Double extractor induction motor: Variational calculation using the HamiltonJacobi-Bellman formalism

1. 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 𝑔 are specified functions, 𝑡₀ and 𝑡_𝑓 are bounded and 𝜏 is a phantom integration variable now it is considered longer problems and subdividing the interval is:

The principle of optimality is:

Where 𝐽∗(𝑥(𝑡+∆𝑡),𝑡+∆𝑡 is the minimum process cost for a time interval 𝑡+∆𝑡≤𝜏≤𝑡𝑓 with an initial state x(t + ∆𝑡). Assuming that the second partial derivative of 𝐽∗ exists and that it is limited 𝐽∗(𝑥(𝑡+∆𝑡),𝑡+∆𝑡 can be expanded, 𝑡 + ∆𝑡 in Taylor series around one point (𝑥 (𝑡),𝑡) is obtained.

Now for little ones ∆𝑡

Where 𝑂(∆𝑡) denotes the terms contained [∆𝑡]² of the highest order of ∆𝑡 that arise from the approximation of the integral and the truncation produced from the expansion in Taylor series, now removing the terms 𝐽^∗(𝑥 (𝑡),𝑡) and 𝐽𝑡^∗(𝑥(𝑡),𝑡).

Minimization is obtained:

Dividing by ∆𝑡 taking the limit when ∆𝑡→ 0 gives:

To find the limits for this differential equation let 𝑡 =𝑡_𝑓 give:

The Hamiltonian is defined as:

Since the minimization of the control will depend on _𝑥,𝐽_𝑥 ^∗, 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].

2. Application problem

The complete dynamic model of the DSIM is [1]:

By notation 𝑊_𝑠=𝜌̇ is the frequency of the motor, 𝑅_𝑠 and 𝑅_𝑟 are the stator and rotor resistors respectively 𝐿𝑠 𝑦 𝐿_𝑟, are the inductances of the stator and rotor respectively. M is the magnetization, 𝐿_𝑠𝑝 is the main cyclic inductance, 𝐽_𝑚 is the moment of inertia of the rotor, 𝐾_𝑙 is the constant torque load, 𝐿_𝑠1𝑑, 𝐿_𝑠2𝑑, 𝐿_𝑠1𝑞 y 𝐿_𝑠2𝑞 are the direct and quadrature current of stator 1 and stator 2 respectively. 𝑣_𝑠1𝑑, 𝑣_𝑠2𝑑, 𝑣_𝑠1𝑞 y 𝑣_𝑠2𝑞 are respectively the voltages of each stator on the dq axes, 𝑝 is the number of poles, 𝛾 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 <𝜖 <1 and 𝑈 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

The factors 𝛽₁, 𝛽₂, 𝛽₃, 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 𝑐₀ > 0.

2.1 Determination of the Hamilton-Jacobi-Bellman equation

To increase the readability of subsequent equations it will be denoted 𝑥1 = 𝜑𝑟 and 𝑥2= 𝛺.

The dynamics of the system and the cost of the function are defined as:

And

Therefore, the function 𝑣 (𝑇,𝑥₁), which necessarily satisfies the boundary condition 𝑣 (𝑇_,𝑥₁),= 𝑘 (𝑥₁(𝑡)), where 𝑘 (𝑥₁(𝑡)) is the final state:

The optimal co-state 𝜆∗(𝑡) corresponds to the gradient of the cost function to be optimized:

Where is continuously differentiable with respect to 𝑥₁ and 𝑥₁=𝑥₁∗ along the admissible path 𝑢₁(.) and 𝑢₁∗(.) corresponding sub-optimal cost function is:

It is given as:

According to the following differential equation:

Thus:

This is an equation for the optimal cost function, which is called (HJB) [4] [5]. So, we found an optimum for the variable 𝑢∗(𝑡,𝑥₁), with the subsequent state variable 𝑥₁∗(𝑡), considering of the following equation.

Which transfers the initial state 𝛷_𝑟(0)= 𝛷_𝑟0 and a final state 𝑥1(0)=𝛷_𝑟(𝑇) with the limit:

This determines the HJB equation:

The model described in (34) was simulated numerically and the results are shown below.

4. 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.

References

Additional information

How to cite: F. Mesa, C. Ramírez, J. Barba-Ortega, “Double extractor induction motor: Variational calculation using the Hamilton-Jacobi-Bellmanformalism,” Rev. UIS Ing., vol. 19, no. 3, pp. 97-102, 2020. doi: https://doi.org/10.18273/revuin.v19n3-2020010

Revista UIS Ingenierías
ISSN: 1657-4583
Vol. 19
Num. 3
Año. 2020

Double extractor induction motor: Variational calculation using the HamiltonJacobi-Bellman formalism

FernandoCarlos AJosé José MesaRamírezBarba-Ortega
Universidad Tecnológica de PereiraUniversidad Nacional de ColombiaUniversidad Nacional de Colombia,ColombiaColombiaColombia