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Observer-based adaptive neural control for switched stochastic pure-feedback systems with input saturation
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Observer-Based Adaptive Neural Control for Switched Stochastic Pure-feedback Systems with Input Saturation Zahra Namadchian , Modjtaba Rouhani PII: DOI: Reference:
S0925-2312(19)31290-1 https://doi.org/10.1016/j.neucom.2019.09.028 NEUCOM 21283
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Neurocomputing
Received date: Revised date: Accepted date:
28 January 2019 13 August 2019 3 September 2019
Please cite this article as: Zahra Namadchian , Modjtaba Rouhani , Observer-Based Adaptive Neural Control for Switched Stochastic Pure-feedback Systems with Input Saturation, Neurocomputing (2019), doi: https://doi.org/10.1016/j.neucom.2019.09.028
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Observer-Based Adaptive Neural Control for Switched Stochastic Pure-feedback Systems with Input Saturation Zahra Namadchiana, and Modjtaba Rouhanib,1 a
Department of Electrical Engineering, Islamic Azad University of Gonabad, Khorasan Razavi, Iran b Department of Computer Engineering, Ferdowsi University of Mashhad, Mashhad, Iran
Abstract The purpose of this paper is to investigate the problem of adaptive neural network (NN) output-feedback tracking control for input saturated switched stochastic nonlinear systems in pure-feedback form with unmeasured states. In order to facilitate the controller design process, a dead zone-based model of saturation is implemented and radial basis function NNs (RBFNNs) are employed to approximate unknown nonlinear functions and to construct an NN switched nonlinear observer to cope with difficulties raised by the presence of immeasurable state variables. Based on the adaptive backstepping technique and Lyapunov function method, an adaptive NN output feedback control scheme is developed. Furthermore, it is proved that the proposed controller can provide that, under arbitrary deterministic switching, all signals in the closed-loop system are semiglobally uniformly ultimately bounded in probability and the tracking error converges to a small neighborhood of the origin. Finally, simulation examples are presented to validate the effectiveness of the proposed adaptive NN control approach. Keywords: input saturation, Adaptive neural control, output-feedback control, switched nonlinear systems, stochastic purefeedback system.
1. Introduction In recent years, the stability analysis and observer-based control design problems for nonlinear systems with unmeasured states have received much attention, since in many practical nonlinear systems (such as power industry, aerospace, chemical engineering applications and thermal processes [1]) the information of state variables are often unavailable. Recent studies highlight that approximation-based adaptive backstepping output-feedback control approach has great potential to cope with immeasurable states difficulty in which different kinds of observers (i.e. linear state observer, k-filters state observer, input-driven observer, and fuzzy/NN nonlinear state observers) have been successfully designed to estimate unavailable state variables. For example, based on a high-gain observer, an adaptive NN output feedback controller was designed for the robotic manipulators and the flapping wing micro aerial vehicle (FWMAV) in [2] and [3], respectively. Regarding this concern, the control problem of nonlinear stochastic systems could be an important issue, both in practice and theory [4]-[8]. For example, With the help of a fuzzy observer and adaptive backstepping technique with dynamic surface control (DSC), an observer-based adaptive fuzzy controller for stochastic nonlinear systems in nonstrict-feedback form has been constructed in [5]. An approximation-based adaptive output-feedback control scheme for strict-feedback nonlinear systems with stochastic disturbances, unmodeled dynamics, and unknown dead zone input has been investigated in [7]. Based on fuzzy logic 1
Corresponding author: Modjtaba Rouhani (e-mail: [email protected]).
system and the Nussbaum gain function in [8], an adaptive fuzzy output-feedback prescribed performance control (PPC) problem for a class of multi-input and multi- output (MIMO) stochastic nonlinear nonstrict-feedback systems with unknown dead zone and unknown control direction was solved. As is well known, pure-feedback nonlinear systems have no affine appearance of the state variables to be used as virtual controls or actual control [9]. In addition, without the direct measurements of state variables, the control design of nonlinear pure-feedback systems has become more and more complicated, some results have been proposed in [9]-[17]. By applying DSC approach and RBFNNs, Li et al. [9] developed an adaptive NN output-feedback tracking control scheme for MIMO purefeedback nonlinear systems with backlash-like hysteresis and unmeasured states. Yang et al. [10] focused on decentralized adaptive NNs fault-tolerant output-feedback control design for a class of large-scale MIMO nonlinear nonaffine pure-feedback systems with actuator failures. Also, an adaptive fuzzy backstepping output-feedback control problem in [13] and [15] has been studied for uncertain pure-feedback nonlinear systems with immeasurable states. To overcome algebraic loop and immeasurable states problems, Tong et al. [14] applied filtered signals and a fuzzy state observer to propose an adaptive fuzzy output-feedback fault-tolerant control approach for a class of MIMO stochastic pure-feedback nonlinear systems. Based on the backstepping technique and a constructed fuzzy state observer, an adaptive fuzzy output-feedback control approach for a class of stochastic nonlinear systems in pure-feedback form has been investigated in [11] and [16]. On the other hand, in the last few years, a growing body of research has developed the adaptive output-feedback control approaches for nonlinear switched systems [18]-[27]. For example, the problem of adaptive output-feedback fault-tolerant tracking control for nonlinear switched large-scale systems in pure-feedback form with unknown dead zones has been described by [18]. By employing the average dwell time approach and Nussbaum gain function, a new observer-based fuzzy adaptive backstepping control algorithm has been explained for the switched MIMO nonlinear system in non-triangular form under considerations contain unknown backlash hysteresis and unknown control directions [20]. Tong et al. [23] studied an adaptive fuzzy decentralized control design for a class of switched nonlinear large-scale systems with dead zones and immeasurable states. To overcome the unmeasured states difficulty in nonstrict-feedback switched nonlinear systems with dead zone input, Tong et al. [24] constructed a switched fuzzy state observer to present an adaptive fuzzy output-feedback tracking control. An adaptive fuzzy output-feedback partial tracking errors constrained problem has been considered in [25] for switched stochastic nonlinear systems with average dwell time switching. Huo et al. [27] studied an observer-based fuzzy adaptive stabilization problem of uncertain switched stochastic nonlinear systems with unmeasured states and quantized input signals quantized by a hysteretic quantizer. The most important non-smooth nonlinearity, input saturation, often occurs in practical engineering and is the main limitation of the system performance and even could be the leading cause of instability. Therefore, in the field of control problem, the design and development of practical controllers for systems with saturation input is a serious challenge. Moreover, there is still considerable concern about the stability and control problems of nonlinear systems with saturation nonlinearity and unmeasured states which has been considered in recent research [28]-[36]. Sui et al. [36] used Butterworth low-pass filter and adaptive backstepping techniques to design the adaptive fuzzy output-feedback tracking controller and the fuzzy state observer for MIMO stochastic nonlinear system in purefeedback form with input saturation. The problem of adaptive fuzzy output-feedback control for a class of nonlinear strict-feedback stochastic systems with saturation nonlinearity and output constraint was investigated by Xu et al. in [28]. Yang et al. [34] focused on decentralized adaptive NN output-feedback control design for a class of large-scale time-delay systems in pure-feedback form with saturating inputs, based on DSC technique and NN approximator. Quite recently, few publications have addressed the control problem of nonlinear switched systems with input saturation [37]-[42]. Yu et al. [40] proposed an adaptive quantized tracking scheme for a class of switched stochastic strict-feedback nonlinear systems with asymmetrical input saturation. Furthermore, based on an inputdriven observer, NN approximation ability, and the backstepping technique, an adaptive output-feedback tracking control problem of switched nonstrict feedback nonlinear systems with unknown control direction and asymmetric saturation has been investigated in [43]. By employing MT-filters and RBFNNs, Li et al. [44] presented an observer-based NN adaptive PPC for nonlinear switched systems with input saturation. Motivated by the aforementioned discussions, this paper investigates the observer-based NN output-feedback controller design for a class of switched stochastic nonlinear pure-feedback systems under arbitrary switching signal while both unmeasured states and input saturation are simultaneously considered. With the help of presented method in ([11], [13], [15], and [16]), the problem of switched pure feedback form is solved and an NN switched observer will be successfully designed on the basis of universal approximation ability of RBFNNs for input saturated
systems. Afterward, an adaptive NN output-feedback control scheme is proposed in the framework of adaptive backstepping method and Lyapunov stability theory that ensures all signals in a closed-loop system are semiglobally uniformly ultimately bounded in probability. The main advantages of this paper are summarized as follows. 1) The proposed control method can deal with the output-feedback tracking control problem of switched stochastic nonlinear pure-feedback systems with unknown input saturation. The presented procedure is practically the same as the ones explained in ([11], [13], [15], and [16]) and also is more or less identical to the studies done in ([9], [10], [12], [14], and [17]). Despite these similarities, these works were restricted to nonswitched pure-feedback systems or nonswitched stochastic systems in pure-feedback form without input nonlinearities (except [9] with input hysteresis). Therefore, the tracking control investigation of nonlinear switched stochastic pure-feedback systems with unmeasured state variables and unknown saturation problems simultaneously will become a challenging issue. 2) Compared with the previous works ([28]-[36]) which only studied the adaptive output-feedback control for nonswitched nonlinear systems with known input saturation, in this paper, the proposed control approach has been applied to a class of switched nonlinear systems with unknown saturation. The chosen model of input saturation, dead zone-based model, is the same as the one proposed in [45] which is applied to reduce the control process complexity, while has beneficial features to model diverse kinds of saturation and adjusts the balance between modeling accuracy and computation. In addition, unlike the result in [37]-[40], no knowledge about the upper and lower bounds of unknown functions of input saturation is needed for the control design. 3) In contrast to studies ([37]-[42] and [46]-[49]) that the states of switched systems with input saturation and switched nonlinear stochastic systems in pure-feedback form are required to be all available for measurements, respectively, the presented adaptive control approach based on the obtained NN switched observer can cancel this condition. In addition, to deal with control problem of switched pure-feedback nonlinear systems, the Butterworth low-pass filter technique is utilized to avoid the algebraic loop issue and eliminate the existence of known bounds and the known signs of nonlinear functions assumptions. This paper is divided into six sections. Section 2 gives preliminaries and problem formulation. In Section 3, the NN switched observer is designed. Section 4 analyzes the stability and adaptive output-feedback control problems for the suggested system. Simulation examples are given in Section 5. Finally, the conclusions are drawn in Section 6. 2.
Preliminaries and problem formulation
2.1 System Information and Assumption Consider a class of switched stochastic pure-feedback nonlinear systems given as follows: ( )
{
(
( )
(
) )
( ) ( )
( )
( )
(1)
, , ), where ,( , and are the state vector and output of the system, respectively. are unmeasurable, the only output variable is measurable. * + is the arbitrary deterministic switching signal, which takes value in the compact ( ) , ) set . If ( ) , kth subsystem will be active. For any and , () and () denote the unknown smooth nonlinear functions. is an r-dimensional independent standard Brownian motion defined on the complete probability space, with the incremental covariance { } ( ) ( ) . presents the saturation nonlinearity to be described as follows:
( )
{ ( )
( ) ( ) ( )
where and are the unknown upper and lower bounded parameters of the input saturation, . As shown in (2), there are two sharp corners when and . So the backstepping technique
(2)
cannot be directly utilized to design control input signal. As considered in [45], the dead zone-based model of saturation will be employed to tackle this problem and defined as: ( ) ( )
(3)
( )
∫
( )
(4)
where ∫ ( ) is an unknown constant, in which ( ) is supposed to be unknown. ( ) is a density function satisfying ( ) , . When , ( ) . ( ) is the dead-zone operator defined by ( ) ( ( )) Remark 1: To model a saturation function, different types of density functions, can be chosen as follows ( )
{
(5)
Moreover, the saturation calculated by [45]:
accesses the saturated region on
. From (3), we obtain the saturated value
∫
( ) (6)
( ) ( ) Let ( ) ( ) ,( ) and . Then, based on (3), system (1) is equivalent to the following switched stochastic nonlinear pure-feedback system:
{
.
(
.
( )
( )
)/
( ))/
(
( ) (7)
Let define: (
)
(
(̂ ̂
)
(̂
)
)
(
( ))
́
(8) (
) (9)
where and
́ (̂ ) (̂ ) which will be defined by state observer designed later. ̂
̂
( )̂
(
)
́
(
)
( ),
( ) ; ̂ and ̂ are the estimates of state vectors and are the filtered signals defined by:
( )
(10)
̂,
where ( ) is a Butterworth low-pass filter (LPF). The parameters of Butterworth LPF can be obtained in [36]. Then, system (7) can be rewritten as: . {
where
.
(̂ ̂ (̂
)/ )/
( ) ( )
́ (̂ ) (̂ ) ( ). Now, we can express (11) in the following form:
(11)
,
∑,
(̂ ̂
)-
(
(̂
)
)
( )
-
(12) [
where
[⏟
],
⏟
] ,
[
,
],
[ ],
[
( )
],
,
-,
-.
Furthermore, for and , with appropriate choice of coefficients , one can be assured that a strict Hurwitz matrix. Thus, for any given matrix , there exists a positive definite matrix satisfying
is
(13) Before presenting some definitions, lemmas, and assumptions to facilitate the adaptive control design, we first consider the following stochastic nonlinear system: ( )
( )
(14)
( ) where ( ) and ( ) are locally Lipschitz functions in and satisfy ( ) . Definition 1([4], [8], and [37]): For any function ( ), related with (14), define the differential operator as follows ( )
( )
2 ( )
( ) is the trace of
where
( )
( )3
and the term
(15) 2 ( )
( )
( )3 is called Itô correction term, in which the
( )
second-order differential makes the controller design much more difficult than that of the deterministic system. + of the stochastic system (14) is said to be semiglobally Definition 2 ([28] and [40]): The trajectory * ( ) uniformly ultimately bounded in the th moment, if for some compact set and any initial state ( ), there exist a constant and a time constant ( ) such that (| ( )| ) for all . Especially, when , it is usually called semiglobally uniformly ultimately bounded in mean square. Assumption 1 ([25], [36], [43]): There exists a set of known constant , for , such that the following inequality holds: |
(
)
(
)|
‖
‖
Assumption 2 ([9], [12], [13], and [36]): There exist unknown constants ) and | | . Assumption 3 ([14], [25], [36]): The disturbance covariance [
̅
such that | ̂
| |
|
,(
|
,
̅ is bounded, where
] . Assumption 4 ([9], [13]): There exist unknown constants and such that | | . Lemma 1 (Young’s inequality [5]): The following inequality holds for ( ) : | |
̂
and |
(16)
where , , , and ( Lemma 2 ([13], [43]): For any
)( and
)
. ,|
|
(
⁄)
is satisfied.
2.2 RBF neural network The radial basis function neural networks (RBFNNs) are usually used to approximate unknown smooth nonlinear functions due to their simple structure and good capabilities in function approximation. Therefore, in this paper, The RBFNNs are employed to approximate unknown smooth nonlinear functions ( )( ) over a compact set . The RBFNNs can be described as ( ) where , and are the weight vector ( )and the number of NN nodes, respectively. ( ) , ( ) is the vector of known basis function and ( ) have the following form: (
( )
) (
)
(17)
where [ ] is the center of the receptive field and is the width of the Gaussian function. Based on the results of [50], with sufficiently large , the RBFNNs can approximate any continuous function to any desired accuracy over a compact set . Lemma 3([49]): For any accuracy , there exists an NN ( ) such that | ( )
( )|
(18)
The objective of this paper is to design an adaptive NN output-feedback controller and adaptive laws for system (1) in presence of input saturation such that all signals in the closed-loop remain bounded in probability under arbitrary deterministic switching signal. Furthermore, the output of system tracks a desired reference signal in the mean square sense.
Neural network switched observer design
3.
Throughout this section, the RBFNN switched observer will be constructed to solve the unmeasured state variables problem for system (11), and also its stability will be investigated. Design RBFNN switched observer as follows: ̂̇ { ̂̇
̂
̂
̂
̂ (̂ ̂ ̂ (̂ )
̂
)
( (
̂ ) ̂ ) (19)
where |̂ )
̂ (̂ ̂ ̂ (̂
|̂ )
̂ ̂
(̂ ̂ (̂
)
)
(20)
Based on Lemma 3, the unknown nonlinear functions in (12) can be approximated by the above RBFNNs. Define the optimal weight values and of RBFNNs: ̂
,
(̂ ̂
0
)
| ̂ (̂ ̂ | ̂ (̂
|̂ )
|̂ )
(̂ ̂ (̂
)|1
are bounded compact regions for ̂ and ( ̂ ̂
).
̂
where
0
(̂
)
)|1
,
are bounded compact sets for ̂ and
(̂
), respectively. The NN minimum approximation errors
and NN approximation errors
(̂ ̂
)
̂ (̂ ̂
| )
,
(̂ ̂
)
̂ (̂ ̂
|̂ )
,
(̂ (̂
are defined as: ̂ (̂
) )
|
) |̂ )
̂ (̂
(21) Denote | Rewrite (19) as:
|
̂̇
,
̂
̂
, by Assumption 4, there is an unknown constant
such that
.
| ̂ )-
̂ (̂ ̂
∑,
|̂ )
̂ (̂
(̂ )
̂
(22) ̂ be an observer error vector. Then, from (12), (21), and (22), one has
Let ,
̃ -
( )
(23)
where [ ] Consider the following Lyapunov candidate
: (24)
The time derivative of ̇
is as follows: ,
-
̃
*
+
(25)
According to Assumption 1, 2 and Lemma 1, one has , |
̃ |
‖ ‖
‖ ‖ ‖
‖
‖ ‖
‖ .|
‖
|
‖ |
‖ |̃ | | /
(26) ‖ ‖
‖
‖ ∑
‖ ‖
‖
‖ ∑ (27)
Based on Assumption 3, the following inequality hold: *
| ̅
+
̅ |
‖
‖
(28)
Substituting (26)-(28) into (25) yields to: ̇
‖ ‖ (
where (
and 4.
‖ )
‖ |̃ | ‖
(29) ‖ ∑
,
) is the smallest eigenvalue of matrix
‖
‖ ‖
‖
‖
‖ ∑
‖
‖
| ̅
̅ |
.
Adaptive neural network control design and stability analysis
In this section, with the help of the above constructed NN observer (19) and Butterworth LPF, the adaptive NN output-feedback controller design for switched stochastic nonlinear pure-feedback systems will be developed in the framework of backstepping technique. Moreover, the stability of the resulting closed-loop system will be analyzed on the basis of Theorem 1. The proposed control procedure will be detailed as follows.
Step 1: Define the following coordinate transformation: ̂
,
,
.
(30)
where denotes the tracking error and is the virtual control input which will be defined later. According to Itô’s differential rule (see [14], [25], [35], [36], and references therein), the differential of can be obtained as (
(̂ ̂ .
) ̂
( )
̇ ) (̂ ̂
)
̃ ̇
(̂ ̂
( )
)/
(31) Choose the Lyapunov function as follows: ̃ ̃
̅
̃
(32)
where and ̅ are design parameters. ̂ and ̂ are the estimates of ̃ ̂ ̂ and . By using (30) and (31), the derivative of can be expressed as follows: ̂
. ̃ ̃̇
̅
̃
(̂ ̂
)
̃ ̇
(̂ ̂
and
. Also
̃
)/
̃̇
(33)
Based on Assumptions 1-3 and Young’s inequalities, one has (
)
‖ ‖ |
(34) ‖ ‖
| | ̅
(35)
̅ |
(36)
By substituting (29), (34)-(36) into (33) and application of Lemma 2, we have
̂̇ )
̅
‖ ‖
.
̂
̃ ( ̅
. /
̇̂ )
(̂ ̂
)
̇
̂
‖
‖ |̃ |
| ̅
̅ |
̃ (
. //
(̂ ̂
)
(37) where
and
Define the first virtual control input ̇ ̂̇ ̇̂
and parameter of adaptive laws ̂̇ and ̇̂
(̂ ̂
)
̂
. /
as (38)
)
̂
(39)
. /
̅ ̂
(40)
(̂ ̂ ̅
̂
.
where , , and ̅ are positive design parameters. Insertion (38)-(40) into (37) yields to ‖ ‖ Step i (
̃ ̂
̅ ̅
̃
̂
‖
‖ |̃ | .
): Differentiation of both sides of (30) is described by (42)
(41)
̂̇ ̇
where ̅
(
̂
̂ )
( )
∑
̃
̅
0̂
(̂ ̂
(̂ ̂
)
∑
)
̂ ( (̂ ̂
.̂
(
̂̇
̂
( )
)1
̇̂
̂
∑
̇̂
̂
∑
̂
. (42)
̂̇
)/.
Define the following stochastic Lyapunov function: ̃ ̃
̃
̅
(43)
and ̅ are positive design constants. ̂ and ̂ are the estimates of ̂ . As done previously in (33) and according to Lemma 2, we have
where ̃
̅
[ ̃ (
(̂ ̂
̂̇ )
)
(
)
̃ ( ̅
̅
̂
̂̇ )
. /
. Also ̃
and
̂
. / (
and
]
)
(44)
By applying Lemma 1 and Assumptions 1-3, the following inequalities hold: (
)
.
/
.
/
.
Based on inequality
‖ ‖
‖
.
̅
̃ ̂
∑
. /
(45) /
.
|
/
|
(46)
and substituting (41), (45), and (46) into (44):
∑
̂
)‖ ‖
(
/
̅
.
̃
̅
∑ ̂
̅
̃ (
/ )
̃
̂
̅
(
(̂ ̂
̂̇ )
)
̅
. /.
̃ ( ̅
/ ̇̂ )
. /
‖ |̃ |
(47) (
where
)(
(
),
̂̇
(̂ ̂
̂̇ ̅
where
. /.
/
̂
|
/
and adaptive laws ̂̇ and ̂̇
The intermediate control function ̅
).
. /
|
∑
.
are chosen as follows
.
/
.
/
(48)
̂
) . /
(49)
̅ ̂
(50)
, and ̅ are positive design parameters. By inserting (48)-(50) into (47), we conclude
, , ‖ ‖
∑
̃ ̂
∑
̅ ̅
̃
̂
∑
̅ ̅
̃
̂
‖
‖ |̃ |
Step n: In the final step, the control input design process will be presented. The time derivative of ̂̇ ̇
0̂
̅
̃
(̂
)
(
)1
( )
(51) is (52)
where ̅
( ( )
∑
̂
̂ )
(̂
∑
)
̂ ( (̂ ̂
.̂
̂̇
̂
̂̇
̂
∑
̂̇
̂
∑
̂
̂̇
)/.
Consider the Lyapunov function candidate as ̃ ̃
̃
̃
̃
̅
(53)
where ̃ ̂ , ̂ is the estimate of ( ⁄ ), and , ̅ , , and Now, we choose the control input law and adaptive laws as follows:
are positive design constants.
̂ ̅
(54) ̅
̅
. /.
̂̇
/
̂
. /
.
/
.
/
(55)
̂
̂̇ ̅
(56)
̂
(57)
And also ̂̇ and ̂̇ ̂̇
(̂
̇̂
are defined as
)
̅
. /
̂
(58)
̅ ̂
(59)
where , , , , , , , and ̅ are positive design parameters. Substituting (54)-(59) and into time derivative of gets:
‖ ‖
∑
∑ (
where
̃ ̂
)(
̅ ̅
̃
∑ ̂ (
) and
̅ ̅
).
̃
̂
̃ ̂ ‖
̃ ̂ /
|
|
∑
‖ |̃ |
(60)
.
To discuss the stability of overall system, the following theorem is considered. Theorem 1: Consider the switched stochastic pure-feedback nonlinear system (1) with unknown input saturation under Assumption 1-4 and arbitrary deterministic switching signal, the proposed controller (54) with the NN switched observer (19), the intermediate virtual signals (38) and (48), and the adaptive laws (39), (40), (49), (50), and (55)-(59) can ensure that all the signals are semiglobally uniformly ultimately bounded in probability and the observer and tracking errors converge to a small neighborhood of the origin by suitable choice of the design parameters. Proof: Based on Young’s inequality, we have ̃ ̂
̃ (
̃)
̃ ̃
̃ ̂
̃ (
̃ )
̃
(62)
̃)
̃
(63)
̃ ̂ ̅ ̅
̃
̅ ̅
̃(
̃
̂
̂
̅
̃
̅ ̅ ̅
̃
̅ ̅ ̅ ̅
Invoking (61)-(65), we deduce
‖
‖
(61)
(64) (65)
‖ ‖
∑ ∑
where
‖
‖
̅
̃ ̂
∑
̃
̅
̅
̅
̅
̅
∑
̅ ̅
̃
‖
.
‖
/̃
̃
(66)
. {
To investigate the stability analysis, define
(
)
̅
(
‖ ‖
)
}
Thus, (67) can be expressed by (67) Applying Itô formula to
leads to
(
( )
(
) ( )
where
) ( )
∑
(68) ( )
(
∑
( )) .
Based on (67) and (68), we obtain (
( )
)
Integration of equation (69) over , ( )
( )
∫
(69) - is given by ( )
(70)
Now, by tacking expectation on (70), we have , ( )-
, ( )-
(71)
Based on (71) and the results reported by [9], [13], [36] and some references therein, it can be found that for ̂ ̂ , the signals ̂ ̂ ̂ , and ̂ are semiglobally uniformly ultimately bounded in probability. Moreover, by increasing (in fact, the design parameters ̅ , and ) and decreasing ̅ , and the observer errors and tracking errors can be converge to a small region. Despite these findings, if ̅ , and are chosen larger and becomes smaller, the control energy will be higher. Consequently, it is required to adjust the control parameters appropriately in practical applications to reach satisfactory transient performance and control action.
5.
Fig. 1. Switching signal 𝜎(𝑡) Simulation examples
Fig. 2. Trajectories of 𝑦 and 𝑦𝑑 in Example 1.
In this section, two numerical examples are considered to confirm the practical and effective application of our methodology.
Example 1 [46]: Consider the following switched stochastic nonlinear system in pure-feedback form:
{
(
)
(
( )
( )) , -
( )
))
(72)
) )) ( ), (( ) ( ) ( ) ( ), , ( ) ( ) ( )) . Furthermore, the output of saturation nonlinearity ( ) is obtained by ( ( ) (3) with the parameter The desired reference is defined as: ( ). In the simulation, ⁄ the Butterworth LPF is chosen as ( ) and all constructed basis functions ( ̂ ̂ ) and ( ̂ ) are the Gaussian functions with 7 and 20 nodes and centers and evenly spaced in , , - and , - , - , -, and widths , , respectively. Let set the parameters , for the following designed NN switched observer (19): ((
where
,
̂ ̂
(
(
̂ (̂ ̂ ) ̂ (̂ )
̂ ̂
)
(
( (
̂ ) ̂ )
(73)
Based on Theorem 1, choose the virtual control law, the actual control signal, and the adaptive laws for ( ) as ̂ ̇
(̂ ̂
)
̂
. /
̂ ̅ ̅ ̅
( )(
̂̇ ̂̇ ̂̇
)
̂
4 5 ̇̂
̂ , ̅
. / (̂ ̂
)
4
̅ ̂ ,
̂̇
̂,
̂̇
̅
5
(
)
̂
̅
. / (̂
)
̅ ̂ ̂
where and ̂ . The simulation is carried out with the control parameters as: , , , ̅ , ̅ , ̅ ̅ , , , ( ) , and the initial conditions are ( ) ̂ ( ) ̂ ( ) , ̂ ( ) ̂( ) , ̂ ( ) ̂ ̂ ̂ ( ) , ( ) , ( ) , under deterministic switching signal which is described in Fig. 1. Fig. 2 shows trajectories of the system output and the given reference signal , as it can be seen, the system output can track the desired reference signal as close as possible. The state signals and , and their estimates ̂ and ̂ are illustrated in Fig. 3. It is obvious that the state variables can be tracked with a small error. Fig.4 highlights the observer errors. The control signal and input saturation are indicated in Fig. 5 and Fig. 6, respectively. It is apparent from Fig. 2-5 that all signals of the resulting closed-loop system are bounded.
Fig. 4. Observer errors in Example 1.
Fig. 3. Trajectories of states and observer states in Example 1.
Example 2: Niu et al. in [46] has extensively explained the continuous stirred tank reactor system with two modes feed stream and modeled it as a switched stochastic pure-feedback nonlinear system which will be applied in this paper as a practical example with input saturation: ( {
)
( )
( )
(74) ,
where
(
-
)
,
(
)
,
(
)
(
)
,
( )
,
( )
, and the reference signal is ( ). The parameter of input saturation is chosen as . In this simulation, the parameters of the aforementioned observer, parameters design and initial conditions are selected as: , , , , , , ̅ ̅ , , , ̅ ̅ ,, ( ) , , , , , and the initial conditions are ( ) ̂ ( ) ̂ ( ) , ̂ ( ) ̂( ) , ̂ ( ) ̂ ( ) , ̂ ( ) , ̂ ( ) . The properties of - , - and , -, and designed basis functions are as: 8 and 5 nodes, centers and evenly spaced in , widths , , respectively. Similarly, according to these parameters and Theorem 1, the observer-based adaptive NN controller is designed to provide the simulation results in Fig. 7-10 under arbitrary deterministic switching signal, Fig. 1. As shown in Fig. 7, the system output signal follow the given reference signal . It is clear that the given reference signal can be tracked as closely as possible. The bounded state signals and , and the bounded observer state signals ̂ and ̂ are represented in Fig 8 which illustrates that the state variables can be observed effectively. In addition, Fig.9 shows the observer errors. Fig. 10 highlights the control signal. As demonstrated, the simulation results emphasize the validity of our approach so that the system output signal and the observer states can effectively follow the reference signal and state variables by choosing design parameters appropriately. Moreover, all signals, the observer errors and the tracking errors remain bounded. Remark 2: Since the above switched stochastic nonlinear pure feedback systems (72) and (74) involve the unknown input saturation and unmeasurable variable states, the existing control approaches for switched nonlinear stochastic pure feedback systems in ([46]-[49]) cannot be applied to control design problem of these systems. Also, the proposed output feedback control method is essentially the same as that used in [11], [13], [15], and [16] for nonlinear nonswitched pure feedback systems, but with some changes that will extend this control method to switched nonlinear systems with pure feedback form and unknown saturation input. In this way, the proposed method in this paper can simultaneously deal with the unmeasured states and unknown input saturation issues for switched nonlinear stochastic systems in pure feedback form.
6.
Conclusion
This paper analyzes the adaptive NN-based output-feedback control design issue for a class of switched stochastic nonlinear pure-feedback systems contain unknown saturation input and immeasurable state variables. To simplify the control procedure, input saturation is characterized by a dead zone-based model. In addition, integrating an NN switched observer and adaptive backstepping approach solves the adaptive output-feedback NN tracking control problem. In conclusion, it is demonstrated that the proposed controller can guarantee all signals in a closedloop system are semiglobally uniformly ultimately bounded in probability also the tracking and observer errors coverage to a neighborhood of the origin with appropriate choice of the design parameters. To future our research we intend to study adaptive NN output feedback control problem for switched stochastic large-scale nonlinear systems with input saturation.
Fig. 5. Control input 𝜐 in Example 1.
Fig. 6. Input saturation in Example 1.
Fig. 7. Trajectories of 𝑦 and 𝑦𝑑 in Example 2. Fig. 7. Trajectories of 𝑦 and 𝑦𝑑 in Example 2.
Fig. 8. Trajectories of states and observer states in Example 2.
Fig. 9. Observer errors in Example 2.
Fig. 10. input 𝜐 in Example 2.
Declaration of interests
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.
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Biography of author 1 Zahra Namadchian received the M.S. degree in control theory and control engineering from the Islamic Azad University, Gonabad branch, Gonabad, Iran, in 2012, where she is currently pursuing the Ph.D. degree with the department of Electrical Engineering. Her current research interests include adaptive control, neural network control, and nonlinear control system. Biography of author 2 Modjtaba Rouhani received the B.Sc. and M.Sc. degrees in electrical engineering from the Ferdowsi University of Mashhad, Mashhad, Iran, in 1992 and 1994, respectively, and the Ph.D. degree in automatic control and robotics from the Amirkabir University of Technology, Tehran, Iran, in 2002. From 2002 to 2014, he was with the Islamic Azad University of Gonabad, Gonabad, Iran. He is currently an Assistant Professor with the Department of Computer Engineering, Ferdowsi University of Mashhad. His current research interests include nonlinear control, neural networks, and computational intelligence.
Observer-Based Adaptive Neural Control for Switched Stochastic Pure-feedback Systems with Input Saturation Zahra Namadchian , Modjtaba Rouhani PII: DOI: Reference:
S0925-2312(19)31290-1 https://doi.org/10.1016/j.neucom.2019.09.028 NEUCOM 21283
To appear in:
Neurocomputing
Received date: Revised date: Accepted date:
28 January 2019 13 August 2019 3 September 2019
Please cite this article as: Zahra Namadchian , Modjtaba Rouhani , Observer-Based Adaptive Neural Control for Switched Stochastic Pure-feedback Systems with Input Saturation, Neurocomputing (2019), doi: https://doi.org/10.1016/j.neucom.2019.09.028
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Observer-Based Adaptive Neural Control for Switched Stochastic Pure-feedback Systems with Input Saturation Zahra Namadchiana, and Modjtaba Rouhanib,1 a
Department of Electrical Engineering, Islamic Azad University of Gonabad, Khorasan Razavi, Iran b Department of Computer Engineering, Ferdowsi University of Mashhad, Mashhad, Iran
Abstract The purpose of this paper is to investigate the problem of adaptive neural network (NN) output-feedback tracking control for input saturated switched stochastic nonlinear systems in pure-feedback form with unmeasured states. In order to facilitate the controller design process, a dead zone-based model of saturation is implemented and radial basis function NNs (RBFNNs) are employed to approximate unknown nonlinear functions and to construct an NN switched nonlinear observer to cope with difficulties raised by the presence of immeasurable state variables. Based on the adaptive backstepping technique and Lyapunov function method, an adaptive NN output feedback control scheme is developed. Furthermore, it is proved that the proposed controller can provide that, under arbitrary deterministic switching, all signals in the closed-loop system are semiglobally uniformly ultimately bounded in probability and the tracking error converges to a small neighborhood of the origin. Finally, simulation examples are presented to validate the effectiveness of the proposed adaptive NN control approach. Keywords: input saturation, Adaptive neural control, output-feedback control, switched nonlinear systems, stochastic purefeedback system.
1. Introduction In recent years, the stability analysis and observer-based control design problems for nonlinear systems with unmeasured states have received much attention, since in many practical nonlinear systems (such as power industry, aerospace, chemical engineering applications and thermal processes [1]) the information of state variables are often unavailable. Recent studies highlight that approximation-based adaptive backstepping output-feedback control approach has great potential to cope with immeasurable states difficulty in which different kinds of observers (i.e. linear state observer, k-filters state observer, input-driven observer, and fuzzy/NN nonlinear state observers) have been successfully designed to estimate unavailable state variables. For example, based on a high-gain observer, an adaptive NN output feedback controller was designed for the robotic manipulators and the flapping wing micro aerial vehicle (FWMAV) in [2] and [3], respectively. Regarding this concern, the control problem of nonlinear stochastic systems could be an important issue, both in practice and theory [4]-[8]. For example, With the help of a fuzzy observer and adaptive backstepping technique with dynamic surface control (DSC), an observer-based adaptive fuzzy controller for stochastic nonlinear systems in nonstrict-feedback form has been constructed in [5]. An approximation-based adaptive output-feedback control scheme for strict-feedback nonlinear systems with stochastic disturbances, unmodeled dynamics, and unknown dead zone input has been investigated in [7]. Based on fuzzy logic 1
Corresponding author: Modjtaba Rouhani (e-mail: [email protected]).
system and the Nussbaum gain function in [8], an adaptive fuzzy output-feedback prescribed performance control (PPC) problem for a class of multi-input and multi- output (MIMO) stochastic nonlinear nonstrict-feedback systems with unknown dead zone and unknown control direction was solved. As is well known, pure-feedback nonlinear systems have no affine appearance of the state variables to be used as virtual controls or actual control [9]. In addition, without the direct measurements of state variables, the control design of nonlinear pure-feedback systems has become more and more complicated, some results have been proposed in [9]-[17]. By applying DSC approach and RBFNNs, Li et al. [9] developed an adaptive NN output-feedback tracking control scheme for MIMO purefeedback nonlinear systems with backlash-like hysteresis and unmeasured states. Yang et al. [10] focused on decentralized adaptive NNs fault-tolerant output-feedback control design for a class of large-scale MIMO nonlinear nonaffine pure-feedback systems with actuator failures. Also, an adaptive fuzzy backstepping output-feedback control problem in [13] and [15] has been studied for uncertain pure-feedback nonlinear systems with immeasurable states. To overcome algebraic loop and immeasurable states problems, Tong et al. [14] applied filtered signals and a fuzzy state observer to propose an adaptive fuzzy output-feedback fault-tolerant control approach for a class of MIMO stochastic pure-feedback nonlinear systems. Based on the backstepping technique and a constructed fuzzy state observer, an adaptive fuzzy output-feedback control approach for a class of stochastic nonlinear systems in pure-feedback form has been investigated in [11] and [16]. On the other hand, in the last few years, a growing body of research has developed the adaptive output-feedback control approaches for nonlinear switched systems [18]-[27]. For example, the problem of adaptive output-feedback fault-tolerant tracking control for nonlinear switched large-scale systems in pure-feedback form with unknown dead zones has been described by [18]. By employing the average dwell time approach and Nussbaum gain function, a new observer-based fuzzy adaptive backstepping control algorithm has been explained for the switched MIMO nonlinear system in non-triangular form under considerations contain unknown backlash hysteresis and unknown control directions [20]. Tong et al. [23] studied an adaptive fuzzy decentralized control design for a class of switched nonlinear large-scale systems with dead zones and immeasurable states. To overcome the unmeasured states difficulty in nonstrict-feedback switched nonlinear systems with dead zone input, Tong et al. [24] constructed a switched fuzzy state observer to present an adaptive fuzzy output-feedback tracking control. An adaptive fuzzy output-feedback partial tracking errors constrained problem has been considered in [25] for switched stochastic nonlinear systems with average dwell time switching. Huo et al. [27] studied an observer-based fuzzy adaptive stabilization problem of uncertain switched stochastic nonlinear systems with unmeasured states and quantized input signals quantized by a hysteretic quantizer. The most important non-smooth nonlinearity, input saturation, often occurs in practical engineering and is the main limitation of the system performance and even could be the leading cause of instability. Therefore, in the field of control problem, the design and development of practical controllers for systems with saturation input is a serious challenge. Moreover, there is still considerable concern about the stability and control problems of nonlinear systems with saturation nonlinearity and unmeasured states which has been considered in recent research [28]-[36]. Sui et al. [36] used Butterworth low-pass filter and adaptive backstepping techniques to design the adaptive fuzzy output-feedback tracking controller and the fuzzy state observer for MIMO stochastic nonlinear system in purefeedback form with input saturation. The problem of adaptive fuzzy output-feedback control for a class of nonlinear strict-feedback stochastic systems with saturation nonlinearity and output constraint was investigated by Xu et al. in [28]. Yang et al. [34] focused on decentralized adaptive NN output-feedback control design for a class of large-scale time-delay systems in pure-feedback form with saturating inputs, based on DSC technique and NN approximator. Quite recently, few publications have addressed the control problem of nonlinear switched systems with input saturation [37]-[42]. Yu et al. [40] proposed an adaptive quantized tracking scheme for a class of switched stochastic strict-feedback nonlinear systems with asymmetrical input saturation. Furthermore, based on an inputdriven observer, NN approximation ability, and the backstepping technique, an adaptive output-feedback tracking control problem of switched nonstrict feedback nonlinear systems with unknown control direction and asymmetric saturation has been investigated in [43]. By employing MT-filters and RBFNNs, Li et al. [44] presented an observer-based NN adaptive PPC for nonlinear switched systems with input saturation. Motivated by the aforementioned discussions, this paper investigates the observer-based NN output-feedback controller design for a class of switched stochastic nonlinear pure-feedback systems under arbitrary switching signal while both unmeasured states and input saturation are simultaneously considered. With the help of presented method in ([11], [13], [15], and [16]), the problem of switched pure feedback form is solved and an NN switched observer will be successfully designed on the basis of universal approximation ability of RBFNNs for input saturated
systems. Afterward, an adaptive NN output-feedback control scheme is proposed in the framework of adaptive backstepping method and Lyapunov stability theory that ensures all signals in a closed-loop system are semiglobally uniformly ultimately bounded in probability. The main advantages of this paper are summarized as follows. 1) The proposed control method can deal with the output-feedback tracking control problem of switched stochastic nonlinear pure-feedback systems with unknown input saturation. The presented procedure is practically the same as the ones explained in ([11], [13], [15], and [16]) and also is more or less identical to the studies done in ([9], [10], [12], [14], and [17]). Despite these similarities, these works were restricted to nonswitched pure-feedback systems or nonswitched stochastic systems in pure-feedback form without input nonlinearities (except [9] with input hysteresis). Therefore, the tracking control investigation of nonlinear switched stochastic pure-feedback systems with unmeasured state variables and unknown saturation problems simultaneously will become a challenging issue. 2) Compared with the previous works ([28]-[36]) which only studied the adaptive output-feedback control for nonswitched nonlinear systems with known input saturation, in this paper, the proposed control approach has been applied to a class of switched nonlinear systems with unknown saturation. The chosen model of input saturation, dead zone-based model, is the same as the one proposed in [45] which is applied to reduce the control process complexity, while has beneficial features to model diverse kinds of saturation and adjusts the balance between modeling accuracy and computation. In addition, unlike the result in [37]-[40], no knowledge about the upper and lower bounds of unknown functions of input saturation is needed for the control design. 3) In contrast to studies ([37]-[42] and [46]-[49]) that the states of switched systems with input saturation and switched nonlinear stochastic systems in pure-feedback form are required to be all available for measurements, respectively, the presented adaptive control approach based on the obtained NN switched observer can cancel this condition. In addition, to deal with control problem of switched pure-feedback nonlinear systems, the Butterworth low-pass filter technique is utilized to avoid the algebraic loop issue and eliminate the existence of known bounds and the known signs of nonlinear functions assumptions. This paper is divided into six sections. Section 2 gives preliminaries and problem formulation. In Section 3, the NN switched observer is designed. Section 4 analyzes the stability and adaptive output-feedback control problems for the suggested system. Simulation examples are given in Section 5. Finally, the conclusions are drawn in Section 6. 2.
Preliminaries and problem formulation
2.1 System Information and Assumption Consider a class of switched stochastic pure-feedback nonlinear systems given as follows: ( )
{
(
( )
(
) )
( ) ( )
( )
( )
(1)
, , ), where ,( , and are the state vector and output of the system, respectively. are unmeasurable, the only output variable is measurable. * + is the arbitrary deterministic switching signal, which takes value in the compact ( ) , ) set . If ( ) , kth subsystem will be active. For any and , () and () denote the unknown smooth nonlinear functions. is an r-dimensional independent standard Brownian motion defined on the complete probability space, with the incremental covariance { } ( ) ( ) . presents the saturation nonlinearity to be described as follows:
( )
{ ( )
( ) ( ) ( )
where and are the unknown upper and lower bounded parameters of the input saturation, . As shown in (2), there are two sharp corners when and . So the backstepping technique
(2)
cannot be directly utilized to design control input signal. As considered in [45], the dead zone-based model of saturation will be employed to tackle this problem and defined as: ( ) ( )
(3)
( )
∫
( )
(4)
where ∫ ( ) is an unknown constant, in which ( ) is supposed to be unknown. ( ) is a density function satisfying ( ) , . When , ( ) . ( ) is the dead-zone operator defined by ( ) ( ( )) Remark 1: To model a saturation function, different types of density functions, can be chosen as follows ( )
{
(5)
Moreover, the saturation calculated by [45]:
accesses the saturated region on
. From (3), we obtain the saturated value
∫
( ) (6)
( ) ( ) Let ( ) ( ) ,( ) and . Then, based on (3), system (1) is equivalent to the following switched stochastic nonlinear pure-feedback system:
{
.
(
.
( )
( )
)/
( ))/
(
( ) (7)
Let define: (
)
(
(̂ ̂
)
(̂
)
)
(
( ))
́
(8) (
) (9)
where and
́ (̂ ) (̂ ) which will be defined by state observer designed later. ̂
̂
( )̂
(
)
́
(
)
( ),
( ) ; ̂ and ̂ are the estimates of state vectors and are the filtered signals defined by:
( )
(10)
̂,
where ( ) is a Butterworth low-pass filter (LPF). The parameters of Butterworth LPF can be obtained in [36]. Then, system (7) can be rewritten as: . {
where
.
(̂ ̂ (̂
)/ )/
( ) ( )
́ (̂ ) (̂ ) ( ). Now, we can express (11) in the following form:
(11)
,
∑,
(̂ ̂
)-
(
(̂
)
)
( )
-
(12) [
where
[⏟
],
⏟
] ,
[
,
],
[ ],
[
( )
],
,
-,
-.
Furthermore, for and , with appropriate choice of coefficients , one can be assured that a strict Hurwitz matrix. Thus, for any given matrix , there exists a positive definite matrix satisfying
is
(13) Before presenting some definitions, lemmas, and assumptions to facilitate the adaptive control design, we first consider the following stochastic nonlinear system: ( )
( )
(14)
( ) where ( ) and ( ) are locally Lipschitz functions in and satisfy ( ) . Definition 1([4], [8], and [37]): For any function ( ), related with (14), define the differential operator as follows ( )
( )
2 ( )
( ) is the trace of
where
( )
( )3
and the term
(15) 2 ( )
( )
( )3 is called Itô correction term, in which the
( )
second-order differential makes the controller design much more difficult than that of the deterministic system. + of the stochastic system (14) is said to be semiglobally Definition 2 ([28] and [40]): The trajectory * ( ) uniformly ultimately bounded in the th moment, if for some compact set and any initial state ( ), there exist a constant and a time constant ( ) such that (| ( )| ) for all . Especially, when , it is usually called semiglobally uniformly ultimately bounded in mean square. Assumption 1 ([25], [36], [43]): There exists a set of known constant , for , such that the following inequality holds: |
(
)
(
)|
‖
‖
Assumption 2 ([9], [12], [13], and [36]): There exist unknown constants ) and | | . Assumption 3 ([14], [25], [36]): The disturbance covariance [
̅
such that | ̂
| |
|
,(
|
,
̅ is bounded, where
] . Assumption 4 ([9], [13]): There exist unknown constants and such that | | . Lemma 1 (Young’s inequality [5]): The following inequality holds for ( ) : | |
̂
and |
(16)
where , , , and ( Lemma 2 ([13], [43]): For any
)( and
)
. ,|
|
(
⁄)
is satisfied.
2.2 RBF neural network The radial basis function neural networks (RBFNNs) are usually used to approximate unknown smooth nonlinear functions due to their simple structure and good capabilities in function approximation. Therefore, in this paper, The RBFNNs are employed to approximate unknown smooth nonlinear functions ( )( ) over a compact set . The RBFNNs can be described as ( ) where , and are the weight vector ( )and the number of NN nodes, respectively. ( ) , ( ) is the vector of known basis function and ( ) have the following form: (
( )
) (
)
(17)
where [ ] is the center of the receptive field and is the width of the Gaussian function. Based on the results of [50], with sufficiently large , the RBFNNs can approximate any continuous function to any desired accuracy over a compact set . Lemma 3([49]): For any accuracy , there exists an NN ( ) such that | ( )
( )|
(18)
The objective of this paper is to design an adaptive NN output-feedback controller and adaptive laws for system (1) in presence of input saturation such that all signals in the closed-loop remain bounded in probability under arbitrary deterministic switching signal. Furthermore, the output of system tracks a desired reference signal in the mean square sense.
Neural network switched observer design
3.
Throughout this section, the RBFNN switched observer will be constructed to solve the unmeasured state variables problem for system (11), and also its stability will be investigated. Design RBFNN switched observer as follows: ̂̇ { ̂̇
̂
̂
̂
̂ (̂ ̂ ̂ (̂ )
̂
)
( (
̂ ) ̂ ) (19)
where |̂ )
̂ (̂ ̂ ̂ (̂
|̂ )
̂ ̂
(̂ ̂ (̂
)
)
(20)
Based on Lemma 3, the unknown nonlinear functions in (12) can be approximated by the above RBFNNs. Define the optimal weight values and of RBFNNs: ̂
,
(̂ ̂
0
)
| ̂ (̂ ̂ | ̂ (̂
|̂ )
|̂ )
(̂ ̂ (̂
)|1
are bounded compact regions for ̂ and ( ̂ ̂
).
̂
where
0
(̂
)
)|1
,
are bounded compact sets for ̂ and
(̂
), respectively. The NN minimum approximation errors
and NN approximation errors
(̂ ̂
)
̂ (̂ ̂
| )
,
(̂ ̂
)
̂ (̂ ̂
|̂ )
,
(̂ (̂
are defined as: ̂ (̂
) )
|
) |̂ )
̂ (̂
(21) Denote | Rewrite (19) as:
|
̂̇
,
̂
̂
, by Assumption 4, there is an unknown constant
such that
.
| ̂ )-
̂ (̂ ̂
∑,
|̂ )
̂ (̂
(̂ )
̂
(22) ̂ be an observer error vector. Then, from (12), (21), and (22), one has
Let ,
̃ -
( )
(23)
where [ ] Consider the following Lyapunov candidate
: (24)
The time derivative of ̇
is as follows: ,
-
̃
*
+
(25)
According to Assumption 1, 2 and Lemma 1, one has , |
̃ |
‖ ‖
‖ ‖ ‖
‖
‖ ‖
‖ .|
‖
|
‖ |
‖ |̃ | | /
(26) ‖ ‖
‖
‖ ∑
‖ ‖
‖
‖ ∑ (27)
Based on Assumption 3, the following inequality hold: *
| ̅
+
̅ |
‖
‖
(28)
Substituting (26)-(28) into (25) yields to: ̇
‖ ‖ (
where (
and 4.
‖ )
‖ |̃ | ‖
(29) ‖ ∑
,
) is the smallest eigenvalue of matrix
‖
‖ ‖
‖
‖
‖ ∑
‖
‖
| ̅
̅ |
.
Adaptive neural network control design and stability analysis
In this section, with the help of the above constructed NN observer (19) and Butterworth LPF, the adaptive NN output-feedback controller design for switched stochastic nonlinear pure-feedback systems will be developed in the framework of backstepping technique. Moreover, the stability of the resulting closed-loop system will be analyzed on the basis of Theorem 1. The proposed control procedure will be detailed as follows.
Step 1: Define the following coordinate transformation: ̂
,
,
.
(30)
where denotes the tracking error and is the virtual control input which will be defined later. According to Itô’s differential rule (see [14], [25], [35], [36], and references therein), the differential of can be obtained as (
(̂ ̂ .
) ̂
( )
̇ ) (̂ ̂
)
̃ ̇
(̂ ̂
( )
)/
(31) Choose the Lyapunov function as follows: ̃ ̃
̅
̃
(32)
where and ̅ are design parameters. ̂ and ̂ are the estimates of ̃ ̂ ̂ and . By using (30) and (31), the derivative of can be expressed as follows: ̂
. ̃ ̃̇
̅
̃
(̂ ̂
)
̃ ̇
(̂ ̂
and
. Also
̃
)/
̃̇
(33)
Based on Assumptions 1-3 and Young’s inequalities, one has (
)
‖ ‖ |
(34) ‖ ‖
| | ̅
(35)
̅ |
(36)
By substituting (29), (34)-(36) into (33) and application of Lemma 2, we have
̂̇ )
̅
‖ ‖
.
̂
̃ ( ̅
. /
̇̂ )
(̂ ̂
)
̇
̂
‖
‖ |̃ |
| ̅
̅ |
̃ (
. //
(̂ ̂
)
(37) where
and
Define the first virtual control input ̇ ̂̇ ̇̂
and parameter of adaptive laws ̂̇ and ̇̂
(̂ ̂
)
̂
. /
as (38)
)
̂
(39)
. /
̅ ̂
(40)
(̂ ̂ ̅
̂
.
where , , and ̅ are positive design parameters. Insertion (38)-(40) into (37) yields to ‖ ‖ Step i (
̃ ̂
̅ ̅
̃
̂
‖
‖ |̃ | .
): Differentiation of both sides of (30) is described by (42)
(41)
̂̇ ̇
where ̅
(
̂
̂ )
( )
∑
̃
̅
0̂
(̂ ̂
(̂ ̂
)
∑
)
̂ ( (̂ ̂
.̂
(
̂̇
̂
( )
)1
̇̂
̂
∑
̇̂
̂
∑
̂
. (42)
̂̇
)/.
Define the following stochastic Lyapunov function: ̃ ̃
̃
̅
(43)
and ̅ are positive design constants. ̂ and ̂ are the estimates of ̂ . As done previously in (33) and according to Lemma 2, we have
where ̃
̅
[ ̃ (
(̂ ̂
̂̇ )
)
(
)
̃ ( ̅
̅
̂
̂̇ )
. /
. Also ̃
and
̂
. / (
and
]
)
(44)
By applying Lemma 1 and Assumptions 1-3, the following inequalities hold: (
)
.
/
.
/
.
Based on inequality
‖ ‖
‖
.
̅
̃ ̂
∑
. /
(45) /
.
|
/
|
(46)
and substituting (41), (45), and (46) into (44):
∑
̂
)‖ ‖
(
/
̅
.
̃
̅
∑ ̂
̅
̃ (
/ )
̃
̂
̅
(
(̂ ̂
̂̇ )
)
̅
. /.
̃ ( ̅
/ ̇̂ )
. /
‖ |̃ |
(47) (
where
)(
(
),
̂̇
(̂ ̂
̂̇ ̅
where
. /.
/
̂
|
/
and adaptive laws ̂̇ and ̂̇
The intermediate control function ̅
).
. /
|
∑
.
are chosen as follows
.
/
.
/
(48)
̂
) . /
(49)
̅ ̂
(50)
, and ̅ are positive design parameters. By inserting (48)-(50) into (47), we conclude
, , ‖ ‖
∑
̃ ̂
∑
̅ ̅
̃
̂
∑
̅ ̅
̃
̂
‖
‖ |̃ |
Step n: In the final step, the control input design process will be presented. The time derivative of ̂̇ ̇
0̂
̅
̃
(̂
)
(
)1
( )
(51) is (52)
where ̅
( ( )
∑
̂
̂ )
(̂
∑
)
̂ ( (̂ ̂
.̂
̂̇
̂
̂̇
̂
∑
̂̇
̂
∑
̂
̂̇
)/.
Consider the Lyapunov function candidate as ̃ ̃
̃
̃
̃
̅
(53)
where ̃ ̂ , ̂ is the estimate of ( ⁄ ), and , ̅ , , and Now, we choose the control input law and adaptive laws as follows:
are positive design constants.
̂ ̅
(54) ̅
̅
. /.
̂̇
/
̂
. /
.
/
.
/
(55)
̂
̂̇ ̅
(56)
̂
(57)
And also ̂̇ and ̂̇ ̂̇
(̂
̇̂
are defined as
)
̅
. /
̂
(58)
̅ ̂
(59)
where , , , , , , , and ̅ are positive design parameters. Substituting (54)-(59) and into time derivative of gets:
‖ ‖
∑
∑ (
where
̃ ̂
)(
̅ ̅
̃
∑ ̂ (
) and
̅ ̅
).
̃
̂
̃ ̂ ‖
̃ ̂ /
|
|
∑
‖ |̃ |
(60)
.
To discuss the stability of overall system, the following theorem is considered. Theorem 1: Consider the switched stochastic pure-feedback nonlinear system (1) with unknown input saturation under Assumption 1-4 and arbitrary deterministic switching signal, the proposed controller (54) with the NN switched observer (19), the intermediate virtual signals (38) and (48), and the adaptive laws (39), (40), (49), (50), and (55)-(59) can ensure that all the signals are semiglobally uniformly ultimately bounded in probability and the observer and tracking errors converge to a small neighborhood of the origin by suitable choice of the design parameters. Proof: Based on Young’s inequality, we have ̃ ̂
̃ (
̃)
̃ ̃
̃ ̂
̃ (
̃ )
̃
(62)
̃)
̃
(63)
̃ ̂ ̅ ̅
̃
̅ ̅
̃(
̃
̂
̂
̅
̃
̅ ̅ ̅
̃
̅ ̅ ̅ ̅
Invoking (61)-(65), we deduce
‖
‖
(61)
(64) (65)
‖ ‖
∑ ∑
where
‖
‖
̅
̃ ̂
∑
̃
̅
̅
̅
̅
̅
∑
̅ ̅
̃
‖
.
‖
/̃
̃
(66)
. {
To investigate the stability analysis, define
(
)
̅
(
‖ ‖
)
}
Thus, (67) can be expressed by (67) Applying Itô formula to
leads to
(
( )
(
) ( )
where
) ( )
∑
(68) ( )
(
∑
( )) .
Based on (67) and (68), we obtain (
( )
)
Integration of equation (69) over , ( )
( )
∫
(69) - is given by ( )
(70)
Now, by tacking expectation on (70), we have , ( )-
, ( )-
(71)
Based on (71) and the results reported by [9], [13], [36] and some references therein, it can be found that for ̂ ̂ , the signals ̂ ̂ ̂ , and ̂ are semiglobally uniformly ultimately bounded in probability. Moreover, by increasing (in fact, the design parameters ̅ , and ) and decreasing ̅ , and the observer errors and tracking errors can be converge to a small region. Despite these findings, if ̅ , and are chosen larger and becomes smaller, the control energy will be higher. Consequently, it is required to adjust the control parameters appropriately in practical applications to reach satisfactory transient performance and control action.
5.
Fig. 1. Switching signal 𝜎(𝑡) Simulation examples
Fig. 2. Trajectories of 𝑦 and 𝑦𝑑 in Example 1.
In this section, two numerical examples are considered to confirm the practical and effective application of our methodology.
Example 1 [46]: Consider the following switched stochastic nonlinear system in pure-feedback form:
{
(
)
(
( )
( )) , -
( )
))
(72)
) )) ( ), (( ) ( ) ( ) ( ), , ( ) ( ) ( )) . Furthermore, the output of saturation nonlinearity ( ) is obtained by ( ( ) (3) with the parameter The desired reference is defined as: ( ). In the simulation, ⁄ the Butterworth LPF is chosen as ( ) and all constructed basis functions ( ̂ ̂ ) and ( ̂ ) are the Gaussian functions with 7 and 20 nodes and centers and evenly spaced in , , - and , - , - , -, and widths , , respectively. Let set the parameters , for the following designed NN switched observer (19): ((
where
,
̂ ̂
(
(
̂ (̂ ̂ ) ̂ (̂ )
̂ ̂
)
(
( (
̂ ) ̂ )
(73)
Based on Theorem 1, choose the virtual control law, the actual control signal, and the adaptive laws for ( ) as ̂ ̇
(̂ ̂
)
̂
. /
̂ ̅ ̅ ̅
( )(
̂̇ ̂̇ ̂̇
)
̂
4 5 ̇̂
̂ , ̅
. / (̂ ̂
)
4
̅ ̂ ,
̂̇
̂,
̂̇
̅
5
(
)
̂
̅
. / (̂
)
̅ ̂ ̂
where and ̂ . The simulation is carried out with the control parameters as: , , , ̅ , ̅ , ̅ ̅ , , , ( ) , and the initial conditions are ( ) ̂ ( ) ̂ ( ) , ̂ ( ) ̂( ) , ̂ ( ) ̂ ̂ ̂ ( ) , ( ) , ( ) , under deterministic switching signal which is described in Fig. 1. Fig. 2 shows trajectories of the system output and the given reference signal , as it can be seen, the system output can track the desired reference signal as close as possible. The state signals and , and their estimates ̂ and ̂ are illustrated in Fig. 3. It is obvious that the state variables can be tracked with a small error. Fig.4 highlights the observer errors. The control signal and input saturation are indicated in Fig. 5 and Fig. 6, respectively. It is apparent from Fig. 2-5 that all signals of the resulting closed-loop system are bounded.
Fig. 4. Observer errors in Example 1.
Fig. 3. Trajectories of states and observer states in Example 1.
Example 2: Niu et al. in [46] has extensively explained the continuous stirred tank reactor system with two modes feed stream and modeled it as a switched stochastic pure-feedback nonlinear system which will be applied in this paper as a practical example with input saturation: ( {
)
( )
( )
(74) ,
where
(
-
)
,
(
)
,
(
)
(
)
,
( )
,
( )
, and the reference signal is ( ). The parameter of input saturation is chosen as . In this simulation, the parameters of the aforementioned observer, parameters design and initial conditions are selected as: , , , , , , ̅ ̅ , , , ̅ ̅ ,, ( ) , , , , , and the initial conditions are ( ) ̂ ( ) ̂ ( ) , ̂ ( ) ̂( ) , ̂ ( ) ̂ ( ) , ̂ ( ) , ̂ ( ) . The properties of - , - and , -, and designed basis functions are as: 8 and 5 nodes, centers and evenly spaced in , widths , , respectively. Similarly, according to these parameters and Theorem 1, the observer-based adaptive NN controller is designed to provide the simulation results in Fig. 7-10 under arbitrary deterministic switching signal, Fig. 1. As shown in Fig. 7, the system output signal follow the given reference signal . It is clear that the given reference signal can be tracked as closely as possible. The bounded state signals and , and the bounded observer state signals ̂ and ̂ are represented in Fig 8 which illustrates that the state variables can be observed effectively. In addition, Fig.9 shows the observer errors. Fig. 10 highlights the control signal. As demonstrated, the simulation results emphasize the validity of our approach so that the system output signal and the observer states can effectively follow the reference signal and state variables by choosing design parameters appropriately. Moreover, all signals, the observer errors and the tracking errors remain bounded. Remark 2: Since the above switched stochastic nonlinear pure feedback systems (72) and (74) involve the unknown input saturation and unmeasurable variable states, the existing control approaches for switched nonlinear stochastic pure feedback systems in ([46]-[49]) cannot be applied to control design problem of these systems. Also, the proposed output feedback control method is essentially the same as that used in [11], [13], [15], and [16] for nonlinear nonswitched pure feedback systems, but with some changes that will extend this control method to switched nonlinear systems with pure feedback form and unknown saturation input. In this way, the proposed method in this paper can simultaneously deal with the unmeasured states and unknown input saturation issues for switched nonlinear stochastic systems in pure feedback form.
6.
Conclusion
This paper analyzes the adaptive NN-based output-feedback control design issue for a class of switched stochastic nonlinear pure-feedback systems contain unknown saturation input and immeasurable state variables. To simplify the control procedure, input saturation is characterized by a dead zone-based model. In addition, integrating an NN switched observer and adaptive backstepping approach solves the adaptive output-feedback NN tracking control problem. In conclusion, it is demonstrated that the proposed controller can guarantee all signals in a closedloop system are semiglobally uniformly ultimately bounded in probability also the tracking and observer errors coverage to a neighborhood of the origin with appropriate choice of the design parameters. To future our research we intend to study adaptive NN output feedback control problem for switched stochastic large-scale nonlinear systems with input saturation.
Fig. 5. Control input 𝜐 in Example 1.
Fig. 6. Input saturation in Example 1.
Fig. 7. Trajectories of 𝑦 and 𝑦𝑑 in Example 2. Fig. 7. Trajectories of 𝑦 and 𝑦𝑑 in Example 2.
Fig. 8. Trajectories of states and observer states in Example 2.
Fig. 9. Observer errors in Example 2.
Fig. 10. input 𝜐 in Example 2.
Declaration of interests
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.
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Biography of author 1 Zahra Namadchian received the M.S. degree in control theory and control engineering from the Islamic Azad University, Gonabad branch, Gonabad, Iran, in 2012, where she is currently pursuing the Ph.D. degree with the department of Electrical Engineering. Her current research interests include adaptive control, neural network control, and nonlinear control system. Biography of author 2 Modjtaba Rouhani received the B.Sc. and M.Sc. degrees in electrical engineering from the Ferdowsi University of Mashhad, Mashhad, Iran, in 1992 and 1994, respectively, and the Ph.D. degree in automatic control and robotics from the Amirkabir University of Technology, Tehran, Iran, in 2002. From 2002 to 2014, he was with the Islamic Azad University of Gonabad, Gonabad, Iran. He is currently an Assistant Professor with the Department of Computer Engineering, Ferdowsi University of Mashhad. His current research interests include nonlinear control, neural networks, and computational intelligence.