Recently, Ott, Grebogi and Yorke (OGY) [6] found an effective method to control chaotic systems to unstable fixed points by using only small control forces; however, OGY's method is based on and limited to a linear theory and requires considerable knowledge of the dynamics of the system to be controlled. In this paper we use two radial basis function networks: one as a model of an unknown plant and the other as the controller. The controller is trained with a recurrent learning algorithm to minimize a novel objective function such that the controller can locate an unstable fixed point and drive the system into the fixed point with no a priori knowledge of the system dynamics. Our results indicate that the neural controller offers many advantages over OGY's technique.

Recently, Ott, Grebogi and Yorke (OGY) [6] found an effective method to control chaotic systems to unstable fixed points by using onlysmall control forces; however, OGY's method is based on and limited to a linear theory and requires considerable knowledge of the dynamics of the system to be controlled. In this paper we use two radial basis function networks: one as a model of an unknown plant and the other as the controller. The controller is trained with a recurrent learning algorithm to minimize a novel objective function such that the controller can locate an unstable fixed point and drive the system into the fixed point with no a priori knowledge ofthe system dynamics. Our results indicate that the neural controller offers many advantages over OGY's technique.

Recently, Ott, Grebogi and Yorke (OGY) [6] found an effective method to control chaotic systems to unstable fixed points by using only small control forces; however, OGY's method is based on and limited to a linear theory and requires considerable knowledge of the dynamics of the system to be controlled. In this paper we use two radial basis function networks: one as a model of an unknown plant and the other as the controller. The controller is trained with a recurrent learning algorithm to minimize a novel objective function such that the controller can locate an unstable fixed point and drive the system into the fixed point with no a priori knowledge of the system dynamics. Our results indicate that the neural controller offers many advantages over OGY's technique.

We proposed a model of Time Warping Invariant Neural Networks (TWINN) to handle the time warped continuous signals. Although TWINN is a simple modification ofwell known recurrent neural network, analysis has shown that TWINN completely removestime warping and is able to handle difficult classification problem. It is also shown that TWINN has certain advantages over the current available sequential processing schemes: Dynamic Programming(DP)[I], Hidden Markov Model( HMM)[2], Time Delayed Neural Networks(TDNN) [3] and Neural Network Finite Automata(NNFA)[4]. Wealso analyzed the time continuity employed in TWINN and pointed out that this kind of structure can memorize longer input history compared with Neural Network FiniteAutomata (NNFA). This may help to understand the well accepted fact that for learning grammatical reference with NNFA one had to start with very short strings in training set. The numerical example we used is a trajectory classification problem. This problem, making a feature of variable sampling rates, having internal states, continuous dynamics,heavily time-warped data and deformed phase space trajectories, is shown to be difficult to other schemes. With TWINN this problem has been learned in 100 iterations. For benchmark we also trained the exact same problem with TDNN and completely failed as expected.

We proposed a model of Time Warping Invariant Neural Networks (TWINN) to handle the time warped continuous signals. Although TWINN is a simple modification of well known recurrent neural network, analysis has shown that TWINN completely removes time warping and is able to handle difficult classification problem. It is also shown that TWINN has certain advantages over the current available sequential processing schemes: Dynamic Programming(DP)[I], Hidden Markov Model( HMM)[2], Time Delayed Neural Networks(TDNN) [3] and Neural Network Finite Automata(NNFA)[4]. We also analyzed the time continuity employed in TWINN and pointed out that this kind of structure can memorize longer input history compared with Neural Network Finite Automata (NNFA). This may help to understand the well accepted fact that for learning grammatical reference with NNF A one had to start with very short strings in training set. The numerical example we used is a trajectory classification problem. This problem, making a feature of variable sampling rates, having internal states, continuous dynamics, heavily time-warped data and deformed phase space trajectories, is shown to be difficult to other schemes. With TWINN this problem has been learned in 100 iterations. For benchmark we also trained the exact same problem with TDNN and completely failed as expected.

The two well known learning algorithms of recurrent neural networks are the back-propagation (Rumelhart & el al., Werbos) and the forward propagation (Williamsand Zipser). The main drawback of back-propagation is its off-line backward path in time for error cumulation. This violates the online requirement in many practical applications. Although the forward propagation algorithmcan be used in an online manner, the annoying drawback is the heavy computation load required to update the high dimensional sensitivity matrix(0(fir) operations for each time step). Therefore, to develop a fast forward algorithm is a challenging task.

The two well known learning algorithms of recurrent neural networks are the back-propagation (Rumelhart & el al., Werbos) and the forward propagation (Williams and Zipser). The main drawback of back-propagation is its off-line backward path in time for error cumulation. This violates the online requirement in many practical applications. Although the forward propagation algorithm can be used in an online manner, the annoying drawback is the heavy computation load required to update the high dimensional sensitivity matrix (0( fir) operations for each time step). Therefore, to develop a fast forward algorithm is a challenging task.

J.B. Pollack, Implications of Recursive Distributed Representations, Advances in Neural Information Systems 1, D.S. Touretzky (ed), Morgan Kaufmann, San Mateo, Ca, p. 527 (1989).

J.B. Pollack, Implications of Recursive Distributed Representations, Advances in Neural InformationSystems 1, D.S. Touretzky (ed), Morgan Kaufmann, San Mateo, Ca, p. 527 (1989).

We propose a new scheme to construct neural networks to classify patterns. The new scheme has several novel features: 1. We focus attention on the important attributes of patterns in ranking order.