ermore, if (3.10) holds, it follo ws from the results from [25, 26], that the optimal con trol f or (3.30)-(3.33) exists and is u nique. This completes the pr o of of Theorem 3.2. It follo ws th at if the optimization prob lem (3.30)-(3.33) has an optimal solution then (3.9) m us t h old. F urthermore if (3.10) holds, it follo ws that the solution exists and is unique (up to to within equiv alence). Hence the pr o of for Theorem 3.1 follo ws from Theorem 3.2. Remark 3.1. If C = 0 in the pr oblem (2.1 ) - (2.4) , then the r e quir ement that the system (2.3) - (2.4) b e stable in the me an-squar e sense f or u t = 0 wil l b e e quivalent to r e quiring that the matrix A satisfy ρ ( A ) < 1 . In addition, if we set Θ = G then (3.21) holds. Th er efor e Condition 6.1 is satisfie d and F ( x , u ) = x ∗ Gx + 2 Re x ∗ γ x + u ∗ Γ u , and The or em 3.1 wil l b e the same as the r esults fr om [25, 26] with x 0 = E a . 3.2 Numerical Algorithm In this section we p ro vide a Matlab co de that tak es m atrices G , A and C as inpu ts, then c hecks if the sy s tem is stable in the m ean-squ are sense. I f it is stable, the p rogram calculates matrices Θ and H . %------- -------- ----------------------------------------------------- %FILE NAME: numeric s.m %DESCRIP TION: Chec k if the discrete-ti me Linear Quadrat ic Cont rol % Problem is Solvable. That is, we calculate H and Theta % that satisfy: {A’HA-H +Theta=0 , Th eta-C’HC -G=0} %INPUTS: Matric es G, A, C %OUTPUT: Matirx Theta, H %------- -------- ----------------------------------------------------- function [Theta , H] = nume rics1(G, A, C) %Verify that the inputs are all square matrices of the same dimensi on s1=size( G); s2=size(A); s3=si ze(C); if((s1(1 )~=s2(1) )|(s1(1)~=s3(1))|(s2(1)~=s3(1))|... 8 (s1(2)~= s2(2))|( s1(2)~=s3(2))|(s2(2)~=s3(2))|... (s1(1)~= s1(2))|( s2(1)~=s2 (2))|(s3(1)~=s3(2))) disp(’ER ROR! Dimension Mismatch’); return; end; %------- -------- ----------------------------------------------------- %Get the symmet ric part of G G=0.5*(G +G’); %------- -------- ----------------------------------------------------- %Verify that the spectral radius of A is less than 1 if(max(a bs(eig(A ))) >= 1) disp(’Ma trix A is not convergent’ ); return; end; %------- -------- ----------------------------------------------------- %Verify that the system is Exponen tial Bound ed in the mean-square sense Big_A = kron(A, A)+kron(C ,C); if(max(a bs(eig(B ig_A))) >= 1) disp(’Th e system is not EMS stable’); return; end; %------- -------- ----------------------------------------------------- %Solve the system, i.e. Calculate matrice s H and Th eta A1=kron( A’,A’)-e ye(size(A’).^2); B1=eye(s ize(A’). ^2); A2=-kron (C’,C’); B2=eye(s ize(A’). ^2); M=[A1, B1; A2, B2]; v=[zeros (size(G( :)));G(:)]; solution =M\v; theta=so lution(l ength(solution)/2+1:length(solution)); h=soluti on(1:len gth(solution)/2); %------- -------- ----------------------------------------------------- 9 %Return outputs H and Theta and Terminate the program Theta=re shape(th eta,size(A)); H=reshap e(h,size (A)); References [1] V. 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