1. Introduction

2. Problem Statement. From Classical Physical Modeling of Energy Conversion Flow to Data Free Model Control

3. The DDC of AC drive. Illustrative Case Study: The DDC of a PMSM Drive System

3.2. DDC of the PMSM Drive System

The PMSM drives are largely used in the adjustable applications that require high performance control performances and high efficiency. A conventional structure used in practice includes a two level power electronic inverter as in Figure 6. On this input of the power electronic inverter is usually ensured a constant DC bus voltage U_DC which is provided by a DC bus circuit that contains a DC link voltage and a voltage rectifier. The main feature of this power structure is that the discrete logic commnad S_abc={S_a,S_b,S_c} applied on the top switches {1,3,5} is negated applied to the bottom switches {4,6,2}.

The power inverter discrete command is provided by the control structure that is developed in the following.

3.2.1. Plant Modeling: PMSM and Power Inverter

Naturally, the model of PMSM is developed in the three-phase coodinates (a,b,c). This model has a complex structure, being difficult to be used in the practical close-loop control applications. A first approach to avoid this drawback is to use ortogonal fixed coordinate (α,β) that offer some improvments. At last, the ortogonal transfomation in a moving system reference frame (d,q) provides the highest benefits of the model of PMSM for close loop applications. Indeed, the usual FOC tehnique appied to PMSM is done by align the space vector of PM flux ${\overline{\psi }}_{PM}$ with a north pole of the rotor of PMSM.

Figure 7. Space vector diagrame of a PMSM with rotor system frame orientation .

Figure 7. Space vector diagrame of a PMSM with rotor system frame orientation .

In addition, the stator current components (i_d,i_q) of space vector current $\overline{i}$ admits to be colinear and perpendicular with the PM flux direction. The both space vectors as PM flux ${\overline{\psi }}_{PM}$ and current $\overline{i}$ rotates with the electrical speed ω_e.

For this kind of application, the pair of flow and effort mention above in the privious section it corresponds to torque and angular speed:

$$(e,s)\mapsto (m,{\omega }_m).$$

(34)

The PMSM model obtained by FOC principle is composed by the currents and the motion equations [8]:

$$\begin{array}{l}{\displaystyle \frac{d{i}_d(t)}{dt}}={\displaystyle \frac{1}{L_d}}(u_d(t)-R_s{i}_d(t)+L_q{\omega }_e(t)i_q(t))\\ {\displaystyle \frac{d{i}_q(t)}{dt}}={\displaystyle \frac{1}{L_q}}(u_q(t)-R_s{i}_q(t)-L_d{\omega }_e(t)i_d(t)-{\omega }_e(t){\psi }_{PM})\end{array}.$$

(35)

$${\displaystyle \frac{d{\omega }_m(t)}{dt}}={\displaystyle \frac{1}{J}}\left(m(t)-{\displaystyle \frac{{\mu }_f}{z_p}}{\omega }_m(t)-m_{\mathcal{l}}(t)\right)$$

(36)

where: (u_d,u_q),(i_d,i_q) and (L_d,L_q) are voltages, currents and inductances, all expressed in (d,q) coordinates, (ω_e, ω_m) are electrical and mechanical angular speed, respectively, R_s denoted the stator resistance, ψ_PM represents the permanent magnet flux, m and m_ℓ are the electromagnetic and load torques, respectively, and the group of constants represented by z_p,J and μ_f that are poles pairs, total inertia of drive and friction factor, respectively.

The system (35) – (36) was completed by introducing the expressions of the electromagnetic torque, magnetic flux components, input and output power, respectively, as follows:

$$m(t)={\displaystyle \frac{3}{2}}{z}_p\left[(L_d-L_q)i_d(t)i_q(t)+{\psi }_{PM}{i}_q(t)\right]$$

(37)

$$\begin{array}{l}{\psi }_d(t)=L_d{i}_d(t)+{\psi }_{PM}\\ {\psi }_q(t)=L_q{i}_q(t)\end{array}$$

(38)

$$\begin{array}{l}{p}_1(t)=\mathrm{Re}\left\{\underset{\_}{u}{\underset{\_}{i}}^{*}\right\}=\mathrm{Re}\left\{(u_d(t)+j{u}_q(t))(i_d(t)-j{i}_q(t))\right\}\\ p_2(t)=m(t)\cdot {\omega }_m(t)\end{array}$$

(39)

where u and i are the phasors of voltage and current, respectively, i^* represents the complex-conjugate phasor of current, and by Re{◦} is denoted the real part of the applied quantity.

The equivalent circuit scheme of the PMSM based on the dq current model (35) is depicted in Figure 8, where has been introduced the total ElectroMotive Force (EMF) force components:

$$\begin{array}{l}{e}_d(t)=-L_q{\omega }_e(t)i_q(t)\\ e_q(t)=L_d{\omega }_e(t)i_d(t)+{\omega }_e(t){\psi }_{PM}\end{array}$$

(40)

The Equations (34) – (38) describe the so-called dq model of the PMSM drive system. Based on the dq model there is designed the control structure in a propper maner. However, the physical operation of PMSM is done in the natural three-phase system reference frame. To obatin the desired quantities there are used the Clarke and Park direct/indirect transformations, respectively, of the quantity λ=u,i,ψ, as follows:

$$\begin{array}{c}{\lambda }_{\alpha \beta }=T_{\alpha \beta }{\lambda }_{abc}\\ {\lambda }_{abc}=T_{\alpha \beta }^{-1}{\lambda }_{\alpha \beta }\end{array}$$

(41)

$$\begin{array}{c}{\lambda }_{dq}=T_{dq}{\lambda }_{\alpha \beta }\\ {\lambda }_{\alpha \beta }=T_{dq}^{-1}{\lambda }_{dq}\end{array}$$

(42)

where vector-matrix quantities used are:

$$\begin{array}{l}{\lambda }_{abc}={\left[\begin{array}{ccc}{\lambda }_a& {\lambda }_b& {\lambda }_c\end{array}\right]}^T,\\ {\lambda }_{\alpha \beta }={\left[\begin{array}{cc}{\lambda }_d& {\lambda }_q\end{array}\right]}^T,\\ {\lambda }_{dq}={\left[\begin{array}{cc}{\lambda }_d& {\lambda }_q\end{array}\right]}^T.\end{array}$$

(43)

and the involved matrix-tranformations are given by:

$$\begin{array}{c}{T}_{\alpha \beta }={\displaystyle \frac{2}{3}}\left[\begin{array}{ccc}1& -{\displaystyle \frac{1}{2}}& -{\displaystyle \frac{1}{2}}\\ 0& {\displaystyle \frac{\sqrt{3}}{2}}& -{\displaystyle \frac{\sqrt{3}}{2}}\end{array}\right];T_{\alpha \beta }^{-1}={\displaystyle \frac{2}{3}}\left[\begin{array}{ccc}1& 0& 1\\ 0& -{\displaystyle \frac{1}{2}}& {\displaystyle \frac{\sqrt{3}}{2}}\end{array}\right];\\ T_{dq}={\displaystyle \frac{2}{3}}\left[\begin{array}{ccc}\cos {\alpha }_m& \cos \left({\alpha }_m-{\displaystyle \frac{2\pi }{3}}\right)& \cos \left({\alpha }_m+{\displaystyle \frac{2\pi }{3}}\right)\\ \sin {\alpha }_m& \sin \left({\alpha }_m-{\displaystyle \frac{2\pi }{3}}\right)& \sin \left({\alpha }_m+{\displaystyle \frac{2\pi }{3}}\right)\end{array}\right];T_{dq}^{-1}={\displaystyle \frac{2}{3}}\left[\begin{array}{ccc}\cos {\alpha }_m& \sin {\alpha }_m& 1\\ \cos \left({\alpha }_m-{\displaystyle \frac{2\pi }{3}}\right)& \sin \left({\alpha }_m-{\displaystyle \frac{2\pi }{3}}\right)& {\displaystyle \frac{\sqrt{3}}{2}}\end{array}\right].\end{array}$$

(44)

The power invertor command will be supplied by the finite set values S_abc which contains the optimal logical values requited for obtaining inverter control.

Based on finite set values S_abc the model of the two-level power inverter provides the three-phase voltages:

$$u_{abc}={\displaystyle \frac{1}{3}}{U}_{DC}{T}_I{S}_{abc}$$

(45)

where the power inverter matrix model is:

$$T_I=\left[\begin{array}{ccc}2& -1& -1\\ -1& 2& -1\\ -1& -1& 2\end{array}\right]$$

(46)

3.2.2. MPC with Finite Set

The DDC method is developed by using the main features of the MPC strategy that provides a command law which corresponds to voltage of the inverter, computed by a switching functions set. In this strategy, the main variables as current, voltage and electromagnetic torque are discretized on every time instant $T_s$ by forward Euler forward method, that gives to a generic variable z(t) the next fist-time derivative linear approximation:

$${\left.{\displaystyle \frac{dz(t)}{dt}}\right|}_{z=z(k+1)}\approx {\displaystyle \frac{z(k+1)-z(k)}{T_s}}$$

(47)

Taking into account Euler forward approximation method, the first-time derivate of current (35) becomes:

$$\begin{array}{l}{i}_d(k+1)=\left(1-{\displaystyle \frac{T_s{R}_s}{L_d}}\right)i_d(k)+{\displaystyle \frac{L_q}{L_d}}{T}_s{\omega }_e(k)i_q(k)+{\displaystyle \frac{T_s}{L_d}}{u}_d(k)\\ i_q(k+1)=\left(1-{\displaystyle \frac{T_s{R}_s}{L_q}}\right)i_q(k)-{\displaystyle \frac{L_d}{L_q}}{T}_s{\omega }_e(k)i_d(k)-{\displaystyle \frac{T_s}{L_q}}{\omega }_e(k){\psi }_{PM}+{\displaystyle \frac{T_s}{L_q}}{u}_q(k)\end{array}$$

(48)

Now, from the first-time derivate of current (37), the predictive electromagnetic torque results:

$$m(k+1)={\displaystyle \frac{3}{2}}{z}_p\left[(L_d-L_q)i_d(k+1)i_q(k+1)+{\psi }_{PM}{i}_q(k+1)\right]$$

(49)

The one-step-ahead MPC law is an optimal solving problem defined as follows:

$$\begin{array}{c}{g}^{*}=\underset{S_{abc}}{\min g_i}\\ \mathrm{s}.\mathrm{t}.\left\{\begin{array}{l}{g}_i=\left\{\begin{array}{l}\left|m^{ref}-m(k+1)\right|+i_d(k),{\omega }_m\le {\omega }_m^{rated}\\ \left|m^{ref}-m(k+1)\right|+\left|em{f}^{rated}-({\psi }_{PM}+L_d{i}_d(k))\right|,{\omega }_m>{\omega }_m^{rated}\end{array}\right.\\ i=1÷8; k=1÷N\end{array}\right. \end{array}$$

(50)

where emf^rated is the rated back motion EMF computed by:

$$emf=k_e{\omega }_m$$

(51)

with the back EMF constant:

$$k_e=z_p{\psi }_{PM}$$

(52)

the index i corresponds to the combination of switching state S_abc (see below the Table 1).

A better ilustration of the voltage space vector of the power inverter and their hexagonal bounds is done in Figure 9.

In the dq system reference frame, the components of the space vector $\underset{¯}{v}$ for the eight possible states are obtained in a vector-matrix representation:

$$\begin{array}{l}{v}_d=\left[\begin{array}{cccccccc}0& 2& 1& -1& 2& -1& 1& 0\end{array}\right]{\displaystyle \frac{U_{DC}}{3}}\\ v_q=\left[\begin{array}{cccccccc}0& 0& 1& -1& 0& -1& -1& 0\end{array}\right]{\displaystyle \frac{\sqrt{3}}{3}}{U}_{DC}\end{array}$$

(53)

Further, in Figure 10 is presented the cascade control structure, with a PI speed control in an outer loop and a MPC multivariable inner current control loop for the electronic power converter which supply the PMSM. The PI speed control is designed via the classical pole placement method [45].

The electrical energy provided from the power grid is of fixed values of voltage and frequency. By and adevate osition there is osition a DC voltage. The DC-bus voltage U_DC is filtred by a passive Low Pass Filter (LPF) represented by a high value capacitor C, obtaining a constant value of the DC-bus voltage U_DC=cst.

The PI controller is used for osition the mechanical speed ω_m^MPC its reference ω_m^ref-MPC. For the MPC formulation of the inner loop, there are measured the three-phase current (a,b,c), and then by the Park/Clarke transformations are successively obtained the dq MPC current i_dq^MPC(k)=[i_d^MPC(k) i_q^MPC(k)]^T . Further, based on (48) is obtained the current prediction i_dq^MPC(k+1)=[i_d^MPC(k+1) i_q^MPC(k+1)]^T. Now,the MPC problem defined according to (50) will give optimal switching combinations S_abc^MPC. For the future computations, there is additionally calculated the voltages u_dq^MPC(k)=[u_d^MPC(k) u_q^MPC(k)]^T based on a switch-voltage transformation S/u.

It can be mentioned that the mecanical speed ω_m^MPC is not directly measured. Firstly, there is measured the osition of the rotor α_m^MPC(k) by an encoder (E), and then by derivation is obtained the actual mechanical speed ω_m^MPC(k).

3.2.3. DDC Law of PMSM

The main idea of DDC strategy is to learn from a well-known MBC method whose performances has been already proved. In this sense, the DDC method of a PMSM drive is developed by selecting a DB containing the data provided by MPC law. Following the metodology developed for DDC in the sens of the maximum energy conversion process objective, for PMSM the output is given by speed and torque $y(k)\equiv \left[\begin{array}{cc}{\omega }_m^{MPC}(k)& m^{MPC}(k)\end{array}\right]$, while the control by dq voltage $u(k)\equiv u_{dq}^{MPC}(k)$. In this conext, the training sequence becomes:

$$d^{MPC}(k)={\left[\begin{array}{ccc}{u}_{dq}^{MPC}(k)& {\omega }_m^{MPC}(k)& m^{MPC}(k)\end{array}\right]}^T,$$

(54)

Also, for the PMSM drive the training matrix of the RBF-NN is composed by speed and dq volage of the reduced model (high efficiency points extraction):

$$\begin{array}{l}{M}_F=[\begin{array}{cc}\stackrel{⌣}{y}& \stackrel{⌣}{u}\end{array}]T,\\ \stackrel{⌣}{y}=[\begin{array}{cc}{\stackrel{⌣}{\omega }}_m^{MPC}(k)& {\stackrel{⌣}{m}}^{MPC}(k)\end{array}]T,\\ \stackrel{⌣}{u}={\stackrel{⌣}{u}}_{dq}^{MPC}(k),\end{array}$$

(55)

The training phase of the DDC controller is depicted in Figure 11. As mention above, the entire metodology is based on the the learning of the experience of the high-value efficiency steady-state data provided by MPC strategy.

The set of aprioric informations $K_{DB}$ includes rated data of PMSM, power inverter load and other components, envirovment and design requirments of EDSs.

Once training is done, the DDC control structure may be done as in Figure 12. The output of the DDC controller is given by the dq voltage $u_{dq}^{DDC}(k)$. The optimal set of the switching functions combinations $S_{abc}^{DDC}$is then obtained by a command shaping block.

To have a succinct description, in Figure 13 is depicted a schematic representation of the DDC control structure. The main controllers as PI and DDC are highlighted. The DDC algorithm is represented in a sequential structure, where the command shaping has an internal computing loop. The DDC algorithm has a complex structure. A first stage is the searching on the DB the corresponding command$u_{dq}^{ref-DDC}$. Once finded, an internal computing programming loop is designed. Further, there are compared the command$u_{dq}^{ref-DDC}(k)$ and the feedback $v_{dq}^{opt-DDC}(k)$ (that is also an optimal value) , resulting an error which is then integrated obtaining the the voltage control $v_{dq}^{c-DDC}(k)$ that acts as input in the searching block. Finally, there is obtained switching functions develivred by DDC method $S_{abc}^{DDC}$that will feed the gate of the inverter.

More detailes will presented later in the pseudocode format of the DDC method.

3.2.4. Design of the DDC Database

The input data in DB is initially done from the MPC operation, selected from the high efficiency values. After that, the data are processed according to various criteria. Hence, the acronym MPC will be substituted, making the distinction between MPC and DDC whenever is necessary.

The operation of a PMSM drive depends on the required mechanical speed domain limits. Generally, there are possible three situation that show how torque and outer power varies vs. mechanical speed:

$${\omega }_m=\left\{\begin{array}{l}\mathrm{I}. {\omega }_m\in (0,{\omega }_{m,b})\Rightarrow {\omega }_m\nearrow \Rightarrow \left.\left\{\begin{array}{l}m=cst.\\ P_2\nearrow \end{array}\right.\right|\mathrm{Constant} \mathrm{torque}\\ \mathrm{II}. {\omega }_m\in ({\omega }_{m,b},{\omega }_{m,\max })\Rightarrow {\omega }_m\nearrow \Rightarrow \left.\left\{\begin{array}{l}m\searrow \\ P_2=cst.\end{array}\right.\right|\mathrm{Constant} \mathrm{power}\\ \mathrm{III}. {\omega }_m>{\omega }_{m,\max }\Rightarrow {\omega }_m\nearrow \Rightarrow \left.\left\{\begin{array}{l}m\searrow \\ P_2\searrow \end{array}\right.\right|\mathrm{Critical} \mathrm{operation}\end{array}\right..$$

(56)

where: ω_m,b is the based mechanical speed which corresponds to the rated one (ω_m,b=ω_mN), and ω_m,max represent the maximal mechanical speed.

In Figure 14 are illustrated the torque – speed characteristics of a PMSM drive system. Until base speed ω_m < ω_m,b, the operation of PMSM is done on constant torque in region I. In the region II, where the power becomes constant, there is used field-weakening technique. At last, the critical operation done for ω_m < ω_m,max must be avoided (region III) for ensuring a safety operation.

Let be Ω,M, P and H the sets of sets of torque, power, and their orresponding efficiency:

$$\Omega =\left\{{{\omega }_m}_j\right|j=\overline{1,N_{\omega }}\}$$

(57)

$$M=\left\{m_j\right|j=\overline{1,N_m}\}$$

(58)

$$P=\left\{P_{2j}\right|j=\overline{1,N_p}\}$$

(59)

$$H=\left\{{\eta }_j\right|j=\overline{1,N_h}\}$$

(60)

where N_ω/m/p/h is the number of points of each set.

As can be easlier oserver from Figure 14, some of the different paris of speed-torque (ω_mj,m_j) or speed – power (ω_mj,P_2j) corresponds to the same value of efficiency. That means that the solution is not unique.

Considering that all the sets (57) – (60) has the same number of points $N_{\omega }=N_p=N_h=N$, there are introduced the next sets obtained by cartesian product:

$${\Lambda }_m=\{{\omega }_{mj}\times m_j|j=\overline{1,N}\}$$

(61)

$${\Lambda }_p=\{{\omega }_{mj}\times P_{2j}|j=\overline{1,N}\}$$

(62)

Now, there can be definited the following multi-valued maps:

$$f_m:{\Lambda }_m\to H$$

(63)

$$f_p:{\Lambda }_p\to H$$

(64)

whose points are contained in the sets:

$${\Gamma }_m=\{({\Lambda }_m(j),{\eta }_j)|j=\overline{1,N}\},{\Gamma }_m\subseteq {\Lambda }_m\times H$$

(65)

$${\Gamma }_p=\{({\Lambda }_p(j),{\eta }_j)|j=\overline{1,N}\},{\Gamma }_p\subseteq {\Lambda }_p\times H$$

(66)

To avoid the efficiency multi-valued representation, we select the high efficiency points that corresponds to each point, defining the functions:

$$f_m^{*}:{\Lambda }_m\to H^{\max }$$

(67)

$$f_p^{*}:{\Lambda }_p\to H^{\max }$$

(68)

where the maximal efficiency vector is :

$$H^{\max }={[\begin{array}{ccccc}{\eta }_1^{\max }& {\eta }_2^{\max }& {\eta }_3^{\max }& \cdots & {\eta }_N^{\max }\end{array}]}^T$$

(69)

and the contained sets are given by:

$${\Gamma }_m^{*}=\{({\Lambda }_m(j),{\eta }_j^{\max })|j=\overline{1,N}\},{\Gamma }_m^{*}\subseteq {\Lambda }_m\times H^{\max }$$

(70)

$${\Gamma }_p^{*}=\{({\Lambda }_p(j),{\eta }_j^{\max })|j=\overline{1,N}\},{\Gamma }_p^{*}\subseteq {\Lambda }_p\times H^{\max }$$

(71)

An illustrative representation of the transformation of te multi-valued map of the efficiency vs torque/power into a single-valued functions of a PMSM drive characteristics is depicted in Figure 15.

An essential objective is the design of the DB. Generally, there are few steps that must be followed in the order to obtain high computation accuracy as it is suggested in Figure 16.

Succinctly, the main followed steps can be described as:

I.

Storage grid of high efficiency data. At this step there are collected the main data set in the steady-state regime and knowledge which corresponds to a high efficiency value stored in $D_F=W\{K_{db};d^{°}(k)\}$

$$\begin{array}{l}{d}^{°}(k)={\left[\begin{array}{cccc}{u^{°}}_{dq}(k)& {i^{°}}_{dq}(k)& m^{°}(k)& {{\omega }^{°}}_m(k)\end{array}\right]}^T, \\ \mathrm{s}.\mathrm{t}. \eta (k)={\eta }_{\max }\end{array}$$

(72)

II.

Database grid learning. The database sets are extended by a learning process, adding new points contained in the set:

$$\begin{array}{l}{d}^{°e}(k)\supseteq d^{°}(k)\\ \mathrm{s}.\mathrm{t}. \eta (k)={\eta }_{\max }\end{array}$$

(73)

III.

Application to data grid searching. It is checked that the added new points corresponds to the real operation for open loop applications.

IV.

Close loop data grid control. Finally, the designed database grid is introduced in the desired close loop application from Figure 16.

These steps can be changed/improved depending on the desired objectives. For example, exogenous quantities may have a high influence on the system which are not on thermodynamic equilibrium.

Numerical computations of the quantities delivered by DB are affected by errors and noises. The DDC strategy is a searching discrete – interpolative method of the best solution within learning data bounds.

In the order to improve the acuarcy of data processing there is used a moving window filter, whose output is given by:

$$y^{ftr}[n]=\frac{1}{L_w}{\displaystyle \sum _{k=0}^{L_w-1}{u}^{nftr}[n-k]}$$

(74)

where L_w is the filter length, n is the current sample, and the input samples (nonfiltered) and output samples (filtred), respectively, are:

$$u^{nftr}={\left[\begin{array}{ccc}{u}_{dq}^{nftr}(k)& m^{nftr}(k)& {\omega }_m^{nftr}(k)\end{array}\right]}^T$$

(75)

$$y^{ftr}={\left[\begin{array}{ccc}{u}_{dq}^{ftr}(k)& m^{ftr}(k)& {\omega }_m^{ftr}(k)\end{array}\right]}^T$$

(76)

where $u_{dq}^{nftr/ftr}(k)={\left[\begin{array}{cc}{u}_d^{nftr/ftr}(k)& u_q^{nftr/ftr}(k)\end{array}\right]}^T$.

The moving window filter structure is represented in Figure 17. The input samples are introducd by using the unit-delay operator Δ. On the output is obtained an average value of the moving samples.

As can be observed from (75) – (76) the filtred qunatities are voltage, torque and mechanical speed. Based on this, further there result the filering of both input and output powers of PMSM. Finally, the filtred powers are used to compute the efficincy, that do not contains high numerical noises and errors.

3.2.5. DDC Matlab Implementation

The DDC of the PMSM is implemented in Matlab simulation environment, tacking into account the main programming features of this software pakage.

To obtain a high analysis, in the paper is made up a comparasion between the MPC method and DDC strategy of the PMSM drive.

MPC Algorithm

The basic idea of the model predictive control for the PMSM fed by the simplest inverter with two voltage levels, having eight available state voltage switching vectors v_i (Table 1) is the following: the torque and speed can be controlled by direct selecting a proper switching invertor’s voltage vector based on the machine parametric predictive model and an appropriate cost function g.

The objective of the Algorithm 1 represented in pseudocode format from Figure 12 is to give to reader the necessary tools to understand the simulations results and to replicate the implementation of the exposed algorithms thinking in Matlab environment.

The switching vector v^opt , that was predicted to provide the most favourable process behaviour, i. e, the one that minimize the objective function g, is considered to be optimal and is generated by the inverter, applying directly the corresponding switch functions svs=[ S_a, S_b, S_c ].

The parameters of the motor and the driven process are included in the vector ${\mathcal{P}}_{em}$(electromagnetic parameters). After initialisations of the finite states for the swithching vector (lines 1,2) and the rated values and state variables, the main iteration loop are opend (line 8).

The model requires the actual state variables $x=[\begin{array}{ccc}{i}_d& i_q& {\omega }_m\end{array}]$ to be recalled from the previous sampling period state$x^{old}=[\begin{array}{ccc}{i}_d^{old}& i_q^{old}& {\omega }_m^{old}\end{array}]$.

Figure 18. The pseudocode format of the MPC algorithme of a PMSM drive control.

Figure 18. The pseudocode format of the MPC algorithme of a PMSM drive control.

The electrical and mechanical parameters respectively,${\mathcal{P}}_e$ and ${\mathcal{P}}_m$, of the plant are grouped in the electromechanically set of parameters${\mathcal{P}}_{em}$, as follows:

$${\mathcal{P}}_{em}(t)==[\underset{{\mathcal{P}}_e}{\underbrace{\begin{array}{cccc}{R}_s& L_d& L_q& {\psi }_{PM}\end{array}}}\underset{{\mathcal{P}}_m}{\underbrace{\begin{array}{ccc}{m}_{\mathcal{l}}& J& {\mu }_f\end{array}}}]={\mathcal{P}}_e\cup {\mathcal{P}}_m$$

(77)

The MPC algorithm starts with the initializing the main data used in the computing loop.

A first programing syntax is developed for implement the PI speed controller:

$$\underset{\mathrm{Output}}{\underbrace{\left[m^{ref}\right]}}=PI\_control\underset{\mathrm{Input}}{\underbrace{({\omega }^{ref},{\omega }^{old} m^{max}, K_p, K_i)}}$$

(78)

where ${\omega }^{ref/old}$the reference /actual mechanical speed is,$m^{max}$denotes the maximal value of the electromagnetic torque, and K_p/i are the parameters of the PI controller. The PI_control generates the required value of torque reference (set point) m^ref.

The relation (78) describes a typical programing syntax that contain the Input/Output sets of quantities. This type of programing syntax structure was fallowed in the rest of the paper.

The MPC strategy takes as input the voltages v_d/q, the previous measured currents $i_{d/q}^{old}$, the electromecanical set of parameters ${\mathcal{P}}_{em}(t)$ and sampling time instant of MPC calculated around ten times higher then the simulation discretizing one:

$$\left[i_d^p\left(i\right),i_q^p\left(i\right),m^p\left(i\right),p_1\left(i\right)\right]=MPC\_PMSM\_model(v_d\left(i\right),v_q\left(i\right),i_d^{old},i_q^{old},{\mathcal{P}}_{em}(t),10\cdot T_s),$$

(79)

Thinking in the Matlab environment implementation, the function which simulate the PMSM model has as the input the controls (u_d,u_q) - the components of the supply voltage, the old states, the load torque and the electomagnetique parameters and as the outputs - the current components (i_d,i_q), the electomagnetique torque m and the angular velocity ω_m:

$$\underset{\mathrm{Output}}{\underbrace{[i_d,i_q,m,{\omega }_m]}}=Plant\_PMSM\_model\underset{\mathrm{Input}}{\underbrace{[u_d^{old},u_q^{old},i_d^{old},i_q^{old},{\omega }_m^{old},m_{\mathcal{l}},{\mathcal{P}}_{em}(t),T_s]}}$$

(80)

Finally, the programs were grouped and compacted in a software application of the proposed control structure.

The equations of the physical model (80) are in the continuous time domain. The variables of the model take values in the real domain$ℝ$. In fact the PMSM is supplied by the power electronic inverter which provide in the simplest case, a eight finite switching state vector:

$$v_{dq}=\{v_d(i),v_q(i)\},i=1,\mathrm{...},8$$

(81)

The components (u_d, u_q) takes values on the set v_dq. For this reason, these quantities must be shaped. According to the principle algorithm of the power inverter command, within each sampling period T_s, the components (v_d^opt,v_q^opt) and the duration d (0 <d> T_s) are calculated so that the machine currents resulting from switching supply voltage $v_{dq}^{opt}=[v_d^{opt},v_q^{opt}]$has to be as close as possible to the ideal sinusoidal form of the three-phase currents:

$$Find: v_{dq}^{opt} \mathrm{until} \Rightarrow THD{i}_{abc}=\min$$

(82)

which admits the syntax:

$$[v_d^{opt},v_q^{opt}]=Inverter\_Command\_Shaping\left[u_d(k),u_q(k),T_s\right]$$

(83)

The command algorithm used by MPC control is implicit, that is different from classical version. Within each sampling period T_s, the command algorithm looks for the optimal components (v_d^opt,v_q^opt), which minimizes the g criterion function:

$$[v_d^{opt},v_q^{opt}]=MPC\_Command\_Shaping\left[g\left(v_d(k),v_q(k),m,m^{ref},i_d(k),p_1(k),T_s\right)\right]$$

(84)

This function provides the variables of the process at the current sampling period. The filter is implemented by the function Mouving_Average, having the input the switching voltage vector components (v_d^opt,v_q^opt), length of the window of the filter L and the sample time period T_s, providing the output of the moving window that corresponds to the filtered inputs (u_d^ftr, u_q^ftr):

$$\left[{u_d}^{ftr},{u_q}^{ftr}\right]=Mouving\_Average[{v_d}^{opt},{v_q}^{opt},L,T_s]$$

(85)

The numerical data calculated at each step T_s are stored in the DB named Data_Base_1, which will continue to be used in DDC Algorithm presented in the next section.

3.4.2. DDC Algorithm

The DDC controls (u_d, u_q) are directly obtained from DB with $M_F$ (Figure 11) by the interpolation techniques RBF – NN exposed in the seconnd section. The control low was generated by means of a known set $M_F$ of operating points of the process (55) through an interpolation technique $A_F$(25). The resulting Algorithm 2 of the DDC strategy in pseudocode format is depicted in Figure 19 that schematically is given by Figure 13. The main common programing syntax from MPC and DDC has the same significance.

In first line there is loaded the data from database DB1, created with the MPC algorithm (Figure 18), where the support points of the control surfaces are described by:

$$\begin{array}{l}u(a)=F(mm(a),\omega \omega (a)),\\ u={\left[\begin{array}{cc}{u_d}^{ftr}(a)& {u_q}^{ftr}(a)\end{array}\right]}^T,\\ F={\left[\begin{array}{cc}{F}_d& F_q\end{array}\right]}^T,\\ a=1,\dots ,N_{obs}.\end{array}$$

(86)

As mention before, the control system is designed in two stages. In the the first, by MPC software are simulated a set of motion trajectories. In database DB1 are stored in a set of grided operation points of the process. Based on DB1 is simulated the DDC algorithm with which they were obtained the results presented in the paper.

There is a particularity of the electric drive systems, because the controls u_dq(k) must be shaped by the switching vector v_dq of the power inverter (81). The set$M_F$of operating points of the process (55) is essential for the efficiency of the energy conversion system with power electronic converter and electric machine. In that aim we selected from a large database a few set of the optimal energy operation points defined by the speed, torque, votltge control of power converter, input and output powers.

The parameters P_em (line 5) are used for the offline simulation of the process. The online operation of the control algorithm is parameters free. These is the main advantage of the DDC method.

In the main loop there are called the following functions:

• The PI_control which calculate the actual torque set point m^ref using the measured past speed ω_m^old and the speed set point ω_m^ref, and the tunning parameters K_p and K_i ;
• The matlab RBF_Interpolant function which interpolate the control surfaces (24) by scattered support points contained in DB 1;
• The function Process_Model which calculate the actual process variables (i_d,i_q,m,ω_,) and the powers p₁,p₂.

The control u(a)= u_d(a)+ju_q(a) are the query of the actual points (ω_m^old, m^ref) for the surfaces F_d and F_q . These controls are sampled from the continuous time domain. The inverter is controlled by the finite states switching functions svs . The transformation of the controls u_d and u_q from continuous time domain in the finite states domain of the inverter’s switching vectors v_d and v_q is a synthesis process.

The actual control u(a) at the step a is synthetized by a finite set of the switching vectors v=v_d+jv_q as follows

$$\begin{array}{l}u(a)=\{{v_k}^{opt}(r),{v_k}^{opt}(r+1),\dots {v_k}^{opt}(r+H_m)\},\\ r=1,\mathrm{...},H_m,\\ a=1,\mathrm{...},N_{obs}.\end{array}$$

(87)

where H_m is the horizon of modulation and each v^opt(k) vectors are determinates with a searching procedure, minimizing the g criteria function. The dimension of the H_m horizon was chosen by analogy with the classical modulation.

4. Illustrative Case Study of DDC of a PMSM Drive System

5. Conclusions

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