Data-Driven Adaptive Sliding Mode Control of Nonlinear ... - IEEE Xplore

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS: SYSTEMS

1

Data-Driven Adaptive Sliding Mode Control of Nonlinear Discrete-Time Systems With Prescribed Performance Dong Liu and Guang-Hong Yang , Senior Member, IEEE

Abstract—This paper deals with the issue of data-driven adaptive sliding mode control for a family of discrete-time nonlinear processes with tracking error constraint. More specifically, a novel transformed error algorithm together with a new sliding mode control framework is investigated to ensure the tracking error converges to a predefined zone all the time. Moreover, the proposed controller can guarantee a convergence rate and steady-state behavior for the tracking error depending only on the input/output measurement data, which is more effective in the complex industrial processes. Simulations are given to validate the theoretical results. Index Terms—Model-free adaptive control, prescribed performance control (PPC), sliding mode control.

I. I NTRODUCTION ODERN control theory contains various control design methods, such as robust control, zero-pole assignment, optimal control, and so on. Most of the considering approaches can be seen as typical model-based control methodologies where a priori quantitative or qualitative knowledge about the process is employed to design a controller [1]–[5]. These approaches are extensively used and acquire remarkable results in many industrial processes. However, in some cases, it is hard to identify the accurate model of a nonlinear plant. Hence, the data-driven method becomes an alternative way to attain control objectives, which does not need the accurate knowledge of the plant and depends merely on the input/output (I/O) data [6]–[10].

M

Manuscript received July 14, 2017; revised September 17, 2017 and November 4, 2017; accepted November 28, 2017. This work was supported in part by the National Natural Science Foundation of China under Grant 61621004 and Grant 61420106016, in part by the Fundamental Research Funds for the Central Universities under Grant N130604005, and in part by the Research Fund of State Key Laboratory of Synthetical Automation for Process Industries under Grant 2013ZCX01. This paper was recommended by Associate Editor M. Dotoli. (Corresponding author: Guang-Hong Yang.) D. Liu is with the College of Information Science and Engineering, Northeastern University, Shenyang 110819, China (e-mail: [email protected]). G.-H. Yang is with the College of Information Science and Engineering, Northeastern University, Shenyang 110819, China, and also with the State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang 110819, China (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TSMC.2017.2779564

Recently, data-driven algorithms have attracted many attention. The data-driven approaches have been applied to wide practical areas, such as three tank water systems, wood/berry distillation column, and steam-water heat exchanger systems. Among these frameworks, the model-free adaptive control has drawn much attention since it is using only the on-line I/O data of the controlled process and without using explicit or implicit information of the controlled system. Model-free adaptive control was first initiated by Hou [11]. It is an effective work for discrete-time nonlinear systems with unknown dynamics. The extensions were given by [12] and [13]. Dynamic linearization techniques were further explored to give different looks in [14]. Neural network learning adaptive control methods had been reported in several papers, such as [15]–[17]. On the other hand, adaptive observer-based control schemes were studied in [18] and [19]. The multidegree-of-freedom robotic exoskeletons were considered with general sliding mode adaptive control method in [20]. Furthermore, the event-based observer design and event-triggered controller design were considered in [21] and [22], respectively. On another research forefront, a novel method named prescribed performance control (PPC) was studied in [23] for the first time. One of the main characteristics of PPC, for which it has gained so wide acceptance, is that it can ensure the tracking error within a small enough of the domain by appropriately defining the tracking error transformation. This problem is solved by constructing unconstrained equivalent transformed error instead of commonly constrained tracking error. Subsequently, the controller is designed to stabilize the unconstrained transformed error. Up to now, there has been numerous literature studying the problem of PPC in continuous-time case. It is of great practical application to investigate the PPC method since the practical engineering systems often suffer from the constraints in some ways, for example, the steady-state error, the overshoot, and the convergence rate performance. These constraints are not only originating from system characteristics but also from security considerations. Consequently, PPC has become one of the hot technical issues in control theory. In [24], adaptive control problem was studied for strict feedback single-input–single-output (SISO) nonlinear processes with prescribed performance bounds. Moreover, adaptive fuzzy tracking control framework was discussed for nonlinear systems with a prescribed performance in [25]. In addition, a

c 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. 2168-2216 See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 2

IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS: SYSTEMS

low-complexity adaptive output feedback control framework was given in [26]. However, the major PPC results are modelbased [27]–[36]. The plant structures or dynamics information were usually have to be known (either explicitly or implicitly) in [23]–[25]. In many practical applications, the accurate model cannot always be obtained. Consequently, model-free continuous control law was developed to guarantee tracking error constraint [37], [38]. To the best of our knowledge, no study has yet been considered on the data-driven adaptive sliding mode control for nonlinear discrete-time processes with prescribed performance function. Inspired by the above discussions, this paper further considers the data-driven adaptive sliding control problem for a family of discrete-time nonlinear processes with tracking error constraint. The works of this paper have the following characteristic in contrast with the existing results. 1) A novel transformed error algorithm together with a new sliding mode control framework is investigated to insure the convergence of the tracking error within a predefined zone. 2) The tracking error constraint is considered in the design framework, the proposed method can guarantee a prescribed steady-state error and convergence rate behavior for the tracking error depending merely on the I/O data. 3) Compared with the PPC methods in [23]–[27], the design framework has no connection with the plant model. This paper is constructed as follows. Section II illustrates the preliminary work. The data-driven adaptive sliding mode control with PPC is shown in Section III. In Section IV, simulation examples confirm the theoretical results. Section V gives the conclusion. To facilitate the description of the proposed control method, the following notations are first presented. The superscript “T” represents for matrix transposition. For a square matrix A ∈ Rn×n , λmin (A) and λmax (A) express the minimum and maximum eigenvalues of matrix A, and tr{A} is the trace operator of matrix A. For a vector b = (b1 , . . . , bk )T , ||b|| indicates the 2-norm of b. Unless otherwise specified, the function f (·) is interpreted as f . II. P ROBLEM F ORMULATION A. System Model The SISO discrete-time nonlinear system is considered as follows: (1) yk+1 = f uk , . . . , uk−du , yk , . . . , yk−dy where f (·) ∈ R is an unknown nonlinear function, du , dy ∈ Z are the unknown orders, uk and yk are the system input and output at the time k, respectively. The plant output could be made converge to the desired trajectory, if an ideal nonlinear controller exists (2) uk = C ek+1 , . . . , ek−de , uk−1 , . . . , uk−dc where ek = yk − yd,k is the tracking error, yd,k is the desired trajectory, C(·) is an unknown nonlinear function, and de , dc ∈ Z are the unknown orders.

Assumption 1: The partial derivatives of f (·) in relation to the control signals uk , uk−1 , . . . , uk−du are continuous for k ∈ N, where du ∈ N is the length constant to be designed. Assumption 2: The system (1) is generalized Lipschitz, which implies that there exists a positive constant C1 , thus |yk+1 | ≤ C1 σk , yk+1 = yk+1 − yk , where σkT = [uk , . . . , uk−du +1 ], uk = uk − uk−1 , and du ∈ N is the linearization length constant of the plant. According to the above assumptions, the lemma for the nonlinear system is given to promote our analysis. Lemma 1 [13]: Because the nonlinear system (1) satisfies Assumptions 1 and 2, we define the vector φk , called pseudo-gradient, such that the system (1) can be represented as follows: yk+1 = σkT φk

(3)

where φk = [φ1,k , . . . , φdu ,k ]T , du ∈ N is the linearization length constant of the plant. Remark 1: Note that, Lemma 1 indicates that φk contains the nonlinear dynamics of the plant (1). The critical issue of achieving the goal is to find an appropriate method to estimate φk . The cost function given in [12] for parameter estimation is used in this paper as 2 2 J φˆ k = yk − yk−1 − σkT φˆ k + χ φˆ k − φˆ k−1 . (4) Using optimal condition ∂J/∂ φˆ k = 0, we can derived the estimation algorithm as follows: φˆ k = φˆ k−1 +

T φˆ κσk−1 (yk − σk−1 k−1 )

χ + σk−1 2

(5)

where χ > 0, κ ∈ (0, 2) is a positive constant. Remark 2: With the help of the estimation law (5), the data yk is employed to update φˆ k . The benefits of the algorithm are that the convergence property can be demonstrated in [12] and [13]. It can directly estimate φˆ k without requiring the plant model. B. Prescribed Performance Function The idea of using PPC methodology to design the controller to guarantee tracking error constraint seems to have been originated in [23]. In this section, the knowledge of PPC will be introduced in detail. Inspired by [27], we consider the discrete-time positive decreasing boundary and smooth function ρk − ρk < ek < ρk ρk+1 = (1 − ϑ)ρk + ϑρ∞ lim ρk = ρ∞

k→∞

(6) (7) (8)

with initial value ρ0 > ρ∞ > 0 and convergent rate relate to ϑ ∈ (0, 1). The purpose is to design a control algorithm that accounts for the prescribed performance of the tracking error, such that it is constrained in the sense of (6). In order to handle the constrained control problem (6), the tracking error ek is constructed as unconstrained equivalent transformed error form.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. LIU AND YANG: DATA-DRIVEN ADAPTIVE SLIDING MODE CONTROL OF NONLINEAR DISCRETE-TIME SYSTEMS

A transformed error τk is introduced to the strictly increasing function (τk ). More specifically, we define ek = ρk (τk ).

3

we define one-step ahead tracking error ek+1 as follows: ek+1 = yd − yk+1

(9)

= yd − yk −

For the analysis of the developed approaches, the following properties are given: (τk ) ∈ (−1, 1), for all real numbers τk lim (τk ) = 1 and lim (τk ) = −1.

τk →+∞

τk →−∞

(10) (11)

Furthermore, due to the properties of (τk ) and ρk ≥ ρ∞ > 0, the inverse transformation can be obtained by ek . (12) τk = −1 ρk For any initial tracking error e0 , if ρ0 is selected such that −ρ0 < e0 < ρ0 and τk is bounded, then (τk ) ∈ (−1, 1) holds, following that (6) holds. Here, we introduce a strictly increasing function for control design (τk ) =

eτk − e−τk . eτk + e−τk

= ek −

σkT φˆ i,k

(17)

i=1

where yd is the desired trajectory. Using (16) and (14), one obtains ρk+1 + ek+1 1 sk+1 = sk + ατk + ln . 2 ρk+1 − ek+1 With the help of the following reaching law:

(18)

sk = sk+1 − sk = 0.

(19)

From (17) and (18), the corresponding equation (19) can be given as du

ek − φˆ i,k uk−i+1 (1 + ζk ) + (1 − ζk )ρk+1 = 0 (20) where (21)

uM k

(14)

The control signal can be represented in the following form: du

φˆ 1,k −1 (1 − ζk ) M ek − φˆ i,k uk−i+1 + φˆ 1,k uk = ρk+1 . 2 (1 + ζk ) ˆ λ + φ1,k i=2 (22) We design the control input uk as the following form: S uk = uk − uk−1 = uM k + uk .

a,k

For the nonlinear system (1), consider the following first order sliding function:

(16)

1,k k

c,k

υa,k = 2ρk+1 1 υb,k = 1 + ζk υc,k = (1 + ζk ). 2 ιφˆ 1,k 2 λ + φˆ 1,k

ek −

du

(25) φˆ i,k uk−i+1 .

(26)

i=2

Then, we give φˆ 1,k s sign(sk ). uSk = 2 λ + φˆ 1,k

(15)

where α ∈ (0, 1) is a design parameter. Then, the sliding function sk+1 is given as follows:

a,k

where

Ma,k = 1 −

A. MFAC Sliding Mode Control Design

(23)

Substituting (17) into (16) and using (22) and (23), we can get

ˆ 1,k uS υb,k − M + φ υ a,k a,k k 1

(24) sk+1 − sk = ln 2 υ + M + φˆ uS υ

For simplicity

III. MFAC S LIDING M ODE C ONTROL W ITH PPC

sk+1 = sk + τk+1 + ατk

i=1 du

ζk = e−2ατk .

Remark 3: The increasing function (τk ) is employed to handle the tracking error constraint, such that an error transformation is capable of transforming the original nonlinear system, with the constrained [see (6)] tracking error behavior, into an equivalent unconstrained one. Different from the traditional output constraint problem, where barrier Lyapunov function is used in control design [2]. It should be pointed out that tan-type barrier-function [39] is not suitable for the discrete-time nonlinear systems. This paper employs the function (τk ), which is convenient to design the controller. Remark 4: For the PPC, the parameters ρ0 , ρ∞ , and ϑ are the key factors affecting the performance. This can be validated by (6) and (7). More specifically, the parameter ρ∞ determines the maximum allowable size of the tracking error at the steady state. The parameter ϑ gives the convergence speed of the tracking error. A higher value of ϑ results in a fast convergence rate. The parameter ρ0 represents the maximum undershoot during the inception phase.

sk = sk−1 + τk + ατk−1

σkT φˆ i,k

i=1

(13)

Then, τk is represented as

ρk + ek 1 . τk = ln 2 ρk − ek

du

(27)

2 ]/[λ + There exists ι ∈ (0, 1], λ > 0, such that |1 − ([ιφˆ 1,k ∈ (0, 1). According to [12] and [13], the convergence

2 ])| φˆ 1,k



of the Ma,k can be proved. Thus, the fact that Ma,k is a bounded quantity since φˆ 1,k is bounded. It is easy to see that |Ma,k | ≤ Ma . Now, we assume that s > 0 satisfy the following conditions: ∞ |Ma | < s < υρb,k , for sk > 0 (28) ρ∞ |Ma | < s < υc,k , for sk < 0. The final control laws are summarized as du

φˆ i,k uk−i+1 uk = uk−1 + ιmb ek − i=2

−1 (1 − ζk ) + φˆ 1,k ρk+1 + mb s sign(sk ) (29) (1 + ζk ) 2 )]). where mb = ([φˆ 1,k ]/[(λ + φˆ 1,k Remark 5: Since φˆ k is bounded, when λ > λmin > 0, we can get ˆ ˆ φ1,k φ1,k 1 0.5 ≤ √ < √ |mb | = < < 1. 2 C1 2 λ ˆ ˆ min λ + φ1,k 2 λφ1,k

B. Stability Analysis Theorem 1: Consider the SISO discrete-time nonlinear processes (1) with sliding function (15). If the sliding mode variable will reach a boundary under the control law (29), sk and τk are bounded such that the predefined tracking performances can be guaranteed. Proof: sk+1 − sk =

1 υa,k − (Ma + mb s sign(sk ))υb,k ln . 2 υa,k + (Ma + mb s sign(sk ))υc,k

1 = υa,k − (Ma + mb s sign(sk ))υb,k 2 = υa,k + (Ma + mb s sign(sk ))υc,k . If sk > 0 1 = υa,k − (Ma + mb s )υb,k 2 = υa,k + (Ma + mb s )υc,k . With the help of (28), we have

Case 1: If sk out of the region , i.e., 1 υa,k + 3s υb,k sk > ln 2 υa,k − 3s υc,k or sk
sk > (1/2) ln([(υa,k −3s υb,k )/(υa,k +3s υc,k )])}, with the help of (34), it follows that: υa,k − 3s υb,k 1 ln 2 υa,k + 3s υc,k 1 υa,k − (Ma − mb s sign(sk ))υb,k + ln < sk+1 2 υa,k + (Ma − mb s sign(sk ))υc,k 1 υa,k − (Ma − mb s sign(sk ))υb,k < ln . (35) 2 υa,k + (Ma − mb s sign(sk ))υc,k

Let

Then,

Using (30) and (31), one has 1 υa,k − 3s υb,k υa,k + 3s υb,k 1 < sk+1 − sk < ln . ln 2 υa,k + 3s υc,k 2 υa,k − 3s υc,k (32)

(31)

(36)

Combining (35) and (36), it is known that the sliding mode variable is bounded. 2) If 0 < sk < (1/2) ln([(υa,k + 3s υb,k )/(υa,k − 3s υc,k )]), and owing to (35) 1 υa,k − (Ma + mb s sign(sk ))υb,k ln < sk+1 2 υa,k + (Ma + mb s sign(sk ))υc,k υa,k + 3s υb,k 1 < ln 2 υa,k − 3s υc,k 1 υa,k − (Ma + mb s sign(sk ))υb,k + ln . 2 υa,k + (Ma + mb s sign(sk ))υc,k

(37)


Due to s > |Ma |, it follows that: υa,k − 3s υb,k 1 ln 2 υa,k + 3s υc,k 1 υa,k − (Ma + mb s sign(sk ))υb,k < ln < 0. 2 υa,k + (Ma + mb s sign(sk ))υc,k

5

(38)

Using (37) and (38), we can get υa,k − 3s υb,k υa,k + 3s υb,k 1 1 ln < sk+1 < ln . 2 υa,k + 3s υc,k 2 υa,k − 3s υc,k (39)

Fig. 1.

Desired trajectory yd,k and different measurement outputs yk .

Fig. 2.

Tracking error with prescribed performance.

Fig. 3.

Comparison of tracking error ek .

The tracking trajectory is given as yd (k + 1) = 1 + 0.2 sin(2πk/50) + sin(2πk/100) + sin(2πk/150) . (41)

Fig. 4.

Control inputs uk .

As a matter of fact, it is assumed that (40) is merely made to produce the output data. The sampling time Ts = 0.1. The parameters to be decided for the system are du = 4, α = 0.1, ϑ = 0.08, ρ∞ = 0.02, ρ0 = 1.5, s = 0.01, χ = 0.6, κ = 1, ι = 1, λ = 10, and φˆ 0 = [0.6, 0, 0, 0]T . The control input uk is limited by key factor λ in MFAC algorithm [12], [13]. In

the simulation, λ = 0.5, λ = 10 in [12] and λ = 10 in [20] are given to compared with the presented method. Obviously, the presented method has better tracking accuracy. The simulation results are shown in Figs. 1–4. Fig. 1 exhibits a comparison of desired trajectory yd,k and different measurement outputs yk . Figs. 1 and 3 compare the presented

According to the above analysis, we conclude that the sliding mode variable sk is bounded. From (15), we have sk − sk−1 = τk + ατk−1 . There exists α ∈ (0, 1), such that τk is bounded, which implies that ek is satisfied by the condition (6). This completes the proof. Remark 6: In contrast with the considering MFAC algorithms such as in [12]–[16], [18], and [19], a PPC framework is investigated for a family of discrete-time nonlinear processes. It is of great practical requirement to research the PPC method since the practical engineering systems often face to the constraints in some ways. Fortunately, the proposed method can always guarantee a prescribed convergence rate and steady-state error converges to the predefined values all the time. Remark 7: Different from the PPC approaches in [23]–[27], a transformed error algorithm together with a new sliding mode control framework is studied to ensure the tracking error converge to a predefined zone. More specifically, instead of requiring either the dynamics model or any approximation structures, in this scheme, the proposed controller can guarantee a prescribed steady-state error and convergence rate depending merely on the I/O measurement data, which is more practical in the complex industrial processes. IV. S IMULATION In this section, simulation results will be constructed to confirm the validity of the obtained controller design approach. Example 1: The controlled nonlinear plant is y(k + 1) = sin y(k) + u(k) 5 + cos y(k)u(k) . (40)



Fig. 5.

Fig. 6.

Fig. 7.

Tracking error with prescribed performance.

Fig. 8.

Comparison of tracking error ek .

Fig. 9.

Control inputs uk .

Heat exchanger.

Desired trajectory yd and measurement outputs yk .

adaptive sliding mode control with PPC and traditional methods without PPC [12], [20]. With the same initial conditions, the proposed method achieves a better tracking performance. Fig. 2 presents the tracking error with prescribed performance. The comparison of tracking errors ek in the simulation is exhibited by Fig. 3. Fig. 4 illustrates the control input (29) compared with other methods for the system. Example 2: A steam-water heat exchanger is used to validate the presented data-driven MFAC algorithm. The engineering flow sheet of the steam-water heat exchanger is shown in Fig. 5. The heat exchanger dynamics is represented by a Hammerstein model and it can represented as [40] x(k) = 1.5u(k) − 1.5u(k)2 + 0.5u(k)3 y(k + 1) = 0.6y(k) − 0.1y(k − 1) + 1.2x(k) − 0.1x(k − 1). (42) As a matter of fact, it is assumed that (42) is merely made to produce the output data. In the example, λ = 0.5 and λ = 4 in [12] are illustrated to compared with the proposed approach. The sampling time Ts = 0.1. The parameters to be chosen for the system are du = 4, α = 0.1, ϑ = 0.08, ρ∞ = 0.01, ρ0 = 3, s = 0.01, χ = 0.6, κ = 1, ι = 1, λ = 15, and φˆ 0 = [0.8, 0, 0, 0]T . The simulations are shown in Figs. 6–9. Fig. 6 provides a comparison of desired trajectory yd,k and different measurement outputs yk . From Fig. 6, we could observe that, the developed method could achieve faster convergence rate and less overshoot, while other methods need more time to adjust its parameters. The tracking error with prescribed performance is given in Fig. 7. The comparison of ek is illustrated in Fig. 8. Fig. 9 gives the control inputs for the system.

V. C ONCLUSION This paper dealt with a data-driven adaptive sliding mode control method for discrete-time nonlinear processes, dedicated to guarantee the tracking error constraint. Additionally, a new transformed error algorithm together with a novel sliding mode control framework was investigated to ensure the tracking error converges to a predefined zone. Finally, it was demonstrated, by means of examples, that the obtained adaptive sliding mode control method could be constructed and efficiently implemented. It is stressed that only SISO systems are considered in this paper, multiple-input–single-output systems and multiple-input–multiple-output systems will be investigated in the future works.

R EFERENCES [1] Y.-J. Liu, S. C. Tong, C. L. P. Chen, and D.-J. Li, “Neural controller design-based adaptive control for nonlinear MIMO systems with unknown hysteresis inputs,” IEEE Trans. Cybern., vol. 46, no. 1, pp. 9–19, Jan. 2016. [2] Y.-J. Liu and S. C. Tong, “Barrier Lyapunov functions for Nussbaum gain adaptive control of full state constrained nonlinear systems,” Automatica, vol. 76, pp. 143–152, Feb. 2017. [3] J.-W. Zhu, G.-H. Yang, H. Wang, and F. L. Wang, “Fault estimation for a class of nonlinear systems based on intermediate estimator,” IEEE Trans. Autom. Control, vol. 61, no. 9, pp. 2518–2524, Sep. 2016.


[4] Y.-X. Li and G.-H. Yang, “Adaptive asymptotic tracking control of uncertain nonlinear systems with input quantization and actuator faults,” Automatica, vol. 72, pp. 177–185, Oct. 2016. [5] Y.-X. Li and G.-H. Yang, “Model-based adaptive event-triggered control of strict-feedback nonlinear systems,” IEEE Trans. Neural Netw. Learn. Syst., to be published, doi: 10.1109/TNNLS.2017.2650238. [6] S. Yin, C. M. Yang, J. X. Zhang, and Y. C. Jiang, “A data-driven learning approach for nonlinear process monitoring based on available sensing measurements,” IEEE Trans. Ind. Electron., vol. 64, no. 1, pp. 643–653, Jan. 2017. [7] H. G. Zhang, D. R. Liu, Y. H. Luo, and D. Wang, Adaptive Dynamic Programming for Control: Algorithms and Stability. London, U.K.: Springer-Verlag, 2013. [8] F. L. Lewis and D. Vrabie, “Reinforcement learning and adaptive dynamic programming for feedback control,” IEEE Circuits Syst. Mag., vol. 9, no. 3, pp. 32–50, Aug. 2009. [9] Q. L. Wei and D. R. Liu, “Data-driven neuro-optimal temperature control of water–gas shift reaction using stable iterative adaptive dynamic programming,” IEEE Trans. Ind. Electron., vol. 61, no. 11, pp. 6399–6408, Nov. 2014. [10] Z. S. Hou and S. T. Jin, Model Free Adaptive Control: Theory and Applications. Boca Raton, FL, USA: CRC Press, 2013. [11] Z. S. Hou, “The parameter identification, adaptive control and model free learning adaptive control for nonlinear systems,” Ph.D. dissertation, College Inf. Sci. Eng., Northeastern Univ., Shenyang, China, 1994. [12] Z. S. Hou and S. T. Jin, “A novel data-driven control approach for a class of discrete-time nonlinear systems,” IEEE Trans. Control Syst. Technol., vol. 19, no. 6, pp. 1549–1558, Nov. 2011. [13] Z. S. Hou and S. T. Jin, “Data-driven model-free adaptive control for a class of MIMO nonlinear discrete-time systems,” IEEE Trans. Neural Netw., vol. 22, no. 12, pp. 2173–2188, Dec. 2011. [14] Z. S. Hou and Y. M. Zhu, “Controller-dynamic-linearization-based model free adaptive control for discrete-time nonlinear systems,” IEEE Trans. Ind. Informat., vol. 9, no. 4, pp. 2301–2309, Nov. 2013. [15] Y. M. Zhu and Z. S. Hou, “Data-driven MFAC for a class of discretetime nonlinear systems with RBFNN,” IEEE Trans. Neural Netw. Learn. Syst., vol. 25, no. 5, pp. 1013–1020, May 2014. [16] Y. M. Zhu, Z. S. Hou, F. Qian, and W. L. Du, “Dual RBFNNsbased model-free adaptive control with aspen HYSYS simulation,” IEEE Trans. Neural Netw. Learn. Syst., vol. 28, no. 3, pp. 759–765, Mar. 2017. [17] Y. M. Zhu and Z. S. Hou, “Controller dynamic linearisation-based model-free adaptive control framework for a class of non-linear system,” IET Control Theory Appl., vol. 9, no. 7, pp. 1162–1172, May 2015. [18] D. Z. Xu, B. Jiang, and P. Shi, “A novel model-free adaptive control design for multivariable industrial processes,” IEEE Trans. Ind. Electron., vol. 61, no. 11, pp. 6391–6398, Nov. 2014. [19] D. Z. Xu, B. Jiang, and P. Shi, “Adaptive observer based data-driven control for nonlinear discrete-time processes,” IEEE Trans. Autom. Sci. Eng., vol. 11, no. 4, pp. 1037–1045, Oct. 2014. [20] X. F. Wang, X. Li, J. H. Wang, X. K. Fang, and X. F. Zhu, “Datadriven model-free adaptive sliding mode control for the multi degree-offreedom robotic exoskeleton,” Inf. Sci., vol. 327, pp. 246–257, Jan. 2016. [21] D. Liu and G.-H. Yang, “Event-based model-free adaptive control for discrete-time non-linear processes,” IET Control Theory Appl., vol. 11, no. 15, pp. 2531–2538, Sep. 2017, doi: 10.1049/iet-cta.2016.1672. [22] D. Liu and G.-H. Yang, “Neural network-based event-triggered MFAC for nonlinear discrete-time processes,” Neurocomputing, vol. 272, pp. 356–364, Jan. 2018, doi: 10.1016/j.neucom.2017.07.008. [23] C. P. Bechlioulis and G. A. Rovithakis, “Robust adaptive control of feedback linearizable MIMO nonlinear systems with prescribed performance,” IEEE Trans. Autom. Control, vol. 53, no. 9, pp. 2090–2099, Oct. 2008. [24] C. P. Bechlioulis and G. A. Rovithakis, “Adaptive control with guaranteed transient and steady state tracking error bounds for strict feedback systems,” Automatica, vol. 45, no. 2, pp. 532–538, Feb. 2009. [25] T. Wang et al., “Performance-based adaptive fuzzy tracking control for networked industrial processes,” IEEE Trans. Cybern., vol. 46, no. 8, pp. 1760–1770, Aug. 2016. [26] J.-X. Zhang and G.-H. Yang, “Fuzzy adaptive output feedback control of uncertain nonlinear systems with prescribed performance,” IEEE Trans. Cybern., to be published, doi: 10.1109/TCYB.2017.2692767. [27] J.-X. Zhang and G.-H. Yang, “Prescribed performance fault-tolerant control of uncertain nonlinear systems with unknown control directions,” IEEE Trans. Autom. Control, vol. 62, no. 12, pp. 6529–6535, Dec. 2017, doi: 10.1109/TAC.2017.2705033.

7

[28] D. Zhai, C. J. Xi, L. W. An, J. X. Dong, and Q. L. Zhang, “Prescribed performance switched adaptive dynamic surface control of switched nonlinear systems with average dwell time,” IEEE Trans. Syst., Man, Cybern., Syst., vol. 47, no. 7, pp. 1257–1269, Jul. 2017. [29] W. Wang, D. Wang, Z. H. Peng, and T. S. Li, “Prescribed performance consensus of uncertain nonlinear strict-feedback systems with unknown control directions,” IEEE Trans. Syst., Man, Cybern., Syst., vol. 46, no. 9, pp. 1279–1286, Sep. 2016. [30] H. Y. Li, L. Bai, Q. Zhou, R. Q. Lu, and L. J. Wang, “Adaptive fuzzy control of stochastic nonstrict-feedback nonlinear systems with input saturation,” IEEE Trans. Syst., Man, Cybern., Syst., vol. 47, no. 8, pp. 2185–2197, Aug. 2017. [31] Q. Zhou, H. Y. Li, C. W. Wu, L. J. Wang, and C. K. Ahn, “Adaptive fuzzy control of nonlinear systems with unmodeled dynamics and input saturation using small-gain approach,” IEEE Trans. Syst., Man, Cybern., Syst., vol. 47, no. 8, pp. 1979–1989, Aug. 2017. [32] H. Y. Li, P. Shi, and D. Y. Yao, “Adaptive sliding-mode control of Markov jump nonlinear systems with actuator faults,” IEEE Trans. Autom. Control, vol. 62, no. 4, pp. 1933–1939, Apr. 2017. [33] M. Wang and A. Yang, “Dynamic learning from adaptive neural control of robot manipulators with prescribed performance,” IEEE Trans. Syst., Man, Cybern., Syst., vol. 47, no. 8, pp. 2244–2255, Aug. 2017, doi: 10.1109/TSMC.2016.2645942. [34] L. L. Zhang and G.-H. Yang, “Adaptive fuzzy prescribed performance control of nonlinear systems with hysteretic actuator nonlinearity and faults,” IEEE Trans. Syst., Man, Cybern., Syst., to be published, doi: 10.1109/TSMC.2017.2707241. [35] M. L. Nguyen, X. Chen, and F. Yang, “Discrete-time quasi-slidingmode control with prescribed performance function and its application to piezo-actuated positioning systems,” IEEE Trans. Ind. Electron., vol. 65, no. 1, pp. 942–950, Jan. 2018, doi: 10.1109/TIE.2017.2708024. [36] S. El-Ferik, H. A. Hashim, and F. L. Lewis, “Neuro-adaptive distributed control with prescribed performance for the synchronization of unknown nonlinear networked systems,” IEEE Trans. Syst., Man, Cybern., Syst., to be published, doi: 10.1109/TSMC.2017.2702705. [37] Y. Karayiannidis and Z. Doulgeri, “Model-free robot joint position regulation and tracking with prescribed performance guarantees,” Robot. Auton. Syst., vol. 60, no. 2, pp. 214–226, Feb. 2012. [38] Y. Karayiannidis, D. Papageorgiou, and Z. Doulgeri, “A model-free controller for guaranteed prescribed performance tracking of both robot joint positions and velocities,” IEEE Robot. Autom. Lett., vol. 1, no. 1, pp. 267–273, Jan. 2016. [39] X. Jin, “Adaptive fault-tolerant control for a class of output-constrained nonlinear systems,” Int. J. Robust Nonlin. Control, vol. 25, no. 18, pp. 3732–3745, 2015. [40] E. Eskinat, S. H. Johnson, and W. L. Luyben, “Use of Hammerstein models in identification of nonlinear systems,” AIChE J., vol. 37, no. 2, pp. 255–268, Feb. 1991. Dong Liu received the B.S. degree in mathematics and applied mathematics and the M.S. degree in operational research and cybernetics from Shenyang Normal University, Shenyang, China, in 2009 and 2012, respectively. He is currently pursuing the Ph.D. degree with the College of Information Science and Engineering, Northeastern University, Shenyang. His current research interests include data-driven control, model-free adaptive control, event-triggered control, and reinforcement learning control. Guang-Hong Yang (SM’04) received the B.S. and M.S. degrees in mathematics and the Ph.D. degree in control theory and control engineering from Northeast University, Shenyang, China, in 1983, 1986, and 1994, respectively. From 2001 to 2005, he was a Research Scientist/Senior Research Scientist with the National University of Singapore, Singapore. He is currently a Professor and the Dean with the College of Information Science and Engineering, Northeastern University. His current research interests include fault-tolerant control, fault detection and isolation, cyber physical systems, and robust control. Dr. Yang is an Associate Editor of the International Journal of Control, Automation and Systems, the International Journal of Systems Science, the IET Control Theory and Applications, and the IEEE T RANSACTIONS ON F UZZY S YSTEMS.