Observer design for one-sided Lipschitz descriptor systems

Applied Mathematical Modelling 40 (2016) 2301–2311

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Observer design for one-sided Lipschitz descriptor systems Ali Zulfiqar, Muhammad Rehan∗, Muhammad Abid Department of Electrical Engineering, Pakistan Institute of Engineering and Applied Sciences (PIEAS), P.O. Box 45650, Islamabad, Pakistan

a r t i c l e

i n f o

Article history: Received 25 October 2013 Revised 31 July 2015 Accepted 23 September 2015 Available online 9 October 2015 Keywords: One-sided Lipschitz condition Quadratic inner boundedness Descriptor systems Linear matrix inequality Observer design

a b s t r a c t This paper describes the design of a nonlinear observer for a class of nonlinear descriptor systems with one-sided Lipschitz nonlinearities. To provide a general framework applicable to a large class of systems, nonlinearities and disturbances are considered at the state as well as at the output equations. Non-strict and strict bilinear matrix inequality (BMI)-based robust observer synthesis schemes are provided by utilizing the one-sided Lipschitz condition, the concept of quadratic inner boundedness, the generalized Lyapunov theory for singular systems, and the L2 gain reduction. The BMI-based condition is converted into the linear matrix inequality (LMI)-based condition by utilizing change of variables for straightforward computation of the observer gain matrix. In contrast to the traditional observer schemes for one-sided Lipschitz nonlinear systems, the resultant scheme is applicable to singular systems, capable of dealing with nonlinearities at the output equation and appropriate for treating with disturbances. Two simulation examples for the one-sided Lipschitz and the singular one-sided Lipschitz nonlinear systems are provided to validate the proposed observer synthesis methodologies. © 2015 Elsevier Inc. All rights reserved.

1. Introduction Descriptor systems, composed of both the ordinary differential equations (ODEs) and the algebraic equations, appearing from a more precise inherent modeling methodology, representing behavior of a more general class of systems (than the traditional state-space models), have emerged in many scientific and engineering disciplines such as biological systems, chemical processes, physical systems, electrical circuits, power systems and mechanical structures in addition to the social and economic fields [1–9]. Descriptor systems are also called singular systems, for which the initial condition always satisfies some algebraic constraints in contrast to the traditional ODE-based models. Observer design for these descriptor systems to estimate their state vectors has vast applications in controller synthesis, filtering, and fault diagnosis [2,3,6,10–13]; however, extra efforts are required to apply the generalized Lyapunov theory based on LaSalle’s principle to ensure convergence of the estimation error. Considerable research activities concerning observer design for linear (see for example, [1–3,10,12,14]) as well as nonlinear (see, for example, [3,6,11,13]) descriptor systems have been carried out in the past decade. In the recent years, one-sided Lipschitz nonlinear systems have become an important class of nonlinear systems due to their ability in representing a more general form compared to the Lipschitz systems and owing to superiority of the one-sided Lipschitz constant than the Lipschitz constant. Thanks to their numerous applications, the observer design problem has been studied in the works [15–19] for the one-sided Lipschitz systems. In [15], conditions for existence of an observer and a preliminary methodology for the observer gain selection are presented. The concept of quadratic inner boundedness was introduced in [16], which was

∗

Corresponding author. Tel.: +92 51 2207381x3443; fax: +92 51 2208070. E-mail addresses: alizulfi[email protected] (A. Zulfiqar), [email protected] (M. Rehan), [email protected] (M. Abid).

http://dx.doi.org/10.1016/j.apm.2015.09.056 S0307-904X(15)00590-9/© 2015 Elsevier Inc. All rights reserved.

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found to be interesting for linear matrix inequality (LMI)-based and Riccati equation based full-order and reduced-order observer synthesis [17,18]. Further, a discrete-time observer design approach, providing Lyapunov-based stability formulizations, has been presented in the recent work [19,20]. Note that these observer design schemes cannot be applied to the descriptor systems, containing both ordinary differential and algebraic equations. Further, the approaches [15–19] assume presence of nonlinear dynamics in the state equation only; though, incorporation of a nonlinear function into the output equation provides a more general description of a system and, hence, a less conservative observer design framework. Moreover, the effects of perturbations and disturbances, to ensure robust state estimation, have not been focused in these studies. In this paper, a novel robust observer design methodology for the one-sided Lipschitz descriptor systems in the presence of both the input and the output nonlinearities in addition to disturbances is considered. By utilizing the one-sided Lipschitz condition, the quadratic inner boundedness, the generalized Lyapunov stability theory and the L2 gain reduction, first a nonstrict bilinear matrix inequality (BMI)-based condition is developed, which is further transformed into strict BMIs using a proper change of variables. Furthermore, an LMI-based formulation, suitable for computing the observer gain matrix, is provided herein by utilizing an exceptional change of variables to ensure asymptotic convergence of the state estimation error to origin and to accomplish robustness against L2 norm bounded disturbances. The main contribution of the paper is summarized below: (1) To the best of authors’ knowledge, the observer design problem for the one-sided Lipschitz descriptor systems is addressed for the first time. (2) The present work provides a less conservative observer design framework for the one-sided Lipschitz nonlinear systems by considering nonlinearities in the output as well as the state equations. (3) Robustness against perturbations is addressed by considering disturbance at the state equation as well as at the output equation in contrast to the works [15–20]. Simulation results of the observer design methodologies for the one-sided Lipschitz and the singular one-sided Lipschitz nonlinear systems are provided to verify effectiveness of the proposed schemes. This paper is organized as follows. System description is provided in Section 2. Section 3 presents observer design schemes utilizing BMIs and LMIs. Simulation results for the state estimation are detailed in Section 4. Section 5 draws conclusions of the study. Standard notation is used throughout the paper. The matrix inequality Z > 0 (or Z ≥ 0) for a symmetric matrix Z implies that Z is a positive-definite (or semi-positive-definite) matrix. Transpose of a matrix Z is represented by ZT . a, b represents the inner product of two vectors a and b in Rn space. The L2 gain of a nonlinear system with input and output  vectors w and z (of ∞ 2 appropriate dimensions) under zero initial condition is defined as supw2 =0 (z2 /w2 ), where  · 2 = 0  ·  dt and  ·  represent the L2 norm and the Euclidian norm, respectively, for the vectors w and z. 2. System description Consider a nonlinear descriptor system, having dynamics

E

dx = Ax + ϕ1 (x, u) + Bd, dt

y = Cx + H ϕ2 (x, u) + Dd,

(1)

(2)

where u ∈ Rm , x ∈ Rn , y ∈ Rp , and d ∈ Rq represent the input, the state, the output, and the disturbance vectors for the above system. The nonlinear functions ϕ 1 (x, u) ∈ Rn and ϕ 2 (x, u) ∈ Rn denote the nonlinearities in the state and the output equations, respectively. The matrices A ∈ Rn × n , B ∈ Rn × q , C ∈ Rp × n , D ∈ Rp × q , E ∈ Rn × n and H ∈ Rp × n have constant entries. The matrix E, called the mass matrix, can be singular to represent a descriptor system. Further, the matrix E can be non-symmetric to denote a more general form of singular systems. Remark 1. Usually, number of outputs is less than or equal to number of states for a system, that is p ≤ n. If p ≤ n is verified, any arbitrary nonlinear function f(x, u) ∈ Rp can be represented by Hϕ 2 (x, u), without loss of generality, through appropriate selection of H and ϕ 2 (x, u). Even if p > n, the nonlinearity Hϕ 2 (x, u) can represent f(x, u), for instance, if some of the entries of f(x, u) are zero. Assumption 1. (a) The nonlinear functions ϕ 1 (x, u) and ϕ 2 (x, u) satisfy the one-sided Lipschitz condition, given by:

ϕi (x1 , u) − ϕi (x2 , u), x1 − x2  ≤ ρi x1 − x2 2 , i = 1, 2,

(3)

for real scalars ρ 1 and ρ 2 and x1 , x2 ∈ Rn , where ρ 1 and ρ 2 are the one-sided Lipschitz constants for the corresponding nonlinearities.

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(b) The nonlinear functions ϕ 1 (x, u) and ϕ 2 (x, u) satisfy the quadratic inner boundedness, given by: T (ϕi (x1 , u) − ϕi (x2 , u)) (ϕi (x1 , u) − ϕi (x2 , u)) 2 ≤ δi x1 − x2  + i x1 − x2 , ϕi (x1 , u) − ϕi (x2 , u), i = 1, 2,

(4)

for δ 1 , δ 2 , 1 , 2 ∈ R. The one-sided Lipschitz condition in Assumption 1(a) and the quadratic inner boundedness criteria in Assumption 1(b) are satisfied for all Lipschitz nonlinearities but the converse is not true (see [16–18,20]). It is worth mentioning that the one-sided Lipschitz constants ρ 1 and ρ 2 are always less than or equal to the corresponding Lipschitz constants for the Lipschitz nonlinearities. Owing to this fact, the one-sided Lipschitz condition in (3) can be employed to design observer for Lipschitz nonlinear systems with relatively large Lipschitz constants. Hence, the one-sided Lipschitz condition (3) is less conservative than the conventional Lipschitz condition. Additionally, the quadratic inner boundedness in (4) produces the traditional Lipschitz condition as a specific case for i = 0. Moreover, the inequality (4), recently investigated for the one-sided Lipschitz nonlinear systems, facilitates the observer synthesis owing to its specific tractable characteristics. Assumption 2. The disturbance d has bounded L2 norm. The proposed observer dynamics are given by:

E

dxˆ ˆ u) + L(y − yˆ), = Axˆ + ϕ1 (x, dt

(5)

ˆ u), yˆ = C xˆ + H ϕ2 (x, Rn

(6)

Rp

where xˆ ∈ and yˆ ∈ represent the state and the output vectors for the observer. The matrix L ∈ gain matrix to be calculated for state estimation.

Rn × p

represents the observer

3. Observer synthesis Now the objective of the present work is to formulate conditions for designing an observer (5) and (6) for the system (1) and (2) under Assumptions 1 and 2. Defining e = x − xˆ and e1 = y − yˆ and further using (1) and (2) and (5) and (6), we obtain

E

de ˆ u) − LH (ϕ2 (x, u) − ϕ2 (x, ˆ u)) + (B − LD)d, = (A − LC )e + ϕ1 (x, u) − ϕ1 (x, dt

(7)

ˆ u)) + Dd. e1 = Ce + H (ϕ2 (x, u) − ϕ2 (x,

(8)

Theorem 1. Consider a one-sided Lipschitz nonlinear system of the form (1) and (2) satisfying Assumptions 1 and 2. Suppose there exist scalars γ , ɛ1 , ɛ2 , ɛ3 and ɛ4 and matrices P ∈ Rn × n and L ∈ Rn × p , such that the inequalities

P T E = E T P ≥ 0,

(9)

γ > 0, ε1 > 0, ε2 > 0, ε3 > 0, ε4 > 0, ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

AT P + P T A − P T LC − C T LT P + ε1 ρ1 I +ε2 δ1 I + ε3 ρ2 I + ε4 δ2 I ∗ ∗ ∗ ∗

 

(10)

 

P T − ε1 I/2 +ε2 1 I/2 −ε2 I ∗ ∗ ∗

⎤



−P T LH − ε3 I/2 +ε4 2 I/2 0 −ε4 I ∗ ∗

P (B − LD) T

0 0 −γ I ∗

I

⎥ ⎥

0 ⎥ < 0, ⎥ 0 ⎥ 0 ⎦ −γ I

(11)

are satisfied under rank(P T E ) = rank(E ). The nonlinear observer (5) and (6) ensures (a) asymptotic convergence of xˆ to x, if d = 0; (b) the L2 gain from the disturbance d to the state estimation error e = x − xˆ less than γ for all time. Proof. Consider a Lyapunov function candidate

V (t, e) = eT γ E T Pe,

(12)

where E T P = P T E ≥ 0, rank(P T E ) = rank(E ), and γ > 0. Note that V (t, e) = 0, if and only if Ee = 0, and V(t, e) > 0, otherwise (see the generalized Lyapunov stability theory [6,21,22]). Consider the inequality



J = γ −1 V˙ + eT e − γ 2 dT d < 0.

(13)

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Under d = 0, V˙ (t, e) < 0 is implied, that is to say, the error converges to origin asymptotically. Integrating (13) from 0 to ∞, it is implicit to conclude that e2 /d2 < γ [23,24]. Explicitly, the L2 gain from the disturbance d to the state estimation error e is less than γ . Taking the time-derivative of (12) along (7), we obtain T ˆ u)) Pe + dT (B − LD)T Pe V˙ (t, e) = eT (A − LC )T Pe + (ϕ1 (x, u) − ϕ1 (x,

ˆ u)) H T LT Pe + eT P T (A − LC )e −(ϕ2 (x, u) − ϕ2 (x, T

ˆ u)) + eT P T (B − LD)d +eT P T (ϕ1 (x, u) − ϕ1 (x, ˆ u)). −eT P T LH (ϕ2 (x, u) − ϕ2 (x,

Using (13) and (14) reveals that J =



ψT

ψ = eT (ϕ1 (x, u) − ϕ1 (x, ˆ u))

T

1ψ

(14)

< 0, where

ˆ u)) (ϕ2 (x, u) − ϕ2 (x,

T

dT

T

,

(15)

⎡ ⎤ (A − LC )T P + PT (A − LC ) + γ −1 I PT −PT LH PT (B − LD) ⎢ ⎥ ∗ 0 0 0 1 = ⎣ ⎦. ∗ ∗ 0 0 ∗ ∗ ∗ −γ I

(16)

ˆ u)) ≥ 0, for i = 1, 2, which further can be written as ψ T 2 ψ ≥ 0 and Assumption 1(a) reveals that ρi eT e − eT (ϕi (x, u) − ϕi (x, 3 ψ ≥ 0, respectively, where

ψ T

⎡ ⎤ ⎡ ⎤ ε1 ρ1 I −ε1 I/2 0 0 ε3 ρ2 I 0 −ε3 I/2 0 0 0 0⎥ 0 0 0⎥ ⎢ ∗ ⎢ ∗ 2 = ⎣ , 3 = ⎣ . ∗ 0 0 0⎦ ∗ 0 0 0⎦ ∗

0

0

∗

0

0

0

(17)

0

The quadratic inner boundedness condition in Assumption 1(b) implies ψ T 4 ψ ≥ 0 and ψ T 5 ψ ≥ 0, where

⎡ ⎤ ⎡ ⎤ ε2 δ1 I ε2 1 I/2 0 0 ε4 δ2 I 0 ε4 2 I/2 0 0 0⎥ −ε2 I 0 0 0⎥ ⎢ ∗ ⎢ ∗ 4 = ⎣ , 4 = ⎣ . 0⎦ ∗ 0 0 0⎦ ∗ 0 −ε4 I ∗

0

∗

0 0

Combining the matrix inequalities produces

ψ T

⎡

(A − LC )T P + PT (A − LC ) + γ −1 I ⎢ +ε1 ρ1 I + ε2 δ1 I + ε3 ρ2 I + ε4 δ2 I ⎢ ⎢ ∗ ⎣ ∗ ∗

1ψ

< 0,

 

ψ T

0 2ψ

0 ≥ 0,

ψ T

 

P T − ε1 I/2 +ε2 1 I/2 −ε2 I ∗ ∗

(18)

0 3ψ

≥ 0, ψ T 4 ψ ≥ 0 and ψ T 5 ψ ≥ 0 using S-procedure



−P T LH − ε3 I/2 +ε4 2 I/2 0 −ε4 I ∗

⎤ P T (B − LD) 0 0 −γ I

⎥ ⎥ ⎥ < 0. ⎦

(19)

If (19) holds, it implies that J = ψ T 1 ψ < 0. The BMI (11) is obtained by applying Schur’s complement to the inequality (19), which completes the proof of Theorem 1.  The BMI-based inequalities provided in Theorem 1 cannot be solved effectively using existing numerical algorithms due to presence of the non-strict inequalities. The conditions in Theorem 1 can be converted into the strict BMIs (or LMIs) by application of change of variables [25]. Theorem 2. Consider a one-sided Lipschitz nonlinear system of the form (1) and (2) satisfying Assumptions 1 and 2. Suppose there exist scalars γ , ɛ1 , ɛ2 , ɛ3 and ɛ4 , symmetric matrix X ∈ Rn × n and matrices L ∈ Rn × p and Y ∈ Rn × n such that the BMI-based strict inequalities (10),

X > 0,

⎡



⎢Z1 ⎢ ⎢∗ ⎢ ⎢∗ ⎣∗ ∗

(20)

 

E T X + Y E⊥ −ε1 I/2 + ε2 1 I/2 −ε2 I ∗ ∗ ∗

 

−E T XLH − Y E⊥ LH −ε3 I/2 + ε4 2 I/2 0 −ε4 I ∗ ∗

⎤



E T XB + Y E⊥ B −E T XLD − Y E⊥ LD 0 0 −γ I ∗

I

⎥ ⎥

0 ⎥ < 0, ⎥ 0 ⎥ 0 ⎦ −γ I

(21)

are satisfied, where E⊥ is the orthogonal complement of E and

Z1 = AT XE + E T XA + AT E⊥T Y + Y E⊥ A − E T XLC − C T LT XE

−Y E⊥ LC − C T LT E⊥T Y + ε1 ρ1 I + ε2 δ1 I + ε3 ρ2 I + ε4 δ2 I.

(22)

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The nonlinear observer (5) and (6) ensures (a) asymptotic convergence of xˆ to x, if d = 0; (b) the L2 gain from the disturbance d to the state estimation error e = x − xˆ less than γ for all time. T Y to (11) implies (21). Note that V (t, e) = eT γ E T Pe = eT γ E T XEe ≥ 0 because Proof. Applying the transformation P = XE + E⊥ E T P = E T XE for matrices X > 0 and Y. X > 0 implies that V˙ (t, e) < 0 and V˙ (t, e) = 0 for Ee = 0 and Ee = 0, respectively. The remaining part of the proof is same as for the proof of Theorem 1. 

To simplify the numerical computations of the design constraints, a LMI-based observer synthesis strategy is provided in the following theorem. Theorem 3. Consider a one-sided Lipschitz nonlinear system of the form (1) and (2) satisfying Assumptions 1 and 2. Suppose there exist scalars γ , ɛ1 , ɛ2 , ɛ3 and ɛ4 , a symmetric matrix X ∈ Rn × n and a matrix Y ∈ Rn × n such that the LMIs (10), (20),





I − E T X − Y E⊥ > 0, I

I ∗

⎡



⎢Z2 ⎢ ⎢∗ ⎢ ⎢∗ ⎣∗ ∗

(23)

 

E T X + Y E⊥ −ε1 I/2 + ε2 1 I/2 −ε2 I ∗ ∗ ∗

 

−QH − ε3 I/2 +ε4 2 I/2 0 −ε4 I ∗ ∗

E T XB + Y E⊥ B −QD 0 0 −γ I ∗

⎤

 I

⎥ ⎥

0 ⎥ < 0, ⎥ 0 ⎥ 0 ⎦ −γ I

(24)

are satisfied, where E⊥ is the orthogonal complement of E and

Z2 = AT XE + E T XA + AT E⊥T Y + Y E⊥ A − QC − C T Q T + ε1 ρ1 I + ε2 δ1 I + ε3 ρ2 I + ε4 δ2 I.

(25)

The nonlinear observer (5) and (6) with observer gain matrix

L = (E T X + Y E⊥ )−1 Q,

(26)

ensures (a) asymptotic convergence of xˆ to x, if d = 0; (b) the L2 gain from the disturbance d to the state estimation error e = x − xˆ less than γ for all time. Proof. : Applying the transformation Q = P T L to (11) reveals

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

AT P + PA − QC − C T Q T + ε1 ρ1 I +ε2 δ1 I + ε3 ρ2 I + ε4 δ2 I ∗ ∗ ∗ ∗

 

 

P T − ε1 I/2 +ε2 1 I/2 −ε2 I ∗ ∗ ∗

⎤



−QH − ε3 I/2 +ε4 2 I/2 0 −ε4 I ∗ ∗

P B − QD T

0 0 −γ I ∗

I

⎥ ⎥ 0 ⎥ < 0. ⎥ 0 ⎥ 0 ⎦ −γ I

(27)

The matrix P T = E T X T + Y T E⊥ is required to be invertible for solving (26). To ensure the matrix P invertible, we require the following [6]:



I ∗

I − PT I



> 0.

(28)

By applying the transformation P = XE + E⊥ T Y into (27) and (28) and by solving rigorous matrix algebra, we obtain LMIs (23) and (24). The state feedback gain matrix can be computed by solving L = (P T )−1 Q, which further implies (26) for symmetric matrices X and Y. The remainder of the proof is same as for the proof of Theorem 2.  Remark 2. An LMI-based methodology can be developed by applying the transformation PL = Q under P = P T > 0 in a traditional way (like [24]), which can constrain the matrix E to be symmetric. However, the attractiveness of the proposed LMI-based methodology in Theorem 3 is that it does not require the matrix P to be symmetric or positive-definite owing to utilization of T Y is used the two transformations, P T L = Q and P = XE + E⊥ T Y , simultaneously (also see [25]). The transformation P = XE + E⊥ to convert the non-strict inequalities, arising for singular systems, into the strict inequalities, while the other transformation P T L = Q is supportive to derive LMIs in the presence of the output nonlinearity and the output disturbance (that is, ϕ 2 (x, u) and Dd). It is worth mentioning that the resultant strict LMI-based methodology in Theorem 3 for singular one-sided Lipschitz nonlinear systems does not require the matrix E to be symmetric.

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Remark 3. The proposed methodologies by Theorems 1–3 can be utilized for non-singular one-sided Lipschitz nonlinear systems having an invertible matrix E. In that case, the proposed methodologies are advantageous, as compared to the traditional approaches like [15–20], owing to the incorporation of nonlinear dynamics ϕ 2 (x, u) and disturbance d for the state estimation. Remark 4. LMI constraint (23) in Theorem 2 has been employed to obtain the matrix E T X + Y E⊥ invertible. This LMI can be ignored, if an invertible solution of the matrix E T X + Y E⊥ can be determined, which can be suitable, for robust observer design, to attain a feasible solution for a small value of L2 gain γ . 4. Simulation results Two simulation examples are provided in this section to illustrate advantages of our schemes compared with the available methodologies for the observer design of one-sided Lipschitz nonlinear systems and to demonstrate effectiveness of the proposed observer synthesis approach for the singular one-sided Lipschitz systems. In Example 1, state estimation under disturbances (in state equation as well as output equation) and output nonlinearity is addressed. Example 2 is devoted to the modeling of synchronous systems as singular systems and state estimation of the one-sided Lipschitz descriptor systems. 4.1. Example 1: Application to a one-sided Lipschitz system Consider the dynamics of a moving particle, given by





1 0 E= , 0 1

 B=



0 , 0





1 1 A= , −1 1



1 0 0 1

ϕ1 (x, u) =



D= 0



0

 T

,

(29)



1 ,



−x1 x21 + x22

, −x2 x21 + x22

1 0

C=H=

(30)

 ϕ2 (x, u) =



0.001x21 . 0

(31)

In contrast to [16–19], the output nonlinearity ϕ 2 (x, u) is considered to incorporate nonlinear characteristics of the sensor. In the traditional observer design techniques [16–19] for one-sided Lipschitz nonlinear systems, one of the plant state x1 is available as output. However, in the presence of the output nonlinearity ϕ 2 (x, u), none of the states x1 and x2 is exactly measureable, for which state vector estimation is challenging. It has been derived in [18] (see also [16]) that ϕ 1 (x, u) satisfies (3) and (4) for ρ1 = 0, δ1 = −99 and 1 = −100 under x ≤ 5. Further, it can be verified that ϕ 2 (x, u) satisfies (3) for ρ1 = 0.0045 because T 2 ˆ u)) = 0.001(x1 + xˆ1 )(x1 − xˆ1 ) , (x − xˆ) (ϕ2 (x, u) − ϕ2 (x,

√ 2 ≤ 0.002 5(x1 − xˆ1 ) ,

(32)

is valid for the region x ≤ 5. Furthermore, ϕ 2 (x, u) satisfies (4) for δ1 = 0.005 and 1 = 0 for x ≤ 5. The optimal value of γ was obtained as γopt = 100.01 by solving the LMI constraints in Theorem 3 using Robust Control Toolbox of Matlab. However, γ = 150 is selected in the present simulation study due to slow response of the observer for the optimal case. Note that the LMIs (23) and (24) are feasible for all γ ≥ γ opt . By solving the LMIs in Theorem 3 for γ = 150, we obtain ε1 = 0.0015, ε2 = 0.0363, ε3 = 0.0356, ε4 = 1.24, and L = [ 0.6755 0.0153 ]T . Figs. 1 and 2 plot states of the plant and the observer against time for initial conditions x(0) = [2 1.2] and xˆ(0) = [0.5 −1.5] under disturbance, given by:



T

d = 0.15 sin (10t ) 0.12 sin (15t ) 0.13 sin (20t ) .

(33)

The states of the proposed observer are converging to the plant states. Fig. 3 plots the estimation errors between the plant and the observer states, which are converging near the origin under effect of the input and the output disturbances. Hence, the proposed observer design framework can be applied to estimate the state vector of the one-sided Lipschitz nonlinear systems with output nonlinearity and disturbances. 4.2. Example 2: Application to a one-sided Lipschitz descriptor system FitzHugh–Nagumo (FHN) systems are widely used to model neuronal behavior, chemical reaction kinetics, artificial neural networks and electronic oscillators and to study external stimulation (such as deep brain stimulation) therapies for effective treatment of brain disorders (see [26,27] and references therein). Consider the following two coupled synchronous FHN neurons with both the unidirectional and the bidirectional gap junctions in the medium between them [27,28] under disturbances.

x˙ 11 = x11 (x11 − 1)(1 − rx11 ) − v1 x12 − g1 (x11 − x21 ) + (a/ω) cos ωt + d1 , x˙ 12 = b1 x11 + d2 , x˙ 21 = x21 (x21 − 1)(1 − rx21 ) − v2 x22 − g2 (x21 − x11 ) + (a/ω) cos ωt + d3 , x˙ 22 = b2 x21 + d4 .

(34)

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2

State x1 Estimation of x

Actual and estimated states

1.5

1

1

0.5

0

-0.5

-1

-1.5 0

5

10

15

20

25

30

Time (sec) Fig. 1. Convergence of the observer state xˆ1 to the plant state x1 in Example 1.

1.5

State x2 Estimation of x

Actual and estimated states

1

2

0.5

0

-0.5

-1

-1.5

0

5

10

15

20

25

30

Time (sec) Fig. 2. Convergence of the observer state xˆ2 to the plant state x2 in Example 1.

where x11 and x21 represent normalized activation potentials of the master and the slave neurons, respectively, x12 and x22 are the recovery variables for the master and the slave neurons, respectively, (a/ω)cos ωt is the external stimulation input, d1 , d2 , d3 and d4 are disturbances, and v1 , v2 , b1 , b2 and r are the parameters of the FHN systems. The activation potentials are measureable; however, recovery variables cannot be measured through sensors. By considering the states x11 and x21 measureable, an observer for the coupled FHN systems can be easily designed for estimation of x12 and x22 . While studying behavior of the coupled neurons in a neural network, it is almost impossible in vivo to record activation potentials of every neuron owing to biological restrictions. It can be interesting to study estimation of both the activation and the recovery variables of coupled synchronous neurons in a neural network when activation potentials of all of the neurons of a network are not measureable. Therefore, the present study considers availability of the activation potential x11 of the master FHN system. Let es = x11 − x21 denotes the synchronization error. In a recent study, it has been shown that neurons with different recovery variables can have identical activation potentials, that is, x11 − x21 → 0, owing to the coupling between the neurons. Therefore, the coupled neuronal model (34) can be rewritten as:

x˙ 11 = −rx311 + (1 + r)x211 − x11 − v1 x12 − g1 (x11 − x21 ) + (a/ω) cos ωt + d1 , x˙ 12 = b1 x11 + d2 , x˙ 21 = −rx321 + (1 + r)x221 − x21 − v2 x22 − g2 (x21 − x11 ) + (a/ω) cos ωt + d3 , x˙ 22 = b2 x21 + d4 , 0e˙ s = x11 − x21 − es .

(35)

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3

Estimation error e1 Estimation error e

2.5

2

Estimation errors

2

1.5

1

0.5

0

-0.5 0

5

10

15

20

25

30

Time (sec) Fig. 3. Convergence of the state estimation errors e1 and e2 in Example 1.

1 Estimation error e 1 Estimation error e 2

0.8

Estimation error e 3 Estimation error e 4

Estimation errors

0.6

0.4

0.2

0

-0.2

-0.4

0

0.5

1

1.5

2

2.5 3 Time (sec)

3.5

4

4.5

5

Fig. 4. Convergence of the state estimation errors e1 , e2 , e3 and e4 in Example 2.

Under the assumption of availability of x11 , the model (35) can be rewritten in the form of (1) by selecting

⎡1 0 0 0 0⎤ ⎡0 0 0 0 0⎤ 0 1 0 0 0 ⎢ ⎥ ⎢0 0 0 0 0⎥ E = ⎢0 0 1 0 0⎥, E⊥ = ⎢0 0 0 0 0⎥, ⎣ ⎦ ⎣ ⎦ 0 0

0 0

0 0

1 0

0 0

⎡−g − 1 −v g1 0 1 1 0 0 0 ⎢ b1 0 −g2 − 1 −v2 A = ⎢ g2 ⎣ 0 1

0 0

(36)

0 0 0 0 0 0 0 0 0 1

b2 −1

0 0

⎤

0 0⎥ 0 ⎥, ⎦ 0 −1

⎡1⎤T ⎢0 ⎥ C = ⎢0 ⎥ , ⎣ ⎦ 0 0

(37)

A. Zulfiqar et al. / Applied Mathematical Modelling 40 (2016) 2301–2311

Actual and estimated activation potentials

1.2

2309

Activation potential x 11 Estimation of x 11

1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 0

20

40

60

80

100 120 Time (sec)

140

160

180

200

Fig. 5. Convergence of the observer state xˆ1 to the neural system state x1 in Example 2.

Actual and estimated recovery variables

3 Recovery variable x 12 Estimation of x 12

2.5

2

1.5

1

0.5

0

-0.5 0

20

40

60

80

100 120 Time (sec)

140

160

180

200

Fig. 6. Convergence of the observer state xˆ2 to the neural system state x2 in Example 2.

⎡1 0 0 0⎤ ⎡x ⎤ ⎡ x ⎤ ⎡ ⎤ 1 11 0 x2 ⎥ ⎢x12 ⎥ ⎢0 1 0 0⎥ ⎢ ⎢0⎥ B = ⎢0 0 1 0⎥, DT = ⎣ ⎦, x = ⎢x3 ⎥ = ⎢x21 ⎥, H = 0, ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ 0 0 0

0 0

0 0

1 0

0

x4 x5

(38)

x22 es

⎡−rx3 + (1 + r)x2 + (a/ω) cos ωt ⎤ 1

1

0 ⎢ ⎥ 3 2 ⎥ ϕ1 (x, u) = ⎢ ⎣−rx3 + (1 + r)x3 + (a/ω) cos ωt ⎦, ϕ2 (x, u) = 0. 0 0

(39)

2310

A. Zulfiqar et al. / Applied Mathematical Modelling 40 (2016) 2301–2311

Actual and estimated activation potentials

1.2

Activation potential x 21 Estimation of x 21

1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 0

20

40

60

80

100 120 Time (sec)

140

160

180

200

Fig. 7. Convergence of the observer state xˆ3 to the neural system state x3 in Example 2.

Actual and estimated recovery variables

3 Recovery variable x 22 Estimation of x 22

2.5

2

1.5

1

0.5

0

-0.5 0

20

40

60

80

100 120 Time (sec)

140

160

180

200

Fig. 8. Convergence of the observer state xˆ4 to the neural system state x4 in Example 2.

We choose r = 10, a = 1, ω = 0.26π , g1 = 0.7, g2 = 1, b1 = 0.8, b2 = 1.2, v1 = b−1 , and v2 = b−1 for the state simulation pur1 2 pose. Note that the nonlinearity ϕ 1 (x, u) is locally Lipchitz and the activation potentials of neurons are always bounded; consequently, their normalized values x1 and x3 remain bounded between −0.4 and 1 as seen in the studies [26–28]. Due to the bounded activation potentials, the function ϕ 1 (x, u) can be readily treated as a globally Lipchitz function. As stated earlier, the conditions in (3) and (4) are valid for a Lipschitz function. To design a controller, we fix 1 = 0.10 and select the scalars ρ 1 and δ 1 as much large as possible so that maximum tolerance can be attained against the nonlinearity ϕ 1 (x, u) for observer synthesis objective. The proposed observer design scheme in Theorem 3 is applied to determine the observer gain matrix for ρ1 = 0.14, δ1 = 0.20, 1 = 0.10, and ρ2 = δ2 = 2 = 0 to robust stability of the estimation error e. The observer gain matrix LT = 105 × [8.73 −4.40 0.19 −0.07 −0.03] and the parameters ε1 = 2.2, ε2 = 1.8, ε3 = 3.85, ε4 = 253.42, and γ = 100 are

A. Zulfiqar et al. / Applied Mathematical Modelling 40 (2016) 2301–2311

2311

obtained by solving the LMIs of Theorem 3. For simulation study, the coupled model (34), rather than (35), and the corresponding state observer are coded in Matlab. The disturbances are selected as:

d1 = 0.1 sin 15t, d2 = 0.12 sin 20t, d3 = 0.13 sin 25t, d4 = 0.11 sin 32t.

(40)

State estimation errors e1 = x1 − xˆ1 , e2 = x2 − xˆ2 , e3 = x3 − xˆ3 and e4 = x4 − xˆ4 are plotted in Fig. 4. Note that the state estimation errors are converging in the presence of disturbances as demonstrated by Fig. 4. Figs. 5–8 demonstrate the activation potentials and recovery variables of the coupled neurons along with the corresponding estimated states. It is notable that the observer states converge to the states of coupled FHN neuronal system. Hence, the proposed methodology can be utilized for robust state estimation of the singular one-sided Lipschitz nonlinear systems. 5. Conclusions This paper presented the BMI-based and the LMI-based observer design strategies for estimating state vector of the one-sided Lipschitz descriptor systems. Nonlinearities and disturbances at the state as well as the output equations were incorporated for the class of singular nonlinear systems. 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