International Journal of Control Vol. 80, No. 5, May 2007, 749–762
Design of an extended unknown input observer with stochastic robustness techniques and evolutionary algorithms M. WITCZAK* and P. PRE¸ TKI Institute of Control and Computation Engineering, University of Zielona Go´ra, ul. Podgo´rna 50, 65–246 Zielona Go´ra, Poland, (Received 12 July 2006; in final form 18 December 2006) The paper deals with the problem of designing an unknown input observer for discrete-time non-linear systems. In particular, with the use of the Lyapunov method, it is shown that the proposed observer is convergent under certain, non-restrictive conditions. Based on the achieved results, a general solution for increasing the convergence rate is proposed and implemented with the use of stochastic robustness techniques. In particular, it is shown that the problem of increasing the convergence rate of the observer can be formulated as a stochastic robustness analysis task that can be transformed into a structure selection and parameter estimation problem of a non-linear function, which can be solved with the B-spline approximation and evolutionary algorithms. The final part of the paper shows an illustrative example based on a two phase induction motor. The presented results clearly exhibit the performance of the proposed observer.
Notation
1. Introduction xk , x^ k 2 Rn yk , y^k 2 Rm ek 2 Rn "k 2 Rm uk 2 R r dk 2 R q fk 2 Rs gðÞ, hðÞ Ek 2 Rnq
L1, k 2 Rns , L2, k 2 Rms p 2 Rnp
state vector and its estimate output vector and its estimate state estimation error output error input vector unknown input vector, qm fault vector non-linear functions unknown input distribution matrix fault distribution matrices design parameter vector of the observer
*Corresponding author. Email:
[email protected]
A large amount of knowledge on designing observers for non-linear systems has been accumulated through the literature since the beginning of the 1970s. Indeed, observers are commonly used in both control and fault diagnosis schemes for non-linear systems (Alcorta Garcia and Frank 1997, Chen and Patton 1999, Witczak et al. 2002, Califano et al. 2003, Witczak 2003, Korbicz et al. 2004). Although the problem has been attacked from many different angles, there is no one general solution that can effectively be applied for all classes of non-linear systems. One of the possible approaches can be implemented by a suitable non-linear change of coordinates to bring the original system into a linear (or pseudo-linear) one (see, e.g. Bastle and Zeitz (1983) and Keller (1987)). As can be expected, the main drawback of such an approach is related with strong design conditions that limit its application to a particular class of non-linear systems. The class of non-linear Lipschitz systems has also been investigated in a number of papers
International Journal of Control ISSN 0020–7179 print/ISSN 1366–5820 online 2007 Taylor & Francis http://www.tandf.co.uk/journals DOI: 10.1080/00207170601178108
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M. Witczak and P. Pre¸tki
(Rajamani 1998, Rajamani and Cho 1998, Aboky et al. 2002, Pertew et al. 2005, Witczak and Korbicz 2006). In this case, the observer is designed, based on the linear part, and by imposing certain conditions on the nonlinear part. Thus, the non-linearities can be tolerated through sufficiently large gain. However, all the above approaches inherit a common drawback that is related to the fact that the observer convergence conditions are difficult to be satisfy for large values of the Lipschitz constant. The application of the extended kalman filter (EKF) (Anderson and Moore 1979) to the state estimation of non-linear deterministic systems has also received considerable attention during the last two decades (see Boutayeb and Aubry (1999) and the references therein). This is mainly because the EKF can be directly applied to a large class of non-linear systems. Moreover, it is possible to show that the convergence of such a deterministic observer is ensured under certain conditions. The present paper deals with the problem of designing observers that can be used for robust fault detection and isolation (FDI) (Chen and Patton 1999, Korbicz et al. 2004) for non-linear systems. Undoubtedly, the most common approach that can be used to tackle such a problem is to employ an unknown input observer (UIO) (Chen and Patton 1999, Seliger and Frank 2000, Huis and Zak 2005), which can tolerate a degree of model uncertainty and hence increases the reliability of fault diagnosis. Although the origins of UIOs can be traced back to the early 1970s (cf. the seminal work of Wang et al. 1975) the problem of designing such observers is still of paramount importance both from the theoretical and practical viewpoints. A large amount of knowledge on using such techniques for model-based fault diagnosis has been accumulated through the literature for the last three decades (see Chen and Patton (1999), Korbicz et al. (2004) and the references therein). Generally, the design problems regarding UIOs for non-linear systems can be divided into the three distinct categories. (i) Non-linear state-transformation-based techniques. Apart from a relatively large class of systems for which they can be applied, even if the non-linear transformation is possible it leads to another nonlinear system and hence the observer design problem remains open; see Alcorta Garcia and Frank (1997) Seliger and Frank (2000) and the references therein). (ii) Linearization-based techniques. Such approaches are based on a similar strategy as that for the extended Kalman filter (Korbicz et al. 2004). In Witczak et al. (2002) and Witczak (2003) the authors proposed an extended unknown input
observer (EUIO) for non-linear systems. They also proved that the proposed observer is convergent under certain conditions. (iii) Observers for particular classes of non-linear systems. For example UIOs for polynomial and bilinear systems (Hac 1992, Ashton et al. 1999) or UIOs for Lipschitz systems (Koenig and Mammer 2001, Pertew et al. 2005). Taking into account the presented state-of-the-art regarding observers and unknown input observers for non-linear systems, the number of real world applications (not only simple simulated systems) of non-linear observers should proliferate. Unfortunately, this is not the case. The main reason of such a situation is related with a relatively high design complexity of non-linear observers (Zolghadri et al. 1996 Alcorta Garcia and Frank 1997). This does not encourage engineers to apply them in an industrial reality. Indeed, apart from the theoretically large potential of the observer-based schemes, their computer implementation causes serious problems for engineers that are, usually, not fluent in a complex mathematical description involved in the theoretical developments. Therefore the design techniques, which are based on either the non-linear state-transformation or designs for restricted classes of non-linear systems seem less attractive from the implementation complexity viewpoint than the linearization-based strategies. Indeed, Witczak et al. (2002) proposed an EUIO, whose design procedure is almost as simple as the one for linear systems. Unfortunately, the main drawback of this technique is related to the convergence of such a linearization-based strategy. Apart from the fact that the authors developed the convergence conditions of the EUIO, the achieved results are based on restrictive assumptions. Moreover, their approach to increasing the convergence is also restrictive. Indeed, it is impossible to show that the technique proposed in Witczak et al. (2002) will work properly for all possible initial conditions of the non-linear system. Thus, the main contribution of this paper is to propose an alternative structure and design procedure of the EUIO, which overcome the above-mentioned drawbacks. The paper is organized as follows. Section 2 presents two different approaches that can be used for determining the final structure of the EUIO. In x 3, the main drawbacks regarding the existing convergence conditions (Witczak et al. 2002) of the EUIO are outlined. Moreover, an alternative way of determining the convergence conditions (Guo and Zhu 2002) of the EUIO is presented and criticised. Section 4 presents the proposed extended unknown input observer (EUIO) and its convergence conditions that do not involve the
Design of an extended unknown input observer drawbacks outlined in x 3. In x 5, a general solution to the problem of increasing the convergence rate of the EUIO is proposed. Section 6 presents a practical implementation of the above strategy, which makes it possible to maximize the performance of the EUIO for a sufficiently described set of possible initial conditions of the non-linear system. Section 7 shows the experimental results concerning state estimation and fault detection of an induction motor with the proposed EUIO. Finally, the last part is devoted to conclusions.
2. UIO structures Let us consider the following non-linear system with unknown input: xkþ1 ¼ gðxk Þ þ hðuk Þ þ Ek dk , ykþ1 ¼ Ckþ1 xkþ1 :
rankðCkþ1 Ek Þ ¼ rankðEk Þ ¼ q,
where g ðÞ ¼ G k gðÞ, h ðÞ ¼ G k hðÞ, G k ¼ I Ek Hkþ1 Ckþ1 , Ek ¼ Ek Hkþ1 : Thus, the unknown input observer for (1) and (2) can be given as follows: x^ kþ1 ¼ g ðx^ k Þ þ h ðuk Þ þ Ek ykþ1 þ Kkþ1 ð yk Ck x^ k Þ:
xkþ1 ¼ Ak xk þ Bk uk þ Ek dk , ykþ1 ¼ Ckþ1 xkþ1 ,
zkþ1 ¼ Fkþ1 zk þ Tkþ1 Bk uk þ K1, kþ1 yk , x^ kþ1 ¼ zkþ1 þ H1, kþ1 ykþ1 ,
K1, kþ1 ¼ Kkþ1 þ K2, kþ1 ,
ð10Þ
Ek ¼ H1, kþ1 Ckþ1 Ek ,
ð11Þ
Tkþ1 ¼ I H1, kþ1 Ckþ1 , Fkþ1 ¼ Tkþ1 Ak Kkþ1 Ck ,
xkþ1 ¼ g ðxk Þ þ h ðuk Þ þ Ek ykþ1 ,
ð12Þ ð13Þ ð14Þ
Substituting (9) into (8) and then using (12), (13) and (14), it can be shown that x^ kþ1 ¼ Ak x^ k þ Bk uk H1, kþ1 Ckþ1 Ak x^ k þ Bk uk Kkþ1 Ck x^ k Fkþ1 H1, kþ1 yk þ ½Kkþ1 þ Fkþ1 H1, kþ1 yk þ H1, kþ1 ykþ1 ,
ð15Þ
or equivalently
ð4Þ
Substituting (4) into (1) gives
ð8Þ ð9Þ
where
K2, kþ1 ¼ Fkþ1 H1, k :
dk ¼ Hkþ1 ykþ1 Ckþ1 gðxk Þ þ hðuk Þ :
ð7Þ
that can be described as follows Chen and Patton (1999):
ð3Þ
see Chen and Patton (1999, p. 72, Lemma 3.1) for a comprehensive explanation. The objective of this section is to show that the EUIO structure being considered can be derived in two different ways, which lead to the same final result. Let us start with the second approach. If condition (3) is satisfied, then it is possible 1 to calculate Hkþ1 ¼ ðCkþ1 Ek Þþ ¼ ðCkþ1 Ek ÞT Ckþ1 Ek ðCkþ1 Ek ÞT , where ðÞþ stands for the pseudo-inverse of its argument. Thus, by multiplying (2) by Hkþ1 and then inserting (1) it is straightforward to show that
ð6Þ
Now, let us consider the first of the above-mentioned approaches that can be used for designing the EUIO Chen and Patton (1999). For the sake of notational simplicity, let us start with the UIO for linear discretetime systems
ð1Þ ð2Þ
In the existing approaches, the unknown input is usually treated in two different ways. The first one (see, e.g., Chen and Patton (1999)) relies on introducing an additional matrix into the state estimation equation, which is then used for decoupling the effect of an unknown input on the state estimation error (and consequently on the residual signal). In the second approach (see e.g. (Keller and Darouach 1999)), the system with an unknown input is suitably transformed into a system without it. In both cases the necessary condition for the existence of a solution to the unknown input decoupling problem is
751
x^ kþ1 ¼ x^ kþ1=k þ H1, kþ1 ð ykþ1 Ckþ1 x^ kþ1=k Þ þ Kkþ1 ð yk Ck x^ k Þ,
ð16Þ
x^ kþ1=k ¼ Ak x^ k þ Bk uk :
ð17Þ
where ð5Þ
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Substituting the solution of (11), i.e. H1, kþ1 ¼ Ek Hkþ1 into (16) yields x^ kþ1 ¼ ½I Ek Hkþ1 Ckþ1 x^ kþ1=k þ Ek Hkþ1 ykþ1 þ Kkþ1 ð yk Ck x^ k Þ:
ð19Þ
Finally, by substituting (19) into (18) and then comparing it with (6) it can be seen that the observer structures being considered are identical. On the other hand, it should be clearly pointed out that they were designed in a significantly different way.
3. Preliminaries The preceding section shows two equivalent observer structures (6) and (16) that can be used for the purpose of decoupling an unknown input. Apart from the relative simplicity of the presented approaches, the task of designing either (6) or (16) boils down to designing an observer for non-linear deterministic systems. As was mentioned, the application of the EKF to the state estimation of non-linear deterministic systems has received considerable attention during the last two decades; see Boutayeb and Aubry (1999) and the references therein. This is mainly because the EKF can be directly applied to a large class of non-linear systems. Moreover, it is possible to show that the convergence of such a deterministic observer is ensured under certain conditions. Motivated by the results presented in Boutayeb and Aubry (1999), Witczak et al. (2002) developed an EUIO for the following class of non-linear systems: xkþ1 ¼ gðxk Þ þ hðuk Þ þ L1, k fk þ Ek dk , ykþ1 ¼ Ckþ1 xkþ1 þ L2, kþ1 fkþ1 ,
@gðxk Þ Ak ¼ : @xk xk ¼x^ k
ð18Þ
Thus, in order to use (18) for (1) and (2) it is necessary to change (17) into x^ kþ1=k ¼ gðx^ k Þ þ hðuk Þ:
It should also be pointed out that the matrix Ak used in the design procedure of (22) and (23) is defined by
Witczak et al. (2002) proposed the convergence conditions of such an EUIO. They also indicated the the proposed condition is based on a restrictive assumption under which the matrix A1, kþ1 ¼ Tkþ1 Ak is invertible. Unfortunately, as will also be shown in the subsequent part of this paper, it is clear that the matrix A1, kþ1 is singular when Ek 6¼ 0. Thus, the convergence conditions can formally be obtained only when Ek ¼ 0. It seems that one possible approach to overcome the above-mentioned restrictive assumptions is to use a different strategy for the convergence analysis. An alternative approach to the one presented in Boutayeb and Aubry (1999) and Witczak et al. (2002) was proposed by Guo and Zhu (2002). For the sake of simplicity, let us assume that Ek ¼ 0. Instead of using the approach introduced by Boutayeb and Aubry (1999), Guo and Zhu (2002) proposed the following approach:
where
ð22Þ x^ kþ1=k ¼ gðx^ k Þ þ hðuk Þ, ^xkþ1 ¼ x^ kþ1=k þ H1, kþ1 ð ykþ1 Ckþ1 x^ kþ1=k Þ þ Kkþ1 ð yk Ck x^ k Þ: ð23Þ
ekþ1=k ¼ gðxk Þ g ðx^ k Þ ¼ Ak ek ,
ð25Þ
@gðxÞ Ak ¼ : @x x¼x^ k þDk
ð26Þ
where Dk 2 Rn , and x^ k þ Dk is between xk and x^ k . This means that there exists a scalar 2 ½0, 1 such that x^ k þ Dk ¼ xk þ ð1 Þx^ k . Unfortunately, in general, the above approach is, however, incorrect as demonstrated by the following counterexample. Let us consider the an exemplary structure of gðxÞ ðx ¼ ½x1 , x2 T )
ð20Þ ð21Þ
where gðxk Þ is assumed to be continuously differentiable with respect to xk . Similarly to the case of the EKF (Anderson and Moore 1979), the UIO presented in Chen and Patton (1999, pp. 98–108) (i.e., an observer given by (16)) was extended to the class of non-linear systems (20) and (21). This led to the following structure of the EUIO:
ð24Þ
T gðxÞ ¼ x21 , ex1 þx2 :
ð27Þ
Moreover, let us assume that xk ¼ 0 and x^ k ¼ 1 for which gðxk Þ ¼ gð0Þ ¼ ½0, 1T
Thus
and
gðx^ k Þ ¼ gð1Þ ¼ ½1, e2 T : ð28Þ
T gðxk Þ gðx^ k Þ ¼ 1, 1 e2
ð29Þ
T Ak ek ¼ 2ð1 Þ, 2e2ð1Þ :
ð30Þ
and
753
Design of an extended unknown input observer This means that there should exists a scalar 2 ½0, 1 such that
T T 1, 1 e2 ¼ 2ð1 Þ, 2e2ð1Þ :
ð31Þ
From the first equation of the above set of equations it is clear that ¼ 1=2 which is not valid for the second equation, i.e. 1 e2 6¼ 2e. This counterexample clearly shows that the observer convergence conditions described in Guo and Zhu (2002) are invalid.
4. An alternative EUIO and its convergence analysis Taking into account all the difficulties exposed in x 3, the main objective of this section is to propose an alternative structure of the EUIO and to derive its convergence conditions. As has been shown in x 2, the state equation (1) can be transformed into (5) but instead of using the observer structure (6) it is proposed to use its minor modification that can be given as x^ kþ1 ¼ x^ kþ1=k þ Kkþ1 ð ykþ1 Ckþ1 x^ kþ1=k Þ,
ð32Þ
where x^ kþ1=k
¼ g ðx^ k Þ þ h ðuk Þ þ Ek ykþ1 :
ð33Þ
As will be shown in the subsequent part of this paper, the minor modification of (32) makes it possible to provide convergence conditions of the EUIO under less restrictive assumption than those used in Witczak et al. (2002). The algorithm used for the state estimation of (20) and (21) is given as follows: x^ kþ1=k ¼ g ðx^ k Þ þ h ðuk Þ þ Ek ykþ1 , Pkþ1=k ¼ Ak Pk Ak T þ Qk , 1 Kkþ1 ¼ Pkþ1=k CTkþ1 Ckþ1 Pkþ1=k CTkþ1 þ Rkþ1 , x^ kþ1 ¼ x^ kþ1=k þ Kkþ1 ð ykþ1 Ckþ1 x^ kþ1=k Þ, Pkþ1 ¼ ½I Kkþ1 Ckþ1 Pkþ1=k ,
convergence of EUIO strongly depends on the instrumental matrices Qk and Rk . Moreover, the fault-free mode is assumed, i.e. fk ¼ 0. Using (37), the state estimation error can be given as ekþ1 ¼ xkþ1 x^ kþ1 ¼ ½I Kkþ1 Ckþ1 ekþ1=k ,
ð40Þ
where ekþ1=k ¼ xkþ1 x^ kþ1=k ¼ g ðxk Þ g ðx^ k Þ ¼ ak Ak ek ð41Þ and there exists an unknown diagonal matrix ak ¼ diagð1, k , . . . , n, k Þ such that (41) is satisfied. Thus, using (41), equation (40) becomes ekþ1 ¼ ½I Kkþ1 Ckþ1 ak Ak ek :
ð42Þ
It is clear from (41) that ak represents the linearization error. This means that the convergence of the proposed observer is strongly related to the admissible bounds of the diagonal elements of ak . Thus, the main objective of further consideration is to show that these bounds can be controlled with the use of the instrumental matrices Qk and Rk . First let us start with the convergence conditions, which require the following assumptions. Assumption 1: Following Boutayeb and Aubry (1999), it is assumed that the system given by (5) and (2) is locally uniformly rank observable. This guarantees that the matrix Pk is bounded (Boutayeb and Aubry 1999 and the references therein), i.e. there exists positive scalars > 0 and > 0 such that I P1 k I:
ð43Þ
Assumption 2: The matrix Ak is uniformly bounded and there exists A1 k . Moreover, let us define
ð34Þ k ¼ max jj, k j,
ð35Þ ð36Þ ð37Þ ð38Þ
j¼1,..., n
k ¼ min jj, k j: j¼1,..., n
ð44Þ
Theorem 1: If ðAk Þ2 ðCkþ1 Þ2 ðAk Pk Ak T þ Qk Þ kþ1 Pkþ1=k CTkþ1 þ Rkþ1 Þ ðC !1=2 ð1 ÞðAk Pk Ak T þ Qk Þ þ , 2 Ak Þ ðP kÞ ð
k 2k where @gðxk Þ k @gðxk Þ ¼ G ¼ Gk Ak : Ak ¼ @xk xk ¼x^ k @xk xk ¼x^ k
ð39Þ
The main objective of the subsequent part of this section is to present the convergence conditions of the proposed EUIO. In particular, the main aim is to show that the
ð45Þ
where 0 < < 1, then the proposed extended unknown input observer is locally asymptotically convergent. Proof:
See appendix.
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Remark 1: As can be observed by a straightforward comparison of (45) and (46), the convergence condition (45) is less restrictive than the solution obtained with the approach proposed in Boutayeb and Aubry (1999), which can be written as ð1 ÞðAk Pk Ak T þ Qk Þ k 2 kÞ ðA k Þ ðP
!1=2 :
ð46Þ
However, (45) and (46) become equivalent when Ek 6¼ 0, i.e., in all cases when unknown input is considered. This is because of the fact that the matrix Ak is singular when Ek 6¼ 0, which implies that ðAk Þ ¼ 0. Indeed, from (39) h 1 Ak ¼ Gk Ak ¼ I Ek ðCkþ1 Ek ÞT Ckþ1 Ek i ðCkþ1 Ek ÞT Ckþ1 Ak ,
ð47Þ
down to the classical approach with constant Qk ¼ 1 I. It is, of course, possible to set Qk ¼ 1 I with 1 large enough. As was mentioned, the more accurate (near ‘‘true’’ values) the covariance matrices, the better the convergence rate. This means that, in the deterministic case, both the matrices should be zero ones. To tackle this problem, a compromise between the convergence and the convergence rate should be established. This can be easily done by setting Qk as Qk ¼ "Tk "k þ 1 I,
"k ¼ yk Ck x^ k ,
ð53Þ
with > 0 and 1 > 0 large and small enough, respectively. Since the form of Qk is established, then it is possible to obtain Rk in such a way as to minimize Ckþ1 Pkþ1=k CTkþ1 þ Rkþ1 :
ð54Þ
and under Assumption 2, it is evident that Ak is singular when
To tackle this problem, let us start with the solution proposed in Boutayeb and Aubry (1999) Guo and Zhu (2002)
1 Ek ðCkþ1 Ek ÞT Ckþ1 Ek ðCkþ1 Ek ÞT Ckþ1
Rkþ1 ¼ Ckþ1 Pkþ1=k CTkþ1 þ 2 I,
ð48Þ
is singular. The singularity of the above matrix can be easily shown with the use of (3), i.e., 1 rank Ek ðCkþ1 Ek ÞT Ckþ1 Ek ðCkþ1 Ek ÞT Ckþ1 min½rankðEk Þ, rankðCkþ1 Þ ¼ q:
ð49Þ
Taking into account the fact that q < n, the singularity of Ak becomes evident. Remark 2: It is clear from (45) that the bound of k can be maximized by suitable settings of the instrumental matrices Qk and Rk . Indeed, Qk should be selected in such a way as to maximize Ak Pk Ak T þ Qk :
ð50Þ
To tackle this problem, let us start with a similar solution to the one proposed in Guo and Zhu (2002), i.e., Qk ¼ Ak Pk Ak T þ 1 I,
ð51Þ
where 0 and 1 > 0. Substituting, (51) into (50) and taking into account that ðAk Þ ¼ 0, it can be shown that ð1 þ Þ Ak Pk Ak T þ 1 I ¼ 1 I:
ð52Þ
Indeed, singularity of Ak , causes ðAk Pk Ak T Þ ¼ 0, which implies the final result of (52). Thus, this solution boils
ð55Þ
with 0 and 2 > 0. Substituting (55) into (54) gives ð1 þ Þ Ckþ1 Pkþ1=k CTkþ1 þ 2 I:
ð56Þ
Thus, in (55) should be set so as to minimize (56), which implies Rkþ1 ¼ 2 I,
ð57Þ
with 2 small enough.
5. Increasing the convergence rate of EUIO Although the settings of Qk and Rk determined by (53) and (57) are very straightforward and easy to implement, it is possible to increase the convergence rate further. Indeed, in Witczak et al. (2002) the authors proposed to set the instrumental matrices as follows Qk1 ¼ 2 ð"k1 ÞI þ 1 I,
Rk ¼ r2 ð"k ÞI þ 2 I,
ð58Þ
where ð"k1 Þ and rð"k Þ are non-linear functions of the output error "k (the squares are used to ensure the positive definiteness of Qk1 and Rk ). Moreover, they developed a genetic programming-based technique to determine an optimal structure of ð"k1 Þ and rð"k Þ. A completely different approach was proposed in Guo and Zhu (2002). In particular the authors suggested to
755
Design of an extended unknown input observer use a neural network-based strategy for the estimation of an initial condition of the system. In Guo and Zhu (2002) and Witczak et al. (2002) the authors demonstrated significant advantages that can be gained by using the proposed strategies. However, both these approaches inherit a common drawback that they are designed for a particular setting of the initial condition of the system. This means that there is no guarantee that the proposed strategies will have the same performance when the initial condition will be significantly different than the one used for the design purposes. The main objective of this section is to present an alternative solution which overcomes the abovementioned drawbacks. Following Witczak et al. (2002), let us assume that Qk ¼ ð"k ÞI þ 1 I,
ð59Þ
but Rk is determined by (57), and ð"k Þ is a positive definite function. Contrary to Witczak et al. (2002), to approximate ð"k Þ, it is suggested to use spline functions (de Boor 1978) instead of genetic programming. Such a choice is not accidental and it is dictated by the computational complexity related to the proposed design procedure, which will be carefully described in the subsequent part of this paper. Indeed, many researchers working with genetic programming clearly indicate that its main drawback is related to its computational burden. Similarly, the application of neural networks Korbicz et al. (2004) reduces to the non-linear parameter estimation task, which usually involves a high computational burden. Contrary to neural networks and genetic programming, the suggested spline approximation technique is less computationally demanding apart from their excellent approximation capabilities. The solution presented in the subsequent part of this section is based on the general idea of stochastic robustness method originally used by Marrison and Stengel (1997) for designing robust control systems. In other words, stochastic robustness analysis is a practical method for quantifying robustness of control systems. In this paper, a stochastic robustness metric characterizes an EUIO (denoted by O) with the probability that the observer will have an unacceptable performance in the presence of possible variations of the initial condition of the system x0 2 X Rn . The probability P can be defined as the integral of an indicator function over the space of expected variations of the initial condition Z I½Sðx0 Þ, Oð pÞprðx0 Þdx0 ,
Pð pÞ ¼ X
ð60Þ
where S stands for the system structure, p denotes the design parameter vector of the observer O, i.e., the parameter vector of ð"k Þ in (59), and prðx0 Þ is the probability density function. Note that there is no need for restricting prðx0 Þ and it can be set arbitrarily by the designer. Moreover, the binary indicator function is defined as follows: I½Sðx0 Þ, Oð pÞ ¼
0
keT k2 ke0 k2 ,
1
otherwise:
ð61Þ
where T > 0 and 0 < < 1 are the parameters set by the designer. Thus, by considering (61), it can be seen that performance of the observer O for x0 is acceptable when kek k2 , k ¼ 0, . . . , T decreases in time T such that keT k2 =ke0 k2 . Unfortunately, (60) cannot be integrated analytically. A practical alternative is to use Monte Carlo evaluation (MCE) with prðx0 Þ shaping the random values of x0 that can be denoted by xi0 . When N random xi0 , i ¼ 1, . . . , N are generated, then the estimate of P can be given as N X ^ pÞ ¼ 1 Pð I S xi0 , Oð pÞ , N i¼1
ð62Þ
where P^ approaches P in the limit as N ! 1. Moreover, it should be pointed out that the initial condition of the observer remains fixed for all xi0 , i ¼ 1, . . . , N and it should be set by the designer. Since the initial condition of the system is unknown, the need for designing the EUIO that works properly for any initial condition of the system under one fixed initial state estimate is fully justified. It is obvious that it is impossible to set N ¼ 1. Thus, the problem is to select N in such a way as to obtain P^ (which is a random variable) with standard deviation P^ less than a predefined threshold. Since the stochastic metric (61) is binary then it is clear that P^ has a binomial distribution with sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pð pÞ Pð pÞ2 P^ ¼ : N
ð63Þ
Since P is unknown then the only feasible way is to calculate the upper bound of (63) which gives 1 1 P^ pffiffiffiffi : 2 N
ð64Þ
Using (64), it can be shown that N can be calculated as follows:
1 2 N¼ , 4 P^
ð65Þ
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where de is a rounding operator returning an integer value that is not smaller than its argument. Since all the ingredients of the algorithm are given, then its outline can be presented. Step 1:
select T, , x^ 0 and P^ .
Step 2:
Calculate N according to (65).
Step 3: Determine ð"k Þ (by estimating p) in (59) in such a way as to minimize the cost function (62).
6. Increasing the convergence rate with spline approximation and evolutionary optimization In the literature, there are two commonly used ways to represent a polynomial spline: the piecewise polynomial function ( pp-form) and its irredundant representation, the so-called basis-splines (B-form) (de Boor 1978). Given the knot sequence tx ¼ ft1 , t2 , . . . , tn g, the B-form that describes a univariate spline f(x) can be expressed as a weighted sum: fðxÞ ¼
n X
ci Bi, k ðxÞ,
i1 ¼1 i2 ¼1
...
Nn X
i ¼ 1, . . . , np :
ð68Þ
7. Experimental results
where Bi, k ðxÞ is a non-negative piecewise-polynomial function of degree k, that is non-zero only on the interval ½ti , tiþk . The simplest method of obtaining multivariate interpolation and approximation routines is to take univariate methods and form a multivariate method via tensor products (de Boor 1978): N1 X N2 X
pi, new ¼ jpi, kþ1 j,
ð66Þ
i¼1
fðxÞ ¼
is based on a probably the simplest selection-mutation model of the Darwinian’s evolution (Obuchowicz and Pre¸ tki 2004). It should be stressed here that the original algorithm is improved by applying a directional mutation (Pre¸ tki and Obuchowicz 2006) that considerably increases effectiveness of the algorithm for multi-dimensional problems (Pre¸ tki and Obuchowicz 2006). The algorithm being considered is carefully described in Obuchowicz and Pre¸ tki (2006) and Pre¸ tki and Obuchowicz (2006). Moreover, to meet the requirement for positive definiteness of the function ð"k Þ, the evolutionary algorithm is supplied with an auxiliary correction algorithm. Let us notice that the function (67) used to approximate ð"k Þ is positive if all spline coefficients ci1 , i2 ,..., in are non-negative. Therefore, the purpose of the correction procedure is to maintain each individual p in a feasible set. In the theory of the evolutionary computation, many techniques of dealing with constraints can be found (Michaelewicz 1996), while in this paper a simple reflection procedure is applied
ci1 , i2 ,..., in Bi1 , k1 , t1 ðx1 Þ
The purpose of this section is to show the reliability and effectiveness of the proposed EUIO. The numerical example considered here is a fifth-order two-phase non-linear model of an induction motor, which has already been the subject of a large number of various control design applications; see Boutayeb and Aubry (1999) and the references therein.
in ¼1
Bi2 , k2 , t2 ðx2 Þ . . . Bin , kn , tn ðxn Þ,
ð67Þ
where each variable xi possess its own knot sequence ti of length Ni and order ki. The problem of determining coefficients ci1 , i2 ,..., in can be solved efficiently by repeatedly solving univariate interpolation problems as described in de Boor (1978). Thus, (67) is used to represent ð"k Þ in (59), i.e. ð"k Þ ¼ fð"k Þ, and the parameter vector p of ðÞ consists of all coefficients ci1 , i2 ,..., in , i1 ¼ 1, . . . , N1 , . . . , in ¼ 1, . . . , Nn . Unfortunately, (62) cannot be differentiated with respect to p and hence gradient-based algorithm cannot be employed. Indeed, calculation of (62) involves N runs of the algorithm (34)–(38). Moreover, the cost function (62) can be even multi-modal and hence global optimization techniques should be preferred rather than local optimization tools. In this paper, it is proposed to use evolutionary search with soft selection (ESSS) algorithm (Galar 1989, Karcz-Dule¸ ba 2004 and Obuchowicz and Pre¸ tki 2004). The ESSS algorithm
7.1 Description of the application The complete discrete-time model in a stator-fixed (a, b) reference frame is
K 1 x1,kþ1 ¼x1,k þ h x1k þ x3k þ Kpx5k x4k þ u1k , Tr Ls ð69Þ
K 1 u2k , x2,kþ1 ¼x2,k þ h x2k Kpx5k x3k þ x4k þ Tr Ls
M 1 x1k x3k px5k x4k , x3,kþ1 ¼x3,k þ h Tr Tr
M 1 x2k þ px5k x3k x4k , x4,kþ1 ¼x4,k þ h Tr Tr
pM TL ðx3k x2k x4k x1k Þ x5,kþ1 ¼x5,k þ h , JLr J y1,kþ1 ¼x1,kþ1 , y2,kþ1 ¼ x2,kþ1 :
ð70Þ ð71Þ ð72Þ ð73Þ ð74Þ
Design of an extended unknown input observer where xk ¼ ½x1, k , . . . , xn, k T ¼ ½isak , isbk , rak , rbk , !k T represents the currents, the rotor fluxes, and the angular speed, respectively, while uk ¼ ½usak , usbk T is the stator voltage control vector, p is the number of the pairs of poles, and TL is the load torque. The rotor time constant Tr and the remaining parameters are defined as Lr M2 , ¼1 , Rr Ls Lr M Rs Rr M2 K¼ , ¼ þ , Ls Lr Ls Ls L2r
Tr ¼
ð75Þ
where Rs, Rr and Ls, Lr are stator and rotor per-phase resistances and inductances, respectively, and J is the rotor moment inertia. The numerical values of the above parameters are as follows: Rs ¼ 0:18, Rr ¼ 0:15, M ¼ 0:068 H, Ls ¼ 0:0699 H, Lr ¼ 0:0699 H, J ¼ 0:0586 kgm2 , TL ¼ 10 Nm, p ¼ 1, and h ¼ 0:1 ms. The input signals are u1,k ¼ 300 cosð0:03kÞ,
u2,k ¼ 300 sinð0:03kÞ:
ð76Þ
Let X be a bounded hypercube denoting the space of the possible variations of the initial condition x0 X ¼ ½276, 279 ½243, 369 ½15, 38 ½11, 52 ½11, 56 R5 :
ð77Þ
Let us assume that each initial condition of the system x0 is equally probable, i.e., 8 < 1 for x0 2 X prðx0 Þ ¼ mðXÞ : 0 otherwise, where mðAÞ is the Lebesgue measure of the set A.
Figure 1.
757
7.2 Observer design First let us start with the unknown input-free case, i.e., Ek ¼ 0. In order to completely define the indicator function (61), the following values of its parameters were chosen: T ¼ 30 and ¼ 0.001. This means that the main objective is to obtain the design parameter vector p in such a way as to minimize the probability (estimated by (62)) that the observer O will not converge in such a way that ke30 k2 =ke0 k2 0:001 for any initial condition x0 2 X where the initial condition for the observer is given by x^ 0 ¼ ½1:5, 63, 11:5, 20:5, 22:5T , which is the centre of X. ^ pÞ was selected as The standard deviation of Pð P^ ¼ 0:005 and hence, according to (65), N ¼ 10000. As can be observed from (74), the dimension of "k is m ¼ 2. Thus, the two-dimensional spline function of the degree kx1 ¼ kx2 ¼ 3 used in the experiment to approximate ð"k Þ is defined by the knot sequence 400 20 , tx1 ¼ tx2 ¼ 200 þ ði 1Þ 19 i¼1 consequently N1 ¼ N2 ¼ 20. The parameters related to the ESSS algorithm were selected as follows: the initial parameter vector p00 was randomly generated; the population size was nm ¼ 50; the maximum number of iterations tmax ¼ 1000, while the mutation parameters were ¼ 0.1, ¼ 1.5 and ¼ 0.01; see Pre¸ tki and Obuchowicz (2006) for details. Finally, the instrumental matrix Rk was Rk ¼ 103 I. As a results of using the ESSS algorithm for the minimization of (62) with respect to the design parameter vector p it was determined that P^ ¼ 0:0819. The shape of the resulting function ð"k Þ is presented in figure 1. In particular, figure 1b presents its shape in the whole domain of "k being considered, while figure 1a exhibits its small part around "k ¼ 0. It can
Shape of the obtained ð"k Þ.
758
Figure 2.
M. Witczak and P. Pre¸tki
(a) Trajectories of kek k2 for the proposed observer; (b) and the observer described in Boutayeb and Aubry (1999).
be observed in figure 1a that the value of ð"k Þ increases rapidly when "k starts to diverge from 0, which is consistent with the theoretical analysis performed in x 4. For the sake of comparison, the selection strategy of Qk proposed in Boutayeb and Aubry (1999) was employed, i.e., Qk ¼ 103 "Tk "k I þ 103 I. As a results, it was figured out that P^ ¼ 0:8858. This means that for the total number of N ¼ 10000 initial conditions the considered observer cannot provide an acceptable performance for 8858 cases. In the case of the proposed observer there are 819 unacceptable cases, which is definitely a better result. This situation is clearly exhibited in figure 2 (successful runs are denoted by the dark colour), which shows the trajectories of kek k2 (for all N ¼ 10000 cases) for the proposed observer figure 2a and the observer described in Boutayeb and Aubry (1999) figure 2b).
observer is designed, it is possible to check its performance with respect to an unknown input decoupling. For that purpose let us assume that the unknown input is given by dk ¼ 3:0 sinð0:5 kÞ cosð0:03 kÞ:
ð79Þ
Figure 3 presents the residual (output error) "k for the observer designed in x 7. From this figure, it is clear that the unknown input influences the residual signal and hence it may cause an unreliable fault detection (and, consequently, fault isolation). On the contrary, figure 4 shows the residual with unknown input decoupling. In this case the residual is almost zero, which confirms the importance of an unknown input decoupling. 7.4 Fault detection
7.3 Unknown input decoupling The objective of presenting the next example is to show the abilities of an unknown input decoupling. First, let us assume that the unknown input distribution matrix is E ¼ ½1:2, 0:2, 2:4, 1, 1:6T :
ð78Þ
Thus, the system (1)–(2) is described using (69)–(74) and (78). Since the system description is given it is possible to design the extended unknown input observer in the same way and with the same parameter values as that presented in x 7. As dk is unknown it is impossible to use it in the design procedure. On the other hand, only the knowledge regarding (78) is necessary and hence any form of dk can be used for the design purposes. In this paper, the following setting is used: dk ¼ 0. As a result of using the proposed approach, the function ð"k Þ was obtained for which P^ ¼ 0:0834. Since the
The objective of presenting the next example is to show the effectiveness of the proposed observer as a residual generator in the presence of an unknown input. For that purpose, the following fault scenarios were considered. Case 1: An abrupt fault of y1,k sensor f1, k ¼
0:1y1,k ,
140 < k < 500,
0,
otherwise,
ð80Þ
and f2,k ¼ 0. Case 2: An abrupt fault of u1, k actuator f2,k ¼ and f1,k ¼ 0.
0:2u1,k ,
140 < k < 500,
0,
otherwise:
ð81Þ
759
Design of an extended unknown input observer (a)
1
(b)
0.5
e2,k
0.5
e1,k
1
0
0
–0.5
–0.5
–1
–1 0
200
400
600
800
0
1000
200
Discrete time
Figure 3. (a)
600
800
1000
800
1000
Discrete time
Residuals for an observer without unknown input decoupling.
1
(b)
1
0.5
e2,k
0.5
e1,k
400
0
–0.5
0
–0.5
–1
–1 0
200
400
600
800
1000
Discrete time
Figure 4.
0
0 0 0 1 0 ¼ : 0 0
0
0
0
400
600
Residuals for an observer with unknown input decoupling.
0 0
L1,k ¼ L2,k
1 Ls
200
Discrete time
Thus, the system is now described by (20) and (21) with (69)–(74), (78), fk ¼ ½ f1, k , f2, k T , and "
0
#T ,
ð82Þ ð83Þ
From figures 5 and 6 it can be observed that the residual signal is sensitive to the faults under consideration. This, together with unknown input decoupling, implies that the process of fault detection becomes a relatively easy task.
8 Conclusions The main objective of this paper was to propose a novel structure and design procedure of an extended unknown
input observer. In particular, the paper analyzes two approaches to designing an UIO and shows their equivalence. The EUIO structure proposed in Witczak et al. (2002) is reminded and carefully analyzed with special attention on its convergence analysis. As a result of the presented discussion, a novel EUIO structure is proposed and its convergence is carefully analyzed with the Lyapunov method. It is shown that the achieved convergence condition is less restrictive than the one resulting from the approach described in Boutayeb and Aubry (1999). Another objective of the present paper was to propose an approach that can be used for increasing the convergence of the EUIO. To tackle such a challenging problem, the stochastic robustness technique was utilized to form a stochastic robustness metric describing an unacceptable performance of the EUIO. The paper proposes and describes a design procedure that can be used for minimizing such an unacceptable performance. In particular, it was shown that the observer performance can be significantly improved with appropriate
760
M. Witczak and P. Pre¸tki 0.1
(b) 0.1
0.05
0.05
0
0
e2,k
e1,k
(a)
–0.05
–0.05 Faulty mode
Faulty mode –0.1
–0.1 0
100
200
300
400
500
600
700
0
100
200
Discrete time
400
500
600
700
Discrete time
Figure 5. (a)
300
Residuals for a sensor fault.
0.2
(b) 0.1
0.15 0.1
0.05
0
e2,k
e1,k
0.05 0
–0.05 –0.1
–0.05 Faulty mode
Faulty mode
–0.15 –0.2
–0.1 0
100
200
300
400
500
600
700
0
100
Discrete time
Figure 6.
200
300
400
500
600
700
Discrete time
Residuals for an actuator fault.
selection of the instrumental matrix Qk . For that purpose, the B-spline approximation technique and evolutionary algorithms were utilized. The main advantage of the proposed EUIO is that its convergence rate is maximized for the whole set of possible initial conditions X but not only for a single and fixed initial condition as this was the case in Boutayeb and Aubry (1999) and Witczak et al. (2002). This superiority was clearly exemplified on an example regarding state estimation of an induction motor. Indeed, the performed experiments clearly show the profits that can be gained while using the proposed approach. The experiments also show the abilities of the proposed approach with respect to an unknown input decoupling. In particular, it was shown that the residual obtained without unknown input decoupling may impair the abilities of fault diagnosis. On the contrary, the residual provided by the proposed EUIO was almost zero during the normal operating mode of the system and considerably
different than zero when actuator and sensor faults occur.
Appendix: proof of Theorem 1 Proof: The main objective of further deliberation is to determine conditions for which the sequence fVk g1 k¼1 , defined by the Lyapunov candidate function Vkþ1 ¼ eTkþ1 P1 kþ1 ekþ1 ,
ð84Þ
is a decreasing one. Substituting (42) into (84) gives Vkþ1 ¼ eTk Ak T ak I CTkþ1 KTkþ1 P1 kþ1 ½I Kkþ1 Ckþ1 ak Ak ek :
ð85Þ
Design of an extended unknown input observer Using (38), it can be shown that
Applying (92) and (93) to (91) and then using (35) gives
I CTkþ1 KTkþ1 ¼ P1 kþ1=k Pkþ1 :
ð86Þ
Inserting (36) into ½I Kkþ1 Ckþ1 yields ½I Kkþ1 Ckþ1 ¼
h
Pkþ1=k P1 kþ1=k
þ Rkþ1 Þ1 Ckþ1 :
761
CTkþ1
Ckþ1 Pkþ1=k CTkþ1 ð87Þ
Substituting (86) and (87) into (85) gives h T T Vkþ1 ¼ eTk Ak T ak P1 kþ1=k Ckþ1 Ckþ1 Pkþ1=k Ckþ1 i 1 þRkþ1 Ckþ1 ak Ak ek :
2 Ak ðCkþ1 Þ2 Ak Pk Ak T þ Qk Ckþ1 Pkþ1=k CTkþ1 þ Rkþ1 ð1 Þ Ak Pk Ak T þ Qk þ , 2 kÞ Ak ðP
ðak Þ2 ðak Þ2
ð94Þ
which is equivalent to (45). Thus, if the condition (45) is satisfied then fVk g1 k¼1 is a decreasing sequence and hence, under Assumption 1, the proposed observer is locally asymptotically convergent. œ
ð88Þ Acknowledgments
The sequence fVk g1 k¼1 is decreasing when there exists a scalar , 0 < < 1, such that Vkþ1 ð1 ÞVk 0:
ð89Þ
Using (84) and (88), the inequality (89) can be written as h h T T eTk Ak T ak P1 kþ1=k Ckþ1 Ckþ1 Pkþ1=k Ckþ1 i þRkþ1 Þ1 Ckþ1 ak Ak : ð1 ÞP1 k ek 0:
References
ð90Þ
Using the bounds of the Rayleigh quotient for X 0, i.e. ðXÞ eTk Xek =eTk ek ðXÞ, inequality (90) can be transformed into the following form: T T T Ak T ak P1 kþ1=k ak Ak Ak k Ckþ1 Ckþ1 Pkþ1=k Ckþ1 1 þRkþ1 Ckþ1 ak Ak ð1 Þ P1 0: ð91Þ k It is straightforward to show that 2 2 1 a Þ Ak T ak P1 ðA Pkþ1=k , a A k kþ1=k k k k
ð92Þ
and 1 Ak T ak CTkþ1 Ckþ1 Pkþ1=k CTkþ1 þ Rkþ1 Ckþ1 ak Ak 2 2 2 1 Ckþ1 Pkþ1=k CTkþ1 þ Rkþ1 ak Ak Ckþ1 2 2 2 ak Ak Ckþ1 : ¼ Ckþ1 Pkþ1=k CTkþ1 þ Rkþ1
The authors would like to express their sincere gratitude to the referees, whose efforts have significantly improved the quality of the paper. The work was partially supported by the Ministry of Science and Higher Education in Poland.
ð93Þ
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