Dynamical observers design for discrete-time descriptor linear systems

Transcription

1 Memorias del XVI ongreso Latinoamericano de ontrol Automático LA 24 Octubre ancún Quintana Roo México Dynamical observers design for discrete-time descriptor linear systems G.-L. Osorio-Gordillo M. Darouach.-M. Astorga-Zaragoza L. Boutat-Baddas Tecnológico Nacional de México - NIDT Interior Internado Palmira S/N ol. Palmira uernavaca Mor. México ( gloriaosorio@cenidet.edu.mx astorga@cenidet.edu.mx). RAN-NRS (UMR 739) Université de Lorraine IUT Longwy 86 Rue de Lorraine 544 osnes et Romain France ( mohamed.darouach@univ-lorraine.fr). Abstract: This paper addresses the problem of observer design for discrete-time descriptor systems by using a new concept of dynamical observer. The advantage of this new concept is the structure of this dynamical observer which is more general than the Proportional-Integral observer and the Proportional observers. The sufficient and necessary conditions for the stability of the proposed dynamical observer are given in a set of linear matrix inequalities (LMIs). A numerical example is given to show the applicability of the present approach. eywords: Dynamical observer Disrete-time descriptor systems LMI.. INTRODUTION This paper concerns the dynamical observers design for discrete-time descriptor systems. An observer is a dynamical system which uses the available information on the inputs and outputs to reconstruct the unmeasured states of the system. A new structure of the observers known as dynamical observers was developed by Goodwin and Middleton (989) and by Marquez (23). This structure presents an alternative state estimation which can be considered as more general than Proportional Observers (PO) and Proportional-Integral Observers (PIO). Which can be only considered as particular cases of this structure. The idea of including additional dynamics in the observer was presented by Goodwin and Middleton (989). Observers design for discrete-time descriptor systems were presented by Wu et al. (29) by using the PIO approach. In Darouach et al. (2) and Wang et al. (22) a PO design for discrete-time nonlinear descriptor systems was treated by using a linear matrix inequality (LMI) approach. In Lin (22) a PO design for discrete-time rectangular descriptor systems with time varying delay was presented. In Liying and Zhaolin (24) the authors present a PO design for discrete-time non linear descriptor systems with unknown input where the simultaneous state and unknown input estimation is possible by extending the state vector with the unknown input vector. Descriptor systems also known as singular or differentialalgebraic systems are used to describe several real systems having complex interrelations between the state variables. These systems were introduced by Luenberger (977) from a control theory point of view and since great efforts have been made to investigate singular systems theory and its applications (see Müller and Hou (993); Müller (25); Liu et al. (28); Boulkroune et al. (29); Darouach (29); Zhou and Lu (29); Darouach (22); Araujo et al. (22)). The main contribution of this paper is that this new observer structure is more general than those presented in Darouach et al. (2) and Wu et al. (29). This observer is composed by both a dynamical part and a statical part. A numerical example is given to illustrate our approach. 2. PRLIMINARIS In this section we shall present some basic results which are used in the sequel of this paper. We shall use the following notations: The symbol A T denotes the transpose of the matrix A. A + denotes any generalized inverse of the matrix A i.e. it verifies AA + A = A. I n denotes an identity matrix with dimensions n n I denotes an identity matrix with appropriate dimensions denotes a scalar or matrix zero with appropriate dimensions. The symbol ( ) denotes the transpose elements in the symmetric positions ones nm denotes a matrix with dimensions n m with all elements equal to one. The symbol denotes a maximal row rank matrix such that =. The Schur complement Lemma is needed to deduce an LMI feasible problem Lemma. (Schur complement) Given a symmetric matrix S S S = 2 S2 T the following conditions are equivalent: S 22 () S < ; (2) S < S 22 S2S T S 2 < ; (3) S 22 < S S 2 S22 ST 2 <. In Section 5 we use the following theorem to solve LMIs Theorem. (Skelton et al. 998) Let matrices B D = D T be given then the following statements are equivalent: 23

2 LA 24 Octubre ancún Quintana Roo México () There exists a matrix X satisfying BX + (BX ) T + D <. (2) The following two conditions hold B DB T < or BB T > T D T T < or T >. Suppose that the statement 2 holds. Let r b and r c be the ranks of B and respectively and (B l B r ) and ( l r ) be any full rank factors of B and i.e. B = B l B r = l r. Then matrix X in statement is given by X = B r + + l + Z B + r B r Z l + l where Z is an arbitrary matrix and = R Bl T ϑr T ( r ϑr T ) + S /2 L( r ϑr T ) /2 S =R R Bl T ϑ ϑ T r ( r ϑr T ) r ϑ B l R where L is an arbitrary matrix such that L < and R is an arbitrary positive definite matrix such that ϑ = (B r R Bl T D) >. 3. PROBLM FORMULATION onsider the linear discrete-time descriptor system of the form x(k + ) =Ax(k) + Bu(k) () =x(k) where x(k) R n is the semi state vector u(k) R m is the input and R p represents the measured output vector. Matrices R n n A R n n B R n m and R p n. Let rank() = r n i.e. matrix may be singular and let R r n be a full row rank matrix such that = in this case r = n r. Assumption. rank = n Remark. Assumption is equivalent to the impulse observability (Dai ). i.e. A rank = rank() + n. Now let us consider the following dynamical observer for system () ζ(k + ) =Nζ(k) + Hv(k) + F + Ju(k) (2) v(k + ) =Sζ(k) + Lv(k) + M (3) ˆx(k) =P ζ(k) + Q (4) where ζ(k) R q represents the state vector of the observer v(k) R v is an auxiliary vector and ˆx(k) R n is the estimate of x(k). Remark 2. a The observer (2)-(4) is in a general form and generalizes the existing ones In fact: For H = S = M = and L a stability matrix let matrices F and Q be partitioned according to the partition of as F = F a and Q = Q a respectively then the following observer is obtained ζ(k) =Nζ(k) + F a + Ju(k) ˆx(k) =P ζ(k) + Q a which is the form used for the PO for descriptor systems (Darouach et al. 2). For L = and let S = and M = Q + I then the following observer is obtained ζ(t) =Nζ(t) + Hv(t) + F v(k) = ˆx(k) ˆx(k) =P ζ(k) + Q y(t) y(t) + Ju(k) which is in the form used for the unknown input PIO for descriptor systems. The order of the observer is q n when q = n p the reduced order observer is obtained and for q = n the full order one. Now we can give the following lemma Lemma 2. There exists an observer of the form (2)-(4) for the system () if and only if the following two statements hold I. There exists a matrix T of appropriate dimensions such that the following conditions are satisfied (a) NT + F T A = (b) J = T B (c) M + ST = (d) P Q T = I n N H II. The matrix is a stability matrix. S L Proof. Let T R q n be a parameter matrix and define ε(k) = ζ(k) T x(k) then its dynamic is given by ( ) ε(k + ) =Nε(k) + NT T A + F x(k) (5) + (J T B)u(k) + Hv(k) by using the definition of ε(k) equations (3) and (4) can be written as ( ) v(k + ) =Sε(k) + M + ST x(k) + Lv(k) (6) ˆx(k) =P ε(k) + P Q T x(k) (7) Now if conditions (a)-(d) of Lemma 2 are satisfied the following observer error dynamics is obtained from (5) and (6) 24

3 LA 24 Octubre ancún Quintana Roo México and from (7) ε(k + ) N H ε(k) = v(k + ) S L v(k) }{{}}{{}}{{} ϕ(k+) A ϕ(k) (8) ˆx(k) x(k) =P ε(k) e(k) =P ε(k) (9) in this case if A is a stability matrix then lim e(k) =. k The problem of the dynamical observer is reduced to determine the matrices N F J H L M S P Q and T such that Lemma 2 is satisfied. 4. PARAMTRIZATION OF TH DYNAMIAL OBSRVR In this section we shall present a parameterization of the dynamical observer matrices by solving the constrained Sylvester equations (a)-(d) of Lemma 2. Define the following matrices: Ω = Iq P = Σ + Iq Iq N = T AΣ + N 2 = T 2 AΣ + T = RΩ + In Iq N 3 = (I n+r +p ΣΣ + ) T 2 = (I n+r +p ΩΩ + In ) 2 = (I n+r +p ΩΩ + ) = RΩ + I r+p = T AΣ + 2 = T I 2 AΣ + r+p 3 = (I q+r+p ΣΣ + ) F 2 = T 2 AΣ + Q = Σ + F I = T AΣ + r+p F 3 = (I q+r+p ΣΣ + ) and let R R q n be a full rank matrix and define the matrix Σ = R such that rank(σ) = n. The following lemma gives the general form of the matrices T S M P Q N and F. Lemma 3. Under assumption the general form of matrices T S M P Q N and F are T =T Z T 2 () S = Y N 3 () M = Y F 3 (2) P =P Y 2 N 3 (3) Q =Q Y 2 F 3 (4) N =N Z N 2 Y 3 N 3 (5) F =F Z F 2 Y 3 F 3 (6) where Z Y Y 2 and Y 3 are arbitrary matrices of appropriate dimensions. Proof. onditions (c) and (d) can be rewritten as S M T P Q = I n (7) the necessary and sufficient condition for (7) to have a solution is rank T T = rank = n (8) I n From the definition of matrices Σ and Ω and by considering rank = rank(ω) condition (8) is equivalent to the Ω Σ existence of two matrices T R q n and R q (r+p) such that T + = R (9) where R is an arbitrary matrix such that rank(σ) = n. quation (9) can also be rewritten as T Ω = R (2) Ω since rank = rank(ω) the general solution to (2) is R given by T =T Z T 2 (2) = Z 2 (22) where Z is an arbitrary matrix of appropriate dimensions. By considering (9) we can get T Iq = Σ (23) I p replacing (23) in (7) we obtain S M P Q Iq Σ = I p I n (24) Since Σ is of full column rank the general solution of (24) is given by S = Y N 3 (25) M = Y F 3 (26) P =P Y 2 N 3 (27) Q =Q Y 2 F 3 (28) where Y and Y 2 are arbitrary matrices of appropriate dimensions. Replacing T from (9) in condition (a) we obtain N ( R ) + F =T A NR + =T A (29) where = F N equation (29) can be expressed also as N Σ = T A (3) the general solution to (3) is given by N =N Z N 2 Y 3 N 3 (3) = Z 2 Y 3 3 (32) where Z and Y 3 are arbitrary matrices of appropriate dimensions. As N T and are known matrices we can deduce the form of F as follows F = F Z F 2 Y 3 F 3 (33) 25

4 LA 24 Octubre ancún Quintana Roo México 5. DYNAMIAL OBSRVR DSIGN In this section a synthesis method for a dynamical observer given by (2)-(4) is presented. This method is obtained such that the matrix A given in (8) is a stability matrix. By using () and (5) the observer error dynamics (8) can be rewritten as ϕ(k + ) = (A YA 2 )ϕ(k) (34) N Z where A = N 2 Y3 H Y = and A Y L 2 = N3. I v The following theorem gives the LMI conditions that allow the determination of all dynamical observer matrices and which guarantee the stability of matrix A. Theorem 2. Under Assumption there exist two parameter matrices Y and Z such that the system (34) is asymptotically stable if there exist some symmetric positive definite matrices X and X 2 and a matrix W such that the following LMIs are satisfied X 2 X > (35) N3 T X N3 T T ( ) ( ) X N N3 T T W N 2 N3 T T X ( ) < (36) X N N3 T T W N 2 N3 T T X X 2 ϑ =(B r R Bl T D) > (39) S =R R Bl T ϑ ϑ T r ( r ϑr T ) r ϑ B l R (4) X ( ) ( ) ( ) X with D = X 2 X N W N 2 X ( ) B = I q+v X N W N 2 X X 2 N3 = and matrices L R and Z are arbi- I v trary matrices of appropriate dimensions satisfying R > and L <. Matrices l r B l and B r are full rank matrices such that = l r and B = B l B r respectively. Proof. onsider the following Lyapunov function V (ϕ(k)) = ϕ(k) T Xϕ(k) (4) X X where X = > then by using the complement X X 2 Schur lemma since X = X T > we obtain X 2 X >. Now the time difference of V (ϕ(k)) along the solution of (2)-(4) is V (ϕ(k)) =ϕ(k) T (A YA 2 ) T X(A YA 2 )ϕ(k) ϕ(k) T Xϕ(k) (42) The inequality V (ϕ(k)) < holds for all ϕ(k) if and only if (A YA 2 ) T X(A YA 2 ) X <. (43) By using Lemma for (43) gives X > and X (A YA 2 ) T X <. (44) X(A YA 2 ) X which can be rewritten as BX + (BX ) T + D < (45) where X = XY B = = A I 2 and D = q+v X A T X which are the same as the ones defined in XA X Theorem 2. According to Theorem the inequality (45) is equivalent to B DB T < (46) T D T T < (47) with B = I q+v and T N T = 3. By using I q+v the definition of matrices B and D the inequality (46) is equivalent to (35) and by using matrices T D and W (47) is equivalent to (36). The following algorithm allows the determination of all the observer matrices to summarize the above results. Algorithm. step. hoose the observer order q and a matrix R R q n In this case Z = X W such that rank(σ) = n. and matrix Y is parametrized step 2. ompute matrices N as follows N 2 N 3 T T 2 2 Y = X (B r + + l + Z B r + B r Z l + l ) (37) 2 3 and P defined in Section 4. step 3. Solve the LMI (35) and (36) to find X and Z. where step 4. Find R > such that (39) be positive definite. = R Bl T ϑr T ( r ϑr T ) + S /2 L( r ϑr T ) /2 (38) step 5. Find L and Z such that L < to solve equations (38) and (4) then obtain matrix Y as in (37). step 6. ompute the matrices of the dynamical observer (2)- (4): N H F J S L M P and Q by using (5) to compute N (37) to compute H and L ()-(4) to compute S M P and Q by taking Y 2 = F is defined in (6) and J is defined in Lemma 2 6. NUMRIAL XAMPL In order to illustrate our results let us consider the following descriptor system described by () where.7.4 = A =.5 B = and =. By following the algorithm of Section 5 we have step. An observer with order q = 2 was chosen with a matrix R = such that rank(σ) = 4. step 2. The following matrices were calculated N = T.8.9 = N 2 = T 2 =

5 LA 24 Octubre ancún Quintana Roo México N 3 = P = = = = = and = step 3. By using the LMI toolbox of MATLAB we solved the inequalities (35) and (36) to get Z = and X = step 4. By considering R = I 4. to solve (39) as ϑ = step 5. By considering L = ones 43. and Z = to solve (38) and (4) we obtain Y = step 6. Finally we compute the matrices of the dynamical observer by considering Y 2 = then we obtain the following dynamical observer In order to evaluate the performance of our observer a measurement noise w(k) was considered in the output then the disturbed outputs become y (k) = x 2 (k)+x 3 (k)+ w(k) and y 2 (k) = x 4 (k) + w(k). The results of the simulation are depicted in Fig. -6. Fig. shows the input u(k) behavior. Fig. 2 shows the measurement noise behavior. Figs. 3-6 show the system states and their estimations. It can be seen that the observer have an acceptable performance despite the measurement noise Fig.. Input u(k) behavior Fig. 2. Measurement noise w(k) behavior Fig. 3. stimation x (k) (black line: original state; blue u(k) w(k) x(k) ˆx(k) x2(k) ˆx2(k) ζ(k + ) = ζ(k) + v(k) u(k) v(k + ) = ζ(k) + v(k) ˆx(k + ) = ζ(k) Fig. 4. stimation x 2 (k) (black line: original state; blue Fig. 5. stimation x 3 (k) (black line: original state; blue x3(k) ˆx3(k) 27

6 LA 24 Octubre ancún Quintana Roo México Fig. 6. stimation x 4 (k) (black line: original state; blue 7. ONLUSION In this paper a dynamical observers design for disretetime descriptor linear systems have been presented. Since regularity assumption is not required the design procedure can be applied to square systems but also to rectangular systems which is a more general representation of linear systems and square systems can be viewed as a particular case. The stability of the observer was proved by the Lyapunov method and it is guaranteed by solving a set of LMIs. An algorithm is described in order to facilitate the observer matrices computation. A numerical example was presented to show the performances of our approach. RFRNS Araujo J.M. Barros P.R. and Dorea..T. (22). Design of observers with error limitation in discretetime descriptor systems: A case study of a hydraulic tank system. I Transactions on ontrol Systems Technology Boulkroune B. Darouach M. Zasadzinski M. Gill S. and Fiorelli D. (29). A nonlinear observer design for an activated sludge wastewater treatment process. Journal of Process ontrol Boyd S. Ghaoui L. Feron. and Balakrishnan V. (994). Linear Matrix Inequalities in System and ontrol Theory. Society for Industrial and Applied Mathematics (SIAM). Dai L. (988). Observers for discrete singular systems. I Transactions on Automatic ontrol Dai L. (989). Singular control systems. In Lecture Notes in ontrol and Information Sciences. Springer. Darouach M. (29). H unbiased filtering for linear descriptor systems via LMI. I Transactions on Automatic ontrol Darouach M. (22). On the functional observer for linear descriptor systems. Systems & ontrol Letters Darouach M. Boutat-Baddas L. and Zerrougui M. (2). H observers design for a class of discrete time nonlinear singular systems. In 8th Mediterranean onference on ontrol & Automation. Goodwin G.. and Middleton R.H. (989). The class of all stable unbiased state estimators. Systems & ontrol Letters Lin Y. (22). Observer design for rectangular discretetime singular systems with time-varying delay. International Journal of the Physical Sciences Liu P. Zhang. Yang X. and Yang L. (28). Passivity and optimal control of descriptor bilogical complex systems. I Transactions on Automatic ontrol x4(k) ˆx4(k) Liying S. and Zhaolin. (24). Observer design for discrete-time descriptor systems with unknown inputs and inputs estimation. In 5th Asian ontrol onference. Luenberger D.G. (977). Dynamic equations in descriptor form. I Transactions on Automatic ontrol Marquez H.J. (23). A frequency domain approach to state estimation. Journal of the Franklin Institute Müller P.. (25). Modelling and control of mechatronic systems by the descriptor approach. Journal of Theorical and Applied Mechanics Müller P.. and Hou M. (993). On the observer design for descriptor systems. I Transactions on Automatic ontrol Skelton R.. Iwasaki T. and Grigoriadis. (998). A unified algebraic approach to linear control design. Taylor & Francis. Wang Z. Shen Y. Zhang X. and Wang Q. (22). Observer design for discrete-time descriptor systems: An LMI approach. Systems & ontrol Letters Wu A.G. Duan G.R. Dong J. and Fu Y.M. (29). Design of proportional-integral observers for discretetime descriptor linear systems. IT ontrol Theory and Applications Zhou L. and Lu G. (29). Detection and stabilization for discrete-time descriptor systems via a limited capacity communication channel. Automatica