applied sciences

Article New Stable Non-Vector Control Structure for Induction Motor Drive

Pavol Fedor, Daniela Perdukova * , Peter Bober and Marek Fedor

Department of Electrical Engineering and Mechatronics, Technical University of Kosice, Letna 9, 04200 Kosice, Slovakia; [email protected] (P.F.); [email protected] (P.B.); [email protected] (M.F.) * Correspondence: [email protected]; Tel.: +421-55-602-2254

Abstract: The article focuses on a design and experimental verification of continuous nonlinear systems control based on a new control structure based on a linear reference model. An application of Lyapunov’s second method ensures its asymptotic stability conditions. The basic idea in the development of the control structure consists of utilizing additional information from a newly introduced state variable. The structure is applied for angular speed control of an induction motor (IM) drive representing a higher-order nonlinear system. The developed control algorithm helps to achieve the zero steady-state control deviation of the IM drive angular speed. Simulations and experiments performed in various operating states of the IM drive confirm the advantages of the new control structure. Except for set dynamics, the method ensures that the system is stable, invariant to disturbances, and is robust against variations of the parameters. When comparing the obtained control structure of the IM control with the classical vector control, the proposed control structure is


simpler. In addition, the proposed control structure is linear, robust against variation in important
 parameters and invariant against external disturbances. The main advantage over conventional Citation: Fedor, P.; Perdukova, D.; control techniques consists of the fact that the controller design does not require any exact knowledge Bober, P.; Fedor, M. New Stable of the system parameters and, moreover, it does not suffer from system stability problems. The Non-Vector Control Structure for method will find a wide applicability not only in the field of AC controlled drives with IM but also Induction Motor Drive. Appl. Sci. generally in control of industry applications. 2021, 11, 6518. https://doi.org/ 10.3390/app11146518 Keywords: control system synthesis; induction motor; motion control; nonlinear control systems; variable speed drives; Lyapunov’s second method Academic Editors: Javier Poza, Gaizka Ugalde and Gaizka Almandoz

Received: 10 June 2021 1. Introduction Accepted: 13 July 2021 Published: 15 July 2021 Induction motors (IMs) are robust and reliable, and due to their low cost and main- tenance, they find a wide utilization in industrial applications. A problem consists of Publisher’s Note: MDPI stays neutral their control requiring more complex control circuitry due to variable frequency, complex with regard to jurisdictional claims in dynamics, and parameter variations [1–3]. The IM itself presents a typical example of a published maps and institutional affil- nonlinear and considerably oscillating system with incorporated positive feedback. The iations. accuracy of its speed control is significantly influenced by unknown external disturbances and variable motor parameters. Several control methods of the IM are known. The simplest one is the scalar speed control method. It has a simple control structure [4,5] due to which is suitable for simpler Copyright: © 2021 by the authors. industrial applications. A better drive performance of the scalar control method requires Licensee MDPI, Basel, Switzerland. using on account of a more expensive and less reliable solution. This article is an open access article A precise drive performance is obtained by field oriented control (FOC) employing distributed under the terms and classical cascade PI controllers [6]. Various modifications of the FOC usually requiring conditions of the Creative Commons transformation of the IM variables into a rotating reference frame have been developed Attribution (CC BY) license (https:// which enable to control the IM in a similar manner like a separately excited DC motor [7–9]. creativecommons.org/licenses/by/ A drawback of such a solution consists of increased complexity of the control scheme 4.0/).

Appl. Sci. 2021, 11, 6518. https://doi.org/10.3390/app11146518 https://www.mdpi.com/journal/applsci Appl. Sci. 2021, 11, 6518 2 of 20

and in necessity of using powerful computational means (digital signal processors) for its implementation. From the point of view of the modeling and control the IM principally presents a nonlinear dynamic system with uncertain parameters. Many design techniques for nonlinear control have already been applied for the control of IM drives providing better performance than the FOC [10,11]. Various sensorless methods to measure the rotor position of electrical drives with IMs [12–14] have been proposed in order to decrease the hardware complexity and cost and simultaneously to increase higher mechanical robustness and by this a reliability of the drive performance. From the field of applications of nonlinear control methods to the speed and position control of the IMs, a sliding mode control should be mentioned [15–17]. Soft computing methods like fuzzy logic, artificial neural networks, evolutionary algorithms and their combinations have also been applied for the IM drive’s control [18–23]. However, again, their drawbacks consists of powerful real-time calculation processes and, moreover, they can incline to the stability problems of the system (as is often the Z\case of fuzzy control theory methods). An overview of several methods of the IM control at inaccurate determined parameters (the rotor resistance, mains failure and load torque) is given in [24]. It follows that the quality of the IM control depends in principle on the accuracy of the IM model used for all methods. However, the exact model of the drive is not available as at a large volume of production, the exact motor parameters can vary significantly. Several methods have been proposed to solve the problem of IM control with an inaccurate model. A disadvantage of an adaptive version of the sliding mode [25,26] is that it guarantees the robustness only within a range of the uncertainties, and it still suffers from a chattering problem. The predictive control [27–29] can be complemented by a parameter observer to estimate the uncertain model parameters, but the stability for such schemes is usually not guaranteed. A backstepping control method [30], that has appeared recently, allows the design of the control law and the estimation of the motor parameters. However, the proposed method is suitable for a limited set of adaptive parameters only. Summarizing the review, it is clear that the high quality of the IM control should take into account the following criteria: a nonlinear and oscillating character of the IM dynamics, variations of motor parameters, the influence of external disturbances, and a simple implementation of the control algorithm. In this article, a new robust control structure with a reference model is designed to control the angular speed of an IM drive where the system stability is derived on basis of the Lyapunov’s second method [31]. Its main advantage consists of the fact that that the design of the control structure does not require any accurate knowledge of values of the IM parameters. The resulting structure yields optimal dynamic properties in terms of the minimum control deviation and minimum input energy [32] criteria, which are normally used for evaluation of the control efficiency. The proposed method consists of an extension of the control algorithm by aadditional information. This is easily obtained from the system output variable which ensures that the steady-state of the output variable is zero. If the control algorithm for the extended system is designed in such a way that the system would be asymptotically stable with the dynamics prescribed by the reference model, it will reach the goal of control both in the steady and in the transition states. The properties of the proposed control structure have been verified by simulations and experimental measurements on the IM laboratory model. The proposed control structure is considerably simpler than an FOC structure. It is stable, linear, robust, and it has identical dynamical properties without any necessity for knowledge of an exact mathematical model of the IM. These features will increase the implementation potential of this strategy in industrial applications. Appl. Sci. 2021, 11, 6518 3 of 20

The paper is organized as follows: after the ntroduction, the design of the linear model reference control structure is presented in Section2. Section3 describes the mathematical model of IM, its parameters and properties, which in Section4 are further used to design a control for the IM drive’s angular speed control. The proposed control method is verified by simulation in various IM operating states and by experimental measurements on a laboratory model in Sections5 and7. Section6 describes a comparison of the proposed control structure with the vector control structure (FOC). Finally, basic characteristics of the novel control structure are presented in Sections8 and9.

2. Design of the Linear Model Reference Control Structure The desired system dynamics of a controlled system are very often described by a reference model. When this is a linear system, it can be optimally designed using standard methods of the optimal control theory [33]. The state-space reference model for the controlled system with n state variables and p inputs is described by the state equation:

dx M = A x + B w (1) dt M M M

where xM (n × 1) is the state vector of the reference model, AM (n × n) is the state matrix of the reference model, BM (n × p) is the input matrix of the model and w (p × 1) is the vector of the desired values. The controlled system is described in state space as a nonlinear continuous system with parametric and with additive disturbances (or deviations from the reference model) in the form: dx = (A + ∆A)x + (B + ∆B)u + v = A x + B u + (∆Ax + ∆Bu + v) (2) dt M M M M where x (n × 1) is the state vector of the controlled structure, u (p × 1) the vector of input variables, ∆A (n × n), ∆B (n × p)—the matrices of the parametric disturbances and v (n × 1)—the vector of additive disturbances. In this case, the goal of the electric drive control is twofold: (1) To reach the zero state of the state vector x. (2) To reach the zero state of all deviation of state variables from the desired values. For this reason, the deviations of the state vector components from the desired values are suitable to be chosen as the state variables of the controlled system. The system’s stability is investigated with regard to these deviations. Let us calculate the deviation between the reference model and controlled system:

e = xM − x (3)

where e (n × 1) is the vector of deviations between the state variables of the model xM according to (1) and of the system x, defined in Equation (2). By differentiating this vector of deviations one gets: de dx dx = M − (4) dt dt dt After inserting Equations (1) and (2), the expanded system is:

de = A (x − x) + B w − B u − ∆Ax − ∆Bu − v (5) dt M M M M de = A e − B u + f (6) dt M M Appl. Sci. 2021, 11, 6518 4 of 20

where the vector f (n × 1) presents a generalized disturbance vector comprising all para- metrical and additive disturbances affecting the system with regard to its reference model:

f = −∆Ax − ∆Bu − v + BMw (7)

The goal of the controller design is to find such mathematical formulation for deter- mining the input vector u for which the zero solution of the system (6) is asymptotically stable, i.e., lim e = 0. In order to investigate the asymptotic stability of the system (6) t→∞ according to the Lyapunov criterion, the positive definite Lyapunov function is chosen in the weighted quadratic form of the system states:

V = eTPe (8)

The derivation of the Lyapunov function (8) after inserting (1), (2), (3), (6) and per- forming simple modifications is:

dV   h i h i = eT AT P + PA e + 2 fTz − (B u)Tz = eTQe + 2 fTz − (B u)Tz (9) dt M M M M where the vector z (n × 1) is the weighted state deviation vector:

z = Pe (10)

In (8)–(10) the matrix P (n × n) is a symmetric positively definite matrix which satisfies the Lyapunov matrix equation:

T AMP + PAM = −Q (11)

where Q (n × n) is also a symmetric positive definite matrix. Choosing the reference model according to (1) one can avoid solving (11). If the state matrix of the reference model is in the controllability form, then based on the optimal control theory [32] it is possible to determine the elements of the matrix P analytically as follows: Q = −αP (12) where the parameter α allows to set an optimal dynamics of the controlled variable satisfy- ing the criteria of the minimum control deviation and of the minimum input energy. The model dynamics are inversely proportional to the value of the parameter α. The system (6) is asymptotically stable if the derivation of Lyapunov function (9) is a negative definite function. Based on (10)–(12), the derivation of the Lyapunov function is:

dV h i = −αeTPe + 2 fTz − (B u)Tz (13) dt M dV h i = −αeTz + 2 fTz − (B u)Tz (14) dt M Here, the expression eTz = eTPe (where z = Pe) is always positive and as a result, the term −αeTz in (14) is negative. Then, the system (6) will be asymptotically stable, i.e., its derivation will be negative, if for the input u it holds that:

u = Kz (15)

where K (n × n) is an optional constant matrix of positive parameters. The matrix BM is a constant matrix and its influence can be generally included in the values of the optional elements of the matrix K, when modifying Equation (14) with respect to Equation (15). Appl. Sci. 2021, 11, 6518 5 of 20

Then, the second expression on the right side in (14) will be negative if for each component of the vector f the following inequality is met:

Appl. Sci. 2021, 11, x FOR PEER REVIEW kTz ≥ | f | pre i = 1 . . . n 5 (16)of 20 i i

T where ki is the i-th row of the matrix K. Let us note that for a single-input system instead T ofwhere the matrix𝐤 is theK the i-th row row vector of thek matrixis used. K The. Let inequality us note that (16) for is ensured a single-input if the optional system positiveinstead of parameters the matrix in K the the matrix row Kvectorwill havekT is sufficientlyused. The inequality large values. (16) is ensured if the optionalIn order positive to achieve parameters the zero in the control matrix deviation, K will have i.e., thesufficiently difference large between values. the output variablesIn order of the to referenceachieve the model zero control and the deviati controlledon, i.e., system the difference (yM − y) inbetween steady-state, the output the first component of the vector e will be chosen as an integral of this difference: variables of the reference model and the controlled system (𝑦 −𝑦) in steady-state, the first component of the vector e will be chosenZ as an integral of this difference: xext = (yM − y)dt (17) 𝑥 =(𝑦 −𝑦)d𝑡 (17) By introducing this integral, the reference model (1) is extended by the new state variableBy introducingxext. Now, the this extended integral, deviation the referenc vectore model (3) e* will(1) be:is extended by the new state variable xext. Now, the extended deviation vector (3) e* will be:  x  e∗ = ext𝑥 (18) 𝐞∗ = e 𝐞 (18) The referencereference modelmodel extensionextension doesdoes notnot affectaffect thethe stabilitystability ofof thethe designeddesigned controlcontrol structure providedprovided that that the the conditions conditions in Equations in Equations (10)–(12) (10)–(12) are valid are also valid for thealso extended for the system.extended system. Maximum values ofof thethe disturbancedisturbance vectorvector componentscomponents ||ff|| usuallyusually areare physicallyphysically limited.limited. The limitation can be ensured byby aa relevantrelevant increaseincrease ofof thethe valuesvalues ofof thethe optionaloptional parametersparameters inin thethe matrixmatrix KK—the—the conditioncondition (16).(16). The blockblock diagramdiagram ofof thethe designeddesigned controlledcontrolled systemsystem withwith thethe extendedextended referencereference model derived accordingaccording toto thethe aboveabove theorytheory isis shownshown inin FigureFigure1 1..

Figure 1. The structure of the designeddesigned controlledcontrolled system.system.

The following basicbasic featuresfeatures resultresult fromfrom thethe controlcontrol structurestructure design:design: • The control structure design and controller parameters do not depend on accurate • The control structure design and controller parameters do not depend on accurate values of the nonlinear system parameters and thus, they do not depend on an exact values of the nonlinear system parameters and thus, they do not depend on an exact description and the values of the nonlinear controlled system parameters; description and the values of the nonlinear controlled system parameters; • The controlled system dynamics are prescribed by the reference model (1), where • The controlled system dynamics are prescribed by the reference model (1), where both the reference model and the controlled system are of the same order. The linear both the reference model and the controlled system are of the same order. The linear reference model can be designed using standard methods of linear control theory in reference model can be designed using standard methods of linear control theory in order to set an optimal motion dynamics of the controlled system; order to set an optimal motion dynamics of the controlled system; • The reference model dynamics can be set using a single parameter α. By its introduc- • The reference model dynamics can be set using a single parameter α. By its tion, the matrix P satisfies the Lyapunov matrix Equation (11) and according to [32] it introduction, the matrix P satisfies the Lyapunov matrix Equation (11) and does not have to be solved; according to [32] it does not have to be solved; • The stability of the control is ensured by condition (16), in which the positive elements (gains) of the matrix K present the optional parameters at the controller design. • The values of the disturbance vector f components in technical systems usually are physically limited. It means that at sufficiently large values of the matrix K elements, the control deviation of the output (controlled) variable always converges to zero in steady-state. Appl. Sci. 2021, 11, 6518 6 of 20

• The stability of the control is ensured by condition (16), in which the positive elements (gains) of the matrix K present the optional parameters at the controller design. • The values of the disturbance vector f components in technical systems usually are physically limited. It means that at sufficiently large values of the matrix K elements, the control deviation of the output (controlled) variable always converges to zero in steady-state.

3. Mathematical Model of Induction Motor The complete mathematical model of the IM presents a nonlinear dynamic system of the 5th order. In the following, it is assumed that the current control loops are involved in circuits of a standard frequency converter. Due to their high dynamics, their influence is neglected. Then, the current–flux model of the IM presents the 3rd order system. Its current–flux model [8] is described in the {x, y} reference system by components of the stator currents and rotor fluxes: dψ 2x = −ω ψ + L ω i + (ω − ω)ψ (19) dt g 2x m g 1x 1 2y

dψ2y = −ω ψ + L ω i − (ω − ω)ψ (20) dt g 2y m g 1y 1 2x p   Tmech = Lm ψ2xi1y − ψ2yi1x (21) L1 J dω m = T − T (22) p dt mech L where the notation of the parameters and variables is as follows:

i1x, i1y components of the stator current space vector i1; ωm mechanical angular speed of the rotor; ω1 angular frequency of the stator voltage; ω2 slip angular speed ω2 = ω1–ωm; r2 rotor phase resistance; r1 stator phase resistance; ψ2x, ψ2y stator and rotor magnetic flux components; Lm main inductance; L1,L2 leakage inductances; Ωg constant ωg = R2/L2; Tmech mechanical motor torque; TD dynamic motor torque; p number of pole pairs; J moment of inertia; TL load torque. From Equations (19)–(22) it is obvious that the IM presents a strongly nonlinear higher-order controlled system of an oscillating character. The values of the motor parameters used for simulation are specified in Table1.

Table 1. Induction motor parameters for modeling and experimentation.

PN = 3 kW U1N = 220 V I1N = 6.9 A 2 J = 0.1 kgm r1 = 1.8 Ω R1 = 2/3 r1 = 1.2 Ω nN = 1430 rev./min r2 = 1.85 Ω R2 = 2/3 r2 = 1.23 Ω −1 p = 2 L1 = L2 = 0.2106 H ωg = R2/L2 = 5.84 s TN = 20 Nm

The characteristics in Figure2 show the motor model torque and angular speed responses at the step change on the motor inputs in time t = 0 s, when the motor is Appl. Sci. 2021, 11, 6518 7 of 20

supplied from a current converter in which the current loop time constant already is Appl. Sci. 2021, 11, x FOR PEER REVIEWcompensated. Let us note that according to (19)–(22) in this case, the values of the current7 of 20 Appl. Sci. 2021, 11, x FOR PEER REVIEW 7 of 20 vector components of the IM model used for simulation are i1x = 0 A, i1y = 15 A. At this supply, the angular speed reaches the value of ω1 = 200 rad/s.

Figure 2. TimeTime courses courses of of the the IM IM model model torque torque and and sp speedeed at at motor motor starting starting when when supplied supplied by by the the current vector. current vector.

4. Design Design of of the the IM IM Drive Drive Control Control The objective of the IM drive control cons consistsists of control of its angular speed wherewhere thethe motor dynamics are prescribed by a linearlinear referencereference modelmodel toto bebe designed.designed. For thethe control structure design according toto Section2 2,, it it isis notnot necessarynecessary toto knowknow anan exactexact modelmodel of thethe controlled controlled system. system. The The electromagnetic electromagnetic phenomena phenomena within within the motor the are motor consider- are considerablyably faster than faster the than mechanical the mechanical phenomena phen onomena the motor on the shaft motor and shaft the and electrical the electrical circuits circuitscomprise comprise all nonlinearities. all nonlinearities. The electromagnetic The electromagnetic phenomena dynamicsphenomena are replaceddynamics by theare replaced1st order proportionalby the 1st order system proportional and the mechanical system phenomena and the mechanical are similarly phenomena replaced by theare 1st order system. The simplified model of the IM presents a nonlinear dynamic system of similarly replaced by the 1st order system. The simplified model of the IM presents a the 2nd order as shown in Figure3. nonlinear dynamic system of the 2nd order as shown in Figure 3.

Figure 3. Simplified model of the IM presenting a 2nd order nonlinear system. Figure 3. Simplified model of the IM presenting a 2nd order nonlinear system. The static nonlinearity in generating the mechanical torque of the T motor is usu- The static nonlinearity in generating the mechanical torque of themech Tmech motor is The static nonlinearity in generating the mechanical torque of the Tmech motor is usuallyally described described by the by Kloss the formula Kloss withformula the 1st with order the dynamics 1st order and thedynamics electromechanical and the usuallylinear subsystem described is describedby the Kloss by the equationformula ofwith the torquethe 1st equilibrium order dynamics on the drive and shaft. the electromechanical linear subsystem is described by the equation of the torque Let us select the state variables of the IM as shown in Figure3: x1 = y = ωm (the rotor equilibrium on the drive shaft. angular speed) and x2 = dx1/dt = dωm/dt (the rotor acceleration corresponding to its motor Let us select the state variables of the IM as shown in Figure 3: x1 = y = ωm (the rotor Let us select the state variables of the IM as shown in Figure 3: x1 = y = ωm (the rotor dynamic torque TD). angular speed) and x2 = dx1/dt = dωm/dt (the rotor acceleration corresponding to its motor angularAs thespeed) IM isand considered x2 = dx1/d ast = a d nonlinearωm/dt (the 2nd rotor order acceleration system, according corresponding to [32] to its its required motor dynamic torque TD). dynamicdynamics torque are prescribed TD). by the 2nd order linear reference model: As the IM is considered as a nonlinear 2nd order system, according to [32] its required dynamics are prescribed.  " by the 2nd order# linear reference" # model: required dynamics are xprescribed1M by0 the 2nd 1 orderx 1linearM reference0 model: . = α2 + α2 w (23) x2M − 01−α x2M 0 𝑥 2 01 𝑥 0 2 𝑥= α 𝑥+α𝑤 (23) 𝑥 = α 𝑥 +α 𝑤 (23) 𝑥 − −α 𝑥 2 2 where the state variables x1M and x2M prescribe the dynamic behavior of the where the state variables x1M and x2M prescribe the dynamic behavior of the corresponding state variables of the controlled system, i.e.,: x1M = yM = ωmM and x2m = corresponding state variables of the controlled system, i.e.,: x1M = yM = ωmM and x2m = dx1M/dt = dωmM/dt. dx1M/dt = dωmM/dt. The optimal dynamics of the controlled variable can be set by the optional positive parameter α in the reference model (23). Appl. Sci. 2021, 11, 6518 8 of 20

where the state variables x1M and x2M prescribe the dynamic behavior of the correspond- ing state variables of the controlled system, i.e., x1M = yM = ωmM and x2m = dx1M/dt = dωmM/dt. The optimal dynamics of the controlled variable can be set by the optional positive parameter α in the reference model (23). To ensure the zero control deviation of the angular speed of the controlled system and the reference model at steady-state, the reference model is extended by a new state variable xext. To create a deviation between the state variable of the reference model and the system according to (17), the following relation is used: Z Z xext = (x1M − x1)dt = (yM − y)dt (24)

Now the extended vector of the deviation will be in the form:   xext ∗ e =  e1  (25) e2

This means the extended reference model of the IM drive presents the 3rd order system. According to the optimization theory [32], its state matrices are in the form:

 0 1 0   0  AM =  0 0 1 ; bM =  0  (26) α3 3α2 3α α2 − 2 − 2 − 2 2

where bM is the input vector in the case of the one-input model. The optimal matrices P and Q, that satisfy the Lyapunov matrix Equation (11), are:

 α5 4 α3  2 α 2 P =  α4 5α3 3α2 ; Q = −αP (27)  2 2  α3 3α2 3α 2 2 2 The IM as a controlled system will follow the extended reference model with lim e∗ = t→∞ 0, i.e., the controlled system will be asymptotically stable, if the input u is calculated using (15):   z1   u = k1 k2 k3  z2  (28) z3 According to (16), the elements for the vector kT presenting gains must be positive and large enough to ensure the asymptotic stability of the controlled system. On the other hand, the value of the parameters in the vector kT is limited by physical constraints in the controlled system. In the case of an electric drive, it is the electric motor current, the dynamics of real power converter, etc. According to (10), the components of the vector z are:

z1 = p11xext + p12e1 + p13e2 (29)

z2 = p21xext + p22e1 + p23e2 (30)

z3 = p31xext + p32e1 + p33e2 (31) The elements of the matrix P are known from evaluating Equation (27). The resulting control scheme for controlling the angular speed of the IM drive in accordance with the derived control structure presented in Figure1 is shown in Figure4. Appl. Sci. 2021 11 Appl. Sci. 2021,, 11,, 6518x FOR PEER REVIEW 99 ofof 2020

Figure 4. IM drive block diagram.

In order for the IMIM toto operateoperate inin aa stablestable partpart ofof itsits torquetorque characteristic,characteristic, thethe blockblock diagram in Figure4 4 has has beenbeen completedcompleted byby aa conversionconversion ofof thethe actionaction variablevariable to to slipslip that that is completed by its limitation (like it is case in aa realreal motor).motor). The The slip slip limiter, however, does not change the validity of the structure shownshown inin FigureFigure1 1.. TheThe limitationlimitation of of the the state state variables of the drive system (the current and torque) can be realized inside the reference model without any affecting the control loop stability. This is due to the fact thatthat thethe reference model is linear and the limitationslimitations do not influenceinfluence thethe positionposition ofof thethe linearlinear system poles. system poles. 5. Simulation of the IM Drive in Basic Operation States 5. Simulation of the IM Drive in Basic Operation States The verification of the control structure properties was carried out by simulation in MATLABThe verification with the of motor the control parameters structure specified properties in Table was1 carried. The desired out by simulation value of the in MATLAB with the motor parameters specified in Table 1. The desired value of the mechanical angular speed of the IM was set in the reference model to ωmref = 150 rad/s mechanical angular speed of the IM was set in the reference model to ωmref = 150 rad/s and and the stator current vector components were i1x = 0 A, i1y = 20 A. the statorThe proposedcurrent vector controller components structure were is i based1x = 0 A, on i1y Lyapunov’s = 20 A. second method, which definesThe the proposed range of itscontroller optional structure parameters, is base i.e.,d values on Lyapunov’s of the elements second of the method, matrix whichP and thedefines positive the range optional of vectorits optional of parameters parameters,kT fori.e., which values the of controlled the elements system of the is stable.matrix P and Thethe positive elements optional of the matrix vectorP areof computedparameters from kT for the which Lyapunov the matrixcontrolled Equation system (11), is wherestable. the positive definite matrix Q must be chosen. The values of elements of the matrix Q (andThe thus elements of the of matrix the matrixP) influence P are computed the speed from of the decreasing Lyapunov the matrix Lyapunov Equation function (11), (Equationwhere the (8)),positive which definite means matrix a deceleration Q must be rate chosen. of the The control values deviation of elementse. The of thesystem matrix (6) willQ (and be stablethus of for the any matrix arbitrarily P) influence selected the elements speed of of decreasing the matrix theQ Lyapunovwhen fulfilling function the condition(Equation of(8)), its which positive means definiteness. a decelera However,tion rate of if wethe choosecontrol thedeviation reference e. The model system which (6) ensureswill be thestable optimal for any dynamic arbitrarily properties selected of the elements controlled of the system matrix according Q when to thefulfilling criteria the of minimalcondition control of its positive deviation definiteness. and minimum Howe inputver, energy, if we choose then it isthe possible reference to avoidmodel solving which theensures Lyapunov the optimal matrix dynamic Equation properties (11), because of the elementscontrolled of system the matrix accordingP can be to determined the criteria analytically,of minimal control based on deviation Equation and (12) minimum from the matrix input (27)energy, (see [then32]): it is possible to avoid solving the Lyapunov matrix Equation (11), because the elements of the matrix P can be   determined analytically, based on Equation1562.5 (12) 625 from 62.5 the matrix (27) (see [32]): P = 625 312.5 37.5 (32)  1562.5 625 62.5 𝐏=62.5625 37.5 312.5 7.5 37.5 (32) 62.5 37.5 7.5 The designed controlled structure also includes the controller gain, which is repre- sentedThe by designed optional elementscontrolled of thestructure vector kalsoT in includes Equation the (28) controller for calculating gain, the whichinput uis. T Theserepresented elements by mustoptional be positive elements and of their the valuevector must k in be Equation chosen in (28) such for a way calculating that during the theinput operation u. These the elements IM must must not reachbe positive the current and th limit.eir value To fulfil must all be requirements, chosen in such the vectora way kthatT in during (28) has beenthe operation set to the valuethe kIMT =must [0.0031 not 0.0019 reach 0.00038]. the current These valueslimit. wereTo obtainedfulfil all T T byrequirements, a gradual increasingthe vector thek in values (28) has of individualbeen set to elementsthe value ofk the= [0.0031 vector 0.0019kT so 0.00038]. that the These values were obtained by a gradual increasing the values of individual elements of Appl. Sci. 2021, 11, x FOR PEER REVIEW 10 of 20

the vector kT so that the dynamics of decreasing the Lyapunov function would be the largest one at considering physical limits of the control system. An advantage of the described settings of parameters against the classical approaches consists of the fact they can be determined from measurements of responses Appl. Sci. 2021, 11, 6518 and form the knowledge of the system limits without knowledge of its precise10 of 20 parameters (e.g., of the rotor resistance, moment of inertia, etc.). The dynamics of the reference model setting (using the parameter α) must take into account the physical properties of the drive, which in our case are presented by the currentdynamics (torque) of decreasing overloading the Lyapunov of the IM. functionThe dynamic would properties be the largest of the one IM at are considering described byphysical unit step limits characteristics of the control shown system. in Figure 2. The value of the parameter α is inversely proportionalAn advantage to the of model the described dynamics. settings This ofmeans parameters that based against on thethe classical Shannon–Kotelnik approaches theoremconsists offor the stabilizing fact they the can angular be determined speed of from the measurements IM within 1 s, of the responses value of andthe formoptional the parameterknowledge is of chosen the system as α = limits 5. without knowledge of its precise parameters (e.g., of the rotorThe resistance, operation moment cycle of inertia,the IM etc.).drive consists of three phases: starting the IM drive, runningThe at dynamics a constant of speed, the reference and stopping model (Figure setting (using5a). An the external parameter (additive)α) must disturbance take into account the physical properties of the drive, which in our case are presented by the current of the IM drive—the load torque TL = TN = 20 Nm—was also introduced in time t = 3 s. This(torque) is valid overloading for all simulation of the IM. experiments. The dynamic properties of the IM are described by unit step characteristics shown in Figure2. The value of the parameter α is inversely proportional The time course of the angular speed ωm (the output variable) for the considered operationto the model cycle dynamics. is shown in This Figure means 5. It that is obvious based onthat the the Shannon–Kotelnik angular speed of the theorem IM drive for stabilizing the angular speed of the IM within 1 s, the value of the optional parameter is practically tracks the reference angular speed prescribed by the reference model (Figure chosen as α = 5. 5a) during the entire operation cycle, even during the step disturbance TL as is shown in The operation cycle of the IM drive consists of three phases: starting the IM drive, Figure 5b. This is also observed on changes of the variable dωm/dt ≈ TD (corresponding to running at a constant speed, and stopping (Figure5a). An external (additive) disturbance the acceleration) in Figure 5b, where the considered disturbance torque TL is settled with of the IM drive—the load torque T = T = 20 Nm—was also introduced in time t = 3 s. high dynamics. This experiment verifiesL N that the proposed control structure is invariant This is valid for all simulation experiments. against the additive disturbances.

(a) (b)

Figure 5. Time courses of the IM control during the operation cycle: ( a) prescribed prescribed by the reference model, ( b) achieved by the motor at its starting andand loading.loading.

The robustness time course of of the the proposed angular control speed ω structurem (the output has been variable) verified for at the the considered change of theoperation two most cycle important is shown inparameters Figure5. Itof is the obvious controlled that the system angular significantly speed of the affecting IM drive its properties.practically tracksIn the thecase reference of IM, they angular are the speed rotor prescribed resistance by R the2 and reference the moment model of (Figure inertia5 aJ. ) duringIn the the majority entire operation of the control cycle, systems even during for IM, the the step precise disturbance speed controlTL as isdepends shown on in knowledgeFigure5b. This of the is also rotor observed resistance on changesvalue that of the usually variable is identified d ωm/dt ≈ byT Dvarious(corresponding types of observers.to the acceleration) Figure 6 inshows Figure the5b, time where response the considered when the disturbance rotor resistance torque RT2L wasis settled increased with tohigh twice dynamics. its original This value. experiment The dynamics verifies thatof the the controlled proposed system control are structure almost isidentical invariant to theagainst dynamics the additive of the disturbances.reference model during the entire operation cycle (Figure 5b). The time Thecourses robustness of the variables of the proposed in Figure control 6 confirm structure the controlled has been system verified robustness at the change for the of consideredthe two most variation important of the parameters parameter ofR2. the controlled system significantly affecting its properties. In the case of IM, they are the rotor resistance R2 and the moment of inertia J. In the majority of the control systems for IM, the precise speed control depends on knowledge of the rotor resistance value that usually is identified by various types of observers. Figure6 shows the time response when the rotor resistance R2 was increased to twice its original value. The dynamics of the controlled system are almost identical to the dynamics of the reference model during the entire operation cycle (Figure5b). The time courses of the variables in Figure6 confirm the controlled system robustness for the considered variation of the parameter R2. Appl. Sci. 2021, 11, x FOR PEER REVIEW 11 of 20 Appl.Appl. Sci.Sci. 20212021,, 1111,, 6518x FOR PEER REVIEW 1111 ofof 2020

Figure 6. The effect of change of rotor resistance R2 = 2 R2N on control dynamics at motor starting R2 R2N andFigure loading. 6. The effect of change of rotor resistance R 2 == 22 R 2 Nonon control control dynamics dynamics at atmotor motor starting starting and loading. Another parameter that considerably influences the motor dynamics is the moment Another parameterparameter thatthat considerablyconsiderably influences influences the the motor motor dynamics dynamics is theis the moment moment of of inertia J applied to the motor shaft. Its value can vary during the operation cycle, inertiaof inertiaJ applied J applied to the to motorthe motor shaft. shaft. Its value Its value can vary can duringvary during the operation the operation cycle, whichcycle, which is the case for many industrial applications of the IM drives (e.g., winding iswhich the caseis the for case many for industrial many industrial applications applications of the IM of drives the IM (e.g., drives winding (e.g., machines, winding machines, robotic and transportation systems). As it follows up from the time courses in roboticmachines, and robotic transportation and transportation systems). As systems). it follows As up it fromfollows the up time from courses the time in Figure courses7a,b, in Figure 7a,b, the increase of the moment of additional inertia to a half/or double that of the theFigure increase 7a,b, the of theincrease moment of the of moment additional of inertiaadditional to a inertia half/or to doublea half/or that double of the that motor’s of the motor’s nominal moment of inertia does not influence the motor dynamics. This fact nominalmotor’s momentnominal ofmoment inertia doesof inertia not influence does no thet influence motor dynamics. the motor This dynamics. fact presents This a veryfact presents a very significant advantage in the control of nonlinear systems with an significantpresents a advantagevery significant in the control advantage of nonlinear in the systemscontrol withof nonlinear an oscillating systems character. with The an oscillating character. The effect of the load torque TL is again compensated with high effectoscillating of the character. load torque TheTL effectis again of compensatedthe load torque with TL high is again dynamics compensated also at a significantwith high dynamics also at a significant change of the motor inertia, which confirms the high changedynamics of thealso motor at a significant inertia, which change confirms of the the motor high robustnessinertia, which against confirms the parameter the high robustness against the parameter variations in the proposed control structure. variationsrobustness in against the proposed the parameter control variatio structure.ns in the proposed control structure.

(a) (b) (a) (b) FigureFigure 7. 7.TheThe effect effect of ofchange change of of additional additional moment moment of of inertia inertia to to the the value: value: (a ()a )J J= =0.5 0.5 JNJ;N (;(b)b J) =J 2= J 2N JonN on control control dynamics dynamics at at motorFigure starting 7. The and effect loading. of change of additional moment of inertia to the value: (a) J = 0.5 JN; (b) J = 2 JN on control dynamics at motormotor startingstarting andand loading.loading. TheThe simulations simulations have have confirmed confirmed that that the the pr proposedoposed controller controller ensures ensures a high a high quality quality The simulations have confirmed that the proposed controller ensures a high quality angularangular speed speed control control of ofthe the IM IM drive drive during during the the entire entire operation operation cycle cycle according according to tothe the angular speed control of the IM drive during the entire operation cycle according to the prescribedprescribed dynamics. dynamics. Moreover, Moreover, the the controller controller satisfies satisfies all all basic basic control control objectives: objectives: the the prescribed dynamics. Moreover, the controller satisfies all basic control objectives: the drivedrive is isinvariant invariant against against the the disturbances disturbances (the (the load load torque) torque) and and robust robust against against changes changes of of drive is invariant against the disturbances (the load torque) and robust against changes of thethe motor motor parameters parameters (the (the rotor rotor resistance resistance an andd additional additional moment moment of ofinertia inertia connected connected to to the motor parameters (the rotor resistance and additional moment of inertia connected to thethe rotor rotor shaft). shaft). the rotor shaft). Appl. Sci. 2021, 11, x FOR PEER REVIEW 12 of 20

Appl. Sci. 2021, 11, 6518 12 of 20

6. Comparison of the Proposed Control Structure with the Vector Control Structure 6. ComparisonThe vector of drive the Proposed control of Control the IM Structure presents with one the of Vectorthe most Control common Structure control methodsThe vectorin technical drive controlpractice of today. the IM Its presents specific onedynamic of the properties most common depend control on methodsthe used inmodification technical practice and are today. described Its specific by a large dynamic amount properties of the literature, depend on e.g., the [6,10]. used modification At first, the andproposed are described control properties by a large amountwill be compared of the literature, with those e.g., [of6, 10the]. Atvector first, control the proposed on the controlbasis of propertiesa comparison will of be their compared structures. with thoseSecondly, of the the vector performance control on comparison the basis of of a these com- parisontwo controlled of their structures structures. will Secondly, be given. the performance comparison of these two controlled structuresThe basic will bestructures given. of both control methods are shown in Figures 8–10. In both structures,The basic the upper structures control of bothlevel controlcontrols methods the desired are mechanical shown in Figuresspeed ω8–m 10of. the In drive, both structures,and eventually the upper it sets control the excitation level controls magnet theic desiredflux magnitude mechanical of speedthe motorωm ofthrough the drive, its andcurrent eventually components it sets in the the excitation rectangular magnetic rotating flux reference magnitude frame of the{x, motory} rotating through at the its currentangular componentsspeed ω1. in the rectangular rotating reference frame {x, y} rotating at the angular speed ω1.

Appl. Sci. 2021, 11, x FOR PEER REVIEW (a) (b) 13 of 20

Figure 8. The upper control level: ( a) of the designed control structure; ((b)) ofof thethe vectorvector controlcontrol structure.structure.

At the upper control level (Figure 8a,b), there are two basic differences: 1. The first difference relates to the control structures: both structures are linear. They differ in their internal interconnection and by the control structure order; 2. The second difference consists of the type of the output action variable. In the vector control, they are variable setpoints of the stator current components i1x, i1y. In the proposed control structure the angular speed of the stator current vector presents the action variable and, in principle, the components i1x, i1y are kept constant. The lower control level (Figure 9) contains the controllers of the stator current components i1x, i1y and the transformations between the stator system {a, b, c} and the reference frame {x, y}. At the lower control level, the two structures being compared are practically identical. A significant difference consists of the stator currents transformations into the reference frame {x, y}. While in the vector control, this system most often tied to the position of the rotor flux vector γ, in the proposed control structure it is firmly tied to the Figure 9. The lower control level of the proposed control structure, where γ1 is the instantaneous positionFigure 9. ofThe the lower stator control current level vectorof the proposed i1 (which control in the structure, fact presents where γan1 is integralthe instantaneous from the position of the stator voltage vector and u , u are the components of the stator voltage space angularposition speedof the stator ω1). In voltage the case vector of andthe vectoru1x1,x u1y 1 areycontrol the components (Figure 10), of the statorproblems voltage arise space when vectorusingvector uthe uxx.. Park transformation. It requires the measurability of the rotor flux position, the dependence of this position estimation on the unknown and variable value of the rotor At the upper control level (Figure8a,b), there are two basic differences: resistance, problems at low mechanical angular speeds, etc. These uncertainties can 1.significantlyThe first affect difference the quality relates of to the the control. control structures: both structures are linear. They differ in their internal interconnection and by the control structure order; 2. The second difference consists of the type of the output action variable. In the vector control, they are variable setpoints of the stator current components i1x, i1y. In the proposed control structure the angular speed of the stator current vector presents the action variable and, in principle, the components i1x, i1y are kept constant. The lower control level (Figure9) contains the controllers of the stator current compo- nents i1x, i1y and the transformations between the stator system {a, b, c} and the reference

Figure 10. The lower level of the vector control structure, where γ is the actual position of the rotor flux vector.

In the proposed control structure, the value of the stator angular velocity ω1 inputs from the higher control level. Its value is always precisely known which guarantees the accuracy of this transformation and its independence from the current states or other parameters of the IM. A basic comparison of the drive’s behavior with closed control loops for exactly known IM parameters is shown in Figure 11. The first two figures from the left compare the response of the closed control loop to a step of the controlled system setpoint and to a disturbance. Here, the motor is starting, running to the angular speed ωm = 150 rad/s, then in time t = 3 s it is loaded by the nominal load torque TL = TN = 20 Nm and finally, it is stopping. The next two figures show a detail of the course of the first and second state variables of the controlled system, i.e., the angular velocity ωm and the dynamic torque ωm/dt when the IM drive is loaded by the nominal load torque in time t = 3 s. In the case of vector control, the dynamic torque of the motor at its starting and stopping is higher. Its course has a non-linear shape, while in the proposed structure, the IM drive behaves as a linear 2nd order system. The time response to the load torque in the dynamic state is practically the same, which is shown in Figure 11 in details of individual courses of the motor state variables. Appl. Sci. 2021, 11, x FOR PEER REVIEW 13 of 20

Appl. Sci. 2021, 11, 6518 13 of 20

Figure 9. The lower control level of the proposed control structure, where γ1 is the instantaneous positionframe { xof, ythe}. Atstator the voltage lower controlvector and level, u1x, theu1y are two the structures components being of the compared stator voltage are practically space vectoridentical. ux.

Figure 10. The lower level of the vector control structure, where γ is the actual position of the rotor Figure 10. The lower level of the vector control structure, where γ is the actual position of the rotor fluxflux vector. A significant difference consists of the stator currents transformations into the refer- In the proposed control structure, the value of the stator angular velocity ω1 inputs ence frame {x, y}. While in the vector control, this system most often tied to the position from the higher control level. Its value is always precisely known which guarantees the of the rotor flux vector γ, in the proposed control structure it is firmly tied to the position accuracy of this transformation and its independence from the current states or other of the stator current vector i1 (which in the fact presents an integral from the angular parameters of the IM. speed ω1). In the case of the vector control (Figure 10), the problems arise when using the ParkA transformation. basic comparison It requires of the drive’s the measurability behavior with of the closed rotor control flux position, loops for the exactly depen- knowndence of IM this parameters position estimation is shown onin Figure the unknown 11. The and first variable two figures value from of the the rotor left resistance, compare theproblems response at low of the mechanical closed control angular loop speeds, to a st etc.ep of These the controlled uncertainties system can significantly setpoint and affect to a disturbance.the quality of Here, the control. the motor is starting, running to the angular speed ωm = 150 rad/s, then in time t = 3 s it is loaded by the nominal load torque TL = TN = 20 Nm and finally, it is In the proposed control structure, the value of the stator angular velocity ω1 inputs stopping.from the higher control level. Its value is always precisely known which guarantees the accuracyThe next of this two transformation figures show anda detail its independence of the course from of the currentfirst and states second or otherstate variablesparameters of ofthe the controlled IM. system, i.e., the angular velocity ωm and the dynamic torque ωm/dtA when basic the comparison IM drive is of loaded the drive’s by the behavior nominal withload torque closed in control time tloops = 3 s. forIn the exactly case ofknown vector IM control, parameters the dynamic is shown torque in Figure of the 11 motor. The first at its two starting figures and from stopping the left is compare higher. Itsthe course response has of a thenon-linear closed controlshape, while loop to in a the step proposed of the controlled structure, system the IM setpoint drive behaves and to asa disturbance.a linear 2nd order Here, system. the motor The is time starting, response running to the to theload angular torque speedin the dynamicω = 150 state rad/s, is Appl. Sci. 2021, 11, x FOR PEER REVIEW m 14 of 20 practicallythen in time thet = same, 3 s it iswhich loaded is shown by the nominalin Figure load 11 in torque detailsTL of= TindividualN = 20 Nm courses and finally, of the it motoris stopping. state variables.

Figure 11.11. ComparisonComparison of of dynamic dynamic properties properties of of the the controlled controlled drive dr withive with the inductionthe induction motor: motor: the proposed the proposed and the and vector the controlvector control structures. structures.

FigureThe next 12 twocompares figures the show effect a detail of an ofunknown the course change of the of first the and rotor second resistance state R variables2 on the dynamicsof the controlled of closed system, control i.e., loops the angular for two velocity cases: forωm theand designed the dynamic control torque structureωm/dt and when for the vector IM drive control. is loaded While by the the dynamics nominal of load the torque motor inwith time thet =proposed 3 s. In the control case ofstructure vector remaincontrol, almost the dynamic unchanged, torque in of the the case motor of atvector its starting control, and the stopping change of is higher.the parameter Its course R2 duringhas a non-linear starting and shape, stopping while inleads the proposedto a visible structure, destabilization the IM drive of the behaves motor as dynamic a linear torque,2nd order both system. during The the time starting response and especial to the loadly during torque its in loading. the dynamic For the state reduced is practically value of the rotor resistance (R2 = 0.5 R2N), the dynamics of the vector control are worse than for the proposed control.

Figure 12. Comparison of the proposed and the vector control structures for change of the rotor resistance to the values R2 = 0.5 R2N and R2 = 2 R2N.

The effect of an unknown change of the moment of inertia on the motor shaft on the dynamics of the closed control loops is compared in graphs in Figure 13. While the dynamics of the proposed control structure during the motor starting and stopping behaves like a linear 2nd order system, the vector control significantly changes its dynamic properties (mainly the settling time). From the point of view of superior control, one has to look at the vector control as on a system with variable dynamics (having variable parameters), which complicates the design of the IM control. The dynamics of the load torque compensation in the case with a increased moment of inertia J = 2 JN in time t = 3 s has better quality at the vector control. In the proposed control structure, the change of the moment of inertia does not affect the quality of its compensation. Appl. Sci. 2021, 11, x FOR PEER REVIEW 14 of 20

Appl. Sci. 2021, 11, 6518 14 of 20

Figure 11. Comparison of dynamic properties of the controlled drive with the induction motor: the proposed and the vector control structures. the same, which is shown in Figure 11 in details of individual courses of the motor state variables. Figure 12 compares the effect of an unknown change of the rotor resistance R2 on the Figure 12 compares the effect of an unknown change of the rotor resistance R2 on the dynamics of closed control loops for two cases:cases: for the designed control structure and for the vector control. While the dynamics of thethe motor with the proposedproposed controlcontrol structurestructure remain almost unchanged, in the case of vector control, the change of the parameter R2 remain almost unchanged, in the case of vector control, the change of the parameter R2 during startingstarting and and stopping stopping leads leads to ato visible a visible destabilization destabilization of the of motor the dynamicmotor dynamic torque, bothtorque, during both theduring starting the starting and especially and especial duringly during its loading. its loading. For the For reduced the reduced value ofvalue the 2 2N rotorof the resistancerotor resistance (R2 = 0.5(R R= 20.5N), R the), dynamics the dynamics of the of vector the vector control control are worseare worse than than for thefor proposedthe proposed control. control.

Figure 12. Comparison of the the proposed proposed and and the the vector vector control control struct structuresures for for change change of of the the roto rotorr resistance resistance to tothe the values values R2 = 0.5 R2N and R2 = 2 R2N. R2 = 0.5 R2N and R2 = 2 R2N.

The effect of of an an unknown unknown change change of of the the moment moment of ofinertia inertia on onthe the motor motor shaft shaft on the on thedynamics dynamics of the of theclosed closed control control loops loops is co ismpared compared in graphs in graphs in Figure in Figure 13. 13While. While the thedynamics dynamics of the of theproposed proposed control control structur structuree during during the the motor motor starting starting and stoppingstopping behaves likelike a a linear linear 2nd 2nd order order system, system, the vector the vector control control significantly significantly changes itschanges dynamic its propertiesdynamic properties (mainly the(mainly settling the time).settling Fromtime). the From point the ofpoint view of ofview superior of superior control, control, one hasone tohas look to atlook the at vector the vector control control as on aas system on a withsystem variable with variable dynamics dynamics (having variable(having parameters),variable parameters), which complicates which complicates the design the of design the IM of control. the IM Thecontrol. dynamics The dynamics of the load of torquethe load compensation torque compensation in the case in with the acase increased with a momentincreased of moment inertia J =of 2 inertiaJN in time J = 2t =JN3 in s hastime better t = 3 s quality has better at the quality vector at control. the vector In the contro proposedl. In the control proposed structure, control the structure, change of the momentchange of of the inertia moment does of not inertia affect does the qualitynot affect of itsthe compensation. quality of its compensation. The comparison between the proposed and the vector control structures shows that the vector control structure exhibits a nonlinear behavior, while the proposed structure always shows linear transient responses. In terms of the dynamics, the responses of these structures are very similar to the basic setting of the parameters of the controlled system. However, the robustness of the proposed control structure is significantly better with changes of unknown parameters. Appl. Sci. 2021, 11, x FOR PEER REVIEW 15 of 20

Appl. Sci. 2021, 11, 6518 15 of 20 Appl. Sci. 2021, 11, x FOR PEER REVIEW 15 of 20

Figure 13. Comparison of proposed control structure and vector control for moment of inertia changed to J = 0.5 JN and J = 2 JN.

The comparison between the proposed and the vector control structures shows that the vector control structure exhibits a nonlinear behavior, while the proposed structure always shows linear transient responses. In terms of the dynamics, the responses of these structures are very similar to the basic setting of the parameters of the controlled system. Figure 13. Comparison of proposed control structure and vector control for moment of inertia changed to J = 0.5 JN and J = 2 Figure 13. Comparison of proposedHowever, control the structurerobustness and of vector the proposed control for momentcontrol structure of inertiachanged is significantly to J = 0.5 JbetterN and with JN. J = 2 JN. changes of unknown parameters. The comparison between the proposed and the vector control structures shows that 7.7. ExperimentalExperimental VerificationVerification of of the the IM IM Drive Drive Speed Speed Control Control on on a a Laboratory Laboratory Model Model the vector control structure exhibits a nonlinear behavior, while the proposed structure The linear model reference control structure is verified on the three-phase induction alwaysThe shows linear linear model transient reference responses. control structure In terms ofis verifiedthe dynamics, on the thethree-phase responses induction of these motor TYPE 1AV3104B (400 V, 2.2 kW, 1465 RPM, 14.3 Nm) driven by the VQFREM 400 structuresmotor TYPE are 1AV3104B very similar (400 to V,the 2.2 basic kW, setting 1465 RPM,of the 14.3 parameters Nm) driven of the by controlled the VQFREM system. 400 004–4MA power converter (400 V, 11 A), manufactured by VONSCH Co, Brezno, Slovakia. However,004–4MA thepower robustness converter of the(400 proposed V, 11 A) cont, manufacturedrol structure isby significantly VONSCH Co,better Brezno, with The structure of the experimental setup is shown in Figure 14. changesSlovakia. of The unknown structure parameters. of the experimental setup is shown in Figure 14.

7. Experimental Verification of the IM Drive Speed Control on a Laboratory Model The linear model reference control structure is verified on the three-phase induction motor TYPE 1AV3104B (400 V, 2.2 kW, 1465 RPM, 14.3 Nm) driven by the VQFREM 400 004–4MA power converter (400 V, 11 A), manufactured by VONSCH Co, Brezno, Slovakia. The structure of the experimental setup is shown in Figure 14.

FigureFigure 14.14. TheThe structurestructure ofof thethe experimentalexperimental setupsetup withwith VQFREMVQFREM 400400 004–4MA004–4MA powerpower converter.converter.

TheThe controlcontrol boardboard ofof thethe powerpower converterconverter hashas twotwo processorsprocessors DSPDSP TMS320F2406.TMS320F2406. TheThe firstfirst DSPDSP processesprocesses signalssignals fromfrom thethe current,current, DC-link,DC-link, andandposition positionsensors. sensors. ItIt alsoalso performsperforms thethe ClarkeClarke andand ParkPark transformationstransformations andand usesuses PIPI typetype controllerscontrollers withwith aa samplesample µ i i x y timetime ofof 5050 µss toto controls controls the the stator stator currents currents 1ix1xand and 1iy1y inin thethe {{x,, y}} referencereference frame.frame. TheThe secondsecond DSPDSP implements implements the the speed speed control contro and communicatesl and communicates with a higher-level with a higher-level controller Figure 14. The structure of the experimental setup with VQFREM 400 004–4MA power converter. andcontroller operator and console. operator The console. speed The control speed sample control time sample is 1 ms.time Theis 1 fastms. The current fast controlcurrent loop is written in assembler language and the rest of the code is written in C. The Texas controlThe loop control is written board inof assembler the power language converte anr dhas the two rest processors of the code DSP is written TMS320F2406. in C. The Instruments software development toolchain is used for programming DSPs. The setup TheTexas first Instruments DSP processes software signals development from the current toolchain, DC-link, is used andfor programmingposition sensors. DSPs. It also The with the input choke, power converter, and brake resistors is shown in Figure 15. performssetup with the the Clarke input and choke, Park power transformations converter, andand brakeuses PI resistors type controllers is shown within Figure a sample 15. To verify experimentally the proposed control structure in Figure4, several measure- timements of were 50 µs performed to controls on the a laboratorystator currents model i1x withand i IM1y in at the its starting{x, y} reference to therated frame. speed The second DSP implements the speed control and communicates with a higher-level ωm = 150 rad/s in time t = 0.5 s. An influence of the optional positive parameter in the controllerreference modeland operatorα on the console. motor dynamics The speed was control investigated. sample Fortime example, is 1 ms. Figure The fast 16 acurrent shows controlthe control loop dynamics is written for in theassembler parameter languageα = 5 (Equation and the rest (23)) of and the thecode values is written of the in optional C. The Texasvector InstrumentskT elements softwarekT = [0.0031 development 0.0019 0.00038]. toolchain is used for programming DSPs. The setup with the input choke, power converter, and brake resistors is shown in Figure 15. Appl. Sci. 2021, 11, x FOR PEER REVIEW 16 of 20

Appl.Appl. Sci.Sci.2021 2021,,11 11,, 6518 x FOR PEER REVIEW 1616 ofof 2020

Figure 15. The experimental setup with VQFREM 400 004–4MA converter.

To verify experimentally the proposed control structure in Figure 4, several measurements were performed on a laboratory model with IM at its starting to the rated speed ωm = 150 rad/s in time t = 0.5 s. An influence of the optional positive parameter in the reference model α on the motor dynamics was investigated. For example, Figure 16a shows the control dynamics for the parameter α = 5 (Equation (23)) and the values of the optionalFigureFigure 15. 15.vector TheThe experimentalexperimentalkT elements setupsetupkT = [0.0031 withwith VQFREMVQFREM 0.0019 0.00038]. 400400 004–4MA004–4MA converter.converter.

To verify experimentally the proposed control structure in Figure 4, several measurements were performed on a laboratory model with IM at its starting to the rated speed ωm = 150 rad/s in time t = 0.5 s. An influence of the optional positive parameter in the reference model α on the motor dynamics was investigated. For example, Figure 16a shows the control dynamics for the parameter α = 5 (Equation (23)) and the values of the optional vector kT elements kT = [0.0031 0.0019 0.00038].

(a) (b)

FigureFigure 16. 16. MotorMotor starting starting at: at: (a (a) )αα == 5 5and and kkT T= =[0.0031 [0.0031 0.0019 0.0019 0.00038]; 0.00038]; (b ()b α) α= 5= and 5 and kTk =T 0.25= 0.25 [0.0031 [0.0031 0.0019 0.0019 0.00038]. 0.00038].

InIn the the graphs, graphs, the the time time courses courses of the of the model model state state variables variables x1M,x x12MM,, prescribingx2M, prescribing the requiredthe required dynamics dynamics for the for motor the motor angular angular speed speedand acceleration and acceleration are compared are compared with their with their measured values. The time courses of the measured quantities show a very good measured values. The time courses of the measured quantities show a very good agreementagreement with with the the reference reference model. model. Figure(aFigure) 16b 16 bshows shows time time similar similar courses courses wh whenen decreasing decreasing(b) the the optional optional gain gain of of the the T Figure 16. Motor startingelementselements at: (a) α of = of 5the and the vector k vectorT = [0.0031 kTk toto 0.001925% 25% of 0.00038]; of their their original original(b) α = 5value. and value. k TIt = It is0.25 isobvious obvious[0.0031 that0.0019 that the the0.00038]. introduced introduced changechange influences influences the timetime coursecourse of of variables variables in dynamicin dynamic states states (e.g., (e.g., here thehere state the variable state variablex2 oscillatesIn x the2 oscillates graphs, less, but less, the it timebut has it acourses has larger a larger absoluteof the absolute model deviation statedeviation variables from from the x referencethe1M, referencex2M, prescribing model). model). It theis It importantrequiredis important dynamics that that again again for a the substantial a substantial motor angular agreement agreement speed between and between acceleration the the measured measured are compared quantities quantities with and and their the thereferencemeasured reference model values.model outputs outputsThe hastime has been coursesbeen reached. reached. of the measured quantities show a very good agreementTheThe time time with courses courses the referencein in Figure Figure model.17a 17 ashow show the the dynamics dynamics of of the the IM IM drive drive starting starting at at 300% 300% T changechangeFigure of of the the controller16b controller shows optional optionaltime similar gain gain k courseskT againstagainst wh its itsen original originaldecreasing value. value. the Due Due optional to to the the substantial substantialgain of the T x increaseincreaseelements of of theof the the gain gain vector in in the the k control controlto 25% circuit, of circuit, their the theoriginal state state variable variablevalue. Itx 2is (presenting2 (presentingobvious that the the the acceleration acceleration introduced proportionalproportionalchange influences to to the the motor the motor time dynamic dynamic course torque) torque)of variab is isvisiblyles visibly in moredynamic more oscillating, oscillating, states (e.g., while while here the the outputthe output state variable x (IM speed) practically follows the reference model. variablevariable x1 x(IM12 oscillates speed) practicallyless, but it hasfollows a larger the referenceabsolute deviation model. from the reference model). It is Theimportant starting that of theagain motor a substantial from the non-energized agreement between state and the its measured stopping quantities for the values and ofthe the reference control parametersmodel outputs valid has for been Figure reached. 16a are shown in Figure 17b (here the simulation was performedThe time courses for the in motor Figure starting 17a show in time thet dynamics= 0). One canof the observe IM drive here starting that the at motor 300% followschange theof the reference controller model optional both duringgain kT startingagainst its and original stopping, value. i.e., Due it behaves to the substantial as a linear system according to the reference model (23). Therefore, the proposed control structure is increase of the gain in the control circuit, the state variable x2 (presenting the acceleration able to ensure the quality of the angular speed IM control within the whole control range. proportional to the motor dynamic torque) is visibly more oscillating, while the output The experimental measurements confirm that the controlled drive behaves with high variable x1 (IM speed) practically follows the reference model. accuracy as its reference model, where for this case an optimal 2nd order linear system was chosen. The reference model dynamics are determined by the optional parameter α and the gain vector kT. Appl. Sci. 2021, 11, x FOR PEER REVIEW 17 of 20 Appl. Sci. 2021, 11, 6518 17 of 20

(a) (b)

FigureFigure 17. 17. MotorMotor starting starting at: at: (a ()a )αα = =5 5and and kTk =T 3= [0.0031 3 [0.0031 0.0019 0.0019 0.00038]; 0.00038]; (b ()b α) α= 5= and 5 and kTk =T [0.0031= [0.0031 0.0019 0.0019 0.00038]. 0.00038].

8. DiscussionThe starting of the motor from the non-energized state and its stopping for the values of the control parameters valid for Figure 16a are shown in Figure 17b (here the Based on the simulation and experimental results and on comparison of the designed simulation was performed for the motor starting in time t = 0). One can observe here that control structure with the vector control structure, the proposed new control structure with thethe motor reference follows model the isreference characterized model by both the during following starting properties: and stopping, i.e., it behaves as a linear system according to the reference model (23). Therefore, the proposed control • In general, the order of the reference model must be the same one as the order of structure is able to ensure the quality of the angular speed IM control within the whole the controlled system. In practice, it is often possible to reduce the controlled system control range. order (e.g., in the case of electrical machines, the influence of fast control loops can be The experimental measurements confirm that the controlled drive behaves with neglected, the dynamics of electromagnetic phenomena can also be replaced by the 1st high accuracy as its reference model, where for this case an optimal 2nd order linear order system, etc.). In the case of the IM, the number of the reference model state vari- system was chosen. The reference model dynamics are determined by the optional ables has been the same as the order of the reduced controlled system. The reference parameter α and the gain vector kT. model of the IM presents a nonlinear 2nd order system (Figure3 ). This significantly simplifies the control structure while maintaining its required properties which has 8. Discussion been proved by the simulation on the 3rd order IM model and by experimentation on Baseda real on drive the with simulation IM (Sections and5 andexperimental7). results and on comparison of the designed• For control the new structure control with structure, the vector no accurate control values structure, of the the controlled proposed systemnew control param- structureeters with are the required. reference This model fact is isch describedaracterized in by Section the following6, where properties: the influence of an • Inunknown general, the change order of of the the motor reference important model parameters must be the (R same2, J) on one the as dynamics the order is of shown the controlledin Figures system. 11–13 andIn practice, compared it is performance often possible of theto designedreduce the control controlled structure system and ordervector (e.g., control in the structure. case of electrical This fact machin is shownes, and the describedinfluence inof morefast control detail inloopsSection can6 , bewhere neglected, the influence the dynamics of the of unknown electromag changenetic ofphenomena important can parameters also be replaced (R2, J) on by the thecontrol 1st order dynamics system, of etc.). the In designed the case andof the vector IM, the control number structures of the reference are compared model in stateFigures variables 11–13 has. been the same as the order of the reduced controlled system. The • referenceThe parameter’s model of settingthe IM is presents performed a nonlinear through two2nd constantorder system parameters (Figure (the 3). gains).This significantlyThe first parameter simplifies (the the parameter controlα structurein the description while maintaining of the reference its modelrequired (23)) propertiesdetermines which the has overall been control proved dynamics. by the simulation It can be on determined the 3rd order from IM the model measured and bystandardized experimentation responses on a real of the drive controlled with IM system (Sections (e.g., 5 based and 7). on unit step characteristics, T • ForFigure the 2)new using control the Shanon–Kotelnik structure, no accu theorem.rate values The elements of the ofcontrolled the vector systemk that parametersensure the are transient required. phenomena This fact is anddescribed stability in presentSection 6, further where optional the influence parameters. of an They must comply with condition (16). The setting of the mentioned parameters unknown change of the motor important parameters (R2, J) on the dynamics is showncan be in realized Figures without 11–13 anyand exactcompared knowledge performance of the controlledof the designed nonlinear control system structureparameters. and Theyvector must control be positive structure. and This are tuned fact is by shown a gradual and increase described of their in more values detailup to in physical Section limitations 6, where ofthe the influence controlled of system, the unknown i.e., up to thechange current of limitationimportant of the IM. parameters (R2, J) on the control dynamics of the designed and vector control • Comparing with the speed vector control structure, the proposed control structure structures are compared in Figures 11–13. is a simpler one that concerns the interconnection and lower order of the control structure). The use of the necessary Park transform in the proposed control structure Appl. Sci. 2021, 11, 6518 18 of 20

is independent from the exact knowledge of the IM parameters which increases the quality of control (as shown in Section6). • From the view of the superior control, the novel structure exhibits dynamic proper- ties of an optimal linear dynamic system satisfying the criteria of minimum control deviation and minimum input energy [32]. Thus the drive properties are practically identical to the properties of a high quality vector control of the IM drive. • In the proposed control structure, the controlled nonlinear system is complemented by linear subsystems only (Figure8a). For this reason, it can be implemented by cheap conventional hardware. • The stability of the proposed control structure is ensured by calculating the parameters (the elements of the positive definite matrix P) by means of the Lyapunov matrix Equation (11) or in the case of an optimal chosen model according to [32], they can be calculated from (27). • The proposed control structure is robust against considerable variations changes not only of the important parameters of IM (e.g., of the rotor resistance R2 (Figure6) or of the moment of inertia—Figure7a,b), but also to inaccurate known parameters of the nonlinear controlled system. Here, the control system follows with high accuracy the reference model output which is confirmed by the simulations in Figures 11–13 and also by the measurements in Figures 16 and 17. • The proposed control structure is invariant against the external disturbances (e.g., of the load torque equal to the nominal torque: Figures5–7) where the dynamics of the disturbance is very fast. • The proposed control of the IM angular speed solves also the problems occurring at low speed. The controlled drive follows its reference model with high accuracy also in the range of low speeds up to zero speed.

9. Conclusions The designed new stable control method can be applied generally for any continuous nonlinear system. In this article, a controlled AC drive with an induction motor was chosen for verification of its properties, where the IM represents a highly oscillating nonlinear system and its parameters (e.g., the rotor resistance and moment of inertia) can vary during the motor operation. Both simulation results and experimental measurements performed for basic operating states of the IM drive have confirmed advantages of the proposed controller concerning simplicity at the design and implementation and the excellent performance of the controlled system. Application of the new control method considerably contributes to improving dynamic properties and simultaneously ensures the drive stability, invariance against disturbances and robustness against the motor parameters variations. Compared to the control structures of the vector control of IM, the new control structure is significantly simpler; it exhibits features of an optimal linear system and simultaneously it achieves almost identical dynamic control performance. When comparing with the classical vector control structure of the IM, the designed control structure is much more robust against changes of the important motor parameters. The main advantage of the novel control structure consists of the fact that it does not require any knowledge of controlled system parameters. On the other side, the precise control at the majority of the used control methods strongly depends on precise knowledge of system parameters. The control quality is also guaranteed by the independence of the Park transformation calculation from the exact knowledge of the IM parameters. The control structure ensures high quality of the IM speed control within the whole control range, up to zero speeds. The proposed control structure is suitable for the control of continuous nonlinear sys- tems with unknown and time-varying parameters. The controller is suitable especially for any drive system including control of robotic systems control having a precise hierarchical control structure. Therefore, its broad utilization in industrial applications can be assumed. Appl. Sci. 2021, 11, 6518 19 of 20

Author Contributions: Conceptualization, P.F. and D.P.; methodology, P.F. and D.P.; writing— original draft preparation, P.F.; writing—review and editing, D.P.; formal analysis, P.B.; software, P.B. and M.F.; validation, P.B. and M.F. All authors have read and agreed to the published version of the manuscript. Funding: This work was supported by the Slovak Research and Development Agency under the contract No. APVV-19-0210. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Not applicable. Conflicts of Interest: The authors declare no conflict of interest.

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