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Model based control and MFAC, which is better in simulation? *
13th IFAC Symposium on Large Scale Complex Systems: Theory and Applications July 7-10, 2013. Shanghai, China
Model based control and MFAC, which is better in simulation? * Zhongsheng Hou*1 and Yuanming Zhu** Advanced Control Systems Lab, School of Electronic and Information Engineering, Beijing Jiaotong University, Beijing, China * (e-mail: [email protected]) ** (e-mail: [email protected]) Abstract: This work presents a new data driven model free adaptive controller by virtue of the gradient information of the available plant model. Our approach is novel in the sense that we use the measured data to directly design the controller, and predict the system output using the plant model if the plant model is available. Whereas, the traditional model based control approach uses the measured data to identify the plant model firstly, and then uses the model to design and analyze the controller design based on this identified model. The main contribution is that this novel scheme opens a way for the data driven control method utilizing the plant model if the plant model is available. The theoretical analysis and the simulation results demonstrate the effectiveness and superiorities of the proposed method. complicated. Modeling the process becomes more and more difficult, and the traditional MBC theory may fail to deal with these kinds of processes since the accurate model is unavailable. However, a huge number of process data, containing valuable information of operations and equipments, is generated and stored. In this case, how to use these huge amount process data, both on-line and off-line, to directly design controller for industrial processes, would has great significance when the accurate process models are unavailable. Therefore, the establishment and development of the data driven control theory (DDC) are the urgent issues in both theories and applications.
I. INTRODUCTION The modern control theory, which is also called model based control (MBC), is derived from the state-space model introduced by Kalman in 1960 and the optimal control (Kalman 1960; Kalman 2009). As a main branch of MBC approach, adaptive control method is developed to deal with the controlled system with unknown constant or time varying parameters. Traditional adaptive control theory uses the available measured input and output data of the controlled plant to identify the parameters firstly, and then the controller is designed based on the plant model by using the “certainty equivalence principle” with the faith that the plant model could represent the real system. Explicit identification of the plant parameters leads to indirect adaptive control whereas direct identification of the controller parameters leads to direct adaptive control (Astrom and Wittenmark 1994). However, in either case, the model based controller design would not work well if the plant model does not fall into the assumed model set. Thus, the resulting controller based on the plant model will definitely lead to unsatisfied performance. Rohr’s counterexample (Rohrs et al. 1982; Rohrs et al. 1985) has demonstrated that the reported stable adaptive control system based on some assumptions made about the plant model could lead to unstable behavior when there is an unmodelled dynamics. Securing safe adaptive control is far from straightforward this well-known model based adaptive control (Anderson and Dehghani 2008).
So far, there are over 10 different kinds of DDC methods in literature (Hou and Wang 2013), such as model free adaptive control (MFAC) (Hou and Huang 1997), lazy learning control (LL) (Bontempi and Birattari 2005), virtual reference feedback tuning (VRFT) (Sala 2007), iterative feedback tuning (IFT) (Hjalmarsson, 2005), etc.. In terms of the methods on how to design the controller structure, there are two fundamental categories of DDC methods. One is that, the controller structure with some unknown parameters is supposed to contain the optimal controller for the controlled plant. It could come from experimental knowledge on the plant, or be derived from the structure of the plant. Then the controller design issue is transformed into the direct identification problem for the controller parameter. Most of DDC methods, such as VRFT and IFT, are in this category. The other one is that, the general controller is designed based on certain function approximations or some equivalent descriptions on the original controlled plant, such as neural networks, Taylor approximation, or other equivalent representation, and then the controller parameters are tuned by minimizing a specified performance criterion using the
As the development of information science and technology, the practical process, for instance, chemical industry, metallurgy, machinery, electronics, electricity, transportation, has been undergone significant changes. The scale of the industry becomes large, and the production technology, equipment and production process become more and more Resrach supported by NSFC under granted No. 61120106009. Hou Zhongsheng is the corresponding author.
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The essential idea of MFAC is using an equivalent dynamic linearization data model and a novel concept pseudo partial derivative (PPD) at current operation point to represent the general discrete time nonlinear system. Then the optimal controller is derived based on estimation of the PPD on-line solely using the input and output data of the controlled plant.
I/O data, including off-line and on-line data. Typical ones are MFAC, LL, etc.. The MFAC prototype is firstly proposed for a class of unknown general discrete-time SISO nonlinear systems (Hou and Huang 1997). 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 dynamic linearization technique. The resulting time-varying parameter pseudo-partialderivative is estimated using only I/O data of the plant. Obviously, MFAC falls into the second category. The stability and convergence of MFAC scheme for regulation problem are proposed in (Hou and Jin 2011), and it has been successfully implemented in many applications. Further, to determine the controller structure without involving neither the dynamic model nor the structure information of the plant, a new type of MFAC method, controller compact form dynamic linearization based MFAC (CFDLc-MFAC), is proposed for a class of unknown nonlinear system (Zhu and Hou 2012). The feature of this method is that, the two equivalent dynamic linearization data models, one is on the unknown ‘ideal’ controller to a general discrete-time nonlinear system and the other is on the controlled plant, are used to design and analyze the control system.
There are three different kinds of dynamic linearization data models in MFAC method, including compact form dynamic linearization (CFDL) data model, partial form dynamic linearization (PFDL) data model, and full form dynamic linearization (FFDL) data model (Hou 1999). For the sake of brevity and readability, throughout this paper, only the CFDL method is considered. If a system satisfied the generalized Lipschitz condition, that is, Δy (k + 1) ≤ κ Δu (k ) for any fixed k and Δu (k ) ≠ 0 , equation (1) can be equivalently expressed as following, which is called CFDLp, y (k + 1) = y (k ) + φ (k )Δu (k ) ,
where φ (k ) is the PPD of controlled plant (Hou and Jin 2011). The proof of derivation of (2) is given in (Hou and Jin 2011). The proposed dynamic linearization method is novel as the following:
Note that, MBC methods have strong ability to control the plant when its accurate model is available and have a series of designing and analysis tools, but the DDC methods do have a better performance when the plant model is not available but lacking of systematic designing procedures or analysis means. Under this observation, a new type of MFAC, gradient-based MFAC (GbMFAC), is proposed to combine the advantages of both MBC and DDC method to make full use the knowledge of the controlled plant. Different from the CFDLc-MFAC, the controller parameter in GbMFAC is tuned using the available gradient information. The control performance is expected to be improved since we have more information to guide the controller tuning.
1) It is an equivalent dynamic linearization data model rather than an approximation model, which only serves for the controller design. 2) The dynamic linearization model has time-varying incremental form with very simple structure and very few parameters, in which the order, time delay and the model structure of the controlled plant is not required. The introduction of pseudo orders can facilitate the controller design. 3) The behavior of pseudo partial derivate of the dynamic linearization model is not sensitive to the time varying parameter, the time varying structure and the delay of the controlled plant.
The outline of this paper is as follows. Section II briefly introduces the MFAC method, including the MFAC prototype, CFDLc-MFAC and the GbMFAC. Section III discusses the main difference and relationship between DDC method and the MBC method. Simulation comparison results are shown in Section IV, Section V concludes the work.
4) This dynamic linearization method could be easily extended to the cases of MISO, MIMO nonlinear systems without any difficulty (Hou and Jin 2011).
Throughout the paper, definition Δx(k ) = x(k ) − x(k − 1) is used.
To design an optimal controller based on CFDL, an estimation algorithm for time varying parameter should be considered to obtain the value of PPD in CFDL data model firstly. Thus, the following modified criterion function is used,
II. MODEL FREE ADAPTIVE CONTROL
J (φ (k )) = (Δy (k ) − φ (k )Δu (k − 1)) 2 + μ (φ (k ) − φˆ(k − 1)) 2 (3)
The nonlinear controlled plant is
y (k + 1) = f (y (k ), , y (k − n y ), u (k ), , u (k − nu )) ,
(2)
(1) where μ > 0 is weighting factor. Minimizing (3) with respect to φ (k ) gives the following PPD estimation algorithm,
where n y and nu are the unknown orders of output and input, respectively, and f () is an unknown nonlinear function.
Δφˆ(k ) =
A. MFAC prototype
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η k Δu (k − 1) (Δy (k ) − φˆ(k − 1)Δu (k − 1)) , (4) μ + Δu (k − 1) 2
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φˆ(k ) = φˆ(1), if φˆ(k ) < σˆ or φˆ(k ) > κ
merely using the I/O data of the controller and of the controlled plant.
(5)
where, η k is step-size sequence which is added to increase flexibility of the algorithm, σˆ is a small positive constant.
For nonlinear system (1), assume that there exists an ideal nonlinear controller with form (8), which can stabilize the plant (1) and drive the plant output to track the desired signal asymptotically.
Next, consider the following weighted one step-ahead control input cost function, J (u (k )) = ( yd (k + 1) − y (k + 1)) 2 + λΔu (k ) 2
u (k ) = C (u (k − 1), , u (k − nc ), e(k + 1), , e(k − ne )) , (8)
(6)
where C (⋅) is a smooth unknown nonlinear function, nc , ne are the two unknown orders of the controller.
where yd (k + 1) is desired output of controlled system at the time instant k + 1 , λ is a weighting factor.
Similar process as the derivation of CFDLp, if system (8) satisfies the generalized Lipschitz condition, that is, Δu (k ) ≤ b Δe(k + 1) for any fixed k and Δe(k + 1) ≠ 0 ,
Substituting (2) into (6) and solving the optimal problem, a control law can be given as follows Δu (k ) =
ρ k φˆ(k ) λ + φˆ(k )
2
( yd (k + 1) − y (k )) ,
then equation (8) can be equivalently expressed as following dynamic linearization data model
(7)
Δu (k ) = ψ (k )Δe(k + 1) ,
where, ρ k is the step-size sequence which is added to increase flexibility of the controller.
(9)
where ψ (k ) is the PPD of controller (Zhu and Hou 2012). Since controller (9) is the equivalent form of the ideal controller (8) for controlled plant (1), a perfect control performance, that is e(k + 1) = 0 , is expected to be able to achieved by using an qualified parameter estimation algorithm. However, it does not mean the actual output tracking error will vanish at one-step ahead consequently due to the uncertainties or estimation error in practice. A practical controllers based on the equivalent form (9) of the ideal controller (8), called CFDLc, is as follows
According to (3)-(7), the controller design does not rely on the mathematic model and the order of the controlled plant, and only the measured I/O data of the plant is utilized. The scheme can deal with the adaptive control problem of the plant with time-varying parameter and time-varying structure. Because only one parameter needs to be tuned in this control scheme, the algorithm can be executed on line and suitable for real time control. Extensive simulation studies shows that the pseudo partial derivative φ (k ) is a slow time-varying parameter, so almost all the traditional time-varying parameter estimation algorithm can be applied to calculate it. The resetting algorithm (5) can strengthen the tracking ability of estimated algorithm.
Δu (k ) = −ψ (k )e(k ) .
(10)
To tune the parameter ψ (k ) , consider the following cost function with an additional penalty on the abrupt rate of estimated parameter
The convergence result of MFAC is show in (Hou and Jin 2011), which is listed briefly as follows.
J (ψ (k )) = (yd (k + 1) − y (k + 1)) 2 + λk (ψ (k ) −ψˆ (k − 1)) 2 , (11)
Assumption 1 The system (1) is a smooth nonlinear function, and the partial derivatives of f (⋅, ⋅, ⋅) with respect to the control input u (⋅) are continuous.
where λk > 0 is weighting factor series. Noting that y (k + 1) is unavailable at instant k since model of controlled plant is unknown. Thus we need to resort to the CFDL data model (2).
Assumption 2 System (1) is generalized Lipschitz.
Substituting (2) and (10) into (11), differentiating (11) with respect to ψ (k ) and setting it to be zero, yields the following update law
Assumption 3 φ (k ) > σˆ , ∀k ∈ Ζ , where σˆ is a small positive constant. +
Δψˆ (k )= −
Theorem 1 The plant described by (1) satisfying assumptions 1-3 is controlled by the MFAC scheme (4)(5)(7) for regulator yd (k + 1) = const . Then there exists a λmin > 0 , and if λ > λmin , it guarantees: 1) lim k →∞ yd (k + 1) − y (k + 1) = 0 , and 2) { y (k )} and {u (k )} are bound sequences.
φ (k )e(k )(yd (k + 1) − y(k ) + φ (k )ψˆ (k − 1)e(k )) λk + (φ (k )e(k ))2
(12)
Thus, the control law algorithm Δu (k ) = −ψˆ (k )e(k ) ,
(13)
the estimation algorithm of controller PPD ψ (k ) B. CFDLc-MFAC Δψˆ (k )= −
Different from the MFAC prototype, this method is based on the equivalent dynamic linearization method both on the ideal controller and on the controlled plant, which can be realized 84
φˆ(k )e(k )( yd (k + 1) − y (k ) + φˆ(k )ψˆ (k − 1)e(k )) ,(14) λk + (φˆ(k )e(k )) 2
IFAC LSS 2013 July 7-10, 2013. Shanghai, China
ψˆ (k )= − b if ψˆ (k ) < − b or ψˆ (k ) > −b
Obviously, if the ∂y (k + 1) / ∂u (k ) is known a prior, the ∂y (k + 1) / ∂ψ (k ) can be calculated directly. Thus, if this gradient information can be used in the CFDLc-MFAC, the control performance should be improved since ∂y (k + 1) / ∂u (k ) is accurately. In this GbMFAC, we use the available plant model to calculate ∂y (k + 1) / ∂u (k ) since the plant model is assumed to be known in this work.
(15)
and estimation algorithm of plant PPD φ (k ) , (4) and (5), gives the system control scheme of CFDLc-MFAC for the nonlinear system (1). The sign of ψˆ (k ) is always negative and φˆ(k ) is positive according to the resetting conditions in algorithm (4) and (5). The reason why we reset the ψˆ (k )= − b as (15) is that the controller gain of ψˆ (k ) should not be too small or too large. Otherwise, it cannot be implementable or energy saving device.
Setting the ∂J ∂ψ (k ) to zero, we can obtain the estimation of ψ (k ) . Substituting it into the control law (13) gives the GbMFAC. It should be noted that, noise and disturbance always exist in reality, some steps should be taken to reduce these influence to get an almost unbiased derivative. A simple way is using a moving average algorithm or some filter to preprocess the calculated value, which can block the high frequency noise or disturbance.
The controller in CFDLc-MFAC is directly derived from the ideal controller and does not rely on plant model or other data model of the plant. The convergence result of CFDLc-MFAC is show in (Zhu and Hou 2012), which is listed briefly as follows.
III. MODEL BASED CONTROL VS DATA DRIVEN CONTROL
Assumption 4 The controller (8) is a smooth nonlinear function, and the partial derivatives of C (⋅) with respect to error input signals e(⋅) are continuous.
A. Difference between MBC and DDC methods The main difference between MBC and DDC methods is that, one is the controller design approach which relies on the accurate model of the plant, and the other is that the controller design is independent of the plant model. Since the controller is designed only by using the I/O data of the plant, the DDC methods are born with many merits:
Assumption 5 The controller (8) is generalized Lipschitz.. Assumption 6 ψ (k ) ≤ −b ,∀k ∈ Ζ + , where b is a small positive constant. Theorem 2 The plant described by (1) satisfying Assumptions 1-5 is controlled by scheme (4)(5)(13)(14)(15) for regulation problem yd (k + 1) = const . Then the tracking error of closedloop system monotonically converges to zero, that is lim e(k + 1) = 0 , if λk > c(φˆ(k ) − κ )φˆ(k )e(k ) 2 (κ b − c) ≥ 0 ,
1) The controller design in DDC methods does not rely on the plant model. Thanks to dynamic linearization model, the controller structure is determined merely by the on-line I/O data in MFAC.
k →∞
2) The unmodelled dynamics and robustness in traditional MBC theory does not exist in DDC method since plant model is not required in DDC controller design.
0 < σˆ ≤ φˆ(k ) ≤ κ and κ b < c < 2 , c > 1 .
C. Proposed Gradient-based MFAC
3) Stability and convergence analysis of DDC approaches do not depend on the accurate plant model. Different from the MBC method, such as, direct adaptive control, which uses the system dynamics and system structure information, the data driven MFAC has a series theoretical results that only requires the upper bound of the pseudo partial derivative.
The partial derivative or gradient information of system output with respect to control input ∂y (k + 1) / ∂u (k ) is crucial for the controller design in DDC approach, since the plant model is unknown, instead a huge of measured I/O data of the controlled plant. In CFDLc-MFAC method, projection algorithm is used to calculate the partial derivate ∂y (k + 1) / ∂ψ (k ) by virtue of the data model. It requires an additional estimation algorithm to get the parameter of the data model. Even though the estimation accuracy does not influence the stability of the control system from theoretical analysis, it definitely affects the control performance if the estimation PPD is not good enough. Note that, from the cost function (11), the partial derivate is
In MFAC methods, the controller is independent of the plant model, an equivalent data model of a controlled plant or equivalent controller data model to an ideal controller is derived by rigorous mathematical theory, and only the online/offline measurement I/O dada is used, thus it can make the control system safe and reliable.
B. Relationship between MBC and DDC methods
∂J ∂ψ (k ) = 2( y (k + 1) − yd (k + 1) (∂y (k + 1) ∂ψ (k )) + 2λΔψ (k ) . = 2( y (k + 1) − yd (k + 1)
Both MBC and DDC methods aim to design the controller to drive the output of the controlled plant to track the desired signal or to satisfy a specified criterion. Moreover, each control method, no matter it is a MBC or a DDC method, does have its own advantages and disadvantages. MBC methods can do well for the control problem if accurate plant model is
(16)
∂y (k + 1) ∂u (k ) + 2λΔψ (k ) ∂u (k ) ∂ψ (k )
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θ1 (k ) = 1 + 0.2sin(2kπ 400) , θ 2 (k ) = 0.5 + 0.2k 400
available and have a systematic designing and analysis tools; but DDC methods are superior when the accurate plant model is not available. As well known to all, when we design the controller, some information about the plant may be known to us. Thus, we can use this information to design the data driven controller to improve the control performance, like the GbMFAC method.
θ3 (k ) = e
(18)
− k 400
are time varying parameters of the plant, and v(k ) is random normal distribution output noise with mean 0 and standard derivation 0.01.
The brief introduction on how to design a control system in a complementary modularized manner using MBC and DDC methods or among DDC methods is discussed in (Hou and Wang 2013).
The setup of the MFAC methods is as follows, MFAC: ρ k = 1, λ = 10,ηk = 0.1, μ = 0.01 . CFDLc-MFAC: ρk = 1, λ = 10,ηk = 0.1, μ = 0.01,ψˆ (1) = −0.03 .
C. Key issue in DDC methods
GbMFAC: ρk = 1, λ = 10,ηk = 0.1, μ = 0.01,ψˆ (1) = −0.03 .
In DDC methods, the tracking error e(k + 1) is not available since the plant model is unknown, instead of a huge amount of measured I/O data of the controlled system. Thus, how to get the one-step ahead tracking error for the controller implementation becomes the key issue to DDC methods.
The least square approach is used in AC method to estimate the parameters of the plant. Since there are 3 parameters in the plant model, the weighting factor matrix is set as 30000 ⋅ I 3*3 , and the initial parameters are set as θ = [0.7, 0.7, 0.7]T .
Since the controller structure is independent of the plant model in DDC methods, predict the one-step-ahead output of the plant for the controller parameter tuning using some optimization approaches only using the I/O data is necessary. Theoretically speaking, any existing prediction methods can serve as the predictor, including the data-based model, dynamics model based model, rule-based model, neural networks based model, and so on. Thus, data-based modeling is of great significance for the development of DDC theories. In MFAC prototype, an equivalent dynamic linearization data model for the plant is introduced. With this equivalent data model of the plant, the tracking error is obtained and the controller is designed by projection algorithm. In CFDLcMFAC, the controller is derived from the equivalent dynamic linearization model on the ideal controller and then the controller parameter is tuned by using the data model of the plant. In GbMFAC, supposing the ∂y (k + 1) / ∂ψ (k ) is known, this partial derivative information is used to tuned the controller parameter.
Figure 1. Control Performance
IV. SIMULATION STUDY In this subsection, numerical example is given to show the comparisons between MFAC, CDFLc-MFAC, GbMFAC and traditional adaptive control method, respectively. Note that, in DDC methods, the following model is only used to generate the I/O data, and no dynamics of the plant is required in the controller design. The controlled nonlinear plant is y (k + 1) = θ1 (k )
0.1 + y (k − 1) + θ 2 (k )( y ( k ) + 1)u (k ) 1 + y (k − 1) 2 (17)
Figure 2. Control Input
The control performance of the four methods is shown in Fig.1, and the corresponding control inputs are shown in Fig. 2. Figure 3 shows the time-varying parameters of the plant. 50 simulations are carried out and the average of the numerical indexes of ITAE of these 50 simulations, as (19), are listed in Table I.
+θ 3 (k )u (k − 1)3 +v(k )
where
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From Fig.1, one can see that all the DDC methods are effective even though there are measurement noise in the output and time varying parameters in the plant. It illustrates that these three kinds of MFAC method is born with a strong robustness and adaptation since it does not rely on the plant model. While, the control performance of AC is not satisfactory with bad transient and large offset. The reason is that the plant parameters change ‘fast’ and traditional AC can only deal with the system with constant parameters or slow time varying parameters.
Extensive simulations have shown that MFAC method can replace the traditional model based adaptive controls with a much better control performances both for linear and nonlinear discrete-time systems. REFERENCES Kalman, R. E. (2009). Contributions to the theory of optimal control, Boletin de la Sociedad Matematica Mexicana, 5, 102-119. Kalman, R. E. (1960). A new approach to linear filtering and prediction problems, Transactions ASME, Series D, J. Basic Eng., 82, 34-45. Astrom, K. J. and Wittenmark, B. (1994). Adaptive control: Addison-Wesley Longman Publishing Co., Inc., pp. 19– 22. Rohrs, C. E., Valavani, L., Athans, M., and Stein, G. (1982). Robustness of adaptive control algorithms in the presence of unmodeled dynamics, in Proc. the 21st IEEE conference on decision and control, Orlando. Rohrs, C. E., Valavani, L., Athans, M., and Stein, G. (1985). Robustness of continuous-time adaptive control algorithms in the presence of unmodeled dynamics, IEEE Transactions on Automatic Control, 30 (9), 881889. Anderson, B. D. O., Dehghani, A. (2008). Challenges of adaptive control- past, permanent and future, Annual Reviews in Control, 32, 123-135. Hou, Z. S., Wang, Z. (2013). From model-based control to data-driven control: Survey, classification and perspective, Information Sciences, vol. 235, pp. 3-35 Hou, Z. S. and Huang, W. H. (1997). "The model-free learning adaptive control of a class of SISO nonlinear systems," in the 1997 IEEE American Control Conference, Albuquerque, USA, pp. 343-344. Bontempi, G. and Birattari, M. (2005). "From Linearization to Lazy Learning: A Survey of Divide-and-Conquer Techniques for Nonlinear Control," International Journal of Computational Cognition, vol. 3, pp. 56-73. Sala, A. (2007). "Integrating virtual reference feedback tuning into a unified closed-loop identification framework," Automatica, vol. 43, pp. 178-183. Hjalmarsson, H. (2005). "From experiment design to closedloop control," Automatica, vol. 41, pp. 393-438. Hou, Z. S. and Jin, S.T. (2011). "A novel data-driven control approach for a class of discrete-time nonlinear systems," Control Systems Technology, IEEE Transactions on, vol. 19, pp. 1549-1558. Zhu, Y. M. and Hou, Z. S. (2012). Controller Compact Form Dynamic Linearization Based Model Free Adaptive Control. 51st IEEE Conference on Decision and Control. Maui, Hawaii, USA, pp.4817-4822. Hou, Z. S. (1999). Nonparametric Models and Its Adaptive Control Theory. Beijing: Science Press. Hou, Z. S. and Jin, S. T. (2011). Data driven model-free adaptive control for a class of MIMO nonlinear discretetime systems, IEEE Transactions on Neural Networks, 22(12), 2173-2188.
Figure 3. Time-varying Parameters
In Fig.2, we can see that, the control inputs of the MFAC methods are smooth and the control input of AC has abrupt change at the time when the operation point is changed. It is because that a penalty factor is added in the criterion when the MFAC type controllers are designed, which can limit the rate of controller input; and the controller of traditional AC relies on the “certainty equivalence principle”, in which the change in desired output reflects directly the control input. From the indexes of these four methods, the data driven MFAC methods are much better than the traditional model based AC method. The control performance of MFAC prototype and CFDLc-MFAC are similar and the performance of GbMFAC is the best since the gradient information is utilized to tune the controller parameter. eITAE =
1 50 2 n i 50
TABLE I.
eITAE
(
200 k =1
)
k en,i (k ) .
(19)
COMPARISION OF INDEXES
MFAC
CFDLc-MFAC
GbMFAC
AC
438.9826
432.9906
422.9404
1327.0023
V. CONCLUSION In this paper, we have presented a new type of MFAC approach that can use the known gradient information to tune the controller parameters in order to improve the control performance. The dynamic linearized controller is a novel representation of the unknown ‘ideal’ nonlinear controller, which does not rely on the plant model. Therefore, the problems of unmodeled dynamics and conventional robustness in traditional MBC framework are avoided. 87
Model based control and MFAC, which is better in simulation? * Zhongsheng Hou*1 and Yuanming Zhu** Advanced Control Systems Lab, School of Electronic and Information Engineering, Beijing Jiaotong University, Beijing, China * (e-mail: [email protected]) ** (e-mail: [email protected]) Abstract: This work presents a new data driven model free adaptive controller by virtue of the gradient information of the available plant model. Our approach is novel in the sense that we use the measured data to directly design the controller, and predict the system output using the plant model if the plant model is available. Whereas, the traditional model based control approach uses the measured data to identify the plant model firstly, and then uses the model to design and analyze the controller design based on this identified model. The main contribution is that this novel scheme opens a way for the data driven control method utilizing the plant model if the plant model is available. The theoretical analysis and the simulation results demonstrate the effectiveness and superiorities of the proposed method. complicated. Modeling the process becomes more and more difficult, and the traditional MBC theory may fail to deal with these kinds of processes since the accurate model is unavailable. However, a huge number of process data, containing valuable information of operations and equipments, is generated and stored. In this case, how to use these huge amount process data, both on-line and off-line, to directly design controller for industrial processes, would has great significance when the accurate process models are unavailable. Therefore, the establishment and development of the data driven control theory (DDC) are the urgent issues in both theories and applications.
I. INTRODUCTION The modern control theory, which is also called model based control (MBC), is derived from the state-space model introduced by Kalman in 1960 and the optimal control (Kalman 1960; Kalman 2009). As a main branch of MBC approach, adaptive control method is developed to deal with the controlled system with unknown constant or time varying parameters. Traditional adaptive control theory uses the available measured input and output data of the controlled plant to identify the parameters firstly, and then the controller is designed based on the plant model by using the “certainty equivalence principle” with the faith that the plant model could represent the real system. Explicit identification of the plant parameters leads to indirect adaptive control whereas direct identification of the controller parameters leads to direct adaptive control (Astrom and Wittenmark 1994). However, in either case, the model based controller design would not work well if the plant model does not fall into the assumed model set. Thus, the resulting controller based on the plant model will definitely lead to unsatisfied performance. Rohr’s counterexample (Rohrs et al. 1982; Rohrs et al. 1985) has demonstrated that the reported stable adaptive control system based on some assumptions made about the plant model could lead to unstable behavior when there is an unmodelled dynamics. Securing safe adaptive control is far from straightforward this well-known model based adaptive control (Anderson and Dehghani 2008).
So far, there are over 10 different kinds of DDC methods in literature (Hou and Wang 2013), such as model free adaptive control (MFAC) (Hou and Huang 1997), lazy learning control (LL) (Bontempi and Birattari 2005), virtual reference feedback tuning (VRFT) (Sala 2007), iterative feedback tuning (IFT) (Hjalmarsson, 2005), etc.. In terms of the methods on how to design the controller structure, there are two fundamental categories of DDC methods. One is that, the controller structure with some unknown parameters is supposed to contain the optimal controller for the controlled plant. It could come from experimental knowledge on the plant, or be derived from the structure of the plant. Then the controller design issue is transformed into the direct identification problem for the controller parameter. Most of DDC methods, such as VRFT and IFT, are in this category. The other one is that, the general controller is designed based on certain function approximations or some equivalent descriptions on the original controlled plant, such as neural networks, Taylor approximation, or other equivalent representation, and then the controller parameters are tuned by minimizing a specified performance criterion using the
As the development of information science and technology, the practical process, for instance, chemical industry, metallurgy, machinery, electronics, electricity, transportation, has been undergone significant changes. The scale of the industry becomes large, and the production technology, equipment and production process become more and more Resrach supported by NSFC under granted No. 61120106009. Hou Zhongsheng is the corresponding author.
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The essential idea of MFAC is using an equivalent dynamic linearization data model and a novel concept pseudo partial derivative (PPD) at current operation point to represent the general discrete time nonlinear system. Then the optimal controller is derived based on estimation of the PPD on-line solely using the input and output data of the controlled plant.
I/O data, including off-line and on-line data. Typical ones are MFAC, LL, etc.. The MFAC prototype is firstly proposed for a class of unknown general discrete-time SISO nonlinear systems (Hou and Huang 1997). 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 dynamic linearization technique. The resulting time-varying parameter pseudo-partialderivative is estimated using only I/O data of the plant. Obviously, MFAC falls into the second category. The stability and convergence of MFAC scheme for regulation problem are proposed in (Hou and Jin 2011), and it has been successfully implemented in many applications. Further, to determine the controller structure without involving neither the dynamic model nor the structure information of the plant, a new type of MFAC method, controller compact form dynamic linearization based MFAC (CFDLc-MFAC), is proposed for a class of unknown nonlinear system (Zhu and Hou 2012). The feature of this method is that, the two equivalent dynamic linearization data models, one is on the unknown ‘ideal’ controller to a general discrete-time nonlinear system and the other is on the controlled plant, are used to design and analyze the control system.
There are three different kinds of dynamic linearization data models in MFAC method, including compact form dynamic linearization (CFDL) data model, partial form dynamic linearization (PFDL) data model, and full form dynamic linearization (FFDL) data model (Hou 1999). For the sake of brevity and readability, throughout this paper, only the CFDL method is considered. If a system satisfied the generalized Lipschitz condition, that is, Δy (k + 1) ≤ κ Δu (k ) for any fixed k and Δu (k ) ≠ 0 , equation (1) can be equivalently expressed as following, which is called CFDLp, y (k + 1) = y (k ) + φ (k )Δu (k ) ,
where φ (k ) is the PPD of controlled plant (Hou and Jin 2011). The proof of derivation of (2) is given in (Hou and Jin 2011). The proposed dynamic linearization method is novel as the following:
Note that, MBC methods have strong ability to control the plant when its accurate model is available and have a series of designing and analysis tools, but the DDC methods do have a better performance when the plant model is not available but lacking of systematic designing procedures or analysis means. Under this observation, a new type of MFAC, gradient-based MFAC (GbMFAC), is proposed to combine the advantages of both MBC and DDC method to make full use the knowledge of the controlled plant. Different from the CFDLc-MFAC, the controller parameter in GbMFAC is tuned using the available gradient information. The control performance is expected to be improved since we have more information to guide the controller tuning.
1) It is an equivalent dynamic linearization data model rather than an approximation model, which only serves for the controller design. 2) The dynamic linearization model has time-varying incremental form with very simple structure and very few parameters, in which the order, time delay and the model structure of the controlled plant is not required. The introduction of pseudo orders can facilitate the controller design. 3) The behavior of pseudo partial derivate of the dynamic linearization model is not sensitive to the time varying parameter, the time varying structure and the delay of the controlled plant.
The outline of this paper is as follows. Section II briefly introduces the MFAC method, including the MFAC prototype, CFDLc-MFAC and the GbMFAC. Section III discusses the main difference and relationship between DDC method and the MBC method. Simulation comparison results are shown in Section IV, Section V concludes the work.
4) This dynamic linearization method could be easily extended to the cases of MISO, MIMO nonlinear systems without any difficulty (Hou and Jin 2011).
Throughout the paper, definition Δx(k ) = x(k ) − x(k − 1) is used.
To design an optimal controller based on CFDL, an estimation algorithm for time varying parameter should be considered to obtain the value of PPD in CFDL data model firstly. Thus, the following modified criterion function is used,
II. MODEL FREE ADAPTIVE CONTROL
J (φ (k )) = (Δy (k ) − φ (k )Δu (k − 1)) 2 + μ (φ (k ) − φˆ(k − 1)) 2 (3)
The nonlinear controlled plant is
y (k + 1) = f (y (k ), , y (k − n y ), u (k ), , u (k − nu )) ,
(2)
(1) where μ > 0 is weighting factor. Minimizing (3) with respect to φ (k ) gives the following PPD estimation algorithm,
where n y and nu are the unknown orders of output and input, respectively, and f () is an unknown nonlinear function.
Δφˆ(k ) =
A. MFAC prototype
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η k Δu (k − 1) (Δy (k ) − φˆ(k − 1)Δu (k − 1)) , (4) μ + Δu (k − 1) 2
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φˆ(k ) = φˆ(1), if φˆ(k ) < σˆ or φˆ(k ) > κ
merely using the I/O data of the controller and of the controlled plant.
(5)
where, η k is step-size sequence which is added to increase flexibility of the algorithm, σˆ is a small positive constant.
For nonlinear system (1), assume that there exists an ideal nonlinear controller with form (8), which can stabilize the plant (1) and drive the plant output to track the desired signal asymptotically.
Next, consider the following weighted one step-ahead control input cost function, J (u (k )) = ( yd (k + 1) − y (k + 1)) 2 + λΔu (k ) 2
u (k ) = C (u (k − 1), , u (k − nc ), e(k + 1), , e(k − ne )) , (8)
(6)
where C (⋅) is a smooth unknown nonlinear function, nc , ne are the two unknown orders of the controller.
where yd (k + 1) is desired output of controlled system at the time instant k + 1 , λ is a weighting factor.
Similar process as the derivation of CFDLp, if system (8) satisfies the generalized Lipschitz condition, that is, Δu (k ) ≤ b Δe(k + 1) for any fixed k and Δe(k + 1) ≠ 0 ,
Substituting (2) into (6) and solving the optimal problem, a control law can be given as follows Δu (k ) =
ρ k φˆ(k ) λ + φˆ(k )
2
( yd (k + 1) − y (k )) ,
then equation (8) can be equivalently expressed as following dynamic linearization data model
(7)
Δu (k ) = ψ (k )Δe(k + 1) ,
where, ρ k is the step-size sequence which is added to increase flexibility of the controller.
(9)
where ψ (k ) is the PPD of controller (Zhu and Hou 2012). Since controller (9) is the equivalent form of the ideal controller (8) for controlled plant (1), a perfect control performance, that is e(k + 1) = 0 , is expected to be able to achieved by using an qualified parameter estimation algorithm. However, it does not mean the actual output tracking error will vanish at one-step ahead consequently due to the uncertainties or estimation error in practice. A practical controllers based on the equivalent form (9) of the ideal controller (8), called CFDLc, is as follows
According to (3)-(7), the controller design does not rely on the mathematic model and the order of the controlled plant, and only the measured I/O data of the plant is utilized. The scheme can deal with the adaptive control problem of the plant with time-varying parameter and time-varying structure. Because only one parameter needs to be tuned in this control scheme, the algorithm can be executed on line and suitable for real time control. Extensive simulation studies shows that the pseudo partial derivative φ (k ) is a slow time-varying parameter, so almost all the traditional time-varying parameter estimation algorithm can be applied to calculate it. The resetting algorithm (5) can strengthen the tracking ability of estimated algorithm.
Δu (k ) = −ψ (k )e(k ) .
(10)
To tune the parameter ψ (k ) , consider the following cost function with an additional penalty on the abrupt rate of estimated parameter
The convergence result of MFAC is show in (Hou and Jin 2011), which is listed briefly as follows.
J (ψ (k )) = (yd (k + 1) − y (k + 1)) 2 + λk (ψ (k ) −ψˆ (k − 1)) 2 , (11)
Assumption 1 The system (1) is a smooth nonlinear function, and the partial derivatives of f (⋅, ⋅, ⋅) with respect to the control input u (⋅) are continuous.
where λk > 0 is weighting factor series. Noting that y (k + 1) is unavailable at instant k since model of controlled plant is unknown. Thus we need to resort to the CFDL data model (2).
Assumption 2 System (1) is generalized Lipschitz.
Substituting (2) and (10) into (11), differentiating (11) with respect to ψ (k ) and setting it to be zero, yields the following update law
Assumption 3 φ (k ) > σˆ , ∀k ∈ Ζ , where σˆ is a small positive constant. +
Δψˆ (k )= −
Theorem 1 The plant described by (1) satisfying assumptions 1-3 is controlled by the MFAC scheme (4)(5)(7) for regulator yd (k + 1) = const . Then there exists a λmin > 0 , and if λ > λmin , it guarantees: 1) lim k →∞ yd (k + 1) − y (k + 1) = 0 , and 2) { y (k )} and {u (k )} are bound sequences.
φ (k )e(k )(yd (k + 1) − y(k ) + φ (k )ψˆ (k − 1)e(k )) λk + (φ (k )e(k ))2
(12)
Thus, the control law algorithm Δu (k ) = −ψˆ (k )e(k ) ,
(13)
the estimation algorithm of controller PPD ψ (k ) B. CFDLc-MFAC Δψˆ (k )= −
Different from the MFAC prototype, this method is based on the equivalent dynamic linearization method both on the ideal controller and on the controlled plant, which can be realized 84
φˆ(k )e(k )( yd (k + 1) − y (k ) + φˆ(k )ψˆ (k − 1)e(k )) ,(14) λk + (φˆ(k )e(k )) 2
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ψˆ (k )= − b if ψˆ (k ) < − b or ψˆ (k ) > −b
Obviously, if the ∂y (k + 1) / ∂u (k ) is known a prior, the ∂y (k + 1) / ∂ψ (k ) can be calculated directly. Thus, if this gradient information can be used in the CFDLc-MFAC, the control performance should be improved since ∂y (k + 1) / ∂u (k ) is accurately. In this GbMFAC, we use the available plant model to calculate ∂y (k + 1) / ∂u (k ) since the plant model is assumed to be known in this work.
(15)
and estimation algorithm of plant PPD φ (k ) , (4) and (5), gives the system control scheme of CFDLc-MFAC for the nonlinear system (1). The sign of ψˆ (k ) is always negative and φˆ(k ) is positive according to the resetting conditions in algorithm (4) and (5). The reason why we reset the ψˆ (k )= − b as (15) is that the controller gain of ψˆ (k ) should not be too small or too large. Otherwise, it cannot be implementable or energy saving device.
Setting the ∂J ∂ψ (k ) to zero, we can obtain the estimation of ψ (k ) . Substituting it into the control law (13) gives the GbMFAC. It should be noted that, noise and disturbance always exist in reality, some steps should be taken to reduce these influence to get an almost unbiased derivative. A simple way is using a moving average algorithm or some filter to preprocess the calculated value, which can block the high frequency noise or disturbance.
The controller in CFDLc-MFAC is directly derived from the ideal controller and does not rely on plant model or other data model of the plant. The convergence result of CFDLc-MFAC is show in (Zhu and Hou 2012), which is listed briefly as follows.
III. MODEL BASED CONTROL VS DATA DRIVEN CONTROL
Assumption 4 The controller (8) is a smooth nonlinear function, and the partial derivatives of C (⋅) with respect to error input signals e(⋅) are continuous.
A. Difference between MBC and DDC methods The main difference between MBC and DDC methods is that, one is the controller design approach which relies on the accurate model of the plant, and the other is that the controller design is independent of the plant model. Since the controller is designed only by using the I/O data of the plant, the DDC methods are born with many merits:
Assumption 5 The controller (8) is generalized Lipschitz.. Assumption 6 ψ (k ) ≤ −b ,∀k ∈ Ζ + , where b is a small positive constant. Theorem 2 The plant described by (1) satisfying Assumptions 1-5 is controlled by scheme (4)(5)(13)(14)(15) for regulation problem yd (k + 1) = const . Then the tracking error of closedloop system monotonically converges to zero, that is lim e(k + 1) = 0 , if λk > c(φˆ(k ) − κ )φˆ(k )e(k ) 2 (κ b − c) ≥ 0 ,
1) The controller design in DDC methods does not rely on the plant model. Thanks to dynamic linearization model, the controller structure is determined merely by the on-line I/O data in MFAC.
k →∞
2) The unmodelled dynamics and robustness in traditional MBC theory does not exist in DDC method since plant model is not required in DDC controller design.
0 < σˆ ≤ φˆ(k ) ≤ κ and κ b < c < 2 , c > 1 .
C. Proposed Gradient-based MFAC
3) Stability and convergence analysis of DDC approaches do not depend on the accurate plant model. Different from the MBC method, such as, direct adaptive control, which uses the system dynamics and system structure information, the data driven MFAC has a series theoretical results that only requires the upper bound of the pseudo partial derivative.
The partial derivative or gradient information of system output with respect to control input ∂y (k + 1) / ∂u (k ) is crucial for the controller design in DDC approach, since the plant model is unknown, instead a huge of measured I/O data of the controlled plant. In CFDLc-MFAC method, projection algorithm is used to calculate the partial derivate ∂y (k + 1) / ∂ψ (k ) by virtue of the data model. It requires an additional estimation algorithm to get the parameter of the data model. Even though the estimation accuracy does not influence the stability of the control system from theoretical analysis, it definitely affects the control performance if the estimation PPD is not good enough. Note that, from the cost function (11), the partial derivate is
In MFAC methods, the controller is independent of the plant model, an equivalent data model of a controlled plant or equivalent controller data model to an ideal controller is derived by rigorous mathematical theory, and only the online/offline measurement I/O dada is used, thus it can make the control system safe and reliable.
B. Relationship between MBC and DDC methods
∂J ∂ψ (k ) = 2( y (k + 1) − yd (k + 1) (∂y (k + 1) ∂ψ (k )) + 2λΔψ (k ) . = 2( y (k + 1) − yd (k + 1)
Both MBC and DDC methods aim to design the controller to drive the output of the controlled plant to track the desired signal or to satisfy a specified criterion. Moreover, each control method, no matter it is a MBC or a DDC method, does have its own advantages and disadvantages. MBC methods can do well for the control problem if accurate plant model is
(16)
∂y (k + 1) ∂u (k ) + 2λΔψ (k ) ∂u (k ) ∂ψ (k )
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θ1 (k ) = 1 + 0.2sin(2kπ 400) , θ 2 (k ) = 0.5 + 0.2k 400
available and have a systematic designing and analysis tools; but DDC methods are superior when the accurate plant model is not available. As well known to all, when we design the controller, some information about the plant may be known to us. Thus, we can use this information to design the data driven controller to improve the control performance, like the GbMFAC method.
θ3 (k ) = e
(18)
− k 400
are time varying parameters of the plant, and v(k ) is random normal distribution output noise with mean 0 and standard derivation 0.01.
The brief introduction on how to design a control system in a complementary modularized manner using MBC and DDC methods or among DDC methods is discussed in (Hou and Wang 2013).
The setup of the MFAC methods is as follows, MFAC: ρ k = 1, λ = 10,ηk = 0.1, μ = 0.01 . CFDLc-MFAC: ρk = 1, λ = 10,ηk = 0.1, μ = 0.01,ψˆ (1) = −0.03 .
C. Key issue in DDC methods
GbMFAC: ρk = 1, λ = 10,ηk = 0.1, μ = 0.01,ψˆ (1) = −0.03 .
In DDC methods, the tracking error e(k + 1) is not available since the plant model is unknown, instead of a huge amount of measured I/O data of the controlled system. Thus, how to get the one-step ahead tracking error for the controller implementation becomes the key issue to DDC methods.
The least square approach is used in AC method to estimate the parameters of the plant. Since there are 3 parameters in the plant model, the weighting factor matrix is set as 30000 ⋅ I 3*3 , and the initial parameters are set as θ = [0.7, 0.7, 0.7]T .
Since the controller structure is independent of the plant model in DDC methods, predict the one-step-ahead output of the plant for the controller parameter tuning using some optimization approaches only using the I/O data is necessary. Theoretically speaking, any existing prediction methods can serve as the predictor, including the data-based model, dynamics model based model, rule-based model, neural networks based model, and so on. Thus, data-based modeling is of great significance for the development of DDC theories. In MFAC prototype, an equivalent dynamic linearization data model for the plant is introduced. With this equivalent data model of the plant, the tracking error is obtained and the controller is designed by projection algorithm. In CFDLcMFAC, the controller is derived from the equivalent dynamic linearization model on the ideal controller and then the controller parameter is tuned by using the data model of the plant. In GbMFAC, supposing the ∂y (k + 1) / ∂ψ (k ) is known, this partial derivative information is used to tuned the controller parameter.
Figure 1. Control Performance
IV. SIMULATION STUDY In this subsection, numerical example is given to show the comparisons between MFAC, CDFLc-MFAC, GbMFAC and traditional adaptive control method, respectively. Note that, in DDC methods, the following model is only used to generate the I/O data, and no dynamics of the plant is required in the controller design. The controlled nonlinear plant is y (k + 1) = θ1 (k )
0.1 + y (k − 1) + θ 2 (k )( y ( k ) + 1)u (k ) 1 + y (k − 1) 2 (17)
Figure 2. Control Input
The control performance of the four methods is shown in Fig.1, and the corresponding control inputs are shown in Fig. 2. Figure 3 shows the time-varying parameters of the plant. 50 simulations are carried out and the average of the numerical indexes of ITAE of these 50 simulations, as (19), are listed in Table I.
+θ 3 (k )u (k − 1)3 +v(k )
where
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From Fig.1, one can see that all the DDC methods are effective even though there are measurement noise in the output and time varying parameters in the plant. It illustrates that these three kinds of MFAC method is born with a strong robustness and adaptation since it does not rely on the plant model. While, the control performance of AC is not satisfactory with bad transient and large offset. The reason is that the plant parameters change ‘fast’ and traditional AC can only deal with the system with constant parameters or slow time varying parameters.
Extensive simulations have shown that MFAC method can replace the traditional model based adaptive controls with a much better control performances both for linear and nonlinear discrete-time systems. REFERENCES Kalman, R. E. (2009). Contributions to the theory of optimal control, Boletin de la Sociedad Matematica Mexicana, 5, 102-119. Kalman, R. E. (1960). A new approach to linear filtering and prediction problems, Transactions ASME, Series D, J. Basic Eng., 82, 34-45. Astrom, K. J. and Wittenmark, B. (1994). Adaptive control: Addison-Wesley Longman Publishing Co., Inc., pp. 19– 22. Rohrs, C. E., Valavani, L., Athans, M., and Stein, G. (1982). Robustness of adaptive control algorithms in the presence of unmodeled dynamics, in Proc. the 21st IEEE conference on decision and control, Orlando. Rohrs, C. E., Valavani, L., Athans, M., and Stein, G. (1985). Robustness of continuous-time adaptive control algorithms in the presence of unmodeled dynamics, IEEE Transactions on Automatic Control, 30 (9), 881889. Anderson, B. D. O., Dehghani, A. (2008). Challenges of adaptive control- past, permanent and future, Annual Reviews in Control, 32, 123-135. Hou, Z. S., Wang, Z. (2013). From model-based control to data-driven control: Survey, classification and perspective, Information Sciences, vol. 235, pp. 3-35 Hou, Z. S. and Huang, W. H. (1997). "The model-free learning adaptive control of a class of SISO nonlinear systems," in the 1997 IEEE American Control Conference, Albuquerque, USA, pp. 343-344. Bontempi, G. and Birattari, M. (2005). "From Linearization to Lazy Learning: A Survey of Divide-and-Conquer Techniques for Nonlinear Control," International Journal of Computational Cognition, vol. 3, pp. 56-73. Sala, A. (2007). "Integrating virtual reference feedback tuning into a unified closed-loop identification framework," Automatica, vol. 43, pp. 178-183. Hjalmarsson, H. (2005). "From experiment design to closedloop control," Automatica, vol. 41, pp. 393-438. Hou, Z. S. and Jin, S.T. (2011). "A novel data-driven control approach for a class of discrete-time nonlinear systems," Control Systems Technology, IEEE Transactions on, vol. 19, pp. 1549-1558. Zhu, Y. M. and Hou, Z. S. (2012). Controller Compact Form Dynamic Linearization Based Model Free Adaptive Control. 51st IEEE Conference on Decision and Control. Maui, Hawaii, USA, pp.4817-4822. Hou, Z. S. (1999). Nonparametric Models and Its Adaptive Control Theory. Beijing: Science Press. Hou, Z. S. and Jin, S. T. (2011). Data driven model-free adaptive control for a class of MIMO nonlinear discretetime systems, IEEE Transactions on Neural Networks, 22(12), 2173-2188.
Figure 3. Time-varying Parameters
In Fig.2, we can see that, the control inputs of the MFAC methods are smooth and the control input of AC has abrupt change at the time when the operation point is changed. It is because that a penalty factor is added in the criterion when the MFAC type controllers are designed, which can limit the rate of controller input; and the controller of traditional AC relies on the “certainty equivalence principle”, in which the change in desired output reflects directly the control input. From the indexes of these four methods, the data driven MFAC methods are much better than the traditional model based AC method. The control performance of MFAC prototype and CFDLc-MFAC are similar and the performance of GbMFAC is the best since the gradient information is utilized to tune the controller parameter. eITAE =
1 50 2 n i 50
TABLE I.
eITAE
(
200 k =1
)
k en,i (k ) .
(19)
COMPARISION OF INDEXES
MFAC
CFDLc-MFAC
GbMFAC
AC
438.9826
432.9906
422.9404
1327.0023
V. CONCLUSION In this paper, we have presented a new type of MFAC approach that can use the known gradient information to tune the controller parameters in order to improve the control performance. The dynamic linearized controller is a novel representation of the unknown ‘ideal’ nonlinear controller, which does not rely on the plant model. Therefore, the problems of unmodeled dynamics and conventional robustness in traditional MBC framework are avoided. 87