Double extractor induction motor: Variational calculation using the Hamilton-Jacobi-Bellman formalism Motor de inducción de doble extractor: Cálculo variacional usando el formalismo Hamilton-Jacobi-Bellman

This contribution presents optimal control over a double extractor induction motor using formalism through variational model. The criterion is subject to non-stationary equations of a reduced order (Dynamics equations of a reduced order model (DSIM)). As is well know, in this model the state variables are the rotor flow and motor speed in a circuit mechanical process. For non-stationary and stationary states, based on the theory of optimal control, this limit provides a high expensive function given as a weighted contribution of a DSIM theory. To order to acquire a lowest energy rotor flow path, the idea is to minimize the function to a dynamic of two equations of the motor speed and rotor flow. This problem is solved using with the Hamilton-Jacobi-Bellman equation and a time dependent solution for the rotor flow is determined analytically.


Introduction
Variational problems have stimulated, in the last decade, theoretical and analytical research in several topics in mathematical physics.The variational model has disadvantages when the method is applied for analysis in high order plants.However, unrestricted design may not work properly in practice.Due to the limited variables and the restricted control could cause serious performance deterioration.The controllers must be synthesized to achieve the desired one.A great attempt was made in non-linear minimization and development on this system attracting a big attention of the scientific community.The objective is to design a controller for a type of systems with limitations on states and control.The model gives a practical model in which the designer can resume the limited controllers to get the goals of the synthesis below is the Jacobi-Bellman equation [6].
Let a systems be simulated by the equation: The measure to be minimized is: ℎ and  are specified functions,  0 and   are bounded and  is a phantom integration variable now it is considered longer problems and subdividing the interval is: The principle of optimality is: Where (∆) denotes the terms contained [∆] 2 of the highest order of ∆ that arise from the approximation of the integral and the truncation produced from the expansion in Taylor series, now removing the terms  * ( (), ) and   * ((), ).
Minimization is obtained: Dividing by ∆ taking the limit when ∆ → 0 gives: To find the limits for this differential equation let  =   give: The Hamiltonian is defined as: Since the minimization of the control will depend on ,   * , and t using those definitions the Hamilton-Jacobi equation is obtained: This equation is the analog in continuous time to the Bellman's recurrence equation, however it will refer to (12) as the Hamilton-Jacobi-Bellman equation [7].

Application problem
The complete dynamic model of the DSIM is [1]: By notation   = ̇ is the frequency of the motor,   and   are the stator and rotor resistors respectively      , are the inductances of the stator and rotor respectively.M is the magnetization,   is the main cyclic inductance,   is the moment of inertia of the rotor,   is the constant torque load,  1 ,  2 ,  1 y  2 are the direct and quadrature current of stator 1 and stator 2 respectively. 1 ,  2 ,  1 y  2 are respectively the voltages of each stator on the dq axes,  is the number of poles,  is the electromagnetic torque [2].
To eliminate the non-linear terms of (13), the order of the DSIM model of an integral proportional control (PI) is reduced to a simple one with proportional gain and the following is defined: Where 0 <  < 1 and  is a system command, the usual form of reduced model is obtained as follows: the cost of the function can be defined as: The index corresponding to the weighted sum is (  ,   ,   , Ω) =  1   +  2   + 3   (15) The factors  1 ,  2 ,  3 , are weighted and are used to scale the energy power while minimizing the cost function minimizes the stored magnetic energy and minimize of losses in the winding increasing the efficiency of the machine [3].The power in the rotating frame (d-q) is: The system (13) has that the input power is given by: The relationship between the stator and rotor current is given by: The active power is: The equation ( 17) is defined as the derivative of the stored magnetic field is given as follows: Joule's losses are given by: By using equation (13) and equation ( 21), Power losses can be as: In this document, the operations of the mechanical process are restricted to Acceleration modes to limit the transitional study.Then the engine speed can be expressed as follows: With  0 > 0.

Determination of the Hamilton-Jacobi-Bellman equation
To increase the readability of subsequent equations it will be denoted  1 =   and  2 = .
The dynamics of the system and the cost of the function are defined as: And Therefore, the function  (,  1 ), which necessarily satisfies the boundary condition  (,  1 ), =  ( 1 ()), where  ( 1 ()) is the final state: The optimal co-state  * () corresponds to the gradient of the cost function to be optimized: Where is continuously differentiable with respect to  1 and  1 =  1 * along the admissible path  1 (. ) and  1 * (. ) corresponding sub-optimal cost function is: Which transfers the initial state   (0) =  0 and a final state  1 (0) =   () with the limit: This determines the HJB equation: (, The model described in (34) was simulated numerically and the results are shown below.

Conclusions
We present an optimal control over a double stator induction motor using an energy minimization model via a DSIM model.For to obtain a minimum energy rotor flow path, the process is to minimize this limited function to a no-static of two equations set of the rotor flow and motor speed.The optimum control system is analyzed by the HJB equation and the rotor flow solution is determined in a mathematical manner which varies over time.