2009 American Control Conference Hyatt Regency Riverfront, St. Louis, MO, USA June 10-12, 2009
WeB16.3
Sliding-Mode Observers for Uncertain Systems ˙ Karanjit Kalsi, Jianming Lian, Stefen Hui and Stanislaw H. Zak
Abstract— Sliding-mode observer design is considered for linear systems with unknown inputs when the so-called observer matching condition is not satisfied. To circumvent the restriction imposed by the observer matching condition, the method of utilizing auxiliary outputs generated by high-order slidingmode exact differentiators in the sliding-mode observer design has been proposed in the literature. In this paper, an alternative approach is proposed to use high-gain approximate differentiators of simpler architecture instead of high-order slidingmode exact differentiators. The capability of reconstructing the unknown inputs using the proposed high-gain approximate differentiator based sliding-mode observer is also discussed and then illustrated with a numerical example.
I. I NTRODUCTION Unknown input observer (UIO) has been developed to estimate the states of the system with inputs that are unknown or partially known. Linear UIO architectures that have been developed for linear system are presented in [1]–[5]. UIO architectures for non-linear systems with unknown inputs have been reported in [6], [7]. Motivated by the design of slidingmode controllers, first-order sliding mode based UIOs have been discussed in [5], [8]–[10]. The main advantage of sliding-mode observers over their linear counterparts is that while in sliding, they are insensitive to the unknown inputs, and, moreover, they can be used to reconstruct unknown inputs which could be a combination of system disturbances, faults or non-linearities. The reconstruction of unknown inputs has found impressive applications in fault-detection and isolation [4], [9], [10]. The necessary and sufficient conditions for the existence of most of the unknown input observers proposed thus far are that the observer matching condition is satisfied and the invariant zeros of the system involving unknown input are in the open left half complex plane. However, the observer matching condition seriously restrict the applicability of sliding mode observers. Recently, high-order sliding mode based unknown input observers [11]–[14] have been developed to overcome this restrictive condition. In [13], a change of coordinates is performed using a constructive algorithm to transform the system into a quasi-block triangular observable form. Then a step-by-step second order slidingmode observer is constructed for the transformed system. In [14], auxiliary outputs are defined so that the conventional unknown input sliding-mode observer proposed in [9] can ˙ are with the School Karanjit Kalsi, Jianming Lian and Stanislaw H. Zak of Electrical and Computer Engineering, Purdue University, IN 47907, USA. {kkalsi,jlian,zak}@purdue.edu. Stefen Hui is with the Department of Mathematical Sciences, San Diego State University, San Diego, CA 92182, USA.
[email protected].
978-1-4244-4524-0/09/$25.00 ©2009 AACC
be constructed for systems that do not satisfy the observer matching condition. In this paper, we design the sliding-mode observer presented in [8] for systems that do not satisfy the observer matching condition. We adopt the idea of auxiliary outputs used in [14], but propose an alternative approach for the generation of auxiliary outputs. We use high-gain observers rather than high-order sliding-mode observers to obtain the estimates of auxiliary outputs. The high-gain observer is often referred to as approximate differentiator [15]. The proposed high-gain approximate differentiator based slidingmode observer can achieve good state estimation performance. The advantage of our developed technique is that the overall observer architecture is simpler than the highorder sliding-mode exact differentiator based sliding-mode observer proposed in [14]. We also discuss the capability of the proposed high-gain approximate differentiator based sliding-mode observer in the unknown input reconstruction, which is then illustrated with a numerical example. II. S YSTEM D ESCRIPTION AND P ROBLEM S TATEMENT We consider the following class of linear time-invariant systems with unknown inputs x˙ = Ax + B 1 u1 + B 2 u2 (1) y = Cx, where x ∈ Rn , y ∈ Rp , u1 ∈ Rm1 and u2 ∈ Rm2 are the state, output, known and unknown input vectors, and B 1 ∈ Rn×m1 , B 2 ∈ Rn×m2 and C ∈ Rp×n are known constant matrices. For the above system, we assume that 1) B 2 and C are of full rank, that is, rank B 2 = m2 and rank C = p, and m2 ≤ p; 2) there is ρ > 0 such that ku2 (t)k ≤ ρ for all t, where k·k denotes the standard Euclidean norm; 3) the invariant zeros of the system model given by the triple (A, B 2 , C) are in the open left-hand complex plane, or equivalently, sI n − A B 2 rank = n + m2 . (2) C O for all s such that ℜ(s) ≥ 0. It follows from [8] that, if the so-called observer matching condition [13] is also satisfied for the system modeled by (1), that is, rank B 2 = rank(CB 2 ) = m2 , (3) ˙ sliding-mode observer, we can construct the Walcott-Zak
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ˆ ) − B 2 E(y, y ˆ , η) x ˆ˙ = Aˆ x + B 1 u1 + L (y − y
(4)
ˆ = Cx ˆ and with y F (yˆ −y ) η F yˆ −y )k k ( ˆ , η) = E(y, y 0
B. High-Gain Observer Construction if F (ˆ y − y) 6= 0
(5)
if F (ˆ y − y) = 0,
where η is a positive design parameter, L ∈ Rn×p and F ∈ Rm2 ×p are matrices such that ⊤
(A − LC) P + P (A − LC) = −2Q < 0 and F C = B ⊤ 2 P for some symmetric positive definite P ∈ Rn×n and Q ∈ Rn×n . The design procedures for the matrices Li , F i and P oi is given in [5]. However, many physical systems that can be modeled by (1) do not satisfy the observer matching condition (3). The observer matching condition (3) is sometimes too restrictive in practical applications. III. H IGH -G AIN A PPROXIMATE D IFFERENTIATOR In this section, we propose a high-gain approximate differentiator based sliding-mode observer for the systems that do not satisfy the observer matching condition. A. Auxiliary Output Signals We first define as in [14] the auxiliary outputs that are then used to construct the sliding-mode observer. Let ci be the ith row of the output matrix C. Recall that the relative degree of the i-th output yi with respect to the unknown input u2 is defined to be the smallest positive integer ri such that ci Ak B 2 = 0, ci Ari −1 B 2 6= 0.
k = 0, . . . , ri − 2
In [14], high-order sliding-mode observers have been employed to obtain the auxiliary outputs in y a . We propose to use high-gain observers to estimate the auxiliary outputs instead. The reason behind this is because they have simpler architectures than high-order sliding-mode observers. To proceed, we let yij = ci Aj−1 x, i = 1, . . . , p and ⊤ ⊤ j = 1, . . . , γi . Thus, we have y a = [y ⊤ a1 · · · y ap ] , where ⊤ y ai = [yi1 · · · yiγi ] . If γi > 1, the dynamics of y ai , i = 1, . . . , p, are given by ¯i1 fi (x, u2 ) + ¯ ¯ i y ai + b y˙ ai = A bi2 u1 (6) yi1 = ¯ ci y ai ,
¯i1 ) is in canonical controllable form ¯ i, b where the pair (A representing the chain of γi integrators, fi (x, u2 ) = ci Aγi x + ci Aγi −1 B 1 u2 ,
¯i2 = [ci B 1 · · · ci Aγi −1 B 1 ]⊤ and ¯ b ci = [1 0 · · · 0]. We assume, as in [14], that x and x˙ are bounded and |yij | ≤ dij , which implies that u1 is bounded. If γi > 1, we construct the following high-gain observers, yˆ˙ i1 = yˆi2 + αǫi1 (yi1 − yˆi1 ) + ci B 1 u1 .. . α i −1) yˆ˙ i(γi −1) = yˆiγi + ǫi(γ (yi1 − yˆi1 ) + ci Aγi −2 B 1 u1 γi −1 α γi −1 i ˙yˆiγ = iγ (y − y ˆ ) + c A B u , γ i1 i1 i 1 1 i ǫ i (8) where ǫ ∈ (0, 1) is a design parameter and αij , j = 1, . . . , γi , are selected so that the roots of the equation, sγi +αi1 sγi −1 + · · · + αi(γi −1) s + αiγi = 0, have negative real parts. Let y hi = [ˆ yi1 · · · yˆiγi ]⊤ and li = [αi1 /ǫ · · · αiγi /ǫγi ]⊤ . We can rewrite (8) as ¯i2 u1 . ¯ i y hi + li ¯ ci (y ai − y hi ) + b y˙ hi = A
We can choose integers γi (1 ≤ γi ≤ ri ) such that c1 .. . c1 Aγ1 −1 .. Ca = . cp .. .
(7)
(9)
If γi = 1, we do not need to construct the above high-gain observer (9) because of the availability of yi1 . In such a case, we have y hi = y ai = yi1 . To proceed, let ζ i = 0 if γi = 1 and let ζ i = [ζi1 · · · ζiγi ]⊤ if γi > 1, where yij − yˆij , j = 1, . . . , γi . ǫγi −j It follows from (6) and (9) that if γi > 1, we have ζij =
cp Aγp −1
¯i1 fi (x, u2 ), ¯ ci ζ i + ǫb ǫζ˙ i = A
is of full rank with rank(C a B 2 ) = rank B 2 . It is proved in [14] that the system zeros of the system model given by the triple (A, B 2 , C a ) are in the open left-hand complex plane if the triple (A, B 2 , C) satisfies (2). Thus, we can construct the sliding-mode observer of the form (4) for the following system model x˙ = Ax + B 1 u1 + B 2 u2 y a = C a x, if the output y a = C a x is available. However, some components of the vector y a are not measurable and, therefore, additional observers are needed to estimate them.
(10)
(11)
¯ ci = ǫD−1 (A ¯ i − li c ¯i )D i is a Hurwitz matrix where A i independent of ǫ. Applying the method in [16], we can prove the following proposition. Proposition: For the high-gain observer (9), there exists a finite time Ti (ǫ) such that kζ i (t)k ≤ βi ǫ for some positive constant βi and t ≥ t0 + Ti (ǫ). Moreover, Ti (ǫ) approaches zero when ǫ approaches to zero, that is, limǫ→0+ Ti (ǫ) = 0. It follows from (10) that y ai − y hi = Di ζ i , where D i = ⊤ ⊤ diag[ǫγi −1 ǫγi −2 · · · 1]. Let y h = [y ⊤ h1 · · · y hp ] , D = ⊤ ⊤ ⊤ diag[D 1 · · · Dp ] and ζ = [ζ 1 · · · ζ p ] . We have
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y a − y h = Dζ.
(12)
Note that the induced Euclidean norm of D is 1, that is, kDk = 1. Let βi = 0 and Ti (ǫ) = 0 if γi = 1. Thus, itPfollows from the proposition that kζk ≤ βǫ, where β = 1 p ( i=1 βi2 ) 2 , after a finite time T (ǫ) = max1≤i≤p Ti (ǫ), and limǫ→0 T (ǫ) = 0.
e(t) is bounded for t0 ≤ t ≤ t0 + T (ǫ). For t ≥ t0 + T (ǫ), because y sh (t) = y h (t) and y h = y a − Dζ, the dynamics of the state estimation error (15) become ˆ a ) − B 2 u2 − B 2 E a (y h , y ˆ a , η) e˙ = Ae + La (y h − y = (A − La C a ) e − La Dζ − B 2 u2 ˆ a , η). − B 2 E a (y h , y (16)
IV. S TATE E STIMATION P ERFORMANCE A NALYSIS In order to eliminate the peaking phenomena that accompanies the operation of the above high-gain observer [17], we introduce the saturation of the signal y h such that y sh = [y sh1 ⊤ · · · y shp ⊤ ]⊤ , where y shi = y ai = yi1 if γi = 1 and ⊤ yˆiγi yˆi1 s · · · Siγi sat y hi = Si1 sat Si1 Siγi
Consider the Lyapunov function candidate V = 21 e⊤ P a e for t ≥ t0 + T (ǫ). Evaluating the time derivative of V on the solutions of (16), we obtain V˙ = e⊤ P (A − La C a ) e − e⊤ P a La Dζ ˆ a , η) − e⊤ P a B 2 u2 − e⊤ P a B 2 E a (y h , y
= −e⊤ Qa e − e⊤ P a La Dζ ˆ a , η) − (F a C a e)⊤ u2 − (F a C a e)⊤ E a (y h , y
with Sij > dij if γi > 1. Then we construct the following sliding-mode observer,
= −e⊤ Qa e − e⊤ P a La Dζ − (F a C a e + F a Dζ)⊤ u2 ˆ a , η) − (F a C a e + F a Dζ)⊤ E a (y h , y ˆ a , η). + (F a Dζ)⊤ u2 + (F a Dζ)⊤ E a (y h , y
ˆ a )−B 2 E a (y sh , y ˆ a , η), (13) x ˆ˙ = Aˆ x +B 1 u1 +La (y sh − y ˆ and ˆa = Cax where y F a (yˆ a −y sh ) η F yˆ −y s if F a (ˆ y a − y sh ) 6= 0 s k a ( a h )k ˆ a , η) = E a (y h , y 0 if F a (ˆ y a − y sh ) = 0.
If F a (C a e + Dζ) = 0, then −(F a C a e + F a Dζ)⊤ u2
− (F a C a e + F a C a e)⊤ E a = 0.
where La ∈ Rn×γ and F a ∈ Rm2 ×γ are matrices such that ⊤
On the other hand, if F a (C a e + Dζ) 6= 0, then
(A − La C a ) P a + P a (A − La C a ) = −2Qa < 0
−(F a C a e + F a Dζ)⊤ u2 − (F a C a e + F a C a e)⊤ E a = −(F a C a e + F a Dζ)⊤ u2 F a C a e + F a Dζ − η(F a C a e + F a Dζ)⊤ kF a C a e + F a Dζk ≤ −(η − ρ)kF a C a e + F a Dζk ≤ 0. (18)
and F aC a =
B⊤ 2 Pa
(17)
(14)
for some symmetric positive definite P a ∈ Rn×n and Qa ∈ Rn×n . It follows from (1) and (13) that ˆ a ) − B 2 u2 − B 2 E a (y sh , y ˆ a , η). (15) e˙ = Ae + La (y sh − y In the following, we analyze the performance of the proposed high-gain approximate differentiator based sliding-mode observer given by (13). Theorem 1: For the dynamical system (1) and the associated sliding-mode observer (13) with high-gain approximate differentiators (9), there exists a constant ǫ∗ ∈ (0, 1) such that if ǫ ∈ (0, ǫ∗ ) and η ≥ ρ, then the state estimation error e(t) is uniformly ultimately bounded. Specifically, after a finite time Tf (ǫ), we have ke(t)k ≤ κ(ǫ), where s p κ1 ǫ + κ21 ǫ2 + 4µa κ1 ǫ 2 κ(ǫ) = 2µa λmin (P a ) for positive constants µa , κ1 and κ2 . Proof: It follows from the proposition that kζ(t)k ≤ βǫ for t ≥ t0 + T (ǫ). Then, it follows from (12) that kya (t) − y h (t)k ≤ βǫ for t ≥ t0 +T (ǫ). There exists a constant ǫ¯ such that if kya (t) − y h (t)k ≤ β¯ ǫ, then y h (t) is not saturated, that is, y sh (t) = y h (t). Thus, we can choose ǫ∗ = min{¯ ǫ, 1} such that if ǫ ∈ (0, ǫ∗ ), then kζ(t)k ≤ βǫ and y sh (t) = y h (t) after a finite time T (ǫ). For t0 ≤ t ≤ t0 + T (ǫ), it is guaranteed that the observer ˆ (t) in (13) is bounded because u1 , y sh and state vector x s ˆ E a (y h , y a , η) are bounded and A−La C a is Hurwitz. Thus,
It follows from (17) and (18) that in both cases we have V˙ ≤ −e⊤ Qa e − e⊤ P a La Dζ
ˆ a , η) + (F a Dζ)⊤ u2 + (F a Dζ)⊤ E a (y h , y 2 ≤ −λmin (Qa )kek + βǫkP a La kkek
+ (η + ρ)βǫkF a k √ (19) = −2µa V + κ1 ǫ V + κ2 ǫ, q √ where κ1 = 2βkP a La k/ λmax(P a ) and κ2 = (η +
ρ)βkF a k. It follows from (19) that √ V˙ ≤ −µa V − µa V + κ1 ǫ V + κ2 ǫ √ √ = −µa V − V − R− V − R+ ,
where
R− =
κ1 ǫ −
p κ21 ǫ2 + 4µa κ2 ǫ 0. 2µa
and R+ =
(20)
We conclude from (20) that V˙ < 0 when kek > R+ . Thus, the state estimation error e is uniformly ultimately bounded with respect to any closed ball of radius greater than R+ .
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√ 2 Hence, as V > R+ , that is, V > R+ , we √ as long √ have ( V − R− )( V − R+ ) < 0. Therefore, if V (t0 + 2 2 T (ǫ)) = V (e(t0 + T (ǫ))) > R+ and V (t) > R+ for t ≥ t0 + Tf (ǫ), then V˙ ≤ −µa V , which implies that V (t) ≤ exp (−µa (t − t0 − T (ǫ))) V (t0 + T (ǫ)). Thus, we 2 can find a finite time Tf (ǫ) such that V (t) ≤ R+ for t ≥ t0 + T (ǫ), where Tf (ǫ) is the solution to the equation 2 V (t0 + T (ǫ)) exp(−µa (Tf (ǫ) − T (ǫ))) = R+ as 1 V (t0 + T (ǫ)) Tf (ǫ) = T (ǫ) + . ln 2 µa R+ 2 2 On the other hand, if V (t0 + T (ǫ)) ≤ R+ , then V (t) ≤ R+ for t ≥ t0 + T (ǫ). In such a case, we can choose Tf (ǫ) = T (ǫ). Therefore, there exists a finite time Tf (ǫ) such that 2 V (t) ≤ R+ for t ≥ t0 + Tf (ǫ), which implies that ke(t)k ≤ κ(ǫ). The proof of the theorem is complete. Remark: It follows from Theorem 1 that the state estimation error enters the closed ball {e : kek ≤ κ(ǫ)} after a finite time Tf (ǫ). It is easy to verify that ∞ if V (t0 ) 6= 0 lim Tf (ǫ) = 0 if V (t0 ) = 0, ǫ→0+
because limǫ→0+ T (ǫ) = 0 and limǫ→0+ R+ = 0. Moreover, the radius of the above closed ball can be adjusted by the design parameter ǫ and because limǫ→0+ κ(ǫ) = 0, the state estimation error e converges to the origin as ǫ goes to zero. Theorem 2: For sufficiently large η, the sliding surface, {(e, ζ) : σ = F a (C a e + Dζ) = 0} is invariant in the (e, ζ)-space and is reached in finite time. ⊤ ⊤ Proof: Let ζ = [ζ ⊤ 1 · · · ζ p ] . Using (11) for γi > 1 and the fact that ζ i = 0 if γi = 1, we obtain ¯ c ζ + ǫB ¯ 1 f (x, u2 ), ǫζ˙ = A (21) ¯ c = diag[A ¯ c1 · · · A ¯ cp ], B ¯ 1 = diag[b ¯11 · · · b ¯p1 ] where A ¯ ¯ with Aci = O and bi1 = 0 if γi = 1 and f (x, u2 ) = [f1 (x, u2 ) · · · fp (x, u2 )]⊤ . Because x and u2 are bounded, we have kf (x, u2 )k ≤ β1 for some β1 > 0. For t ≥ t0 + Tf (ǫ), it follows from (16) and (21) that σ ⊤ σ˙ = σ ⊤ F a C a e˙ + F a D ζ˙ ⊤ F a C a (A − La C a )e − F a C a La Dζ =σ − F a C a B 2 u2 − F a C a B 2 E a 1 ¯ c ζ + F aDB ¯ 1 f (x, u2 ) + F aDA ǫ ≤ κ(ǫ)kF a C a (A − La C a )kkσk ¯ 1 kkσk + βǫkF a C a La kkσk + β1 kF a kkB
where
+ λmax (B ⊤ 2 P a B 2 )ku2 kkσk ¯ − ηλmin (B ⊤ a kkAc kkσk 2 P a B 2 )kσk + βkF κ3 + κ4 + κ5 + κ6 + κ7 κ8 kσk, (22) =− η− κ8
κ3 = κ(ǫ)kF a C a (A − La C a )k, κ4 = βǫkF a C a La k, ¯ c k, κ5 = ρλmax (B ⊤ κ6 = βkF a kkA 2 P a B 2 ), ¯ 1 kkf(x, u2 )k, κ8 = λmin (B ⊤ P a B 2 ). κ7 = kF a kkB 2
It follows from (22) that if we choose η such that κ3 + κ4 + κ5 + κ6 + κ7 + ε, η≥ κ8 where ε is a small positive constant, then σ ⊤ σ˙ ≤ −εkσk,
(23)
which implies the above hyperplane is invariant. Let Ts denote the time the sliding surface is reached. Using the same arguments as in [9, p. 53], we obtain kσ(t0 + Tf (R))k . ε Thus, the proof of the theorem is complete. Ts ≤ t0 + Tf (R) +
V. U NKNOWN I NPUT R ECONSTRUCTION It follows from Theorem 2 that the manifold {(e, ζ) : σ = F a (C a e+Dζ) = 0} is invariant and is reached after a finite time. Therefore, we have σ˙ = F a C a (A − La C a )e − F a C a La Dζ − F a C a B 2 u2 ˆ a , η) + F a D ζ˙ = 0. (24) − F a C a B 2 E a (y sh , y Substituting (14) into (24) and performing simple manipulations, we obtain −1 F a C a (A − La C a )e + F a D ζ˙ u2 = B ⊤ 2 P aB2 ˆ a , η).. F a C a La Dζ − E a (y sh , y (25)
By the proposition, we have kζ(t)k ≤ βǫ for t ≥ t0 + T (ǫ). By Theorem 1, we have ke(t)k ≤ κ(ǫ) for t ≥ t0 + Tf (ǫ), where limǫ→0+ κ(ǫ) = 0. Therefore, for sufficiently small ǫ, kζ(t)k and ke(t)k becomes negligible after a finite time. If, ˙ in addition, kζ(t)k becomes negligible for sufficiently small ǫ, it follows from (25) that after a finite time, ˆ a , η). u2 ≈ −E a (y sh , y
(26)
That is, we can use the proposed architecture to estimate the unknown input u2 for sufficiently small ǫ. ⊤ ⊤ ˙ Recall that ζ = [ζ ⊤ 1 · · · ζ p ] and ζ i = ζ i = 0 if γi = 1. Thus, in order to obtain (26), it remains to show that if γi > 1, then kζ˙ i (t)k becomes negligible for sufficiently small ǫ. We first rewrite (11) as 1¯ (27) ζ˙ i (t) = A ci ζ i (t) + v i (t), ǫ ¯i1 fi (x(t), u2 (t)). Because x(t) and u2 (t) where v i (t) = b are bounded, it follows from (7) that fi (x(t), u2 (t)) is bounded. Thus, v i (t) is bounded. To proceed, we define two notions regarding the function v i (t). Definition 1: A function v i (t) is left-continuous if limǫ→0+ v i (t − ǫ) = v i (t) for all t. Definition 2: A function v i (t) defined on S ⊂ R is weakly uniformly continuous if for every ν > 0, there exists a δ > 0 such that for each interval Ω ⊂ S with length less than δ, kv i (s) − v i (t)k < ν for s, t ∈ Ω. In the following, we use S1 + S2 , where S1 , S2 ⊂ R, to denote the set {s1 + s2 : s1 ∈ S1 , s2 ∈ S2 }. If S1 or S2 is empty, then S1 + S2 is defined to be empty.
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Let J denote the set of points at which v i (t) is discontinuous and let τ > t0 > 0. It can be shown, as in [18], that if v i (t) is left-continuous, then limǫ→0+ ζ˙ i (t) = 0 for each t > t0 ≥ 0. Moreover, if v i (t) is also weakly uniformly continuous on [τ, ∞)\J, then the convergence of ζ˙ i (t) to 0 as ǫ → 0+ is uniform on [τ, ∞)\(J + (0, ξ)) for each ξ > 0. In particular, if v i (t) is uniformly continuous, then the convergence is uniform on [τ, ∞). The detailed proof of this result can be found in [18], where a more general case regarding v i (t) is also considered.
1.5 y 12 y12 estimate
1 0.5 0 −0.5 −1 −1.5 0
2
4 6 time (sec)
8
10
1.5
VI. N UMERICAL E XAMPLE
y 13 y13 estimate
1
In this section, we illustrate the effectiveness of our proposed high-gain approximate differentiator based slidingmode observer with a numerical example. Our simulations demonstrate that its performance is quite similar to that of the high-order sliding-mode exact differentiator based sliding-mode observer. Due to lack of space, we only show simulations with the high-gain approximate differentiator based sliding-mode observer. We consider a linear time invariant system modeled by 0 1 0 0 0 0 0 1 0 0 0 0 1 0 A= 0 , 0 0 0 0 1 −1 −5 −10 −10 −5 0 0 0 0 1 0 0 0 0 B 2 = 0 −1 , C = . 0 0 0 1 0 1 0 0 0
We do not consider B 1 , because we set u1 = 0 for simplicity. The initial condition is selected to be x(0) = [0.5 0.5 0.5 −0.5 −0.5]⊤ . The unknown input u2 consists of a square wave of amplitude 1 and frequency 1Hz, and a sawtooth signal of amplitude 2 and frequency 1Hz. It is easy to check that for this system rank(CB 2 ) 6= rank B 2 because c1 B 2 = 0. Thus, we choose γ1 = r1 = 3 such that c1 1 0 0 0 0 c1 A 0 1 0 0 0 Ca = c1 A2 = 0 0 1 0 0 0 0 0 1 0 c2
0.5 0 −0.5 −1 −1.5 0
Fig. 1.
2
4 6 time (sec)
8
10
True and estimated auxiliary outputs.
η = 50 to obtain 6 1 0 0 0 7 0 0 0 La = 0 0 2.0659 , 0 0 0 2.0659 0 0 0 0
⊤ 0 0 0 0 Fa = 0 −1 . 1 0
We set the initial states of the sliding-mode observer to be ˆ (0) = 0, and select S11 = S12 = S13 = 1.5. zero, that is, x In Fig. 2, we show the state estimation performance. The unknown inputs reconstruction is illustrated in Fig. 3. VII. C ONCLUSIONS A novel sliding-mode observer has been proposed for systems with unknown inputs, where the observer matching condition is not satisfied. High-gain approximate differentiators were employed to estimate auxiliary outputs that are then used by the sliding-mode observer to estimate the states and reconstruct the unknown inputs. The proposed observer has simple architecture and performs comparably to the highorder sliding-mode exact differentiator based sliding-mode observer in [14].
is of full rank with rank(C a B 2 ) = rank B 2 . We employ a high-gain observer to estimate the auxiliary outputs y12 = c1 Ax and y13 = c1 A2 x. The design parameters of the highgain observer are selected to be α11 = 3, α12 = 3, α13 = 1 and ǫ = 0.001. The estimated and true values of the auxiliary outputs are shown in Fig. 1. Now we use the estimates of the auxiliary outputs to construct the sliding-mode observer described by (13). Following the algorithm given in [5], we use κ = 2.0659 and
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R EFERENCES [1] M. Hou and P. M¨uller, “Design of observers for linear systems with unknown inputs,” IEEE Trans. Autom. Control, vol. 37, no. 6, pp. 871–875, Jun. 1992. [2] M. Darouach, M. Zasadzinski, and S. Xu, “Full-order observers for linear systems with unknown inputs,” IEEE Trans. Autom. Control, vol. 39, no. 3, pp. 606–609, Mar. 1994. [3] M. Corless and J. Tu, “State and input estimation for a class of uncertain systems,” Automatica, vol. 34, no. 6, pp. 757–764, 1998. [4] J. Chen and R. Patton, Robust Model-Based Fault Diagnosis for Dynamical Systems. Norwell, Massachusetts: Kluwer Academic Publishers, 1999.
3 u 21 u21 estimate
1.5 x 1 x1 estimate
2
1
1 0
0.5 −1 0
−0.5 0
−2 −3 0 2
4 6 time (sec)
8
2
10
4 6 time (sec)
8
10
4 u 22 u22 estimate
1 x 2 x2 estimate
3 2 1
0.5
0 −1 0
−2 −3
−0.5 0
−4 0 2
4 6 time (sec)
8
10
Fig. 3.
1
0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 2
4 6 time (sec)
8
10
1 x 4 x4 estimate
0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 0
2
4 6 time (sec)
8
10
2 x5 x estimate
1.5
5
1 0.5 0 −0.5 −1 −1.5 0
Fig. 2.
4 6 time (sec)
8
10
Unknown input reconstruction.
x 3 x3 estimate
0.8
−1 0
2
2
4 6 time (sec)
8
10
True and estimated system states.
˙ [5] S. Hui and S. H. Zak, “Observer design for systems with unknown inputs,” Int. J. Appl. Math. Comput. Sci, vol. 15, no. 4, pp. 431–446, 2005. [6] J.-J. Slotine, J. K. Hedrick, and E. A. Misawa, “On sliding observers for nonlinear systems,” ASME J. Dyn. System Measurement Control, vol. 109, pp. 245–252, 1987. [7] Y. Xiong and M. Saif, “Sliding mode observer for nonlinear uncertain systems,” IEEE Trans. Autom. Control, vol. 46, no. 12, pp. 2012–2017, Dec. 2001. ˙ [8] B. Walcott and S. H. Zak, “State observation of nonlinear uncertain dynamical systems,” IEEE Trans. Autom. Control, vol. 32, no. 2, pp. 166–170, Feb. 1987. [9] C. Edwards and S. K. Spurgeon, Sliding Mode Control: Theory and Applications. London, UK: Taylor and Francis Group, 1998. [10] C. Edwards, S. K. Spurgeon, and R. J. Patton, “Sliding mode observers for fault detection and isolation,” Automatica, vol. 36, pp. 541–553, 2000. [11] L. Fridman, A. Levant, and J. Davila, “Observation of linear systems with unknown inputs via higher order sliding modes,” Int. J. Systems Science, vol. 38, no. 10, pp. 773–791, Oct. 2007. [12] F. J. Bejarano, L. Fridman, and A. Poznyak, “Exact state esimation for linear systems with unknown inputs based on hierarchical super twisting algorithm,” Int. J. Robust Nonlinear Contr., vol. 17, no. 18, pp. 1734–1753, Mar. 2007. [13] T. Floquet and J. P. Barbot, “A canonical form for the design of unknown input sliding mode observers,” in Advances in Variable Structure and Sliding Mode Control, C. Edwards, E. F. Colet, and L. Fridman, Eds. Berlin: Springer, 2006, vol. 334. [14] T. Floquet, C. Edwards, and S. K. Spurgeon, “On sliding mode observers for systems with unknown inputs,” Int. J. Adapt. Control Signal Process., vol. 21, pp. 638–656, 2007. [15] H. K. Khalil, “High gain observers in nonlinear feedback control,” in Lecture Notes in Control and Information Sciences. Berlin: SpringerVerlag, 1999, vol. 244. [16] N. A. Mahmoud and H. K. Khalil, “Asymptotic regulation of minimum phase nonlinear systems using output feedback,” IEEE Trans. Autom. Control, vol. 41, no. 10, pp. 1402–1412, Oct. 1996. [17] F. Esfandiari and H. K. Khalil, “Output feedback stabilization of fully linearizable systems,” Int. J. Contr., vol. 56, pp. 1007–1037, Nov. 1992. ˙ [18] K. Kalsi, J. Lian, S. Hui, and S. H. Zak, “Sliding-mode observers for linear systems with unknown inputs,” revised and re-submitted to IEEE Trans. Autom. Control, 2008.
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