Suykens J.A.K., De Moor B., Vandewalle J., "Static and dynamic stabilizing neural controllers, applicable to transition between equilibrium points," Neural Networks, Vol.7, No.5, pp.819-831, 1994.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....and, moreover, significant theoretical difficulties and benefits can already be found at this level. 3 Recurrent Neural Networks Recurrent networks are a natural tool in any domain where time plays a role, such as speech recognition, control, or time series prediction, to mention just a few [8,9,25,41]. They are also used for the classification of symbolic data such as DNA sequences [31] Turing Capabilities The fact that their inputs and outputs may be sequences suggests the comparison to other mechanisms operating on sequences, such as classical Turing machines. One can consider the internal ....

J. Suykens, B. DeMoor, and J. Vandewalle. Static and dynamic stabilizing neural controllers applicable to transition between equilibrium point. Neural Networks, 7(5), 1994.

.... Stabilizing a pendulum which is free to rotate, and 2 initially may be pointing downward, is therefore a more challenging nonlinear control problem [2] Here, stabilization of an unstable stationary state, and destabilization of a stable stationary state have to be realized by one controller [9] [10]. In section 3 we will show that this problem is easily solved by evolved neural network solutions if the controller has access to the full phase space information. Two minimal solutions of the feedforward type are presented, although also recurrent networks were generated by the algorithm. If ....

....seems to be a much harder problem. In fact, we can not present an evolved neurocontroller, which acts as successfully for all initial conditions of the physical system as, for instance, controller w 1 . But it was quite easy to evolve a controller which is able to solve the swinging up problem [10], i.e. starting the cart rotator system from initial conditions x 0 = 0, 0 = The architecture of this controller w 3 is shown in figure 5, and its weights are given as follows: w 3 3 = Gamma0:22; Gamma9:86; 0:36; Gamma0:79; 1:42; 4:59; Gamma4:49) w 3 4 = Gamma0:45; Gamma1:31; ....

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Suykens, J. A. K., De Moor, B. L. R., and Vandewalle, J. P. L. (1994), Static and dynamic stabilizing neural controllers, applicable to transition between equilibrium points, Neural Networks, 7, 819--831.

....how dynamic backpropagation can be considerably simplified by performing a truncation in time. This simplification is further generalized to the regulation of processes with arbitrary time delay. Stability analysis of this simplified algorithm has been done based on Lyapunov stability theory. In [29] methods for incorporating linear controller design results are discussed, such that e.g. a transition between working points can be realized with guaranteed local stability at the target point. Another formalism for calculating the gradients has been proposed by Werbos [33] This is done by ....

Suykens J.A.K., De Moor B., Vandewalle J. (1994). Static and dynamic stabilizing neural controllers, applicable to transition between equilibrium points, Neural Networks, Vol.7, No.5, pp.819-831.

....static optimization of an universal approximator whenever this latter is used in a dynamical context i.e. with the error computed over a given temporal horizon. This algorithm has been derived and used by Werbos (1990) Nguyen Widrow (1990) Parisini Zoppoli (1991) Plumer (1993) and Suykens, De Moor Vandewalle (1994) for the automatic tuning of neurocontroller aiming at minimizing a control cost either given over a certain temporal horizon or delayed in time. As the sections will show, the easiest and cleanest way to derive the algorithm is by introducing Lagrange multipliers and performing a gradient descent ....

....minimal cost on the control actions. We assume a statefeedback class of controller with the control law given by u(n) U[x(n) x0] where U is a differentiable vector function. Similar problems have been addressed by Werbos (1990) Nguyen Widrow (1990) Parisini Zoppoli (1991) Plumer (1993) Suykens, De Moor Vandewalle (1994) and Saerens, Renders Bersini (1995) The purpose of the learning algorithm is to train fuzzy controllers to provide the optimal control law for all initial conditions in a given operating region u(n) N[x(n) x0;w] where N is the mapping computed by the universal approximator and w its parametric ....

Suykens J., De Moor B. & Vandewalle J., 1994, Static and dynamic stabilizing neural controllers, applicable to transitions between equilibrium points. In Neural Networks, 7 (5) - pp. 819-831.

....approximation. When gradient based optimisation algorithms are used, derivatives are required which entails in turn the need for an action derivable cost and, in the case of model based control, the knowledge of the process Jacobians. Back Propagation Through Time (BPTT) 4] 5] 12] 13] 17][19][21] algorithm is the temporal extension of backpropagation algorithm when the error to minimize is, like it is indeed the case for optimal control, given over a temporal horizon or delayed in time. A clean way to derive BPTT is by introducing Lagrange multipliers and performing a gradient descent ....

Suykens J., De Moor B. & Vandewalle J., 1994, Static and dynamic stabilizing neural controllers, applicable to transitions between equilibrium points. In Neural Networks, 7 (5) - pp. 819-831.

.... Stabilizing a pendulum which is free to rotate, and 2 initially may be pointing downward, is therefore a more challenging nonlinear control problem [2] Here, stabilization of an unstable stationary state, and destabilization of a stable stationary state have to be realized by one controller [9] [10]. In section 3 we will show that this problem is easily solved by evolved neural network solutions if the controller has access to the full phase space information. Two minimal solutions of the feedforward type are presented, although also recurrent networks were generated by the algorithm. If ....

....seems to be a much harder problem. In fact, we can not present an evolved neurocontroller, which acts as successfully for all initial conditions of the physical system as, for instance, controller w 1 . But it was quite easy to evolve a controller which is able to solve the swinging up problem [10], i.e. starting the cart rotator system from initial conditions x 0 = 0, 0 = The architecture of this controller w 3 is shown in gure 5, and its weights are given as follows: w 3 3 = 0:22; 9:86; 0:36; 0:79; 1:42; 4:59; 4:49) w 3 4 = 0:45; 1:31; 6:49; 7:48; 0:36; 1:2; 0) w 3 5 ....

[Article contains additional citation context not shown here]

Suykens, J. A. K., De Moor, B. L. R., and Vandewalle, J. P. L. (1994), Static and dynamic stabilizing neural controllers, applicable to transition between equilibrium points, Neural Networks, 7, 819-831.

....algorithms and the search algorithm of Hooke and Jeeves (Hooke and Jeeves, 1961) were used in order to optimize the rulebase, applied to continuous processes and supporting transitions between several working or equilibrium points. This behaviour can also be achieved by a pure neural approch (Suykens et al. 1994). In the meantime several implementation and representation specific optimization strategies with increased performance were developed and examined. These results will be published separately. Fuzzy Controller Fuzzy Pilot Control Rule Optimization Expected Process Reaction Process Model ....

Suykens, Johan A. K., Bart L. R. de Moor and Joos Vandewalle (1994). Static and dynamic stabilizing neural controllers, applicable to transition between equilibrium points. Neural Networks 7(5), 819--831.

.... regularization methods like stopped training (Sjoberg Ljung 1992b, Sjoberg, McKelvey Ljung 1993) default models (Thompson Kramer 1994, Su, Bhat McAvoy 1992, Kramer et al. 1992, Johansen Foss 1992, Sjoberg Ljung 1992a) constraints (Joerding Meador 1991, Thompson Kramer 1994, Suykens, De Moor Vandewalle 1994), penalty on parameter magnitude (Weigend, Huberman Rumelhart 1990) and smoothness regularization (Bishop 1991) has recently been suggested. The relationship to Bayesian estimation is also being pursued (MacKay 1991, Williams 1995) The optimization framework presented here will be useful for ....

Suykens, J. A. K., De Moor, B. L. R. &Vandewalle, J. (1994), `Static and dynamic stabilizing neural controllers, applicable to transition between equilibrium points', Neural Networks 7, 819--831.

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Suykens J.A.K., De Moor B., Vandewalle J., "Static and dynamic stabilizing neural controllers, applicable to transition between equilibrium points," Neural Networks, Vol.7, No.5, pp.819-831, 1994.

....thereby relating the Lagrange multiplier sequence to the backpropagation algorithm. Parisini Zoppoli (1994) assumed a linear structure preserving principle for the state tracking problem with application to control of a space robot. Plumer (1996) studied the case of optimal nal time. In Suykens et al. 1994)(1996) the problem of swinging up of an inverted pendulum and double inverted pendulum with stabilization at the endpoint has been formulated as a parametric optimization problem in the unknown weights of feedforward or recurrent neural controllers. In this paper we discuss N stage optimal control ....

....not the number of weights in the primal space, which can be in nite dimensional) We illustrate the LS SVM control method on a number of simulation examples including swinging up an inverted pendulum with local stabilization at the endpoint and a tracking problem for a ball and beam system. In Suykens et al. 1994) methods for realizing a transition between two states with local stabilization at the target point was discussed. This was illustrated on swinging up an inverted pendulum with local stabilization at the endpoint. For the full static state feedback case a Linear Quadratic Regulator (LQR) Franklin ....

[Article contains additional citation context not shown here]

Suykens J.A.K., De Moor B. & Vandewalle J. (1994). Static and dynamic stabilizing neural controllers, applicable to transition between equilibrium points, Neural Networks, Vol.7, No.5, pp.819-831.

....Fig.6 for 20 runs. In dashed line the minimal value, obtained after random search, according to the same 20 initial pdfs is shown, for an equivalent amount of function evaluations. 6.3 Example 3 In this example we consider a neural optimal control problem for swinging up an inverted pendulum. In Suykens et al. 1994) this problem was solved as follows: after parameterizing the control signal by means of a neural network architecture the optimal control problem is formulated as a nonlinear optimization problem in the unknown interconnection weights. In order to obtain a locally stabilizing controller at the ....

.... The cost function is calculated from the simulation result of the closed loop system dynamics of the inverted pendulum, controlled either by a feedforward (static feedback) or recurrent neural network (dynamic feedback) For a precise formulation of the problem the reader is referred to Suykens et al. 1994). Although we were able in that paper to find satisfactory results to the swinging up problem by means of a multi start local optimization algorithm, a comparison will be made here with the FP machine in terms of the quality of the obtained solutions and the number of function evaluations needed. ....

[Article contains additional citation context not shown here]

Suykens J.A.K., De Moor B.L.R., Vandewalle J. (1994). Static and dynamic stabilizing neural controllers, applicable to transition between equilibrium points, Neural Networks, Vol.7, No.5, pp.819-831.

....and finally to apply QN for fast convergence to the local optimum. Furthermore the FP algorithm was successfully applied to the neural optimal control problem of swinging up an inverted pendulum system together with local stabilization at its upright position (see Figure 1) a problem described in [10] and solved by means of FP in [11] Again FP QN was considerably better than multistart QN here. VI. CONCLUSIONS In this paper a new method for nonconvex optimization is proposed by considering the Fokker Planck equation related to continuous simulated annealing. After parametrizing the ....

Suykens J.A.K., De Moor B.L.R., Vandewalle J., Static and dynamic stabilizing neural controllers, applicable to transition between equilibrium points, Neural Networks, Vol.7, No.5, pp.819-831, 1994.

....t tanh(V y) 3) with interconnection weights w 2 R nh , V 2 R nh Thetam . Such parametrizations make sense because any continuous nonlinear function may be approximated by a multilayer neural network with one or more hidden layers [3] 4] 6] For the dynamic output feedback case we refer to [11]. The design procedure for the feedforward and recurrent neural net then basically works as follows: 1. Design a linear controller around the target equilibrium point x eq based on the linearized model. The nonlinear control law then becomes the linearized controller around x eq and is a ....

....such as specifications on the transition from x(0) to x eq . This results into optimization problems constrained by the linear controller design results of step 1. We will treat only the static feedback case in the sequel, but the method completely extends to the dynamic output feedback case (see [11]) 2.2 Linearization around the target point Using the feedback law (3) we obtain the closed loop system x = f(x) g(x) ff tanh(w t tanh(V h(x) 4) In order to derive conditions for local stability around x = 0 the Jacobian matrix J is calculated and evaluated at x = 0: J(x) f x g x ....

Suykens J., B. De Moor, J. Vandewalle, Static and dynamic stabilizing neural controllers, applicable to transition between equilibrium points, accepted for publication in Neural Networks.

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