Model Predictive Control of Nonlinear Systems With ... - IEEE Xplore
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Model Predictive Control of Nonlinear Systems With Unmodeled Dynamics Based on Feedforward and Recurrent Neural Networks Zheng Yan, Student Member, IEEE, and Jun Wang, Fellow, IEEE
Abstract—This paper presents new results on a neural network approach to nonlinear model predictive control. At first, a nonlinear system with unmodeled dynamics is decomposed by means of Jacobian linearization to an affine part and a higher-order unknown term. The unknown higher-order term resulted from the decomposition, together with the unmodeled dynamics of the original plant, are modeled by using a feedforward neural network via supervised learning. The optimization problem for nonlinear model predictive control is then formulated as a quadratic programming problem based on successive Jacobian linearization about varying operating points and iteratively solved by using a recurrent neural network called the simplified dual network. Simulation results are included to substantiate the effectiveness and illustrate the performance of the proposed approach. Index Terms—Feedforward neural networks, model predictive control (MPC), real-time optimization, recurrent neural networks, supervised learning, unmodeled dynamics.
I. INTRODUCTION
M
ODEL predictive control (MPC) is an optimization-based control method which has been widely recognized and appreciated in academia and industries. MPC applies real-time optimization to a system model over a finite horizon of predicted future. By taking the current state as an initial state, an objective function is optimized in each sample interval, and the calculation is repeated at the next computation time interval with new state information over a moving time window. As a combination of control and planning, a sequence of optimal control actions are calculated at each step, but only the one for next step is implemented. Compared with other control techniques, MPC has several desirable features, such as the ability to directly take into account input and output constraints as well as the capability to carry out multivariable control. Linear MPC is considered to be a mature technique and achieved great success in process industry (see [3], [5], [13], [20], [27], and [40]–[42]). However, as most industrial pro-
Manuscript received September 15, 2011; revised January 04, 2012; accepted February 22, 2012. Date of publication June 21, 2012; date of current version October 18, 2012. This work was supported by the Research Grants Council of the Hong Kong Special Administrative Region, China, under Grant CUHK417209E. Paper no. TII-11-525. The authors are with the Department of Mechanical and Automation Engineering, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong (e-mail: [email protected], [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/TII.2012.2205582
cesses are inherently nonlinear, linear MPC is not often suitable for the cases where applications involve model changes or adaptations because of changing of operating points and safety-criticality [28], [41]. As a result, nonlinear model predictive control (NMPC) techniques are needed to meet tight performance specifications. The majority of existing NMPC techniques require solving nonconvex optimization problems [5]. However, there are no reliable optimization procedures for solving such problems which would find exact solutions reliably and quickly [27]. Real-time nonconvex optimization is computationally intractable. In order to address the issue of NMPC, several studies were carried out recently. For example, by exploiting multiparametric programming techniques, explicit model predictive control allows one to solve off-line optimization problems for a given range of operating conditions [1]. Optimal control actions are calculated as an explicit function of the state and reference vectors, so that on-line computation reduces to simple function evaluation. In [4], function approximation techniques are applied to improve NMPC efficiency. In [67], by incorporating sensitivity-based strategy with an interior-point nonlinear programming algorithm, an advanced step NMPC technique is proposed to enable the computation of fast approximate solutions to dynamic optimization problems arising in NMPC. In [68], an efficient output feedback nonlinear predictive controller is designed based on – fuzzy modeling and LMI optimization techniques. There are two major drawbacks for these NMPC techniques, first, some controllers are too conceptual to be implemented in practice. Second, many NMPC schemes heavily rely on the structure of underlying processes so that they can not be applied for general nonlinear cases. Hence, further investigations on efficient and implementable NMPC are necessary as well desirable. As MPC technique is model-based, its control reliability is largely determined by the accuracy of prediction model. One critical issue for MPC is how to represent underlying processes with accurate models so that the process outputs over a future horizon can be accurately predicted. There are two main classes of modeling approaches. One is to use a full nonlinear model for prediction without simplifications. A direct consequence is that a high nonlinear (most likely nonconvex as well) optimization needs to be solved during each sampling interval. An alternative is to find linear approximations of the nonlinear process. A benefit of linearization is that MPC optimization task becomes quadratic programming. But MPC with linear model is inherently suboptimal, especially for cases there unmodeled dynamics exist. Improving the accuracy of the model used for prediction is vital for optimizing the overall control performance
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YAN AND WANG: MODEL PREDICTIVE CONTROL OF NONLINEAR SYSTEMS WITH UNMODELED DYNAMICS
of MPC. Explicitly taking into account the residual results from linearization and unmodeled dynamics can improve the modeling accuracy. Real-time optimization is another critical issue for MPC implementation. The effectiveness and efficiency of the optimization method determines whether the MPC technique can be implemented successfully in practice. Classical optimization techniques such as active-set method or interior-point method may not be competent here due to problem dimensionality and stringent requirement on computational time. In the past two decades, neurodynamic optimization based on recurrent neural networks (RNNs) emerged as a promising computational approach for solving various optimization problems. There are several advantages in neurodynamic optimization: first, recurrent neural networks can serve as parallel computational models to solve problems with time-varying parameters; secondly, they can deal with large-scale problems due to their inherent parallelism; thirdly, they are hardware implementable [11]. Tank and Hopfiled first proposed a neural network for linear programming [51]. Kennedy and Chua developed a neural network which extended the results of [51] to general nonlinear programming problems [18]. Their work inspired many researchers to develop neural network models for solving linear and nonlinear optimization problems. Many neural network models have been proposed in recent two decades. In particular, based on the duality and projection methods, several neural networks for solving linear programming, quadratic programming, general convex programming and pseudoconvex programming were developed, e.g., [15], [22]–[24], [58]–[62]. These recurrent neural networks improved performance in terms of global convergence and reduced model complexity. Neural network approaches to system control have been widely investigated. In [38], [50], [56], and [57], comprehensive perspectives on neural networks based control were presented. Topics such as the theoretical foundations on neural networks for systems modeling and control, the model-based neural control design, the stability proofs, as well as the suitability of using neural network models directly within model predictive control strategies were discussed in detail. In [9], [12], [19], [21], [29], [30], [37], [39], [43], [45]–[47], and [65], feedforward neural networks were used for systems modeling. In [17], [25], [36], [44], [53], [54], and [66], recurrent neural networks were used for systems modeling. In [6], [8], [31]–[36], [55], [63], and [64], recurrent neural networks were used for solving the optimization problems involved in MPC. Most existing neural network based model predictive controllers are designed for system models with particular structures, study on a neural network based MPC for more general nonlinear systems is still attractive. In this paper, an NMPC approach based on neural networks to nonlinear systems with unmodeled dynamics is presented. The NMPC problem is first formulated to a quadratic optimization problem with some unknown parameters. The unknown parameters are then modeled off-line based on a feedforward neural network via supervised learning. Then a one-layer recurrent neural network called the simplified dual neural network is applied for solving the quadratic optimization problem to generate the optimal control variables.
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The rest of the paper is organized as follows. In Section II, the NMPC problem is formulated. In Section III, neural networks are applied to model and predict the unknown parameters and to solve the involved optimization problems. In Section IV, simulation results are provided. Finally, Section V concludes this paper. II. PROBLEM FORMULATION Consider a nonlinear discrete-time and time-invariant system with unmodeled dynamics (1) with the following constraints:
(2) is the state vector, is the conwhere trol input vector, is the control increment vector, is the output vector, is a continuous differen, and tiable nonlinear function with represents the unmodeled dynamics of the system, which can be unindentified structures or unknown parameters. It is assumed that is time-invariant and . , and are lower and upper bounds. All state variables of the system are assumed to be measurable. By using Taylor series, the nonlinear function can be decomposed about an operating point into an affine system plus an unknown term at each time instant
(3) and are respectively the Jacobians of where with respect to and , and is the higher-order residual of the corresponding Taylor series. By denoting
setting lated as follows:
, and , system (1) can be reformu-
(4) MPC is an iterative optimization technique: at each sampling time , it measures or estimates the current state, and obtains the optimal input vector by optimizing a cost function. For model
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(1), the following cost function with a finite horizon quadratic criterion is commonly used
The predicted output
is then expressed as follows: (9)
where (5) .. . where denotes the reference vector, denotes the predicted output vector, and denotes the input increment vector, and are prediction horizon and control horizon respectively, and are appropriate weighting matrices, and denote the weighted Euclidean norms defined as and , respectively. The first term in (5) represents the error function between the predicted outputs and the references while the second term is concerned with the control effort. According to the reformulated model (4), the cost function (5) can be rewritten as follows:
.. .
.. .
(6)
..
.. .
.. .
..
.
.. .
.. .
.
Define the following vectors: Therefore, the original optimization problem in (5) becomes
(10) (7) As the linearization is performed once for a given sampling time instant , and the same local linear model is used within the and are written as and for prediction horizon, brevity. According to (3)
(11) where
.. .
.. .
..
.
.. .
(12)
.. . By defining , problem (6) can be rewritten as a quadratic programming (QP) problem
(13) where (8)
YAN AND WANG: MODEL PREDICTIVE CONTROL OF NONLINEAR SYSTEMS WITH UNMODELED DYNAMICS
If and are positive definite, then is also positive definite. So the objective function in (13) is strictly convex. As the feasible region defined by the constraints is a closed convex set, the solution to (13) is unique and satisfies the Karush–Kauhn–Tucker (KKT) optimality conditions [2]. The solution to the QP problem (13) gives control increment vector which can be used to calculate the optimal control vector. III. NEURAL NETWORK APPROACH In [24], Liu and Wang developed a one-layer recurrent network called the simplified dual network for solving quadratic programming problems by utilizing dual variables. The neural network has shown good performance with low computational complexity. The neural network model can be described as follows. • State equation
• Output function (14) where is a positive constant, vector, is the output vector, and linear activation function defined as follows:
is the state is a piecewise
(15) We employ the simplified dual network here for solving (13). The simplified dual network has a single-layer structure with totally neurons. According to the convergence analysis in [24], it is Lyapunov stable and globally convergent to the optimal solution of any strictly convex QP problem. However, it is worth noting that the term in (9) is still unknown. Consequently, the parameters and in (13) are also unknown. is estimated. The QP problem (13) can be solved only after Neural networks are universal approximators and can be used as black-boxes in model identification. Over decades, many feedforward neural networks have been developed with superior capabilities for data regression problems, such as multilayer perceptron networks (MLP-BP) [14] and extreme learning machines (ELM) [16]. In this study, the two neural networks and least squares support vector machines (LS-SVM) [49] were applied to model unknown parameters off-line and their performances were evaluated.
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Let be a global function that is a combination of the higherorder term resulted from Jacobian linearization and unmodeled dynamics of original plant. Given a data set , we can measure a corresponding target data from original plant. Once obtaining two group of sample data sets, we can use a feedforward neural network to map between data sets by treating as the input vector and as the target vector. A well-trained neural network after validation can be used to estimate the value of at each sampling time within the prediction horizon as the global function is valid at any sampling instant. The NMPC scheme based on the feedforward and recurrent neural networks is summarized as follows. 1) Let . Set control time terminal , prediction horizon , control horizon , sampling period , weight matrices and . 2) Model by means of off-line supervised learning based on a feedforward neural network. 3) Estimate using the trained neural network in Step 2. Calculate process model matrices and neural . network matrices 4) Solve the convex optimization problem (13) by using neural network (14) to obtain the optimal input increment vector . 5) Calculate the optimal input vector and implement . , set , go to Step 3; otherwise end. 6) If The advantage of the proposed NMPC scheme is twofold. First, it uses a simple structured prediction model with improved accuracy. By successive linearization, a local linear model plus unknown terms is obtained at each sampling instant. The residual resulted from the Jacobian linearization, together with time-invariant unmodeled dynamics, is estimated off-line by using a feedforward neural network. The explicit structure of estimated underlying function remains unknown, only numerical compensation is taken into account. Thus, only quadratic programming problems are solved on-line. A good compromise of accuracy and efficiency is achieved without confronting nonconvexity. Second, the proposed scheme is implementable and efficient for practical industrial applications. The simplified dual network is capable of solving quadratic optimization problems in microsecond scale as shown in [24]. Due to the inherent parallelism of recurrent neural networks, the proposed scheme would not result in lower computational efficiency even if the optimization problem is of large size. In contrast, most existing NMPC algorithms are computationally demanding when dealing with large-scale multi-variable industrial processes. IV. SIMULATION RESULTS In this section, four application examples are provided to substantiate effectiveness and demonstrate performance of the proposed NMPC approach. Example 1: Consider the following nonaffine continuous stirred tank reactor system [10] with the following model:
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TABLE I COMPARISON OF LEARNING RESULTS IN EXAMPLE 1
Fig. 3. First output response in Example 1.
Fig. 1.
-norm of training errors in Example 1.
Fig. 4. Second output response in Example 1.
Fig. 2.
-norm of testing errors in Example 1.
(16) where and are the concentration and temperature of a tank, respectively; is the coolant flow rate, , and are estimated parameters. The true values are assumed to be , and . is known precisely. The unmodeled dynamics results from parameter errors is represented as . The control objective is to track the reference
(17)
The initial conditions are and . Set , and s. and The constraints are , where the constants and are the maximum values of the coolant flow rate and the tank temperature, respectively. To apply the proposed method, the map from to was first modeled via supervised learning. The learning results of MLP and ELM based on 500 training data and 500 testing data are summarized in Table I and illustrated in Figs. 1–2 where are randomly generated under a uniform distribution. LS-SVM was also applied for comparison purpose. After modeling , the quadratic optimization problem (13) was solved by using the simplified dual network. The simulation results are shown in Figs. 3–6. Compared with the nonlinear controller presented in [7], the proposed method results in shorter settling time and much reduced overshoot. To demonstrate the effectiveness of the proposed method, the method presented in [35] and the linear MPC presented in [31] were also applied on the exact nonlinear model of the plant without unmodeled dynamics. Despite a linear model of the exact nonlinear system was used, the outputs of linear MPC can not track
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Fig. 7.
-norm of training errors in Example 2.
Fig. 8.
-norm of testing errors in Example 2.
Fig. 5. Control input in Example 1.
Fig. 6. Control cost in Example 1.
The nonlinear function
the reference. The total actual control cost over the control ter, for the method minal, herein with ELM, the method in [35] and linear MPC are 2289.9, 7396.2, and 21300, respectively. Example 2: Consider the cement milling circuit presented in [26] as modeled as follows:
(18) where is the product flow rate (tons/h), is the load in the mill is the tailing flow rate (tons/h), is the output (tons), flow rate of the mill (tons/h), is the feed flow rate (tons/h), is the classifier speed (rpm), is the hardness of the material inside the mill. The estimated nonlinear function is
where the actual
is
is
The unmodeled dynamics results from functional errors is represented as . Define and . The reference set point at is , and at is . The initial state is , and the initial input is . Let , and sampling period be 2 min. The constraints are: . The learning results based on 500 training data and 500 testing data are summarized in Table II. The -norms of training errors and testing errors are depicted in Figs. 7–8, respectively. The optimization problem (13) was then solved by using the simplified dual network. The simulation results are illustrated in Figs. 9–12. The tracking performance based on three models are all superior. Compared with linear MPC on the exact model, the proposed approach results in significantly faster responses for set-points transition in presence of unmodeled dynamics. The total actual control cost over the control terminal, , for
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Fig. 12. Control cost in Example 2. Fig. 9. First output response in Example 2. TABLE II COMPARISON OF REGRESSION RESULTS IN EXAMPLE 2
Example 3: Consider a nonlinear automatic flight control system described in [7]. The objective is to control the angle of attack of an F-8 at and an altitude of 30 000 ft. The equations of motion representing the dynamics of the aircraft are
Fig. 10. Second output response in Example 2.
(19) where is the angle of attack, is the pitch angle, is the pitch rate, and is the control input provided by the tail deis unmodeled dynamics where flection angle, and . The control reference is to track the commanded angle of attack [48]:
(20)
Fig. 11. Control inputs in Example 2.
the method herein with ELM, the method in [35] and linear MPC are 44.44, 42.21, and 206.92, respectively.
The initial state is , and the initial input is . Let , and sampling frequency be 10 Hz. The constraints are: , and . The learning results based on 400 training data and 400 testing data are summarized in Table III and illustrated in Figs. 13–14. The tracking results are illustrated in Figs. 15–17. The tracking results by applying three models
YAN AND WANG: MODEL PREDICTIVE CONTROL OF NONLINEAR SYSTEMS WITH UNMODELED DYNAMICS
Fig. 13.
-norm of training errors in Example 3.
Fig. 14.
-norm of testing errors in Example 3.
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Fig. 16. Control input in Example 3.
Fig. 17. Cost function in Example 3. TABLE III COMPARISON OF REGRESSION RESULTS IN EXAMPLE 3
with ELM and linear MPC are 5.1712, 4.2085, and 21.7660, respectively. Example 4: Consider the nonlinear X-cell 50 helicopter model discussed in [52]. The dynamics of the helicopter is modeled as follows:
Fig. 15. Output tracking in Example 3.
are all superior. In contrast, linear MPC on exact model resulted in undesirable tracking. The total cost within control terminal, , for the method herein, the method in [35]
(21)
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Fig. 18. Altitude tracking in Example 4.
where denotes its altitude, denotes its altitude rate, denotes its rotor speed, denotes its collective pitch angle, and denotes its collective pitch rate. Assume to be a constant input to the throttle, and be the input to the collective servomechanisms. The and are given to be: nominal values for constants
Fig. 19. Altitude rate and collective pitch rate in Example 4.
, and . is unmodeled dynamics of the system and . The control where objective is to force the altitude to track the reference while maintaining zero collective pitch angle rate at steady altitude; i.e.,
(22) Fig. 20. Control input in Example 4.
The initial state is tial input is
, and the ini. The simulation parameters are set as: . The sampling frequency is 20 Hz. The supervised learning results based on 500 training data and 500 testing data are summarized in Table IV. The optimization problem (13) was iteratively solved by employing the simplified dual network. The altitude of the helicopter is depicted in Fig. 18, the other two outputs are shown in Fig. 19, and the control input is shown in Fig. 20. The simulation results show a fast and stable tracking performance. The overall tracking performance for the proposed method based on three models are all superior. In contrast, linear MPC leads to longer rise time. Among the four simulation applications, the ELM network is the most efficient model in terms of training time. With the same training data set and the same number of hidden neurons/features, ELM learns thousands of times faster and generates fairly accurate output. On the other hand, MLP-BP and LS-SVM generate outputs with higher accuracy. With the same training data set and the same number of hidden neurons/features, the training
TABLE IV COMPARISON OF REGRESSION RESULTS IN EXAMPLE 4
MSE of MLP-BP and LS-SVM is usually much smaller than that of ELM. Besides using the proposed approach, we also applied the interior-point algorithm provided by the MATLAB Optimization Toolbox directly on the original models of the simulation examples. The resulting tracking performances were much poorer than the proposed neural network approach. V. CONCLUSION This paper proposed a model predictive control approach to nonlinear systems with unmodeled dynamics based on feedforward and recurrent neural networks. With off-line training of
YAN AND WANG: MODEL PREDICTIVE CONTROL OF NONLINEAR SYSTEMS WITH UNMODELED DYNAMICS
multilayer perceptron or extreme learning machine, unknown parameters due to unmodeled dynamics and linearization errors in the formulated quadratic optimization problems can be accurately estimated and compensated. Effective control actions can be generated by using the simplified dual network via on-line optimization. The simulation results showed superior performance of the proposed nonlinear model predictive control approach. Although the recurrent neural network approach is exploited to solve quadratic programming problems on-line in this paper, the method is by no means complete. It is possible to use other methods, such as simultaneous perturbation stochastic approximation, for solving the quadratic programming problems. Our further investigations aim at development of robust MPC based on neural networks. REFERENCES [1] A. Alessio and A. Bemporad, “A survey on explicit model predictive control,” in Nonlinear Model Predictive Control, LNCIS, L. Magni, Ed. Berlin, Germany: Springer–Verlag, 2009, et al.. [2] M. Bazaraa, H. Sherali, and C. Shetty, Nonlinear Programming: Theory and Algorithms. New York: Wiley, 1993. [3] E. Camacho and B. Bordons, Model Predictive Control. New York: Springer, 2004. [4] M. Canale, L. Fagiano, and M. Milanese, “Efficient model predictive control for nonlinear systems via function approximation techniques,” IEEE Trans. Autom. Control, vol. 55, no. 8, pp. 1911–1916, Aug/ 2010. [5] M. Cannon and B. Kouvaritakis, Non-Linear Predictive Control: Theory & Practice. London, U.K.: The Institution of Engineering and Technology, 2001. [6] L. Cheng, Z. Hou, and M. Tan, “Constrained multi-variable generalized predictive control using a dual neural network,” Neural Comput. Appl., vol. 16, no. 6, pp. 505–512, 2007. [7] T. Cimen and S. Banks, “Global optimal feedback control for general nonlinear systems with nonquadratic performance criteria,” Syst. Control Lett., vol. 53, pp. 327–346, 2004. [8] K. Dalamagkidis, K. Valavanis, and L. Piegl, “Nonlinear model predictive control with neural network optimization for autonomous autoratation of small unmanned helicopters,” IEEE Trans. Control Syst. Technol., vol. 19, no. 4, pp. 818–831, Jul. 2011. [9] A. Draeger, S. Engell, and H. Ranke, “Model predictive control using neural networks,” IEEE Control Syst. Mag., vol. 15, no. 5, pp. 61–66, 1995. [10] S. Ge, J. Zhang, and T. Lee, “Adaptive MNN control for a class of non-affine NARMAX systems with disturbances,” Syst. Control Lett., vol. 53, pp. 1–12, 2004. [11] A. Gomperts, A. Ukil, and F. Zurfluh, “Development and implementation of parameterized FPGA-based general purpose neural networks for online applications,” IEEE Trans. Ind. Informat., vol. 7, no. 1, pp. 78–89, Feb. 2011. [12] D. Gu and H. Hu, “Neural predictive control for a car-like mobile robot,” Robot. Auton. Syst., vol. 39, no. 2, pp. 73–86, 2002. [13] M. Henson, “Nonlinear model predictive control: Current status and future directions,” Comput. Chem. Eng., vol. 23, no. 2, pp. 187–202, 1998. [14] K. Hornik, M. Stinchcombe, and H. White, “Multilayer feedforward networks are universal approximators,” Neural Netw., vol. 2, no. 5, pp. 359–366, 1989. [15] X. Hu and J. Wang, “An improved dual neural network for solving a class of quadratic programming problems and its k-winners-take-all application,” IEEE Trans. Neural Netw., vol. 19, no. 12, pp. 2022–2031, Dec. 2008. [16] G. Huang, Q. Zhu, and C. Siew, “Extreme learning machine: Theory and application,” Neurocomputing, vol. 70, pp. 489–501, 2005. [17] J. Huang and F. Lewis, “Neural network predictive control for nonlinear dynamic systems with time-delay,” IEEE Trans. Neural Netw., vol. 14, no. 2, pp. 377–389, Feb. 2003. [18] M. Kennedy and L. Chua, “Neural networks for nonlinear programming,” IEEE Trans. Circuits Syst., vol. 35, no. 5, pp. 554–562, May 1988.
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[45] D. Soloway and P. Haley, “Neural generalized predictive control,” in Proc. 1996 IEEE Int. Symp. Intell. Control, 1996, pp. 277–282. [46] Y. Song, Z. Chen, and Z. Yuan, “New chaotic PSO-based neural network predictive control for nonlinear process,” IEEE Trans. Neural Netw., vol. 18, no. 2, pp. 595–601, Feb. 2007. [47] P. Sorensen, M. Norgarrd, O. Ravn, and N. Poulsen, “Implementation of neural network based nonlinear predictive control,” Neurocomputing, vol. 28, pp. 37–51, 1999. [48] B. Stevens and F. Lewis, Aircraft Control and Simulation. Hoboken: Wiley, 1992. [49] J. Suykens, T. Gestel, J. Brabanter, B. DeMoor, and J. Vandewalle, Least Squares Support Vector Machines. Singapore: World Scientific, 2002. [50] J. Suykens, J. Vandewalle, B. DeMoor, and J. Vandewalle, Artificial Neural Networks for Modeling and Control of Non-Linear Systems. New York: Springer, 1996. [51] D. Tank and J. Hopfield, “Simple neural optimization networks: An A/D converter, signal decision circuit, and a linear programming circuit,” IEEE Trans. Circuits Syst., vol. CAS-33, pp. 533–541, May 1986. [52] K. Tee, S. Ge, and F. Tay, “Adaptive neural network control for helicopters in vertical flight,” IEEE Trans. Control Syst. Technol., vol. 16, no. 4, pp. 753–762, Jul. 2008. [53] K. Temeng, P. Schnelle, and T. McAvoy, “Model predictive control of an industrial packed bed reactor using neural networks,” J. Process Control, vol. 5, no. 1, pp. 19–27, 1995. [54] C. Venkateswarlu and K. V. Rao, “Dynamic recurrent radial basis function network model predictive control of unstable nonlinear processes,” Chem. Eng. Sci., vol. 60, no. 23, pp. 6718–6732, 2005. [55] L. Wang and F. Wan, “Structured neural networks for constrained model predictive control,” Automatica, vol. 37, no. 8, pp. 1235–1243, 2001. [56] D. White and D. Sofge, Handbook of Intelligent Control: Neural, Fuzzy, and Adaptive Approaches. New York: Van Nostrand Reinhold, 1992. [57] M. Wills, G. Montague, C. Massimo, M. Tham, and A. Morris, “Artificial neural networks in process estimation and control,” Automatica, vol. 28, no. 6, pp. 1181–1187, 1992. [58] Y. Xia, G. Feng, and J. Wang, “A primal-dual neural network for online resolving constrained kinematic redundancy in robot motion control,” IEEE Trans. Syst., Man Cybern. B, Cybern., vol. 35, no. 1, pp. 54–64, Feb. 2005. [59] Y. Xia, G. Feng, and J. Wang, “A novel recurrent neural network for solving nonlinear optimization problems with inequality constraints,” IEEE Trans. Neural Netw., vol. 19, no. 8, pp. 1340–1353, Aug. 2008. [60] Y. Xia, H. Leung, and J. Wang, “A projection neural network and its application to constrained optimization problems,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 49, no. 4, pp. 447–458, Apr. 2002. [61] Y. Xia and J. Wang, “A general projection neural network for solving optimization and related problems,” IEEE Trans. Neural Netw., vol. 15, pp. 318–328, 2004. [62] Y. Xia and J. Wang, “A recurrent neural network for nonlinear convex optimization subject to nonlinear inequality constraints,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 51, no. 7, pp. 1385–1394, Jul. 2004. [63] Z. Yan and J. Wang, “Model predictive control of nonlinear affine systems based on the general projection neural network and its application to a continuous stirred tank reactor,” in Proc. Int. Conf. Inf. Sci. Technol., Nanjing, China, 2011, pp. 1011–1015. [64] Z. Yan and J. Wang, “Robust model predictive control of nonlinear affine systems based on a two-layer recurrent neural network,” in Proc. Int. Joint Conf. Neural Netw., San Jose, CA, 2011, pp. 24–29. [65] D. Yu and J. Gomm, “Implementation of neural network predictive control to a multivariable chemical reactor,” Control Eng. Practice, vol. 11, no. 11, pp. 1315–1323, 2003. [66] J. Zamarreno and P. Vega, “Neural predictive control. Application to highly non-linear system,” Eng. Appl. Art. Intell., vol. 12, no. 2, pp. 149–158, 1999.
[67] V. Zavala and L. Biegler, “The advanced step NMPC controller: Optimality, stability and robustness,” Automatica, vol. 45, no. 1, pp. 86–93, 2009. [68] T. Zhang, G. Feng, and X. Zeng, “Output tracking of constrained nonlinear processes with offset-free input-to-state stable fuzzy predictive control,” Automatica, vol. 45, no. 4, pp. 900–909, 2009.
Zheng Yan (S’11) received the B.Eng. degree in automation and computer-aided engineering from the Chinese University of Hong Kong, Shatin, Hong Kong, in 2010. He is currently working toward the Ph.D. degree in mechanical and automation engineering at the Chinese University of Hong Kong, Shatin, Hong Kong. His current research interests include computational intelligence and model predictive control.
Jun Wang (S’89–M’90–SM’93–F’07) received the B.S. degree in electrical engineering and the M.S. degree in systems engineering from the Dalian University of Technology, Dalian, China, in 1982 and 1985, respectively, and the Ph.D. degree in systems engineering from Case Western Reserve University, Cleveland, OH, in 1991. He is a Professor in the Department of Mechanical and Automation Engineering, The Chinese University of Hong Kong, Shatin, Hong Kong. He held various academic positions at Dalian University of Technology, Case Western Reserve University, and the University of North Dakota. He also held various short-term or part-time visiting positions at the U.S. Air Force Armstrong Laboratory in 1995; RIKEN Brain Science Institute in 2001, Universite Catholique de Louvain in 2001, Chinese Academy of Sciences in 2002, Huazhong University of Science and Technology from 2006 to 2007, Shanghai Jiao Tong University as a Cheung Kong Chair Professor from 2008 to 2011, and the Dalian University of Technology as a National Thousand-Talent Chair Professor since 2011. His current research interests include neural networks and their applications. Dr. Wang has served as an Associate Editor of the IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B since 2003, an Editorial Advisory Board Member of the International Journal of Neural Systems since 2006, and Editorial Board Member of the Neural Networks since 2012. He also served as an Associate Editor of the IEEE TRANSACTIONS ON NEURAL NETWORKS from 1999 to 2009 and the IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART C from 2002 to 2005. He was a Guest Editor of special issues of the European Journal of Operational Research in 1996, International Journal of Neural Systems in 2007, and Neurocomputing in 2008. He served as the President of the Asia–Pacific Neural Network Assembly (APNNA) in 2006, the General Chair of the 13th International Conference on Neural Information Processing in 2006, and IEEE World Congress on Computational Intelligence in 2008. In addition, he has served on many committees, such as the IEEE Fellow Committee. He is an IEEE Computational Intelligence Society Distinguished Lecturer for 2010–2012. He was the recipient of the Research Excellence Award from The Chinese University of Hong Kong for 2008–2009, two Natural Science Awards (first class) respectively from Shanghai Municipal Government in 2009 and Ministry of Education of China in 2011, the Outstanding Achievement Award from Asia Pacific Neural Network Assembly, the IEEE TRANSACTIONS ON NEURAL NETWORKS Outstanding Paper Award (with Q. Liu) from IEEE Computational Intelligence Society in 2011.
IEEE TRANSACTIONS ON INDUSTRIAL INFORMATICS, VOL. 8, NO. 4, NOVEMBER 2012
Model Predictive Control of Nonlinear Systems With Unmodeled Dynamics Based on Feedforward and Recurrent Neural Networks Zheng Yan, Student Member, IEEE, and Jun Wang, Fellow, IEEE
Abstract—This paper presents new results on a neural network approach to nonlinear model predictive control. At first, a nonlinear system with unmodeled dynamics is decomposed by means of Jacobian linearization to an affine part and a higher-order unknown term. The unknown higher-order term resulted from the decomposition, together with the unmodeled dynamics of the original plant, are modeled by using a feedforward neural network via supervised learning. The optimization problem for nonlinear model predictive control is then formulated as a quadratic programming problem based on successive Jacobian linearization about varying operating points and iteratively solved by using a recurrent neural network called the simplified dual network. Simulation results are included to substantiate the effectiveness and illustrate the performance of the proposed approach. Index Terms—Feedforward neural networks, model predictive control (MPC), real-time optimization, recurrent neural networks, supervised learning, unmodeled dynamics.
I. INTRODUCTION
M
ODEL predictive control (MPC) is an optimization-based control method which has been widely recognized and appreciated in academia and industries. MPC applies real-time optimization to a system model over a finite horizon of predicted future. By taking the current state as an initial state, an objective function is optimized in each sample interval, and the calculation is repeated at the next computation time interval with new state information over a moving time window. As a combination of control and planning, a sequence of optimal control actions are calculated at each step, but only the one for next step is implemented. Compared with other control techniques, MPC has several desirable features, such as the ability to directly take into account input and output constraints as well as the capability to carry out multivariable control. Linear MPC is considered to be a mature technique and achieved great success in process industry (see [3], [5], [13], [20], [27], and [40]–[42]). However, as most industrial pro-
Manuscript received September 15, 2011; revised January 04, 2012; accepted February 22, 2012. Date of publication June 21, 2012; date of current version October 18, 2012. This work was supported by the Research Grants Council of the Hong Kong Special Administrative Region, China, under Grant CUHK417209E. Paper no. TII-11-525. The authors are with the Department of Mechanical and Automation Engineering, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong (e-mail: [email protected], [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/TII.2012.2205582
cesses are inherently nonlinear, linear MPC is not often suitable for the cases where applications involve model changes or adaptations because of changing of operating points and safety-criticality [28], [41]. As a result, nonlinear model predictive control (NMPC) techniques are needed to meet tight performance specifications. The majority of existing NMPC techniques require solving nonconvex optimization problems [5]. However, there are no reliable optimization procedures for solving such problems which would find exact solutions reliably and quickly [27]. Real-time nonconvex optimization is computationally intractable. In order to address the issue of NMPC, several studies were carried out recently. For example, by exploiting multiparametric programming techniques, explicit model predictive control allows one to solve off-line optimization problems for a given range of operating conditions [1]. Optimal control actions are calculated as an explicit function of the state and reference vectors, so that on-line computation reduces to simple function evaluation. In [4], function approximation techniques are applied to improve NMPC efficiency. In [67], by incorporating sensitivity-based strategy with an interior-point nonlinear programming algorithm, an advanced step NMPC technique is proposed to enable the computation of fast approximate solutions to dynamic optimization problems arising in NMPC. In [68], an efficient output feedback nonlinear predictive controller is designed based on – fuzzy modeling and LMI optimization techniques. There are two major drawbacks for these NMPC techniques, first, some controllers are too conceptual to be implemented in practice. Second, many NMPC schemes heavily rely on the structure of underlying processes so that they can not be applied for general nonlinear cases. Hence, further investigations on efficient and implementable NMPC are necessary as well desirable. As MPC technique is model-based, its control reliability is largely determined by the accuracy of prediction model. One critical issue for MPC is how to represent underlying processes with accurate models so that the process outputs over a future horizon can be accurately predicted. There are two main classes of modeling approaches. One is to use a full nonlinear model for prediction without simplifications. A direct consequence is that a high nonlinear (most likely nonconvex as well) optimization needs to be solved during each sampling interval. An alternative is to find linear approximations of the nonlinear process. A benefit of linearization is that MPC optimization task becomes quadratic programming. But MPC with linear model is inherently suboptimal, especially for cases there unmodeled dynamics exist. Improving the accuracy of the model used for prediction is vital for optimizing the overall control performance
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YAN AND WANG: MODEL PREDICTIVE CONTROL OF NONLINEAR SYSTEMS WITH UNMODELED DYNAMICS
of MPC. Explicitly taking into account the residual results from linearization and unmodeled dynamics can improve the modeling accuracy. Real-time optimization is another critical issue for MPC implementation. The effectiveness and efficiency of the optimization method determines whether the MPC technique can be implemented successfully in practice. Classical optimization techniques such as active-set method or interior-point method may not be competent here due to problem dimensionality and stringent requirement on computational time. In the past two decades, neurodynamic optimization based on recurrent neural networks (RNNs) emerged as a promising computational approach for solving various optimization problems. There are several advantages in neurodynamic optimization: first, recurrent neural networks can serve as parallel computational models to solve problems with time-varying parameters; secondly, they can deal with large-scale problems due to their inherent parallelism; thirdly, they are hardware implementable [11]. Tank and Hopfiled first proposed a neural network for linear programming [51]. Kennedy and Chua developed a neural network which extended the results of [51] to general nonlinear programming problems [18]. Their work inspired many researchers to develop neural network models for solving linear and nonlinear optimization problems. Many neural network models have been proposed in recent two decades. In particular, based on the duality and projection methods, several neural networks for solving linear programming, quadratic programming, general convex programming and pseudoconvex programming were developed, e.g., [15], [22]–[24], [58]–[62]. These recurrent neural networks improved performance in terms of global convergence and reduced model complexity. Neural network approaches to system control have been widely investigated. In [38], [50], [56], and [57], comprehensive perspectives on neural networks based control were presented. Topics such as the theoretical foundations on neural networks for systems modeling and control, the model-based neural control design, the stability proofs, as well as the suitability of using neural network models directly within model predictive control strategies were discussed in detail. In [9], [12], [19], [21], [29], [30], [37], [39], [43], [45]–[47], and [65], feedforward neural networks were used for systems modeling. In [17], [25], [36], [44], [53], [54], and [66], recurrent neural networks were used for systems modeling. In [6], [8], [31]–[36], [55], [63], and [64], recurrent neural networks were used for solving the optimization problems involved in MPC. Most existing neural network based model predictive controllers are designed for system models with particular structures, study on a neural network based MPC for more general nonlinear systems is still attractive. In this paper, an NMPC approach based on neural networks to nonlinear systems with unmodeled dynamics is presented. The NMPC problem is first formulated to a quadratic optimization problem with some unknown parameters. The unknown parameters are then modeled off-line based on a feedforward neural network via supervised learning. Then a one-layer recurrent neural network called the simplified dual neural network is applied for solving the quadratic optimization problem to generate the optimal control variables.
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The rest of the paper is organized as follows. In Section II, the NMPC problem is formulated. In Section III, neural networks are applied to model and predict the unknown parameters and to solve the involved optimization problems. In Section IV, simulation results are provided. Finally, Section V concludes this paper. II. PROBLEM FORMULATION Consider a nonlinear discrete-time and time-invariant system with unmodeled dynamics (1) with the following constraints:
(2) is the state vector, is the conwhere trol input vector, is the control increment vector, is the output vector, is a continuous differen, and tiable nonlinear function with represents the unmodeled dynamics of the system, which can be unindentified structures or unknown parameters. It is assumed that is time-invariant and . , and are lower and upper bounds. All state variables of the system are assumed to be measurable. By using Taylor series, the nonlinear function can be decomposed about an operating point into an affine system plus an unknown term at each time instant
(3) and are respectively the Jacobians of where with respect to and , and is the higher-order residual of the corresponding Taylor series. By denoting
setting lated as follows:
, and , system (1) can be reformu-
(4) MPC is an iterative optimization technique: at each sampling time , it measures or estimates the current state, and obtains the optimal input vector by optimizing a cost function. For model
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(1), the following cost function with a finite horizon quadratic criterion is commonly used
The predicted output
is then expressed as follows: (9)
where (5) .. . where denotes the reference vector, denotes the predicted output vector, and denotes the input increment vector, and are prediction horizon and control horizon respectively, and are appropriate weighting matrices, and denote the weighted Euclidean norms defined as and , respectively. The first term in (5) represents the error function between the predicted outputs and the references while the second term is concerned with the control effort. According to the reformulated model (4), the cost function (5) can be rewritten as follows:
.. .
.. .
(6)
..
.. .
.. .
..
.
.. .
.. .
.
Define the following vectors: Therefore, the original optimization problem in (5) becomes
(10) (7) As the linearization is performed once for a given sampling time instant , and the same local linear model is used within the and are written as and for prediction horizon, brevity. According to (3)
(11) where
.. .
.. .
..
.
.. .
(12)
.. . By defining , problem (6) can be rewritten as a quadratic programming (QP) problem
(13) where (8)
YAN AND WANG: MODEL PREDICTIVE CONTROL OF NONLINEAR SYSTEMS WITH UNMODELED DYNAMICS
If and are positive definite, then is also positive definite. So the objective function in (13) is strictly convex. As the feasible region defined by the constraints is a closed convex set, the solution to (13) is unique and satisfies the Karush–Kauhn–Tucker (KKT) optimality conditions [2]. The solution to the QP problem (13) gives control increment vector which can be used to calculate the optimal control vector. III. NEURAL NETWORK APPROACH In [24], Liu and Wang developed a one-layer recurrent network called the simplified dual network for solving quadratic programming problems by utilizing dual variables. The neural network has shown good performance with low computational complexity. The neural network model can be described as follows. • State equation
• Output function (14) where is a positive constant, vector, is the output vector, and linear activation function defined as follows:
is the state is a piecewise
(15) We employ the simplified dual network here for solving (13). The simplified dual network has a single-layer structure with totally neurons. According to the convergence analysis in [24], it is Lyapunov stable and globally convergent to the optimal solution of any strictly convex QP problem. However, it is worth noting that the term in (9) is still unknown. Consequently, the parameters and in (13) are also unknown. is estimated. The QP problem (13) can be solved only after Neural networks are universal approximators and can be used as black-boxes in model identification. Over decades, many feedforward neural networks have been developed with superior capabilities for data regression problems, such as multilayer perceptron networks (MLP-BP) [14] and extreme learning machines (ELM) [16]. In this study, the two neural networks and least squares support vector machines (LS-SVM) [49] were applied to model unknown parameters off-line and their performances were evaluated.
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Let be a global function that is a combination of the higherorder term resulted from Jacobian linearization and unmodeled dynamics of original plant. Given a data set , we can measure a corresponding target data from original plant. Once obtaining two group of sample data sets, we can use a feedforward neural network to map between data sets by treating as the input vector and as the target vector. A well-trained neural network after validation can be used to estimate the value of at each sampling time within the prediction horizon as the global function is valid at any sampling instant. The NMPC scheme based on the feedforward and recurrent neural networks is summarized as follows. 1) Let . Set control time terminal , prediction horizon , control horizon , sampling period , weight matrices and . 2) Model by means of off-line supervised learning based on a feedforward neural network. 3) Estimate using the trained neural network in Step 2. Calculate process model matrices and neural . network matrices 4) Solve the convex optimization problem (13) by using neural network (14) to obtain the optimal input increment vector . 5) Calculate the optimal input vector and implement . , set , go to Step 3; otherwise end. 6) If The advantage of the proposed NMPC scheme is twofold. First, it uses a simple structured prediction model with improved accuracy. By successive linearization, a local linear model plus unknown terms is obtained at each sampling instant. The residual resulted from the Jacobian linearization, together with time-invariant unmodeled dynamics, is estimated off-line by using a feedforward neural network. The explicit structure of estimated underlying function remains unknown, only numerical compensation is taken into account. Thus, only quadratic programming problems are solved on-line. A good compromise of accuracy and efficiency is achieved without confronting nonconvexity. Second, the proposed scheme is implementable and efficient for practical industrial applications. The simplified dual network is capable of solving quadratic optimization problems in microsecond scale as shown in [24]. Due to the inherent parallelism of recurrent neural networks, the proposed scheme would not result in lower computational efficiency even if the optimization problem is of large size. In contrast, most existing NMPC algorithms are computationally demanding when dealing with large-scale multi-variable industrial processes. IV. SIMULATION RESULTS In this section, four application examples are provided to substantiate effectiveness and demonstrate performance of the proposed NMPC approach. Example 1: Consider the following nonaffine continuous stirred tank reactor system [10] with the following model:
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TABLE I COMPARISON OF LEARNING RESULTS IN EXAMPLE 1
Fig. 3. First output response in Example 1.
Fig. 1.
-norm of training errors in Example 1.
Fig. 4. Second output response in Example 1.
Fig. 2.
-norm of testing errors in Example 1.
(16) where and are the concentration and temperature of a tank, respectively; is the coolant flow rate, , and are estimated parameters. The true values are assumed to be , and . is known precisely. The unmodeled dynamics results from parameter errors is represented as . The control objective is to track the reference
(17)
The initial conditions are and . Set , and s. and The constraints are , where the constants and are the maximum values of the coolant flow rate and the tank temperature, respectively. To apply the proposed method, the map from to was first modeled via supervised learning. The learning results of MLP and ELM based on 500 training data and 500 testing data are summarized in Table I and illustrated in Figs. 1–2 where are randomly generated under a uniform distribution. LS-SVM was also applied for comparison purpose. After modeling , the quadratic optimization problem (13) was solved by using the simplified dual network. The simulation results are shown in Figs. 3–6. Compared with the nonlinear controller presented in [7], the proposed method results in shorter settling time and much reduced overshoot. To demonstrate the effectiveness of the proposed method, the method presented in [35] and the linear MPC presented in [31] were also applied on the exact nonlinear model of the plant without unmodeled dynamics. Despite a linear model of the exact nonlinear system was used, the outputs of linear MPC can not track
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Fig. 7.
-norm of training errors in Example 2.
Fig. 8.
-norm of testing errors in Example 2.
Fig. 5. Control input in Example 1.
Fig. 6. Control cost in Example 1.
The nonlinear function
the reference. The total actual control cost over the control ter, for the method minal, herein with ELM, the method in [35] and linear MPC are 2289.9, 7396.2, and 21300, respectively. Example 2: Consider the cement milling circuit presented in [26] as modeled as follows:
(18) where is the product flow rate (tons/h), is the load in the mill is the tailing flow rate (tons/h), is the output (tons), flow rate of the mill (tons/h), is the feed flow rate (tons/h), is the classifier speed (rpm), is the hardness of the material inside the mill. The estimated nonlinear function is
where the actual
is
is
The unmodeled dynamics results from functional errors is represented as . Define and . The reference set point at is , and at is . The initial state is , and the initial input is . Let , and sampling period be 2 min. The constraints are: . The learning results based on 500 training data and 500 testing data are summarized in Table II. The -norms of training errors and testing errors are depicted in Figs. 7–8, respectively. The optimization problem (13) was then solved by using the simplified dual network. The simulation results are illustrated in Figs. 9–12. The tracking performance based on three models are all superior. Compared with linear MPC on the exact model, the proposed approach results in significantly faster responses for set-points transition in presence of unmodeled dynamics. The total actual control cost over the control terminal, , for
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Fig. 12. Control cost in Example 2. Fig. 9. First output response in Example 2. TABLE II COMPARISON OF REGRESSION RESULTS IN EXAMPLE 2
Example 3: Consider a nonlinear automatic flight control system described in [7]. The objective is to control the angle of attack of an F-8 at and an altitude of 30 000 ft. The equations of motion representing the dynamics of the aircraft are
Fig. 10. Second output response in Example 2.
(19) where is the angle of attack, is the pitch angle, is the pitch rate, and is the control input provided by the tail deis unmodeled dynamics where flection angle, and . The control reference is to track the commanded angle of attack [48]:
(20)
Fig. 11. Control inputs in Example 2.
the method herein with ELM, the method in [35] and linear MPC are 44.44, 42.21, and 206.92, respectively.
The initial state is , and the initial input is . Let , and sampling frequency be 10 Hz. The constraints are: , and . The learning results based on 400 training data and 400 testing data are summarized in Table III and illustrated in Figs. 13–14. The tracking results are illustrated in Figs. 15–17. The tracking results by applying three models
YAN AND WANG: MODEL PREDICTIVE CONTROL OF NONLINEAR SYSTEMS WITH UNMODELED DYNAMICS
Fig. 13.
-norm of training errors in Example 3.
Fig. 14.
-norm of testing errors in Example 3.
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Fig. 16. Control input in Example 3.
Fig. 17. Cost function in Example 3. TABLE III COMPARISON OF REGRESSION RESULTS IN EXAMPLE 3
with ELM and linear MPC are 5.1712, 4.2085, and 21.7660, respectively. Example 4: Consider the nonlinear X-cell 50 helicopter model discussed in [52]. The dynamics of the helicopter is modeled as follows:
Fig. 15. Output tracking in Example 3.
are all superior. In contrast, linear MPC on exact model resulted in undesirable tracking. The total cost within control terminal, , for the method herein, the method in [35]
(21)
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Fig. 18. Altitude tracking in Example 4.
where denotes its altitude, denotes its altitude rate, denotes its rotor speed, denotes its collective pitch angle, and denotes its collective pitch rate. Assume to be a constant input to the throttle, and be the input to the collective servomechanisms. The and are given to be: nominal values for constants
Fig. 19. Altitude rate and collective pitch rate in Example 4.
, and . is unmodeled dynamics of the system and . The control where objective is to force the altitude to track the reference while maintaining zero collective pitch angle rate at steady altitude; i.e.,
(22) Fig. 20. Control input in Example 4.
The initial state is tial input is
, and the ini. The simulation parameters are set as: . The sampling frequency is 20 Hz. The supervised learning results based on 500 training data and 500 testing data are summarized in Table IV. The optimization problem (13) was iteratively solved by employing the simplified dual network. The altitude of the helicopter is depicted in Fig. 18, the other two outputs are shown in Fig. 19, and the control input is shown in Fig. 20. The simulation results show a fast and stable tracking performance. The overall tracking performance for the proposed method based on three models are all superior. In contrast, linear MPC leads to longer rise time. Among the four simulation applications, the ELM network is the most efficient model in terms of training time. With the same training data set and the same number of hidden neurons/features, ELM learns thousands of times faster and generates fairly accurate output. On the other hand, MLP-BP and LS-SVM generate outputs with higher accuracy. With the same training data set and the same number of hidden neurons/features, the training
TABLE IV COMPARISON OF REGRESSION RESULTS IN EXAMPLE 4
MSE of MLP-BP and LS-SVM is usually much smaller than that of ELM. Besides using the proposed approach, we also applied the interior-point algorithm provided by the MATLAB Optimization Toolbox directly on the original models of the simulation examples. The resulting tracking performances were much poorer than the proposed neural network approach. V. CONCLUSION This paper proposed a model predictive control approach to nonlinear systems with unmodeled dynamics based on feedforward and recurrent neural networks. With off-line training of
YAN AND WANG: MODEL PREDICTIVE CONTROL OF NONLINEAR SYSTEMS WITH UNMODELED DYNAMICS
multilayer perceptron or extreme learning machine, unknown parameters due to unmodeled dynamics and linearization errors in the formulated quadratic optimization problems can be accurately estimated and compensated. Effective control actions can be generated by using the simplified dual network via on-line optimization. The simulation results showed superior performance of the proposed nonlinear model predictive control approach. Although the recurrent neural network approach is exploited to solve quadratic programming problems on-line in this paper, the method is by no means complete. It is possible to use other methods, such as simultaneous perturbation stochastic approximation, for solving the quadratic programming problems. Our further investigations aim at development of robust MPC based on neural networks. REFERENCES [1] A. Alessio and A. Bemporad, “A survey on explicit model predictive control,” in Nonlinear Model Predictive Control, LNCIS, L. Magni, Ed. Berlin, Germany: Springer–Verlag, 2009, et al.. [2] M. Bazaraa, H. Sherali, and C. Shetty, Nonlinear Programming: Theory and Algorithms. New York: Wiley, 1993. [3] E. Camacho and B. Bordons, Model Predictive Control. New York: Springer, 2004. [4] M. Canale, L. Fagiano, and M. Milanese, “Efficient model predictive control for nonlinear systems via function approximation techniques,” IEEE Trans. Autom. Control, vol. 55, no. 8, pp. 1911–1916, Aug/ 2010. [5] M. Cannon and B. Kouvaritakis, Non-Linear Predictive Control: Theory & Practice. London, U.K.: The Institution of Engineering and Technology, 2001. [6] L. Cheng, Z. Hou, and M. Tan, “Constrained multi-variable generalized predictive control using a dual neural network,” Neural Comput. Appl., vol. 16, no. 6, pp. 505–512, 2007. [7] T. Cimen and S. Banks, “Global optimal feedback control for general nonlinear systems with nonquadratic performance criteria,” Syst. Control Lett., vol. 53, pp. 327–346, 2004. [8] K. Dalamagkidis, K. Valavanis, and L. Piegl, “Nonlinear model predictive control with neural network optimization for autonomous autoratation of small unmanned helicopters,” IEEE Trans. Control Syst. Technol., vol. 19, no. 4, pp. 818–831, Jul. 2011. [9] A. Draeger, S. Engell, and H. Ranke, “Model predictive control using neural networks,” IEEE Control Syst. Mag., vol. 15, no. 5, pp. 61–66, 1995. [10] S. Ge, J. Zhang, and T. Lee, “Adaptive MNN control for a class of non-affine NARMAX systems with disturbances,” Syst. Control Lett., vol. 53, pp. 1–12, 2004. [11] A. Gomperts, A. Ukil, and F. Zurfluh, “Development and implementation of parameterized FPGA-based general purpose neural networks for online applications,” IEEE Trans. Ind. Informat., vol. 7, no. 1, pp. 78–89, Feb. 2011. [12] D. Gu and H. Hu, “Neural predictive control for a car-like mobile robot,” Robot. Auton. Syst., vol. 39, no. 2, pp. 73–86, 2002. [13] M. Henson, “Nonlinear model predictive control: Current status and future directions,” Comput. Chem. Eng., vol. 23, no. 2, pp. 187–202, 1998. [14] K. Hornik, M. Stinchcombe, and H. White, “Multilayer feedforward networks are universal approximators,” Neural Netw., vol. 2, no. 5, pp. 359–366, 1989. [15] X. Hu and J. Wang, “An improved dual neural network for solving a class of quadratic programming problems and its k-winners-take-all application,” IEEE Trans. Neural Netw., vol. 19, no. 12, pp. 2022–2031, Dec. 2008. [16] G. Huang, Q. Zhu, and C. Siew, “Extreme learning machine: Theory and application,” Neurocomputing, vol. 70, pp. 489–501, 2005. [17] J. Huang and F. Lewis, “Neural network predictive control for nonlinear dynamic systems with time-delay,” IEEE Trans. Neural Netw., vol. 14, no. 2, pp. 377–389, Feb. 2003. [18] M. Kennedy and L. Chua, “Neural networks for nonlinear programming,” IEEE Trans. Circuits Syst., vol. 35, no. 5, pp. 554–562, May 1988.
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Zheng Yan (S’11) received the B.Eng. degree in automation and computer-aided engineering from the Chinese University of Hong Kong, Shatin, Hong Kong, in 2010. He is currently working toward the Ph.D. degree in mechanical and automation engineering at the Chinese University of Hong Kong, Shatin, Hong Kong. His current research interests include computational intelligence and model predictive control.
Jun Wang (S’89–M’90–SM’93–F’07) received the B.S. degree in electrical engineering and the M.S. degree in systems engineering from the Dalian University of Technology, Dalian, China, in 1982 and 1985, respectively, and the Ph.D. degree in systems engineering from Case Western Reserve University, Cleveland, OH, in 1991. He is a Professor in the Department of Mechanical and Automation Engineering, The Chinese University of Hong Kong, Shatin, Hong Kong. He held various academic positions at Dalian University of Technology, Case Western Reserve University, and the University of North Dakota. He also held various short-term or part-time visiting positions at the U.S. Air Force Armstrong Laboratory in 1995; RIKEN Brain Science Institute in 2001, Universite Catholique de Louvain in 2001, Chinese Academy of Sciences in 2002, Huazhong University of Science and Technology from 2006 to 2007, Shanghai Jiao Tong University as a Cheung Kong Chair Professor from 2008 to 2011, and the Dalian University of Technology as a National Thousand-Talent Chair Professor since 2011. His current research interests include neural networks and their applications. Dr. Wang has served as an Associate Editor of the IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B since 2003, an Editorial Advisory Board Member of the International Journal of Neural Systems since 2006, and Editorial Board Member of the Neural Networks since 2012. He also served as an Associate Editor of the IEEE TRANSACTIONS ON NEURAL NETWORKS from 1999 to 2009 and the IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART C from 2002 to 2005. He was a Guest Editor of special issues of the European Journal of Operational Research in 1996, International Journal of Neural Systems in 2007, and Neurocomputing in 2008. He served as the President of the Asia–Pacific Neural Network Assembly (APNNA) in 2006, the General Chair of the 13th International Conference on Neural Information Processing in 2006, and IEEE World Congress on Computational Intelligence in 2008. In addition, he has served on many committees, such as the IEEE Fellow Committee. He is an IEEE Computational Intelligence Society Distinguished Lecturer for 2010–2012. He was the recipient of the Research Excellence Award from The Chinese University of Hong Kong for 2008–2009, two Natural Science Awards (first class) respectively from Shanghai Municipal Government in 2009 and Ministry of Education of China in 2011, the Outstanding Achievement Award from Asia Pacific Neural Network Assembly, the IEEE TRANSACTIONS ON NEURAL NETWORKS Outstanding Paper Award (with Q. Liu) from IEEE Computational Intelligence Society in 2011.
