Sliding-mode Observers for Nonlinear Systems with ... - Springer Link

International Journal of Control, Automation, and Systems (2013) 11(5):903-910 DOI 10.1007/s12555-012-0463-9

ISSN:1598-6446 eISSN:2005-4092 http://www.springer.com/12555

Sliding-mode Observers for Nonlinear Systems with Unknown Inputs and Measurement Noise Junqi Yang, Fanglai Zhu*, and Wei Zhang Abstract: This paper deals with the design of observers for Lipschitz nonlinear systems with not only unknown inputs but also measurement noise when the observer matching condition is not satisfied. First, an augmented vector is introduced to construct an augmented system, and an auxiliary output vector is constructed such that the observer matching condition is satisfied and then a high-gain sliding mode observer is considered to get the exact estimates of both the auxiliary outputs and their derivatives in a finite time. Second, for nonlinear system with both unknown inputs and measurement noise, an adaptive robust sliding mode observer is developed to asymptotically estimate the system’s states, and then an unknown input and measurement noise reconstruction method is proposed. Finally, a numerical simulation example is given to illustrate the effectiveness of the proposed methods. Keywords: Auxiliary outputs, measurement noise reconstruction, nonlinear system, sliding mode observer, unknown input reconstruction.

1. INTRODUCTION Designing an unknown input observer (UIO) for nonlinear systems with both unknown inputs and measurement noise is much more challenging and important in the modern control theory. Engineering applications for UIOs include the disturbance estimation in control systems, cutting force estimation in machine tools, transmitted signal estimation in communication systems, and the evaluation of incipient failure of plant components in fault detection. So far, many UIO design problems without considering the measurement noise have been discussed. For instance, the necessary and sufficient __________ Manuscript received October 25, 2012; revised March 6, 2013 and April 17, 2013; accepted May 27, 2013. Recommended by Editorial Board member Juhoon Back under the direction of Editor Hyungbo Shim. This work is supported by National Nature Science Foundation (NNSF) of China under Grant 61074009. This work is also supported by the Research Fund for the Doctoral Program of Higher Education of China under Grant 20110072110015, Guangxi Key Laboratory of Manufacturing System and Advanced Manufacturing Technology under Grant PF110289, the Fundamental Research Funds for the Central Universities, Shanghai Leading Academic Discipline Project under Grant B004, the Program of Natural Science of Henan Provincial Education Department under Grant 13B413035 and 13B413028, and Shanghai Municipal Natural Science Foundation under Grant 12ZR1412200. Junqi Yang is with the College of Electronics and Information Engineering, Tongji University, Shanghai 201804, China; and the College of Electrical Engineering and Automation, Henan Polytechnic University, Jiaozuo 454000, China (e-mail: [email protected]. cn). Fanglai Zhu is with the College of Electronics and Information Engineering, Tongji University, Shanghai 201804, China (e-mail: [email protected]). Wei Zhang is with the Laboratory of Intelligent Control and Robotics, Shanghai University of Engineering Science, Shanghai 201620, China (e-mail: [email protected]). * Corresponding author. © ICROS, KIEE and Springer 2013

conditions for the existence of the observer are given in [1]. In [2], the simultaneous estimation of the state and input for a class of nonlinear systems is discussed. Trinh et al. deal with the simultaneous estimation of state and input based on reduced-order observers in [3]. Recently, UIOs for switched linear or nonlinear systems are considered in [4] and [5]. Based on the geometric approach, an UIO design for state affine system is considered, and a necessary and sufficient condition is given for the existence of UIO [6]. In [7], by assuming that the unknown inputs can be approximated by a polynomial over a local time interval, a finite-time observer is proposed to achieve joint state and input estimation. An adaptive sliding mode observer is developed to identify unknown parameters under the assumption that the time derivatives of some outputs are measurable [8]. A disturbance observer is designed in order to estimate the properties of the innate backlash for electromechanical systems in [9]. By designing suitable multiple observers, the parameter and state estimation problems are discussed for uncertain linear time-invariant systems [10]. The problem of dealing with measurement noise is discussed [11-13]. A high-gain observer with a gain adapted on-line to cope with measurement noise and uncertainties is proposed in [12]. A high-gain PIobserver which can estimate unknown inputs and system states is constructed in [13]. For the UIO design, most work deals with the unknown inputs without considering measurement noise [1,3,5,8,14,15]. In paper [16], the problems of state estimation and unknown information reconstruction for a class of linear systems with unknown inputs and measurement noise are considered and based reduced-order observer in which the unknown inputs and measurement noise are coped with by selecting a special reduced-order observer gain. The present paper deals with the similar issues for Lipschitz nonlinear systems based on full-order adaptive robust

Junqi Yang, Fanglai Zhu, and Wei Zhang

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sliding mode observer, while the Lipschitz nonlinear term and the unknown information are dealt with by designing an adaptation law and a robust sliding mode control law, respectively. The necessary and sufficient conditions for developing UIOs are that the invariant zeros of the system lie in open left half complex plane, and the so-called observer matching condition is satisfied. Although for some practical control systems such as some mechanical systems, observer matching condition may be satisfied, it is truly a harsh condition for many physical systems. For linear systems with only unknown inputs, the paper [15] discusses how to break through the restriction of the observer matching condition. In this paper, for nonlinear systems with both unknown inputs and measurement noise, we adopt the idea of generating auxiliary outputs from [15] to deal with the observer matching condition issue. By designing a high-gain sliding mode observer and an adaptive robust sliding mode observer, a kind simultaneous reconstruction method of unknown input and measurement noise is developed. The main purpose of the present paper is that we try to improve the main results in [15] to be suitable for: (1) a class of Lipschitz nonlinear systems; (2) systems with not only unknown inputs but also measurement noise. To reach these goals, first, a new treatment to the Lipschitz nonlinear term of the system is proposed. In fact, the Lipschitz constant can be unknown in our design because it can be adjusted adaptively by an adaption law. Second, a different unknown information reconstruction method is developed, and with this method, the simultaneous unknown input and measurement nose reconstruction goal is reached. The paper is organized as follows. In Section 2, for nonlinear system with both unknown inputs and measurement noise, the method of the state estimation based on the constructed auxiliary outputs is developed. In Section 3, a kind simultaneous reconstruction method of unknown input and measurement noise is developed. In Section 4, a numerical simulation example is given to illustrate the effectiveness of the proposed methods. Some conclusions are given in Section 5. 2. UIO DESIGN 2.1. Model description Consider a class of Lipschitz nonlinear systems with both unknown inputs and measurement noise as follows:

⎧ x = Ax + Bf ( x, u ) + Dη , ⎨ ⎩ y = Cx + Fd ,

(1)

where x ∈ » n , y ∈ » p and u ∈ » m are the state, measured output and known input vectors, respectively. η (t ) ∈ » l stands for the unknown input vector. The nonlinear function f : » n × » m → » m is a real-valued vector function. d (t ) ∈ » q is the measurement noise vector. A, B, C, D and F are all known constant matrices. We assume that rankC = p, rankE = m + l and n ≥ p ≥ w, where E = [ B D] and w = m + l + q.

Assumption 1: For system (1), ⎡ sI − A E rank ⎢ 0 ⎣ C

0⎤ = n+w F ⎥⎦

(2)

holds for all complex number s with Re( s ) ≥ 0. Assumption 2: The system state x(t), unknown input vector η(t), measurement noise d(t) and their derivatives, are all bounded in norm by some unknown constant. Assumption 3: The nonlinear function f (x, u) satisfies the Lipschitz condition, i.e.,

f ( x, u ) − f ( xˆ , u ) ≤ L f x − xˆ , ∀x,xˆ ∈ » n , u ∈ » m (3) holds for some Lipschitz constant Lf. If we introduce a new state vector z ∈ » p satisfying the following differential equation z = Ad z − Ad y,

(4)

where Ad ∈ » p× p is selected to be Hurwitz and nonsingular, and use x = [ xT zT ]T as an augmented state vector, the original system (1) can then be expanded into an augmented system described by ⎪⎧ x = Ax + Bf ( Kx , u ) + Dϕ , ⎨ ⎪⎩ z = Cx ,

(5)

0⎤ ⎡ A ⎡B⎤ where A = ⎢ , B = ⎢ ⎥ , C = [0 I p ], D = ⎥ ⎣0⎦ ⎣ − Ad C Ad ⎦ 0 ⎤ ⎡D ⎡η ⎤ ⎢ 0 − A F ⎥ , K = [ I n 0n× p ] and ϕ = ⎢ d ⎥ is the new ⎣ ⎦ d ⎦ ⎣ unknown input of system (5). According to Assumption 2, there exists a positive constant ρ such that ϕ (t ) ≤ ρ .

Lemma 1: The system (5) is minimum phase, i.e., the invariant zeros of the triple { A, C , E} are all in the open left-hand complex plant, or ⎡ sI − A E ⎤ rank ⎢ ⎥ = n+ p+w 0⎦ ⎣ C

(6)

holds for all complex number s with Re( s ) ≥ 0 if and only if (2) holds for all complex number s with Re( s ) ≥ 0, where E = [ B D]. Proof: Since the matrix Ad is selected to be nonsingular, we obtain 0 ⎤ ⎡ sI − A E rank ⎢ ⎥ ⎣ Ad C 0 − Ad F ⎦ 0 ⎤ ⎡ sI − A E 0 ⎤ ⎡I = rank ⎢ n ⎥⎢ 0 − F ⎥⎦ ⎣ 0 Ad ⎦ ⎣ C ⎡ sI − A E 0 ⎤ ⎡ sI − A E = rank ⎢ = rank ⎢ ⎥ 0 −F ⎦ 0 ⎣ C ⎣ C

0⎤ . F ⎥⎦

So (2) holds for all complex number s with Re( s ) ≥ 0 if and only if

Sliding-mode Observers for Nonlinear Systems with Unknown Inputs and Measurement Noise T T Ca = [CaT1  CaiT  Cap ] ,

0 ⎤ ⎡ sI − A E rank ⎢ ⎥ = n+w ⎣ Ad C 0 − Ad F ⎦ holds for all complex number s with Re( s ) ≥ 0. Because ⎡ sI − A E ⎤ rank ⎢ ⎥ 0⎦ ⎣ C ⎡ sI − A 0 ⎢ = rank ⎢ Ad C sI − Ad ⎢ 0 Ip ⎣

B D 0 0

905

0 0

0 ⎤ ⎥ − Ad F ⎥ 0 ⎥⎦

0 ⎤ ⎡ sI − A B D = rank ⎢ ⎥+ p ⎣ Ad C 0 0 − Ad F ⎦ 0 ⎤ ⎡ sI − A E = rank ⎢ ⎥ + p, ⎣ Ad C 0 − Ad F ⎦

we conclude that (2) holds for all complex number s with Re( s ) ≥ 0 if and only if ⎡ sI − A E ⎤ rank ⎢ ⎥ = n+ p+w 0⎦ ⎣ C

⎧⎪( A − La Ca )T Pa + Pa ( A − La Ca ) = −Qa ⎨ T ⎪⎩ E Pa = Ga Ca

(8)

hold, where Ga1 ∈ » m×γ and Ga 2 ∈ » (l + q )×γ . Lemma 3 (Barbalat Lemma [18]): If Φ : » → » + is a uniformly positive function for t ≥ 0 and the limit of t

holds for all complex number s with Re( s ) ≥ 0. This is the end of proof. 0 ⎤ ⎡E Because rankE = rank ⎢ ⎥ = m + l + q = w and ⎣ 0 − Ad F ⎦ rank(CE ) = rank[ 0 − Ad F ] = q, we conclude that the observer matching condition

rankE = rank(CE )

where Cai = [ciT (ci A)T  (ci Aγ i −1 )T ]T , Ca is of full rank and the so-called observer matching condition rankE = rank(Ca E ) holds, where γ = γ 1 + γ 2 +  + γ p is called the total relative degree. Lemma 2 [14]: The invariant zeros of the triples { A, C , E} and { A, Ca , E} are identical. Remark 1 [17]: Lemma 1 and the observer matching condition rankE = rank(Ca E ) hold if and only if for some symmetric positive definite matrix Qa ∈ » ( n + p )×( n + p ) , there exist matrices La ∈ » ( n + p )×γ , Ga = [GaT1 GaT2 ]T ∈ » ( m + l + q )×γ and a symmetric positive definite matrix Pa ∈ » ( n + p )×( n + p ) such that

(7)

is not satisfied for the augmented system (5). 2.2. The construction of auxiliary outputs and state estimation We have verified that the observer matching condition (7) is not satisfied for the augmented system (5). In this section, based on the work of [15], we first construct an auxiliary output vector which may satisfy the observer matching condition (7). Under the assumption that auxiliary output vector is known, an adaptive robust sliding mode observer is developed to asymptotically estimate system states. Definition 1: Suppose ri (i = 1, 2, , p) is the smallest integer such that 

⎧⎪ci A E = 0, for  = 0,1, , ri − 2, ⎨ r −1 ⎪⎩ci A i E ≠ 0, then the system (5) is said to have vector relative degree (r1 , r2 , , rp ) with respect to the nonlinear term f ( Kx , u ) and unknown input vector ϕ , where the vector ci ∈ 1×( n + p ) (i = 1, 2, , p) stands for the i-th row vector of the output matrix C . Assumption 4: Suppose that we can choose integers γ i ( 1 ≤ γ i ≤ ri , i = 1, 2, , p ) such that

the integral, lim ∫ Φ(τ )dτ , exists and is finite, then t →∞ 0

lim Φ(t ) = 0.

t →∞

Now consider the following system which has the same state equation as (5) but with an auxiliary output ⎪⎧ x = Ax + Bf ( Kx , u ) + Dϕ , ⎨ ⎪⎩ za = Ca x ,

(9)

where Ca ∈ »γ ×( n + p ) is given in Assumption 4, and za is the auxiliary output vector. Theorem 1: If the auxiliary output za is measureable, the state estimation xˆ from the adaptive robust sliding mode observer determined by (10), (11) and (12) converges to the actual state x asymptotically.

 xˆ = Axˆ + Bf ( Kxˆ , u ) + La ( za − Ca xˆ ) 1ˆ ˆ ˆ + kBG a1 ( za − Ca x ) + α ( za , x , t ), 2

(10)

⎪⎧ x = Ax + Bf ( Kx , u ) + Dϕ , ⎨ ⎪⎩ za = Ca x ,

(11)

⎧⎪ x = Ax + Bf ( Kx , u ) + Dϕ , ⎨ ⎪⎩ za = Ca x ,

(12)

where lk is a positive constant, and λ is selected to be large enough to satisfy λ > ρ. Proof: The observer error dynamic system between (9) and (10) can be obtained as follows x = ( A − La Ca ) x + Bf + Dϕ

(13) 1 ˆ ˆ ) − α ( z , xˆ , t ), − kBG ( z − C x a1 a a a 2 where x = x − xˆ , f = f ( Kx , u ) − f ( Kxˆ , u ). Consider the

Junqi Yang, Fanglai Zhu, and Wei Zhang

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1 Lyapunov function V = x T Pa x + 2 lk−1k 2 , where k = k −kˆ, k = L2f / ε is a constant which will be adjusted by the adaptive law (11), and ε is a positive scalar selected to be small enough. The derivative of V along with the error system (13) is

V = x T [( A − La Ca )T Pa + Pa ( A − La Ca )] x ˆ − x T P kBG ( z − C xˆ ) a

a1

a

a

 . + 2 x T Pa Bf + 2 x T Pa Dϕ − 2 x T Paα ( za , xˆ , t ) + lk−1kk

Based on the Assumption 3 we have

f = f ( Kx , u ) − f ( Kxˆ , u ) ≤ L f K ( x − xˆ ) ≤ L f K x − xˆ = L f x − xˆ so, according to (8) and (12) we obtain 2 x T Pa Bf ≤ 2 B T Pa x

f ≤ 2 L f B T Pa x x

≤ ( L2f / ε ) Ga1Ca x =k Ga1Ca x

2

2

+ ε x

2

zai ,2 ⎡ ⎤ ⎢ ⎥  ⎢ ⎥ =⎢ ⎥. zai ,γ i ⎢ ⎥ ⎢c A γ i −1 [ Ax + Bf ( Kx , u ) + Dϕ ]⎥ ⎣ i ⎦

2 x T Pa Dϕ ≤ 2 DT Pa x ϕ ≤ 2 ρ Ga 2Ca x ,

x T Pa DGa 2 ( za − Ca xˆ ) G ( z − C xˆ ) a2

a

a

zai ,γ i +1 = ψ i ( x , ϕ ) = ci A

So, we have 2

+ (k − kˆ) Ga1Ca x

2

+ k Ga1Ca x

2

 ˆ. − 2(λ − ρ ) Ga 2Ca x − lk−1kk

Considering the fact that the λ is selected to be large enough to satisfy λ > ρ and using the adaptation law (11), we find that the above inequality can be enlarged to 2 V ≤ (−λmin (Qa ) + ε ) x . So if we choose ε small enough to satisfy ε ≤ λmin (Qa ), we obtain

V + Φ(t ) ≤ 0,

γ i −1

[ Ax + Bf ( Kx , u ) + Dϕ ] ,

and regard zi1 = zi as the output equation, we obtain an augmented state-space description as follows

2

 ˆ − 2(λ − ρ ) Ga 2Ca x − lk−1kk = (−λmin (Qa ) + ε ) x

(15)

If we introduce a new variable

= 2λ Ga 2 Ca x .

V = (−λmin (Qa ) + ε ) x

2.3. The estimation of the auxiliary outputs and their derivatives In Theorem 1, we assume that za is measurable. However, the real output signal is z instead of za. za contains not only z but also some other unknown variables. In this section, we consider a high-gain sliding mode observer to get the exact estimates of both the auxiliary output vector za and its derivative in a finite time. After this goal has been reached, the state estimation can be completed by the replacement of za with the estimate of za in Theorem 1. T T ] , zai = Cai x Denote za = Ca x = [ zaT1 zaT2  zap T = [ zai,1 zai,2  zai ,γ i ] (i = 1, 2, , p). Based on Definition 1 and Assumption 4, differentiating zai with respect to time t, we have zai = Cai x = Cai Ax + Cai Bf ( Kx , u ) + Cai Dϕ

2 + ε x ,

ˆ ˆ ˆ  2 x T Pa kBG a1 ( za − Ca x ) = k Ga1Ca x ,

2 x T Paα ( za , xˆ , t ) = 2λ

Remark 2: The proof procedure of Theorem 1 shows that the Lipschitz constant can be unknown and the magnitude of it has no limitation because it is injected into the constant k which can then be adjusted adaptively by the adaption law (11).

(14)

2 where Φ(t ) = (λmin (Qa ) − ε ) x > 0. Based on Lyapunov stability theory, the inequality (14) means that the equilibrium points, x = 0 and k = 0, of error dynamic

(13) are stable. Now integrating (14) from zero to t yields t V (t ) + ∫ Φ(t ) ≤ V (0). As t → ∞, the above integral is 0 always less than or equal to V(0). So by Lemma 3, we have lim Φ(t ) = 0 which implies lim x (t ) = 0. This is t →∞ t →∞ the end of proof.

⎧ zai,1 = zai ,2 , ⎪  ⎪ ⎪ ⎨ zai,γ i = zai ,γ i +1 , ⎪ ⎪ zai,γ i +1 = ψ i , ⎪z = z , i ⎩ i1

(16)

where ψ i is unknown but bounded in norm by some unknown constant because of Assumption 2. For system (16), the following high-gain sliding mode observer is considered based on the work of [19] ⎧ zˆai ,1 = zˆai ,2 − wi ,1 , ⎪  ⎪⎪ ⎨ ⎪ zˆai ,γ i = zˆai ,γ i +1 − wi ,γ i , ⎪ ⎪⎩ zˆai ,γ i +1 = − wi ,γ i +1 ,

(17)

⎧ wi ,0 = zˆai ,1 − zi1 , ⎪ where ⎨ and (γ i − j +1) /(γ i − j + 2) ⋅ sign(wi , j −1 ) ⎪⎩ wi , j = κ ai , j wi , j −1 κ ai , j > 0 ( j = 1, 2, γ i + 1) are the gains of the observer.

Sliding-mode Observers for Nonlinear Systems with Unknown Inputs and Measurement Noise

Theorem 2: Under Assumption 2, the system (17) is a high-gain sliding mode observer of system (16) which is able to exactly estimate not only zai but also the derivative of zai in a finite time. More specifically, zˆai = [ zˆai ,1  zˆai ,γ i ]T is the exact estimate of zai = [ zai ,1  zai ,γ i ]T in a finite time, and ζ ai = [ zˆai ,2  zˆai,γ i +1 ]T is the exact estimate of zai in a finite time. Proof: The error dynamic system between system (16) and (17) is governed by ⎧eai ,1 = eai ,2 − wi,1 , ⎪  ⎪ ⎨e = eai,γ i +1 − wi ,γ i , ⎪ ai ,γ i ⎪e  ⎩ ai ,γ i +1 = − wi,γ i +1 −ψ i ,

where eai , j = zˆai , j − zai , j ( j = 1, 2, , γ i + 1). Now by a similar way to [19], we can show that a sliding mode appears on the manifold eai ,1 = eai ,2 =  = eai ,γ i +1 = 0 in a finite time by choosing the gains of κ ai , j properly. That is, in a finite time, zˆai , j is the exact estimate of zai , j . Based on (16), we obtain that zˆai and ζ ai are the estimates of zai and zai , respectively. This is the end of proof. Remark 3: A similar high-gain sliding mode observer to (17) is also used to deal with auxiliary output estimating problems in [14] and [20]. In [21] and [22], it is used as unknown input observers directly for affine nonlinear systems. Considering the fact that the zˆa is the exact estimate of the auxiliary output vector za in a finite time, we give Theorem 3 as follows based on Theorem 1. Theorem 3: Under Assumption 1-4, the following adaptive robust sliding mode observer  xˆ = Axˆ + Bf ( Kxˆ , u ) + La ( zˆa − Ca xˆ ) 1ˆ ˆ ˆ + kBG a1 ( zˆa − Ca x ) + α ( zˆa , x , t ) 2

(18)

2

a

)

(21)

ηˆ = ⎡⎣ I k

0k ×q ⎤⎦ ϕˆ ,

(22)

I q ⎤⎦ ϕˆ ,

(23)

dˆ = ⎡⎣ 0q×k

T T where U = Ca D, ζ a = [ζ aT1 ζ aT2  ζ ap ] is the exact T estimate of za = [ za1 za 2  zap ] and ζ ai (i = 1, 2,  p ) is provided by the high-gain sliding observer (17). ϕˆ is the estimate of augmented unknown input ϕ = [η T d T ]T , so ηˆ and dˆ are the estimates of the unknown input η and measurement noise d, respectively. Proof: Based on (9), we obtain

za = Ca x = Ca ⎡⎣ Ax + Bf ( Kx , u ) + Dϕ ⎤⎦

or U ϕ = za − Ca ⎡⎣ Ax + Bf ( Kx , u ) ⎤⎦ . Since rankU = rankCa D = rankD = l + q, so U T U is invertible because U is with full column rank. So the unknown vector ϕ satisfies ϕ = (U T U )−1U T ⎡⎣ za − Ca ( Ax + Bf ( Kx , u ) ) ⎤⎦ .

(24)

The error equation between (21) and (24) is ϕ = (U T U )−1U T ⎡⎣ζa − Ca ( Ax + Bf ) ⎤⎦ ,

where ϕ (t ) = ϕ (t ) − ϕˆ (t ), ζa (t ) = za (t ) − ζ a (t ), x (t ) = x (t ) − xˆ (t ) and f = f ( Kx , u ) − f ( Kxˆ , u ). So we have lim ϕ (t ) = 0

since

lim ζa (t ) = 0, lim x (t ) = 0

t →∞

t →∞

and

lim f (t ) = 0. This is the end of proof.

(19)

DGa 2 ( zˆa − Ca xˆ ) , G ( zˆ − C xˆ ) a2

(

ϕˆ = (U T U ) −1U T ⎡ζ a − Ca Axˆ + Bf ( Kxˆ , u ) ⎤ , ⎣ ⎦

t →∞

and a sliding mode control law of α ( zˆa , xˆ , t ) = λ

system states and the derivative of the auxiliary output vector. Theorem 4: Under Assumption 2~4, the reconstruction of the unknown inputs and measurement noise is constructed as follows

t →∞

with an adaptation law of  kˆ(t ) = lk Ga1 ( zˆa − Ca xˆ )

907

(20)

a

is an adaptive robust sliding mode observer of the system (5) which is able to asymptotically estimate the states of the system (5) , where zˆa is provided by the high-gain sliding observer (17) and lk is a positive constant. 3. THE RECONSTRUCTION OF UNKNOWN INPUTS AND MEASUREMENT NOISE In this section, we will provide an unknown input reconstruction method based on the estimates of both

4. SIMULATION In this section, a numerical simulation is used to verify the effectiveness of the proposed scheme. We consider the laboratory model of a single-link flexible joint robot defined in form of (1) as follows [23,24] 1 0 0 ⎤ ⎡ 0 ⎢ −3.75 −0.0015 3.75 ⎥ 0 ⎥, A=⎢ ⎢ 0 0 0 1 ⎥ ⎢ ⎥ 0 −3.75 −0.0013⎦⎥ ⎣⎢ 3.75 ⎡ 0 ⎤ ⎡1 0 0 0 ⎤ ⎢ −1.1104 ⎥ ⎥ , C = ⎢0 0 0 1 ⎥ , B=⎢ ⎢ ⎥ ⎢ 0 ⎥ ⎢⎣ 0 0 1 0 ⎥⎦ ⎢ ⎥ ⎢⎣ 0 ⎥⎦

Junqi Yang, Fanglai Zhu, and Wei Zhang

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⎡ 0 ⎤ ⎡0 ⎤ ⎢0.02 ⎥ ⎢ ⎥ , F = ⎢⎢1 ⎥⎥ D= ⎢ 0 ⎥ ⎢ ⎥ ⎣⎢0 ⎦⎥ ⎣⎢0.04 ⎦⎥

and nonlinear function f ( x, u ) = sin( x1 ). The system state vector is with the form of x = [ x1 x2 x3 x4 ]T = [θ m ω m θl ωl ]T , where θm and θl are the angular rotations of the motor and the link, respectively, with ωm and ωl being their angular velocities. The unknown input and measurement noise vectors are supposed as η = 2 cos(4t + 3.6) and d = 0.4sin(4t + 2.8) + 0.5cos(2t +4.5), respectively. In the unknown input term, the bounded functions η2 = 0.02η and η4 = 0.04η are the perturbations in the system including the motor disturbance. d is the noise signal in the output channel. For the differential equation (4), if we select Ad = − I 3 , then it is easy to obtain the matrices A, B , D, C , K and E = [ B D]. It is not difficult to verify that the observer matching condition (7) is not satisfied for augmented system (5). That is rankE ≠ rank(C E ). The augmented system (5) has vector relative degree (r1 , r2 , r3 ) = (3,1,3) with respect to the new unknown inputs ϕ which consists of unknown input η and the measurement noise d. If we choose γ 1 = r1 = 3, γ 2 = r2 = 1 and γ 3 = r3 = 3, we have ⎡ c1 ⎤ ⎡ 0 ⎢cA⎥ ⎢ ⎢ 1 ⎥ ⎢1 ⎢ c A 2 ⎥ ⎢ −1 ⎢ 1 ⎥ ⎢ Ca = ⎢ c2 ⎥ = ⎢ 0 ⎢ c ⎥ ⎢0 ⎢ 3 ⎥ ⎢ ⎢ c3 A ⎥ ⎢ 0 ⎢ ⎥ ⎢ 2 ⎣⎢c3 A ⎦⎥ ⎢⎣ 0

0 0 0 0 1 0 0 0 0 0 0 1 0 −1

0 1 0 −1 0 1 0 0 0 0 0 0 1 0

= [ za1,1 = [ z1

za1,2 za1,2

zaT2

Fig. 2. State estimated errors.

zaT3 ]T

za1,3 za1,3

Fig. 1. Estimated error of auxiliary output.

0 0⎤ 0 0 ⎥⎥ 0 0⎥ ⎥ 1 0 ⎥. 0 1⎥ ⎥ 0 −1⎥ 0 1 ⎥⎥⎦

Now it is easy to check that the observer matching condition is satisfied with respect to the auxiliary output za = Ca x , that is rankE = rank(Ca E ). For the auxiliary output za = Ca x = [ zaT1

states and state estimation are set as x(0) = [0.8 0.4 0.2 0.5]T and xˆ(0) = [1.0 0.6 1.0 1.2]T , respectively, then the state estimation can be obtained by Theorem 3. The state estimated errors are plotted in Fig. 2. From Fig. 2 we see that the state estimation effect is satisfactory. Fig. 3 gives the adaptation for the constant k determined by (19).

z2

za 3,3 ]T

za 2,1

za 3,1

za 3,2

z3

za3,2

za 3,3 ]T ,

we find that it contains four unmeasured variables, za1,2 , za1,3 , za 3,2 and za3,3 , which need to be estimated by the high-gain sliding mode observer (17). The initial values are set as zˆa1,2 (0) = 0.2, zˆa1,3 (0) = 0.15, zˆa3,2 (0) = 0.5, zˆa 3,3 (0) = 0.3, and the estimated errors for them are shown in Fig. 1 which shows that the estimating effects are satisfactory. In the simulation, the known input u is set as 0. If we set lk = 1 and λ = 30, the initial value of adaptation law is given as kˆ(0) = −1.2, and the initial

Fig. 3. The adaptation for constant k.

Sliding-mode Observers for Nonlinear Systems with Unknown Inputs and Measurement Noise

[1]

[2]

[3]

[4]

Fig. 4. Reconstruction of η.

[5]

[6]

[7]

[8]

Fig. 5. Reconstruction of d. Now the state estimation xˆ from adaptive robust T T observer (18) together with ζ a = [ζ aT1 ζ aT2  ζ ap ] can give the reconstruction of the unknown inputs and measurement noise by (22)~(24). Here, ζ ai (i = 1, 2, p) is provided by the high-gain sliding observer (17), and ζ a is the estimation of za = [ za1 za 2  zap ]T . The reconstruction results of both η and d are shown in Figs. 4 and 5, respectively. 5. CONCLUSIONS This paper considers the problems of state estimation and simultaneous unknown input and measurement noise reconstruction for Lipschitz nonlinear system when the observer matching condition is not satisfied. A high-gain sliding mode observer is constructed to exactly estimate not only the auxiliary outputs but also their derivatives. An adaptive robust sliding mode observer is designed to estimate the system states. Because an adaption law which can adaptively adjust the Lipschitz constant is introduced, the Lipschitz constant can be unknown in the design. A different unknown information reconstruction method which can reconstruct simultaneously unknown inputs and measurement nose is developed.

[9]

[10]

[11]

[12]

[13]

[14]

909

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Junqi Yang, Fanglai Zhu, and Wei Zhang

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Junqi Yang received his M.S. degree in Control Theory and Control Engineering from China Three Gorges University in 2005. He is currently a Ph.D. candidate in Control Theory and Control Engineering at Tongji University, China. His research interests include observer design, model-based fault detection, and faulttolerant control. Fanglai Zhu received his Ph.D. degree in Control Theory and Control Engineering from Shanghai Jiao Tong University in 2001. Now he is a professor of Tongji University, China. His research interests include nonlinear observer design, chaotic synchronization based on observer, system identification, and model-based fault detection and isolation. Wei Zhang received his Ph.D. degree in Control Theory and Control Engineering from Shanghai Jiao Tong University, Shanghai, China, in 2010. He is currently an associate professor with the Laboratory of Intelligent Control and Robotics, Shanghai University of Engineering Science, Shanghai, China. His current research interests lie in the areas of nonlinear observer design, robust control and networked control systems.