Design of a full order H filter using a polynomial approach

Transcription

1 American ontrol onference Marriott Waterfront, Baltimore, M, USA June -July, WeA7. esign of a full order H filter using a polynomial approach M. Ezzine, H. Souley Ali, M. arouach and H. Messaoud Abstract The paper deals with a frequency domain solution to the standard H filtering problem for linear time-invariant multivariable systems where all measurements are affected by disturbances using a polynomial approach. The design procedure is first obtained by considering time domain solution which is related to a Riccati equation, and then due to the connecting relationship that parameterizes the dynamics behavior between time and frequency domain, given by Hippe, the full order filter representation in the frequency domain is obtained. The filter is easy to calculate as it requires the computation of a single gain and it is easily implementable also. I. INTROUTION The problem of observing the state vector of deterministic or stochastic linear time-invariant multivariable system has been a great deal of investigation during the last decades. So, observing or filtering a state vector, centres on the estimation of the states or a linear combination of the states of a system using the input and output measurements. Two performance measures in filtering problem are widely used, the H and H norms. The H filtering performance is chosen in this paper as, in contrast to Kalman filtering, it does not require any knowledge of the noise : it consists on minimizing the energy of the estimation error for the worst possible bounded energy disturbance. Unfortunately, there is less literature on the state estimation in frequency domain, compared to time domain,,,..., although it is the basis for most analysis performed on control systems, 6, 7. And, it 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 8, and it was demonstrated in 9 that the optimal state feedback law and the optimal filter can be characterized by polynomial matrices that directly parameterize the state feedback control and the observer in the frequency domain. In addition, it is important to recall that polynomial matrices method, was first applied to linear control systems by Popov and Rosenbrock. The main reason of formulating the results of the time domain in the frequency one is the advantages that it presents M. Ezzine is with Ecole Nationale d Ingénieurs de Monastir, Avenue Ibn El Jazzar, 9 Monastir, Tunisie and with RAN Nancy Université, IUT de ongwy, 86 Rue de orraine, osnes et Romain, France montassar.ezzine@iut-longwy.uhp-nancy.fr H. Souley Ali and M. arouach are with RAN Nancy Université, IUT de ongwy, 86 Rue de orraine, osnes et Romain, France souley@iut-longwy.uhp-nancy.fr, darouach@iut-longwy.uhp-nancy.fr H. Messaoud is with Ecole Nationale d Ingénieurs de Monastir, Avenue Ibn El Jazzar, 9 Monastir, Tunisie hassani.messaoud@enim.rnu.tn for the observer-based control 7. In fact, in this case, the compensator is driven by the input and the output of the system. So only the input-output behavior of the compensator (characterized by its transfer function) influences the properties of the closed-loop system. The additional degrees of freedom given by the frequency approach can then be used for robustness purpose for example 7. Motivated by this fact, we intend in this paper to solve the H filtering in the frequency domain by the use of a polynomial approach, where all measurements y(t) of the considered system are corrupted by the disturbance w(t). In fact, we propose to find the optimal frequency domain filter that is stable and minimizes the energy gain of the process from disturbance w(t) to estimation error e(t). The style and notation adopted here are similar to those used by. However, the reader will only require the form of the filtering Riccati equation that stems from this work, since all other results and notation are explained in the following. The present work is organized as follows. The next section present the problem that we propose to solve by introducing the H filtering problem. Section III gives a time domain solution for this filtering problem, deduced from. In section IV we first give the frequency domain representation of the full order filter in terms of polynomial matrices. This representation is based on the time domain solution with fictitious error feedback, by the use of the connecting relation between time and frequency domain parametrization of the full order filter. Second, the polynomial matrix equation of the full order H filter is derived from a full order Algebraic Riccati Equation (ARE). Section V presents a numerical example and section VI concludes the paper. II. PROBEM STATEMENT onsider the linear time-invariant multivariable system described by ẋ(t) Ax(t) + Bw(t) () y(t) x(t) + w(t) () z(t) x(t) () where x(t) R n and y(t) R m are the state vector and the output vector of the system, w(t) R q represents the disturbance vector, and z(t) R m z is the unmeasurable outputs to be estimated. The matric has the dimension (m,q) such that all measurements y(t) are affected by the disturbances w(t). Further, it is assumed through the paper that : Assumption : i) T > //$6. AA 9

2 ( ) ii),a is detectable, and (A,B) is stabilisable. The initial condition x() is considered to be known and, without loss of generality, assumed to be zero. Problem : Our aim is to design an H filter that generates an estimate ẑ(t) for z(t) using the measurements y(t) in the frequency domain. For this purpose, consider the following left coprime (polynomial) matrix fraction description (MF) (si A) (s) N x (s) () where the two polynomial matrices (s) and Nx (s) have the specification to be left coprime with (s) row reduced (see ). So, the frequency domain description of the disturbance behaviour of the system (()-()) can be given by z(s) ( (si A) B + )w(s) () )w(s) ( (s) N w (s) + with N w (s) N x.b As the purpose of an H filtering problem is to find a filter to minimize the worst case estimation error e over all bounded energy disturbances w, let s define the (worst case) performance measure as follows J sup w, ) z ẑ w F ew (6) where w and F ew (s) e(s) w(s) is the filter transfer matrix from the finite energy disturbances w(t) to the estimation error e(t) z(t) ẑ(t). Therefore the filtering problem is equivalent to the following norm minimization problem : H filtering problem: Assume that R T > Given a γ >, find a stable filter if it exists such that J < γ. III. FU ORER H FITER IN THE TIME OMAIN Among the techniques existing in the litterature for solving the H filtering problem, we use the time domain solution of Grimble, where the feedback gain of the estimator is obtained by solving an Algebraic Riccati equation (ARE). Therefore, a full order filter for the system (()-()) can be given by ˆx(t) Aˆx(t) + K f (y(t) ŷ(t)) (7) ẑ(t) ˆx(t) (8) with ŷ(t) ˆx(t) (9) The full order estimation error is given by ε(t) x(t) ˆx(t) () Then, the dynamic of this estimation error is ε(t) ẋ(t) ˆx(t) () (A K f )ε(t) + (B K f )w(t) () Our aim therefore, is to minimize the effect of the disturbance w(t) on the estimation error e(t), and this can be ensured by solving the H filtering problem using the following lemma. emma : onsider the system (()-()) and let the assumption be satisfied. Then the H filtering problem is solved by the full order filter ((7)-(8)) with the feedback gain K f (Y T + B T )R () where Y Y T is a stabilizing solution of the following Algebraic Riccati Equation (A B T R )Y + Y (A B T R ) T + Y (γ T T R )Y + B(I T R )B T () The scalar γ represents an upper bound on the H norm of the filtering error. The minimum value of γ corresponds to the optimum value of the H norm and it is found by iteration. Proof : : The filter gain () and the ARE () are deduced from Grimble. ontrary to, we can easily note that the full order filter ((7)-(8)) is implementable, since one has not to differentiate the measurements y(t). In the sequel we shall give a frequency domain solution of the H filtering problem. IV. FU ORER H FITER IN THE FREQUENY OMAIN A. Polynomial MF of the full order H filter Here, a frequency domain representation of the full order filter ((7)-(8)) with a fictitious error feedback (z(t) ẑ(t)) (as z(t) is not measured), is given. So, one can write ((7)-(8)) as z(t) ẑ(t) ˆx(t) Aˆx(t) + K fz K f y(t) ŷ(t) ŷ(t) ˆx(t) ẑ(t) ˆx(t) () or equivalently ˆx(t) (A K fz K f )ˆx(t) z(t) + K fz K f y(t) ẑ(t) ˆx(t) (6) 9

3 Note that this fictitious error feedback, and therefore, the fictitious feedback gain K fz is needed to derive the polynomial matrix equation of the full order H filter design in the frequency domain. This fictitious feedback gain is given by the following equation: K fz γ Y T (7) In fact, as Y is a stabilizing solution of( the ARE ) () (see lemma ) and from assumption,,a is detectable, then Y ensures the stability of the matrix A B T R + Y (γ T T R ) A γ Y T K f its readily follows that (7) holds. In addition, it is interesting to recall that (see ), the time and the frequency domain parameterization of the full order filter are related due to the following equation: (s) f (s) I mz+m + (si A) K fz K f (8) where, (s) is given by () and the polynomial matrix f (s) parameterizes the dynamics of the fictitious filter (6) in the frequency domain. So, the filter poles are the solution of det f (s) (9) The frequency domain representation of the implementable filter ((7)-(8)) is given by the next lemma : emma : : A frequency domain representation of the full order filter ((7) (8)) of order n in terms of polynomial matrices is given by ẑ(s) (s) ( (s) Imz+m ŷ(s) (s)) () where ẑ(s) is the estimate of the unmeasurable outputs, ŷ(s) is an estimate of the measurements and (s) is the filter denominator matrix. Proof : : In view of (), and by setting K fz (in fact, is the fictitious feedback matrix that we can omit for the implementable filter), we have z(t) ẑ(t) ˆx(t) Aˆx(t) + K f y(t) ŷ(t) Aˆx(t) + K f (ϕ(t) ˆϕ(t)) where ϕ(t) ˆϕ(t) ẑ(t) ŷ(t) z(t) y(t),. ˆx(t) ϕ ˆx(t), giving () ˆϕ(s) ϕ ˆx(s) ϕ (si A) K f ϕ(s) ϕ (si A) K f ˆϕ(s) () The connecting relation for the filter ((7) (8)) is obtained from (8) by setting K fz. So, (s) (s) Imz+m + () (si A) K f So, we can easily write, that ϕ (si A) K f (s) (s) Imz+m () Using (), the relation () becomes ˆϕ(s) ( (s) (s) Imz+m)ϕ(s) ( (s) (s) Imz+m)ˆϕ(s) () but, with ϕ(s) since z(s) is not measurable. When solving () for ˆϕ(s), we obtain ˆϕ(s) (s)( (s)imz+m (s)) which yields to (). More, the filter denominator matrix (s) and the denominator matrix f (s) of the fictitious filter (6), are related according to the following lemma: emma : The matrices (s) and f (s) are related by the following equation (s) (s) + N x (s) K f (6) mz f (s) + I m (s) Imz (7) m Proof : The proof is omitted due to lack of place. Following (6) and (7), the denominator matrix (s) of the filter can be obtained by, solving the full order ARE () or computing the fictitious denominator matrix f (s), which is resulting from a polynomial matrix equation, that is the subject of the next section. B. Polynomial matrix equation of the full order H filter in the frequency domain In this section, a polynomial matrix equation for the fictitious denominator f (s) is given. But, before and according to (8), one can see that f (s) is related to the full order system (), so a full order ARE is needed for the derivation 9

4 of the desired polynomial matrix equation, as we will see in the sequel. We will adopt the following notation: P (s) P T ( s) for P {, Nw, f or R γ }. emma : The useful full order ARE to derive the polynomial matrix equation for computing the denominator matrix of the fictitious filter f (s) is given by γ I mz R Y A T + AY K fz K f K T fz K T f + BB T (8) Proof : By taking into account (), (), (7) and R T >, it is readily shown that (8) holds. Therefore, the polynomial matrix equation is stated by the following theorem. Theorem : onsider the system () with MFs (), (), and let the assumptions for the H filtering problem be satisfied. Assume that there exists a fictitious denominator matrix f (s) related to a full order H filter of order n solving the following polynomial matrix equation f (s) R γf (s) (s) Rγ (s) + N w (s) N w(s) + N w (s) T (s) + (s) N w(s) (9) such that det f (s) and det (s) have all their roots in the open left half plane and R γ γ I mz T () and postmultiplying it by ( si A T ) T T, gives Y ( si A T ) T T + (si A) Y T T (si A) γ Y T Y ( si A T ) T T + (si A) K f RKf T ( si A T ) T T (si A) BB T ( si A T ) T T () Because of.. K fz K f Rγ Y T T + B T () + (si A) γ Y T Y ( si A T ) T T (si A) K f R Kf T ( si A T ) T T () (si A) K fz K f R γ K fz K f T ( si A T ) T T Then the corresponding H filter solving the H filtering problem is given by () with denominator matrix (6). Proof : With (7), write (8) as AY Y A T γ Y T Y + K f RKf T BB T () then, adding sy sy to () and rearranging yields (si A)Y + Y ( si A T ) γ Y T Y + K f RKf T BB T () Premultiplying this result by (si A).. R γ K fz Y ( si A T ) T T T T K f T ( si A T ) B T ( si A T ) T T (6) (si A) Y T T (si A) ( K fz K f R γ B T ) (7) 96

5 relation () can be written as { } ( (si A) K fz K f R γ + R γ ) { ( + R γ) + R γ + T T + } (si A) K fz K f R γ B T ( si A T ) (si A) BB T (si A) B T ( si A T ) T T (8) Introducing the MF () with N w (s) N x.b and using relation (8) one has (s) f (s) R γf (s)( (s)) R γ + N w(s)( (s)) + (s) N w (s) T + (s) N w (s) N w(s)( (s)) (9) By left multiplication with (s) and right multiplication with (s) we obtain relation (9). The fact, that det f (s) and det (s) have all their roots in the open left half plane, assures the stability of the filter (see also (9)). V. NUMERIA EXAMPE onsider the H filtering problem for the following system ẋ(t) x(t) + w(t) 7 y(t) x(t) + w(t) z(t). x(t) () Since R ( T ) > holds and it is easy to verify that the pair,a is detectable, the H filtering problem is solved, in the time domain, by the full order filter ˆx(t) Aˆx(t) + K f (y(t) ŷ(t)) ẑ(t) ˆx(t) () A frequency domain representation of the filter () of order is given by (see () ) ẑ(s) (s) ( (s) I ŷ(s) (s)) () where the denominator matrix (s) ( see (6) and (7)) can either be obtained by solving the ARE () or by computing the fictitious denominator matrix f (s) from the polynomial matrix equation () and (s) is given by (). First, we intend to find the left coprime polynomial matrices (s), N x (s) with (s) row reduced. Relation () gives: (si A). s+ s+ s (s+)(s+) On the other hand, one can write (si A) s + (s + )(s + ). s + s so, s + (s) (s + )(s + ) and N x (s). s + s () () () (6) Now, our purpose is to find the denominator matrix (s). We choose to find the feedback gain K f. In fact, one has K f (Y T + B T )R (7) where Y Y T is a stabilizing solution of (). Next, we search the upper bound γ on the H norm of the filtering error. We then try in the following, to find the optimal γ > that assures, in the same time, the stability of the filter with the minimization of the worst case performance measure J. In fact, we compute J for several values of γ and we draw in Figure the behavior of this performance measure with respect to γ. We deduce that the optimal γ is equal to.7, indeed, for this value, the filter is stable and J.79. Therefore, setting γ.7, we obtain Y (8) so, K f Thus, the denominator matrix (s) is given by (s) s +.87 s +.7s + 8. (9) () and the corresponding frequency domain representation of the filter solving the H filtering problem is given by : ẑ(s) ŷ(s) (s + )(s +.7s + 8.) s +.7s s + ( ) s + 6. Finally, the figures, and show the time and frequency domain behavior of the filter and so, the effectiveness of our approach. 97

6 J VI. ONUSION A frequency domain solution for the H filtering problem is given in this paper in terms of polynomial matrices. The yielded solution, is of full order one. The design of this filter is first derived from the time domain H filtering solution and due to the relation between the time and the frequency domain parametrization of a system, the full order filter is given in frequency domain. The proposed H filter has been carried on a numerical example and the results were successful. The further works will concern the design of the H filter using a MI approach in order to alleviate some of the hypothesis used here or to take into account more specification as robustness in presence of uncertainty. Our aim is also to use the obtained filter in a control loop for the observer-based control purpose as in 7. REFERENES : Antsaklis.P and Michel.N, inear Systems, New York: McGraw- Hill, 997. : eutscher.j, Frequency domain design of reduced order observer based H controllers - a polynomial approach, Int. J. ontrol, Vol 7.No.,pp 96-,. : Grimble.M.J, Polynomial matrix solution of the H filtering problem and the relationship to Riccati equation state-space results, IEEE Transaction on Sig.Processing,, : Hippe.P, esign of reduced-order optimal estimators directly in the frequency domain,int.j.ontrol, Vol,pp 99-6, 989. : ing.x, Guo. and Frank.M.P, Parameterization of inear Observers and Its Application to Observer esign, IEEE Transaction on Automatic ontrol.processing, 9, : ing.x, Frank.M.P and Guo., Robust Observer esign Via Factorization Approach, Proceedings of the 9th onlerence on ecioion and ontrol Honolulu. Hawall ecember : Russell.W. and Bullock.E.T, A Frequency omain Approach to Minimal-Order Observer esign for Several inear Functions of the State, IEEE Transaction on Automatic ontrol.processing,, : MacFarlane,A.G.J, Return-difference matrix properties for optimal stationary Kalman-Bucy filter, Proceedings of the Institution of Electrical Engineers, 8, : Anderson.B..O and Ku cera.v, Matrix fraction construction of linear compensators, IEEE Transaction on Automatic ontrol.processing,, : arouach M., Zasadzinski M., Xu S.J.,, Full- order observers for linear systems with unknown inputs, IEEE Trans. Automat. ontr., vol A. 9, pp. 6669, 99. : O Reilly.J, Observers for inear Systems, New York, NY: Academic, 98. :. G. uenberger, Observers for multivariable systems, IEEE Trans. Automat. ontr., vol., pp. 997, 966. :. G. uenberger,, An introduction to observers, IEEE Trans. Automat. ontr., vol. 6, pp. 966, 97. : Popov. V. M, Some properties of control systems with irreductible matrice transfer functions, In ecture Notes in Mathematics. Springer- Verlag, Berlin, 969, : Rosenbrock. H. H, State-Space and Multivariable Theory, Nelson. ondon, : oyle, J.., Glover, K., Khargonekar, P. P and Francis, B.A,, State- space solutions to standard H and H control problems, IEEE Trans. Automat. ontr., vol., pp. 886, : Hippe.P and eutscher.j, esign of observer-based compensators : From the time to the frequency domain, Springer-Verlag, ondon, 9. Fig.. Fig.. e(t) Fig.. Amplitude J f (gamma) gamma Fig.. The charateristic J f(γ) 7 6 The behavior of the used disturbance w(t) First component of the estimation error Time (sec) The behaviour of the first component of the estimation error e(t) Amplitude Second component of the estimation error Time (sec) The behaviour of the second component of the estimation error Magnitude (db) Phase (deg) Bode iagram 8 Frequency (rad/sec) Fig.. The Bode iagram 98