An Approximate Internal Model-Based Neural Control for Unknown ...
IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 17, NO. 3, MAY 2006
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An Approximate Internal Model-Based Neural Control for Unknown Nonlinear Discrete Processes Han-Xiong Li, Senior Member, IEEE, and Hua Deng
Abstract—An approximate internal model-based neural control (AIMNC) strategy is proposed for unknown nonaffine nonlinear discrete processes under disturbed environment. The proposed control strategy has some clear advantages in respect to existing neural internal model control methods. It can be used for open-loop unstable nonlinear processes or a class of systems with unstable zero dynamics. Based on a novel input–output approximation, the proposed neural control law can be derived directly and implemented straightforward for an unknown process. Only one neural network needs to be trained and control algorithm can be directly obtained from model identification without further training. The stability and robustness of a closed-loop system can be derived analytically. Extensive simulations demonstrate the superior performance of the proposed AIMNC strategy. Index Terms—Input–output approximation, neural networks, nonaffine nonlinear discrete-time systems, nonlinear internal model control, unstable zero dynamics.
I. INTRODUCTION
F
OR THE control of nonlinear discrete-time processes subject to uncertainties (model mismatch and disturbances), nonlinear internal model control (IMC) using neural networks (NNs) has received much attention [12], [9], [19]. Nonlinear IMC was proposed by Economou et al. [2], as shown in Fig. 1. The key characteristic of this type of control strategy is having the inverse controller and the internal model. In Fig. 1, the model of the nonlinear plant is needed as the internal model. Using this internal model, the effect of uncertainties can be suppressed with the feedback signal generated. In nonlinear IMC, the nonlinear model and its inversion play a crucial role. Fig. 2 gives a basic NN IMC structure [18], which is an extension of nonlinear IMC. For the control of unknown nonlinear discrete systems using the nonlinear IMC structure, NN model is employed as the internal model and NN inverse controller is used to replace the inverse controller, as shown in Fig. 2. The control structure given in Figs. 1 and 2 has been shown to have good robustness against uncertainties [10], [12], [9]. For NN IMC, however, even though the NN model is available, it is still not easy to design the NN inverse controller because of the following reasons. First, the process to be controlled Manuscript received July 4, 2004; revised April 14, 2005. This paper was supported in part under a Grant from RGC of Hong Kong (CityU 1129/03E) and a Grant from the National Science Foundation (NSF) of China for distinguished young scholar (50 429 501). H.-X. Li is with the Department of Manufacturing Engineering and Engineering Management, City University of Hong Kong, Kowloon, Hong Kong (e-mail: [email protected]). H. Deng is with the School of Mechanical and Electrical Engineering, Central South University, Changsha 410083, China. Digital Object Identifier 10.1109/TNN.2006.873277
must be stable in open-loop, which definitely limits many of IMC applications [18]. Second, the process to be controlled is required to have stable zero dynamics because of the use of the model inversion. To our knowledge, little result has been reported about the NN IMC of nonlinear nonaffine discrete systems with unstable zero dynamics. Third, it may not be easy to have the model inversion for control law design due to the nonlinear internal model used. For the third problem mentioned previously, several solutions are introduced under certain conditions. One of the solutions is to train an NN inverse controller with the help of the identified NN model [19], [23]. In this case, dynamic gradient methods, such as dynamic back-propagation [15] and back-propagation through time [22], need to be used since the controller is in the feedback loop of the dynamical system [13]. Compared with static back-propagation, however, all dynamic gradient methods are quite slow and computationally intensive for computing the [13]. It is possible to calculate the inversion control signal at each sample time for the given neural nonlinear model using the numerical inverse approach [12], [23]. Such an iterative algorithm requires the time interval to be long enough for computation, and may not ensure necessary and sufficient conditions for convergence [1]. A suboptimal inverse control is developed [9] where the neural nonlinear model of the plant is linearized into a local linear model by which the control law is derived at each sample time using Kalman’s method. Another alternative method is that an affine model is preassumed and then two NNs are used to approximate a general (nonaffine) nonlinear discrete system. Because the control signal occurs linearly, it can be computed directly from the approximate model [13], [19]. However, since the output of the original system is not linear to the to be small input, the method requires the control signal enough to maintain reasonable accuracy. Besides, it is difficult to analyze the closed-loop stability and performance of nonlinear IMC because of the complexity of general nonlinear processes. So far the stability of the closedloop system was discussed only under the assumption that a perfect model is available. However, there is always a plant/model mismatch in practical situation. Furthermore, if the model inversion is not perfect, it would be even more difficult to analyze the system robustness. One can only try to demonstrate that the proposed neural IMC is one of the best possible methods [19]. Therefore, it is essential to know how the controller should be designed for the closed-loop stability. In this paper, an approximate internal model-based neural control (AIMNC) strategy is proposed for the control of unknown nonaffine nonlinear discrete systems subject to uncertainties. The proposed AIMNC is of clear advantages in respect
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Fig. 1. Basic structure of nonlinear IMC.
Fig. 2.
Basic structure of NN IMC.
to the traditional NN IMC because it removes the following restrictions: 1) The process to be controlled is open-loop stable and must have stable zero dynamics; 2) an NN needs to be trained to learn the inverse dynamics of the process and to act as a nonlinear controller; 3) the sampling time interval should be long enough for control computation; and 4) the process must be linear with regard to its input or the operating region is limited to a sufficiently small neighborhood of the origin. Besides, the stability and the performance of the closed-loop system can be easily analyzed. Only one neural network needs to be trained and control algorithm can be directly obtained from model identification without further training. Because of the innovative approximation, the general training algorithms, such as those available in neural network toolbox in MATLAB, can be used directly. Thus, the method proposed can facilitate the design of a neural IMC and is easy for control engineers to use. Finally, extensive simulations demonstrate the superior performance of the proposed neural IMC strategy.
II. NN APPROXIMATE INPUT–OUTPUT MODELS OF NONLINEAR DISCRETE SYSTEMS A. General Input–Output Representation A general input–output representation for an -dimensional nonlinear discrete system with relative degree is as follows [3]: (2.1) where , and are the input , and and output of the system, respectively, . Without loss of generality, is assumed. Equation (2.1) is the so called nonlinear autoregressive moving average (NARMA) model. For a general discrete-time nonlinear system, the NARMA model is an exact representation
of its input–output behavior over the range in which the system operates [13]. Modeling is the first important step for control of an unknown nonlinear plant. Since NNs are universal approximators, an NN will simply be regarded as a convenient way to model a nonlinear input–output mapping. As long as system (2.1) evolves , there is always an NN caon a compact subset of pable of approximating (2.1) on for every tolerance level [3]. Various NNs have been proposed. Among them, the multilayer NN has been widely used in modeling and control of nonlinear discrete dynamical systems [14], [7], [9], [11], [3], [19] due to its general approximation abilities [5], [6]. In our study, this type of NN is used and trained to represent the dynamic behavior of unknown nonlinear discrete systems, which is often called NN NARMA model as (2.2) (2.2) is an NN and the weight vector of the NN is In (2.2), omitted for simplicity. is an approximation error and with being a small positive number. This NN model can be trained by static back-propagation [14] to approximate the underlying process in terms of the past inputs and outputs of the system. It is well known that finding the best NN (approximator) structure is a difficult and unsolved problem [20]. It is believed that a fairly large NN can deal with relative complex approximation problems. Thus, in our study, a basic NN structure with two hidden layers is fixed as in [14], [11], and [13]. Based on the learning performance, the number of hidden neurons can be roughly determined. Using such two-layered NN structure, which is more general than one-layered NNs, can show that the proposed NN IMC strategy in the paper does not rely on special structure of NNs. It is worth noting that, for the identification of open-loop unstable systems, the closed-loop identification techniques must be used [17]. One possible solution as shown in Fig. 3 is to use a simple feedback controller so as to keep the system inside the range in which it is intended to operate. One may also resort to a
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and and with as a finite positive number. and the uncertainty in After neglecting the reminder (2.4), an NN approximate model in input–output form is derived as follows: (2.6) Fig. 3. NN identification for open-loop unstable systems.
manually tuned PID controller or a human operator for controlling the system if a stabilizing controller is not available beforehand [18]. After identification, the simple feedback controller will not be considered again and the design of NN controllers is based on the identified NN model only. When uncertainties are considered, a more general form of nonlinear discrete systems can be described as follows: (2.3) is modeled as the effect of uncertainties (approximawhere and as a finite tion error and disturbances) with positive number. B. NN Approximation for a Simplified Input–Output Relationship Since the output of the NN model (2.3) is nonlinearly dependent on its input, it is difficult to derive its inversion and related control law. Thus, an approximate model is derived first for the system (2.3) to have a simplified input–output relationship. For the NN NARMA model (2.3), the Taylor expansion of with respect to around is shown as
(2.4)
III. APPROXIMATE INTERNAL MODEL BASED NEURAL CONTROL An AIMNC strategy is described in Fig. 4, which is an extension of traditional NN IMC. The nominal NN controller and uncertainty compensation in Fig. 4 are two major parts. Comparing Fig. 4 with the basic structure of NN IMC as shown in Fig. 2, the nominal NN controller replaced the NN inverse controller and the approximate NN model (2.6) was used in uncertainty compensation. The design of the nominal NN controller and the analysis of uncertainty compensation are discussed below in details. It is worth noting that, because of the complexity of general nonlinear systems, the following discussion concentrates on local models that are valid in a neighborhood of an equilibrium state of the system. A. Development of Nominal Neural Control
and
where
In (2.6), the control increment appears linearly to the output, thus, control law design can be simplified greatly. To is crucial. For derive (2.6), the calculation of the double-layered NN used, the calculation of is presented in details in Appendix A. According to (2.5), should not be too large to limit the approximation error of the model (2.6). In practice, this is reasonable as indicated in in the remainder Assumption 2. On the other hand, may be small such that no strict limitation must be imposed . A special example for this case is that the original on nonlinear nonaffine system can be approximately considered as may approach to zero. an affine system and, thus,
. with as a Assumption 1: finite positive number. with as a finite positive Assumption 2: number. Assumption 3: with . Assumption 1 shows that system (2.4) is of well defined relative degree [3]. Assumption 2 is reasonable in practice because the output of a physical system cannot change too fast within a small time interval due to the “inertia” of the system [4]. Assumption 3 indicates that the possible control action provided by a system should be large enough to suppress the inertia of the system; otherwise, the equipped actuator is incapable of controlling the system. According to Assumption 2, the remainder
The control increment parts as shown below
in (2.6) can be divided into two
(3.1) where
refers to the nominal control increment and is used to compensate the uncertainties. Thus, according to (2.6), one has
(3.2) In nominal case, neglecting in (3.2), one has
in (2.4) and, thus, removing
(3.3) (2.5) is bounded by with
, where as a point between
1) Nominal NN Controller: Though various methods can be used to determine the control increment from (3.3), the most convenient way is from the direct inversion of (3.3) because its
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Fig. 4. Conceptual structure of AIMNC.
output is linear with respect to its input. If (3.3) has a stable inversion (stable zeros dynamics), one can simply choose the direct inversion of (3.3) as an approximate inverse controller. If (3.3) does not have a stable inversion (unstable zeros dynamics), direct inversion of (3.3) will lead to an unstable controller, and, thus, some modification may be needed to avoid such a phenomenon. Based on (3.3), the nominal NN controller is determined as follows:
Linearizing (3.6a) around yields (3.6b) where
(3.4) and where with . where is a reference trajecIn (3.4), tory and refers to the setpoint filter in Fig. 4. The setpoint filter is actually a stable reference model for the desired dynamic behavior of the control system. 2) Zero Dynamics and Bounded Control Law: A stable inversion of (3.3) implies a stable zero dynamics. The method in [3] can be used to derive the input–output zero dynamics of (3.3) around the equilibrium point of (2.1) as follows. be zero in (3.3), By letting one has
The prevoius equation can be rewritten as (3.5) . where For given initial conditions, if (3.5) converges, then (3.3) has is stable zero dynamics and the control law (3.4) with bounded. If (3.5) does not converge, then (3.3) has unstable zero is undynamics. In this case, the control law (3.4) with bounded. However, by suitably choosing , a stable control law (3.4) may be achieved even though (3.3) has unstable zero dynamics. A simple method for choosing is given by examining the poles of the linearized system of (3.4) as follows. be zero in (3.4), By letting one has (3.6a)
Taking -transformation of (3.6b) yields the following polynomial equation: (3.6c) By suitable choosing , all the solutions of (3.6c) may be inside the unit circle, which implies that (3.6b) is stable. According to [21], the stability of linear system (3.6b) is identical to that of nonlinear system (3.6a) around . Thus, a chosen that makes (3.6b) stable will lead to a stable (3.6a), which implies that the nominal control law (3.4) is bounded. One simple nonlinear discrete system with unstable zero dynamics is given in Appendix B to illustrate the above method. 3) Stability of the Nominal NN Control: The nominal stability and performance can be derived by neglecting the uncertainties as follows. , Define control error then
(3.7) In (3.7), as follows:
can be expressed using Taylor series
(3.8)
LI AND DENG: AN APPROXIMATE INTERNAL MODEL-BASED NEURAL CONTROL
Fig. 5.
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Algorithm of the AIMNC.
where
with and . Theorem 1: For given and under Assumptions 1–3, the control law (3.4) ensures that the solutions of the error system (3.7) are with ultimate uniformly ultimately bounded for all , bound and where . . Proof: Define the Lyapunov candidate Using the control law (3.4), one has
1) Uncertainty Compensation: When uncertainties exist, uncertainty attenuation can be achieved via internal model with as shown in Fig. 5. Using (3.1), (2.4) a robustness filter becomes
(3.11) and
Since , it becomes
(3.12)
(3.9a) Substituting (3.8) in to (3.9a) yields
(3.9b) Using Assumptions 1–3, one has
(3.10) From (3.10),
when . On the other hand, if a suitable is chosen so that all the roots of (3.6c) are inside the unit circle, the control signal given by (3.4) is bounded. Therefore, one concludes that the solutions of (3.7) are uniwith ultimate bound formly ultimately bounded for all . B. Uncertainty Compensation in Internal Model Structure In practice, uncertainties are inevitable. Large uncertainties significantly degrade the control performance and, thus, uncertainty attenuation is necessary for a better closed-loop performance. To attenuate the uncertainties, the AIMNC strategy of Fig. 4 is adopted, which is shown in detail in Fig. 5
. where , the model apTherefore, by suitably choosing filter and disturbance can be attenuated to proximation error some extent. As the modeling error is larger at high frequencies, intuitively the addition of a low-pass filter adds robustness characteristics in the control architecture. The robustness filter is typically a first-order one with unit gain and frequency response tuned to eliminate high frequency noise introduced by measurement devices [12]. is necessary for IMC of Remark: The robustness filter can not be realized continuous-time systems because practically [8]. For IMC of discrete-time systems, there is no realization problem. If the measurement noise and the input constraint do not need to be considered, can be simply set to be 1. Under the control architecture as shown in Fig. 5, the equivalent NN control law can be expressed as follows:
(3.13) Control law (3.13) consists of the nominal NN control and uncertainty compensation and, thus, combines the advantages of both inverse control and nonlinear IMC. Since the approximate internal model (2.6) does not exactly match the plant, the error in Fig. 5 is not null. A large error may deteriorate performances or even cause instability. Thus, the robustness of the stability of a closed-loop system should be carefully examined.
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2) Robustness of Stability and Performance: Usually, it is difficult to analyze the robust stability and performance of the nonlinear nonaffine discrete system, particularly when the modeling error exists. However, under the proposed AIMNC, the following result is common as stated in Theorem 2. . Define control error Then
Using (3.17), (3.16) becomes
(3.18) when
From (3.18),
(3.14) Theorem 2: For given and under Assumptions 1–3, the NN control law (3.13) ensures that the solutions of the error system (3.14) are uniwith ultimate bound formly ultimately bounded for all , where
, and
are defined in Theorem 1 and . Proof: Choosing the Lyapunov function as , one has
(3.15) Substituting (3.13) into (3.15), one has
(3.16) where
. Thus, the solutions of (3.14) are uniformly ultimately bounded and the ultimate bound is . According to Theorem 2, the nominal neural controller and the uncertainty compensation can be designed separately. The former provides the nominal control performance and guarantees the stability of the closed-loop system and the later contributes to the uncertainty attenuation. C. Performance of the Closed-Loop System Generally, the following performances can be achieved for the proposed neural control strategy when the internal model has a stable inversion. 1) If the internal model is perfect, i.e., the modeling error in (2.4) and , a perfect control can be achieved with the effect of constant disturbances completely suppressed. 2) If the internal model is not perfect, a perfect control cannot be achieved. However, the transient and steadystate performance can be adjusted by tuning the parameters of the filters. The effect of constant disturbances can still be suppressed in the steady-state. 3) Even for the imperfect internal model, the steady state error can still approach zero when tracking constant setpoints and having constant disturbances. This is bein the steady-state such that the apcause . proximation error 4) For bounded disturbances, the stability of the proposed neural IMC is guaranteed and the control error will be bounded as described in Theorem 2. 5) The above performance can be achieved for both openloop stable and unstable nonlinear plants. When the internal model has an unstable inversion, the perfect control can not be achieved. By choosing a suitable , the control law (3.13) may be bounded and a satisfactory control performance may be achieved. IV. CONTROL SIMULATIONS
and
As shown in Theorem 1, . Thus, one has (3.17)
Simulation will be carried out to test the effectiveness of the proposed AIMNC when a process is unknown with uncertainties. The multilayer NNs are used for identification of the unknown process. The following properties are common and employed in the simulations. (i.e., inputs plus a bias, • Neural structure: 10 nodes plus a bias input in the first and second hidden layers, output nodes and all bias inputs are unity).
LI AND DENG: AN APPROXIMATE INTERNAL MODEL-BASED NEURAL CONTROL
Fig. 6.
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Nominal control performance with different r 1s.
• Training algorithm: the Levenberg–Marquart method (a Hessian-based algorithm [16] available in neural network toolbox in MATLAB). with • Setpoint filter: . with • Robustness filter: .
A. Control of an Open-Loop Stable Unknown Process The system is described by following nonlinear difference equation:
(4.1) and the nonlinear plant is open-loop In this example, stable. Identification . • Neural structure: • Training sequence: a random input sequence with 500 values . uniformly distributed in the interval • Training error: mean square error (MSE) of 6.63e-6 is obtained after training on the training sequences. . • NN NARMA model: • NN approximate model: (2.6) from the previously identified NN NARMA model. 1) Nominal Performance: When there is no disturbance, the NN control law (3.4) with from (3.3) is used. The nominal control performance changes as in filter changes. and , the controlled outputs are shown When to achieve a in Fig. 6. One may choose a proper parameter required nominal control performance.
, the NN 2) The Effect of Disturbance: Choose is used again when there are discontrol law (3.4) with turbances. The control results in Fig. 7 show that the nominal control performance without internal model feedback is unsatisfactory when disturbances happen. 3) Robust Stability and Performance: When there are disturfrom the proposed bances, the NN control law (3.13) with neural IMC structure as shown in Fig. 5 can be used. Choose and . The control performance is quite satisfactory as shown in Fig. 8. B. Control of an Open-Loop Unstable Unknown Process . Since the plant is openThe example (4.1) is used with loop unstable, the closed-loop identification method as shown in Fig. 3 is used. Identification . • Neural structure: • Training sequence: a random input sequence with 500 values plus a feeduniformly distributed in the interval . back 0.6 • Training error: mean square error (MSE) of 8.3e-7 is obtained after training on the training sequences. . • NN NARMA model: • NN approximate model: (3.3) from the above identified NN NARMA model. To test the performance of the neural approximate model (3.3), a test input signal
is inputted to the system and the NN approximate model (3.3), simultaneously. The simulation in Fig. 9 shows that the output y(k) of (3.3) is very close to that of the system under the test input signal. The approximation error e(k) is very small. To control the system subject to disturbance, the NN control law (3.13) with is used. Choose and
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Fig. 7.
Nominal control performance with and without disturbances v .
Fig. 8.
Control of an open-loop stable unknown plant.
. The control performance is satisfactory as shown in Fig. 10 even though the nonlinear plant is open-loop unstable with large disturbance. C. Control of a Third-Order Unknown Process With Stable Zero Dynamics The system is described by following difference equations:
(4.2)
Fig. 9.
Evaluation of the neural approximate model (3–3).
Fig. 10.
Control of an open-loop unstable unknown plant.
Let dynamics Identification
and
. In this case, the zero is stable.
• Neural structure: . • Training sequence: a random input sequence with 1000 values uniformly distributed in the interval . • Training error: mean square error (MSE) of 2.31e-5 is obtained after 575 epochs training on the training sequences. • NN NARMA model: . • NN approximate model: Equation (3.3) from the above identified NN NARMA model. The neural control law (3.13) with is used. Choose , . The control results in Fig. 11 shows that the proposed neural control can give quite good control performance even though there are zero dynamics and large disturbance in the nonlinear system.
LI AND DENG: AN APPROXIMATE INTERNAL MODEL-BASED NEURAL CONTROL
Fig. 11.
Control of an unknown plant with stable zero dynamics.
Fig. 12.
Input–output zero dynamics.
D. Control of an Unknown Process With Unstable Zero Dynamics The example (4.2) is used with , and . In this case, the zero dynamics is unstable. The identification of the given unknown nonlinear process can still be performed even though the zero dynamic is unstable. Identification • Neural structure: . • Training sequence: a random input sequence with 1000 . values uniformly distributed in the interval • Training error: mean square error (MSE) of 3.71e-5 is obtained after 600 epochs training on the training sequences.
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• NN NARMA model: • NN approximate model: Equation (3.3) from the above identified NN NARMA model. Equation (3.5) can be used to check if the NN approximate model (2.6) has a stable inversion. The two cases for both and initial value stable and unstable zero dynamics with are shown in Fig. 12. It is clearly shown that (3.5) converges with the stable zero dynamics and diverges with the unstable zero dynamics. This can also be clearly seen from . For the the poles of the linearized system (3.6b) with nonlinear plant with stable zero dynamics, the poles of (3.6b) and 0.2375. However, for the nonlinear plant with are and unstable zero dynamics, the poles of (3.6b) are 0.2504.
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Fig. 13. Control of an unknown plant with unstable zero dynamics.
To control this system, the NN control law (3.13) is chosen and a suitable needs to be determined first. The linearized system (3.6b) is derived as follows:
CALCULATION OF
APPENDIX A FOR A DOUBLE-LAYERED NEURAL NETWORK
A double-layered NN can be expressed as follows: (A.1)
(4.3) If is chosen as , then (4.3) is stable. The poles when is set to be 0.7. of (4.3) will be 0.3087 and The proposed NN control law (3.13) is used with and . The control results in Fig. 13 shows that the proposed AIMNC can give quite good control performance even though the zero dynamics of the system is unstable and large disturbances exist.
where
, , , and are weight matrices in the first hidden layer, second , and hidden layer, and the output layer of the NN, are bias vectors, and is a hyperbolic tangent activation function vector with input vector . Define . Thus
V. CONCLUSION An AIMNC strategy is proposed for control of general unknown nonlinear discrete processes under model mismatch and disturbances. The proposed neural control strategy not only combines the advantages of both inverse control and nonlinear IMC but also removes some of their disadvantages. Based on the new input–output approximation, the neural inverse control law is derived directly and applied straightforward to an unknown process. The stability and the performance of a closed-loop system can be analyzed easily even if the unknown nonlinear process is open-loop unstable or has unstable zero dynamics. The computation is quite small since only one NN needs to be trained for both model approximation and control formulation and no further training (offline or online) is required for the online controller. Extensive simulations demonstrate that the proposed neural control strategy can be used for a wide range of unknown nonlinear discrete dynamical systems under model mismatch and disturbances.
Since the derivative of is -dimensional unit matrix and th hidden layer, one has
with as a as the number of neurons in (A.2)
with .
and
APPENDIX B BOUNDED CONTROL LAW FOR A SYSTEM WITH UNSTABLE ZERO DYNAMICS Consider a nonlinear discrete system as follows:
(B.1)
LI AND DENG: AN APPROXIMATE INTERNAL MODEL-BASED NEURAL CONTROL
In this example the zero dynamics is unstable. The system (B.1) can be expressed in following input–output form:
(B.2) Suppose that an NN is trained to approximate (B.2) perfectly. Thus, an NN NARMA model for (B.2) is obtained as follows: (B.3) where and . From (B.3), an NN approximate model is derived in the form of (2.6) as follows: (B.4) where and . From (B.4), the NN control law in the form of (3.4) is as follows: (B.5) Let
. Then, . Thus, (B.5) becomes
(B.6) where . Linearizing (B.6) around has
and
, one
(B.7) Since
. Thus, (B.7) becomes (B.8) , (B.8) is unstable, which implies that the control law If (B.5) is unbounded. However, by choosing , (B.8) , the control is stable. Therefore, by choosing law (B.5) for the system (B.1) with unstable zero dynamics is bounded.
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ACKNOWLEDGMENT The authors greatly appreciate the useful comments from anonymous referees. REFERENCES [1] R. Boukezzoula, S. Galichet, and L. Foulloy, “Nonlinear internal model control: Application of inverse model based fuzzy control,” IEEE Trans. Fuzzy Syst., vol. 11, no. 6, pp. 814–829, Dec. 2003. [2] C. G. Economou, M. Morari, and B. O. Palsson, “Internal model control. 5. Extension to nonlinear systems,” Ind. Eng. Chem. Process Des. Dev., vol. 25, pp. 25 403–411, 1986. [3] J. B. D. Cabrera and K. S. Narendra, “Issues in the application of neural networks for tracking based on inverse control,” IEEE Trans. Autom. Control, vol. 44, no. 11, pp. 2007–2027, Nov. 1999. [4] S. S. Ge, T. H. Lee, G. Y. Li, and J. Zhang, “Adaptive NN control for a class of discrete-time nonlinear systems,” Int. J. Control, vol. 76, no. 4, pp. 334–354, 2003. [5] K. Hornik, “Approximation capabilities of multilayer feed-forward neural networks,” Neural Netw., vol. 4, pp. 251–257, 1990. [6] K. Hornik, M. Stinchcombe, and H. White, “Multilayer feed-forward networks are universal approximators,” Neural Netw., vol. 2, pp. 359–366, 1989. [7] L. Jin, P. N. Nikiforuk, and M. M. Gupta, “Fast neural learning and control of discrete-time nonlinear systems,” IEEE Trans. Syst., Man Cybern., vol. 25, no. 3, pp. 478–488, Mar. 1995. [8] H. X. Li and P. P. J. Van Den Bosch, “A robust disturbance-based control and its application,” Int. J. Control, vol. 58, no. 3, pp. 537–554, 1993. [9] G. Lightbody and G. W. Irwin, “Nonlinear control structures based on embedded neural system models,” IEEE Trans. Neural Netw., vol. 8, no. 3, pp. 553–567, May 1997. [10] M. Morari and E. Zafiriou, Robust Process Control. Englewood Cliffs, NJ: Prentice-Hall, 1989. [11] S. Mukhopadhyay and K. S. Narendra, “Adaptive control of nonlinear multivariable systems using neural networks,” Neural Netw., vol. 7, no. 5, pp. 737–752, 1994. [12] E. P. Nahas, M. A. Henson, and D. E. Seborg, “Nonlinear internal model control strategy for neural network models,” Comput. Chem. Eng., vol. 16, no. 12, pp. 1039–1057, 1992. [13] K. S. Narendra and S. Mukhopadhyay, “Adaptive control using neural networks and approximate models,” IEEE Trans. Neural Netw., vol. 8, no. 3, pp. 475–485, May 1997. [14] K. S. Narendra and K. Parthasarathy, “Identification and control of dynamical systems using neural networks,” IEEE Trans. Neural Netw., vol. 1, no. 1, pp. 4–27, Jan. 1990. , “Gradient methods for the optimization of dynamical systems con[15] taining neural networks,” IEEE Trans. Neural Netw., vol. 2, no. 2, pp. 252–262, Mar. 1991. [16] S. G. Nash and A. Sofer, Linear and Nonlinear Programming. New York: McGraw-Hill, 1996. [17] O. Nelles, Nonlinear System Identification: From Classical Approaches to Neural Networks and Fuzzy Models. Berlin, Germany: SpringerVerlag, 2001. [18] M. Norgaard, O. Ravn, N. K. Poulsen, and L. K. Hansen, Neural Networks for Modeling and Control of Dynamic Systems. London, U.K.: Springer-Verlag, 2000. [19] I. Rivals and L. Personnaz, “Nonlinear internal model control using neural networks: Application to processes with delay and design issues,” IEEE Trans. Neural Netw., vol. 11, no. 1, pp. 80–90, Jan. 2000. [20] J. T. Spooner, M. Maggiore, R. Ordonez, and K. M. Passino, Stable Adaptive Control and Estimation for Nonlinear Systems. New York: Wiley, 2002. [21] M. Vidyasagar, Nonlinear Systems Analysis. Englewood Cliffs, NJ: Prentice-Hall, 1993. [22] P. J. Werbos, “Backpropagation through time: What it does and how to do it,” Proc. IEEE, vol. 78, no. 10, pp. 1550–1560, Oct. 1990. [23] K. J. Hunt and D. Sbarbaro, “Neural Networks for nonlinear internal model control,” Proc. IEE D-Control Theory and Applications, vol. 138, no. 5, pp. 431–438, Sep. 1991.
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Han-Xiong Li (S’94–M’97–SM’00) received the Ph.D. degree in electrical engineering from University of Auckland, New Zealand, in January of 1997, the M.E. degree in electrical engineering from Delft University of Technology, The Netherlands, in 1991, and the B.E. degree from National University of Defence Technology, China, in 1982. Currently, he is an Associate Professor at the Department of Manufacturing Engineering and Engineering Management, City University of Hong Kong, Hong Kong. He is a “Chang Jiang Scholar” in Central South University, an Honorary Professor endowed by Ministry of Education, China. He was awarded the distinguished young scholar fund by the China National Science Foundation in 2004. In the last 20 years, he worked in different fields including military service, industry, and academia. He has gained industrial experience as a Senior Process Engineer from ASM, a leading supplier for semiconductor assembly equipment. His research interests include intelligent control and learning, modelling and control of industrial process with special interest to distributed parameter systems. Dr. Li serves as an Associate Editor for IEEE TRANSACTIONS ON SYSTEMS, MAN AND CYBERNETICS—PART B: CYBERNETICS.
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Hua Deng received the Ph.D. degree from the Department of Manufacturing Engineering and Engineering Management, City University of Hong Kong, Hong Kong, in 2005, the M.E. degree from Northwestern Polytechnical University, China, in 1988, and the B.E. degree from Nanjing Aeronautical Institute, China, in 1983. He was an Associate Professor at Changsha University of Science and Technology, Changsha, China. Currently, he is a Professor at the School of Mechanical and Electrical Engineering, Central South University, Changsha, China. His research interests include identification and control of complex distributed parameter systems, intelligent control, intelligent manufacturing, fault diagnoses, and computer control system design.
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An Approximate Internal Model-Based Neural Control for Unknown Nonlinear Discrete Processes Han-Xiong Li, Senior Member, IEEE, and Hua Deng
Abstract—An approximate internal model-based neural control (AIMNC) strategy is proposed for unknown nonaffine nonlinear discrete processes under disturbed environment. The proposed control strategy has some clear advantages in respect to existing neural internal model control methods. It can be used for open-loop unstable nonlinear processes or a class of systems with unstable zero dynamics. Based on a novel input–output approximation, the proposed neural control law can be derived directly and implemented straightforward for an unknown process. Only one neural network needs to be trained and control algorithm can be directly obtained from model identification without further training. The stability and robustness of a closed-loop system can be derived analytically. Extensive simulations demonstrate the superior performance of the proposed AIMNC strategy. Index Terms—Input–output approximation, neural networks, nonaffine nonlinear discrete-time systems, nonlinear internal model control, unstable zero dynamics.
I. INTRODUCTION
F
OR THE control of nonlinear discrete-time processes subject to uncertainties (model mismatch and disturbances), nonlinear internal model control (IMC) using neural networks (NNs) has received much attention [12], [9], [19]. Nonlinear IMC was proposed by Economou et al. [2], as shown in Fig. 1. The key characteristic of this type of control strategy is having the inverse controller and the internal model. In Fig. 1, the model of the nonlinear plant is needed as the internal model. Using this internal model, the effect of uncertainties can be suppressed with the feedback signal generated. In nonlinear IMC, the nonlinear model and its inversion play a crucial role. Fig. 2 gives a basic NN IMC structure [18], which is an extension of nonlinear IMC. For the control of unknown nonlinear discrete systems using the nonlinear IMC structure, NN model is employed as the internal model and NN inverse controller is used to replace the inverse controller, as shown in Fig. 2. The control structure given in Figs. 1 and 2 has been shown to have good robustness against uncertainties [10], [12], [9]. For NN IMC, however, even though the NN model is available, it is still not easy to design the NN inverse controller because of the following reasons. First, the process to be controlled Manuscript received July 4, 2004; revised April 14, 2005. This paper was supported in part under a Grant from RGC of Hong Kong (CityU 1129/03E) and a Grant from the National Science Foundation (NSF) of China for distinguished young scholar (50 429 501). H.-X. Li is with the Department of Manufacturing Engineering and Engineering Management, City University of Hong Kong, Kowloon, Hong Kong (e-mail: [email protected]). H. Deng is with the School of Mechanical and Electrical Engineering, Central South University, Changsha 410083, China. Digital Object Identifier 10.1109/TNN.2006.873277
must be stable in open-loop, which definitely limits many of IMC applications [18]. Second, the process to be controlled is required to have stable zero dynamics because of the use of the model inversion. To our knowledge, little result has been reported about the NN IMC of nonlinear nonaffine discrete systems with unstable zero dynamics. Third, it may not be easy to have the model inversion for control law design due to the nonlinear internal model used. For the third problem mentioned previously, several solutions are introduced under certain conditions. One of the solutions is to train an NN inverse controller with the help of the identified NN model [19], [23]. In this case, dynamic gradient methods, such as dynamic back-propagation [15] and back-propagation through time [22], need to be used since the controller is in the feedback loop of the dynamical system [13]. Compared with static back-propagation, however, all dynamic gradient methods are quite slow and computationally intensive for computing the [13]. It is possible to calculate the inversion control signal at each sample time for the given neural nonlinear model using the numerical inverse approach [12], [23]. Such an iterative algorithm requires the time interval to be long enough for computation, and may not ensure necessary and sufficient conditions for convergence [1]. A suboptimal inverse control is developed [9] where the neural nonlinear model of the plant is linearized into a local linear model by which the control law is derived at each sample time using Kalman’s method. Another alternative method is that an affine model is preassumed and then two NNs are used to approximate a general (nonaffine) nonlinear discrete system. Because the control signal occurs linearly, it can be computed directly from the approximate model [13], [19]. However, since the output of the original system is not linear to the to be small input, the method requires the control signal enough to maintain reasonable accuracy. Besides, it is difficult to analyze the closed-loop stability and performance of nonlinear IMC because of the complexity of general nonlinear processes. So far the stability of the closedloop system was discussed only under the assumption that a perfect model is available. However, there is always a plant/model mismatch in practical situation. Furthermore, if the model inversion is not perfect, it would be even more difficult to analyze the system robustness. One can only try to demonstrate that the proposed neural IMC is one of the best possible methods [19]. Therefore, it is essential to know how the controller should be designed for the closed-loop stability. In this paper, an approximate internal model-based neural control (AIMNC) strategy is proposed for the control of unknown nonaffine nonlinear discrete systems subject to uncertainties. The proposed AIMNC is of clear advantages in respect
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Fig. 1. Basic structure of nonlinear IMC.
Fig. 2.
Basic structure of NN IMC.
to the traditional NN IMC because it removes the following restrictions: 1) The process to be controlled is open-loop stable and must have stable zero dynamics; 2) an NN needs to be trained to learn the inverse dynamics of the process and to act as a nonlinear controller; 3) the sampling time interval should be long enough for control computation; and 4) the process must be linear with regard to its input or the operating region is limited to a sufficiently small neighborhood of the origin. Besides, the stability and the performance of the closed-loop system can be easily analyzed. Only one neural network needs to be trained and control algorithm can be directly obtained from model identification without further training. Because of the innovative approximation, the general training algorithms, such as those available in neural network toolbox in MATLAB, can be used directly. Thus, the method proposed can facilitate the design of a neural IMC and is easy for control engineers to use. Finally, extensive simulations demonstrate the superior performance of the proposed neural IMC strategy.
II. NN APPROXIMATE INPUT–OUTPUT MODELS OF NONLINEAR DISCRETE SYSTEMS A. General Input–Output Representation A general input–output representation for an -dimensional nonlinear discrete system with relative degree is as follows [3]: (2.1) where , and are the input , and and output of the system, respectively, . Without loss of generality, is assumed. Equation (2.1) is the so called nonlinear autoregressive moving average (NARMA) model. For a general discrete-time nonlinear system, the NARMA model is an exact representation
of its input–output behavior over the range in which the system operates [13]. Modeling is the first important step for control of an unknown nonlinear plant. Since NNs are universal approximators, an NN will simply be regarded as a convenient way to model a nonlinear input–output mapping. As long as system (2.1) evolves , there is always an NN caon a compact subset of pable of approximating (2.1) on for every tolerance level [3]. Various NNs have been proposed. Among them, the multilayer NN has been widely used in modeling and control of nonlinear discrete dynamical systems [14], [7], [9], [11], [3], [19] due to its general approximation abilities [5], [6]. In our study, this type of NN is used and trained to represent the dynamic behavior of unknown nonlinear discrete systems, which is often called NN NARMA model as (2.2) (2.2) is an NN and the weight vector of the NN is In (2.2), omitted for simplicity. is an approximation error and with being a small positive number. This NN model can be trained by static back-propagation [14] to approximate the underlying process in terms of the past inputs and outputs of the system. It is well known that finding the best NN (approximator) structure is a difficult and unsolved problem [20]. It is believed that a fairly large NN can deal with relative complex approximation problems. Thus, in our study, a basic NN structure with two hidden layers is fixed as in [14], [11], and [13]. Based on the learning performance, the number of hidden neurons can be roughly determined. Using such two-layered NN structure, which is more general than one-layered NNs, can show that the proposed NN IMC strategy in the paper does not rely on special structure of NNs. It is worth noting that, for the identification of open-loop unstable systems, the closed-loop identification techniques must be used [17]. One possible solution as shown in Fig. 3 is to use a simple feedback controller so as to keep the system inside the range in which it is intended to operate. One may also resort to a
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and and with as a finite positive number. and the uncertainty in After neglecting the reminder (2.4), an NN approximate model in input–output form is derived as follows: (2.6) Fig. 3. NN identification for open-loop unstable systems.
manually tuned PID controller or a human operator for controlling the system if a stabilizing controller is not available beforehand [18]. After identification, the simple feedback controller will not be considered again and the design of NN controllers is based on the identified NN model only. When uncertainties are considered, a more general form of nonlinear discrete systems can be described as follows: (2.3) is modeled as the effect of uncertainties (approximawhere and as a finite tion error and disturbances) with positive number. B. NN Approximation for a Simplified Input–Output Relationship Since the output of the NN model (2.3) is nonlinearly dependent on its input, it is difficult to derive its inversion and related control law. Thus, an approximate model is derived first for the system (2.3) to have a simplified input–output relationship. For the NN NARMA model (2.3), the Taylor expansion of with respect to around is shown as
(2.4)
III. APPROXIMATE INTERNAL MODEL BASED NEURAL CONTROL An AIMNC strategy is described in Fig. 4, which is an extension of traditional NN IMC. The nominal NN controller and uncertainty compensation in Fig. 4 are two major parts. Comparing Fig. 4 with the basic structure of NN IMC as shown in Fig. 2, the nominal NN controller replaced the NN inverse controller and the approximate NN model (2.6) was used in uncertainty compensation. The design of the nominal NN controller and the analysis of uncertainty compensation are discussed below in details. It is worth noting that, because of the complexity of general nonlinear systems, the following discussion concentrates on local models that are valid in a neighborhood of an equilibrium state of the system. A. Development of Nominal Neural Control
and
where
In (2.6), the control increment appears linearly to the output, thus, control law design can be simplified greatly. To is crucial. For derive (2.6), the calculation of the double-layered NN used, the calculation of is presented in details in Appendix A. According to (2.5), should not be too large to limit the approximation error of the model (2.6). In practice, this is reasonable as indicated in in the remainder Assumption 2. On the other hand, may be small such that no strict limitation must be imposed . A special example for this case is that the original on nonlinear nonaffine system can be approximately considered as may approach to zero. an affine system and, thus,
. with as a Assumption 1: finite positive number. with as a finite positive Assumption 2: number. Assumption 3: with . Assumption 1 shows that system (2.4) is of well defined relative degree [3]. Assumption 2 is reasonable in practice because the output of a physical system cannot change too fast within a small time interval due to the “inertia” of the system [4]. Assumption 3 indicates that the possible control action provided by a system should be large enough to suppress the inertia of the system; otherwise, the equipped actuator is incapable of controlling the system. According to Assumption 2, the remainder
The control increment parts as shown below
in (2.6) can be divided into two
(3.1) where
refers to the nominal control increment and is used to compensate the uncertainties. Thus, according to (2.6), one has
(3.2) In nominal case, neglecting in (3.2), one has
in (2.4) and, thus, removing
(3.3) (2.5) is bounded by with
, where as a point between
1) Nominal NN Controller: Though various methods can be used to determine the control increment from (3.3), the most convenient way is from the direct inversion of (3.3) because its
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Fig. 4. Conceptual structure of AIMNC.
output is linear with respect to its input. If (3.3) has a stable inversion (stable zeros dynamics), one can simply choose the direct inversion of (3.3) as an approximate inverse controller. If (3.3) does not have a stable inversion (unstable zeros dynamics), direct inversion of (3.3) will lead to an unstable controller, and, thus, some modification may be needed to avoid such a phenomenon. Based on (3.3), the nominal NN controller is determined as follows:
Linearizing (3.6a) around yields (3.6b) where
(3.4) and where with . where is a reference trajecIn (3.4), tory and refers to the setpoint filter in Fig. 4. The setpoint filter is actually a stable reference model for the desired dynamic behavior of the control system. 2) Zero Dynamics and Bounded Control Law: A stable inversion of (3.3) implies a stable zero dynamics. The method in [3] can be used to derive the input–output zero dynamics of (3.3) around the equilibrium point of (2.1) as follows. be zero in (3.3), By letting one has
The prevoius equation can be rewritten as (3.5) . where For given initial conditions, if (3.5) converges, then (3.3) has is stable zero dynamics and the control law (3.4) with bounded. If (3.5) does not converge, then (3.3) has unstable zero is undynamics. In this case, the control law (3.4) with bounded. However, by suitably choosing , a stable control law (3.4) may be achieved even though (3.3) has unstable zero dynamics. A simple method for choosing is given by examining the poles of the linearized system of (3.4) as follows. be zero in (3.4), By letting one has (3.6a)
Taking -transformation of (3.6b) yields the following polynomial equation: (3.6c) By suitable choosing , all the solutions of (3.6c) may be inside the unit circle, which implies that (3.6b) is stable. According to [21], the stability of linear system (3.6b) is identical to that of nonlinear system (3.6a) around . Thus, a chosen that makes (3.6b) stable will lead to a stable (3.6a), which implies that the nominal control law (3.4) is bounded. One simple nonlinear discrete system with unstable zero dynamics is given in Appendix B to illustrate the above method. 3) Stability of the Nominal NN Control: The nominal stability and performance can be derived by neglecting the uncertainties as follows. , Define control error then
(3.7) In (3.7), as follows:
can be expressed using Taylor series
(3.8)
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Fig. 5.
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Algorithm of the AIMNC.
where
with and . Theorem 1: For given and under Assumptions 1–3, the control law (3.4) ensures that the solutions of the error system (3.7) are with ultimate uniformly ultimately bounded for all , bound and where . . Proof: Define the Lyapunov candidate Using the control law (3.4), one has
1) Uncertainty Compensation: When uncertainties exist, uncertainty attenuation can be achieved via internal model with as shown in Fig. 5. Using (3.1), (2.4) a robustness filter becomes
(3.11) and
Since , it becomes
(3.12)
(3.9a) Substituting (3.8) in to (3.9a) yields
(3.9b) Using Assumptions 1–3, one has
(3.10) From (3.10),
when . On the other hand, if a suitable is chosen so that all the roots of (3.6c) are inside the unit circle, the control signal given by (3.4) is bounded. Therefore, one concludes that the solutions of (3.7) are uniwith ultimate bound formly ultimately bounded for all . B. Uncertainty Compensation in Internal Model Structure In practice, uncertainties are inevitable. Large uncertainties significantly degrade the control performance and, thus, uncertainty attenuation is necessary for a better closed-loop performance. To attenuate the uncertainties, the AIMNC strategy of Fig. 4 is adopted, which is shown in detail in Fig. 5
. where , the model apTherefore, by suitably choosing filter and disturbance can be attenuated to proximation error some extent. As the modeling error is larger at high frequencies, intuitively the addition of a low-pass filter adds robustness characteristics in the control architecture. The robustness filter is typically a first-order one with unit gain and frequency response tuned to eliminate high frequency noise introduced by measurement devices [12]. is necessary for IMC of Remark: The robustness filter can not be realized continuous-time systems because practically [8]. For IMC of discrete-time systems, there is no realization problem. If the measurement noise and the input constraint do not need to be considered, can be simply set to be 1. Under the control architecture as shown in Fig. 5, the equivalent NN control law can be expressed as follows:
(3.13) Control law (3.13) consists of the nominal NN control and uncertainty compensation and, thus, combines the advantages of both inverse control and nonlinear IMC. Since the approximate internal model (2.6) does not exactly match the plant, the error in Fig. 5 is not null. A large error may deteriorate performances or even cause instability. Thus, the robustness of the stability of a closed-loop system should be carefully examined.
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2) Robustness of Stability and Performance: Usually, it is difficult to analyze the robust stability and performance of the nonlinear nonaffine discrete system, particularly when the modeling error exists. However, under the proposed AIMNC, the following result is common as stated in Theorem 2. . Define control error Then
Using (3.17), (3.16) becomes
(3.18) when
From (3.18),
(3.14) Theorem 2: For given and under Assumptions 1–3, the NN control law (3.13) ensures that the solutions of the error system (3.14) are uniwith ultimate bound formly ultimately bounded for all , where
, and
are defined in Theorem 1 and . Proof: Choosing the Lyapunov function as , one has
(3.15) Substituting (3.13) into (3.15), one has
(3.16) where
. Thus, the solutions of (3.14) are uniformly ultimately bounded and the ultimate bound is . According to Theorem 2, the nominal neural controller and the uncertainty compensation can be designed separately. The former provides the nominal control performance and guarantees the stability of the closed-loop system and the later contributes to the uncertainty attenuation. C. Performance of the Closed-Loop System Generally, the following performances can be achieved for the proposed neural control strategy when the internal model has a stable inversion. 1) If the internal model is perfect, i.e., the modeling error in (2.4) and , a perfect control can be achieved with the effect of constant disturbances completely suppressed. 2) If the internal model is not perfect, a perfect control cannot be achieved. However, the transient and steadystate performance can be adjusted by tuning the parameters of the filters. The effect of constant disturbances can still be suppressed in the steady-state. 3) Even for the imperfect internal model, the steady state error can still approach zero when tracking constant setpoints and having constant disturbances. This is bein the steady-state such that the apcause . proximation error 4) For bounded disturbances, the stability of the proposed neural IMC is guaranteed and the control error will be bounded as described in Theorem 2. 5) The above performance can be achieved for both openloop stable and unstable nonlinear plants. When the internal model has an unstable inversion, the perfect control can not be achieved. By choosing a suitable , the control law (3.13) may be bounded and a satisfactory control performance may be achieved. IV. CONTROL SIMULATIONS
and
As shown in Theorem 1, . Thus, one has (3.17)
Simulation will be carried out to test the effectiveness of the proposed AIMNC when a process is unknown with uncertainties. The multilayer NNs are used for identification of the unknown process. The following properties are common and employed in the simulations. (i.e., inputs plus a bias, • Neural structure: 10 nodes plus a bias input in the first and second hidden layers, output nodes and all bias inputs are unity).
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Fig. 6.
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Nominal control performance with different r 1s.
• Training algorithm: the Levenberg–Marquart method (a Hessian-based algorithm [16] available in neural network toolbox in MATLAB). with • Setpoint filter: . with • Robustness filter: .
A. Control of an Open-Loop Stable Unknown Process The system is described by following nonlinear difference equation:
(4.1) and the nonlinear plant is open-loop In this example, stable. Identification . • Neural structure: • Training sequence: a random input sequence with 500 values . uniformly distributed in the interval • Training error: mean square error (MSE) of 6.63e-6 is obtained after training on the training sequences. . • NN NARMA model: • NN approximate model: (2.6) from the previously identified NN NARMA model. 1) Nominal Performance: When there is no disturbance, the NN control law (3.4) with from (3.3) is used. The nominal control performance changes as in filter changes. and , the controlled outputs are shown When to achieve a in Fig. 6. One may choose a proper parameter required nominal control performance.
, the NN 2) The Effect of Disturbance: Choose is used again when there are discontrol law (3.4) with turbances. The control results in Fig. 7 show that the nominal control performance without internal model feedback is unsatisfactory when disturbances happen. 3) Robust Stability and Performance: When there are disturfrom the proposed bances, the NN control law (3.13) with neural IMC structure as shown in Fig. 5 can be used. Choose and . The control performance is quite satisfactory as shown in Fig. 8. B. Control of an Open-Loop Unstable Unknown Process . Since the plant is openThe example (4.1) is used with loop unstable, the closed-loop identification method as shown in Fig. 3 is used. Identification . • Neural structure: • Training sequence: a random input sequence with 500 values plus a feeduniformly distributed in the interval . back 0.6 • Training error: mean square error (MSE) of 8.3e-7 is obtained after training on the training sequences. . • NN NARMA model: • NN approximate model: (3.3) from the above identified NN NARMA model. To test the performance of the neural approximate model (3.3), a test input signal
is inputted to the system and the NN approximate model (3.3), simultaneously. The simulation in Fig. 9 shows that the output y(k) of (3.3) is very close to that of the system under the test input signal. The approximation error e(k) is very small. To control the system subject to disturbance, the NN control law (3.13) with is used. Choose and
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Fig. 7.
Nominal control performance with and without disturbances v .
Fig. 8.
Control of an open-loop stable unknown plant.
. The control performance is satisfactory as shown in Fig. 10 even though the nonlinear plant is open-loop unstable with large disturbance. C. Control of a Third-Order Unknown Process With Stable Zero Dynamics The system is described by following difference equations:
(4.2)
Fig. 9.
Evaluation of the neural approximate model (3–3).
Fig. 10.
Control of an open-loop unstable unknown plant.
Let dynamics Identification
and
. In this case, the zero is stable.
• Neural structure: . • Training sequence: a random input sequence with 1000 values uniformly distributed in the interval . • Training error: mean square error (MSE) of 2.31e-5 is obtained after 575 epochs training on the training sequences. • NN NARMA model: . • NN approximate model: Equation (3.3) from the above identified NN NARMA model. The neural control law (3.13) with is used. Choose , . The control results in Fig. 11 shows that the proposed neural control can give quite good control performance even though there are zero dynamics and large disturbance in the nonlinear system.
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Fig. 11.
Control of an unknown plant with stable zero dynamics.
Fig. 12.
Input–output zero dynamics.
D. Control of an Unknown Process With Unstable Zero Dynamics The example (4.2) is used with , and . In this case, the zero dynamics is unstable. The identification of the given unknown nonlinear process can still be performed even though the zero dynamic is unstable. Identification • Neural structure: . • Training sequence: a random input sequence with 1000 . values uniformly distributed in the interval • Training error: mean square error (MSE) of 3.71e-5 is obtained after 600 epochs training on the training sequences.
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• NN NARMA model: • NN approximate model: Equation (3.3) from the above identified NN NARMA model. Equation (3.5) can be used to check if the NN approximate model (2.6) has a stable inversion. The two cases for both and initial value stable and unstable zero dynamics with are shown in Fig. 12. It is clearly shown that (3.5) converges with the stable zero dynamics and diverges with the unstable zero dynamics. This can also be clearly seen from . For the the poles of the linearized system (3.6b) with nonlinear plant with stable zero dynamics, the poles of (3.6b) and 0.2375. However, for the nonlinear plant with are and unstable zero dynamics, the poles of (3.6b) are 0.2504.
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Fig. 13. Control of an unknown plant with unstable zero dynamics.
To control this system, the NN control law (3.13) is chosen and a suitable needs to be determined first. The linearized system (3.6b) is derived as follows:
CALCULATION OF
APPENDIX A FOR A DOUBLE-LAYERED NEURAL NETWORK
A double-layered NN can be expressed as follows: (A.1)
(4.3) If is chosen as , then (4.3) is stable. The poles when is set to be 0.7. of (4.3) will be 0.3087 and The proposed NN control law (3.13) is used with and . The control results in Fig. 13 shows that the proposed AIMNC can give quite good control performance even though the zero dynamics of the system is unstable and large disturbances exist.
where
, , , and are weight matrices in the first hidden layer, second , and hidden layer, and the output layer of the NN, are bias vectors, and is a hyperbolic tangent activation function vector with input vector . Define . Thus
V. CONCLUSION An AIMNC strategy is proposed for control of general unknown nonlinear discrete processes under model mismatch and disturbances. The proposed neural control strategy not only combines the advantages of both inverse control and nonlinear IMC but also removes some of their disadvantages. Based on the new input–output approximation, the neural inverse control law is derived directly and applied straightforward to an unknown process. The stability and the performance of a closed-loop system can be analyzed easily even if the unknown nonlinear process is open-loop unstable or has unstable zero dynamics. The computation is quite small since only one NN needs to be trained for both model approximation and control formulation and no further training (offline or online) is required for the online controller. Extensive simulations demonstrate that the proposed neural control strategy can be used for a wide range of unknown nonlinear discrete dynamical systems under model mismatch and disturbances.
Since the derivative of is -dimensional unit matrix and th hidden layer, one has
with as a as the number of neurons in (A.2)
with .
and
APPENDIX B BOUNDED CONTROL LAW FOR A SYSTEM WITH UNSTABLE ZERO DYNAMICS Consider a nonlinear discrete system as follows:
(B.1)
LI AND DENG: AN APPROXIMATE INTERNAL MODEL-BASED NEURAL CONTROL
In this example the zero dynamics is unstable. The system (B.1) can be expressed in following input–output form:
(B.2) Suppose that an NN is trained to approximate (B.2) perfectly. Thus, an NN NARMA model for (B.2) is obtained as follows: (B.3) where and . From (B.3), an NN approximate model is derived in the form of (2.6) as follows: (B.4) where and . From (B.4), the NN control law in the form of (3.4) is as follows: (B.5) Let
. Then, . Thus, (B.5) becomes
(B.6) where . Linearizing (B.6) around has
and
, one
(B.7) Since
. Thus, (B.7) becomes (B.8) , (B.8) is unstable, which implies that the control law If (B.5) is unbounded. However, by choosing , (B.8) , the control is stable. Therefore, by choosing law (B.5) for the system (B.1) with unstable zero dynamics is bounded.
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ACKNOWLEDGMENT The authors greatly appreciate the useful comments from anonymous referees. REFERENCES [1] R. Boukezzoula, S. Galichet, and L. Foulloy, “Nonlinear internal model control: Application of inverse model based fuzzy control,” IEEE Trans. Fuzzy Syst., vol. 11, no. 6, pp. 814–829, Dec. 2003. [2] C. G. Economou, M. Morari, and B. O. Palsson, “Internal model control. 5. Extension to nonlinear systems,” Ind. Eng. Chem. Process Des. Dev., vol. 25, pp. 25 403–411, 1986. [3] J. B. D. Cabrera and K. S. Narendra, “Issues in the application of neural networks for tracking based on inverse control,” IEEE Trans. Autom. Control, vol. 44, no. 11, pp. 2007–2027, Nov. 1999. [4] S. S. Ge, T. H. Lee, G. Y. Li, and J. Zhang, “Adaptive NN control for a class of discrete-time nonlinear systems,” Int. J. Control, vol. 76, no. 4, pp. 334–354, 2003. [5] K. Hornik, “Approximation capabilities of multilayer feed-forward neural networks,” Neural Netw., vol. 4, pp. 251–257, 1990. [6] K. Hornik, M. Stinchcombe, and H. White, “Multilayer feed-forward networks are universal approximators,” Neural Netw., vol. 2, pp. 359–366, 1989. [7] L. Jin, P. N. Nikiforuk, and M. M. Gupta, “Fast neural learning and control of discrete-time nonlinear systems,” IEEE Trans. Syst., Man Cybern., vol. 25, no. 3, pp. 478–488, Mar. 1995. [8] H. X. Li and P. P. J. Van Den Bosch, “A robust disturbance-based control and its application,” Int. J. Control, vol. 58, no. 3, pp. 537–554, 1993. [9] G. Lightbody and G. W. Irwin, “Nonlinear control structures based on embedded neural system models,” IEEE Trans. Neural Netw., vol. 8, no. 3, pp. 553–567, May 1997. [10] M. Morari and E. Zafiriou, Robust Process Control. Englewood Cliffs, NJ: Prentice-Hall, 1989. [11] S. Mukhopadhyay and K. S. Narendra, “Adaptive control of nonlinear multivariable systems using neural networks,” Neural Netw., vol. 7, no. 5, pp. 737–752, 1994. [12] E. P. Nahas, M. A. Henson, and D. E. Seborg, “Nonlinear internal model control strategy for neural network models,” Comput. Chem. Eng., vol. 16, no. 12, pp. 1039–1057, 1992. [13] K. S. Narendra and S. Mukhopadhyay, “Adaptive control using neural networks and approximate models,” IEEE Trans. Neural Netw., vol. 8, no. 3, pp. 475–485, May 1997. [14] K. S. Narendra and K. Parthasarathy, “Identification and control of dynamical systems using neural networks,” IEEE Trans. Neural Netw., vol. 1, no. 1, pp. 4–27, Jan. 1990. , “Gradient methods for the optimization of dynamical systems con[15] taining neural networks,” IEEE Trans. Neural Netw., vol. 2, no. 2, pp. 252–262, Mar. 1991. [16] S. G. Nash and A. Sofer, Linear and Nonlinear Programming. New York: McGraw-Hill, 1996. [17] O. Nelles, Nonlinear System Identification: From Classical Approaches to Neural Networks and Fuzzy Models. Berlin, Germany: SpringerVerlag, 2001. [18] M. Norgaard, O. Ravn, N. K. Poulsen, and L. K. Hansen, Neural Networks for Modeling and Control of Dynamic Systems. London, U.K.: Springer-Verlag, 2000. [19] I. Rivals and L. Personnaz, “Nonlinear internal model control using neural networks: Application to processes with delay and design issues,” IEEE Trans. Neural Netw., vol. 11, no. 1, pp. 80–90, Jan. 2000. [20] J. T. Spooner, M. Maggiore, R. Ordonez, and K. M. Passino, Stable Adaptive Control and Estimation for Nonlinear Systems. New York: Wiley, 2002. [21] M. Vidyasagar, Nonlinear Systems Analysis. Englewood Cliffs, NJ: Prentice-Hall, 1993. [22] P. J. Werbos, “Backpropagation through time: What it does and how to do it,” Proc. IEEE, vol. 78, no. 10, pp. 1550–1560, Oct. 1990. [23] K. J. Hunt and D. Sbarbaro, “Neural Networks for nonlinear internal model control,” Proc. IEE D-Control Theory and Applications, vol. 138, no. 5, pp. 431–438, Sep. 1991.
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Han-Xiong Li (S’94–M’97–SM’00) received the Ph.D. degree in electrical engineering from University of Auckland, New Zealand, in January of 1997, the M.E. degree in electrical engineering from Delft University of Technology, The Netherlands, in 1991, and the B.E. degree from National University of Defence Technology, China, in 1982. Currently, he is an Associate Professor at the Department of Manufacturing Engineering and Engineering Management, City University of Hong Kong, Hong Kong. He is a “Chang Jiang Scholar” in Central South University, an Honorary Professor endowed by Ministry of Education, China. He was awarded the distinguished young scholar fund by the China National Science Foundation in 2004. In the last 20 years, he worked in different fields including military service, industry, and academia. He has gained industrial experience as a Senior Process Engineer from ASM, a leading supplier for semiconductor assembly equipment. His research interests include intelligent control and learning, modelling and control of industrial process with special interest to distributed parameter systems. Dr. Li serves as an Associate Editor for IEEE TRANSACTIONS ON SYSTEMS, MAN AND CYBERNETICS—PART B: CYBERNETICS.
IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 17, NO. 3, MAY 2006
Hua Deng received the Ph.D. degree from the Department of Manufacturing Engineering and Engineering Management, City University of Hong Kong, Hong Kong, in 2005, the M.E. degree from Northwestern Polytechnical University, China, in 1988, and the B.E. degree from Nanjing Aeronautical Institute, China, in 1983. He was an Associate Professor at Changsha University of Science and Technology, Changsha, China. Currently, he is a Professor at the School of Mechanical and Electrical Engineering, Central South University, Changsha, China. His research interests include identification and control of complex distributed parameter systems, intelligent control, intelligent manufacturing, fault diagnoses, and computer control system design.
