New results on regional observer-based stabilization for locally Lipchitz nonlinear systems

Chaos, Solitons and Fractals 123 (2019) 173–184

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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos

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New results on regional observer-based stabilization for locally Lipchitz nonlinear systems Syeda Rabiya Hamid, Muhammad Shahid Nazir, Muhammad Rehan∗, Haroon ur Rashid Department of Electrical Engineering, Pakistan Institute of Engineering and Applied Sciences (PIEAS), Islamabad, Pakistan

a r t i c l e

i n f o

Article history: Received 22 October 2018 Revised 29 March 2019 Accepted 2 April 2019

Keywords: Locally Lipschitz systems Observer-based control Regional design Exponential stabilization Robustness

a b s t r a c t This paper describes the design of a regional observer-based controller for the locally Lipchitz nonlinear systems, which can be employed successfully to attain both monitoring and control of a wide range of systems. An observer-based control approach has been employed to attain advantages of the traditional state feedback along with the state estimation through an observer. A lot of work has been accomplished for the globally Lipchitz nonlinear systems. However, a less conservative continuity, called the generalized ellipsoidal Lipchitz condition, has been applied in this paper to consider the locally Lipchitz systems. This condition is then incorporated to attain convex routines for computing the local controller and observer gains. The focus of the present study is to investigate conditions for simultaneous design of observer and controller under locally Lipchitz nonlinearities for systems with norm-bounded disturbances that not only guarantees the stabilization and true state estimation but also robustness against external perturbations. Furthermore, the decoupling of the observer and controller design conditions has been worked out for obtainment of a simple design method. Since the resultant control approach applies to a general class of systems, it can be straightforwardly employed to the globally Lipschitz nonlinear systems as a specific case. The approach is tested via a chaotic system and simulation results are provided to validate the effectiveness of resultant control schemes for the locally Lipschitz systems. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction Over the past few decades, state observers for dynamic realworld systems have been intensively studied due to their importance in state filtering, parameter estimation, system identification, fault diagnosis, nonlinearity compensation, and control system design techniques [1–7]. Several theoretical studies, related to the feedback controller design, undertake the availability of all the states of a system for feedback control. Even though, direct measurement of all state variables is almost infeasible due to technical constraints. It can be too costly to develop adequate sensors capable of measuring missing states of a system. Therefore, researchers have widely adopted observer-based controller designs for the control applications, like stabilization, tracking, monitoring, and disturbance rejection [8–13]. Therefore, an observer is incorporated for an output feedback control system to estimate states of a dynamic system. For linear systems, observer design theory is well-established and conditions of observability and detectability to guarantee the asymptotic convergence of the actual as well as

∗

Corresponding author. E-mail addresses: [email protected] (M. Rehan), [email protected] (H. ur Rashid). https://doi.org/10.1016/j.chaos.2019.04.004 0960-0779/© 2019 Elsevier Ltd. All rights reserved.

the estimated states are available for observer-based control. However, nonlinear observer design and observer-based controller synthesis problems are undoubtedly crucial and their theory is still being researched [1–13]. The dynamic responses of numerous engineering and scientific systems are inherently nonlinear in nature. Due to the underlying complexity of nonlinear systems and owing to their different types, output feedback control is still an open and challenging research problem. For a generalized nonlinear system, the wellknown separation principle of simultaneously state estimation and control may not hold; therefore, observer-based controllers, ensuring decoupling of the design constraints, are investigated for special classes of nonlinear systems [8–13]. Lipschitz nonlinearity is one of the popular class of nonlinearities because several physical systems can be transformed into Lipschitz nonlinear systems by application of properties of Lipschitz continuity. Incorporation of the Lipschitz condition in observer and controller designs offers an efficient solution to the nonlinear control problems. Many strategies for observer design have been developed for such systems. However, research efforts are being accomplished to reformulate the Lipschitz condition in a more general condition that can play an important role for feasibility of the attained design conditions [14–19].

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Existing works on design of controller and observer for nonlinear systems usually consider the uniform Lipschitz condition. Incorporation of this global Lipschitz condition into a linear matrix inequality (LMI) offers an efficient solution to the stabilization as well as the state estimation problems, as seen in [8] and [20]. Without uniform Lipschitz condition, the observer-based controller design schemes can be complex in theoretical analysis and in numerical experimentation. Even though, majority of nonlinear physical and biological systems satisfy the Lipschitz condition locally, for example, the modified Chua’s circuit [21], the Lotka–Volterra model [22], the FitzHugh–Rinzel model [23] and Lorenz-like system [24] etc. Recently, some techniques for the controller and observer synthesis of the locally Lipschitz nonlinear systems, for state feedback, model predictive, sliding mode and adaptive controls, have been employed. However, these exceptional works are computationally complex and/or conservative [25–30] in their design and implementation. Therefore, a local observer design schema to realize the region of stability and to ensure convergence of the estimation error, through Lyapunov stability analysis, was proposed in [19,25]. Moreover, for the design of observer-based control, separation principle may hold; consequently, a decoupling condition, necessary and sufficient for the primary design method, was established for the globally Lipschitz nonlinear systems [8,20,31]. In the case of locally Lipschitz nonlinear systems, it is worth to note that a simple regional theoretical framework regarding the synthesis of observer-based output feedback control is still lacking in the existing literature. This paper addresses regional observer-based stabilization of the locally Lipschitz nonlinear systems, where controller and observer are designed by considering an ellipsoidal Lipschitz region for which a nonlinearity satisfies the Lipschitz continuity condition locally. A set of initial conditions is then determined for both plant and observer states for which the closed-loop system states become asymptotically stable in their regions of interest. For output feedback stabilization scheme, an additional constraint ensuring the presence of bounded regions of plant state and estimation error within the ellipsoidal Lipschitz region is considered. A new region in terms of observer states is defined, satisfying the ellipsoidal Lipschitz condition. The regions of stability of actual state as well as estimation error are determined by the application of the Lyapunov redesign theory and regional analysis for observer-based stabilization. In this paper, a regional state estimator has been combined with the state feedback controller to provide a solution of the stabilization problem for the locally Lipschitz nonlinear systems. Motivated by the works in [31–32], the aforementioned regional constraints in [19,25] and the observer-based global control in [8] and [20], conditions for the synthesis of a Luenberger-type observerbased controller, based on nonlinear matrix inequalities, are established to ensure the exponential convergence of plant state and state estimation error to the origin. A necessary and sufficient condition for simultaneous extraction of observer and controller gains is contrived by utilizing the decoupling technique. In order to obtain controller and observer gain matrices, nonlinear matrix inequalities are transformed into convex routines by accounting the matrix inequality procedures and iterative conecomplementary linearization. Further, conditions for the synthesis of observer-based control for the ellipsoidally Lipschitz nonlinear systems under norm-bounded disturbances are also developed to ensure robustness against external perturbations and fast convergence of the systems’ state vector and state estimation error. For attaining the exponential L2 stability of the observer-based control, necessary and sufficient condition has also been derived to simultaneously extract controller and observer gains using iterative convex routines.

The proposed observer-based control strategy for the locally Lipschitz nonlinear systems offers more generalized control treatment in contrast to the conventional schemes [14–18] for nonlinear dynamics. The developed methodology can be employed successfully to the globally Lipschitz nonlinear systems as a special case. Compared to the recent work [25], the present work provides an output feedback control for the locally Lipschitz systems to deal with unavailability of state variables. Our strategy not only ensures the computational simplicity and lesser hardware requirement compared to methods in [26–30] but also provides a clear realization of the region of stability and some other regions of interest. The proposed methodology guarantees the local stability, fast convergence, robustness against perturbations, and easiness in design. The main contribution of this paper is as follows: (1) To the best of authors’ knowledge, a Luenberger-type observer-based control approach for continuous-time ellipsoidally Lipschitz nonlinear systems has been addressed for the first time. For attaining faster convergence, exponential stability investigation with a clear regional understanding is accomplished. (2) Necessary and sufficient condition, solvable via convex routines, for the main design schema by attaining decoupling of the observer-based control method has been provided. (3) The proposed approach is extended to the robust observerbased stabilization, concerning multi-objective synthesis, using exponential L2 control methodology against external disturbances. The effectiveness of the proposed strategies is verified through numerical simulation example for chaos control in the modified Chua’s circuit. This paper is organized as follows: Section 2 describes system dynamics along with assumptions, continuity condition, and regional constraints. In Sections 3 and 4, conditions for the synthesis of observer-based control are devised for locally Lipschitz systems without and with external perturbations, respectively. A numerical example to analyze the performance of designed controller in chaos control is provided in Section 5. Section 6 concludes the article. Notations: The notations and acronyms used are standard throughout. The field of real numbers is denoted by . The set of real matrices of dimension m × n are denoted by m × n . AT denotes the transpose of a matrix A. The symbol ∗ in a matrix is used for the blocks induced by symmetry. The notations P > 0 and P ≥ 0 represent the positive definiteness and positive semi-definiteness of matrix P. I represents the identity matrix of appropriate dimensions. x denotes the Euclidean norm and x2 represents the L2 norm of a vector x. 2. System description Consider a nonlinear system, described by

x˙ (t ) = Ax(t ) + f (t, x ) + Bu(t ) + Bw w(t ), y(t ) = Cx(t ) + Dw w(t ),

(1)

where x(t) ∈ n , u(t) ∈ m , y(t) ∈ p , and w(t) ∈ q represent the nominal plant state, the control input, the output, and the exogenous input (containing disturbance and noise etc.), respectively. A ∈ n × n , B ∈ n × m , C ∈ p × n , and D ∈ p × q denote the linear constant matrices of the system. Bw ∈ n × q and Dw ∈ p × q are the input matrices for exogenous input in state dynamics and output equation, respectively. The nonlinear function f(t, x) ∈ n stands for a time-varying vector to represent nonlinear dynamics of the system. The nonlinearity f(t, x) may depend only on a subset of state variables of the original system rather than all states and may

S.R. Hamid, M.S. Nazir and M. Rehan et al. / Chaos, Solitons and Fractals 123 (2019) 173–184







175



appear in the dynamics of some states, therefore, we can rewrite it in a detailed form as

B¯ w =

f (t, x ) = ( x(t ) ),

   ¯ = diag(, ), h(t, x, xˆ) = φ ((t, x ) )φ (x, xˆ) . 

(2)

where  and  represent the distribution and state dependency matrices of the nonlinear function in the state-space form of the dynamical system, respectively. Utilization of these matrices in designing the observer-based controller provides a less conservative approach, which can guarantee exponential stability and can improve the performance and robustness in many practical systems (see [14]). Let us define the ellipsoidal Lipchitz nonlinearity (see [19] and [25]). Definition 1. A function ( x(t)) is ellipsoidally Lipchitz, if within an ellipsoidal region defined by

x, x¯ ∈ xT R−1 x ≤ 1, R = RT > 0,

∀x ∈ n ,

(3a)

the function satisfies the Lipchitz continuity condition, that is, given by

( x(t ) ) − ( x¯ (t ) ) ≤ (x − x¯ ),

(3b)

where  is a suitable constant matrix. The choice of matrices R, ,  , and  depends on the designer. The matrices  and  depend on each other and, without loss of generality, we can assign  = . Therefore, we will use the following form of (3b) throughout the paper:

( x(t ) ) − ( x¯ (t ) ) ≤ (x − x¯ ).

xˆ˙ (t ) = Axˆ(t ) + f (xˆ, t ) + Bu(t ) + L(y(t ) − C xˆ(t )), xˆ(0 ) = xˆ0 , yˆ(t ) = C xˆ(t ), n ,

(5)  p,

L ∈ n × p

where xˆ ∈ yˆ ∈ and are the state vector, output vector and gain matrix, respectively, for the observer. Defining e(t ) = x(t ) − xˆ(t ) and e1 (t ) = y(t ) − yˆ(t ) as the state estimation and output estimation errors, the error dynamics of state estimation is given by

e˙ (t ) = AL e(t ) +  f (x, xˆ) + Bw w(t ), AL = A − LC,  f (x, xˆ) = φ (x, xˆ),

φ (x, xˆ) = ( x(t )) − ( xˆ(t )).

(6)

An observer-based state feedback controller for the system (1) satisfying Assumptions 1 and 2 is given by

u(t ) = F xˆ(t ).

(7)

where F ∈ m × n is the constant stabilization gain. The closed-loop system dynamics is obtained by substituting (7) into (1) as

From Lemma 1, an augmented ellipsoid region is defined as

ζ (t )H −1 ζ (t ) ≤ 1,  2  0 μR H= . 0 (1 − μ )2 R T

AB = A + BF .

(8)

An augmented system formed by combining Eqs. (6) and (8) is given by

  ζ˙ (t ) = A¯ ζ (t ) + h t, x, xˆ + B¯ w w(t ), 

     φ ((t, x ))  x(t ) f (t, x ) ζ (t ) = , g(t, x, xˆ) = = e(t )  f (x, xˆ) φ (x, xˆ) ¯ h(t, x, xˆ), =

(11)

If (11) holds, both regions xT R−1 x ≤ 1 and xˆT R−1 xˆ ≤ 1 are valid; therefore, the condition (4) remains valid for a locally Lipschitz nonlinear system (1) and (2). The focus of the present study is to explore simultaneous observer and controller design conditions for the locally Lipchitz nonlinear systems in (1), satisfying ellipsoidal Lipchitz condition in Definition 1, such that the exponential stability of the augmented system (9) is ensured. Moreover, the ellipsoidal Lipchitz condition provides a generalized continuity treatment, through which conventional globally Lipchitz systems can be attained as a specific scenario of locally Lipchitz systems. Controller design for the conventional globally Lipchitz systems can also be intensified to the locally Lipchitz systems. 3. Exponential observer-based control To attain the exponential stability of the augmented system described by (9) and (10), the following conditions are established. Theorem 1. Consider a nonlinear system (1) satisfying Assumptions 1 and 2 under w(t ) = 0. The proposed observer-based control law in (5) and (7) ensures the exponential stability of the plant state and estimation error with a decay rate λ = −γ/2 for all initial conditions ζ T (0)Pζ (0) ≤ 1. Moreover, the state of augmented system (9) remains within the ellipsoidal region defined by (11), if there exist symmetric matrices P, Q > 0 and scalars ɛ1 , ɛ2 , η > 0, γ < 0 such that the LMIs



x˙ (t ) = AB x(t ) + f (t, x ) − BF e(t ) + Bw w(t ),

(10)

Lemma 1 [19]. If the closed-loop system trajectories remain bounded in an ellipsoid region xT R−1 x ≤ μ2 and the error trajectories remain in an ellipsoid region eT R−1 e ≤ (1 − μ )2 , then the estimated states remain within xˆT R−1 xˆ ≤ 1.

Assumption 1. The pair (A, C) is detectable.

A Luenberger-type observer with constant observer gain matrix for the system (1) is given by

−BF , A − LC

Note that for a local region (x) containing the origin, within which the nonlinearity ( x(t)) satisfies (4), the elliptical region (3a) can easily be selected as a subset of (x). The matrix  used herein corresponds to a less conservative Lipchitz condition, as observed in works [11,12,14]. If the nonlinear dynamics of a system satisfies (3a) and (4), then the system can be regarded as the ellipsoidally Lipchitz nonlinear system. Moreover, the region (3a) for the locally Lipchitz nonlinearity can be employed to ensure a region of stability for the control system and observer design issues via Lyapunov stability theory because of its ellipsoidal nature. Within the ellipsoidal region defined in (3a) under Assumption 2, if we take an invariant set x ∈ xT R−1 x ≤ μ2 ⊂ xT R−1 x ≤ 1, with a scalar μ, satisfying 0 < μ < 1, for all inputs, then the following lemma holds:

(4)

Assumption 2. The function ( x(t)) satisfies the ellipsoidal Lipschitz condition in Definition 1.

Bw A + BF , A¯ = Bw − LDw 0

2 =



(9)

P ∗

P 0

0 Q



> 0,

(12)



I ≥ 0, H

(13)

and the matrix inequality



A¯ T P +P A¯ +diag{ε1  T  , ε2  T  } ∗





¯ P P <γ ∗ −diag{ε1 I, ε2 I}

0 Q



(14)

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are satisfied. Proof. Consider a Lyapunov function as

V (ζ , t ) = ζ T (t )P ζ (t ).

(15)

Evaluating its time-derivative along (9), under w(t ) = 0, we have

V˙ (ζ , t ) = ζ˙ (t )T P ζ (t ) + ζ (t )T P ζ˙ (t ) = (ζ (t )T A¯ T + gT (t, x, xˆ) + wT (t )B¯ Tw )P ζ (t ) + ζ (t )T P (A¯ ζ (t ) + g(t, x, xˆ) + B¯ w w(t ))    ¯ P ζ (t ) + ζ (t )T P A¯ ζ (t ) = ζ (t )T A¯ T + hT (t, x, xˆ)

Now we develop an equivalent condition to the main design approach in Theorem 1 for computation of the observer and controller gain matrices.



¯ h(t, x, xˆ) . + It can be rewritten as



A¯ T P + P A¯ V˙ (ζ , t ) = ψ2T ∗

ψ = [ζ (t ) T 2

T

(16) Theorem 2. The constraints presented in Theorem 1 can be solved if and only if the following sets of matrix inequalities are satisfied for real matrices (MandN) and symmetric positive-definite matrices (P¯1 , P2 , Q1 , andQ2 ) of appropriate dimensions and for scalar ε¯1 = ε1−1 :

 ¯ P ψ2 , 0

h (t, x, xˆ)]. T

(17)

From condition (4) and scalars ɛ1 , ɛ2 > 0, we have

which leads to



ψ

diag{ε1  T  , ε2  T  } ∗

0

¯

P1 AT +AP¯1 +MT BT +BM − γ P¯1 ∗ ∗

1 =

ζ (t )T diag{ε1  T  , ε2  T  }ζ (t ) −hT (t, x, xˆ)diag{ε1 I, ε2 I}h(t, x, xˆ) ≥ 0, T 2

Remark 2. The proposed approach in Theorem 1 guarantees the exponential stability of the closed-loop system for attaining a faster convergence and for achieving an improved performance. The conventional methods (see [8,9], and [20]) assure asymptotic stability, through which faster convergence of the closed-loop response cannot be established. The presented observer-based control scheme is based on a more general exponential stability treatment and, therefore, it allows a user-defined convergence rate of the closed-loop system’s states.

 ψ2 ≥ 0.

(18)

 2 =

Applying S-procedure to combine (17) and (18), we obtain

V˙ (ζ , t ) ≤ ψ2T 1 ψ2 , T T ¯T ¯

1 = A P + PA + diag{ε1   , ε2   } ∗

< 0,



¯ P . (19) −diag{ε1 I, ε2 I}

V˙ (ζ , t ) + 2λV (ζ , t ) < 0.

AT P2 +P2 A − C T N T −NC −γ P2 ∗ ∗

P2  −ε2 I − γ Q2 ∗

ε2  T

< 0,

0 −ε2 I

(23)

 1 =

For exponential stability, we require

(20)

For P, Q > 0 and the decay rate λ > 0, the following inequality holds:

V˙ (ζ , t ) + 2λV (ζ , t ) ≤ V˙ (ζ , t ) + 2λV (ζ , t ) + hT (t, x, xˆ) ×2λQ × h(t, x, xˆ). Using condition (4), inequalities (14), (18), and λ = −γ/2, we have

V˙ (ζ , t ) + 2λV (ζ , t ) ≤ ψ2T ( 1 − γ 2 )ψ2 .

−ε1 I − γ Q1 ∗

P¯1  T 0 −ε¯1 I

(22)

−diag{ε1 I, ε2 I}





(21)

Condition. (21) is satisfied if 1 − γ 2 < 0, which further yields the inequality (14). The inequality (21) along with 1 − γ 2 < 0 and ζ T (0)Pζ (0) ≤ 1 implicates that the states of the augmented system remain bounded in ζ T (t)Pζ (t) ≤ 1 for all time. The region ζ T (t )H −1 ζ (t ) ≤ 1 can be attained by including ζ T (t)Pζ (t) ≤ 1 into the ellipsoidal ζ T (t )H −1 ζ (t ) ≤ 1. To ensure the presence of the convergence region ζ T (t)Pζ (t) ≤ 1 into the ellipsoidal region ζ T (t )H −1 ζ (t ) ≤ 1, we must take P ≥ H −1 . Rewriting this inequality asP − H −1 ≥ 0 and, further, applying the Schur complement, we obtain (13). It completes the proof.  Remark 1. The traditional LMI-based control schemes presented in the existing literature [8,9], and [20] for the globally Lipschitz nonlinear systems cannot be directly applied to the locally Lipschitz systems due to the lack of consideration of the region of stability in their design procedures. The proposed approach is a generalization of conventional Lipschitz systems, as seen in [19,25]. The proposed design condition in Theorem 1 is developed by incorporating several important regions to provide a gateway of expansion of the global treatment to the locally Lipschitz systems.

 2 =



P¯1 ∗

P¯1 ≥ 0, μ2 R

P2 ∗

I

(24)



(1 − μ )2 R

≥ 0.

(25)

Observer (5) with gain L = P2 −1 N and feedback controller (7) with gain F = MP¯1−1 can be employed for attaining the performance aspects, stated in Theorem 1. Proof. Sufficiency: Applying the Schur complement to inequalities (22) and (23) produces



1 =

P¯1 AT +AP¯1 +MT BT +BM+ε1 P¯1  T  P¯1 − γ P¯1 ∗

 −ε1 I−γ Q1

< 0,

 ¯ 2 =



(26)

AT P2 +P2 A−C T N T − NC +ε2  T  − γ P2 ∗



P2  < 0, −ε2 I−γ Q2 (27)

respectively. Applying the congruence transformation to (26) and (24), using diag(P1 , I) with P1 = P¯1−1 , results respectively into



¯ 1 =

AT P1 +P1 A+F T BT P1 +P BF +ε1  T  − γ P1 ∗

< 0,



¯ 1 = P1  ∗

P1  −ε1 I − γ Q1



(28)

I



μ2 R

≥ 0.

(29)

S.R. Hamid, M.S. Nazir and M. Rehan et al. / Chaos, Solitons and Fractals 123 (2019) 173–184





−P1 BF 0 ¯ 1,  ¯ 2 , and3 and combining  ∗ 0 with a sufficiently large scalar β > 0, it yields Assigning 3 T =



 3 T < 0. β ¯ 2

¯ 1 ∗



(30a)

¯ 1 and2 , it yields Similarly, combining 

¯1 

0

2

∗



≥ 0.

(30b)

Exchanging the rows and columns of inequalities (30a) and (30b) results into the following conditions:

⎡ ⎢ ⎣

(ATB P1 + P1 AB + ε1  T  − γ P1

(

∗ ∗ ∗

⎡

P1 ⎢∗ ⎣∗ ∗

0 P2 ∗ ∗

I 0

μ2 R ∗

ATL P¯2

⎤

0 I ⎥ ⎦ ≥ 0, 0 2 (1 − μ ) R

(32)

where P¯2 = β P2 , Q¯ 2 = β Q2 and ε˜2 = βε2 . Now putting diag(P1 , P¯2 ) = P and diag(Q1 , Q¯ 2 ) = Q in (31) produces (14), and the inequality (32) is equivalent to (13). Necessity: Partitioning P as



P=

P1 ∗

    and P =  ∗

 P2



,

(33)

where non-significant entries of both partitions are represented by . Using (33) into (14) gives

⎡

AT P1 + P1 A + F T BT P1 + P1 BF + ε1  T  ∗ ⎢ ⎣ ∗ ∗

  ∗ ∗

⎤  ⎥ < 0, ⎦ 

P1 



−ε1 I ∗

(34)

⎡





  

AT P2 + P2 A − C T LT P2 − P2 LC + ε2  T  ∗ ∗

⎢∗ ⎣∗ ∗

∗

⎤  P2  ⎥ < 0.  ⎦ −ε2 I

Similarly, using (33) into inequality (13), it produces

P1 ⎢∗ ⎣∗ ∗

 

 ⎢∗ ⎣∗



∗ ∗

⎡

P2 ∗ ∗

∗

⎤   ⎥ ≥ 0, μ2 R ⎦ ∗  ⎤    I ⎥ ⎦ ≥ 0.   2 ∗ (1 − μ ) R I

Let us choose



Y1 =

P1 −1 0 0 0 0 0 I 0



(36)

(37)

 and Y2 =

0 0

I 0

0 0

0 I

Remark 3. The approach in Theorem 1 requires tuning efforts for computing the observer and controller gains, which can complicate the design approach. The condition in Theorem 2 is required for the straightforward computation of the proposed control strategy gains. Compared to Theorem 1, the controller and observer gains can be easily calculated via F = MP¯1−1 and L = P2 −1 N. To the best of our knowledge, this is the first approach, developed through Theorems 1 and 2, for providing a guaranteed regional stability, to

0

⎤

P¯2  ⎥ ⎦ < 0, 0 −ε˜2 I − γ Q¯ 2

(31)

deal with the observer-based control of ellipsoidally Lipschitz nonlinear systems. The ellipsoidally Lipschitz systems are more general than the conventional globally Lipschitz systems and can be used to model a large scenario of practical systems. Remark 4. In the existing works, adaptive and sliding mode control techniques were utilized to deal with the locally Lipschitz systems [26–30]. However, these techniques have limitations depending on the degree of nonlinearity present in a system. Moreover, there are many theoretical problems due to computational complexity, experienced while designing a control scheme [21,27– 28]. Recently, regional observer-based methods, utilizing fuzzy and sample data control, have been presented; however, regional constraints have not been completely considered in existing LMI-based design procedures. Moreover, a clear understanding of various regions of interest like region of attraction was not focused in many studies as seen in [33] and [34]. The proposed technique has considered regional constraints throughout the design procedure and presented an observer-based feedback control scheme. Therefore, our work provides a less conservative and computationally simple output feedback stabilization methodology, compared to the conventional works. 4. Extension to exponential L2 control

(35)

⎡

the inequalities (36) and (38) withY2 and YT2 produces constraints (27) and (25), respectively. Applying the Schur complement to (26) and (27) results into (22) and (23), which concludes the proof. 

P1  0 −ε1 I − γ Q1 ∗

−P1 BF + P¯2 AL + ε˜2  T  − γ P¯2 ∗ ∗

177



In this section, we extend the proposed schemes for dealing with the external disturbances and perturbations. The following condition has been established to attain the exponential L2 stability of the augmented system in (9) and (10). Theorem 3. Consider a nonlinear system (1), satisfying Assumptions 1 and 2. The proposed observer-based control law in (5) and (7) ensures the exponential L2 stability of the plant state and estimation error in the presence of L2 norm-bounded signal w(t), satisfying w(t )22 < ω−1 for all time, where ω−1 represents the tolerable L2 norm of w(t). Moreover, the state of augmented system (9) remains within ellipsoidal region defined by (11) for all initial conditions ζ T (0)Pζ (0) ≤ 1. The L2 gain from exogenous input to the system’s output and from exogenous input to state estimation error re√ √ main bounded by κ1 and κ2 , respectively, if there exist symmetric matrices P, Q > 0 and scalars ɛ1 , ɛ2 , κ 1 , κ 2 > 0, andγ < 0, such that the LMIs



. YTi

By pre- and post-multiplying (35) and (37) with Y1 and and further substituting P1 −1 = P¯1 and F P1 −1 = M, it leads to (26) and (24), respectively. Similarly, by pre- and post-multiplying



P 0 0

λ¯ P ∗

0 Q 0



0 0 T



> 0,

I ≥ 0, H

(38)

(39)

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S.R. Hamid, M.S. Nazir and M. Rehan et al. / Chaos, Solitons and Fractals 123 (2019) 173–184

and the matrix inequality

⎡

A¯ T P + P A¯ + diag{ε1  T  , ε2  T  } ⎢ ∗ ⎣ ∗ ∗

¯ P −diag{ε1 I, ε2 I} ∗ ∗

P B¯ w 0 −I ∗

⎡

⎤

P diag(C T , I ) ⎥ ⎢0 0 ⎦ < γ ⎣0 0 0 −diag(κ1 I, κ2 I )

are satisfied.

0 Q 0 0

0 0 T 0

⎤

0 0⎥ 0⎦ U

(40)

By substituting 2λ = −γ and applying the Schur complement to

Proof. For Lyapunov function (15), suppose that the following inequality holds:

V˙ (ζ , t ) + 2λV (ζ , t )+κ1−1 yT (t )y(t )+κ2−1 eT (t )e(t )−wT (t )w(t ) < 0, (41)

3 < 0, we obtain ⎡

A¯ T P + P A¯ + diag{ε1  T  , ε2  T  } − γ P

¯ P

P B¯ w

∗

−diag{ε1 I, ε2 I} − γ Q

0

∗

∗

−I − γ T

∗

∗

∗

⎢ ⎢ ⎢ ⎢ ⎢ ⎣

which can be rewritten as



  κ1−1C, κ2−1 I .

(42) T

T

Given 0 eT (t )e(t )dt > 0 and 0 yT (t )y(t )dt > 0, the inequality (42) implies the exponential stability (that is, V˙ (ζ , t ) + 2λV (ζ , t ) < 0) of state of augmented system (9), under w(t ) = 0. Taking integration of (42) from 0 to T, we obtain

V ( ζ , T ) − V ( ζ ( 0 ), 0 ) + 2λ  −

0

T

 0

T

V (ζ , t )dt +



T 0

ζ (t )K K ζ (t )dt T

⎥ ⎥ ⎥ ⎥ < 0. ⎥ 0 ⎦ −diag(κ1 I, κ2 I ) 0

(43)

Using w(t )22 < ω−1 , the inequality (43) implies

V (ζ , T ) < V (ζ (0 ), 0 ) + ω−1 .

(44)

Given that initial condition of the augmented system is −1 = 1 + ω −1 , bounded, that is, V (ζ (0 ), 0 ) = ζ T (0 )P ζ (0 ) ≤ 1, and λ ¯ the inequality (44) becomes −1 V (ζ , T ) = ζ T (t )P ζ (t ) < λ ¯ .

(45)

ζ T (t )λPζ (t ) ¯

For a matrix U > 0 of appropriate dimensions, the following relation holds:

⎡

P ⎢0 ⎣0 0

0 Q 0 0

0 0 T 0

⎤

⎡

0 P 0 ⎥ ⎢0 > 0 ⎦ ⎣0 U 0

0 Q 0 0

0 0 T 0

⎤

0 0⎥ . 0⎦ 0

(51)

Using (50) and (51), the matrix inequality (40) is obtained. 

T

wT (t ) w(t )dt < 0.

To include the region

⎤

(50)

V˙ (ζ , t ) + 2λV (ζ , t ) + ζ T (t )K T K ζ (t ) − wT (t )w(t ) < 0, K = diag

diag(C T , I )

≤ 1 into ellipsoidal region

ζ T (t )H −1 ζ (t ) ≤ 1, we must take λ¯ P ≥ H −1 , which implies (39). For

Remark 5. In practical systems, we frequently confront with the uncertainties, perturbations and disturbances due environmental effects and external interferences. The control requirements in such cases are more rigorous, which can be attained by considering robustness of the design. The approaches developed in Theorems 1 and 2 lack in dealing with external disturbances; therefore, a novel approach has been provided in Theorem 3 for achieving robustness against disturbances. This improved control scheme not only assures a faster convergence of the closed-loop response but also guarantees the bounded L2 gains (i) from w(t) to √ √ y(t) bounded by κ1 and (ii) from w(t) to e(t) bounded by κ2 . However, this approach further needs improvement for determining the observer and controller gains, which will be addressed in Theorem 4.

T → ∞, the inequality (43) becomes

K ζ (t )22 < w(t )22 + V (ζ (0 ), 0 ),

(46)

which can be written in detailed form as

κ1−1 y(t )22 + κ2−1 e(t )22 < w(t )22 + V (ζ (0 ), 0 ), √

(47)

√

where κ1 and κ2 define the upper bounds on the L2 gains from the exogenous input to the system state and the exogenous input to the state estimation error, respectively, under zero initial condition. Consider the matrices Q > 0 and T > 0 of appropriate dimensions, such that

V˙ (ζ , t ) + 2λV (ζ , t ) + ζ T (t )K T K ζ (t ) − wT (t )w(t ) + 2λhT (t, x, xˆ)Qh(t, x, xˆ) + 2λwT (t )T w(t ) < 0.

(48)

The inequality (41) for observer-based controller design remains valid if (48) is satisfied. Applying ζ (t )T diag{ε1  T  , ε2  T  }ζ (t ) − hT (t, x, xˆ)diag{ε1 I, ε2 I}h(t, x, xˆ) ≥ 0 and using (14) and (15) into (48) in the same way as seen in the proof of Theorem 1, we have

Remark 6. In Theorem 3, the design scheme for the local observerbased control system provides an estimated state and output feedback-based robust stabilization solution. In comparison to Theorem 1, the approach of Theorem 3 can be applied to obtain multiple design objectives, including local stabilization, fast convergence, maximization of allowable energy of exogenous input, and robustness against external perturbations. Optimization of parameters (like κ1 , lκ2 , lγ , lλ ¯ ) can assist in improving various performance aspects such as reduction of effect of disturbances on system states and its output, enhancement of tolerable disturbance limit, and faster convergence rates for state and estimation error trajectories. The main condition in Theorem 3 is difficult to resolve. Here, we provide a decoupled design methodology for finding the observer and controller gain matrices. Theorem 4. The constraints presented in Theorem 3 can be solved if and only if the following matrix inequalities are satisfied for

V˙ (ζ , t ) + 2λV (ζ , t )+ζ T (t )K T K ζ (t )−wT (t )w(t ) ≤ ψ3T 3 ψ3 < 0,

 ψ3 = ζ T (t )  ¯T

3 =

T

hT (t, x, xˆ) wT (t ) , A P + P A¯ + K T K + 2λP + diag{ε1  T  , ε2  T  } ∗ ∗

¯ P −diag{ε1 I, ε2 I} + 2λQ ∗



P B¯ w . 0 −I + 2λT

(49)

S.R. Hamid, M.S. Nazir and M. Rehan et al. / Chaos, Solitons and Fractals 123 (2019) 173–184

real matrices (MandN) and symmetric positive-definite matrices (P¯1 , P2 , Q1 , andQ2 ) of appropriate dimensions and scalar ε¯1 = ε1−1 > 0:

⎡¯

P1 AT + AP¯1 + MT BT + BM − γ P¯1 ⎢∗ 1 = ⎢ ⎣∗ ∗ ∗

⎡

AT P2 + P2 A − C T N T − NC − γ P2 ⎢∗ 2 = ⎢ ⎣∗ ∗ ∗



λ P¯ 1 = ¯ 1 1  2 =

∗ ∗ ∗

P2  −ε2 I − γ Q2 ∗ ∗ ∗

Bw 0 −I − γ T1 ∗ ∗ P Bw 0 −I − γ T2 ∗ ∗

∗

λ¯ 2 P2

I

(54)



(1 − μ )2 R

≥ 0.

(55)

Observer (5) with gain L = P2 −1 N and feedback controller (7) with gain F = MP¯1−1 can be employed for attaining the performance aspects, stated in Theorem 3. Proof. The proof is analogous to the proof of Theorem 2, therefore, omitted. Remark 7. Compared to Theorem 3, the approach in Theorem 4 can be straightforwardly applied to obtain the gains via L = P2 −1 N and F = MP¯1−1 . It should be noted that the conditions provided in Theorems 2 and 4 are fascinating, as these conditions are necessary and sufficient for the design conditions in Theorems 1 and 3. The developed conditions do not require any additional conservatism for resolving the gain matrices of the proposed observer-based control scheme, compared to the corresponding methods of Theorems 1 and 3. These conditions can be separately considered for attaining the observer and controller gains, which simplifies the design procedure and overcomes computational complexity. Remark 8. The constraints in Theorems 2 and 4 contain some bilinear terms like γ P¯1 , γ Q1 , γ P2 , and γ Q2 , which are difficult to handle for a solution of the proposed observer-based control. For achieving asymptotic stability, these terms can be ignored; however, such terms often appear in the exponential stability case [19]. These terms can be handled through convex routines either for a selection of γ or by considering the inequalities as linear-fractional constraint (LFC) along with generalized eigenvalue minimization approach. The second problem arises due to presence of ɛ1 and its inverse ε¯1 (see, for instance, (52)), which can be settled by application of cone-complementary linearization approach (see [8,20] and references therein) through minimization of product term ε1 ε¯1 by involving an additional constraint, given by



ε1 I ∗

developed methods provide the design of observer-based controller with separate extraction of controller and observer gains to guarantee the stability of the closed-loop system.

P¯1C T 0 0 −κ1 I − γ U1 ∗

⎤ P¯1  T 0 ⎥ ⎥ < 0, 0 ⎦ 0 −ε¯1 I

ε2  T

I 0 0 −κ2 I − γ U2 ∗

(52)

⎤

0 ⎥ ⎥ < 0, 0 ⎦ 0 −ε2 I

(53)



P¯1 ≥ 0, μ2 R

∗

 −ε1 I − γ Q1

179



I ≥ 0. ε¯1 I

Remark 9. Decoupling techniques for observer-based control have been established for linear, fuzzy, and globally Lipschitz systems (see [31,32] and references therein). Note that these conditions were remained unaddressed for the locally Lipschitz nonlinear systems. To solve this dilemma, decoupling techniques are proposed in the present work for the locally Lipschitz systems by incorporating the regional constraints and Lyapunov redesign analysis. In our work, an approach involving convex routines has been formulated in Theorems 2 and 4 for the locally Lipschitz systems. The

Remark 10. A lot of research attention has been focused on the observer-based control of nonlinear systems. However, less control efforts are being paid to attain fast convergence and robustness against external disturbances. For instance, the works in [8] and [20] do not consider the convergence rate for states of nonlinear systems and, therefore, cannot be used for a quick response from the controller. In addition, the results of [20] and [31] are unable to provide robustness against uncertainties and disturbances. Contrastingly, the developed approaches in Theorems 3 and 4 overcome these issues by exploiting the exponential L2 stability of the closed-loop system. In future, this attractive feature can be extended for attaining robustness against parametric uncertainties. 5. Simulation results To analyze the performance of the proposed methods in Theorems 1–4, a nonlinear model of modified chaotic Chua’s circuit is studied. Chua’s circuit has a complex oscillatory behavior, and a stabilization controller design for dealing with this behavior is an interesting control problem. Further design complication arises owing to unavailability of all states of the chaotic circuit for feedback [35–36]. Unlike the conventional Chua’s circuit, containing an absolute nonlinearity, the modified Chua’s circuit has a cubic nonlinearity that can be rewritten in the detailed form, to construct a generalized (locally) Lipchitz model. The conventional strategies for the globally Lipschitz systems, as observed in [8,9], and [20], cannot guarantee stability or state estimation due to involvement of this cubic nonlinearity. Therefore, the ellipsoidal Lipschitz condition approach can be followed to investigate an observer-based control problem using output feedback via LMI method. In this section, the dynamical model of the modified Chua’s circuit and its simulation results are presented to verify the proposed techniques for the observer-based controller design. 5.1. Modified Chua’s circuit model Chua’s circuit system, invented by Leon O. Chua in 1983, is one of the simplest electronic circuits exhibiting chaotic behavior. It contains three energy storage elements (an inductor L and two capacitors C1 and C2 ), a linear resistor R, and a Chua’s diode Dc , as shown in Fig. 1. Chua’s circuit has several applications in secure communication, chaotic behavior analysis, oscillation analysis, and neural behavior inspection due to its ability to produce a variety of dynamical behaviors. The state-space model of a modified Chua’s system has the form





x˙ 1 (t ) = p x1 (t ) + x2 (t ) + f¯(t, x ) , x˙ 2 (t ) = x1 (t ) − x2 (t ) + x3 (t ), x˙ 3 (t ) = −qx2 (t ).

(56)

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S.R. Hamid, M.S. Nazir and M. Rehan et al. / Chaos, Solitons and Fractals 123 (2019) 173–184

√

 = diag( 8.5714x1,max , 0, 0 ), respectively. The distribution matrix  and state-dependency matrix  for the nonlinear function are taken as  = diag(2.857, 0, 0 ) and  = [1 0 0] with ε = 1.05. 5.2. Exponential observer-based stabilization This subsection provides numerical simulation results of the developed schemes for exponential convergence. We choose a sphere of unit radius for considering the region (3a) through r2 = r3 = x21,max = 1. By solving the constraints in Theorem 2, the following gain matrices for the proposed controller and observer are attained:



Fig. 1. Circuit diagram of a Chua’s system.

F = A chaotic attractor is observed due to the cubic nonlinear element f¯ (t, x ) in the response of the circuit. For simulations, the parameters are chosen as p = 10, q = 100/7 and f¯(t, x ) = −2/7x1 3 (t ), which leads to the chaotic behavior [21]. In matrix form, the modified Chua’s system (56) can be written as



A=

 C=

1.4286 1.00 0 1 0

0 0

10.00 −1.00 −14.2857 0 1





0 1.00 , B = 0

, D = 0, f (t, x ) =





1 0 0



0 0 , 1



−2.8571x31 . 0 0

(57)

An observer-based feedback control of (7) can be used to suppress the chaotic trajectories. Considering Assumption 2, the cubic nonlinearity f(t, x) must satisfy the generalized ellipsoidal Lipchitz condition (4). Fig. 2 shows the behavior of the modified Chua’s circuit by taking the initial condition as x(0 ) = [ −0.7 0.1 0.7 ]T . The corresponding phase-portrait shows the chaotic time-evolution of the circuit. For system (1), the ellipsoidal region (3a) is chosen in such a way that, for any positive numbers r2 and r3 , the inequalities x21 ≤ x21,max , x22 ≤ r2 and x23 ≤ r3 hold. The matrix R in (3) and

matrix  in (3b) can be selected as R = diag(x21,max , r2 , r3 ) and

−3.1374 −1.9370

−10.3430 13.3442

24.3779 1.5879 3.9246

−70.8216 −13.2081 . 0.5152

 L=





0.7108 , −0.6747 (58)

State trajectories of the closed-loop system and estimation error trajectories of observer converge exponentially toward the origin with a decay rate of λ = 0.5, under w(t ) = 0, as shown in Figs. 3 and 4, respectively. For comparison with existing method, we consider the approach [20] (see also [8,30]) and the corresponding closed-loop response is shown in Fig. 5. Note that the convergence of states for the case of [20] is slower than the proposed result in Fig. 3. The methodologies in [20] as well as [8] and [30] are based on globally Lipschitz nonlinear system and ensure asymptotic stability of the closed-loop system. There are two drawbacks of these methods, compared to the present work. First, the methods in [8,20] and [30] don’t consider locally Lipschitz nonlinearity; therefore, these schemes do not provide an estimate of the region of stability for the control system. Such methods although can be useful for controlling nonlinear systems; however, they can result into performance degradation (or failure) if initial condition is not taken in the neighborhood of the origin. Second, these control schemes do not incorporate the user-defined information on exponential convergence rate of the closed-loop system. As a result, faster closed-loop response of the Chua’s system states is not

Fig. 2. Chaotic behavior of the modified Chua’s circuit.

S.R. Hamid, M.S. Nazir and M. Rehan et al. / Chaos, Solitons and Fractals 123 (2019) 173–184

181

0.8 state x 1 of closed loop system 0.6

state x 2 of closed loop system state x 3 of closed loop system

Closed-loop Response

0.4 0.2 0 -0.2 -0.4 -0.6 -0.8

0

1

2

3 Time(sec)

4

5

6

Fig. 3. Convergence of states to the origin without disturbance by using the proposed approach in Theorem 2.

1 state estimation error e1

0.8

state estimation error e2

State estimation errors

0.6

state estimation error e3

0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1

0

0.5

1

1.5

2 Time(sec)

2.5

3

3.5

4

Fig. 4. Convergence of estimation errors to the origin without disturbance by using the proposed approach in Theorem 2.

attained through the approach [20], as demonstrated in Fig. 5. The advantage of the proposed observer-based control method is that it can be applied to the state estimation as well as stabilization of the chaotic circuit. In comparison to the conventional works [8,20] and [30], the proposed method effectively deals with the locally Lipschitz systems and it ensures the exponential convergence of the closed-loop response. Hence, the proposed methods in Theorems 1 and 2 can be used to control the chaotic response of the modified Chua’s circuit with a guaranteed region of stability.

Consider a sphere of unit radius for incorporating the region (3a) for the ellipsoidal Lipchitz condition (4). By solving conditions (52)–(55) in Theorem 4, we obtain the following gain matrices for controller and observer:

 F =

−7.2408 1.7424

−10.1451 13.9319

20.4772 3.1168 3.3132

−19.0350 −14.2808 . 5.0918

 L=





0.7069 , −2.1003 (59)

5.3. Exponential L2 observer-based stabilization In this subsection, we analyze the proposed robust observerbased control method for dealing with external disturbances.

The proposed observer-based control law ensures the robust exponential L2 stability of the plant state and estimation error of observer in the presence of L2 norm bounded disturbance w(t). The

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Fig. 5. Observer-based stabilization control using conventional method [20].

Fig. 6. Convergence of states of the system in presence of disturbance by using the proposed approach in Theorem 4.

disturbance vector is selected as



w(t ) =



15 cos 50 π t 16 cos 100 π t . 1.5 cos 500 π t

State-space trajectories of the closed-loop response and estimation error trajectories of the observer converge exponentially for a decay rate of λ = 1. Stability and robustness of the closed-loop system against disturbances can be observed in Figs. 6 and 7. The states of the closed-loop system converge in a region neighborhood to the origin, as depicted from Fig. 6. The estimation errors also converge to a region near the origin as shown in Fig. 7 to ensure the estimation of states of the chaotic

circuit. The response of the closed-loop system and estimation errors is robust against disturbances. The control laws in [20] and [31] do not provide the information of robustness bounds for the disturbance rejection. The beauty of the proposed observerbased control schemes in Theorems 3 and 4 lies in providing a robustness criterion in the form of L2 gain reduction. It should also be noted that the traditional control methods in [25] and [35] fail in the present scenario of chaos control owing to unavailability of the Chua’s system state vector for the control purpose. Hence, the proposed observer-based control approach can employed for robust output feedback stabilization of the locally Lipschitz nonlinear systems for fast convergence and disturbance attenuation.

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183

Fig. 7. Convergence of estimation errors in presence of disturbance by using the proposed approach in Theorem 4.

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