This article was downloaded by: [McGill University Library] On: 26 September 2012, At: 16:21 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK International Journal of Control Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/tcon20 On sliding mode observers via equivalent control approach Ibrahim Haskara Version of record first published: 08 Nov 2010. To cite this article: Ibrahim Haskara (1998): On sliding mode observers via equivalent control approach, International Journal of Control, 71:6, 1051-1067 To link to this article: http://dx.doi.org/10.1080/002071798221470 PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: http://www.tandfonline.com/page/ terms-and-conditions This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae, and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand, or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.
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This article was downloaded by: [McGill University Library]On: 26 September 2012, At: 16:21Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number:1072954 Registered office: Mortimer House, 37-41 Mortimer Street,London W1T 3JH, UK
International Journal ofControlPublication details, including instructions forauthors and subscription information:http://www.tandfonline.com/loi/tcon20
On sliding mode observersvia equivalent controlapproachIbrahim Haskara
Version of record first published: 08 Nov2010.
To cite this article: Ibrahim Haskara (1998): On sliding mode observers viaequivalent control approach, International Journal of Control, 71:6, 1051-1067
To link to this article: http://dx.doi.org/10.1080/002071798221470
PLEASE SCROLL DOWN FOR ARTICLE
Full terms and conditions of use: http://www.tandfonline.com/page/terms-and-conditions
This article may be used for research, teaching, and private studypurposes. Any substantial or systematic reproduction, redistribution,reselling, loan, sub-licensing, systematic supply, or distribution in anyform to anyone is expressly forbidden.
The publisher does not give any warranty express or implied or makeany representation that the contents will be complete or accurateor up to date. The accuracy of any instructions, formulae, and drugdoses should be independently verified with primary sources. Thepublisher shall not be liable for any loss, actions, claims, proceedings,demand, or costs or damages whatsoever or howsoever caused arisingdirectly or indirectly in connection with or arising out of the use of thismaterial.
In this paper, sliding mode observer design principles based on the equivalentcontrol approach are discussed for a linear time invariant system both in contin-uous and discrete time. For the continuous case, the observer is designed using arecursive procedure; however, the observer is eventually expressed as a replica ofthe original system with an additional auxiliary input with a certain nested struc-ture. A direct discrete time counterpart of the sliding mode realization of a reducedorder asymptotic observer using the discrete time equivalent control is also devel-oped. Simulation of a linearized truck-trailer during a manoeuvre illustrates theapproach. Results show the e� ectiveness and the ® nite time convergence character-istics of the proposed discrete and continuous time sliding mode observers.
1. Introduction
The problem of state estimation using sliding modes has been considered forseveral years and uses the same design theory and reasonings with variable structurecontrollers. A standard Luenberger observer reconstructs the original state vectorasymptotically. However, in sliding mode observers, a suitable non-linear outputinjection is used to guarantee ® nite time convergence by a deliberate introductionof sliding mode.
An early sliding mode observer has appeared as the sliding mode realization of areduced order asymptotic observer (Utkin 1977, 1992). In this formulation, the orig-inal system is ® rst transformed into a canonical form in which the output variablesform a part of the state vector. Sliding mode is induced on the known output errorvariables in ® nite time by a discontinuous injection and the remaining observer statesconverge to the associated plant states asymptotically by the equivalent value of thisinjection in sliding mode. Later, the equivalent control methodology has beenexploited from di� erent perspectives and used in the observer design for a linearsystem transformed into the block form using sequential application of state trans-formations (Hashimoto et al. 1990, Drakunov et al. 1992, Drakunov and Utkin1995).
In this paper, ® rst the continuous time sliding mode observer design based on theequivalent control is discussed using a step-by-step procedure as in Drakunov andUtkin (1995). Unlike the previous results, the partial observer equations designed ateach step are gathered so as to express the ® nal observer simply as a replica of theoriginal system with an additional innovation term containing the output error andthe equivalent control ® lters with a certain structure. A discrete time sliding moderealization of a reduced order asymptotic observer via the discrete time equivalentcontrol is proposed. Simulation results are also provided.
0020-7179/98$12.00 Ñ 1998 Taylor & Francis Ltd.
INT. J. CONTROL, 1998, VOL. 71, NO. 6, 1051± 1067
Received April 1998.² The Ohio State University, Department of Electrical Engineering, 2015 Neil Ave. 205
Dreese Labs, Columbus, OH 43210, USA.³ e-mail: ozguner.2@osu. edu
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2. Continuous time sliding mode observers
Consider
Çx = Ax + Bu
y = Cx(1)
where x Î Rn, u Î R
p, y Î Rm , A, B, C are known constant matrices of appropriate
dimensions. The pair {C,A} is observable and rank {C}= m. The problem consid-ered is to estimate the n-dimensional state vector from the m-dimensional outputvector in the framework of sliding mode theory.
The states of the standard Luenberger observer
Çx = Ax + Bu + L (y - Cx) (2)
would converge to the true states asymptotically with a proper observer gain matrixL and the convergence rate would be adjusted by a proper selection of the eigen-values of (A - LC) . However, ® nite time convergence can also be guaranteed viasliding modes as follows.
The system given by equation (1) can be transformed into the following canonicalform
Çy = A11y + A12x1 + B1u
Çx1 = A21y + A22x1 + B2u(3)
by a non-singular transformation matrix
P =CR1
(4)
where R1 Î R(n- m) ´ n is arbitrary as long as P is non-singular and x1 7 R1x.
The corresponding sliding mode observer related to the variable y in equation (3)is written as a replica of the system with an additional nonlinear innovation asfollows:
Çy = A11y + A12x1 + B1u + L 1 sgn (y - y) (5)
where y and x1 denote the observer states associated with the states y and x1 of thetransformed system, L 1 Î R m ´ m is a free parameter to be chosen and the signumfunction is de® ned columnwise, i.e.
sgn
y1
y2
..
.
yl- 1
yl
=
sgn (y1)sgn (y2)
..
.
sgn (yl- 1)sgn (yl)
with
sgn (yi) =1 if yi > 0
- 1 if yi < 0
Subtracting equation (5) from the ® rst part of equation (3), one gets:
Çey = A11ey + A12ex1 - L 1 sgn ey (6)
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where ey = y - y and ex1 = x1 - x1. Selecting L 1 as a positive de® nite, diagonalmatrix with su� ciently large entries, ey and Çey are guaranteed to have di� erentsigns, componentwise, so that sliding mode takes place on the manifold
S = {ex|ey = 0} (7)
in ® nite time. Let the rank of A12 Î R m ´ (n- m) be m1. Clearly m1 < m, m1 < (n - m)and there exists a D1 Î R
m1 ´ m such that rank {D1A12}= m1. At this stage, a newvariable y1 = D1A12x1 with y1 Î R
m1 is de® ned and combined with the second partof equation (3) to form a new system as follows:
Çx1 = A22x1 + A21y + B2u
y1 = D1A12x1
(8)
This system has exactly the same form with the original one, where x1 is the state, y1
is the output and y and u are the inputs. The new system is observable since theobservability of the pair {C,A} implies the observability of the pair {D1A12,A22}.Furthermore, the dimension of the system has reduced to (n - m) . Now, a validquestion is whether the same procedure can be used to also reconstruct some statesof this reduced order system in ® nite time. If so, we will provide the ® nite timeconvergence of more than m states of the original system and hopefully come upwith a full, ® nite time converging observer sequentially following this algorithm. Theanswer to this question depends on the availability of y1. This looks very unrealisticat ® rst glance because y1 is itself a linear combination of the unknown states andhence it is not available directly. However, this information can be extracted fromthe system using the equivalent control method.
The equivalent control is de® ned as the average value of the discontinuous con-trol in sliding mode which is itself enough to keep the state trajectories on themanifold. Mathematically, it is the solution of the dynamic equation associatedwith the generalized control variable for the input which induces sliding modeafter imposing the constraints of sliding mode (Utkin 1977, 1992). In practice,ideal sliding mode cannot exist due to imperfections and trajectories chatter aroundthe manifold instead. The discontinuous input can be considered as a combination ofthe slow equivalent control term of the ideal sliding motion and a high frequencyswitching term. In sliding mode, when this discontinuous input is ® ltered by a lowpass ® lter whose cuto� frequency is less than the switching frequency but higher thanthe maximum frequency of the system dynamics, the high frequency oscillations inthe input are eliminated and the ® lter output yields the equivalent control which is acontinuous state function (Utkin 1977, 1992, Drakunov and Utkin 1995). Whenconsidering the prospective sliding mode observer, the L 1 sgn ey term is the inputwhich enforces sliding mode and its equivalent value should be found. Solvingequation (6) for L 1 sgn ey after replacing ey and Çey by zero produces
[L 1 sgn ey]eq = A12ex1
D1[L 1 sgn ey]eq = ey1
(9)
Therefore, even though y1 is not directly available, the error information in thisvariable is hidden inside the observer auxiliary input. Realizing the equivalentvalue operator as a ® rst order low pass ® lter
On sliding mode observers 1053
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¹1 Çz1 + z1 = L 1 sgn ey (10)
[L 1 sgn ey]eq can be replaced by z1 and equation (9) is rewritten as follows:
D1z1 = ey1(11)
Using the same procedure, equation (8) is ® rst transformed into
Çy1 = A31y1 + A32x2 + A33y + B3u
Çx2 = A41y1 + A42x2 + A43y + B4u(12)
with a non-singular transformation matrix
P1 = D1A12
R2
where R2 Î R(n- m- m1) ´ (n- m) is arbitrary as long as P1 is invertible and x2 7 R2x1.
The sliding mode observer for the ® rst part of equation (12) is built as follows:
where L 2 Î R m1 ´ m1 is a diagonal, positive de® nite matrix. Subtracting equation (13)from the ® rst part of equation (12) provides
Çey1 = A31ey1 + A32ex2 + A33ey - L 2 sgn ey1(14)
where ey1 = y1 - y1, ex2 = x2 - x2 and sliding mode takes place on
S 1 = {ex|ey1 = 0 ´ ey = 0} (15)
in ® nite time choosing L 2 properly to guarantee that ey1 and Çey1 have di� erent signs,componentwise. Using equation (11), equation (13) is rewritten as follows:
with L 1 (ey) = L 1 sgn ey. De® ning a new manifold S k- 1
S k- 1 = {ex|eyk- 1 = 0 ´ S k- 2} (26)
and choosing a diagonal, positive de® nite L k with su� ciently large entries, the errortrajectories are steered to this manifold. At the next step, a new variableyk = DkA2k- 1,2xk is de® ned, a new system of a lower dimension than the one stepbefore is formed, this system is transformed into a new state space realization and soon and so forth.
At each step, the dimension of the state vector of the new system becomes lessthan that of the previous system. Therefore, this process is presumed to terminate ina ® nite number of steps. Since rank{A2k- 1,2}= mk ³ 0 and n - m - k- 1
i=1 mi ³ mk,there exists an integer r such that n = m + r
i=1 mi . At this ® nal stage, de® ning a newvariable yr = DrA2r- 1,2xr with an appropriate Dr Î R mr ´ mr- 1 , the following system isobtained:
Note that, since Dr is chosen such that rank {DrA2r- 1,2}= mr, xr and yr are relatedwith a nonsingular transformation. However, to keep the pattern, the system ofequation (27) is also transformed into
and eyr = yr - yr becomes zero in ® nite time with a proper selection of L r+1 (ey) .To sum up, initially the observer error vector is arbitrarily located in the error
space. If the ® rst observer parameter L 1 is chosen properly, error trajectories aresteered to the ® rst manifold which has been de® ned as S = {ex|ey = 0}and slidingmode takes place on it. After that, the error trajectories are also steered to a subset ofthe previous manifold called the second manifold which has been de® ned asS 1 = {ex|ey = 0 ´ ey1 = 0} passing the ® rst observer input through a suitable lowpass ® lter to get ey1 and choosing L 2 with su� ciently large entries. As the processcontinues, error trajectories eventually converge to the origin of the estimation errorspace travelling from one manifold to the next manifold which is a subspace of theprevious one.
The observer equations designed at each step can be gathered to form the com-plete observer as follows:
The observer is not clearly in exact state space form since it contains xk ’s whichcan be expressed in terms of yk ’ s only if all Dk ’ s are invertible. Even if this is the casethe observer state variables di� er from those of the system considerably. Further-more, the ® nal reconstructed observer state vector will be a transformed version ofthe original state vector. However, it would be more appropriate to design the slidingmode observer and implement it in the original state space realization. This can beachieved by retransforming every single observer equation into the form of theprevious step using the non-singular transformations backward starting from the® nal step.
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Retransforming equation (29) via P- 1r , combining it with the observer equation
for the variable yr- 1 and also retransforming the resulting system by P- 1r- 1 once more,
the last two observer equations can be expressed as follows:
Note that, the initial realization of the step r has now been retrieved. Using the sameprocedure, the auxiliary observer inputs inserted at each step backpropagate andappear as an n-dimensional output innovation with a certain nested structure to theoriginal system replica.
Figure 1 shows the structure of the observer and Theorem 1 summarizes theresult without going into further detail.
Theorem 1: The sliding mode observer which reconstructs the state vector ofequation (1) in ® nite time is as follows:
Çx = Ax + Bu + P- 1
L 1 (ey)
P- 11
L 2 (ey)
P- 12
L 3 (ey)
P- 13
. ..
P- 1r- 1
L r (ey)
P- 1r L r+1 (ey)
(32)
On sliding mode observers 1057
Complete Observer
1
-1-1
1
1
t s + 1s + 1 t t
1
1
-1
.......
Equivalent
Control
Block
sign
+
Discontinuous Observer Input
Plant Input U Plant Output Y
Plant Input UReconstructed State Vector
Replica of the Original System
Original System
Observer Output
sign Low pass filter Low pass f ilter sign
-
Figure 1. Structure of the continuous time sliding mode observer.
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where ey = y - Cx;
P =C
R1
PAP- 1 =A11 A12
A21 A22
P1 =D1A12
R2
P1A22P- 11 =
A31 A32
A41 A42
..
. ...
Pr- 1 =Dr- 1A2r- 3,2
RrPr- 1A2r- 2,2P
- 1r- 1 =
A2r- 1,1 A2r- 1,2
A2r,1 A2r,2
Pr = DrA2r- 1,2
where A2k- 1,2 Î Rmk- 1 ´ n- k- 1
i=0m i , rank{A2k- 1,2}= mk with m0 = m; Dk Î R
mk ´ mk- 1
is chosen such that rank{DkA2k- 1,2}= mk , r is an integer such that n = ri=0 mi, Rk ’s
are arbitrary with appropriate dimensions as long as Pk- 1’ s are invertible and
L k (ey) = L k sgn (Dk- 1zk- 1)
¹k- 1 Çzk- 1 + zk- 1 = L k- 1 (ey)
with L 1 (ey) = L 1 sgn (ey) where L k Î Rmk- 1 ´ mk- 1’s are positive de® nite, diagonal
matrices and ¹k ’s are the equivalent control ® lter time constants.
We also note the following:
� The observer given in Theorem 1 is in the original state space realization of thesystem. Therefore, this theorem can be applied directly without following thestep by step construction procedure for a given system. Once the Pk ’s, Dk ’s andthe integer mk ’ s have been determined as described, the innovation term has acertain structure.
� L k ’s are diagonal, positive de® nite matrices and their entries should be selectedsu� ciently large to induce sliding mode, but without being too conservativeso as not to cause considerable chattering due to a possible discrete timeimplementation. Each equivalent control operator can be implemented by asuitable low pass ® lter, as written explicitly in Theorem 1, and its cuto�frequency should be chosen so as to ® lter out the high frequency switchingterms without eliminating the slow system dynamics.
� The sliding mode observer design has been discussed for a linear time invariantsystem. However, the design idea presented is also applicable for non-linearsystems with certain structures.
� The idea can also be used for derivative estimation in practice. This can behelpful in case several signal derivatives which are not available directly aredesired to incorporate into the controller design.
� If the states of the system are available but there are unknown non-linearitiesand/or disturbances, it is possible to estimate these terms employing asliding mode observer for each state, inducing sliding mode with a properdiscontinuous function of the available state error and extracting the equiva-
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lent value of this discontinuous injection in sliding mode by an equivalentcontrol ® lter.
3. Discrete time sliding mode observers
In the previous section, a continuous time sliding mode observer has beendesigned. However, the discrete time implementation of the observer equationwhich contains discontinuous sgn functions can cause a sampling time orderoscillation around the values of the real states. Theoretically, these high frequencyoscillations can be removed with an in® nite switching frequency. However, practicalimplementation issues of computer control require the extension of continuous timesliding mode design procedures to discrete time. In this section, the sliding moderealization of a reduced order asymptotic observer is proposed with this motivation.
Consideration is given to the following discrete time system:
xk+1 = U xk + C uk
yk = Cxk
(33)
where x Î R n, u Î R p, y Î R m and F , G , C are constant matrices of appropriatedimensions. The pair {C, F } is observable and C has full rank without loss ofgenerality. The problem considered is to reconstruct the state vector from the outputin the discrete time sliding mode framework. First, the system given in equation (33)is transformed into the following canonical form:
yk+1 = U 11yk + U 12x1,k + C 1uk
x1,k+1 = U 21yk + U 22x1,k + C 2uk
(34)
using a proper non-singular state transformation.The corresponding sliding mode observer using the new realization of the system
is written as a replica of equation (34) with an additional auxiliary input term
yk+1 = U 11yk + U 12x1,k + C 1uk - vk
x1,k+1 = U 21yk + U 22x1,k + C 2uk + L vk
(35)
where vk Î Rm and L Î R
(n- m) ´ m . The corresponding error dynamics can be foundsubtracting equation (35) from equation (34)
ey,k+1 = U 11ey,k + U 12ex1,k + vk
ex1,k+1 = U 21ey,k + U 22ex1,k - Lvk
(36)
where ey = y - y and ex1 = x1 - x1.Examining equation (36), the state estimation problem can be modi® ed into an
equivalent problem of ® nding an auxiliary observer input vk in terms of the availablequantities so that the observer error vector components ey and ex1 are steered to zeroin a ® nite number of steps.
For discrete time sliding mode observer design, the sliding manifold is alsode® ned as
S = {ex|ey = 0} (37)
which is simply a subregion of the n-dimensional state space composed by theregions where the error trajectories can travel when the estimation error in the out-
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put variable y is kept at zero. If the error trajectories are on this manifold at eachsampling instance, we say that the observer is in discrete time sliding mode. Betweenthe two sampling instances, trajectories are allowed to deviate from the manifold,however, the deviations do not exceed the order of the sampling time.
According to the chosen manifold, the equivalent control is found by solving theauxiliary observer input with the constraint ey,k+1 = 0 adopting the discrete timeequivalent control de® nition of (Drakunov and Utkin 1989):
vk,eq = - U 11ey,k - U 12ex1,k (38)
If the equivalent control is applied to the system, sliding mode occurs at the next stepand ex1 satis® es the following dynamical relation
ex1,k+1 = ( F 22 + L F 12)ex1,k (39)
and converges to zero choosing a proper L such that the matrix ( U 22 + L U 12) has thedesired eigenvalues. Placing the eigenvalues of the matrix ( U 22 + L U 12) at the desiredlocations requires the observability of the pair {U 12, F 22}. However, the observabil-ity of the pair {C, F } is su� cient for the previous observability requirement asdiscussed before. This is exactly a sliding mode realization of the standard reducedorder asymptotic observer. After sliding mode takes place on the manifold, theobservation error in the variable x1 also converges to zero and our objective isachieved.
However, the equivalent observer auxiliary input cannot be provided since itcontains an unknown term, ex1 . This is the main di� erence of the discrete time slidingmode observer design procedure from the continuous one. In the continuous case,we also do not know the values of some terms at the right-hand side. However, sincethe control aim at that stage is to steer the states towards the sliding manifold,observer parameter L can be chosen large enough so as to suppress the e� ects ofthe unknown terms. As to the discrete time case, a discontinuous term is undesirablebecause of the ® nite switching frequency. Therefore, it becomes necessary to make allthe terms at the right-hand side available to compute the auxiliary input whichbrings error trajectories to the manifold.
To compute the equivalent auxiliary observer input, equation (36) is combinedinto a single equation:
ex1,k = ( U 22 + L U 12)ex1,k- 1 + ( F 21 + L U 11)ey,k- 1 - Ley,k (40)
and the unknown ex1,k term in the equivalent observer input is replaced by zk+1
which is given by
zk+1 = U zk + C (41)
where U = U 22 + L U 12 and C = ( U 21 + L U 11)ey,k- 1 - Ley,k .Note that equation (40) is not a di� erence equation; instead it describes the
constraint for the error variables imposed by the deliberate coupling of the errordynamics. The dynamical equation of zk is exactly the same as that of ex1,k andthe di� erence between zk+1 and ex1,k goes to zero at most at (n - m) steps forany vk independent of the initial error choosing all the eigenvalues of thematrix ( U 22 + L F 12) at the origin. Therefore, the equivalent observer input can beupdated as:
vk,eq = - ( U 11 - F 12 L )ey,k - F 12 ( F 21 + L F 11)ey,k- 1 - U 12 ( U 22 + L F 12)zk (42)
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The estimated equivalent input vk,eq converges to the actual equivalent input veq
after (n - m) steps placing all the eigenvalues of the matrix ( U 22 + L U 12) at theorigin and the discrete time sliding mode takes place at the (n - m + 1)th step. Insliding mode, ex1 which represents the rest of the composite error vector other thaney converges to zero satisfying the relation
ex1,k+1 = ( F 22 + L F 12)ex1,k (43)
Since all the eigenvalues of ( U 22 + L F 12) have already been assigned as zero, ex1 alsoconverges in a ® nite number of steps and the objective is achieved. Note that, insliding mode the total observer motion is decomposed into two independent motionswhich show a zero input characteristic. Furthermore, the eigenvalues of these sub-motions are controlled by the same parameter.
In summary, the complete observer consists of two layers as shown in ® gure 2.The ® rst layer is used to estimate the unknown error terms which are necessary in theequivalent input calculation and the second layer is the layer where the state estima-tion is done. The original system is transformed into a canonical form only once andthere is no ® ltering block as in the continuous time case. Theorem 2 summarizesthe results.
Theorem 2: The discrete time sliding mode observer is
xk+1 = U xk + C uk - P- 1 Im ´ m
- Lvk
where
vk = - ( U 11 - U 12 L )ey,k - U 12 ( F 21 + L U 11)ey,k- 1 - F 12 ( F 22 + L U 12)zk
ey,k = yk - Cxk
P is the similarity transformation matrix;
P =CR
where R Î R(n- m) ´ n is arbitrary provided that P is invertible
PU P- 1 =U 11 U 12
U 21 U 22, PC =
C 1
C 2
with U 11 Î Rm ´ m , U 12 Î Rm ´ (n- m) , U 21 Î R(n- m) ´ m and U 22 Î R(n- m) ´ (n- m) ; zk is thestate of the following error driven system
On sliding mode observers 1061
yu
y
Main Observer Block
^
ey z v
y
x
^
^
Original System
Complete Observer
z- dynamics
( F , G ) ( F , G , L)Equivalent input
( F , G )
Figure 2. Structure of the discrete time sliding mode observer.
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zk+1 = U zk + Gwith
U = ( U 22 + L U 12), C = ( U 21 + L U 11)ey,k- 1 - L ey,k
L is the observer gain matrix which places the eigenvalues of ( U 22 + L U 12) at theorigin.
4. S imulations
In this section, the performance of the proposed continuous and discrete timesliding mode observers is tested in simulations.
The non-linear bicycle truck-trailer system illustrated in ® gure 3 is linearizedaround a constant longitudinal speed with a default parameter set and the followingstate space model is obtained:
: Lateral velocity of the tractor: Yaw rate of the tractor: Yaw rate of the trailer: Articulation angle: Steer angle on the front tyres of the tractor
The problem is to reconstruct the state vector of the system during a manoeuvreusing the articulation angle. For the continuous case, Theorem 1 is directly appliedwith
P =
0 0 0 11 0 0 00 1 0 00 0 1 0
, P1 =0 - 1 11 0 00 1 0
, P2 =13.9852 52.90671 0
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Y
Fy1Fy2
Fx
FyU1
V1
r1
CG1P
CG2
r2
V2
U2
Fy3
a1
b1
b2
d1e2
Tf / Rf
d
Figure 3. Bicycle truck-trailer model.
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On sliding mode observers 1063
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 21.5
1
0.5
0
0.5
1
1.5
time (sec)
Est
imat
ed
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 21.5
1
0.5
0
0.5
1
1.5
time (sec)
Rea
lLateral velocity (m/sec) Continuous case
Figure 4. (a) Continuous time observer (state 1).
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 21
0.5
0
0.5
1
time (sec)
Est
imat
ed
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 21
0.5
0
0.5
1
time (sec)
Rea
l
Tractor yaw rate (rad/sec) Continuous case
Figure 4. (b) Continuous time observer (state 2).
Dow
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1064 ÇI . Haskara et al.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 21
0.5
0
0.5
1
time (sec)
Rea
lTrailer yaw rate (rad/sec) Continuous case
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 21
0.5
0
0.5
1
time (sec)
Est
imat
ed
Figure 4. (c) Continuous time observer (state 3).
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20.4
0.2
0
0.2
0.4
time (sec)
Rea
l
Articulation angle (rad) Continuous case
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20.4
0.2
0
0.2
0.4
time (sec)
Est
imat
ed
Figure 4. (d) Continuous time observer (state 4).
Dow
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16:
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On sliding mode observers 1065
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 21.5
1
0.5
0
0.5
1
1.5
time (sec)
Rea
lLateral velocity (m/sec) Discrete case
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 21.5
1
0.5
0
0.5
1
1.5
time (sec)
Est
imat
ed
Figure 5. (a) Discrete time observer (state 1).
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 21
0.5
0
0.5
1
time (sec)
Rea
l
Tractor yaw rate (rad/sec) Discrete case
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 21
0.5
0
0.5
1
time (sec)
Est
imat
ed
Figure 5. (b) Discrete time observer (state 2).
Dow
nloa
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by [
McG
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Lib
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] at
16:
21 2
6 Se
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2012
1066 ÇI . Haskara et al.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 21
0.5
0
0.5
1
time (sec)
Est
imat
ed
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 21
0.5
0
0.5
1
time (sec)
Rea
lTrailer yaw rate (rad/sec) Discrete case
Figure 5. (c) Discrete time observer (state 3).
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20.4
0.2
0
0.2
0.4
time (sec)
Rea
l
Articulation angle (rad) Discrete case
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20.4
0.2
0
0.2
0.4
time (sec)
Est
imat
ed
Figure 5. (d) Discrete time observer (state 4).
Dow
nloa
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by [
McG
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16:
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6 Se
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2012
P3 = - 163.0964, mi’ s and Di’ s are all 1, the switching frequency (simulation stepsize) is selected as 104 Hz and the equivalent control operators are implemented by® rst-order ® lters with time contants of 0.5, 1 and 5 ms where the ® rst ® lter is for L 1,the second one is for L 2 and so on.
For the discrete case, the continuous time model has been discretized with asampling time of 0.02 s and used in Theorem 2 with the following parameters:
P =
0 0 0 11 0 0 00 1 0 00 0 1 0
, L =672.0839
- 235.4982
- 312.2196
Figures 4 and 5 illustrate the simulation results. The continuous time slidingmode observer reconstructs the state vector within a small time interval withoutfacing chattering problems. The discrete time sliding mode observer also provides® nite time convergence, but at the expense of huge initial oscillations because of thedeadbeat response.
5. Conclusions
The sliding mode observer design methods based on the equivalent control meth-odology have been studied. The continuous time sliding mode observer design hasbeen carried out by a sequential procedure and summarized as a theorem for a directimplementation in the original state space realization of the given plant. A discretetime extension of the sliding mode realization of a reduced order asymptotic obser-ver has been proposed. Simulation results have also been presented to show thevalidity of the theoretical work.
References
Drakunov, S. V., and Utkin, V. I., 1995, Sliding-mode observers. Tutorial Proceedingsof the 34th IEEE Conference on Decision and Control, New Orleans, LA, USA,pp. 3376± 3378.
Drakunov, S. V., Izosimov, D. B., Luk’ayonov, A. G., and Utkin, V. I., 1992, The blockcontrol principle. Automation and Remote Control, 6, 737± 746.
Hashimoto, H.,Utkin, V. I.,Xu, J. X., Suzuki,H., and Harashima, H., 1990, VSS observerfor linear time varying systems. IECON’90, pp. 34± 39.
Haskara, ÇI., Oï zguï ner, Uï ., and Utkin, V. I., 1997, Variable structure control for uncertainsampled data systems. Proceedings of the 36th IEEE Conference on Decision and Con-trol, San Diego, CA, USA, pp. 3226± 3231.
Utkin, V. I., 1977, Variable structure systems with sliding modes. IEEE Transactions onAutomatic Control, AC-22, 212± 222.
Utkin, V. I., 1992, Sliding Modes in Control and Optimization (Berlin, Heidelberg: Springer-Verlag).
Utkin, V. I., and Drakunov, S. V., 1989, On discrete-time sliding mode control, in Proceed-ings of IFAC Symposium on Nonlinear Control Systems (NOLCOS), Capri, Italy,pp. 484± 489.
