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Stabilizing backstepping controller design for arbitrarily switched complex nonlinear system
Applied Mathematics and Computation 369 (2020) 124789
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Stabilizing backstepping controller design for arbitrarily switched complex nonlinear system Qitian Yin a,b, Mao Wang b,∗, He Jing a a b
College of Computer Science and Information Engineering, Harbin Normal University, Harbin 150025, PR China Space Control and Inertial Technology Research Center, Harbin Institute of Technology, Harbin 150001, PR China
a r t i c l e
i n f o
Article history: Received 25 March 2019 Revised 18 July 2019 Accepted 23 September 2019
Keywords: Switched systems Observer Backstepping State feedback Output feedback
a b s t r a c t In this paper, the aim is to design a backstepping controller for switched complex nonlinear system. The nonlinear system is composed of several arbitrarily switched complex nonlinear subsystems, and the nonlinear system does not satisfy the so-called hypothesis of the global Lipschitz-like condition. First an uniform nonlinear decoupling method for switched subsystems is proposed to overcome the double uncertainty from the complex nonlinear structure and arbitrarily switching. Next a states backstepping controller is designed to realize exponentially stability of the switched nonlinear systems. Then based on an uniform gain observer with the uncertainty of the subsystem nonlinearity, an output backstepping controller is construct to realize exponentially stability of the observerbackstepping feedback control system. Two detailed simulations are finally presented to exhibit the effectiveness of the proposed synthesis methods. © 2019 Elsevier Inc. All rights reserved.
1. Introduction Over the past decade, many achievements of the stabilized analysis have been acquired, which are the switched linear systems or the switched nonlinear systems, e.g., [1–14]. Owing to the structural complexity and the arbitrary switching uncertainty, the control stability of the switched nonlinear systems is quite difficult than the stability of switched linear systems. The single lower triangular nonlinear system can be stabilized based on the backstepping method. With the development of the backstepping method, the switched nonlinear systems can also be controlled in [15–19]. At the same time, many achievements of different control scheme have been acquired in [20–24]. Among these, the control scheme based on the Lyapunov function (CLF)is effective to control and stabilize the switched systems in [8,25–29]. In order to improve the control ability, the backstepping controller combining the Lyapunov stability function is proposed to stabilize switched certainty nonlinear systems in [21,30,31] which switch arbitrarily. By extending the backstepping iterative procedure and the Lyapunov stability function, the [30] realizes the stability of the switched strict-feedback form nonlinear systems, combining the common control Lyapunov stability function (CCLF). However, owing to the difficulty to design an uniform coordinate transformation or an uniform compensation for the uncertainty of different subsystems in the backstepping design, few results have been presented over the past decades. In order to make up the gap, many intelligent controller containing fuzzy or neural network approximation have been proposed, with which the backstepping controller ∗
Corresponding author. E-mail address: [email protected] (M. Wang).
https://doi.org/10.1016/j.amc.2019.124789 0 096-30 03/© 2019 Elsevier Inc. All rights reserved.
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stabilizes the switched systems in [32,33]. There is a huge drawback in the application of the segmented systems. The uniform compensation or the rapid approximator is hard to design for the existing of the huge structural differences among the subsystems. In order to overcome the drawback, an effective direct backstepping control method is proposed to stabilize complex nonlinear switched switched system, in which the subsystems are switching with the signal arbitrarily. For the output feedback control problem, the states cannot be acquired or measured directly for various obstacles, the states observe problem usually arises. Over the past decades, the states observer for nonlinear system has been intensively studied, more involved results concerned with the adaptive intelligent observers are also reported in dealing with the dynamical nonlinear system [34–39]. The last decades, there are many intelligent method proposed to construct the observer, such as the paper [35,37,40], which combines the neural network and observer to guarantee the state reconstruction. However, there is a huge drawback that the switched nonlinear system will cause the neural network adjusting to new subsystem time after time, this procedure will cause too much operating time waste. Therefore through the analysis above, the contribution of the paper makes up the gap, the effective uniform gain observer is proposed to redesign the system states. On the other hand, single complex nonlinear system, such as bilinear system [41], affine system [42] or nonaffine systems [43] written as lower triangular form has been controlled through the feedback controller with control Lyapunov function. The above systems are all the states bounded in application and satisfy a hypothesis which has been shown in [44] and [45]. With the development of the switched systems, the nonlinear subsystems with different sates boundedness compose the switched system. The nonlinear subsystems are varying under arbitrary switching signal, and which is continuous and discrete Hybrid Systems in essence. The switched nonlinear system does not satisfy the so-called separation principle, growth condition or uniform boundedness, and the arbitrary switching between subsystems will cause the huge chattering. Then the common backstepping controller cannot be able to suppress the double uncertainty. According to the analysis, the stability of the switched system which is composed of complex nonlinear subsystems under arbitrary switching signal is exceptionally challenging and have never been studied sufficiently. Above all, the motivation of our work is to introduce a uniform decoupling scheme and a backstepping method to suppress and compensate the uncertainty from the arbitrary switching and complex nonlinear structure, further we construct a feedback backstepping controller with an uniform gain observer to realize the control of the switched complex nonlinear system. Several contributions of this paper is listed: (i ) An uniform nonlinear decoupling method for switched system is proposed to overcome the double uncertainty from the complex nonlinear structure and arbitrarily switching. (ii ) A novel backstepping controller based on the full states feedback is proposed to stabilize the switched nonlinear system; (iii ) Based on an uniform observer, a novel output feedback chained backstepping controller is designed to stabilize the control of the switched complex nonlinear system. The rest of this paper is organized as: Preliminaries is given in Section 2. In Section 3, backstepping controller design based on full states feedback is proposed for the considered system. The Section 4 illustrates the output backstepping controller design method based on the uniform gain observer. In Section 5, two simulation results are studied. Finally, the conclusions are drawn in Section 6. 2. Preliminaries In this paper, we deal with the following switched complex nonlinear system, the subsystem is described as the strictfeedback form:
x˙ 1 = x2 + f1,σ (t ) (x¯1 ) .. . x˙ i = xi+1 + fi,σ (t ) (x¯i ) .. . x˙ n−1 = xn + fn−1,σ (t ) (x¯n−1 ) x˙ n = u + fn,σ (t ) (x¯n ))
(1)
The states are x¯i := [x1 , x2 , . . . , xi ]T ∈ Rn . The control input is u ∈ R1 . σ : [0, ∞ ) → {1, . . . , m} is a piecewise constant switching signal that is not observed. The nonlinear function fi,σ (t ) (x¯i ) 1 i n are smooth. The objective is to design a continuous state feedback control law
u = h(x¯ )
(2)
with which the closed-loop control system (1) is able to be global uniformly asymptotically stabilized. It means that for all possible time interval σ (t), the closed-loop backstepping control system (1) is globally asymptotically stable. Note that the designed feedback control law (2) gets rid and independent of the arbitrarily switching signal σ (t). Assumption 1. For i = 1, . . . , n, during every time interval there is a constant γ iσ (t) > 0 make sure that the | fi,σ (t ) (x¯i )| γiσ (t ) (|x1 | + · · · + |xi | ) With this hypothesis, the global exponential stabilization of the single subsystem in the switched nonlinear system during the switching time interval σ (t) has been proven effective using linear state feedback in [45].
Q. Yin, M. Wang and H. Jing / Applied Mathematics and Computation 369 (2020) 124789
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Our research objective is to design a state feedback backstepping controller, and it is able to stabilize the arbitrarily switched complex switched systems. It is know that an unforced switched system is globally uniformly asymptotically stable if and only if there exists a common Lyapunov function for each subsystem [30]. Remark 1. There is a notification should be noticed that, the global stabilization of the single subsystem of the switching system (1) has already been considered in [45]. However this switched system composed of the class of the nonlinear system is placed in a arbitrary switching setting, in order to realize the control, it must be required to design the uniform controller. 3. Backstepping controller design based on full states feedback In order to realize the control, we must design a feedback control law, and with which there exists a common control Lyapunov function V(x) for every closed-loop subsystems in (1). For the case step n = 1. Define
z1 = x1 V1 (x1 ) = 12 z12
(3)
According to the backstepping controller design method for (1), a feasible virtual control function can be constructed as:
α1 (x1 ) = −sgn(z1 )γ1max |z1 |
(4)
and the parameter is defined as
γ1max = max γ1 j
(5)
j=1→σ
where γ 1j is the coefficient of the nonlinear part of the subsystem. Since the nonlinear part f1σ (t ) (x¯1 ) of the subsystems satisfy the Assumption 1, there exist the parameter γ 1max , so we can obtain that | f1σ (t ) (x¯1 )| γ1σ (t ) |x1 | γ1max |x1 | σ (t ) ∈ {1, . . . , m}. Define
z2 = x2 − α1 (x1 )
(6)
By substituting (4) into the derivation of the candidate Lyapunov function V1 in (3) and with the Assumption 1, one has:
V˙ 1 =z1 z2 + z1 α1 (x1 ) + z1 f1σ (t ) (x¯1 ) and taking (4) and (6) into account, such that
V˙ 1 z1 z2 + z1 α1 (x1 ) + |z1 |γ1,σ (t ) |x¯1 | = z1 z2 − (γ1max − γ1,σ (t ) )x¯21 − cz12 + z1 z2
(7)
where γ1max − γ1,σ (t ) = c. Next for the case that n = 2. With the definition of z3 , we have z3 = x3 − α2 (x¯2 ) and choose the 2nd virtual control function below:
α2 (x¯2 ) = −z1 +
∂α1 ∂α1 γ |x | x2 − sgn(z2 )γ2max (|x1 | + |x2 | ) − sgn(z2 ) ∂ x1 ∂ x1 1max 1
and also the parameter is defined as γ2max = max j=1→σ γ2 j and | f2σ (t ) (x¯2 )| γ2 j (|x1 | + |x2 | ) γ2max (|x1 | + |x2 | ). The selected common Lyapunov function is V2 = V1 + 12 z22 , and its time derivation becomes
V˙ 2 = V˙ 1 + z2 z˙ 2 z1 z2 − (γ1max − γ1σ (t ) )z12 + z2 z3 + z2 (α2 (x¯2 ) + f2σ (t ) (x1 , x2 )) − z2 Substituted the α 2 into V˙ 2 with Assumption 1, obtain
V˙ 2 = V˙ 1 + z2 z˙ 2 −(γ1max − γ1σ (t ) )z12 + z2 z3 − z2 sgn(z2 )
∂α1 x˙ ∂ x1 1
γ2max (|x1 | + |x2 | ) − | f2σ (t ) (x1 , x2 )|
∂α1 γ |x | − | f1σ (t ) (x1 )| − z2 sgn(z2 ) ∂ x1 1max 1 −(γ1max − γ1σ (t ) )z12 + z2 z3 = −cz12 + z2 z3
(8)
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For the case ith step for i = 3, . . . , n − 1, design the virtual control error zi = xi − αi−1 (x¯i−1 ), the ith dynamical error system is
z˙ i (x¯i ) = x˙ i − α˙ i−1 (x¯i−1 ) = zi+1 + αi (x¯i ) + fiσ (x¯i ) −
i−1 j=1
∂αi−1 x˙ ∂xj j
(9)
The extended candidate Lyapunov function is Vi (x¯i ) = Vi−1 (x¯i−1 ) + 12 zi2 . We obtain a promising analysis on V˙ i below:
V˙ i (x¯i ) = V˙ i−1 (x¯i−1 ) + zi z˙ i
−cz12
+ zi−1 zi + zi zi+1 + zi αi + |zi || fiα | − zi
i−1 j=1
i−1 ∂αi−1 ∂αi−1 | f | x |z | ∂ x j j+1 j=1 i ∂ x j jσ
(10)
Again, by viewing xi+1 as the virtual control, we can design the ith virtual control input: i−1
∂αi−1 x − sgn(zi )γimax (|x1 | ∂ x j j+1 j=1 i−1 ∂αi−1 γ (|x | + · · · + |x j | ) + · · · + |xi | ) − sgn(zi ) ∂ x j imax 1 j=1
αi (x¯i ) = − zi−1 +
(11)
Define zi+1 = xi+1 − αi (x¯i ) and the time derivative of Vi computed with αi (x¯i ) and Assumption 1 is given by
V˙ i −
cz12
+ zi zi+1 − zi sgn(zi ) γimax (|x1 | + · · · + |xi | ) − fiσ
− zi sgn(zi )
i−1 j=1
−
cz12
∂αi−1 ( γ ( |x | + · · · + |x j | ) − | f j σ | ) |zi | ∂ x j jmax 1
+ zi zi+1
(12)
The nth step. Define
zn =xn − αn−1 (x¯n−1 ) = u + fnσ (x¯n ) −
n−1 j=1
∂αn−1 x˙ ∂xj j
1 Vn =Vn−1 + zn2 2
(13)
From (13), we can see that, the Vn (x) is not only positive definite but also radially unbounded for states x. Combining with the solutions of the nth subsystem in system (1), we obtain that:
V˙ n = V˙ n−1 + zn z˙ n − cz12 + zn−1 zn + zn u + |zn || fnα | − zn
n−1 j=1
n−1 ∂αn−1 ∂αn−1 | f jσ | x j+1 + |zn | ∂xj ∂xj j=1
(14)
At the end of the backstepping, through the similar procedure, the states backstepping controller is designed as:
u = − zn−1 − sgn(zn )γnmax (|x1 | + · · · + |xn | ) +
n−1 j=1
n−1 ∂αn−1 ∂αn−1 x − sgn(zn ) ∂ x j γ jmax (|x1 | + · · · + |x j | ) ∂ x j j+1 j=1
(15)
It can be obtained from V˙ n and u that
V˙ n − cz12 − zn sgn(zn )(γnmax − γn )(|x1 | + · · · + |x j | ) − zn sgn(zn )
∂αn−1 2 ∂ x j (γ jmax − γ j ) · (|x1 | + · · · + |x j | ) −cz1
n−1 j=1
(16)
Now, we are ready to present our first main result in this paper: Theorem 1. Consider the arbitrarily switched complex nonlinear system (1), satisfying Assumption 1, with the full states feedback backstepping controller (15), all the states of the closed-loop stabilized control system will converge to the origin at exponentially rate, and the control signal is bounded.
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Proof. For the arbitrarily switched complex nonlinear system, to address the stability analysis of the resulting closed-loop system (1), let us consider the overall Lyapunov function containing the backstepping controller (15), is described as Vn = n 1 2 i=1 2 zi . Next, taking into account the virtual controller, the derivative of Vn with time t is as (16). Choosing the appropriate γimax , i = 1, 2, . . . , n, the V˙ n satisfies
V˙ n −cz12 0 Consider the parameter c is positive constant. With the selected appropriate parameter c, deeper analysis shows that V˙ n −cVn . Furthermore, based on the Barbalat Lemma, it can be proof that all the signal zi converges to zero at exponential rate, then is further implies that the switched system states converge to zero. Above all, the controlled signals are all bounded. Example 1. In this paper, we present the following three dimensional switched nonlinear system. The system consists of two complex structure subsystem. The subsystems are switching along arbitrarily jumping signal σ (t) ∈ {1, 2}, the simulation results demonstrate the effectiveness of the proposed control scheme: Subsystem 1:
⎧ ⎨x˙ 1 = x2 + 5x1 sin(x21 ) x˙ 2 = x3 +
⎩
x1 +x2 1+x22
x˙ 3 = u + |x2 | 3 |x3 | 3 2
Subsystem
1
2:
x˙ 1 = x2 + x1 − 200x1 sin(ux1 ) 2 1 x˙ 2 = x3 + |x1 | 3 |x2 | 3 x˙ 3 = u + x2 (x1 + x3 )sin(x2 )
With Assumption 1, the nonlinear part of the switched system can be transformed to:
f1σ (x¯1 ) = 200|x1 | f2σ (x¯2 ) = |x1 | + |x2 | f3σ (x¯3 ) = |x1 | + |x2 | + |x3 |
so the parameters γ1max = 200 and γ2max = γ3max = 1 have been acquired. According to previous discussions, let α1 (z1 ) = −205sgn(z1 )|x1 | and by computation, we can choose α2 (z¯2 ) = −z1 − 205sgn(z1 )sgn(x1 ) − 2sgn(z2 )(|x1 ||x2 | ) − 2052 sgn(z2 ). Note that according to (3) and (6), we can choose controller by (15) as
u = − z2 − (1 + 2sgn(z2 ) − 205sgn(z2 ))x2 − (205sgn(z1 ) − 2sgn(z2 )sgn(x2 ))x3 − 2sgn(z3 )(|x1 | + |x2 | + |x3 | ) − 205sgn(z3 )|1 + 2sgn(z2 )sgn(x1 ) + 205sgn(z2 )| − 2sgn(z2 )| − 2sgn(z3 )| · 205sgn(z1 )sgn(x1 ) − 2sgn(z2 )sgn(x2 )(|x1 | + |x2 | ) with this controller, the closed-loop switched system in Example 1 realize the stabilization.
4. Output backstepping control based on the uniform gain observer This paper considers the switched complex nonlinear system with arbitrary switching signal σ (t), described below:
x˙ 1 = x2 + f1,σ (t ) (x¯1 ) .. . x˙ n−1 = xn + fn−1,σ (t ) (x¯n−1 ) x˙ n = u + fn,σ (t ) (x¯n )) y = x1 where y = x1 is the system output and can be acquired directly, we will design an output feedback control scheme to stabilize the switched system (1) with Assumption 1. The design is divided into two steps. First adopts a linear high-gain observer no matter the information of the system nonlinear structure. Second, with the ideas of [45], we construct the chained output feedback dynamic controller and design the uniform gain of the observer.
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4.1. Uniform gain observer design The designed uniform gain observer is described as below:
xˆ˙ 1 = xˆ2 + La1 (y − xˆ1 ) .. . xˆ˙ n−1 = xˆn + Ln−1 an−1 (y − xˆ1 ) xˆ˙ n = u + Ln an (y − xˆ1 )
(17)
The parameter L ≥ 1 is an uniform gain to be designed and aj > 0 j = 1, . . . , n are coefficients, the coefficients satisfy the Hurwitz polynomial p(s ) = sn + a1 sn−1 + · · · + an−1 s + an . Design the observer error as εi =
xi −xˆi Li−1
i = 1, . . . , n. A simple transformation calculation is described as
⎡
⎤ φ1 (x¯1 ) ⎢ 1 φ (x¯ ) ⎥ ε˙ = LAε + ⎣ L 2 2 ⎦ ... 1 φ n (x¯n ) n −1 L
(18)
where
⎡
⎡ ⎤ ε1 ⎢ ε2 ⎥ ε=⎣ ⎦ ... εn
−a1 ⎢ .. A=⎢ . ⎣−a n−1 −an
... .. . ... ...
1 .. . 0 0
⎤
0 .. ⎥ .⎥ ⎦ 1 0
(19)
where A is a Hurwitz matrix. Therefore, we select a positive-definite matrix P = P T > 0 satisfying AT P + PA = −I. Design a Lyapunov function V0 (ε ) = (n + 1 )ε T P ε . Considering Assumption 1, the time derivative of V0 (ε ) is that
⎡
⎤
f1σ (t ) f ⎥ ˙V0 (ε ) = − (n + 1 )Lε2 + 2(n + 1 )ε T P ⎢ ⎣ 2.σ. (.t ) ⎦ fnσ (t )
γ1σ (t ) |x1 | 1 L γ2σ (t ) (|x1 | + |x2 | ) 2 − (n + 1 )Lε + c1 ε . . . n1−1 γnσ (t ) (|x1 | + · · · + |xn | ) L 1 1 2 = − (n + 1 )Lε + c1 ε γ1σ (t ) + γ2σ (t ) + · · · + n−1 γnσ (t ) |x1 | + · · · L
L
1 1 1 γ2σ (t ) + · · · + n−1 γn,σ (t ) |x2 | + · · · n−1 γnσ (t ) |xn | + L
L
where parameter c1 = 2(n + 1 )P . Recall that xi = xˆi + Li−1 εi . Hence
(20)
L
1 |x | Li−1 i
1 |xˆ | Li−1 i
+ |εi | and we choose the max observer uniform coefficient as below:
γmσ (t ) = max γ1σ (t ) + γ2σ (t ) + · · · + γ(n−1)σ (t ) + · · · + γ(n)σ (t )
(21)
with this in mind, we can acquire that
V˙ 0 (ε ) − (n + 1 )Lε2 + c1 γmσ (t )
1 L
ε|xˆ1 | + ε |xˆ2 | + · · · + ε
1 Ln−1
|xˆn |
+ c1 γmσ (t ) ε(|ε1 | + |ε2 | + |ε3 | + · · · + |εn | )
−
(n + 1 )L − c1 γmσ (t )
√ n n+ 2
1 1 ε2 + c1 γmσ (t ) xˆ21 + 2 xˆ22 + · · · + 2
L
1
L
xˆ2 2(n−1 ) n
(22)
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4.2. Output feedback controller design based on observer A set of virtual controllers xˆ∗1 , . . . , xˆ∗n are designed as below:
xˆ∗1 = 0 xˆ∗2 = −Lb1 ξ1 .. . xˆ∗i = −Lbi−1 ξi−1 xˆ∗n = −Lbn−1 ξn−1 with bi > 0 being separated from the gain constant L and the virtual control error:
ξ1 = xˆ1 − xˆ∗1 .. .
ξi = xˆi − xˆ∗i ξn = xˆn − xˆ∗n The recursive backstepping controller is designed as below. Step 1: We select a Lyapunov function as
V1 (ε , ξ1 ) = V0 (ε ) +
1 2 ξ 2 1
(23)
Considering the error system ξ1 = xˆ1 , and define xˆ∗2 being a virtual control xˆ∗2 = −Lb1 ξ1 , then the derivation of ξ 1 is
ξ˙1 = xˆ˙ 1 = xˆ2 + La1 ε1 = ξ2 + xˆ∗2 + La1 ε1 So the dynamical differentiation gives
V˙ 1 (ε , ξ1 ) = V˙ 0 (ε ) + ξ1 ξ˙1 − (n + 1 )L − c1 γmσ (t )
√ n n+ 2
ε2 + c1 γmσ (t )
1 2 1 xˆ + · · · + 2(n−1) xˆ2n 2L4 3 2L
1 1 c1 γmσ (t ) ξ12 + c1 γmσ (t ) 2 ξ22 + c1 b21 γmσ (t ) ξ12 + ξ1 ξ2 − Lb1 ξ12 + La1 ε1 ξ1 2 L √ n 1 2 1 2 ˆ ˆ − nL − c1 γmσ (t ) n+ x + · · · + x ε2 + c1 γmσ (t ) 2 2L4 3 2L2(n−1) n
+
+
1 1 1 c1 γmσ (t ) ξ22 + ξ1 ξ2 + ξ12 c1 γmσ (t ) + La21 + c1 b21 γmσ (t ) − Lb1 2 4 L2
(24)
Observe that ξ1 ε1 ε12 + 14 ξ12 and 12 xˆ22 ξ22 + xˆ∗2 , define the parameter b1 = n + 1 + 14 a21 + 12 , select the uniform observer 2 parameter L > c1 γ mσ (t) with this in mind, we have
V˙ 1 (ε , ξ1 ) − nL − c1 γmσ (t )
√ n n+ 2
1 2 2 xˆ + · · · + ε + c1 γmσ (t ) 4 3 2L
1
2L
xˆ2 2(n−1 ) n
1 + 2 c1 γmσ (t ) ξ22 + ξ1 ξ2 − nL − c1 b21 γmσ (t ) ξ12
(25)
L
Recursive step k, a smooth Lyapunov candidate function Vk (ε , ξ1 , . . . , ξk ) is positive definite, such that
V˙ k (ε , ξk , . . . , ξk ) − (n + 1 − k )L − c1 γmσ (t ) −
k j=1
+
1 L2( j−1)
√ n n+ 2
ε2
(n + 1 − k ) − c1 γmσ (t ) b2j ξ j2 + c1 γmσ (t )
1 xˆ2 2L2(k+1) k+2
+ ··· +
1 xˆ2 2L2(n−1) n
c γ 1 ξk ξk+1 + 1 m2kσ (t ) ξk2+1 L2(k−1) L
(26)
Now, there must be a Lyapunov function existing as
V˙ k+1 (ε , ξ1 , . . . , ξk , ξk+1 ) = Vk (ε , ξ1 , . . . , ξk ) +
ξk+1 = xˆk+1 − xˆ∗k+1
1 2 ξ 2L2k k+1
(27) (28)
According to [45], it is straight forward to show that
ξ˙k+1 = xˆk+2 + Lbk xˆk+1 + L2 bk bk−1 xˆk + · · · + Lk−1 bk−1 bk−2 . . . b1 xˆ2 + Lk+1 (ak+1 + ak bk + · · · + a1 bk . . . b2 b1 )ε1
(29)
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with the virtual controller and the error system, acquire that
d dt
1 2k
1 1 2 ξ = 2k ξk+1 ξ˙k+1 = 2k ξk+1 ξk+2 + 2k k+1
1 dk+1 ξk2+1 L2k−1 1 1 1 + 2k−2 dk ξk ξk+1 + · · · + d1 ξ1 ξk+1 + d0 ε1 ξk+1 L L L L
L
(30)
where d0 , . . . , dk+1 , are suitable real constants. They are not belong to the gain constant L. Consider the selected parameter L > c1 γ mσ (t) and the inequality ξk ξk+1 < Lξk2 + 41L ξk2+1 , we obtain that
V˙ k+1 − (n − k )L − c1 γmσ (t )
+ c1 γmσ (t ) +
1 L2k−1
√
n n+ 2
k ε2 −
1 1 xˆ2 + · · · + 2n−2 xˆ2n 2L 2L2(k+2) k+3
j=1
+
1
L2( j−1)
(n − k ) − c1 γmσ (t ) b2j ξ j2
1 1 ξk+1 ξk+2 + 2(k+2) c1 γmσ (t ) ξk2+2 L2k L
d2 d2 d2 ( 1 + dk )2 c1 c1 − bk+1 + 0 + 1 + · · · + k+1 + + dk+1 + γmσ (t ) + γmσ (t ) b2k+1 4 4 4 4 L L
ξk2+1
(31)
Combining the inequality (31), the linear controller is designed as
xˆ∗k+2 = −Lbk+1 ξk+1
(32)
with
bk+1 = n + 1 +
d02 d2 ( 1 + dk )2 + 1 + ··· + + dk+1 4 4 4
having no relation with the L, so
V˙ k+1 −
(n − k )L − c1 γmσ (t )
+ c1 γmσ (t )
√ n n+ 2
k ε2 − j=1
1 1 xˆ2 + · · · + 2(n−1) xˆ2n 2L 2L2(k+2) k+3
1 ((n − k ) − c1 γmσ (t ) b2j )ξ j2 L2( j−1) (33)
This complete the inductive argument. With the similar inductive procedure, then at the nth step, like [45], acquire the designed recursive backstepping controller
u = − Lbn ξn = − Lbn (xˆn + Lbn−1 (xˆn−1 + · · · + Lb2 (xˆ2 + Lb1 xˆ1 ) . . . ) Where the real constant coefficients are bi > 0 to be redesigned for switched system:
V˙ n − L − c1 γmσ (t ) −
√ n n+ 2
(34)
i = 1, . . . , n. The coefficients having no relation with gain parameters L need
ε2 − (L − c1 γmσ (t ) b21 )ξ12 − · · ·
1 1 (L − c1 γmσ (t ) b2n−1 )ξn2−1 − 2n−2 Lξn2 L2n−4 L
(35)
Theorem 2. Consider the arbitrarily switched complex nonlinear system (1) with the single output y = x1 . If the switched system nonlinear structure satisfy Assumption 1, then under the action of the linear uniform gain observer (17), the output backstepping feedback controller (34), the states of the closed-loop arbitrarily switched complex nonlinear control system will converge to the origin at exponential rate. Proof. There are two parts in the proof. In the first part, a linear uniform gain pre-observer is proposed to overcome the uncertainty from the arbitrarily switching signal and realize the states estimation of the switched system. In the second part, using the states of the pre-observer, a stable controller is designed to stabilize the arbitrarily switched complex nonlinear system. To illustrate the backstepping control scheme, let us consider the conjoint Lyapunove function containing the observer and backstepping control Lyapunov function describing as:
Vn = V0 (ε ) + Vc (ξ1 , . . . , ξn ) = V0 (ε ) +
n i=1
1 ξ2 2L2(i−1) i
Substituting the controller (34) into the derivation of the observer Lyapunov function, we acquire the derivation of the Vn as (35).
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Fig. 1. State feedback.
√ In order to make sure the right-hand of (35) is negative, the uniform gain constant L > L∗ := max{1, ( n + is selected. Therefore, the backstepping controller combining with the uniform gain observer realize the global exponentially stable of the controlled system. n 2 2 2 )c1 γmσ (t ) , c1 γmσ (t ) b1 , . . . , c1 γmσ (t ) bn−1 }
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Fig. 2. Output feedback.
Remark 2. The output feedback control of this class switched system satisfying Assumption 1 can be stabilized using the uniform gain observer and backstepping controller with the reconstructed parameters and no matter how the systems switch under the arbitrary jumping signal, the stability of the switched closed-loop system will be realized.
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Example 2. Another simulation result is presented to validate the effectiveness of our designed output feedback controller. There are two dimensional switched complex nonlinear system which consisting of two subsystems under unknown switching signal σ (t) ∈ {1, 2}: Subsystem 1:
x˙ 1 = x2 +
x1 1+x21
x˙ 2 = u + x1 sin(x2 ) Subsystem
2:
x˙ 1 = x2 + 10x1 sin(x21 ) 2 1 x˙ 2 = u + |x1 | 3 |x2 | 3
and the output of the switched nonlinear system is y = x1 . Due to the presence of the nonlinear part of the switched system is not in a lower-triangular. The discrete jumping complex nonlinear system structure does not satisfy the global Lipschitz condition. In order to cope with the obstacle, the complex structure is decoupled as | fi,σ (t ) (x¯i )| γiσ (t ) (|x1 | + · · · + |xi | ) i = 1, 2 with Assumption 1, So we can acquire the inequality equation with the parameters (21), (32) as below: Subsystem 1:
x1 1+x21
γ11 |x1 |
x1 sin(x2 ) γ12 (|x1 | + |x2 | ) Subsystem
2:
10x1 sin(x21 ) γ21 |x1 | 2 1 |x1 | 3 |x2 | 3 γ22 (|x1 | + |x2 | )
where γ11 = 1, γ21 = 1, and γ12 = 10, γ22 = 1. The perfect gain parameters value is selected as L = c1 = 2(n + 1 )P = 3, so the key coefficient of the output feedback controller are chosen as γmσ (t ) = max{γ1σ (t ) + 1L γ2σ (t ) , γ2σ (t ) } = max{10 + 13 × 1, 1} = 10 31 , according to (34), we design the output feedback controller:
xˆ˙ 1 = xˆ2 + L(y − xˆ1 ) xˆ˙ 2 = u + L2 (y − xˆ1 ) u = −Lb2 (xˆ2 + Lb1 xˆ1 )
(36)
globally exponentially stabilizes the switched nonlinear system.
5. Simulation results In the simulation, according to Theorem 1, the trajectory of the state backstepping controller (15) is plotted in Fig. 1 and according to Theorem 2, the trajectory of the output backstepping controller (34) is plotted in Fig. 2. Under a randomly given switching signal which switches in uneven time interval, the simulation results of the state feedback design in Example 1 with initial value x(0 ) = [50, −20, 2]T , and the output feedback controller in Example 2 with x(0 ) = [5, −2]T are obtained, the control input, states and switching curves are plotted in Figs. 1 and 2, respectively. Intuitively, all the system states converges to zero, this proof that the proposed controllers are all effective to stabilize the switched complex nonlinear system.
6. Conclusion The paper analysis the backstepping control of the arbitrarily switched nonlinear system. The complex nonlinear structure of the switched system does not satisfy the global Lipschitz condition. First a state feedback backstepping controller is designed to realize the stability of the closed-loop switched system. Then based on a designed uniform gain observer, an output feedback backstepping controller is designed to stabilize the arbitrarily switching complex nonlinear systems. According to the method proposed in the paper, our future research directions are concentrate on three aspects: one is designing a suitable function instead of the absolute function in the controller. Second is that we are really hope to design a class of command filter to solve the explosion of complexity problem in the output backstepping controller. Third is that we are considering an efficient intelligent gain tuning scheme instead of fix gain. All the researches in these direction are all under way.
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Qitian Yin was born in Harbin, China, on December 22, 1979. He received the M.S. degree in Communication and Information Systems from College of Information and Communication, Harbin Engineering University in 2009. He is pursuing for the Ph.D. degree in Control Science and Engineering from Space Control and Inertial Technology Research Center, Harbin Institute of Technology. He is now a lecturer in the College of Computer Science and Information Engineering, Harbin Normal University. His research interests include adaptive control, chaos control and synchronization, switched nonlinear systems, intelligent control and their application.
Mao Wang received the B.Eng. degree in automation from Harbin Institute of Technology, in 1985, and the M.Eng. degree in Harbin Engineering University, in 1988, and the Ph.D. degree in Harbin Institute of Technology, in 1992. He joined Harbin Institute of Technology in 1994, where he is currently a professor. His current research interests include adaptive control, sliding mode control, hybrid systems, and inertial technology. He received Chinese National Defense Prize for his progress in science and technology.
He Jing was born in Qingdao, China, on November 1, 1997. She is an excellent student in the College of Computer Science and Information Engineering of Harbin Normal University. She is pursuing her undergraduate degree.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Stabilizing backstepping controller design for arbitrarily switched complex nonlinear system Qitian Yin a,b, Mao Wang b,∗, He Jing a a b
College of Computer Science and Information Engineering, Harbin Normal University, Harbin 150025, PR China Space Control and Inertial Technology Research Center, Harbin Institute of Technology, Harbin 150001, PR China
a r t i c l e
i n f o
Article history: Received 25 March 2019 Revised 18 July 2019 Accepted 23 September 2019
Keywords: Switched systems Observer Backstepping State feedback Output feedback
a b s t r a c t In this paper, the aim is to design a backstepping controller for switched complex nonlinear system. The nonlinear system is composed of several arbitrarily switched complex nonlinear subsystems, and the nonlinear system does not satisfy the so-called hypothesis of the global Lipschitz-like condition. First an uniform nonlinear decoupling method for switched subsystems is proposed to overcome the double uncertainty from the complex nonlinear structure and arbitrarily switching. Next a states backstepping controller is designed to realize exponentially stability of the switched nonlinear systems. Then based on an uniform gain observer with the uncertainty of the subsystem nonlinearity, an output backstepping controller is construct to realize exponentially stability of the observerbackstepping feedback control system. Two detailed simulations are finally presented to exhibit the effectiveness of the proposed synthesis methods. © 2019 Elsevier Inc. All rights reserved.
1. Introduction Over the past decade, many achievements of the stabilized analysis have been acquired, which are the switched linear systems or the switched nonlinear systems, e.g., [1–14]. Owing to the structural complexity and the arbitrary switching uncertainty, the control stability of the switched nonlinear systems is quite difficult than the stability of switched linear systems. The single lower triangular nonlinear system can be stabilized based on the backstepping method. With the development of the backstepping method, the switched nonlinear systems can also be controlled in [15–19]. At the same time, many achievements of different control scheme have been acquired in [20–24]. Among these, the control scheme based on the Lyapunov function (CLF)is effective to control and stabilize the switched systems in [8,25–29]. In order to improve the control ability, the backstepping controller combining the Lyapunov stability function is proposed to stabilize switched certainty nonlinear systems in [21,30,31] which switch arbitrarily. By extending the backstepping iterative procedure and the Lyapunov stability function, the [30] realizes the stability of the switched strict-feedback form nonlinear systems, combining the common control Lyapunov stability function (CCLF). However, owing to the difficulty to design an uniform coordinate transformation or an uniform compensation for the uncertainty of different subsystems in the backstepping design, few results have been presented over the past decades. In order to make up the gap, many intelligent controller containing fuzzy or neural network approximation have been proposed, with which the backstepping controller ∗
Corresponding author. E-mail address: [email protected] (M. Wang).
https://doi.org/10.1016/j.amc.2019.124789 0 096-30 03/© 2019 Elsevier Inc. All rights reserved.
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stabilizes the switched systems in [32,33]. There is a huge drawback in the application of the segmented systems. The uniform compensation or the rapid approximator is hard to design for the existing of the huge structural differences among the subsystems. In order to overcome the drawback, an effective direct backstepping control method is proposed to stabilize complex nonlinear switched switched system, in which the subsystems are switching with the signal arbitrarily. For the output feedback control problem, the states cannot be acquired or measured directly for various obstacles, the states observe problem usually arises. Over the past decades, the states observer for nonlinear system has been intensively studied, more involved results concerned with the adaptive intelligent observers are also reported in dealing with the dynamical nonlinear system [34–39]. The last decades, there are many intelligent method proposed to construct the observer, such as the paper [35,37,40], which combines the neural network and observer to guarantee the state reconstruction. However, there is a huge drawback that the switched nonlinear system will cause the neural network adjusting to new subsystem time after time, this procedure will cause too much operating time waste. Therefore through the analysis above, the contribution of the paper makes up the gap, the effective uniform gain observer is proposed to redesign the system states. On the other hand, single complex nonlinear system, such as bilinear system [41], affine system [42] or nonaffine systems [43] written as lower triangular form has been controlled through the feedback controller with control Lyapunov function. The above systems are all the states bounded in application and satisfy a hypothesis which has been shown in [44] and [45]. With the development of the switched systems, the nonlinear subsystems with different sates boundedness compose the switched system. The nonlinear subsystems are varying under arbitrary switching signal, and which is continuous and discrete Hybrid Systems in essence. The switched nonlinear system does not satisfy the so-called separation principle, growth condition or uniform boundedness, and the arbitrary switching between subsystems will cause the huge chattering. Then the common backstepping controller cannot be able to suppress the double uncertainty. According to the analysis, the stability of the switched system which is composed of complex nonlinear subsystems under arbitrary switching signal is exceptionally challenging and have never been studied sufficiently. Above all, the motivation of our work is to introduce a uniform decoupling scheme and a backstepping method to suppress and compensate the uncertainty from the arbitrary switching and complex nonlinear structure, further we construct a feedback backstepping controller with an uniform gain observer to realize the control of the switched complex nonlinear system. Several contributions of this paper is listed: (i ) An uniform nonlinear decoupling method for switched system is proposed to overcome the double uncertainty from the complex nonlinear structure and arbitrarily switching. (ii ) A novel backstepping controller based on the full states feedback is proposed to stabilize the switched nonlinear system; (iii ) Based on an uniform observer, a novel output feedback chained backstepping controller is designed to stabilize the control of the switched complex nonlinear system. The rest of this paper is organized as: Preliminaries is given in Section 2. In Section 3, backstepping controller design based on full states feedback is proposed for the considered system. The Section 4 illustrates the output backstepping controller design method based on the uniform gain observer. In Section 5, two simulation results are studied. Finally, the conclusions are drawn in Section 6. 2. Preliminaries In this paper, we deal with the following switched complex nonlinear system, the subsystem is described as the strictfeedback form:
x˙ 1 = x2 + f1,σ (t ) (x¯1 ) .. . x˙ i = xi+1 + fi,σ (t ) (x¯i ) .. . x˙ n−1 = xn + fn−1,σ (t ) (x¯n−1 ) x˙ n = u + fn,σ (t ) (x¯n ))
(1)
The states are x¯i := [x1 , x2 , . . . , xi ]T ∈ Rn . The control input is u ∈ R1 . σ : [0, ∞ ) → {1, . . . , m} is a piecewise constant switching signal that is not observed. The nonlinear function fi,σ (t ) (x¯i ) 1 i n are smooth. The objective is to design a continuous state feedback control law
u = h(x¯ )
(2)
with which the closed-loop control system (1) is able to be global uniformly asymptotically stabilized. It means that for all possible time interval σ (t), the closed-loop backstepping control system (1) is globally asymptotically stable. Note that the designed feedback control law (2) gets rid and independent of the arbitrarily switching signal σ (t). Assumption 1. For i = 1, . . . , n, during every time interval there is a constant γ iσ (t) > 0 make sure that the | fi,σ (t ) (x¯i )| γiσ (t ) (|x1 | + · · · + |xi | ) With this hypothesis, the global exponential stabilization of the single subsystem in the switched nonlinear system during the switching time interval σ (t) has been proven effective using linear state feedback in [45].
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Our research objective is to design a state feedback backstepping controller, and it is able to stabilize the arbitrarily switched complex switched systems. It is know that an unforced switched system is globally uniformly asymptotically stable if and only if there exists a common Lyapunov function for each subsystem [30]. Remark 1. There is a notification should be noticed that, the global stabilization of the single subsystem of the switching system (1) has already been considered in [45]. However this switched system composed of the class of the nonlinear system is placed in a arbitrary switching setting, in order to realize the control, it must be required to design the uniform controller. 3. Backstepping controller design based on full states feedback In order to realize the control, we must design a feedback control law, and with which there exists a common control Lyapunov function V(x) for every closed-loop subsystems in (1). For the case step n = 1. Define
z1 = x1 V1 (x1 ) = 12 z12
(3)
According to the backstepping controller design method for (1), a feasible virtual control function can be constructed as:
α1 (x1 ) = −sgn(z1 )γ1max |z1 |
(4)
and the parameter is defined as
γ1max = max γ1 j
(5)
j=1→σ
where γ 1j is the coefficient of the nonlinear part of the subsystem. Since the nonlinear part f1σ (t ) (x¯1 ) of the subsystems satisfy the Assumption 1, there exist the parameter γ 1max , so we can obtain that | f1σ (t ) (x¯1 )| γ1σ (t ) |x1 | γ1max |x1 | σ (t ) ∈ {1, . . . , m}. Define
z2 = x2 − α1 (x1 )
(6)
By substituting (4) into the derivation of the candidate Lyapunov function V1 in (3) and with the Assumption 1, one has:
V˙ 1 =z1 z2 + z1 α1 (x1 ) + z1 f1σ (t ) (x¯1 ) and taking (4) and (6) into account, such that
V˙ 1 z1 z2 + z1 α1 (x1 ) + |z1 |γ1,σ (t ) |x¯1 | = z1 z2 − (γ1max − γ1,σ (t ) )x¯21 − cz12 + z1 z2
(7)
where γ1max − γ1,σ (t ) = c. Next for the case that n = 2. With the definition of z3 , we have z3 = x3 − α2 (x¯2 ) and choose the 2nd virtual control function below:
α2 (x¯2 ) = −z1 +
∂α1 ∂α1 γ |x | x2 − sgn(z2 )γ2max (|x1 | + |x2 | ) − sgn(z2 ) ∂ x1 ∂ x1 1max 1
and also the parameter is defined as γ2max = max j=1→σ γ2 j and | f2σ (t ) (x¯2 )| γ2 j (|x1 | + |x2 | ) γ2max (|x1 | + |x2 | ). The selected common Lyapunov function is V2 = V1 + 12 z22 , and its time derivation becomes
V˙ 2 = V˙ 1 + z2 z˙ 2 z1 z2 − (γ1max − γ1σ (t ) )z12 + z2 z3 + z2 (α2 (x¯2 ) + f2σ (t ) (x1 , x2 )) − z2 Substituted the α 2 into V˙ 2 with Assumption 1, obtain
V˙ 2 = V˙ 1 + z2 z˙ 2 −(γ1max − γ1σ (t ) )z12 + z2 z3 − z2 sgn(z2 )
∂α1 x˙ ∂ x1 1
γ2max (|x1 | + |x2 | ) − | f2σ (t ) (x1 , x2 )|
∂α1 γ |x | − | f1σ (t ) (x1 )| − z2 sgn(z2 ) ∂ x1 1max 1 −(γ1max − γ1σ (t ) )z12 + z2 z3 = −cz12 + z2 z3
(8)
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For the case ith step for i = 3, . . . , n − 1, design the virtual control error zi = xi − αi−1 (x¯i−1 ), the ith dynamical error system is
z˙ i (x¯i ) = x˙ i − α˙ i−1 (x¯i−1 ) = zi+1 + αi (x¯i ) + fiσ (x¯i ) −
i−1 j=1
∂αi−1 x˙ ∂xj j
(9)
The extended candidate Lyapunov function is Vi (x¯i ) = Vi−1 (x¯i−1 ) + 12 zi2 . We obtain a promising analysis on V˙ i below:
V˙ i (x¯i ) = V˙ i−1 (x¯i−1 ) + zi z˙ i
−cz12
+ zi−1 zi + zi zi+1 + zi αi + |zi || fiα | − zi
i−1 j=1
i−1 ∂αi−1 ∂αi−1 | f | x |z | ∂ x j j+1 j=1 i ∂ x j jσ
(10)
Again, by viewing xi+1 as the virtual control, we can design the ith virtual control input: i−1
∂αi−1 x − sgn(zi )γimax (|x1 | ∂ x j j+1 j=1 i−1 ∂αi−1 γ (|x | + · · · + |x j | ) + · · · + |xi | ) − sgn(zi ) ∂ x j imax 1 j=1
αi (x¯i ) = − zi−1 +
(11)
Define zi+1 = xi+1 − αi (x¯i ) and the time derivative of Vi computed with αi (x¯i ) and Assumption 1 is given by
V˙ i −
cz12
+ zi zi+1 − zi sgn(zi ) γimax (|x1 | + · · · + |xi | ) − fiσ
− zi sgn(zi )
i−1 j=1
−
cz12
∂αi−1 ( γ ( |x | + · · · + |x j | ) − | f j σ | ) |zi | ∂ x j jmax 1
+ zi zi+1
(12)
The nth step. Define
zn =xn − αn−1 (x¯n−1 ) = u + fnσ (x¯n ) −
n−1 j=1
∂αn−1 x˙ ∂xj j
1 Vn =Vn−1 + zn2 2
(13)
From (13), we can see that, the Vn (x) is not only positive definite but also radially unbounded for states x. Combining with the solutions of the nth subsystem in system (1), we obtain that:
V˙ n = V˙ n−1 + zn z˙ n − cz12 + zn−1 zn + zn u + |zn || fnα | − zn
n−1 j=1
n−1 ∂αn−1 ∂αn−1 | f jσ | x j+1 + |zn | ∂xj ∂xj j=1
(14)
At the end of the backstepping, through the similar procedure, the states backstepping controller is designed as:
u = − zn−1 − sgn(zn )γnmax (|x1 | + · · · + |xn | ) +
n−1 j=1
n−1 ∂αn−1 ∂αn−1 x − sgn(zn ) ∂ x j γ jmax (|x1 | + · · · + |x j | ) ∂ x j j+1 j=1
(15)
It can be obtained from V˙ n and u that
V˙ n − cz12 − zn sgn(zn )(γnmax − γn )(|x1 | + · · · + |x j | ) − zn sgn(zn )
∂αn−1 2 ∂ x j (γ jmax − γ j ) · (|x1 | + · · · + |x j | ) −cz1
n−1 j=1
(16)
Now, we are ready to present our first main result in this paper: Theorem 1. Consider the arbitrarily switched complex nonlinear system (1), satisfying Assumption 1, with the full states feedback backstepping controller (15), all the states of the closed-loop stabilized control system will converge to the origin at exponentially rate, and the control signal is bounded.
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Proof. For the arbitrarily switched complex nonlinear system, to address the stability analysis of the resulting closed-loop system (1), let us consider the overall Lyapunov function containing the backstepping controller (15), is described as Vn = n 1 2 i=1 2 zi . Next, taking into account the virtual controller, the derivative of Vn with time t is as (16). Choosing the appropriate γimax , i = 1, 2, . . . , n, the V˙ n satisfies
V˙ n −cz12 0 Consider the parameter c is positive constant. With the selected appropriate parameter c, deeper analysis shows that V˙ n −cVn . Furthermore, based on the Barbalat Lemma, it can be proof that all the signal zi converges to zero at exponential rate, then is further implies that the switched system states converge to zero. Above all, the controlled signals are all bounded. Example 1. In this paper, we present the following three dimensional switched nonlinear system. The system consists of two complex structure subsystem. The subsystems are switching along arbitrarily jumping signal σ (t) ∈ {1, 2}, the simulation results demonstrate the effectiveness of the proposed control scheme: Subsystem 1:
⎧ ⎨x˙ 1 = x2 + 5x1 sin(x21 ) x˙ 2 = x3 +
⎩
x1 +x2 1+x22
x˙ 3 = u + |x2 | 3 |x3 | 3 2
Subsystem
1
2:
x˙ 1 = x2 + x1 − 200x1 sin(ux1 ) 2 1 x˙ 2 = x3 + |x1 | 3 |x2 | 3 x˙ 3 = u + x2 (x1 + x3 )sin(x2 )
With Assumption 1, the nonlinear part of the switched system can be transformed to:
f1σ (x¯1 ) = 200|x1 | f2σ (x¯2 ) = |x1 | + |x2 | f3σ (x¯3 ) = |x1 | + |x2 | + |x3 |
so the parameters γ1max = 200 and γ2max = γ3max = 1 have been acquired. According to previous discussions, let α1 (z1 ) = −205sgn(z1 )|x1 | and by computation, we can choose α2 (z¯2 ) = −z1 − 205sgn(z1 )sgn(x1 ) − 2sgn(z2 )(|x1 ||x2 | ) − 2052 sgn(z2 ). Note that according to (3) and (6), we can choose controller by (15) as
u = − z2 − (1 + 2sgn(z2 ) − 205sgn(z2 ))x2 − (205sgn(z1 ) − 2sgn(z2 )sgn(x2 ))x3 − 2sgn(z3 )(|x1 | + |x2 | + |x3 | ) − 205sgn(z3 )|1 + 2sgn(z2 )sgn(x1 ) + 205sgn(z2 )| − 2sgn(z2 )| − 2sgn(z3 )| · 205sgn(z1 )sgn(x1 ) − 2sgn(z2 )sgn(x2 )(|x1 | + |x2 | ) with this controller, the closed-loop switched system in Example 1 realize the stabilization.
4. Output backstepping control based on the uniform gain observer This paper considers the switched complex nonlinear system with arbitrary switching signal σ (t), described below:
x˙ 1 = x2 + f1,σ (t ) (x¯1 ) .. . x˙ n−1 = xn + fn−1,σ (t ) (x¯n−1 ) x˙ n = u + fn,σ (t ) (x¯n )) y = x1 where y = x1 is the system output and can be acquired directly, we will design an output feedback control scheme to stabilize the switched system (1) with Assumption 1. The design is divided into two steps. First adopts a linear high-gain observer no matter the information of the system nonlinear structure. Second, with the ideas of [45], we construct the chained output feedback dynamic controller and design the uniform gain of the observer.
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4.1. Uniform gain observer design The designed uniform gain observer is described as below:
xˆ˙ 1 = xˆ2 + La1 (y − xˆ1 ) .. . xˆ˙ n−1 = xˆn + Ln−1 an−1 (y − xˆ1 ) xˆ˙ n = u + Ln an (y − xˆ1 )
(17)
The parameter L ≥ 1 is an uniform gain to be designed and aj > 0 j = 1, . . . , n are coefficients, the coefficients satisfy the Hurwitz polynomial p(s ) = sn + a1 sn−1 + · · · + an−1 s + an . Design the observer error as εi =
xi −xˆi Li−1
i = 1, . . . , n. A simple transformation calculation is described as
⎡
⎤ φ1 (x¯1 ) ⎢ 1 φ (x¯ ) ⎥ ε˙ = LAε + ⎣ L 2 2 ⎦ ... 1 φ n (x¯n ) n −1 L
(18)
where
⎡
⎡ ⎤ ε1 ⎢ ε2 ⎥ ε=⎣ ⎦ ... εn
−a1 ⎢ .. A=⎢ . ⎣−a n−1 −an
... .. . ... ...
1 .. . 0 0
⎤
0 .. ⎥ .⎥ ⎦ 1 0
(19)
where A is a Hurwitz matrix. Therefore, we select a positive-definite matrix P = P T > 0 satisfying AT P + PA = −I. Design a Lyapunov function V0 (ε ) = (n + 1 )ε T P ε . Considering Assumption 1, the time derivative of V0 (ε ) is that
⎡
⎤
f1σ (t ) f ⎥ ˙V0 (ε ) = − (n + 1 )Lε2 + 2(n + 1 )ε T P ⎢ ⎣ 2.σ. (.t ) ⎦ fnσ (t )
γ1σ (t ) |x1 | 1 L γ2σ (t ) (|x1 | + |x2 | ) 2 − (n + 1 )Lε + c1 ε . . . n1−1 γnσ (t ) (|x1 | + · · · + |xn | ) L 1 1 2 = − (n + 1 )Lε + c1 ε γ1σ (t ) + γ2σ (t ) + · · · + n−1 γnσ (t ) |x1 | + · · · L
L
1 1 1 γ2σ (t ) + · · · + n−1 γn,σ (t ) |x2 | + · · · n−1 γnσ (t ) |xn | + L
L
where parameter c1 = 2(n + 1 )P . Recall that xi = xˆi + Li−1 εi . Hence
(20)
L
1 |x | Li−1 i
1 |xˆ | Li−1 i
+ |εi | and we choose the max observer uniform coefficient as below:
γmσ (t ) = max γ1σ (t ) + γ2σ (t ) + · · · + γ(n−1)σ (t ) + · · · + γ(n)σ (t )
(21)
with this in mind, we can acquire that
V˙ 0 (ε ) − (n + 1 )Lε2 + c1 γmσ (t )
1 L
ε|xˆ1 | + ε |xˆ2 | + · · · + ε
1 Ln−1
|xˆn |
+ c1 γmσ (t ) ε(|ε1 | + |ε2 | + |ε3 | + · · · + |εn | )
−
(n + 1 )L − c1 γmσ (t )
√ n n+ 2
1 1 ε2 + c1 γmσ (t ) xˆ21 + 2 xˆ22 + · · · + 2
L
1
L
xˆ2 2(n−1 ) n
(22)
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4.2. Output feedback controller design based on observer A set of virtual controllers xˆ∗1 , . . . , xˆ∗n are designed as below:
xˆ∗1 = 0 xˆ∗2 = −Lb1 ξ1 .. . xˆ∗i = −Lbi−1 ξi−1 xˆ∗n = −Lbn−1 ξn−1 with bi > 0 being separated from the gain constant L and the virtual control error:
ξ1 = xˆ1 − xˆ∗1 .. .
ξi = xˆi − xˆ∗i ξn = xˆn − xˆ∗n The recursive backstepping controller is designed as below. Step 1: We select a Lyapunov function as
V1 (ε , ξ1 ) = V0 (ε ) +
1 2 ξ 2 1
(23)
Considering the error system ξ1 = xˆ1 , and define xˆ∗2 being a virtual control xˆ∗2 = −Lb1 ξ1 , then the derivation of ξ 1 is
ξ˙1 = xˆ˙ 1 = xˆ2 + La1 ε1 = ξ2 + xˆ∗2 + La1 ε1 So the dynamical differentiation gives
V˙ 1 (ε , ξ1 ) = V˙ 0 (ε ) + ξ1 ξ˙1 − (n + 1 )L − c1 γmσ (t )
√ n n+ 2
ε2 + c1 γmσ (t )
1 2 1 xˆ + · · · + 2(n−1) xˆ2n 2L4 3 2L
1 1 c1 γmσ (t ) ξ12 + c1 γmσ (t ) 2 ξ22 + c1 b21 γmσ (t ) ξ12 + ξ1 ξ2 − Lb1 ξ12 + La1 ε1 ξ1 2 L √ n 1 2 1 2 ˆ ˆ − nL − c1 γmσ (t ) n+ x + · · · + x ε2 + c1 γmσ (t ) 2 2L4 3 2L2(n−1) n
+
+
1 1 1 c1 γmσ (t ) ξ22 + ξ1 ξ2 + ξ12 c1 γmσ (t ) + La21 + c1 b21 γmσ (t ) − Lb1 2 4 L2
(24)
Observe that ξ1 ε1 ε12 + 14 ξ12 and 12 xˆ22 ξ22 + xˆ∗2 , define the parameter b1 = n + 1 + 14 a21 + 12 , select the uniform observer 2 parameter L > c1 γ mσ (t) with this in mind, we have
V˙ 1 (ε , ξ1 ) − nL − c1 γmσ (t )
√ n n+ 2
1 2 2 xˆ + · · · + ε + c1 γmσ (t ) 4 3 2L
1
2L
xˆ2 2(n−1 ) n
1 + 2 c1 γmσ (t ) ξ22 + ξ1 ξ2 − nL − c1 b21 γmσ (t ) ξ12
(25)
L
Recursive step k, a smooth Lyapunov candidate function Vk (ε , ξ1 , . . . , ξk ) is positive definite, such that
V˙ k (ε , ξk , . . . , ξk ) − (n + 1 − k )L − c1 γmσ (t ) −
k j=1
+
1 L2( j−1)
√ n n+ 2
ε2
(n + 1 − k ) − c1 γmσ (t ) b2j ξ j2 + c1 γmσ (t )
1 xˆ2 2L2(k+1) k+2
+ ··· +
1 xˆ2 2L2(n−1) n
c γ 1 ξk ξk+1 + 1 m2kσ (t ) ξk2+1 L2(k−1) L
(26)
Now, there must be a Lyapunov function existing as
V˙ k+1 (ε , ξ1 , . . . , ξk , ξk+1 ) = Vk (ε , ξ1 , . . . , ξk ) +
ξk+1 = xˆk+1 − xˆ∗k+1
1 2 ξ 2L2k k+1
(27) (28)
According to [45], it is straight forward to show that
ξ˙k+1 = xˆk+2 + Lbk xˆk+1 + L2 bk bk−1 xˆk + · · · + Lk−1 bk−1 bk−2 . . . b1 xˆ2 + Lk+1 (ak+1 + ak bk + · · · + a1 bk . . . b2 b1 )ε1
(29)
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with the virtual controller and the error system, acquire that
d dt
1 2k
1 1 2 ξ = 2k ξk+1 ξ˙k+1 = 2k ξk+1 ξk+2 + 2k k+1
1 dk+1 ξk2+1 L2k−1 1 1 1 + 2k−2 dk ξk ξk+1 + · · · + d1 ξ1 ξk+1 + d0 ε1 ξk+1 L L L L
L
(30)
where d0 , . . . , dk+1 , are suitable real constants. They are not belong to the gain constant L. Consider the selected parameter L > c1 γ mσ (t) and the inequality ξk ξk+1 < Lξk2 + 41L ξk2+1 , we obtain that
V˙ k+1 − (n − k )L − c1 γmσ (t )
+ c1 γmσ (t ) +
1 L2k−1
√
n n+ 2
k ε2 −
1 1 xˆ2 + · · · + 2n−2 xˆ2n 2L 2L2(k+2) k+3
j=1
+
1
L2( j−1)
(n − k ) − c1 γmσ (t ) b2j ξ j2
1 1 ξk+1 ξk+2 + 2(k+2) c1 γmσ (t ) ξk2+2 L2k L
d2 d2 d2 ( 1 + dk )2 c1 c1 − bk+1 + 0 + 1 + · · · + k+1 + + dk+1 + γmσ (t ) + γmσ (t ) b2k+1 4 4 4 4 L L
ξk2+1
(31)
Combining the inequality (31), the linear controller is designed as
xˆ∗k+2 = −Lbk+1 ξk+1
(32)
with
bk+1 = n + 1 +
d02 d2 ( 1 + dk )2 + 1 + ··· + + dk+1 4 4 4
having no relation with the L, so
V˙ k+1 −
(n − k )L − c1 γmσ (t )
+ c1 γmσ (t )
√ n n+ 2
k ε2 − j=1
1 1 xˆ2 + · · · + 2(n−1) xˆ2n 2L 2L2(k+2) k+3
1 ((n − k ) − c1 γmσ (t ) b2j )ξ j2 L2( j−1) (33)
This complete the inductive argument. With the similar inductive procedure, then at the nth step, like [45], acquire the designed recursive backstepping controller
u = − Lbn ξn = − Lbn (xˆn + Lbn−1 (xˆn−1 + · · · + Lb2 (xˆ2 + Lb1 xˆ1 ) . . . ) Where the real constant coefficients are bi > 0 to be redesigned for switched system:
V˙ n − L − c1 γmσ (t ) −
√ n n+ 2
(34)
i = 1, . . . , n. The coefficients having no relation with gain parameters L need
ε2 − (L − c1 γmσ (t ) b21 )ξ12 − · · ·
1 1 (L − c1 γmσ (t ) b2n−1 )ξn2−1 − 2n−2 Lξn2 L2n−4 L
(35)
Theorem 2. Consider the arbitrarily switched complex nonlinear system (1) with the single output y = x1 . If the switched system nonlinear structure satisfy Assumption 1, then under the action of the linear uniform gain observer (17), the output backstepping feedback controller (34), the states of the closed-loop arbitrarily switched complex nonlinear control system will converge to the origin at exponential rate. Proof. There are two parts in the proof. In the first part, a linear uniform gain pre-observer is proposed to overcome the uncertainty from the arbitrarily switching signal and realize the states estimation of the switched system. In the second part, using the states of the pre-observer, a stable controller is designed to stabilize the arbitrarily switched complex nonlinear system. To illustrate the backstepping control scheme, let us consider the conjoint Lyapunove function containing the observer and backstepping control Lyapunov function describing as:
Vn = V0 (ε ) + Vc (ξ1 , . . . , ξn ) = V0 (ε ) +
n i=1
1 ξ2 2L2(i−1) i
Substituting the controller (34) into the derivation of the observer Lyapunov function, we acquire the derivation of the Vn as (35).
Q. Yin, M. Wang and H. Jing / Applied Mathematics and Computation 369 (2020) 124789
9
Fig. 1. State feedback.
√ In order to make sure the right-hand of (35) is negative, the uniform gain constant L > L∗ := max{1, ( n + is selected. Therefore, the backstepping controller combining with the uniform gain observer realize the global exponentially stable of the controlled system. n 2 2 2 )c1 γmσ (t ) , c1 γmσ (t ) b1 , . . . , c1 γmσ (t ) bn−1 }
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Fig. 2. Output feedback.
Remark 2. The output feedback control of this class switched system satisfying Assumption 1 can be stabilized using the uniform gain observer and backstepping controller with the reconstructed parameters and no matter how the systems switch under the arbitrary jumping signal, the stability of the switched closed-loop system will be realized.
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11
Example 2. Another simulation result is presented to validate the effectiveness of our designed output feedback controller. There are two dimensional switched complex nonlinear system which consisting of two subsystems under unknown switching signal σ (t) ∈ {1, 2}: Subsystem 1:
x˙ 1 = x2 +
x1 1+x21
x˙ 2 = u + x1 sin(x2 ) Subsystem
2:
x˙ 1 = x2 + 10x1 sin(x21 ) 2 1 x˙ 2 = u + |x1 | 3 |x2 | 3
and the output of the switched nonlinear system is y = x1 . Due to the presence of the nonlinear part of the switched system is not in a lower-triangular. The discrete jumping complex nonlinear system structure does not satisfy the global Lipschitz condition. In order to cope with the obstacle, the complex structure is decoupled as | fi,σ (t ) (x¯i )| γiσ (t ) (|x1 | + · · · + |xi | ) i = 1, 2 with Assumption 1, So we can acquire the inequality equation with the parameters (21), (32) as below: Subsystem 1:
x1 1+x21
γ11 |x1 |
x1 sin(x2 ) γ12 (|x1 | + |x2 | ) Subsystem
2:
10x1 sin(x21 ) γ21 |x1 | 2 1 |x1 | 3 |x2 | 3 γ22 (|x1 | + |x2 | )
where γ11 = 1, γ21 = 1, and γ12 = 10, γ22 = 1. The perfect gain parameters value is selected as L = c1 = 2(n + 1 )P = 3, so the key coefficient of the output feedback controller are chosen as γmσ (t ) = max{γ1σ (t ) + 1L γ2σ (t ) , γ2σ (t ) } = max{10 + 13 × 1, 1} = 10 31 , according to (34), we design the output feedback controller:
xˆ˙ 1 = xˆ2 + L(y − xˆ1 ) xˆ˙ 2 = u + L2 (y − xˆ1 ) u = −Lb2 (xˆ2 + Lb1 xˆ1 )
(36)
globally exponentially stabilizes the switched nonlinear system.
5. Simulation results In the simulation, according to Theorem 1, the trajectory of the state backstepping controller (15) is plotted in Fig. 1 and according to Theorem 2, the trajectory of the output backstepping controller (34) is plotted in Fig. 2. Under a randomly given switching signal which switches in uneven time interval, the simulation results of the state feedback design in Example 1 with initial value x(0 ) = [50, −20, 2]T , and the output feedback controller in Example 2 with x(0 ) = [5, −2]T are obtained, the control input, states and switching curves are plotted in Figs. 1 and 2, respectively. Intuitively, all the system states converges to zero, this proof that the proposed controllers are all effective to stabilize the switched complex nonlinear system.
6. Conclusion The paper analysis the backstepping control of the arbitrarily switched nonlinear system. The complex nonlinear structure of the switched system does not satisfy the global Lipschitz condition. First a state feedback backstepping controller is designed to realize the stability of the closed-loop switched system. Then based on a designed uniform gain observer, an output feedback backstepping controller is designed to stabilize the arbitrarily switching complex nonlinear systems. According to the method proposed in the paper, our future research directions are concentrate on three aspects: one is designing a suitable function instead of the absolute function in the controller. Second is that we are really hope to design a class of command filter to solve the explosion of complexity problem in the output backstepping controller. Third is that we are considering an efficient intelligent gain tuning scheme instead of fix gain. All the researches in these direction are all under way.
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Q. Yin, M. Wang and H. Jing / Applied Mathematics and Computation 369 (2020) 124789
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Qitian Yin was born in Harbin, China, on December 22, 1979. He received the M.S. degree in Communication and Information Systems from College of Information and Communication, Harbin Engineering University in 2009. He is pursuing for the Ph.D. degree in Control Science and Engineering from Space Control and Inertial Technology Research Center, Harbin Institute of Technology. He is now a lecturer in the College of Computer Science and Information Engineering, Harbin Normal University. His research interests include adaptive control, chaos control and synchronization, switched nonlinear systems, intelligent control and their application.
Mao Wang received the B.Eng. degree in automation from Harbin Institute of Technology, in 1985, and the M.Eng. degree in Harbin Engineering University, in 1988, and the Ph.D. degree in Harbin Institute of Technology, in 1992. He joined Harbin Institute of Technology in 1994, where he is currently a professor. His current research interests include adaptive control, sliding mode control, hybrid systems, and inertial technology. He received Chinese National Defense Prize for his progress in science and technology.
He Jing was born in Qingdao, China, on November 1, 1997. She is an excellent student in the College of Computer Science and Information Engineering of Harbin Normal University. She is pursuing her undergraduate degree.