Neural Comput & Applic (2014) 25:693–701 DOI 10.1007/s00521-013-1533-5
ORIGINAL ARTICLE
State estimation in MIMO nonlinear systems subject to unknown deadzones using recurrent neural networks J. Humberto Pe´rez-Cruz • Jose´ de Jesu´s Rubio Jaime Pacheco • Ezequiel Soriano
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Received: 29 June 2013 / Accepted: 9 December 2013 / Published online: 31 December 2013 Ó Springer-Verlag London 2013
Abstract This paper deals with the problem of state observation by means of a continuous-time recurrent neural network for a broad class of MIMO unknown nonlinear systems subject to unknown but bounded disturbances and with an unknown deadzone at each input. With respect to previous works, the main contribution of this study is twofold. On the one hand, the need of a matrix Riccati equation is conveniently avoided; in this way, the design process is considerably simplified. On the other hand, a faster convergence is carried out. Specifically, the exponential convergence of Euclidean norm of the observation error to a bounded zone is guaranteed. Likewise, the weights are shown to be bounded. The main tool to prove these results is Lyapunov-like analysis. A numerical example confirms the feasibility of our proposal. Keywords Neural observer Deadzone Recurrent neural network Exponential convergence
J. H. Pe´rez-Cruz (&) J. J. Rubio J. Pacheco E. Soriano Seccio´n de Estudios de Posgrado e Investigacio´n, ESIME UA-IPN, Av. de las Granjas no. 682, Col. Santa Catarina, C.P. 02250 Mexico, D.F., Mexico e-mail:
[email protected] J. J. Rubio e-mail:
[email protected] J. Pacheco e-mail:
[email protected] E. Soriano e-mail:
[email protected] J. H. Pe´rez-Cruz DEPI, Instituto Tecnolo´gico de Oaxaca, Av. Ing. Victor Bravo Ahuja No. 125, C.P. 68030 Oaxaca, Mexico
1 Introduction Due to the high cost or lack of appropriate sensors, it is common in practice that not all variables associated with a physical system can be measured. However, if these variables are required, it would be necessary to use some method to reconstruct or estimate such unmeasured variables from the measurable ones and from some a priori knowledge or mathematical model of the dynamics of the system. Certainly, as less a priori information about the system is available, the problem of reconstruction of states becomes more complicated. Although the state estimation in linear systems is a problem well-understood and enjoys well-established solutions, the nonlinear case is much more challenging and there is no a universal approach to solve it. In a seminal work [1], Thau presented a Luenberger-like observer for Lipschitz nonlinear systems. Assuming that both linear and nonlinear components of the system are known, he provided a sufficient condition to guarantee the asymptotic convergence of the observation error to zero. Systematic procedures to find the appropriate gain for the observer considered by Thau were provided in [2–5]. All the aforementioned works are based on the assumption that an accurate system model is available. However, the presence of unmodeled dynamics and disturbances is ubiquitous in practical situations. In [6, 7], considering that the system structure is a priori known and the unknown constants are linearly parameterized, the uncertainty associated with the model constants and the disturbances were handled by employing robust adaptive approaches. A still more complex case can occur when the system is nonlinearly parametrized and/or a deficient and partial knowledge of the nominal model is only available. On these conditions, a high-gain observer can still provide an acceptable
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694 Fig. 1 Estimation process: first state, solid line; corresponding estimation, dashed line
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performance due to its capability of estimating the states inclusive without any physical model of the system. Nevertheless, this approach is only valid for systems in Brunovsky canonical form. For more general classes of systems on the aforementioned conditions, the use of artificial neural networks (ANN) has been proposed in [8, 9]. Artificial neural networks can be simply considered as a nonlinear generic mathematical formula whose parameters are adjusted in order to represent the behavior of a static or dynamic system [10]. These parameters are called synaptic weights. Generally speaking, ANN can be classified as feedforward (static) ones, based on the back-propagation technique [11], or as recurrent (dynamic) ones [12]. For the first kind of networks, the adjustment of the weights or learning process is achieved by using methods of local optimization. This could cause some problems such as a slow learning rate and difficulties associated with the global extrema search. In contrast, the second kind of networks converts the learning process into an adequate feedback design [12, 13] avoiding the previous problems and providing a better performance. Taking into account the above, an adaptive observer based on dynamic recurrent neural networks for a class of nonlinear systems was for first time proposed in [8]. In that study, a strictly positive real condition must be satisfied to guarantee the ultimately uniformly boundedness of the observation error. In [9], this constraint was avoided and a wide class of MIMO nonlinear systems subject to disturbances could be considered. The (complex) structure presented in [9] for the neural observer consists of three components: one component is formed by a recurrent multilayer neural network, other component is a correction term as in a classical Luenberger observer, and the last component is an additional time-delay term. Assuming that a matrix Riccati equation formed by the parameters of this
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observer has a unique positive definite solution and using proper differential learning laws for the weights, the asymptotic convergence of the ‘‘average’’ estimation error to a bounded zone was demonstrated. In [14], a sliding mode term was added to the observer structure presented in [9] in order to intend to improve the quality of the estimation process. Taking into account that the difference between the performance provided by a multilayer and a single layer neural network could not be significant and, in contrast, the computational burden is increased considerably in the first case, the hidden layer was removed in [15] and a new learning law with sliding mode terms was proposed. In the same work, the weights boundedness ‘‘on average’’ could be showed. However, this last result does not necessarily imply that the weights belong to L1 . This drawback was overcome in [16] by including a timevarying gain for each weight. These gains are adjusted by its corresponding learning laws in such a way that the boundedness of the weights can now be guaranteed. Notwithstanding, the presence of these time-varying gains could increase the computational cost. In spite of the important contributions presented in [9, 14–16], the following shortcomings are common to all of them: (a) although in those works conditions which guarantee the existence of a solution for a matrix Riccati equation are provided, such conditions could not be always satisfied in practice (besides, the presence of a Riccati equation complicates the design process), (b) their main result deals with the asymptotic convergence of the ‘‘average’’ estimation error to a bounded zone. Nonetheless, in technical literature, it is more common and significant to present the convergence of the estimation error in terms of its Euclidean norm, and (c) they did not consider the case when the deadzone could be present at the input of the system.
Neural Comput & Applic (2014) 25:693–701 Fig. 2 Estimation process: second state, solid line; corresponding estimation, dashed line
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The deadzone is a nonsmooth nonlinearity commonly found in many practical systems such as hydraulic positioning systems [17–19], pneumatic servo systems [20], DC servo motors [21], and so on. When the deadzone is not considered explicitly during the design process, the performance of the control system could be degraded due to an increase in the steady-state error, the presence of limit cycles, or inclusive instability [22–25]. Up to now, the great majority of studies about controllers for uncertain plants with unknown deadzone have considered the case when all states are available for measurement or well, if an observer is utilized, only the case of SISO systems in canonical Brunovsky form [26– 30] is considered. In this paper, using a neural observer with structure similar to [15] and [16] but without the sliding mode term, the estimation of states for a broad class of MIMO nonlinear systems subject to disturbances and with an unknown deadzone at each input is achieved. With respect to others studies [9, 14–16], the main contributions of this paper are (a) using only the property of boundedness corresponding to the sigmoidal functions, the disturbances, and the unmodeled dynamics, the need of a matrix Riccati equation is conveniently avoided and consequently the design process is considerably simplified and (b) a faster convergence is attained since the exponential convergence for Euclidean norm of the estimation error to a bounded zone can now be guaranteed. Likewise, the boundedness of the weights is also guaranteed. The main analytic tool for proving these results is Lyapunovlike analysis. A numerical example corroborates the feasibility of our proposal. This is the first time, up to the best of our knowledge, that recurrent neural networks are utilized in the context of the state estimation for MIMO uncertain systems with each input preceded by an unknown deadzone.
2 Preliminaries 2.1 Notation Throughout this paper, we will use the following notation: Given hðtÞ 2
