Controller-Dynamic-Linearization-Based Model Free ... - IEEE Xplore

IEEE TRANSACTIONS ON INDUSTRIAL INFORMATICS, VOL. 9, NO. 4, NOVEMBER 2013

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Controller-Dynamic-Linearization-Based Model Free Adaptive Control for Discrete-Time Nonlinear Systems Zhongsheng Hou and Yuanming Zhu

Abstract—A new type of model free adaptive control (MFAC) method, including MFAC scheme designs with the compact-form-dynamic-linearization-based controller (CFDLc) and partial-form-dynamic-linearization-based controller (PFDLc), is presented for a class of discrete-time SISO nonlinear systems. The proposed method is a pure data-driven control method since the controller is independent of the model of the controlled plant, and controller parameter tuning is merely based on the measured I/O data of the controlled plant in closed loop. Differing from the MFAC prototype, the proposed method uses the dynamic linearization approach not only on ideal controller but also on the plant. The stability of the CFDLc-MFAC and PFDLc-MFAC is guaranteed by rigorous theoretical analysis, and the effectiveness is evaluated on simulation examples and a three-tank liquid control experimental system. Index Terms—Data-driven control (DDC), discrete-time nonlinear system, dynamic linearization approach, model free adaptive control (MFAC).

I. INTRODUCTION

I

N THE development of information science and technology, the industrial process, for instance, chemical industry, metallurgy, machinery, electronics, and logistics, has undergone significant changes. The scale of industry becomes large, and the production technologies and processes become more and more complex. Modeling the process using the first principles or identification methods becomes more and more difficult, which would lead to that using the traditional model-based control (MBC) methods to tackle the control issues in these kinds of enterprises becomes unpractical. However, thanks to the development of modern electronic technology, a huge amount of process data, containing all of the valuable state information of the process and the equipments in most industrial processes, is capable to be collected and stored. In this case, how to use these data to design controller for the industrial processes directly would have great significance when the accurate process models are unavailable. Manuscript received January 01, 2013; revised March 01, 2013; accepted April 01, 2013. Date of publication April 12, 2013; date of current version October 14, 2013. This work was supported in part by International Cooperation Program (61120106009) of the National Science Foundation of China. Paper no. TII-13-0002. The authors are with the Advanced Control Systems Laboratory, School of Electronic and Information Engineering, Beijing Jiaotong University, Beijing 100044, China (e-mail: [email protected] (corresponding author); yuanm. [email protected]). Digital Object Identifier 10.1109/TII.2013.2257806

Recently, many kinds of data-driven control (DDC) approaches have emerged, such as [1]–[4], especially the data-driven control methods [5]. However, different methods called with different names, such as model free adaptive control (MFAC) [6], lazy learning control (LL) [7], virtual reference feedback tuning (VRFT) [8], and iterative feedback tuning (IFT) [9]. There are two fundamental categories of DDC methods. The first one is that the controller structure is assumed to be known with some unknown parameters, which may come from certain transcendent or experimental knowledge on the plant or derive from the structure of the plant, and then the controller design issue is transformed into the identification problem for the controller parameter. Most of DDC methods follow this line, such as VRFT and IFT. The second one is that the generic controller is designed based on certain function approximations or some equivalent descriptions on the original controlled plant, such as neural networks, Taylor approximation, and other equivalent transformation. Then, the controller parameters are adjusted by minimizing a specified performance criterion using the I/O data, including offline and online data. Typical ones include MFAC and LL. The prototype of the MFAC approach is first proposed for a class of general discrete-time SISO nonlinear systems [6], [10]. Instead of identifying a global nonlinear model of the plant, a virtual equivalent dynamic linearization data model is built along the dynamic operation points of the closed-loop system by using a dynamic linearization technique. The dynamic linearization technique includes two kinds of linearization data models, the compact form dynamic linearization (CFDL) and the partial form dynamic linearization (PFDL). The resulting time-varying parameter in the data model is called a pseudo-partial derivative (PPD) and is estimated using only I/O data of the plant. Obviously, MFAC falls into the second category. The stability and convergence of the MFAC scheme for regulation problem are proposed in [10], and it has been successfully implemented in many applications, e.g., chemical industry [11], artificial heart control [12], injection modelling process and motor control [13], PH value control [14], and so on. The main contribution of this paper is that a new type of MFAC method, including CFDLc-MFAC and PFDLc-MFAC, is proposed for a class of nonlinear system with help of the CFDL data model of the plant, and the stability of the control system is studied. The feature of the proposed method is that the two equivalent dynamic linearization data models (the generic ideal controller to a general discrete-time nonlinear system and

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the other is on the controlled plant) are used to design and analyze the schemes. Neither the dynamic model nor the structure information of the plant is involved in controller design and analysis. The time-varying PPD is tuned by minimizing a given controller designing cost function using online I/O data of the controlled plant in closed loop. The outline of this paper is as follows. Section II introduces the dynamic linearization technique briefly. Section III discusses the dynamic linearization on an ideal controller. Section IV presents controller design and convergence analysis. Simulation comparison results are shown in Section V, followed by the proposed method application on a three-tank liquid control system. Section VI concludes the work. Throughout the paper, denotes the backward difference with respect to the scale and vector without further explanation. II. DYNAMIC LINEARIZATION TECHNIQUE The nonlinear controlled plant is as follows :

capture this complex dynamic behavior. In PFDLp, all of the components of PPD vector share the dynamic behaviors of the system since the change of output caused by the changes of input in a fixed moving interval is explicitly considered, thus the higher quality estimation could be expected with the same estimation algorithm. Finally, the dynamic behavior of pseudo-partial derivative is not sensitive to the time-varying parameter, structure, or delay of a controlled plant, but these factors explicitly formulated in the first principles model or the identified model, in which they are hard to handle. III. DYNAMIC LINEARIZATION ON IDEAL CONTROLLER For nonlinear system (1), an ideal controller should exist for the tracking problem theoretically if one intends to design a controller to drive it to track the desired output. Thus, it is reasonable to assume there exists an ideal nonlinear controller of the form (4), which can stabilize the plant and realize the plant output to asymptotically track the desired signal.

(1) where and are the unknown orders of output and input, respectively, and is an unknown nonlinear function. The dynamic linearization method is proposed in [6] and [10]. Under some mild assumptions, nonlinear system (1) can be equivalently expressed as following two kinds of dynamic linearization data models. 1) CFDL data model of plant (CFDLp): (2) is the PPD of controlled plant with where for any under the case of linearization length constant (LLC) . 2) PFDL data model of plant (PFDLp): (3) , is PPD vector where of controlled plant with for any . With the assumption , detailed proofs of these data models are proposed in [10]. Compared with other linearization methods for nonlinear function, this dynamic linearization method possesses the following features. First, it does not require the mathematics model, order and time delay of the controlled plant. Second, it is an equivalent dynamic linearization data model rather than a static approximation model and no high-order term is dropped. Third, the dynamic linearization model, having time-varying incremental form with very simple structure and very few parameters, is a data model only for the purpose of controller design rather than first principles model. The introduction of linearization length constant can avoid the high-order controller design. Next, the differences between these two kinds of dynamic linearization data models are their complexities. The CFDLp data model is the special case of the PFDLp data model when the linearization length constant is equal to 1. In CFDLp data model, all of the nonlinear properties and the parameter estimation error are fused into the scalar PPD, thus the dynamic behavior of the PPD may be very complicated, and, as a result, it would be difficult to design an algorithm to

(4) is a smooth unknown nonlinear function, is the output tracking error at time instant , and are the desired and actual output of the controlled plant, respectively, and and are two unknown orders of the controller. where

A. CFDL on the Ideal Nonlinear Controller Assumption 1a: The controller (4) is a smooth nonlinear function, and is continuous. Assumption 2a: The controller (4) is generalized Lipschitz, that is, , if , where . Remark 1: From a practical view point, these assumptions are reasonable and acceptable. Assumption 1a is a typical condition for general controller in DDC methods, like VRFT and IFT [8], [9]. Assumption 2a imposes an upper bound limitation on the change rate of controller output driven by the changes of the tracking error. In other word, the controller should be stable. Theorem 1: Consider nonlinear controller (4), satisfying Assumption 1a and Assumption 2a, if , then there exists a time-varying parameter , called PPD, such that system (4) can be transformed into the following equivalent dynamic linearization data model, called CFDLc: (5) with

for any time . Proof: The proof is a special case of Theorem 2. The following assumption will be used in the stability analysis. Assumption 3a: , , where . Remark 2: The PPD in controller (5) is negative since feedback strategy is used. Further, the controller is artificially designed, thus the bound of controller parameters could be known a priori. Using this assumption and Theorem 1, we have (6)

HOU AND ZHU: CONTROLLER-DYNAMIC-LINEARIZATION-BASED MFAC FOR DISCRETE-TIME NONLINEAR SYSTEMS

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B. PFDL on the Ideal Nonlinear Controller Similar to the discussion in Section III-A, another dynamic linearization can be derived analogically. Assumption 1b: The controller (4) is a smooth nonlinear function, and , are continuous, where is the LLC. Assumption 2b: The controller (4) is generalized Lipschitz, that is, , if . With , and . Theorem 2: Consider nonlinear controller (4), satisfying Assumptions 1b and 1b. For a given , if , then there exists a time-varying vector , called pseudo-gradient (PG), such that system (4) can be transformed into the following equivalent dynamic linear data model, called partial form dynamic linearization based controller (PFDLc) (7) for any time . with bounded Proof: Please see Appendix A. Remark 3: It is noted that, to determine a proper controller structure for a nonlinear system is a very difficult task if no mathematical model of the controlled plant is available. In fact, it is also a challenging work even for a known nonlinear system. Fortunately, the simple parameterized dynamic linear controller with time-varying incremental structure, which is equivalent to the ideal nonlinear controller (4), has been obtained by virtue of Theorem 1 and Theorem 2 under two reasonable assumptions. Theoretically speaking, this controller can deal with the tracking problem for the nonlinear system (1) since controller (5) or (7) is an equivalent description of ideal controller (4). In addition, controller (7) could serve as the candidate controller of VRFT or IFT and other DDC methods, in which the controller is prespecified a priori rather than by a systematic approach. Remark 4: From a theoretical point of view, the CFDLc is the special case of PFDLc when the linearization length is 1. If order is known a priori, could be set as . Otherwise, should be set to be a reasonable value according to the complexity of plant. If and are too large, controller (4) would be a high-order controller, while using the controller (5) or (7) with proper choice of could avoid the drawbacks mentioned above without any model reduction or controller order reduction procedure. Remark 5: The controller structure is independent of the plant model, thus the conventional unmodelled dynamics and the traditional robustness under the sense of MBC framework do not exist in an MFAC framework. Therefore, the MFAC control system will be safe and dependable when it is used in practice. Assumption 3b: , where . From Assumption 3b and the result of Theorem 2, we have (8) C. Practical Controller Structure Since controller (5) or (7) is the equivalent form of the ideal controller (4) for the plant (1), the controller with a qualified parameter estimation algorithm is capable to generate a perfect

Fig. 1. Block diagram of the control system.

input signal theoretically to get a perfect control performance, that is, . However, it does not mean that the actual tracking error will vanish one step ahead, consequently, due to the uncertainties or estimation error in practice. To make the ideal controller (5) and (7) have a causal implementable form and to show the convergence analysis without any confusion, we redefine the actual tracking error at instant as . This leads to the practical controllers based on the equivalent form (5) or (7) as (9) (10) .

where IV. CONTROL SYSTEM DESIGN BASED DATA MODEL

ON

CFDLP

The blocked diagram of the proposed method could be set up as depicted in Fig. 1. In order to tune the CFDLc or PFDLc, a CFDLp data model is built to generate the estimation value of unknown time-varying PPD parameter of the plant only using the I/O data of the plant. A. Design and Analysis of CFDLc-MFAC Control System Consider the following cost function with an additional penalty on the abrupt change of estimated parameter:

(11) where is a weighting factor series. Noting that is unavailable at instant since the plant model is unknown. Thus, we resort to the CFDL data model (2). Substituting (2) and (9) into (11) and differentiating (11) with respect to and setting it to be zero yields

(12) To restart the updating of PPD in terms of different operation point and to increase the tracking ability of scheme for the timevarying parameters, a resetting condition is added as follows: or

(13)

Remark 6: Like IFT, VRFT in the other DDC methods, the plant model is unknown instead of a huge amount of measured

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I/O data of the controlled plant. In this case, a remedy or an intuitive strategy is that using a data model to predict the onestep-ahead plant output to estimate the tracking error . To estimate the time-varying value of PPD in CFDLp data model, a modified parameter estimation criterion is considered here:

yield the following updating law of the controller parameters by virtue of matrix inversion lemma [15]:

(20) if

(14) or

where is weighting factor. Minimizing (14) with respect to

(21)

gives where (15)

and .

and denotes the estimation of of (13), we add

. Similar to the purpose

Similar to Section IV-A, integrating control law (22)

or

(16)

Finally, integrating the control law

and estimation algorithm (15) and (20) and resetting condition (16), (21) gives the PFDLc-MFAC control system scheme. For regulation problem, the output tracking error is

(17) and estimation algorithm (12), (13), (15), and (16) gives the system control scheme of CFDLc-MFAC for the nonlinear system (1). Remark 7: The reason for resetting in (13) is to not too small or too large since it is the controller make gain. is negative and is positive according to the resetting conditions in (13). From (2) and (17), the tracking error for the regulation problem is (18) Theorem 3: The plant described by (1) is controlled by scheme (12), (13), (15), (16), (17) for regulation problem . Then the tracking error of closed-loop system monotonically converges to zero, that is , and . Proof: Please see Appendix B. Remark 8: The condition means that the highest gain of controller should not be too large. It depends on Lipschiz constant of plant (1).

(23) Theorem 4: The plant described by (1) is controlled by scheme (15), (16), (20), (21), and (22) for regulation problem . Then, the tracking error of the closed-loop system monotonically converges, that is, , if , and . Proof: Please see Appendix C. Remark 9: The proper selection of and is key issue to the control performance although the estimation accuracy of and will not affect the system stability. Since the plant model is unknown, the analytical solution for and is almost impossible. However, if sufficient data of the controlled system are available, we can use the iterative optimization method like VRFT or IFT [16] or Particle Swarm Optimization algorithm [11] to optimize these parameters.

, if

B. Design and Analysis of PFDLc-MFAC Control System Consider the following cost function:

V. SIMULATION AND EXPERIMENT A. Numerical Example Here, a numerical example is given to show the effectiveness and the advantages of PFDLc-MFAC methods by comparisons with other typical DDC methods, data driven PID (DD-PID), iterative feedback tuning (IFT), and virtual reference feedback tuning (VRFT), respectively. The controlled nonlinear is the Hammerstein system [17]

(19) where is weighting factor series. Substituting (2) and (10) into (19) and minimizing this cost function with respect to

(24)


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TABLE I COMPARISONS OF THE VALUE OF THE CRITERION

Fig. 2. Tracking performance.

where is random normal distribution output noise with mean 0 and standard derivation 0.01, and it is absent in [17]. DD-PID [17]: An elaborated DD-PID controller parameter tuning method is proposed using Lazy Learning and an optimization algorithm. Please see the reference for details. IFT: The controller structure of IFT is set as [18]

Fig. 3. Comparison between the output of controlled plant and the output of CFDLp data model.

(25) . with initial values The step-size of the optimization algorithm is set to 0.35. After nine iterations, the tuned controller parameters are . PFDLc-MFAC: The setup of the scheme is , , . VRFT: Transfer function of the reference model is selected as , where . The candidate controller is as (25). After using “one-shot” minimization, the optimal controller parameter is . The simulation results are shown in Fig. 2. From Fig. 2, one can see that both online DD-PID, PFDLc-MFAC, and offline VRFT algorithms are effective, and the simulation result of IFT is acceptable. Furthermore, the parameter tuning for DD-PID is much more complicated and time-consuming compared with PFDLc-MFAC, which is just an elementarily online iterative algorithm. Finally, the IFT method tunes the controller parameter using an iterative optimization method and requires several experiments at each iteration to get a satisfactory specification, and VRFT sets the controller parameter using a “one-shot” optimization method, which needs a sufficient number of I/O data pairs of the controlled plant in an open-loop or a closed-loop system. Since both IFT and VRFT are offline algorithms, the parameter tuning process is usually time-consuming. The simulations are carried out in MATLAB with i5 dual core processor at 1.2 GHz for period of 400 time instants, and the detailed execution times for online DD-PID and MFAC algorithms are 0.1981s and 0.0374s. The squared sum of tracking error of the four DDC algorithms is shown in Table I. The comparison between the output of controlled plant and output of CFDLp data model is shown in Fig. 3. It can be seen that, using the estimation algorithm (15), the output of data

Fig. 4. Three-tank liquid level control system (a) Block diagram (b) Experiment equipment.

model (2) can almost describe the original system accurately except for interval at the abrupt change operation point. B. Three Water-Tank Experiment Here, the proposed control method was experimentally evaluated on a three-tank system (Fig. 4). The liquid level control system consists of three tanks, which are placed from high to low and not directly connected. Each tank has a manual inlet valve on the top and an outlet valve at the bottom. Rotating the position of the inlet valve can change the flowing channel, and water can flow out from the outlet valve to the lower tank and finally be recycled to the cistern. The amount of water flowing into the tanks is regulated by a control valve with a flow speed from 0 to 32 L/min. The liquid level can be measured by the diffusing the silicon pressure sensor with an accuracy of 0.25 cm.

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VI. CONCLUSION

Fig. 5. Experiment results. (a) Tracking performance. (b) Control input.

TABLE II VALUE OF THE CRITERION

In this work, two novel CFDLc-MFAC and PFDLc-MFAC schemes are proposed for a class of nonlinear systems, and parameters of the dynamic linearization data models on both the controller and the plant model are estimated only using the online measured I/O data of the plant. The correctness and effectiveness of the proposed schemes are verified with rigorous mathematical analysis and illustrated with extensive simulation comparisons and experiment results with three-tank liquid control system. The main feature of the proposed methods is that the control system design only depends on the measured I/O data of the closed-loop system rather than the first principles model or the identified model. Consequently, they are the pure data driven model free control methods, and the problems of unmodeled dynamics and conventional robustness in traditional MBC framework are absent. Our future research will be the stability analysis on the tracking problem and the applications in practice. APPENDIX A PROOF OF THEOREM 2 Proof: Letting

An embedded system is applied to acquire the analog value of the liquid level at a sampling rate of 0.5 s and transmit it to the host computer. Then, the control input value is calculated by using the proposed PFDLc-MFAC and fed back to the embedded system. Finally, the control input is transformed into the 4–20-mA current signal to drive the electrical valve. In the experiment, the manual inlet valve of upper tank (T1) and outlet valves of the three tanks are open, and the inlet valves of middle tank (T2) and lower tank (T3) are closed. The water is pumped to the T1 and flows through the liquid system from top to bottom. The tanks are empty at the beginning. The control task is to maintain the desired liquid level in T3 at 5 cm by manipulating the opening percentage (V%) of electrical control valve. In order to verify the effectiveness of the proposed control method, only the liquid level of the T3 is measured and used to generate the control signal. In such case, the control process will present large delay and high nonlinearity. The parameters of PFDLc-MFAC scheme are , , . In order to show the robustness of the proposed data driven control approach, we add a constant signal to the input of the plant after 960 s with opening percentage when the system is in its stable state. This signal is unknown to the controller, so it is a nonstationary disturbance. Under these circumstances, the total input to the plant is the control input generated by controller plus the disturbance input. The tracking performance is shown in Fig. 5 and numerical indexes are listed in Table II. The preliminarily robustness analysis in MFAC framework has been studied in [19]. From the experimental results, we can see that the proposed method can work well for this three-tank level control system, even if nonstationary disturbance enters the control input.

and from (4), it has

(A1) Then, (A1) can be rewritten as follows by virtue of Assumption 1b and Cauchy differential mean value theorem: (A2) is the partial derivative value of with where respect to tracking error at a certain mean point within the interval and Taking the similar operation as (A1) for times, it has

(A3) Consider the following scalar data equation with vector as a variable for any : (A4)


Since condition holds, (A4) has at least one solution for any . Then, the partial gradient becomes

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This means (B8)

(A5) Thus, we have

Additionally, is an independent variable in (B8). Taking the partial derivative of this term with respect to , and since , it gives

(A6) (B9) The boundedness of 2b.

is the direct result of Assumption is a nonincreasing function with respect to Thus, and it gives

APPENDIX B PROOF OF THEOREM 3

(B10)

Proof: From algorithm (12) and (13), there exists two cases according to the saturation bound. Case 1: The resetting condition (13) is activated. In this case, it has . Since , , there exists a constant such that

Combining (B8) with (B10) leads to (B11) Thus, there exists a constant

such that

(B1) (B12) Thus, it has Using (B5) and (B12), we obtain (B2) Case 2: The resetting condition (13) is inactivated. It has . Since , it gives

(B13) Consequently, it has

.

(B3) Since

, substituting (12) into (18) yields

(B4)

APPENDIX C PROOF OF THEOREM 4 Proof: Case 1: The resetting condition (21) is activated. From (21) and (22), we have (C1)

where (B5)

Substituting (C1) into (23) yields (C2)

Since is set to satisfy and from Theorem 3, we have (B6) Now, we will prove that and Since Theorem 3 and resetting algorithm (16), it has Using (B3) and (B6), we have

Since conditions 4 hold, it gives then there exists a constant

and

in Theorem ,

such that

, with . in terms of .

(C3) Taking the absolute value operation on (C2) gives (C4)

(B7)

Case 2: The resetting condition (21) is inactivated.

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It has


, , with , substituting (20) into (23) yields

. Since

Thus, it has

From (C10), we have (C5) Letting

(C13) Similar with (C11), since have

and (21), we

(C14)

gives Inequality (C14) means

(C15)

(C6) Taking absolute value on both sides of (C6) gives

Note that (C15). Since

is an independent variable of the second term in , we obtain (C16)

(C7) Since

is

set

to

satisfying and

in

Thus, and gives

is a nonincreasing function with respect to

Theorem 4, we have (C17)

(C8) Using resetting condition (21), it has

. Thus

Using (C15) and (C17), there exists a constant

such that

(C9) Since

in (21), we have (C10)

Also, since

(C18) Letting

, using (C8), we have (C19) from (C7), we obtain

(C20)

(C11) Thus, from (C18) and (C20), we have

Inequality (C11) means (C12)

(C21)


Replacing

by

in (C21) and using (C21), we obtain

(C22) , With the same procedure, we have . Since , it is easy to draw a conclusion that the output tracking error is contracted. Consequently, we obtain . REFERENCES [1] T. Y. Chai, Z. S. Hou, F. L. Lewis, A. Hussain, and D. Zhao, “Guest editorial data-based control, modeling, and optimization,” IEEE Trans. Neural Netw., vol. 22, no. 11, pp. 2150–2153, Nov. 2011. [2] S. Zhe and K. Andrew, “Constraint-Based control of boiler efficiency: A data-mining approach,” IEEE Trans. Ind. Inf., vol. 3, no. 1, pp. 73–83, Feb. 2007. [3] D. Wang, “Robust data-driven modeling approach for real-time final product quality prediction in batch process operation,” IEEE Trans. Ind. Inf., vol. 7, pp. 371–377, 2011. [4] G. Chuanhou, J. Ling, L. Xueyi, C. Jiming, and S. Youxian, “Data-Driven modeling based on volterra series for multidimensional blast furnace system,” IEEE Trans. Neural Netw., vol. 22, no. 12, pp. 2272–2283, Dec. 2011. [5] Z. S. Hou and J. X. Xu, “On data-driven control theory: The state of the art and perspective,” Acta Automatica Sinica, vol. 35, pp. 650–667, 2009. [6] Z. S. Hou and W. H. Huang, “The model-free learning adaptive control of a class of SISO nonlinear systems,” in Proc. IEEE Amer. Control Conf., Albuquerque, NM, USA, 1997, pp. 343–344. [7] G. Bontempi and M. Birattari, “From linearization to lazy learning: A survey of divide-and-Conquer techniques for nonlinear control (Invited Paper),” Int. J. Computational Cognition, vol. 3, pp. 56–73, 2005. [8] A. Sala, “Integrating virtual reference feedback tuning into a unified closed-loop identification framework,” Automatica, vol. 43, pp. 178–183, 2007. [9] H. Hjalmarsson, “From experiment design to closed-loop control,” Automatica, vol. 41, pp. 393–438, 2005. [10] 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. [11] L. dos Santos Coelho and A. A. R. Coelho, “Model-free adaptive control optimization using a chaotic particle swarm approach,” Chaos, Solitons & Fractals, vol. 41, pp. 2001–2009, 2009. [12] Y. Chang, B. Gao, and K. Y. Gu, “A model-free adaptive control to a blood pump based on heart rate,” ASAIO J., vol. 57, pp. 262–267, 2011.

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[13] K. K. Tan, T. H. Lee, S. N. Huang, and F. M. Leu, “Adaptive predictive control of a class of SISO nonlinear systems,” Dynam. Control, vol. 11, pp. 151–174, 2001. [14] B. Zhang and W. D. Zhang, “Adaptive predictive functional control of a class of nonlinear systems,” ISA Trans., vol. 45, pp. 175–183, 2006. [15] D. J. Tylavsky and G. R. L. Sohie, “Generalization of the matrix inversion lemma,” Proc. IEEE, vol. 74, no. 5, pp. 1050–1052, May 1986. [16] W. H. Wang, Z. S. Hou, H. B. Huo, and S. T. Jin, “Data-driven based controller design and its parameters tuning method,” J. Syst. Sci. Math. Sci., vol. 30, pp. 792–805, 2010. [17] T. Yamamoto, K. Takao, and T. Yamada, “Design of a data-driven PID controller,” IEEE Trans. Control Syst. Technol., vol. 17, no. 1, pp. 29–39, Jan. 2009. [18] H. Hjalmarsson, “Control of nonlinear systems using iterative feedback tuning,” in Proc. IEEE Amer. Control Conf., Philadephia, PA, USA, 1998, pp. 2083–2087. [19] Z. S. Hou and X. H. Bu, “Model free adaptive control with data dropouts,” Expert Syst. Applications, vol. 38, pp. 10709–10717, 2011. Zhongsheng Hou received the B.S. and M.S. degrees in applied mathematics from Jilin University of Technology, Jilin, China, in 1983 and 1988, respectively, and the Ph.D. degree in control theory from Northeastern University, Shenyang, China, in 1994. In 1997, he joined the Beijing Jiaotong University, Beijing, China, where he is currently a Full Professor and the Head of the Automatic Control Department, and Founding Director of the Advanced Control Systems Laboratory in the School of Electronic and Information Engineering. He has authored or coauthored over 100 journal papers and 100 papers in prestigious conference proceedings. He is the author of two monographs Nonparametric Model and Its Adaptive Control Theory (Science Press of China, 1999) and Model Free Adaptive Control: Theory and Applications (CRC, 2013). His current research interests include data-driven control, model-free adaptive control, learning control, and intelligent transportation systems. Prof. Hou has served as a Committee Member for over 60 international and Chinese conferences and as an associate editor and guest editor for several international and Chinese journals.

Yuanming Zhu received the B.Eng. degree in measurement and control technology and instruments from the Beijing Jiaotong University, Beijing, China, in 2008, where he is currently working toward the Ph.D. degree at the Advanced Control Systems Laboratory. His research interests are in fields of data-driven control, robust control and adaptive control.