Observer-based stabilization of nonlinear discrete-time systems using sliding window of delayed measurements

CHAPTER

18

Observer-based stabilization of nonlinear discrete-time systems using sliding window of delayed measurements

Noussaiba Gasmia , Mohamed Boutayeba , Assem Thabetb , Mohamed Aounb a CRAN

b MACS

UMR CNRS 7039, University of Lorraine, Cosnes et Romain, France Laboratory, National Engineering School of Gabes (ENIG), University of Gabes, Gabes, Tunisia

Chapter outline 1 2 3 4

Introduction....................................................................................... Problem statement............................................................................... Convergence analysis........................................................................... Converting BMI into LMI ........................................................................ 4.1 Particular solution: The case of two measurements ............................ 4.2 Particular solution: The case of M measurements .............................. 5 Discussion on the enhancement .............................................................. 5.1 Classical approach................................................................... 5.2 Sliding window approach vs. classical approach ................................ 6 Illustrative examples ............................................................................ 6.1 Example 1 ............................................................................ 6.2 Example 2 ............................................................................ 7 Conclusion ........................................................................................ 8 Appendix .......................................................................................... 8.1 Differential mean value theorem .................................................. 8.2 Schur’s lemma ....................................................................... References...........................................................................................

367 369 374 375 376 376 377 377 378 379 380 383 383 384 384 384 384

1 Introduction Most of the existing works on control design for linear and nonlinear systems assume that all the components of the state vector are available [1–3]. On the other hand, the Stability, Control and Application of Time-Delay Systems. https://doi.org/10.1016/B978-0-12-814928-7.00018-4 © 2019 Elsevier Inc. All rights reserved.

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CHAPTER 18 Stabilization of nonlinear discrete-time systems

size of the output vector is smaller than the size of the state vector for several reasons (technical implementation, cost, etc.). Therefore, at a given time t, the state cannot be deduced algebraically from the output measurements. In control theory, a state observer is a dynamical system that mirrors the behavior of a physical system. From the output measurements, he/she provides an estimate of the states of the system and gives an approximation as close as possible to reality. For this reasons, observer design for linear and nonlinear systems attracts the attention of many researchers [4–7]. The state observer theory was first introduced by Kalman and Bucy in Ref. [8] for a linear system in a stochastic environment. Then, Luenberger presented a general theory of observers for deterministic linear systems [9]. After that, many works on linear and nonlinear estimation were developed using the famous Luenberger observer [10–12]. The existing works on observer design for Lipschitz systems deal with the continuous case [13–15], and also the discrete case [4,11]. However, there is no result that deals with the design of sliding window filter for this class of nonlinear systems despite the superiority and the good performances proved with the extended Kalman filter in Refs. [16,17]. In the literature, there are many researchers that focus on the observer-based controller design for nonlinear systems. For instance, in Ref. [1], linear matrix inequality (LMI) condition for observer-based stabilization of Lipschitzian systems is given. The authors proposed to compute the controller and observer gains in two steps. On the other hand, in Ref. [18], the proposed design methodology uses a diagonal Lyapunov matrix and allows to compute the observer and controller gains simultaneously via a unique LMI. Then, a recent publication of Kheloufi et al. [19] presents a useful design procedure to synthesize a decentralized observer-based stabilization for nonlinear interconnected systems using a symmetric Lyapunov function. To linearize the obtained constraint, the authors proposed to use a slack variable. In this chapter, the problem of designing H∞ sliding window observer-based controller for nonlinear discrete-time systems in the presence of disturbances is addressed. The considered nonlinearity is assumed to be Lipschitz. The main contribution lies in the use of a sliding window of delayed measurements in the observer structure and a sliding window of delayed states in the control low. Thanks to the introduction of a relevant Lyapunov function with a slack variable technique inspired from Ref. [20], the observer and the controller gains can be computed simultaneously through a less restrictive constraint. In the following, we summarize the improvements with respect to existing results: • First, this chapter investigates the problem of nonlinear control in the presence of disturbances with bounded energy. • One of the practical points guaranteed by the constraint presented in this contribution is the possibility of computing the controller and the observer gains simultaneously through the same LMI, which is not the case in many existing works [1,21]. • Sliding windows of delayed measurements and delayed states are added to the Luenberger observer and the classical feedback control law, respectively, in

2 Problem statement

•

order to ensure better performances and to provide a more general and less restrictive LMI condition. As we know, the problem of using previous measurements with an observer-based controller has not been tackled before. In order to avoid bilinearities caused by the Lyapunov function, a slack variable is introduced to linearize the obtained constraint [19,20].

The remainder of this chapter is organized as follows. In the following section, the problem formulation is introduced. Then, the synthesis procedure of the sliding window H∞ observer-based controller is detailed in Section 3. Section 4 presents an interesting comparison with the classical approach. Finally, two numerical examples are considered to illustrate the effectiveness and the superiority of the proposed design methodology. Notation. The following notation will be used throughout this chapter: ⎛ ⎞T ith  ⎜ n 1 , 0, . . . , 0 ⎟ • en (i) = ⎝ 0, . . . , 0,   ⎠ ∈ R , n  1, is a vector of the canonical n-components basis of Rn . • ST is the transposed matrix of S. • S is a square matrix. The notation S > 0 (S < 0) means that S is positive definite (negative √ definite). • S = ST S is the Euclidean vector norm. • In represents the identity matrix of dimension n. •  S l2 represents the l2 norm of the vector S ∈ Rn and is defined as ∞ 2  S l2 = k=0  Sk  . • In a matrix, the notation () is used for the blocks induced by symmetry.

2 Problem statement This section is interested in providing an efficient new H∞ observer-based controller. The idea consists in adding a sliding window of delayed measurements to the Luenberger observer and to use a new reformulation of the feedback controller including delayed states. The interest of using delayed states and measurements on the feedback controller and the observer’s structure increases the degree of freedom of the constraint to be verified. Consider the class of nonlinear discrete-time systems described by 

x(k + 1) = Ax(k) + Bu(k) + f (x(k)) + W1 ω(k) y(k) = Cx(k) + W2 ω(k)

(1)

where x(k) ∈ Rn , u(k) ∈ Rm , y(k) ∈ Rp , and ω(k) ∈ Rr are the state, the input, the output, and the disturbance vectors, respectively. A ∈ Rn×n , B ∈ Rn×m , C ∈ Rp×n ,

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CHAPTER 18 Stabilization of nonlinear discrete-time systems

W1 ∈ Rn×r , and W2 ∈ Rp×r are constant matrices of adequate dimensions. f : Rn → Rn is the nonlinear function which is assumed to be globally Lipschitz. The pairs (A, B) and (A, C) are assumed to be stabilizable and detectable, respectively. Now, let us consider the following sliding window observer: ⎛

⎞ y(k) − Cˆx(k) ⎜ ⎟ y(k − 1) − Cˆx(k − 1) ⎜ ⎟ xˆ (k + 1) = Aˆx(k) + Bu(k) + f (ˆx(k)) + L ⎜ ⎟ .. ⎝ ⎠ . y(k − N + 1) − Cˆx(k − N + 1)

(2)

coupled with a state estimated feedback controller ⎛

⎞ xˆ (k) N ⎜ xˆ (k − 1) ⎟  ⎜ ⎟ u(k) = Ki xˆ (k − i + 1) = K ⎜ ⎟ .. ⎝ ⎠ . i=1 xˆ (k − N + 1)

(3)

    where xˆ (k), N, L = L1 L2 · · · LN and K = K1 K2 · · · KN represent, respectively, the state estimate, the number of measurements, the observer, and the controller gain matrices. Remark 1. The idea behind using a sliding window approach in the synthesis of the H∞ observer-based controller is to improve the disturbance rejection by using a fixed number of delayed states (x(k), x(k − 1), . . . , x(k − N + 1)) and measurements (y(k), y(k − 1), . . . , y(k − N + 1)). This will increase the accuracy and the robustness of estimation. This new formulation is a significant contribution, contrary to conventional approaches considering only the last available information. The state Eq. (1) can be rewritten in a new form containing the delayed measurements. So, the new state vector is described by X (k + 1) = AX (k) + FBu(k) + Ff (x(k)) + W1  (k)

where ⎛

⎞ x(k) ⎜ x(k − 1) ⎟ ⎜ ⎟ X (k) = ⎜ ⎟ .. ⎝ ⎠ . x(k − N + 1) ⎛ ⎞ ω(k) ⎜ ω(k − 1) ⎟ ⎜ ⎟  (k) = ⎜ ⎟ .. ⎝ ⎠ . ω(k − N + 1)

(4)

2 Problem statement

⎛

A ⎜In ⎜ ⎜ A=⎜ ⎜0 ⎜. ⎝ .. ⎛

0 W1 0 .. . 0

⎜ ⎜ W1 = ⎜ ⎝

0 0 .. . .. . ···

··· ··· .. . .. . 0

··· ···

···

0

..

⎞ 0 0⎟ ⎟ .. ⎟ .⎟ ⎟ .. ⎟ .⎠

. In 0 ⎞ ··· 0 · · · 0⎟ ⎟ .⎟ .. . .. ⎠ 0 0

0 0 .. . ···

and  F = In

0

T

Then, we obtain a new reformulation of the sliding window observer (2) Xˆ (k + 1) = AXˆ (k) + FBu(k) + Ff (ˆx(k)) + F LCε(k) + F LW2  (k)

(5)

where ε(k) = X (k) − Xˆ (k) is the estimation error ⎛ N times ⎞    C = block-diag ⎝C, . . . , C⎠

and

⎛

N times

⎞

   W2 = block-diag ⎝W2 , . . . , W2 ⎠

Then the dynamic of the estimation error ε(k + 1) is given by ε(k + 1) = X (k + 1) − Xˆ (k + 1)

(6)

Using Eqs. (3)–(6), the state estimate and the dynamic of the estimation error can be rewritten as follows: Xˆ (k + 1) = (A + FBK)Xˆ (k) + Ff (ˆx(k)) + F LCε(k) + F LW2  (k)   ε(k + 1) = (A − F LC)ε(k) + F f x(k), xˆ (k) + (W1 − F LW2 ) (k)

with x(k) = F T X (k) xˆ (k) = F T Xˆ (k)

(7) (8)

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CHAPTER 18 Stabilization of nonlinear discrete-time systems

and   f x(k), xˆ (k) = f (x(k)) − f (ˆx(k)) As stated previously, f (.) is globally Lipschitz. We assume that f (0) = 0, then by applying Lemma 1 (see Section 8) to this nonlinear function, we obtain f (ˆx(k)) = Σ1 (Θ)ˆx(k) = Σ1 (Θ)F T Xˆ (k)

(9)

and 



f x(k), xˆ (k) = Σ1 (Θ)(x(k) − xˆ (k)) = Σ1 (Θ)F T ε(k)

(10)

with Σ1 (Θ) =

n,n 

ϕij Hij

i,j=1

Hij = en (i)eTn (j)

The parameter Θ belongs to the bounded convex set Hn , for which the set of vertices is defined by    VHn = ϕij ∈ Rn×n , ϕij ∈ f ij , f ij The observer gain matrix L and the control gain matrix K are unknown matrices to be determined such that xˆ k converges asymptotically to xk . Hence, we can define an augmented system described by the following structure: ¯ x(k) + Σ x¯ (k) + W ¯ (k) x¯ (k + 1) = A¯

(11)

with   Xˆ (k) ε(k)   A + F BK F LC A¯ = 0 A − F LC   T 0 FΣ1 (Θ)F Σ= 0 FΣ1 (Θ)F T

x¯ (k) =

and ¯ = W



F LW2 W1 − F LW2



The synthesis of the sliding window H∞ observer-based controller corresponding to the augmented system (11) return to search the gain matrices L and K that guarantee

2 Problem statement

the convergence of the vector x¯ asymptotically toward zero; that is, we must find the parameters L and K such that  H(x(k) − xˆ (k)) l2 ≤ μ  ω(k) l2

for

x(0) − xˆ (0) = 0

(12)

H is a known matrix and μ > 0 is the disturbance attenuation level that will be minimized. The resolution of this problem returns to search a Lyapunov function V(k) so that ¯ F¯ T x¯ (k) − Λ = V(k) + x¯ T (k)FH

with

 H=

and

0 HT









0 0

H =

0

 F F¯ = 0

μ2 T  (k) (k) < 0 N

0 F

0

(13)



HT H



Then, let us consider the following candidate Lyapunov function for system (1): V(k) = x¯ T (k)P¯x(k)

(14)

where P = PT > 0 is the Lyapunov matrix. Usually, to solve this kind of problem, most existing works on observer-based controller design for linear and nonlinear systems, as Refs. [18–22], use a particular form of the matrix P = diag(P1 , P2 ). This specific form allows to overcome the problem of bilinearities and to simplify the calculation. The resulting LMI remains restrictive due to the use of this particular form. In this contribution, a symmetric Lyapunov matrix P is used to get a more relaxed LMI. For that, consider the following form of the Lyapunov matrix:  P=

P11 PT12

P12 P22

 (15)

Define V(k) = V(k + 1) − V(k). Then, along the solution of the augmented system (11), we have 

x¯ k Λ= ωk

T



x¯ k Ξ ωk

 (16)

where ⎛ Ξ =⎝

¯ F¯ T (A¯ + Σ)T P(A¯ + Σ) − P + FH ()

⎞ ¯ (A¯ + Σ)T PW ⎠ μ2 ¯ − ¯ T PW Ir×N W N

(17)

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CHAPTER 18 Stabilization of nonlinear discrete-time systems

Note that Λ < 0 is satisfied if Ξ < 0 which is equivalent to ⎛

¯ F¯ T −P + FH

0

⎝

⎞

⎠+ μ2 − Ir×N N

0

   (A¯ + Σ)T P A¯ + Σ T ¯ W

 ¯ <0 W

(18)

Applying Schur’s lemma on Eq. (18), we get ⎛

¯ F¯ T −P + FH

⎜ ⎜ ⎝

0 −

() ()

(A¯ + Σ)T

μ2 Ir×N N ()

¯T W −P−1

⎞ ⎟ ⎟<0 ⎠

(19)

It is obvious that inequality (19) does not contain products of unknown terms that may pose bilinearity problems. However, the direct resolution of this matrix inequality remains impossible with the available mathematical tools. This problem is caused by the existence of an unknown matrix with its inverse (P and P−1 ) in the same constraint. To overcome this problem, a very useful design methodology is proposed in the following section.

3 Convergence analysis In this section, the design procedure of the considered H∞ sliding window observerbased controller is detailed and a new enhanced LMI condition is presented. The main contribution of this chapter is introduced in the following corollary. Corollary 1. For a disturbance attenuation level μ > 0, the H∞ sliding window observer-based controller design problem corresponding to system (1), observer (2), and state feedback controller (3) is H∞ asymptotically stabilizable if there exist ˆ 11 , Q22 , L, and K so that the following bilinear definite matrices Pˆ 11 , Pˆ 12 , P22 , Q matrix inequality (BMI) is feasible for all Θ ∈ VHn : min μ subject to ⎛−Pˆ −Pˆ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

11

()

12 F H T H F T − P22

()

()

()

()

()

()

−

0

ˆ 1 (A + F BK + F Σ1 (Θ)F T )T Q

0

(A + F Σ1 (Θ)F T )T

μ2

0

(A − F LC + F Σ1 (Θ)F T )T QT2 ⎟

N ()

W1T

(W1 − F LW2 )T QT2

ˆ1 −Q ˆT Pˆ 11 − Q 1

Pˆ 12

()

()

P22 − Q2 − QT2

Ir×N

⎞ ⎟ ⎟ ⎟< 0 ⎟ ⎠ (20)

Proof. To linearize inequality (19), a slack variable Q is added. Then, multiplying both sides of inequality (19) by block-diag(I, I, Q) and block-diag(I, I, QT ) yields to the following constraint:

4 Converting BMI into LMI

⎛

−P + F¯ HF¯ T

⎜ ⎜ ⎝

−

()

(A¯ + Σ)T QT

0

()

μ2 Ir×N N ()

¯ T QT W −QP−1 QT

⎞ ⎟ ⎟<0 ⎠

(21)

Now, using the inequality −QP−1 QT ≤ P − Q − QT , we obtain ⎛

−P + F¯ HF¯ T

⎜ ⎜ ⎝

()

(A¯ + Σ)T QT

0 −

()

μ2 Ir×N N ()

¯ T QT W

⎞

⎟ ⎟<0 ⎠

(22)

P − Q − QT

Then, we propose the following structure of the slack variable Q: Q=

 Q1 0

Q1 Q2

 (23)

By substituting Eqs. (15), (23) in inequality (22) and after some mathematical developments, the following inequality is obtained: ⎛−P ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

11

()

−P12 F HT HF T − P

22

0

(A + F BK + F Σ1 (Θ)F T )T QT1

0

0

W1T QT1 P11 − Q1 − QT1

P12

()

P22 − Q2 − QT2

()

()

()

()

μ2 − Ir×N N ()

()

()

()

⎞

(A − F LC + F Σ1 (Θ)F T )T QT2 ⎟ ⎟

(A + F Σ1 (Θ)F T )T QT1

(W1 − F LW2 )T QT2

⎟ ⎟<0 ⎟ ⎠ (24)

ˆ 1 , I, I, Q ˆ 1 , I) and Then, premultiplying and postmultiplying Eq. (24) by block-diag(Q −1 ˆ T , I) with Q ˆ 1 = Q and using the notations Pˆ 12 = Q ˆ 1 P12 , ˆ T , I, I, Q block-diag(Q 1 1 1 ˆ 1 P11 Q ˆ T lead to constraint (20). Pˆ 11 = Q 1

4 Converting BMI into LMI To linearize the BMI given by Eq. (20), the following change of variables for the terms coupled with the controller gain K can be defined: ˆ 1 KT Kˆ = Q Then, the controller gain can be computed through the following equation: ˆ −T K = Kˆ T Q 1 However, this is not the case for the terms coupled with the observer gain L because of the matrix F which is a singular matrix. Therefore, a particular solution is proposed to linearize this BMI. To facilitate the task, we start with a specific solution using only two measurements. Afterward, we generalize this solution for the case of M measurements.

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CHAPTER 18 Stabilization of nonlinear discrete-time systems

4.1 Particular solution: The case of two measurements Consider the case of two measurements. Let Q2 has the following particular form:   Q2 = with 0  α < 1, 0  β < 1 and

Q11 2

αQ11 2

βQ11 2

Q22 2

 L = L1

L2



Then, two changing of variables can be done Lˆ 1 = Q11 2 L1 Lˆ 2 = Q11 2 L2

Therefore, the BMI (20) is transformed into LMI which can be easily solved. The gains of the observer are finally given by −1 ˆ L1 = (Q11 2 ) L1 −1 ˆ L2 = (Q11 2 ) L2

4.2 Particular solution: The case of M measurements From the case of two measurements, we can deduce the case of M measurements. Proposition 1. We consider the following particular form of the matrix Q2 : ⎛

Q11 2

⎜ ⎜β1 Q11 2 ⎜ ⎜ . ⎜ . Q2 = ⎜ . ⎜ ⎜ . ⎜ .. ⎝ β1 Q11 2

α1 Q11 2

···

···

Q22 2

α2 Q22 22

···

β2 Q22 2 .. .

..

.

..

.

..

.

..

.

β2 Q22 2

···

α1 Q11 2

⎞

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ N−1,N ⎟ Q2 ⎠

QN,N−1 2

α2 Q22 2 .. .

(25)

QNN 2

where QN−1,N = αN−1 QN−1,N−1 2 2 QN,N−1 = βN−1 QN−1,N−1 2 2

with 0  αi < 1 and 0  βi < 1 for i ∈ {1, . . . , N − 1} and  L = L1

L2

···

LN



Then, we can define the following changes of variables: Lˆ i = Q11 2 Li for i ∈ {1, . . . , N} So, the BMI (20) is transformed into a convex problem.

(26)

5 Discussion on the enhancement

Finally, the observer gains can be computed through the following equations: −1 ˆ Li = (Q11 2 ) Li

for i ∈ {1, . . . , N}

Remark 2. To enhance the feasibility of the proposed LMI procedure, we chose 0  αi < 1 and 0  βi < 1. This choice is established from numerical evaluation of the LMI on many examples. Remark 3. The BMI (20) can be converted to an LMI under Eq. (25), if we fix a priori αi and βi . Hence, we use the gridding technique with respect to αi and βi for each i ∈ {1, . . . , N − 1} to solve Eq. (20) under Eq. (25).

5 Discussion on the enhancement 5.1 Classical approach Usually, it is the standard form of the Luenberger observer that is used xˆ (k + 1) = Aˆx(k) + Bu(k) + f (ˆx(k)) + L(y(k) − Cˆx(k))

(27)

coupled with the following feedback controller: u(k) = K xˆ (k)

(28)

Then, the estimation error is given by e(k) = x(k) − xˆ (k)

(29)

The dynamic of the estimation error is described as follows: e(k + 1) = (A − LC)e(k) + f (x(k), xˆ (k)) + (W1 − LW2 )ω(k)

(30)

Then, the resulting augmented system is given by ¯ x(k) + Σ x¯ (k) + Wω(k) ¯ x¯ (k + 1) = A¯

with   xˆ (k) e(k)  A + BK A¯ = 0  1 (Θ) Σ= 0

x¯ (k) =

and ¯ = W



LC A − LC





0 1 (Θ)

LW2 W1 − LW2



(31)

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CHAPTER 18 Stabilization of nonlinear discrete-time systems

Hence, using the H∞ criterion (12) and an appropriate Lyapunov function (V(k) = eT (k)Pe(k)), we obtain the following constraint: ⎛

−Pˆ 11 ⎜ () ⎜ ⎜ ⎜ () ⎜ ⎝ () ()

−Pˆ 12 T H H − P22 () () ()

0 0 −μ∗ Ir () () 2

ˆ 1 (A + BK + Σ1 (Θ))T Q (A + Σ1 (Θ))T W1T ˆ1 −Q ˆT Pˆ 11 − Q 1

()

⎞ 0 (A − LC + Σ1 (Θ))T QT2 ⎟ ⎟ ⎟ ⎟< 0 (W1 − LW2 )T QT2 ⎟ ⎠ Pˆ 12 P22 − Q2 − QT2 (32)

ˆ 1KT In this case, an LMI can be obtained using a simple changes of variables Kˆ = Q ˆ and L = Q2 L.

5.2 Sliding window approach vs. classical approach The superiority of adding delayed measurements to the observer’s structure and delayed states to the controller is discussed from two different points of view.

5.2.1 Comparison from LMI feasibility point of view In order to compare the proposed H∞ sliding window observer-based controller with the classical one, we consider the case with two measurements N = 2 where         I C 0 A 0 W1 0 W2 , F = n , W1 = , C= , W2 = 0 0 0 0 C 0 In 0     11   ˆ 12 P12 P11 Pˆ 11 Pˆ 12 Pˆ 11 22 22 11 12 P12 , P = , Pˆ 12 = Pˆ 11 = 22 T T 21 22 12 22 ˆ ˆ ˆ 22 P P Pˆ 12 P P P 12 12 11 11 22 22     11 Q12 ˆ 12 ˆ 11 Q Q Q 1 1 2 2 ˆ1 = , Q2 = Q 21 Q22 ˆ 21 Q ˆ 22 Q Q 2 2 1 1     L = L1 L2 , K = K1 K2 

A=

0 W2



Then, the following constraint is obtained from Eq. (20). ⎛

ˆ 12 −Pˆ 11 11 −P11 ⎜ () −Pˆ 22

⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

11

()

()

()

()

()

()

−Pˆ 12 12

0

0

ˆ 11 (1, 7) Q 1

0

−Pˆ 21 −Pˆ 22 12 12 11 T H H − P22 −P12 22 () −P22 22

0

0

ˆ 21 (2, 7) Q 1

0

0

0

(3, 7)

In

0

0

0

0

−(L2 C)T Q11 2

0

W1T

0

(W1 − L1 W2 )T Q11 2

−Pˆ 11 12

()

()

μ2 Ir 2

μ2

⎞

0

⎟ ⎟ ⎟ (3, 10) ⎟ ⎟ T 21 T ⎟ −(L2 C) Q2 ⎟ ⎟ T⎟ 21 T (W1 − L1 W2 ) Q2 ⎟ ⎟<0 T ⎟ −(L2 W2 )T Q21 ⎟ 2 ⎟ Pˆ 12 ⎟ 12 ⎟ ⎟ Pˆ 22 12 ⎟ T ⎟ 12 12 21 P22 − Q2 − Q2 ⎠ 0

(3, 9) T

−(L2 W2 )T Q11 2

T

T

()

()

()

()

()

()

()

()

()

()

Ir 0 0 2 () (7, 7) (7, 8)

()

()

()

()

()

()

()

(8, 8)

()

()

()

()

()

()

()

()

Pˆ 21 12 11 11T P22 − Q11 2 − Q2

()

()

()

()

()

()

()

()

()

Pˆ 11 12

22 22 P22 22 − Q2 − Q2

T

(33)

6 Illustrative examples

with ˆ 11 (A + BK1 + Σ1 (Θ))T + Q ˆ 12 (BK2 )T (1, 7) = Q 1 1 21 T ˆ (A + BK1 + Σ1 (Θ)) + Q ˆ 22 (BK2 )T (2, 7) = Q 1 1 (3, 7) = (A + Σ1 (Θ))T T

T

12 (3, 9) = (A + Σ1 (Θ) − L1 C)T Q11 2 + Q2 T 22T (3, 10) = (A + Σ1 (Θ) − L1 C)T Q21 2 + Q2

ˆ 11 ˆ 11T (7, 7) = Pˆ 11 11 − Q1 − Q1 ˆ 12 − Q ˆ 21T (7, 8) = Pˆ 12 − Q 11

1

1

ˆ 22 ˆ 22T (8, 8) = Pˆ 22 11 − Q1 − Q1

Now, if we consider this particular solution   11  11 ˆ 0 Pˆ 11 ˆ 12 = P12 , P Pˆ 11 = 22 ˆ 0 P11 0    11 11 ˆ 0 Q2 ˆ 1 = Q1 Q ˆ 22 , Q2 = 0 0 Q 1     L = L1 0 , K = K1 0

 0 , Pˆ 22 12  0 Q22 2

P22 =

 11 P22 0

0 P22 22



It can be easily verified that this solution is also solution of Eq. (32) for all positive L1 = L and K1 = K. Hence, we conclude that all the solutions of Eq. (32) are included in the solution of Eq. (33). Then, the H∞ sliding window observerbased controller (even with just two measurements) is less restrictive than the classical one.

5.2.2 Comparison from computational complexity point of view The provided LMIs are solved by using one of the available MATLAB LMI solvers that are based on the interior-point optimization method to return solutions. When this algorithm is applied to a linear convex problem, it gives results in polynomial time. Hence, when we add decision variables, this leads to more iterations before returning solutions. Then the algorithm will take more time, but not significant because of the nature of the interior-point algorithm. Therefore, the computational complexity is not significantly affected.

6 Illustrative examples In this section, two numerical examples are considered to illustrate the performances of the proposed H∞ sliding window observer-based controller.

379

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6.1 Example 1 Link

Motor θm

θl Jm

Jl

FIG. 1 The one-link flexible joint robot.

Consider the one-link flexible joint robot studied in Ref. [23] and represented in Fig. 1. The dynamic of the robot is described by the following equations: ⎧ θ˙m = ωm ⎪ ⎪ ⎨ Kτ τ b ω˙ m = Jm (θl − θm ) − Jm ωm + Jm u

θ˙l = ωl ⎪ ⎪ ⎩ τ

ω˙ l = − J (θl − θm ) − Mgh J sin(θl ) l

l

where θm , ωm , θl , and ωl are, respectively, the positions and velocities of the motor and the link. Jm and Jl represent the inertia of the motor and the link, 2h and M are the length and mass of the link, b represents the viscous friction, and Kτ is the amplifier gain. The measurements are the position and the velocity of the motor. We suppose that the state and the measurement vectors are affected by the same noise ω(t). The system matrices can be computed with the numerical values of the one-link flexible joint robot parameters available in Ref. [24] ⎧ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 0 1 0 0 0 0 1 ⎪ ⎪ ⎪ ⎜−48.6 −1.25 ⎟ ⎜21.6⎟ ⎜ ⎟ ⎜1⎟ 48.6 0 0 ⎪ ⎜ ⎟ x(t) + ⎜ ⎟ ⎜ ⎟ + ⎜ ⎟ ω(t) ⎪ ⎪ ⎝ 0 ⎠ u(t) + ⎝ ⎠ ⎝1⎠ ⎪x˙ (t) = ⎝ 0 0 0 1⎠ 0 ⎨  ⎪ ⎪ ⎪  ⎪ ⎪ y(t) = 1 ⎪ ⎪ ⎩ 

19.5 0

−19.5 0   Ac 0 x(t) +  0.1 ω(t)  0



0

0    Bc



−3.33 sin(x3 (t))   fC (x(t))

1    W1c

W2c

Cc

 where x(t) = θm ωm θl ωl is the state vector. The Euler discretization of this dynamic system gives the following equations: 

⎧ ⎛ 1 T ⎪ ⎪ ⎪ ⎜−48.6T 1 − 1.25T ⎪ ⎜ ⎪ x(k + 1) = ⎝ ⎪ ⎪ 0 0 ⎨ 19.5T

 ⎪ ⎪ ⎪  ⎪ ⎪ y(k) = 1 0 0 ⎪ ⎪ ⎩   C

0





0 48.6T 0 −19.5T

⎛ ⎛ ⎞ ⎛ ⎞ ⎞ ⎞ 0 0 0 T ⎜21.6T ⎟ ⎜ ⎟ ⎜T ⎟ 0⎟ 0 ⎟ +⎜ ⎟ ω(k) ⎟ x(k) + ⎜ ⎟ u(k) + ⎜ ⎝ 0 ⎠ ⎝ ⎠ ⎝T ⎠ T⎠ 0 1 0 −3.33T sin(x3 (k)) T           B

A

0.1 ω(k) 0 x(k) +   W2

where T = 0.01 s is the sample time.

f (x(k))

W1

6 Illustrative examples

According to Lemma 1 (see Section 8), for the Lipschitzian nonlinear function f , we have ⎧⎛ ⎞⎫ 0 0 0 0 ⎪ ⎪ ⎨ ⎬ 0 0⎟ ⎜0 0 V H4 = ⎝ ⎠ 0 0 ⎪ ⎪ ⎩ 0 0 ⎭ 0

0

±3.33T

0

By solving the LMIs given by the sliding window approach for H = 0.4I4 and α1 = β1 = 0.01, we get μ = 1.2934 ⎛ ⎛ ⎞ ⎞ 1.1512 −0.0009 ⎜ 14.9774 ⎟ ⎜−0.0210⎟ ⎜ ⎟ ⎟ L1 = ⎜ ⎝ 1.4281 ⎠ , L2 = ⎝−0.0028⎠ −2.0128 0.0021   K1 = −28.8861 −5.3067 16.4784 −4.1185   K2 = 0.0061 −0.2985 0.0166 −0.0587

 The simulation results are given by Figs. 2–4 for x(0) = 0.1  T xˆ (0) = 1 2 −1 −3 .

0.5 1

FIG. 2 The trajectories of the real state (solid line) and the state estimate (dashed line).

T 0 and

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FIG. 3 Observation error.

FIG. 4 Influence of the control input.

7 Conclusion

It is clear that the observer error is stable in Fig. 3. Also, in Fig. 4, the control signal is stable and very quickly stabilizes the system states θl , ωl , θm , and ωm . Therefore, the obtained results confirm the effectiveness of the proposed approach.

6.2 Example 2 Consider the example studied in Ref. [18]. The corresponding state-space model is described by         ⎧ 0 1 1 α sin(x2 (k)) 0 ⎪ ⎪ x(k + 1) = x(k) + u(k) + + T ω(k) ⎪ 0 0 0 0 0 ⎪ ⎨           B

  ⎪ ⎪ y(k) = 1 0 x(k) +  0 ω(k) ⎪ ⎪ ⎩    A

C

f (x(k))

W1

W2

where α represents the Lipschitz constant. According to Lemma 1 (see Section 8), for the Lipschitzian nonlinear function f , we have   0 ±α V H2 = 0

0

We show the superiority of the proposed design methodology with respect to Refs. [1, 18,25], by calculating the αmax tolerated by each LMI method. The results are given in Table 1 for H = I2 and α1 = β1 = 0.1. Table 1 Different values of αmax Method

αmax

Ibrir [1] Ibrir and Diopt [25] Classical approach Grandvallet et al. [18] Sliding window approach (N = 2)

1 1630 20,423 31,541 6 × 106

According to the values given in Table 1, it is clear that the proposed design methodology, for both classical and sliding window case, presents more degree of freedom then the existing ones.

7 Conclusion In this chapter, new results on H∞ observer-based stabilization design methodology are presented. The considered nonlinear function is assumed to be globally Lipschitz. Less conservative LMI is established using sliding window of delayed measurements

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in the Luenberger observer and sliding window of delayed states in the feedback control law. The obtained constraint is expressed in terms of BMI to be linearized through a useful approach. A comparison with the standard case is analyzed in order to show the superiority of the H∞ observer-based controller. The validity of the proposed design methodology is illustrated through numerical simulations.

8 Appendix 8.1 Differential mean value theorem Lemma 1 (Differential mean value theorem). Let φ: Rn → Rq . Let x1 ∈ Rn , x2 ∈ Rn , suppose that φ is differentiable on Co(x1 , x2 ). Then, there are constant vectors z1 , . . . , zq ∈ Co(x1 , x2 ), zi = x1 , zi = x2 , i = 1, . . . , q such that ⎛ φ(x1 ) − φ(x2 ) = ⎝

q,n 

⎞ eq (i)eTn (j)

i,j=1

∂φi (zj )⎠ (x1 − x2 ) ∂xj

where Co(x1 , x2 ) = {λx1 + (1 − λ)x2 , 0 ≤ λ ≤ 1} is the convex domain of x1 , x2 . Proof. The proof of this theorem is given in Ref. [21].

8.2 Schur’s lemma Lemma 2 (The Schur complement lemma [27]). Given constant matrices M, N, and S of appropriate dimensions where M and S are symmetric, then S > 0 and M + N T S−1 N < 0 if and only if   M NT <0 N

−S

−S NT

N M

or equivalently 

 <0

References [1] S. Ibrir, Static output feedback and guaranteed cost control of a class of discrete-time nonlinear systems with partial state measurements, Nonlinear Anal. Theory Methods Appl. 68 (7) (2008) 1784–1792. [2] C. Wang, Z. Zuo, Z. Lin, Z. Ding, Consensus control of a class of Lipschitz nonlinear systems with input delay, IEEE Trans. Circuits Syst. I Regular Papers 62 (11) (2015) 2730–2738.

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