Stabilization of unknown nonlinear systems using neural networks
Introduction
Stability is one of the most important concepts concerning the design of control strategies. Stabilization of nonlinear plants has been treated with different approaches [1].
The design of linear controllers for stabilizing the nonlinear system around an equilibrium point has been one of these methods. However, the range over which the system will be stable depends upon the system and may be small [5].
Nonlinear techniques based on feedback linearization have been investigated in Ref. [5]. In this reference, the authors proved that stabilization through feedback linearization increases the region of attraction. Whereas, feedback linearization can be applied only for a limited class of systems. For this reason, Levin and Narendra [5] have employed a neural controller that direct stabilizes nonlinear systems. A neural network accomplishing nonlinear control law has been trained to stabilize nonlinear systems in intervals of n steps.
It has been shown that the attraction domain achieved by a nonlinear feedback controller can be significantly greater then one realized by a linear controller and the neural approach presents the advantage that is applicable for a large class of nonlinear systems.
In this study, the problem of stabilization of nonlinear systems has been treated with a neural approach which is different from the one adopted by Levin and Narendra [5]. In Ref. [5], the authors proposed a theoretical and practical aspect to implement control laws based on learning methods that optimize the range over which a contraction stabilizing controller is valid. The developed approach needs to develop a neural state model used to check the rank condition and to train the neural controller. In our approach a neural controller structure, which is different from the proposed in Ref. [5] is developed from an off-line inverse state neural model learning and then used as a feedback states neural controller in order to stabilize the system, around an equilibrium state. In fact, the structure of the neural controller is similar to that proposed in Ref. [7], where a backpropagation neural network has been used to model online an unknown nonlinear system. The same neural network in Ref. [7] has been used to generate the control signals given the measurements of the current states and the desired values of future states.
The addressed problem is how to stabilize the system around a desired state using different methods to optimize the neural controller. For this, a backpropagation neural network is trained to generate a sequence of input control given the current state of the plant and the desired state in the future [7]. The adaptive approaches developed in this paper treats different control objectives where the backpropagation algorithm has been used for adjustment of the neural controller parameters.
This paper is organized as follows: in Section 2, a general description of the considered approach is given. Section 3 presents the architecture of the used neural controller. In Section 4, we state the stabilization method and in Section 5, strategies of optimal control based on minimization of an energy criterion and steady-state error are developed. Finally, simulation results are carried out in Section 6 to prove the effectiveness of the proposed methods.
Section snippets
Problem description
Let us consider a nonlinear system which can be described by the following state model:where x(k) ∈ Rn is the state variables vector at the discrete time k and u(k) ∈ Rm is the control vector. The vector of nonlinear functions f[⋅] is assumed unknown and the state variables are supposed completely measurable.
We can write (1) as follows:which can be rewritten:
This implies that the state at time k + 2 are determined by the state at
Neural controller
To date, most of the work in neural black-box modeling has been performed making the assumption that the process to be modeled can be described accurately by input–output models and using the corresponding input–output neural predictors. Since the state-space constitutes a large class of nonlinear dynamical models, there is an advantage in making the assumption of a state-space description and using the corresponding state-space neural predictors [9], [10], [11], [12].
Commonly, a
Stabilization
After learning procedure, the network emulating the inverse model is connected in cascade with the plant in order to provide the control law to the system. As shown in Fig. 4a, the whole system constitutes a feedback control with a nonlinear controller, which provides to the system N control values at every sampling time.
Optimal stabilization
If the plant to be controlled is unknown and we have only the input–output data, the control task can be considered as an adaptive control problem. The neural controller trained off-line and used in previous section attempts to produce the inverse of the plant over the entire state space, but it can be very difficult to use it alone to provide adequate performance in a practical control application [2]. In order to overcome this problem, we introduce some methods to train the neural controller
Simulation results
The simulation results are given on two nonlinear systems. The first is a third-order system (three states) and the second is a second-order system.
Conclusion
In this paper, two neural control strategies have been proposed for stabilizing unknown nonlinear systems. The second control strategy applied in the sense of receding horizon has shown good performances. The on-line updating of the neural controller has increased the region of attraction substantially, especially when the error state criterion has been considered. These results have been proved by simulation on two unstable nonlinear systems.
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