INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROLInt. J. Robust. Nonlinear Control (2013)Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/rnc.2973
New results in robust functional state estimation using two slidingmode observers in cascade
Chew Yee Kee1, Chee Pin Tan1,*,†, Kok Yew Ng1 and Hieu Trinh2
1School of Engineering, Monash University Sunway Campus, Jalan Lagoon Selatan,Bandar Sunway, 46150, Selangor, Malaysia
2School of Engineering, Deakin University, Waurn Ponds, Geelong, VIC 3217, Australia
Received 27 July 2012; Revised 10 January 2013; Accepted 19 January 2013
KEY WORDS: functional state estimation; sliding mode observer
1. INTRODUCTION
In many practical systems, the input and output of a system can be measured directly but not theinternal states. However, these internal states are often needed for the implementation of the statefeedback controller to ensure stability and to improve system performance. An observer [1] maybe used to estimate the states on the basis of the measurable input and output signals as well asthe model of the system. The state estimated by the observer can be used to implement the feed-back control law or even used to monitor the system condition [2]. However, estimating the entirestate vector may not be practical for high-dimensional system, as it requires high computationalpower and causes the resulting observer to be too sensitive to measurement noise [3, 4]. In manysituations, only part of the state information is required. For example, only the position vectorsare required for the feedback controller design in stabilizing the active magnetic bearing system [5].Apart from that, partial state estimation was found to be more accurate and less restrictive comparedwith the conventional observer [3, 6]. In this case, a functional observer [7], which only estimatesa function or subset of the state vector, can be implemented to reduce the observer complexity. Theproblem of observing a linear function of the states were first reported in [8]. Since then, a sig-nificant amount of research was conducted to obtain the existence conditions for minimum-orderfunctional observer [9–14]. A recent work by Rotella and Zambettakis [15] provides a relatively
*Correspondence to: Chee Pin Tan, School of Engineering, Monash University Sunway Campus, Jalan Lagoon Selatan,Bandar Sunway, 46150, Selangor, Malaysia.
simple solution for minimal-order observer design. However, an iterative process is required if thenecessary and sufficient conditions are not satisfied. Therefore, this approach is not suitable fora high-dimensional system, as it might require a higher number of iteration, which increases thecomputational complexity significantly. Design of functional observer for systems with unknowninput were first considered by Trinh and Ha [16] and Xiong and Saif [17]. Then, Trinh et al. [18]developed a scheme that required less restrictive existence conditions to include the case whereunknown inputs are more than outputs, which could be used in place of the well-known observermatching condition.
As opposed to the linear observers (where the output estimation error is injected linearly intothe observer and converges exponentially), a sliding mode observer receives the output estimationerror via a nonlinear switching function that drives the output estimation error to zero in finite time(whereby sliding motion is said to have occurred thereafter). During sliding motion, a reduced ordermotion takes place, and the estimate of the nonoutput states would converge to the actual nonoutputstates. Motivated by the work of [19], Walcott and Zak [20] formulated an observer design methodwhere under appropriate assumptions, the error system was proven to be quadratically stable despitethe presence of bounded uncertainties. Then, a systematic method were presented by Edwards andSpurgeon [21] to improve the observer design in [20]. The necessary and sufficient conditions (interms of the original system matrices) for the observer to exist and robustly estimate the state inde-pendent of the disturbance were given [22]; this is of particular importance as it firstly enables thedesigner to know the class of systems for which the observer can be implemented upon, and sec-ondly, it provides the basis of comparison for future works. Following the work of Edwards et al.[22], Tan et al. [23, 24] proposed a scheme using two sliding mode observers in cascade, where itwas found that signals from the first observer were analytically found to be the output of a ‘ficti-tious’ system, then a second sliding mode observer was implemented on the fictitious system; it wasfound that the scheme had less restrictive existence conditions compared with Edwards et al. [22].Then, Tan and Edwards [25] further extended the work by using multiple observers in cascade, andinvestigated the existence conditions that were found to be more relaxed. Sliding mode observerswere reported to have better performance compared with traditional linear observer (which includedunknown-input-observers) in comparative studies of different type of observers [26–28] and alsopractical applications, such as induction motor control [29–31], speed estimation for robotic manip-ulator [32], heat estimation for state monitoring purpose [2], and position control for pneumaticactuator [33]. They were found to be more robust towards parameters uncertainties and insensitiveto external noise in the aforementioned studies.
From the survey of literature, the functional observer approach can be used to eliminate redun-dancy possessed in full state estimation [3,4], whereas sliding mode observer has better performancecompared with the linear unknown input observer [28]. Thus, a combination of both observer theo-ries was believed to be able to improve the overall performance of the observer. However, not muchwork has been published on this particular topic. So far, only Fernando et al. [34] and Nikraz et al.[35] have utilized the discontinuous switching function in functional observers. Then, Nikraz et al.[35] extended the result of [34] to include a system with unknown input.
This paper makes two contributions to the area of functional observation, firstly to reformulate thesystem such that the two-observer scheme by Ng et al. [23] can be applied for functional state esti-mation, and secondly to investigate its existence conditions (in terms of the original system matrices)that were found to be less stringent than the one by [35]. From a physical/practical viewpoint, thiscould mean that less sensors for measurement are required, thus reducing cost and complexity ofthe overall physical system. Because the investigated conditions are in terms of the original systemmatrices, it enables the user to know from the outset whether the scheme is applicable or not. Inaddition, they will serve as the basis of comparison for future developments in this area.
This paper is organized as follows: Section 2 provides a general outline of the problem. Section 3presents a state transformations of the system and its reformulation so that the Edwards–Spurgeonobserver [21] can be applied to achieve function state estimation. Section 4 investigates and presentsthe necessary and sufficient conditions of the scheme. Section 5 validates the theory presentedin this paper with a numerical example, and finally, Section 6 concludes the result presentedin this paper.
Consider the linear time-invariant system described by
Px.t/D Ax.t/CDf.t/
y.t/D Cx.t/
´.t/D Fx.t/ (1)
where x.t/ 2 Rn, y.t/ 2 Rp , and f .t/ 2 Rq represent the state, output, and unknown disturbance,respectively, and ´.t/ 2 Rr is the vector to be estimated. A 2 Rn�n, B 2 Rn�m, C 2 Rp�n,D 2Rn�q , and F 2Rr�n are the known constant matrices. Without loss of generality, it is assumed
that rank.C / D p, rank.F / D r < n. Also assume that rank
�F
C
�D r C p; this is not a restric-
tive assumption, as it simply implies that the ‘output states’ will not be estimated because they arealready available for measurement.
The objective of this paper is to achieve functional state estimation, that is, to estimate ´.t/ (whichis a linear function of x) accurately (where O.t/ that is the estimate of ´.t/ will converge asymp-totically to ´.t/) using sliding mode observers. Nikraz et al. [35] have proven that a sliding modefunctional observer exist for system (1) if and only if the following conditions hold
A1. rank
264FA FD
CA CD
C 0
F 0
375D rank
24 CA CD
C 0
F 0
35
A2. rank
24 sF �FA �FD
CA CD
C 0
35D rank
264FA FD
CA CD
C 0
F 0
375 ,8s 2C,Re.s/> 0
This paper seeks to extend the results of Nikraz et al. [35] and provides new results in the area ofrobust functional state estimation by proposing a scheme using two sliding mode observers (similarto [23] and [25]) to estimate ´.t/ accurately (even when condition A1 is not satisfied) and to find itsexistence conditions. A schematic diagram of the scheme is shown in Figure 1.
3. SYSTEM FORMULATION
This section recasts the system (1) into a structure where existing sliding mode observer theorycan be directly applied to attain functional state estimation. This is achieved primarily by using thefollowing lemma.
Lemma 1There exist a linear state transformation x 7! Tox such that the system matrices (A, C , D, F ) from(1) have the following structure
Figure 1. Schematic diagram of the two sliding mode functional observer scheme proposed in this paper.
Because C has full row rank, a nonsingular change of coordinate, Tc D
"N TC
C
#exists, where
NC 2 Rn�.n�p/ spans the null space of C , that is, CNC D 0 such that the system matrices (A, C ,D, F ) can be transformed to be
A 7! Ac D TcAT�1c , D 7!Dc D TcD,
C 7! Cc D CT�1c D
�0 Ip
�, F 7! Fc D F T
�1c D
�Fc,1 Fc,2
�(3)
where Fc,1 2 Rr�.n�p/ has full row rank resulting from the assumption rank
�F
C
�D r C p. The
matrix C at the bottom of Tc will cause the special structure of the aforementioned Cc , whereasN Tc
was placed at the top of Tc to ensure that Tc is invertible. In this coordinate system, Ac andDc haveno specific structures. Then, define another change of coordinates
Tb D
24 N T
F 0
Fc,1 0
0 Ip
35 (4)
where NF 2 R.n�p/�.n�p�r/ is the null space of Fc,1. The matrices in (3) can then be transformedto become
Dc 7!Db D TbDc , Fc 7! Fb D FcT�1b D
�0 Ir Fb,2
�(5)
where Fb,2 2Rr�p . The purpose of this transformation is to achieve the special structure of Fb; the
top left .n�p/� .n�p/ subblock of Tb causes the first n�p columns of Fb , whereas Fb,2 has noparticular structure. Note that Cb D CcT �1b is not affected by this transformation, and Ab , Db have
no particular structures. Define hD rank
�Ab,21
Ab,31
�, and hence h6min¹r Cp,n�p � rº because�
Ab,21
Ab,31
�2R.rCp/�.n�p�r/. Let R1 be an orthogonal matrix such that
�Ab,21
Ab,31
�R�11 D
�0 Aa,22
0 Aa,23
�(6)
where Aa,22 2 Rr�h, Aa,23 2 Rp�h, and
�Aa,22
Aa,23
�has full column rank. The matrix R1 can be
calculated using the QR decomposition of
�Ab,21
Ab,31
�that can be easily performed in MATLAB.
Define Ta WD diag¹R1, IrCpº, then the matrices in (5) can be transformed to have the followingstructures
The matrices in (7) have the same structures as in (2), and the proof is complete. �
In the coordinate system of (2), let the state be transformed and partitioned as follows:
x.t/ 7! Tox.t/D
26664x111.t/
x211.t/
x12.t/
y.t/
37775l n� p � r � hl hl rl p
(8)
where To D TaTbTc . From the structure of F in (2), ´.t/ D x12.t/C F2y.t/; therefore, the statecomponents to be estimated would be x12.t/ and y.t/, whereas x111.t/ and x211.t/ are the ‘unwanted’states, which will be modeled as unknown inputs/disturbances. The system equations (1) in thecoordinates of (2) can be rewritten as follows:�
Px12Py
�„ ƒ‚ …
PNx
D
�A22 A23A32 A33
�„ ƒ‚ …
NA
�x12y
�„ ƒ‚ …
Nx
C
�A221 D2A231 D3
�„ ƒ‚ …
ND
�x211f
�„ ƒ‚ …
Nf
y D�0 Ip
�„ ƒ‚ …NC
�x12y
�„ ƒ‚ …
Nx
(9)
Notice that the system in (9) is in the same form as the system in Edwards and Spurgeon [21];therefore, the Edwards and Spurgeon observer [21] can be implemented on this system to estimateNx.t/ (and hence, x12.t/) if and only if
�NA, NC , ND, NF
�satisfies the following conditions
B1. rank�NC ND
�D rank
�ND�
B2. The invariant zeros of�NA, ND, NC
�(if any) must be stable
Upon further expansion and investigation, it can be shown that conditions A1 and A2 are equiv-alent to B1 and B2, respectively. However, when B1 is not satisfied, that is, b D rank
�NC ND
�<
rank�ND�, it may still be possible to accurately estimate x12 where in the work of Ng et al. [23],
Nx.t/ (and hence, x12 too) can be estimated using two sliding mode observers in cascade if and onlyif the following conditions are satisfied
C1. rank
�NC NA ND NC NDNC ND 0
�D rank
�NC ND
�C rank
�ND�
C2. rank
�sI � NA NDNC 0
�D pC r C rank
�ND�
, 8s 2C,Re.s/> 0
Condition C1 is less restrictive than B1, whereas B2 and C2 are equivalent. It is of interest torecast C1 and C2 in terms of the original system matrices in (2) so that the designer can know at theoutset whether the scheme in this paper is applicable or not.
4. EXISTENCE CONDITIONS
The main result of this paper is stated in the following theorem.
Theorem 1For the case when condition A1 (and B1) is not satisfied, the functional state, ´.t/, can be estimatedby the two-observer scheme given in Figure 1 if and only if the following conditions are satisfied:
The remainder of this section provides a constructive proof of Theorem 1.
To recast C1 and C2 in terms of the original matrices, a further transformation is performed on thematrices in (2) as detailed in the following lemma.
Lemma 2There exist a nonsingular tranasformation for x and f such that x 7! Tsx, f 7! Rsf and thematrices (A, C , D, F ) from (2) will have the following structure
Fs D�0 RF TF
�, Cs D
�0 0 C2
�(10)
where (*) are matrices that play no role in the proceeding analysis; A122, A222, A133, A533, and A933are the square matrices; and g, b, and c are to be defined later, whereas A2031 2 R
c�c , D1012 2 R
g�g ,D021 2R
.b�c/�.b�c/, RF 2Rr�r , and C2 2Rp�p are square and invertible.
ProofFirstly, to achieve the structure of As in (10), let rank
�A231
�D c, where c 6 h < p, whereas R2 and
R3 are the orthogonal matrices such that
R2A231R
�13 D
�0 0
0 A2031
�(11)
where A2031 2Rc�c is the full rank. Then, partition A221 D
�A2211 A2212
�in which A2211 2R
r�.h�c/.Define Tm to be
(12)
Hence, the matrix A in (2) can be transformed such that x 7! xm D Tmx to have the structure in(10) as follows
This transformation of Tn does not affect the special structure of Am in (13) (i.e. A2031 and theblock of zeros are unaltered). Through the transformation of TnTm, the matrix C will experiencea nonsingular scaling in the last p columns to produce C2 that is nonsingular, whereas the blockof zeros in the first n � p columns are unaltered. Then, the matrix F will be scaled in the last pcolumns (but was assumed to have no particular structure), whereas the preceding r columns arescaled nonsingularly to produce a nonsingular RF , and the other column of zeros is unaffected bythis transformation.
Now, An, Cn, Dn, and Fn in (20) are in the structure of (10), and the proof is complete. �
Lemma 3Conditions C1 and D1 are equivalent.
ProofTo firstly investigate the implication of C1, the matrices in (9) are written in the coordinates of (10)as follows:
(21)
(22)
NF D ŒRF TF � , NC D Œ0 C2� (23)
Substituting the matrices from (22) to (23) into NC ND yields
NC ND D TC
24 0 0 0 0 0 0
0 0 0 0 0 D021
0 0 A2031 D1122 D12
22 0
35 , (24)
Because A2031, D021, and D10
12 are the full-rank matrices of rank c, b � c, and g, respectively, it isstraightforward to see that
Lemmas 3 and 4 have proven that conditions D1 and D2 are equivalent to C1 and C2. Hence,if the original system matrices satisfy conditions D1 and D2, then a system in the form of (9) thatsatisfies conditions C1 and C2 can be obtained. A primary observer as illustrated in Figure 1 canbe implemented on system (9) by using the design procedure proposed by Edwards and Spurgeon[21]. However, when B1 is not satisfied, the estimate Ox12 from the primary observer is unable toaccurately track x12. Suppose the estimation error is e12 D Ox12 � x12. A secondary observer [23]can be used in cascade to accurately estimate e12, and hence accurately estimate x12 by the signalOx12 � Oe12 where Oe12 (from the second observer) is the (accurate) estimate of e12.
5. NUMERICAL EXAMPLE
A ninth-order aircraft model from Heck et al. [36] is used to demonstrate the observer schemeproposed in this paper. In the notation of (1), the system states and inputs are the following:
In this study, only the yaw rate and roll rate .rad=s/ need to be estimated. Thus, the matrix F isdefined as
F D
�0 1 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0 0
�(61)
Assume that the bank angle, roll angle, and yaw angle are measurable, and therefore, the matrixC becomes the following:
C D
24 1 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 0 1
35 (62)
It can be easily verified that the system matrices in (59)–(62) satisfy condition D1 and D2. How-ever, condition A1 is not satisfied, which means that the method by Nikraz et al. [35] cannot achieverobust functional state estimation for this system; it actually requires the sideslip angle and yaw rateto be measured as well. Hence, this confirms that the two-observer method in this paper can be usedto achieve robust functional state estimation with fewer sensors.
5.1. State transformation and system formulation
To achieve the structure in (2), the coordinate change in Lemma 1 is performed. Firstly, for Tc , asuitable choice of NC (such that CNC D 0) in Lemma 1 is
Nc D
2666666666664
0 0 0 0 0 0
0 0 0 0 1 0
0 0 0 0 0 1
1 0 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 0 0
0 0 0 0 0 0
3777777777775
(63)
which result in Fc,1 D
�0 0 0 0 1 0
0 0 0 0 0 1
�. Then, to obtain Tb , a suitable choice of NF (such
is chosen to achieve the structure in (6), and Ta is defined. Therefore, the matrices from (59) to (62)after going through the transformation To WD TaTbTc will achieve the structure of (2) are as follows
(67)
The structure of Co shows that the last three states are the output states in this coordinate system,whereas the structure of Fo indicates that the fifth and sixth states are the states to be estimated.
Next, rearrange system (67) into the form of (9) as follows
(68)
where the two observer design in [23] can be implemented directly. It can be easily verified thatsystem (68) does not satisfy condition B1 given in [21], but satisfies conditions C1 and C2, whichconfirms that two cascaded sliding mode observers can be used to accurately estimate Nx (and hence´) despite the disturbances Nf .
5.2. Design of observers
In this paper, the observers will be designed on the basis of the procedure given in [21] where thepoles of the (reduced order) sliding motion are a subset of the observer poles; this design proce-dure is not unique, and there are other available design methods for the observer [37, 38]. For thefirst observer, the poles of the sliding motion are chosen to be ¹�5,�10,�8º, and its associated
Lyapunov matrix is defined as Po D I3. The remaining poles are set to be ¹�3,�1º such that Gnand Gl in the coordinate system of (68) are found to be
Gn D
26664�0.002 �0.002 2.846
0 0 0.2491 0 0
0 1 0
0 0 1
37775 ,Gl D
26664�0.01 �0.021 22.329�0.001 �0.001 2.452
5 0 0.2490 10 0.249
�0.002 0.002 10.846
37775 (69)
As a result, the ‘fictitious’ dynamical system resulting from the sliding motion of the primaryobserver is as follows (see [23] for details):
�Pe12P�f
�„ ƒ‚ …
Pw
D
��3 0
0 �1
�„ ƒ‚ …
MA
�e12�f
�„ ƒ‚ …
w
C
�0 0 �0.68 1.57 0 0
0 0 �0.2 �5.326 0 0
�„ ƒ‚ …
MD
Nf
�N�1�f
�„ ƒ‚ …Ny
D
�0 �1�1 0
�„ ƒ‚ …
MC
�e12�f
�„ ƒ‚ …
w
(70)
where e12 is the estimation error of x12 from the primary observer. Because rank. MC MD/D rank. MD/,a secondary Edwards and Spurgeon observer [21] can be implemented on system (70) to estimatee12, and the state to be estimated, x12 can be obtained by subtracting the estimated x12 from the pri-mary observer and e12 from the secondary observer. From (70), it is clear that the number of states isequal to the number of outputs, and hence, there will be no (reduced order) sliding motion. Similarto the first observer design, the poles of the observer are placed at ¹�0.5,�0.8º, and its associatedLyapunov matrix is defined as Po,2 D I2. The gain matrices, Gn,2 and Gl ,2 for the second observerin the coordinate system of (70) are
The states are assumed to have an initial condition of
x.0/D
2666666666664
0.20
0
0.10.8�0.40.50.1�0.1
3777777777775
(72)
and f were set to be sinusoidal waveforms, where f1.t/ D 0.06 sin.1.81t � 0.03/ and f2.t/ D0.12 sin.1.22tC0.02/. The scalar gain for the nonlinear switching terms (for both observers), � and�2 were selected to be 10. Figure 2 shows the plot of the output estimation error from the primaryobserver, where sliding motion has taken place at around 0.02 s. Figure 3 shows that Ox12 (which isthe estimate of x12 from the primary observer) does not track the actual x12. The output estimationerror for the second observer is shown in Figure 4, where sliding motion happens at 0.05 second.Figure 5 shows Ox12 � Oe12 (where Ox12 is from the first observer and Oe12 from the second observer)that accurately tracks x12.
0 1 2 3 4 5 6 7 8 9 10
−0.25
−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
0.25
0 1 2 3 4 5 6 7 8 9 10
−0.25
−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
0.25
Figure 3. The estimated Ox12.t/ from primary observer does not follow the actual x12 because conditionsA1 and B1 were not satisfied, and the Edwards and Spurgeon observer [21] is not applicable. The dashed
Figure 4. Output estimation error ey2.t/D Ny.t/� ONy.t/ from secondary observer. The sliding motion of thefirst observer takes place at around 0.02 s, whereas sliding motion for the second observer happens at around
0.05 s after which all components of ey2.t/ were forced to zero.
Figure 5. The estimated state obtained from Ox12.t/ � Oe12 (solid) converge to the actual x12 (dotted) afterthe sliding motion occurs for the second observer at around 0.05 s.
6. CONCLUSION
This paper has presented a new scheme in functional state estimation, using two sliding modeobservers in cascade. The original system was expressed in a form such that existing sliding modeobserver theory could be implemented to achieve functional state estimation. Then, the necessaryand sufficient conditions in terms of the original system matrices were investigated and were foundto be less stringent than existing sliding mode functional estimation work, which could mean thatfunctional state observation can be achieved with fewer sensors thus saving cost and reducing com-plexity. The investigated conditions in terms of the original system matrices are very useful, as theyenable the user to know from the outset whether or not the scheme is applicable for their system,and also provides a basis for comparison of further developments in functional state estimation.Finally, simulation results from a ninth-order aircraft model showed the effectiveness of the func-tional observer scheme proposed in this paper. Future work could focus on the use of higher-ordersliding modes [39] to further relax the existence conditions and enhance the scheme.
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