Open access peer-reviewed chapter

Tool of the Complete Optimal Control for Variable Speed Electrical Drives

Written By

Marian Gaiceanu

Submitted: September 19th, 2013 Reviewed: December 10th, 2013 Published: September 8th, 2014

DOI: 10.5772/57521

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1. Introduction

The main objective of the chapter is to use Matlab software in order to develop the optimal control knowledge in relation to the biggest worldwide power consumers: electrical machines. These two major components, optimal control and electrical machines form an optimal drive system. The proposed drive system conducts to the energy efficiency increasing during transient regimes (starting, stopping and reversing), therefore reducing the impact upon the environment. Taking into account the world energy policies, the chapter is strategically oriented towards the compatibility with the priority requirements from world research programmes. This chapter will offer an original Matlab tool in order to implement a highly performant control for electrical drives. The optimal solution provided by the optimal problem is obtained by numerical integration of the matrix Riccati differential equation (MRDE). The proposed optimal control, based on energetic criterion, can be implemented on-line by using a real time experimental platform. The solution has three terms: the first term includes the reference state of the electrical drive system, the second term assures a fast compensation of the disturbance (i.e. the load torque in case of the electrical drives) by using feedforward control, and the third is the state feedback component. Therefore, there is a completed optimal control for linear dynamic systems. The simulation and experimental results will be provided. By using knowledge regarding the electrical machines, electrical and mechanical measurements, control theory, digital control, and real time implementation, a high degree of interdisciplinarity is obtained in the developed Matlab tool The optimal problem statement is without constraints; by choosing adequate weighting matrices, the magnitude constraints of control and of state variable are solved.

Matlab is a widespread software used both in academia and research centres. The author of this chapter facilitates the easy understanding and implementation of the highly efficient electrical drive systems by using Matlab software. The energy problem is one of the most important aspects related to the sustainable growth of the industrial nations. On the one hand, the industrial systems and the increased quality of life require every year a higher quantity of electrical energy produced; on the other hand, the same quality of life requires the generation and the harnessing of this energy with low pollution.

On the one hand, the electrical machines are the main worldwide energy consumer (Siemens), (Tamimi, et al., 2006). Therefore, the focus of the research on this system type is essential in order to minimize the energetical losses. A seemingly insignificant reduction of electricity consumed by each electrical motor has particularly important consequences at national/international level, primarily by reducing consumption of fossil fuels widely used to generate electricity, therefore contributing also to CO2 emissions reduction. Researches in this area are at different levels: I) direct conversion by redesigning the electric motors to obtain high efficiency; II) the power electronic systems by introducing static converters with diminished switching losses; III) control implementation of the conversion process by using the enhanced capabilities of new generation digital signal processor DSP (Veerachary M., 2002).

The efficiency of the motor is high on steady state operation, but very low in transients. On the other hand, the associated equipment for energy conversion purposes, i.e. power converters together with the control subsystem, assures both high static and dynamic performances (related to the settling time, error minimizing and so on), but they are not related to energy expenditure. The analysis of various technical publications shows a yearly increase of energy expenditure of the electrical motors, with a rate of 1,5% in Europe, 2,2% in USA, 3,1% in Japan; these values can even reach the rate of 13%, depending on the development of calculus technology (Matinfar, et al., 2005), (Gattami, et al.,2005), (COM, 2006).

Dynamic programming tool is very useful to obtain the optimal solution, but the values of the torque and speed must be known in advance (Mendes, et al., 1996). In machine building industry the operating cycle is well-known (Lorenz, et al., 1992; 1998) therefore, the dynamic programming is an efficient tool only for these type of applications. But the optimization method fails when the conventional electrical drives are taken into account because the load torque could not be known in advance, i.e. at the final time of the process.

The optimal solution (Gaiceanu, 1997) provided in this chapter overcomes this type of scientific barrier (Rosu, 1985), making possible implementation of the optimal solution to any type of application, without knowing the speed and load torque in advance. As it is nonrecursive, determined on the basis of variational methods (by integrating the Matrix Riccati Differential Equation), the optimal solution can be calculated on-line without memorizing it from the final time to the initial time, as in the recursive method. Therefore, the nonrecursive solution can be implemented to any type of electrical motor with any load variation, for any operating dynamical regimes.

On the other hand, the problem of increasing power conversion efficiency in the drive systems is the subject of numerous worldwide studies. The Siemens Automation (Siemens) claims the optimization based on the speed controller which can be optimized in the time or in the frequency range. In the time range, the symmetrical optimum method can be used. In the frequency range, the optimizing is done according to the amplitude and phase margin rules, known from the control theory, but these are the conventional controls that are used in drives.

The interdisciplinarity of the chapter consists of using specific knowledge from the fields of: energy conversion, power converters, Matlab/Simulink simulation software, real time implementation based on dSPACE platform, electrotechnics, and advanced control techniques.

This chapter is focused on mathematical modelling, optimal control development on power inverter, Matlab implementation and analysis of the optimal control solution in order to minimize the electric input energy of the drive system. The Matlab on-line solution can be downloaded in real-time platforms, as dSPACE or dsPIC Digital Signal Controllers (Gaiceanu, et al., 2008).

The practical implication of using the proposed method is the on-line calculation of the optimal control solution. The main lines of the research methodology are:

  • The optimal control of the dynamical electrical drives problem formulation;

  • The optimal control solution;

  • The numerical simulation results;

  • The implementation of the optimal control.

The strategy of the optimal control electrical drive system is to use both the conventional control during the steady state and the developed optimal control during the transients. The conventional control is based on the rotor field orientated control using Proportional Integral controllers. If the speed error is different from zero, the software switch will enable the optimal control, else will enable the conventional control. Both types of control are developed in the next Sections.

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2. Mathematical model of the variable speed electrical drives (VSED)

There are variable speed DC and AC drives. A DC drive system controlled by armature voltage at constant field is a linear, invariant and controllable dynamic system. Variable speed AC drives are mostly used in connection with induction and synchronous electrical machines. The modern drives contain the energy source, the power converters, the electrical machines, the mechanical power transmission and the load, linked by an adequate control. The well-known mathematical model of the vector controlled AC drive using three phase squirrel cage induction machine in rotor field coordinates supplied from the current inverter, operating at the constant flux (Leonhard, 1996) becomes a linear system (Gaiceanu, 2002). For the electrical drives having the power less than 50 kW, the power loss of the converter could be neglected. In order to compute the optimal solution for synchronous Drive system, the control in rotor field oriented reference frame of surface mounted permanent magnet synchronous motor (PMSM) drives is used. The decoupling of the control loops (torque and flux) is performed in rotor field reference frame. By setting a zero value of direct current stator component, the mathematical model of the PMSM becomes linear. Therefore, there is a common mathematical model of the electrical drives, described by the linear state space standard form representation:

x˙(t)=Ax(t)+Bu(t)+Gw(t)E1

where x(t)is the state vector, u(t) is the control vector, and the perturbation vector is the load torque w(t)=[TL(t)].

Taking into consideration the most used electrical machine, the three-phase squirrel cage induction machine, the electrical drive based on it will be explained as follows. Starting from the nameplate data of the induction machine (Fig.1), the main parameters of the electrical motor can be found (Appendix 1).

Figure 1.

Nameplate data of the electrical machine (three phase induction motor)

The mathematical model of the three phase squirrel cage induction machine (IM) in rotor field reference system operating at constant flux is (Leonhard, 1996):

{Trdimrdt+imr=idsdqdt=ωep+iqsTrimrJpdωedt=TeFpωeTLTe=kmimriqs,(a)whereimrrotormagnetizingcurrent;ids,iqsfluxandtorquecomponentsofthestatorcurrent;Teelectromagnetictorque, TL  loadtorque,p numberofpolepairs,Fviscousforce;Ts,Tr statorandrotortimeconstants;q angleofthemagnetizingcurrentvector;we,welectricalandmechanicalrotorspeed,ωe=pω(b)withLmmagnetizinginductance,srrotormagnetizingdispersionfactor.Km=23pLm1+σr(c)E2

By varying the flux component of the stator current, ids, the magnetizing current response, imr(t), can have signifiant delay. By using an adequate control of power inverter (Gaiceanu, 2002), the rotor magnetizing current, imr, can be maintained at the constant value, and according to the mathematical model of the IM (1a), the flux component of the stator current, ids, equates the rotor magnetizing current): imr=ids=ct.. Therefore, the standard state space form of the IM mathematical model can be obtained (Fig.2):

Figure 2.

Standard space state form characterization of the three phase Induction Motor

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3. Optimal control problem statement

The common objectives of the optimal control drive systems are: the smooth dynamic response, without oscillations; achieving the desired steady state; the fast compensation of the load torque; energy minimization. Taking into consideration the main objectives of the electrical drive system, the following Section describes the chosen quadratic performance criterion.

3.1. The quadratic performance criterion

The functional cost or the quadratic performance criterion is chosen such that the electrical input power, the energy expended in the stator windings of the electrical machines is minimized, and the cost of the control is weighted. Therefore, the performance functional quadratic criterion (Rosu, 1985), (Gaiceanu, 2002), (Athans, 2006), is as follows:

J=12[x(t1)x1]TS[x(t1)x1]+12t0t1[xT(t)Qx(t)+uT(t)Ru(t)]dtE3

in which, taking into consideration a starting of the IM, the initial conditions consists of the initial time t0=0[s] and the initial state vector x(t0)=[ω(t0)q(t0)]T=[00]T,T-being the transpose symbol of the initial state vector. The final condition is the required final state, x1. The components of the x1 are: the desired angular velocity, ω*m, and a free rotor magnetizing angle value, q*=0, i.e x1=[ωm*0];the final time, t1=0.9[s]. The value of the final time is established by the feasibility condition, i.e. the value of it is taken from the obtained final time of the starting process with conventional control (Proportional Integral controllers). In order to exist and to obtain a unique optimal solution, the weighting matrices must be chosen according to: S0, R>0, Q0.The weighted matrix Sminimizes the square error between the reached state x(t1), and the desired state, x1, in the fixed time t1, the weighted matrix Rminimizes the control effort, u(t)=iqs(t), and the weighted matrix Qminimizes the expended energy in the motor windings. The optimal control problem taken into account in this chapter is with free-end point, fixed time and unconstrained. The restrictions of the magnitude for the control vector, and state vectors are managed through the design process, by an adequate choice of the weighting matrices.

3.2. The solution of the optimal control problem.

The existence and the unicity of the optimal control problem solution are based on the controllability and observability of the system (1). The system (1) being controllable and completely observable, the weighting matrices are chosen such that Q, S ≥ 0are positive semidefinite, and matrix R>0is positive definite (Gaiceanu, 2002). In this way, the existence and the unicity of the optimal control problem solution are assured.

By using variational method, the Hamiltonian of the optimal control problem is given by

H(p,x,u,t)=12[xT(t)Qx(t)+uT(t)Ru(t)+yT(t)(Ax(t)+Bu(t)+Gw(t))]E4

in which y(t)∈ℜ2is the costate vector.

The condition R>0assures the inversability of the eigenvectors and that the optimal solution has a minimum (Gaiceanu, 2002). The optimal control minimizing the Hamiltonian (4) is given by

u*(t)=R1BTy(t)E5

By using the canonical system, the solutions of the costate y(t)and state vectors x(t)can be obtained:

[x(t)y(t)]=[ABR1BTQAT][x(t)y(t)]+[G0]w(t)E6

The boundary conditions of the canonical system (5) are:

-the initial state  x(0)=xo;

-the transversality condition of the costate vector:

y(t1)=λ(t)x|t=t1=S[x(t1)x1]E7

y(t1)=S[x(t1)x1]=100[ω(t1)q(t1)][15707,960],

where the weighting matrix S=[100000], and qis the angle of the rotor magnetizing flux.

The integration of the canonical system leads to the well-known matrix Riccati differential equation and the associate vectorial equation (Rosu, 1985), (Gaiceanu, 2002), (Athans, et al., 2006). The integration of these two equations is a very difficult task because the Riccati equation is nonlinear, its solution is recursive (which can be calculated backward in time) (Gattami, et al., 2005), (Jianqiang, et al., 2007), (Rosu, 1985). Moreover, the backward computation needs to know a priori the variation of the perturbation vector w(t) during the control interval [0, t1]. In most electrical drives the last condition cannot be accomplished.

A nonrecursive solution of the Riccati equation has been developed by using two linear transformations (Rosu, et al., 1998). The first transformationchanges actual time tinto t1-t,that is time remaining until the end of the optimal process.

τ=t1t=0.9t,E8

In order to use only the negative eigenvalues for the computation of the fundamental matrix of the system (1), the second transformationis made. Therefore, by defining new vectors p(τ), q(τ), r(τ),

[p(τ)q(τ)]=[ABR1BTQAT][p(τ)q(τ)][G0]r(τ)E9

and by noting the canonical matrix (4×4) of the system (1) as:

[ABR1BTQAT]=ME10

the eigenvalues λi,i=1,4¯of the canonical matrix Mcan be computed with:

det[λIM]=0E11

By using the developed Matlab file from Appendix 3, the appropriate negative eigenvalues position placed on the main diagonal of the eigenvalues matrix, D, with diag(D)={λ1,λ2,λ3,λ4}, can be obtained:

[λ10000λ20000λ10000λ2]=[5.16630000 0.72050000-5.16630000-0.7205]=[Λ00Λ],E12

in which

Λ=[λ100λ2]=[5.1663000.7205].E13

In the Appendix 3 the proper position of the eigenvalues and the eigenvectors of the linear system (1) is provided. By using the program sequence from Appendix 3, the stability of the system is assured and the corresponding matrix of the eigenvectors is obtained:

W=[ 0.9877-0.5906 -0.9740 -0.5702 -0.1567 0.8070-0.2268 -0.8215 0.0004-0.0001  0.0004  0.0000-0.0000 0.0011 0.0000  0.0011]E14

By knowing the inverse of the eigenvector matrix W-1

W1 =1.0e+003 *[ 0.0005  0.00011.35420.31530.0000 0.0006 0.31580.45490.00050.0000  1.3732 0.21780.00000.00060.32710.4469]E15

the new vectors, m(τ), n(τ), p(τ), q(τ) are introduced as follows:

[m(τ)n(τ)]=W1[p(τ)q(τ)],E16
[p(τ)q(τ)]=W[m(τ)n(τ)]E17

By introducing (15) and (16) transformations in (8), the following differential system is obtained:

W[m(τ)n(τ)]=MW[m(τ)n(τ)][G0]r(τ)E18

Therefore, the canonical system can be written in terms of n(τ), and m(τ) vectors:

[m(τ)n(τ)]=W1MW[m(τ)n(τ)]W1[G0]r(τ)E19

By using the well-known relation

W1MW=[Λ00Λ]E20

and by noting the new vector Has:

W1[G0]=H=[  -52.6495    -0.2471    -47.8790      23.0137],E21

the new form of the canonical system can be obtained:

[m(τ)n(τ)]=[Λ00Λ][m(τ)n(τ)]Hr(τ)E22

The solution of the new canonical system (21) can be obtained by using the following equation:

[m(τ)n(τ)]=[eΛτ00eΛτ][m(0)n(0)]0τ[eΛβ00eΛβ]Hr(β)dβE23

As it can be observed the solution of the canonical system (22) solution contains both the positive, eΛτ, and the negative exponentials, eΛτ. It is well-known that the positive exponentials produce the instability of the solution. Moreover, in order to obtain the analytical solution of the system (22) the load torque (r(τ) or w(t)) at the final time, t1, must be known in advance. For the most used electrical drive system this condition is hardly accomplished because the load torque cannot be known in advance, at the final time.

A numerical solution of the optimal problem is proposed. The advantage of maintaining the sampled signal at the same value during the sampled period is that the zero order hold (ZOH) offers a brilliant solution to know the load torque at the final time. Therefore, under this assumption, the solution (22)is computed each sampled time, T,and the perturbation vector rhas constant value during it:

[m(τ)n(τ)]=[eΛτ00eΛτ][m(0)n(0)]+[IeΛτ00IeΛτ][H1H2]rE24

where Iis the identity matrix.

The obtained solution of the canonical system (23) is:

[m1(τ)m2(τ)n1(τ)n2(τ)]=[e(5.1663)τ0000e( 0.7205)τ0000e(5.1663)τ0000e(0.7205)τ][m1(0)m2(0)n1(0)n2(0)]+[1e( 5.1663)τ00001e(0.7205)τ00001e( 5.1663)τ00001e(0.7205)τ][  -52.6495    -0.2471    -47.8790      23.0137]r.E25

The negative exponentials can be obtained by extracting m(0) function on the m(τ). In this way, the first equation of the system (24)becomes:

[m(0)n(τ)]=[eΛτ00eΛτ][m(τ)n(0)]+[I2eΛτ00I2eΛτ][H1H2]rE26

where

I2=[1001]E27

The negative exponentials of the system (25)guarantee the stability of the system (1).

The transversality condition (6), in terms of mand nvectors, is as follows:

n(0)=Em(0)+FxfE28

where,

E=[SW12W22]1[W21SW11]=[  1.0215     -0.0320 -0.0127    -0.9811]E29
F=[SW12W22]1S=[-1.0504         0  0.0404        0]E30

By expressing n(τ) function on m(τ), the following equation can be deducted :

n(τ)=Z(τ)m(τ)+eΛτE(I2eΛτ)H1r+eΛτFx1+(I2eΛτ)H2rE31

where

Z(τ)=eΛτEeΛτE32

Going back through the reverse transformations from m(τ), n(τ) vectors to p(τ), q(τ) vectors, from the used τtime to the current time t, the state x(t)and costate y(t) expressions can be obtained. The costate vector at current time is as follows:

y(t)=P(t1t)x(t)K1(t1t)x1K2(t1t)wE33

where the matrix P(t1t) is the solution of the matrix Riccati diferential equation, and the matricesK1(t1t) and K2(t1-t)vector have the form:

K1(t1t)=v(t1t)eΛ(t1t)F,E34
K2(t1t)=v(t1t)[eΛ(t1t)E(IeΛ(t1t))H1+(IeΛ(t1t))H2],E35

The vmatrix is as follows:

v(τ)=[W22P(τ)W12].E36

The solution of the optimal control problem has three components (Fig.3):

Figure 3.

Optimal control (i*sq) implementation into three phase IM Drive System supplied by the Voltage Power Source Inverter

  1. the state feedback R1BTP(t1t)x(t), in order to assure the stability of the system;

  2. the compensating feedforward load torque R1BTK2(t1t)w, for fast compensation of the main perturbation, i.e. the load torque;

  3. the reference component R1BTK1(t1t)x1, to track accurately the reference signal.

The optimal control, at any moment t, is given by (Gaiceanu, 2002), (Athans, et al., 2006)

u*(t)=R1BTP(t1t)x(t)+R1BTK1(t1t)x1+R1BTK2(t1t)w(t)E37

in whichP(t1-t)is the solution of the matrix Riccati differential equation, MRDE, and the matrices K1 and K2 are calculated via P(t1-t) (Rosu, et al., 1998).

The numerical solution supposes the knowledge of the disturbance w(t)=TL(t), which could be available by using torque estimator, as in (Rosu, et al., 1998), or with torque sensor (more expensive solution that can produce the faults of the electrical drive system).

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4. Simulation and experimental results

In order to implement the optimal control system two electrical drive systems are combined: the conventional vector control and the optimal control drive. Both electrical drive systems operate in parallel. The activation of one of them depends on the speed error: ifthe speed error is zero (steady state operating regime), thenthe conventional vector controlis active, elsethe optimal controldrive is active.

4.1. Optimal control implementation

Optimal control implementation consists of loading nine software modules developed by the author: nameplate motor data (Fig.1), determination of the motor parameters (Appendix 1), determination of the six sectors vector components to be used in Space Vector Modulation (SVM) (Appendix 2), standard space state form characterization of the three phase Induction Motor (Fig.2), the proper arrangement of the eigenvalues and eigenvectors (Appendix 3), optimal control implementation in MATLAB (Appendix 4), the main program (Appendix 5), graphical representation (Appendix 6), Space Vector Modulation –Matlab function svm_mod(Appendix 7). By using the Matlab code from Appendix 8 the graphical representation of the simulation results can be obtained (Figures 4-7).

Figure 4.

Three phase SVM supply voltages

Remark

The SVM increases the DC-link voltage usage, the maximum output voltage based on the space vector theory is 2/3times higher than conventional sinusoidal modulation (Pulse Width Modulation). The purpose of the SVPWM technique is to approximate the reference voltage vector Uoutby a combination of the eight switching patterns. The implemented Matlab sequence for obtaining the pulse pattern of the optimal SVM modulator (Appendix 2, Appendix 7) shown above is based on the developed methodology by the author (Gaiceanu, 2002). In optimal control case, there is no speed overshoot, and smoothly state response (speed and angle of the rotor magnetizing current) have been obtained (Fig.5, Fig.7).

Figure 5.

Optimal control, speed, longitudinal and transversal voltage components

Figure 6.

Voltage loci loculus for a starting process

Figure 7.

Optimal controliqs=f(t), currentimr(t), load torqueTL(t)(elevator), rotor magnetizing angle of the rotor fieldq(t) (left to right)

4.2. Conventional vector control

The conventional vector control consists of the rotor flux orientated drive system. The vector control is implemented in Simulink as following. The control system is in cascaded manner and consists of two major loops: the speed loop and the magnetic flux loop. The torque control loop is the minor loop of the speed control one. The references of the control loop are: the desired angular velocity for the speed loop, ω*, the imposed stator current torque component for the torque loop, i*qs, and the desired stator current flux component for the magnetization flux loop, i*ds.

Tuning of the speed and flux controllers

By knowing the nameplate motor data (Fig.1), the tuning of the PI speed and flux controllers parameters can be done (see the developed Matlab file shown in the Appendix 8). The conventional vector control supposes the use of the two independent control axes: d-axis, along with the magnetizing flux direction, and qaxis, in quadrature with d-axis. The most used field orientated control is that of the rotor orientated magnetizing flux (Fig.8). The magnetization flux and the torque can be controlled independently.

Figure 8.

Simulink implementation of the rotor field orientated vector control of the three-phase induction motor

Figure 9.

Simulink implementation of the speed and flux Proportional-Integral (PI) regulators

The inputs of the speed and flux regulators Simulink block (Fig.9) are the speed reference and the flux reference; the outputs are the q-axis reference current, IQ*, and d-axis reference current, ID*(Fig.10). More details can be obtained by using Fig. 10.

Figure 10.

PI Speed regulator of type1 (kp_om+ki_om/s) and Flux regulator of type 3 (kp_df+ki_df/s), according to Matlab file shown in the Appendix 8

By using the field orientated mechanism implemented in Simulink (Fig.11), based on the input signals (IQ*, fluxand om), the frequency of the stator voltage is delivered.

Figure 11.

The Field Orientated Control (FOC) mechanism implemented in Simulink

The parameters of the speed and flux regulators are delivered by using the Matlab file developed in the Appendix 8.

Figure 12.

Detailed program of the PI control (torque (IQ*) and flux (ID*) control loops), the decoupling of thed,qaxes, and the voltage compensation Simulink block ofFig.7

In the Fig.12. the detailed decoupling and the voltage compensation between d, qaxes are depicted. In order to control the torque current component (IQ) and the flux current component (ID), the PI controllers are involved. By introducing the compensating voltage components, the independent control of flux and torque is obtained (Fig.12).

The reference voltage components pair (VD*, VQ*) is used as input of the Pulse Width Modulation (PWM) transfer function block, shown in the Fig.13.

Figure 13.

The Simulink models of the PWM modulator and of the equivalentd,qmodel of the IM

Figure 14.

The speed control loop

Figure 15.

The rotor flux control loop

The performances of the speed control loop are shown in the Fig. 14, in which the step rotor speed reference (om_ref*) is applied and the actual rotor speed (om) follows the reference with zero steady state error. The performances of the implemented flux control loop are shown in the Fig. 15. Small flux overshoot (up to 11%) and zero steady state error are obtained (Fig.15). The IM starts after t=0,5s (Fig.15), when the rated flux of the motor is attained (Fig.15). In order to implement the optimal control for ac drives, a test bench based on the DS1104 platform has been developed (Fig.16).

Figure 16.

The experimental set-up test bench (Gaiceanuet al., 2008)

Figure 17.

Real time implementation of both electrical drive systems based on the DS1104 real time platform

Both types of control, i.e. conventional and optimal, have been implemented on DS1104 (Fig.17) controller board. The developed ControlDesk interface has two main functions: real time control and signals acquisition. The evaluation of the obtained experimental results reveals the system efficiency improvement through an important decreasing of the windings dissipation power and input power (Fig.18) in optimal control case. Therefore, the power balance provides an increasing of the system efficiency, thus improved thermal and capacity conditions are obtained.

The optimal control (7) of the electrical drive based on the induction machine (1) has been numerically simulated by using Z transform and zero order hold for a starting of a 0.75 [kW], 1480 [rpm] under a rating load torque of 4.77 [Nm]. The initial conditions of the optimal control problem are: to=0[s], xi=[ni=0 q=0]. The free final conditions are: tf=0.9[s], xf=[nf=1480 q=0].

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5. Conclusions

In order to obtain the optimal control, the nonrecursive solution has been used to avoid the main disadvantages of the recursive solution (that is, the solution can be calculated at the current time, not backward in time; the load torque can have any shape, either linear or nonlinear, in admissible limits, due to the fact that the solution is computed at each sample time).

Figure 18.

Conventional and optimal control: comparative evaluation of the energy

A simplified solution is proposed by avoiding the constrained optimal control problem type. By using adequate weighting matricesS, Rand Q, the magnitude constraints have been solved. In this way the states and the control signals are maintained in admissible limits.

The optimal problem shown in this chapter has been formulated for a starting period. In the Fig.18, the energy evaluation for both control methods, conventional and optimal, is depicted. According to Bellman's theorem of optimality, the optimal problem can be extended also on the braking, or reversing periods. The components of the energy of the three-phase IM (Fig.20) are: W1 the input energy, W2 the output energy, WCu1 the energy expended in the stator windings, WCu2 the energy expended in the rotor windings, WFv the viscous friction dissipated energy, WpJ the accumulated energy in inertial mass, W the electromagnetic energy.

From the energetic point of view, it could be noticed that the output energy is the same in both controls (conventional and optimal) while the input energy decreases significantly in the optimal control case due to the decreasing of the stator copper losses, even if the rotor losses slightly increase in the optimal control case (Fig.20).

The proposed optimal control conducts to 8% electrical energy reduction, despite of using the conventional control. The conclusion is: decreasing the input energy for the optimal control system at the same time with maintaining the output energy in both systems, classical and optimal, the efficient electric drive system is obtained.

Due to using only the negative exponentials terms, the system is stable.

The optimal control can be applied in the applications with frequent dynamic regimes. The minimization of the consumed energy conducts to an increasing of the operational period of the electrical drives, or to the electrical motor overload allowance.

The optimal control could be extended to the high phase order induction machine applications, in a decoupled d-qrotor field oriented reference frame and by using adequate control of the power inverter (Gaiceanu, 2002).

Taking into consideration that this chapter is focused on developing and proving the performances of the optimal Electrical Drive Systems (EDS), an immediate effect is represented by the positive impact on the market as an alternative for electrical efficiency. The key of the proposed chapter is the integration of the developed Matlab tool by means of the high technology, i.e. the development of the systems with microprocessor. The chapter leads to drawing up specific research and development methods in the field of electrical drives. It has been underlined that by making EDS in an efficient manner, the CO2 emission reduction will be also obtained.

The achievement of this chapter is the opportunity to attract young MSc. and PhD students in R&D activities that request both theoretical and practical knowledge. The author will have new opportunities for developing international interdisciplinary research, technological transfer, sharing of experience and cooperation in the framework of the specific Research Programmes.

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Appendix 1

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Appendix 2

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Appendix 3

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Appendix 4

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Appendix 5

Appendix 6

Appendix 7

Appendix 8

The Matlab file for tuning of the PI speed and flux controllers parameters in conventional control

Acknowledgments

This work was supported by a grant of the Romanian National Authority for Scientific Research, CNDI– UEFISCDI, project number PN-II-PT-PCCA-2011-3.2-1680.

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Written By

Marian Gaiceanu

Submitted: September 19th, 2013 Reviewed: December 10th, 2013 Published: September 8th, 2014