Time and Frequency domain design of Functional Filters

21 American Control Conference Marriott Waterfront, Baltimore, MD, USA June 3-July 2, 21 WeA17.3 Time and Frequency domain design of Functional Filters M. Ezzine, M. Darouach, H. Souley Ali and H. Messaoud Abstract This paper proposes both time and frequency domain design of functional filters for linear time-invariant multivariable systems all measurements are affected by disturbances. The order of this filter is equal to the dimension of the vector to be estimated. The time procedure design is based on the unbiasedness of the filter using a Sylvester equation; then the problem is expressed in a singular system one and is solved via Linear Matrix nequalities LM to find the optimal gain implemented in the observer design. The frequency procedure design is derived from time domain results by defining some useful Matrix Fractions Descriptions MFDs and mainly, establishing the useful and equivalent form of the connecting relationship that parameterizes the dynamic behavior between time and frequency domain, given by Hippe in the reduced-order case. A numerical example is given to illustrate our approach.. NTRODUCTON The problem of functional filter design is equivalent to find a filter, that estimates a linear combination of the states of a system using the input and output measurements. The filter has the same order as the functional to be estimated. t has been the object of numerous studies in time domain since the original work of Luenberger 17, 18, first appeared. n most case it is related to the constrained or unconstrained Sylvester equations, see 2 and 19. n addition 1, 1, 11 and 12 give different procedures for designing time domain functional observers, one can view how the Sylvester equation was solved. n frequency domain, there is less literature although it is the basis for most analysis performed on control systems 4, 5, 9. t is well known that both linear optimal state feedback and optimal linear filtering problems can be formulated and solved in the frequency domain. A first solution for the filter transfer matrix was presented by MacFarlane 21, and it was demonstrated in 22 that optimal state feedback low and the optimal filter can be characterized by polynomial matrix that directly parameterize the state feedback control and the observer in the frequency domain. n this framework, a time and frequency domain design procedure of functional filters is proposed. n fact, after using the unbiasedness condition, we propose a new method M. Ezzine is Ecole Nationale d ngénieurs de Monastir, Avenue bn El Jazzar, 519 Monastir, Tunisie and CRAN Nancy Université, UT de Longwy, 186 Rue de Lorraine, 544 Cosnes et Romain, France montassar.ezzine@iut-longwy.uhp-nancy.fr M. Darouach and H. Souley Ali are CRAN Nancy Université, UT de Longwy, 186 Rue de Lorraine, 544 Cosnes et Romain, France souley@iut-longwy.uhp-nancy.fr, darouach@iut-longwy.uhp-nancy.fr H. Messaoud is Ecole Nationale d ngénieurs de Monastir, Avenue bn El Jazzar, 519 Monastir, Tunisie hassani.messaoud@enim.rnu.tn to treat the error dynamics in order to avoid the time derivative of the disturbance w in this error see 2, we transform the problem into a singular one problem. Then a LM approach is used to find the optimum gain implemented in the observer design. Then, based on time domain results, we propose a new functional filter the aid of polynomial approach. The main reason of formulating the results of the time domain in the frequency one is the advantages that it presents for the observer-based control 23. n fact, in this case, the compensator is driven by the input and the output of the system. So only the inputoutput behavior of the compensator characterized by its transfer function influences the properties of the closedloop system. The additional degrees of freedom given by the frequency approach can then be used for robustness purpose for example 23. The organization of this paper is as follows. Section gives assumptions used through this paper and states the functional filtering problem that we propose to solve. Section, presents the first contribution of the paper by giving the design procedure of a functional filter in the time domain. Using the unbiasedness condition, the problem is transformed into a singular problem one in order to avoid to have the derivative of the disturbance in the error dynamics. A LM approach is then applied to optimize the gain implemented in the observer. Then section V presents the second result of the paper by giving a frequency domain description of the time domain functional filter designed in section using useful polynomial MFDs. We mainly establish the equivalent connecting relationship between time and frequency domain approach that will be useful for functional filter design. This can be viewed as a generalization of that of 7 given for the reduced order case. Section V gives a numerical example to illustrate our approach and finally section V concludes the paper.. PROBLEM STATEMENT Consider the following linear time-invariant multivariable system ẋt = Axt + But + D 1 wt 1a zt = Kxt 1b yt = Cxt + D 2 wt 1c xt R n and yt R m are the state vector and the measured output vector of the system, wt R q represents the disturbance vector, ut R p is the control input vector of the system and zt R mz is the unmeasurable outputs to be estimated. A, B, D 1, K, C and D 2 are known constant matrices of appropriate dimensions. 978-1-4244-7425-7/1/$26. 21 AACC 599

Further, it is assumed through the paper that : Assumption 1: rank K = m z and rank C = m Problem : We intend in this paper, to give a time and a frequency domain design of functional filters, that generates an estimate ẑt and ẑs for zt and zs, using the input and output measurements ut and yt. The order of these filters m z is equal to the dimension of the vector to be estimated.. TME DOMAN DESGN Our aim is to design a functional filter of order m z for system 1, of the form : ϕt = N ϕt + Jyt + Hut 2a ẑt = ϕt + Eyt 2b ẑt R mz is the estimate of zt. The estimation error is given by et = zt ẑt 3a So, its dynamics can be written as = ψxt ϕt ED 2 wt 3b ψ = K EC 4 ėt = ψẋt ϕt ED 2 ẇt 5a = Net + ψa JC Nψxt + ψb Hut + ψd 1 JD 2 + NED 2 wt ED 2 ẇt 5b The problem of the filter design is to determine N, J, H and E such that 1, 2 i the filter 2 is unbiased if wt = ii the filter 2 is stable, i.e N is Hurwitz. The unbiasedness of the filter is achieved if and only if the following Sylvester equation holds ψa JC Nψ = 6 H = ψb 7 The Sylvester equation 6 can be written by taking into account 4 as Let NK + NE JC + KA ECA = 8 L = J NE 9 Then, equation 8 can be transformed as N L E K C CA = KA 1 For the resolution of 1, let set N L E = X 11 therefore 1 becomes K C CA = Σ 12 KA = Θ 13 XΣ = Θ 14 This equation has a solution X if and only if Σ rank = rank Σ 15 Θ and a general solution for 14, if it exists, is given by X = ΘΣ + Z ΣΣ + 16 Σ + is a generalized inverse of matrix Σ given by 12 and Z is an arbitrary matrix of appropriate dimensions, that will be determined in the sequel using LM approach. Once matrix X is determined, it is easy to give the expressions of matrices N, L and E. n fact, N = X A 11 = ΘΣ + = A 11 ZB 11 17 B 11 = ΣΣ + L = X A 22 = ΘΣ + 18, 19 = A 22 ZB 22 2 B 22 = ΣΣ + E = X A 33 = ΘΣ + 21, and 22 = A 33 ZB 33 23 B 33 = ΣΣ + 24. 25 6

Hence all filter matrices are determined if and only if the matrix Z is known. So, the dynamics of the estimation error 5b reads and ėt = Net + αwt + βẇt 26 α = ψd 1 JD 2 + NED 2 = α 1 Zα 2 27 α 1 = KD 1 ΘΣ + α 2 = ΣΣ + D 2 CD 1 D 2 CD 1 28 29 β = ED 2 3 = β 1 Zβ 2 31 β 1 = A 33 D 2 32 β 2 = B 33 D 2 33 Notice that the error dynamics 26 contains the derivative of wt. This problem is generally solved by adding an additional constraint on the filter matrices see 2 or by choosing a new type of norm for the system. Here we propose another method which consists in rewriting the error system into a descriptor singular form. Then the following descriptor system is given from equation 26: β ėt ρ 1 t = + N α wt et = et ρ 1 t et ρ 1 t 34a 34b et is the estimation error and ρ 1 t is such that ρ 1 t = wt. Before continuing, let us recall the following result on descriptor systems: Lemma 1: 13 Consider a singular system of the form Fẋ = Ax + Bw 35 z = Cx 36 x is the state, w is the exogenous input and z is a controlled output; matrices F, A, B and C are known. The pair F, A is admissible and G < γ G = CsF A 1 B if and only if there exists X R n n such that 1 F T X = X T F 37 2 A T X + X T A + C T C + 1 γ 2 XT BB T X < 38 Before applying this result on the singular system 34, we consider that the gain matrix Z satisfies the following relation Zβ 2 = 39 in order to avoid an unknown to be designed gain matrix β Z in the singular matrix of system 34. So it exists a matrix Z 1 that satisfies Setting A 2 = and ρ 2 = N α Z = Z 1 β 2 β + 2 4 β = β 2 = χt = = β1 A11 ZB 11 α 1 Zα 2 et ρ 1 t the singular system 34 reads 41 42 43 44 ρ 2 χt = A 2 χt + β 2 wt 45a et = χt 45b Now, a LM approach is used in the following theorem to get the gain matrix Z which parameterizes the filter matrices Theorem 1: The pair ρ 2, A 2 is admissible and Ws = sρ 2 A 2 1 β 2 < γ if and only if there exist X 1 R mz mz, X 2 R q q and Y R mz+2m mz such that the following LMs hold 1 X 1 X T β1 T = 1 X1 T β 1 46 X 1 2 P 11 P 12 P 13 P 14 X T 2 X 2 γ 2 θ = β 2 β + 2 T and < 47 P 11 = A T 11X 1 + X T 1 A 11 B T 11θ T Y Y T θ B 11 P 12 = X1 T α 1 Y T θ α 2 P 13 = α1 T X 1 α2 T θ T Y P 14 = X 2 X T 2 Then the gain Z 1 is given by Z 1 = Y X 1 1 T. 61

Proof 1: Using lemma 1 and the Schur lemma 14 on the singular system 45 yields to 1 Φ 1 = Φ T 1 Φ 1 = ρ T 2X 48 2 Φ 2 + Φ T 2 Φ T 3 T Φ 3 γ 2 < 49 Φ 2 = A T 2X 5 Φ 3 = β2x T 51 X1 By taking X = and using 41, 48 is X 2 equivalent to 46. Then, replacing A 2 and β 2, from 49, by their expressions 42, 43 and using 4, the LM 47 holds, θ = β 2 β 2 + T and Y = Z1 T X 1. Design of the time domain functional filter The different steps of the filter computation in the time domain are summarized in the following design method: 1 Compute Σ and Θ from 12, 13. 2 All known matrices implemented in LMs i.e. A 11, B 11, α 1, α 2, B 33, β 2, A 33 and β 1 are computed using respectively 18, 19, 28, 29, 25, 4, 24, 32. So, θ is known. 3 Resolution of the LMs 46, 47 gives the gain Z 1, so from 4, Z is known. 4 Compute the filter matrix N from 17. 5 Matrix E is also obtained by 23. Therefore ψ is known 4 and H is determined as 7 6 Finally, the time domain representation of the functional filter 2 is known, indeed after computing A 22 and B 22 from 21, 22, L can be easily calculated from 2 and therefore matrix J implemented in time design is computed using 9. V. FREQUENCY DOMAN DESGN The next theorem presents the main result of this section by giving a frequency domain description of the functional filter designed in section : Theorem 2: : Consider the following Matrix Fraction Descriptions MFDs i M s N 1 = D 1 s N x s 52 a M and are arbitrary matrix of dimension m z m z and m m z such that matrix M is of full column rank. ii iii b N is the filter 2 matrix of order m z, and the two polynomial matrix Ds and Nx s are of dimensions m z + m m z + m and m z + m m z respectively. They have the specification to be left coprime. These transfer functions can be calculated from the factorization approach presented by Vidyasager 15. M s N 1 H = D 1 s N u s 53 N u s = N x sh 54 D 1 s Ds = mz+m + M s N 1 J55 Ds is given by 52 and the polynomial matrix Ds parameterizes the dynamics of the functional filter 2 in the frequency domain. This left coprime MFD is a generalization of the connecting relationship that parameterizes the dynamics behavior between time and frequency domain given by Hippe 7 for the reduced order case. Then a frequency domain representation of the functional filter 2 of order m z, m z n, related to system 1 is given by, + ẑs = D 1 M s Ds M D 1 s Ds 1 + ys D 1 s N u s us + Ds E ys 56 The filter denominator matrix Ds satisfies Ds = Ds + N x s mz m z J 57 Proof 2: : The Laplace transform of 2a reads as ϕs = s N 1 Jys + s N 1 Hus58 = s N 1 mz 1 mz m z J ys + s N 1 Hus 59 So, the Laplace transform of the estimated vector ẑt 2b, can be written by taking into account59 as ẑs = s N 1 mz 1 mz m z J ys + s N 1 Hus + Eys 6 62

Or, following the proposed left coprime MFDs 55, we have + s N 1 M mz m z J = and from 53 D 1 s Ds mz+m 61 s N 1 H = M + D 1 s N u s 62 the desired estimate of the functional filter reads + M ẑs = D 1 s Ds mz+m mz 1 M + ys m 1 + D 1 s N u s us + E ys 63 Therefore and in view of, D 1 s Ds = mz+m 64 the frequency domain description 56 holds. The polynomial matrix equation of the filter denominator 57 directly follows from equation 55 using 52. V. NUMERCAL EXAMPLE Consider the system presented in section, 5 2 2 1 4 A =, B =, D 2 1 1 = 7 C = 1 1, K =.67 1 - Time domain functional filter design By applying the proposed algorithm of section, we obtain the following results: 1 2 Σ =.67 1 1 2 4 A 11 =.112,B 11 = α 1 =.136, α 2 = B 33 = 3 For γ = 12.333.22.11,, Θ =.134 65, β 2 =.9944.666.333.2998.21.1 66.333.22.11 67 68 A 33 =.666β 1 =.666 69 Z 1 = 1.698 8.6126 158.7342 7 and therefore Z =.89 8.6783 158.713 71 Consequently, the filter matrices are given by N =.112, E =.666, H =.663, L =.1333 and J =.1325. So, the time domain description of the functional filter of order 1, reads ϕt =.112 ϕt +.1325 yt.663 ut 72a ẑt = ϕt +.666 yt 72b 2 - Frequency domain functional filter design For that, let us choose M = 1 and =, so M is of full column rank. By using the factorization approach given in 15 and the algorithms of 16 about realization of RH matrices satisfying the Bezout identity, we obtain and Ds = s+.112 s+2.112 1 s+2.112 1 N s + 2.112 1 x s = 73 74 one can verify that 52 is satisfied. From 54, we have.663s + 2.112 1 N u s = 75 consequently, from 57 the denominator matrix of the functional filter reads as s+.112.8675 Ds = s+2.112 s+2.112 76 1 Therefore, following 63, and by taking into account 73, 75 and 76, ẑs can be rewritten as ẑs = 1.1325 s+.112 1 1 ys 1 1.663 us +.666 ys 77 s +.112 so, its readily follows that the desired frequency description 56 of the functional filter is determined. n other hand, we propose to draw the frequency estimation error. For that, 77 reads.666s +.13322s + 2 ẑs = s +.112s + 2 2.663 s +.112 us +.666s +.1332s2 7s 4 s +.112s + 2 2 ws 78 therefore supposing that u =, and using Laplace transform of 1b, the frequency estimation error satisfies the following equation es = zs ẑs 79 =.666s3 +.82s 2 + 2.142s +.5433 s 3 + 4.11s 2 ws + 4.45s +.448 63

The simulations are done an input u =. For the time domain design, the disturbance wt is taken to be of bounded energy and is given by figure 1. We take e = 2.5 as initial condition. Finally, the figures 2 and 3 show the time and frequency domain behavior of the filter and so, the effectiveness of our approach. Magnitude db 3 2 1 1 2 3 36 Bode diagram of the frequency estimation error es V. CONCLUSON A new time and frequency domain design of functional filter for linear multivariable systems is proposed in this paper. The proposed filter has the same order as the functional to be estimated. The time domain procedure is based on the resolution of Sylvester equation and the use of a singular system approach in order to avoid time derivative of the disturbance. LM approach is then used to find the optimum gain implemented in the observer design. An algorithm that summarizes the different step of resolution is given. Then, based on time domain results, the frequency domain description of the functional filter is derived. n fact, we define some useful matrix fraction descriptions and mainly propose a connecting relation between time and frequency domain parameterizations of functional filter, which is equivalent to that proposed by Hippe 7 in the reduced order case. The proposed design procedure has been applied on numerical example and it shows its effectiveness. Fig. 2..35.3.25.2.15.1.5 1 2 3 4 5 6 7 8 9 1 Fig. 1. Amplitude 3.5 3 2.5 2 1.5 1.5 The behavior of the used disturbance wt 2 4 6 8 1 12 Time sec Evolution of the estimation error et designed in time domain Fig. 3. Phase deg 315 27 225 18 1 4 1 3 1 2 1 1 1 1 1 1 2 1 3 Frequency rad/sec The Bode diagram of the transfert function from ws to es 2 : Darouach.M, Zasadzinski.M and Souley Ali.H, Robust reduced order unbiased filtering via LM. Proceedings of the 6th European Control Conference. Porto, Portugal September 21. 4 : Ding.X, Guo.L and Frank.M.P, Parameterization of Linear Observers and ts Application to Observer Design, EEE Transaction on Automatic Control.Processing, 39, 1648-1652 1994. 5 : Ding.X, Frank.M.P and Guo.L, Robust Observer Design Via Factorization Approach, Proceedings of the 29th Conference on Decision and Control Honolulu. Hawa December 199. 7 : Hippe.P, Design of reduced-order optimal estimators directly in the frequency domain,nt.j.control, Vol 5, pp 2599-2614, 1989. 9 : Russell.W.D and Bullock.E.T, A Frequency Domain Approach to Minimal-Order Observer Design for Several Linear Functions of the State, EEE Transaction on Automatic Control.Processing, 22, 6-63 1977. 1 : Zhang.S, Functional observer and state feedback,nt.j.control, Vol 46, pp 1295-135, 1987. 11 : Moora.B.J and Ledwich.F.G, Minimal-order observers for estimating linear functions of a state vector, EEE Transaction on Automatic Control.Processing, 2, 623-632 1975. 12 : O Reilly.J, Observers for Linear Systems,Mathematics in sciense and engineering, Vol.17, New York, NY: Academic press, 1983. 13 : zumi.masubuchi, Yoshiyuki. Kamitane, Atsumi. Ohara and Nobuhide. Suda, H Control for descriptor Systems : A Matrix nequalities Approach, Automatica, Vol 33, No. 4, pp 669-673, 1997. 14 : Boyd. S, El Ghaoui. L, Feron. E, and BalaKrishnan. V, Linear Matrix nequality in systems and control theory, SAM, Philadelphia, USA, 1994. 15 : Vidyasagar.M, Control Systems Synthesis: A Factorization Approach,M..T,Press, Cambridge, MA,1985. 16 : Francis.B.A, A Course in H Control Theory. Springer-Verlag, Berlin-New York,1987. 17 : D. G. Luenberger, Observers for multivariable systems, EEE Trans. Automat. Contr., vol. 11, pp. 19197, 1966. 18 : D. G. Luenberger,, An introduction to observers, EEE Trans. Automat. Contr., vol. 16, pp. 59662, 1971. 19 : C. Tsui,, A new algorithm for the design of multi-functional observers, EEE Trans. Automat. Contr., vol. 3, pp. 8993, 1985. 2 : P. Van Dooren,, Reduced order observers: A new algorithm and proof, Syst. Contr. Lett., vol. 4, pp. 243251, 1984. 21 : MacFarlane,A.G.J, Return-difference matrix properties for optimal stationary Kalman-Bucy filter, Proceedings of the nstitution of Electrical Engineers, 118, 373-376 1971. 22 : Anderson.B.D.O and Ku cera.v, Matrix fraction construction of linear compensators, EEE Transaction on Automatic Control.Processing, 3, 1112-1114 1985. 23 : Hippe.P and Deutscher.J, Design of observer-based compensators : From the time to the frequency domain, Springer-Verlag, London, 29. REFERENCES 1 : Darouach.M, Existence and Design of Functional Observers for Linear Systems, EEE Transaction on Automatic Control.Processing, 45, 94-943 2. 64