Robust Linear Parameter Varying Output Feedback Control of Permanent Magnet Synchronous Motors
Shahin Tasoujian, Jaecheol Lee, Karolos Grigoriadis, Matthew Franchek
aa r X i v : . [ ee ss . S Y ] J un Robust linear parameter-varyingoutput-feedback control of permanentmagnet synchronous motors
Submitted to Transactions of the Instituteof Measurement and Control(preprint):1–9c (cid:13)
SAGE
Shahin Tasoujian, Jaecheol Lee, Karolos Grigoriadis, and Matthew Franchek
Abstract
This paper investigates the design of a robust output-feedback linear parameter-varying (LPV) gain-scheduled controllerfor the speed regulation of a surface permanent magnet synchronous motor (SPMSM). Motor dynamics is defined inthe α − β stationary reference frame and a parameter-varying model formulation is provided to describe the SPMSMnonlinear dynamics. In this context, a robust gain-scheduled LPV output-feedback dynamic controller is designed tosatisfy the asymptotic stability of the closed-loop system and meet desired performance requirements, as well as,guarantee robustness against system parameter perturbations and torque load disturbances. The real-time impact oftemperature variation on the winding resistance and magnet flux during motor operations is considered in the LPVmodelling and the subsequent control design to address demagnetization effects in the motor response. The controllersynthesis conditions are formulated in a convex linear matrix inequality (LMI) optimization framework. Finally, the validityof the proposed control strategy is assessed in simulation studies, and the results are compared to the results of theconventional field-oriented control (FOC) method. The closed-loop simulation studies demonstrate that the proposedLPV controller provides improved transient response with respect to settling time, overshoot, and disturbance rejectionin tracking the velocity profile under the influence of parameter and temperature variations and load disturbances. Keywords
Permanent magnet synchronous motors (PMSMs), Linear parameter-varying (LPV) systems, Disturbance rejection,Velocity tracking, H ∞ control, Uncertainty and robust control, Gain-scheduled control Introduction
Permanent magnet synchronous motors (PMSMs) areprevalent in industry and in various electromechanicalapplications, such as, electrical appliances, robotic systemsand electric vehicles, due to their compact structure,high torque density, high power density and high-efficiency Boldea and Nasar (1992); Zhong et al. (1997);Yanliang et al. (2001). However, because of the inherentnonlinear dynamics, strong coupling effects and significantsystem parameter variability Pillay and Krishnan (1988);Cai et al. (2017), the precise speed and position control of aPMSM is a challenging task. Traditionally, the field-orientedcontrol (FOC) method has been employed as a vector controlof both magnitude and angle of the flux enabling independentcontrol of torque and speed. Consequently, fast and highprecision motor control can be achieved. For this reason,the motor drives implemented with the FOC method aretypically comprised of two loops in a cascade manner in the d − q rotating reference frame Zhu et al. (2019). The currentcontrol loop is the inner loop for the stator current to followits reference value while the speed control loop is the outerloop taking into account speed error signals and providingreference signals to the inner loop Giri (2013).In the FOC method, a proportional-integral (PI) controlleris typically implemented for both current and speed controldue to its simple design structure. The PI controller gains aretypically determined through nominal motor parameters tosatisfy motor performance specifications Kim et al. (2016).However, PI control is not suitable for applications where high performance and high precision is required. Whenmotor parameter variations and disturbances are present,robustness and stability issues inevitably arise. As anadditional challenge, varying motor temperature has shownto have a significant impact on PMSM speed, currentand torque resulting from the reversible demagnitizationof the permanent magnet (NdFeB or SmCo) and thetemperature-dependence of the stator winding resistance.Hence, traditional PI controllers typically fail to maintainthe desired closed-loop motor response in high performanceapplications. Various robust and nonlinear control methodshave been adopted to address the parameter variability,as well as, to cope with the nonlinearity in the PMSMmodel Zhao and Dong (2019). The sliding mode control(SMC) method has been proposed to assure fast responseand robustness in the presence of nonlinearity in themodel. However, the SMC method inherently causes achattering problem due to the signum function, whichleads to deteriorating performance at steady-state Baik et al.(1998); Kim et al. (2010); Zhang et al. (2012). Disturbance This paper is a preprint of a paper submitted to Transactions of theInstitute of Measurement and Control.Department of Mechanical Engineering at the University of Houston,Houston, TX 77004 U.S.A.
Corresponding author:
Shahin Tasoujian Department of Mechanical Engineering, University ofHouston, Houston, TX 77004 U.S.A.Email: [email protected]
This paper is a preprint of a paper submitted to Transactions of the Institute of Measurement and Control
Submitted to Transactions of the Institute of Measurement and Control (preprint) observers (DOBs) have been proposed to estimate thedisturbance for its compensation. Zhao et al. (2015) studiedthe case with unknown load torque and model parametersand proposed an adaptive observer-based control method forthe speed tracking in the PMSMs. Although DOBs can helpimprove the capability of a motor to reject disturbances,a disadvantage lies in the fact that the methodology isrequired to have full knowledge of the PMSM parametersto ensure the stability of the DOBs Solsona et al. (2000);Chang et al. (2010). Additionally, the fuzzy logic controlmethod has been proposed for the control of PMSMs andhas shown an improved performance regarding robustness todisturbance rejection. However, shortcomings reside in thefact that membership functions rely solely on the designer’sexperience and it demands heavy computations Yu et al.(2007); Chaoui and Sicard (2011).Recently, the use of linear parameter-varying (LPV)gain-scheduling control techniques for the PMSM controlproblem has drawn the attention of researchers due tothe controller’s scheduling nature providing the ability tohandle system parameter variations and nonlinearities ina systematic framework. In this regard, a static fixed-gain state-feedback LPV controller with an estimator hasbeen proposed for PMSM control Lee et al. (2017), wherethe estimator is utilized to provide the state-feedbackcontroller with the full-state information needed to generatethe control input. The authors in Lee et al. (2017) usedthe polytopic LPV description resulting in a relativelyconservative control design, especially for the case of slowparameter variations. The gain-scheduling control techniqueis an extension of the linear control design to handlenonlinear and time variations, where a scheduling parametervector captures the information about the nonlinearitiesor time-varying behavior of the system. The LPV gain-scheduling control methodology was first introduced inShamma and Athans (1991) to overcome the shortcomingof conventional gain-scheduling control techniques, namely,lack of closed-loop stability and performance guarantees.Unlike conventional gain-scheduling design methods whichare based on interpolation between several independentlydesigned LTI controllers for different fixed operating points,LPV gain-scheduling control design provides a direct,efficient, systematic and global control approach, which alsoguarantees closed-loop stability and performance. Stabilityanalysis and control synthesis of LPV systems have beenaddressed extensively in the control literature in the pastdecade Apkarian and Adams (1998); Wu and Grigoriadis(2001); Tasoujian et al. (2019a,b, 2020).In the present paper, first, the α − β stationary referenceframework is considered for the surface permanent magnetsynchronous motors (SPMSMs) modeling. The SPMSMmodel is assumed to be subject to varying parametersand torque load disturbances that impair the response ofthe closed-loop system to track a reference speed profile.Subsequently, we develop a LPV representation to describethe SPMSM dynamics. Resistance and magnetic fluxes inSPMSMs vary with temperature. To this end, temperaturevariation is taken into consideration in the LPV modelingas an LPV scheduling parameter. The presented formulationallows a systematic control design seeking to handle thetemperature-dependent parameter variations and the model uncertainties in SPMSMs. To minimize the conservatism ofthe control design in meeting performance specifications,a parameter-dependent Lyapunov function approach isutilized to design an LPV gain-scheduled dynamic output-feedback controller to track the commanded referencespeed profile and minimize the effect of disturbances andparameter variations over the entire operating envelopeof the motor. The proposed dynamic LPV control designmethod guarantees asymptotic stability and robustnessagainst disturbances and uncertainties in terms of the closed-loop system’s induced L -norm performance index. Alinear matrix inequality (LMI) framework is adopted toformulate the proposed H ∞ control synthesis problem ina convex, computationally tractable setting, which can besolved efficiently using numerical optimization algorithms.Finally, the performance of the proposed method is evaluatedand validated in a computer simulation environment andcompared to the conventional FOC method with a fixed-gainPI controller.The notation to be used in the paper is standard andas follows: R denotes the set of real numbers, and R n and R k × m are used to denote the set of real vectors ofdimension n and the set of real k × m matrices, respectively. M ≻ shows the positive definiteness of the matrix M and the transpose of a real matrix M is shown as M T .Also, S n denotes the set of real symmetric n × n matrix.In a symmetric matrix, terms denoted by asterisk, ⋆ , will beinduced by symmetry as shown below: (cid:20) S + W + J +( ⋆ ) ⋆Q R (cid:21) := (cid:20) S + W + W T + J + J T Q T Q R (cid:21) where S is symmetric. He [ M ] is Hermitian operator definedas He [ M ] , M + M T and C ( J, K ) stands for the set ofcontinuous functions mapping a set J to a set K .The outline of the paper is as follows. Section IIpresents the mathematical modeling for the SPMSMs andthe proposed LPV model formulation. In Section III, theoutput-feedback LPV gain-scheduling control technique isdescribed considering scheduling parameters that capturethe nonlinearity and temperature-dependent variability of theSPMSM model. Section IV outlines the closed-loop resultsand describes the performance evaluation of the proposedLPV controller in a computer simulation environment.Finally, Section V concludes the paper. SPMSM modeling
SPMSM dynamics
We consider a three-phase synchronous motor withpermanent magnets where the magnetic coupling betweenthe phases and the inductance variation due to magneticsaturation are assumed to be negligible. Additionally, themagnetic flux ganerated by the excitation is assumed to havean ideal sinusoidal density distribution. Consequently, thesimplified dynamic model for SPMSMs can be expressed inthe α − β stationary reference frame as follows Hwang et al. Prepared using sagej.cls asoujian et al. (2014): ˙ θ = ω, ˙ ω = 1 J m ( − Bω − K t sin ( pθ ) i α + K t cos ( pθ ) i β − τ L ) , ˙ i α = 1 L s ( − R s i α + pλ pm ω sin ( pθ ) + v α ) , ˙ i β = 1 L s ( − R s i β − pλ pm ω cos ( pθ ) + v β ) , (1)where θ stands for the mechanical rotor angular position[rad], ω is the mechanical rotor speed [rad/sec], v α , v β and i α , i β are the voltages [V] and currents [A] in the α − β stationary reference frame. In this model, L s denotes thestator inductance [H], R s is the stator resistance [ Ω ], λ pm is the magnetic flux of the motor [Wb], J m is the momentof inertia [ kg.m ] , p denotes the number of magnet polepairs, B is the viscous friction coefficient [N · m · sec/rad], τ L is the load torque [N · m], and K t = 32 pλ pm is the torqueconstant [ V · rad/sec ] . To assess the closed-loop SPMSMperformance, the following tracking errors are defined for thequantities of interest: e w = ω ∗ − ω,e z = Z t e ω dx,e α = i ∗ α − i α ,e β = i ∗ β − i β , (2)where ω ∗ is the desired motor speed, and i ∗ α and i ∗ β are thedesired currents in the stationary reference ( α − β ) frame,respectively. Additionally, e z represents the integral of speederror and e α and e β are the current errors in the α − β stationary reference frame, respectively. The desired torque, τ ∗ , the desired currents, i ∗ α and i ∗ β , and the voltage inputs tothe motor, v α and v β are defined as follows τ ∗ = J m ˙ ω ∗ + Bω ∗ ,i ∗ α = − τ ∗ sin ( pθ ) K t ,i ∗ β = τ ∗ cos ( pθ ) K t ,v α = L s ˙ i ∗ α + R s i ∗ α − pλ pm ω ∗ sin ( pθ ) − u α ,v β = L s ˙ i ∗ β + R s i ∗ β + pλ pm ω ∗ cos ( pθ ) − u β , (3)where u α and u β are the control inputs. Hence, the errordynamics can be obtained by combining (1), (2), and (3) asfollows ˙ e z = e w , ˙ e ω = 1 J m ( − Be ω − K t sin ( pθ ) e α + K t cos ( pθ ) e β + τ L ) , ˙ e α = 1 L s ( − R s e α + pλ pm sin ( pθ ) e ω + u α ) , ˙ e β = 1 L s ( − R s e β − pλ pm cos ( pθ ) e ω + u β ) . (4)Subsequently, an LPV representation for the introducedSPMSM error dynamics is developed to enable LPV controldesign: LPV model formulation
In order to be able to implement the proposed LPV controlmethodology to the SPMSM dynamics case study, we firstrewrite the described system (4) as a proper LPV model.LPV systems correspond to a class of linear systems, whosedynamics depend on time-varying parameters, known as thescheduling parameters. Therefore, considering (4), the firsttwo LPV scheduling parameters are defined as follows ρ ( θ ( t )) = pλ pm sin ( pθ ( t )) ,ρ ( θ ( t )) = pλ pm cos ( pθ ( t )) . (5)Since the scheduling parameters in (5) are trigonometricfunctions, they can be bounded as follows − pλ pm ≤ ρ ( θ ( t )) and ρ ( θ ( t )) ≤ pλ pm . (6)It is known that temperature variation has a significanteffect on SPMSM performance. Consequently, we definetemperature as the third scheduling parameter: T ≤ ρ ( t ) = T ( t ) ≤ T , (7)where T , and T are the minimum and maximum motoroperating temperatures in ◦ C , respectively. Resistance andmagnetic fluxes vary considerably throughout the motoroperation as a function of temperature. Embedded insulatetemperature sensors or estimation algorithms can be usedto provide instantaneous measurements or estimates ofstator winding temperature Jun et al. (2018). The followingrelations can be used to obtain empirical expression for thesemotor parameter variations as functions of temperature R s ( ρ ( t )) = R s (cid:18)
235 + ρ ( t )310 (cid:19) ,λ pm ( ρ ( t )) = λ pm (cid:18) α ( ρ ( t ) − (cid:19) , (8)where R s is the resistance value of the windingat 75 ◦ C , λ pm is the flux of the magnet at 30 ◦ C ,and α is the temperature coefficient of the magnetin % / ◦ C Sul (2011). After defining the schedulingparameters, the scheduling parameter vector is representedas, ρ ( t ) = [ ρ ( t ) ρ ( t ) ρ ( t ) ] T . Subsequently, theLPV representation of the SPMSM dynamics takes thefollowing matrix-vector form ˙ e ( t ) = A ( ρ ( t )) e ( t ) + B τ L ( t ) + B u ( t ) , y ( t ) = C e ( t ) , (9)where the augmented state vector is defined as e ( t ) =[ e z ( t ) e ω ( t ) e α ( t ) e β ( t )] T , the control input is u ( t ) =[ u α ( t ) u β ( t ) ] T , y ( t ) is the measured signal vector, andthe state-space matrices of the LPV system (9) are as follows Prepared using sagej.cls
Submitted to Transactions of the Institute of Measurement and Control (preprint) A ( ρ ( t )) = − BJ m − ρ ( t ) J m ρ ( t ) J m ρ ( t ) L s − R s ( ρ ( t )) L s − ρ ( t ) L s − R s ( ρ ( t )) L s , B = , B = L s
00 1 L s , C = (cid:20) (cid:21) . (10)Next, the proposed output-feedback LPV gain-schedulingcontrol design method is described. LPV control design
We aim to design an output-feedback LPV gain-scheduledcontroller for the SPMSM model (9) in the context ofinduced L -norm performance specifications. To this end,we consider a generic LPV open-loop system with thefollowing state-space realization ˙ x ( t ) = A ( ρ ( t )) x ( t ) + B ( ρ ( t )) w ( t ) + B ( ρ ( t )) u ( t ) , z ( t ) = C ( ρ ( t )) x ( t ) + D ( ρ ( t )) w ( t ) + D ( ρ ( t )) u ( t ) , y ( t ) = C ( ρ ( t )) x ( t ) + D ( ρ ( t )) w ( t ) , x (0) = x , (11)where x ∈ R n is the system state vector, w ∈ R n w isthe vector of exogenous disturbances with finite energyin the space L [0 , ∞ ] , u ∈ R n u is the control inputvector, z ( t ) ∈ R n z is the vector of controlled output, y ( t ) ∈ R n y is the vector of measured output, x ∈ R n is theinitial system condition. The state space matrices A ( · ) , B ( · ) , B ( · ) , C ( · ) , C ( · ) , D ( · ) , D ( · ) , and D ( · ) have rational dependence on the time-varying schedulingparameter vector, ρ ( · ) ∈ F ν P , which is also measurable inreal-time. F ν P is the set of allowable parameter trajectoriesdefined as F ν P , { ρ ( t ) ∈ C ( R + , R n s ) : ρ ( t ) ∈ P , | ˙ ρ i ( t ) | ≤ ν i , i = 1 , , . . . , n s } , (12)wherein n s is the number of parameters and P is a compactsubset of R s , i.e. the parameter trajectories and parametervariation rates are assumed bounded as defined. The output-feedback LPV gain-scheduled control design procedureconsists of finding a full-order dynamic LPV controller inthe form of ˙ x K ( t ) = A K ( ρ ( t )) x K ( t ) + B K ( ρ ( t )) y ( t ) , u ( t ) = C K ( ρ ( t )) x K ( t ) + D K ( ρ ( t )) y ( t ) , (13)where x K ( t ) ∈ R n is the controller state vector. By substi-tuting the controller (13) in the open-loop system (11), andassuming x cl ( t ) = [ x ( t ) x K ( t ) ] T , the interconnected closed-loop system ( T zw ) is obtained as follows ˙ x cl ( t ) = A cl ( ρ ( t )) x cl ( t ) + B cl ( ρ ( t )) w ( t ) , z ( t ) = C cl ( ρ ( t )) x cl ( t ) + D cl ( ρ ( t )) w ( t ) , (14)with A cl = (cid:20) A + B D K C B C K B K C A K (cid:21) , B cl = (cid:20) B + B D K D B K D (cid:21) , C cl = (cid:2) C + D D K C D C K (cid:3) , D cl = D + D D K D , where the dependence on the scheduling parameter has beendropped for brevity. The final designed controller should beable to meet the following objectives for the closed-loopsystem: • Input-to-state stability (ISS) of the closed-loop system(14) in the presence of parameter variations anddisturbances, and • Minimization of the worst-case amplification ofthe induced L -norm of the mapping from thedisturbances w ( t ) to the controlled output z ( t ) , givenby k T zw k i, = sup ρ ( t ) ∈ F ν P sup k w ( t ) k =0 k z ( t ) k k w ( t ) k . (15)Accordingly, in this paper, we utilize an extendedform of the Bounded Real Lemma Briat (2014) and aquadratic parameter-dependent Lyapunov functions of theform V ( x cl ( t ) , ρ ( t )) = x T cl ( t ) P ( ρ ( t )) x cl ( t ) to obtain lessconservative results that are valid for arbitrary boundedparameter variation rates Apkarian and Adams (1998). Tothis end, considering the closed-loop system (14), thefollowing result provides sufficient conditions for thesynthesis of a output-feedback LPV controller, which isformulated as convex optimization problems with LMIconstraints. The designed LPV gain-scheduled controllerguarantees closed-loop asymptotic parameter-dependentquadratic (PDQ) stability and a specified performance levelas defined in (15). Theorem 1.
Briat (2014) Considering the given open-loop LPV system (11), there exists a gain-scheduleddynamic full-order output-feedback controller of the form(13) that guarantees the closed-loop asymptotic sta-bility and satisfies the induced L -norm performancecondition k z ( t ) k ≤ γ k w ( t ) k , if there exist continu-ously differentiable parameter-dependent symmetric matri-ces X , Y : R s → S n , parameter-dependent matrices b A ∈ R s → R n × n , b B ∈ R s → R n × n y , b C ∈ R s → R n u × n , b D ∈ R s → R n u × n y , and a scalar γ > such that the following Prepared using sagej.cls asoujian et al. LMI conditions hold for all ρ ∈ F ν P . ˙ X + XA + b B C + ( ⋆ ) ⋆ b A T + A + B b D C − ˙ Y + AY + B b C + ( ⋆ )( XB + b B D ) T ( B + B b D D ) T C + D b D C C Y + D b C⋆ ⋆⋆ ⋆ − γ I ⋆ D + D b D D − γ I ≺ , (16) (cid:20) X II Y (cid:21) ≻ . (17)Subsequently, the LPV control design is expanded to guar-antee robustness against modeling mismatch and parameteruncertainties. To this end, A and B in (11) are consideredto be uncertain system matrices, A ∆ ( ρ ( t )) = A ( ρ ( t )) + ∆A ( t ) , B , ∆ ( ρ ( t )) = B ( ρ ( t )) + ∆B ( t ) , where ∆A ( t ) and ∆B ( t ) are bounded matrices containing parametricuncertainties. The norm-bounded uncertainties are assumedto satisfy the following relation (cid:20) ∆A ( t ) ∆B ( t ) (cid:21) = H∆ ( t ) (cid:20) E E (cid:21) , (18)where H ∈ R n × i , E ∈ R j × n , E ∈ R j × n u are knownconstant matrices and ∆ ( t ) ∈ R i × j is an unknown time-varying uncertainty matrix function satisfying inequality ∆ T ( t ) ∆ ( t ) (cid:22) I . (19)By substituting A ∆ ( ρ ( t )) and B , ∆ ( ρ ( t )) for A and B in(16), the following result presents a condition for ensuringclosed-loop stability and performance in the presence ofnorm-bounded uncertainties via an LPV control design of theform (13). Theorem 2.
There exists a full-order robust output-feedback LPV controller of the form (13), over the sets F ν P with all admissible uncertainties ∆A ( t ) and ∆B ( t ) ofthe form (18) and all ∆ ( t ) satisfying (19), that guaranteesthe closed-loop asymptotic stability and satisfies the induced L -norm performance condition k z ( t ) k ≤ γ k w ( t ) k , ifthere exist continuously differentiable parameter dependentsymmetric matrices X , Y : R s → S n , parameter dependentreal matrices b A ∈ R s → R n × n , b B ∈ R s → R n × n y , b C ∈ R s → R n u × n , b D ∈ R s → R n u × n y , and a positve scalars γ ,and ǫ such that the LMI (20) is feasible. Proof.
By substituting the matrices with additive norm-bounded uncertainties in the LMI condition (16) given byTheorem 1, i.e. , A ∆ ( ρ ( t )) for A and B , ∆ ( ρ ( t )) for B in(16), the new LMI condition will be as follows (16) + He XHH00 ∆ ( t ) (cid:2) E E Y 0 0 (cid:3) ! + He ∆ ( t ) h E b D C E b C E b D D i! ≺ . (21) Finally, using the following inequality Xie (1996) Θ∆ ( t ) Φ + Φ T ∆ T ( t ) Θ T ≤ ǫ − ΘΘ T + ǫ Φ T Φ , (22)which holds for all scalars ǫ > and all constant matrices Θ and Φ of appropriate dimensions, and using the Schurcomplement Boyd et al. (1994), the final LMI condition (20)is obtained.Once the parameter-dependent LMI decision matrices, X , Y , b A , b B , b C , and b D satisfying the LMI conditions (16)and (17) are obtained, the output-feedback LPV controllermatrices can be readily computed following the steps below:1. Determine M and N from the factorization problem I − XY = NM T , (23)where the obtained M and N matrices are square andinvertible in the case of a full-order controller.2. Compute the controller matrices in the following order: D K = b D, C K = ( b C − D K C Y ) M − T , B K = N − ( b B − XB D K ) , A K = − N − ( XAY + XB D K C Y + NB K C Y + XB C K M T − b A ) M − T . (24) Remark 1.
Theorem 1 results in an infinite-dimensionalconvex optimization problem with an infinite number of LMIsand decision variables since the scheduling parameter vectorbelongs to a continuous real vector space, ρ ∈ F ν P . Toaddress this obstacle, the gridding method of the parameterspace is utilized to convert the infinite-dimensionalproblem to a finite-dimensional convex optimization problemApkarian and Adams (1998). In this regard, we choose thematrix parameter functional dependence as M ( ρ ( t )) = M + s P i =1 ρ i ( t ) M i , where M ( ρ ( t )) represents any ofthe parameter-dependent matrices appearing in the LMIconditions (16), and (17). Subsequently, by gridding thescheduling parameter space at appropriate intervals weobtain a finite set of LMIs to be solved for the unknownmatrices and γ . The MATLAB R (cid:13) toolbox YALMIP can beused to solve the introduced optimization problem Lofberg(2004). Also, it should be noted that due to the presence ofderivatives of the parameter-dependent matrices in the LMIcondition (16), i.e. ˙ X , and ˙ Y , the parameter variation rate ˙ ρ , enters affinely in the LMIs, and it is sufficient to check theLMI only at the vertices of the ˙ ρ parameter range. SPMSM LPV control design
We examine the application of the proposed LPV gain-scheduled control design method to the SPMSM speedregulation. The SPMSM error dynamics (9) is formulated inan LPV framework (shown in Section III) which is suitablefor the proposed LPV control design synthesis. Consideringthe generic LPV system state-space realization (11) forthe SPMSM model described in (9), the LPV state-spacematrices of the SPMSM are as shown in (10). Moreover,the vector of the target outputs to be controlled is defined
Prepared using sagej.cls
Submitted to Transactions of the Institute of Measurement and Control (preprint) ˙ X + XA + b B C +( ⋆ ) ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ b A T + A + B b D C − ˙ Y + AY + B b C + ( ⋆ ) ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ( XB + b B D ) T ( B + B b D D ) T − γ I ⋆ ⋆ ⋆ ⋆ ⋆ C + D b D C C Y + D b C D + D b D D − γ I ⋆ ⋆ ⋆ ⋆ H T X H T − ǫ I ⋆ ⋆ ⋆ ǫ E ǫ E Y 0 0 0 − ǫ I ⋆ ⋆ T − ǫ I ⋆ ǫ E b D C ǫ E b C ǫ E b D D − ǫ I ≺ (20) Table 1.
Parameters of the SPMSM.Parameter Value UnitNumber of pole pairs ( p ) 4 -Stator resistance ( R s ) 0.2 Ω Stator inductance ( L s ) 0.4 mHMagnetic flux linkage ( λ pm ) 16.3 mWbMoment of inertia ( J m ) 3.24 × − kg · m Coefficient of friction ( B ) 0.004 N · m · s/rad as follows z T ( t ) = (cid:2) φ · e z ( t ) σ · e ω ( t ) ξ · e α ( t ) ψ · e β ( t ) η · u α ( t ) µ · u β ( t ) (cid:3) . (25)The velocity tracking error which is included in the secondstate x ( t ) = e w ( t ) is penalized by the weighting scalar σ and the control efforts u α ( t ) , u β ( t ) are penalized by theweighting scalars η and µ , respectively. The choice of theweighting scalars φ , σ , ξ , ψ , η , and µ determine the relativeweighting in the optimization scheme and depends on thedesired performance objectives that the designer seeks toachieve Lee et al. (2015). Now, based on the definition of thedesired controlled vector z ( t ) , the output-feedback controlleris designed for the SPMSM to minimize the induced L gain(or H ∞ norm) (15) of the closed-loop LPV system (14).The design objective is to guarantee closed-loop stability andminimize the worst case disturbance amplification over theentire range of model parameter variations.In order to demonstrate the improved performance of theproposed control with respect to the desired velocity profiletracking and load torque disturbance rejection, closed-loop simulations are performed in the MATLAB/Simulinkenvironment. The model parameters of the SPMSM arelisted in Table 1. For comparison purposes, we evaluate theclosed-loop tracking performance of the proposed controlleragainst the FOC method with fixed gains. The FOC tunedPI controller transfer functions are selected as follows Kim(2017): G cs ( s ) = 0 .
533 + 61 . s ,G cc ( s ) = 1 .
38 + 691 s , (26)where G cs ( s ) and G cc ( s ) indicate the speed controller inthe q axis and the current controllers in the d and q axisrespectively. The gains of these controllers are obtainedbased on the nominal parameters of the motor and the desiredbandwidth of the controllers. Figure 1.
Desired velocity reference profile.
Figure 2.
Closed-loop velocity tracking performance of the LPVcontroller and the fixed structure PI controller with nodisturbance during acceleration period.
In order to evaluate the closed-loop tracking performanceof the proposed LPV method, we consider a desired velocityreference profile, as shown in Figure 1. Figures 2 and3 present the magnified plots of the tracking error resultof the LPV and PI controllers in the absence of anydisturbances, for the first step change, when the velocityreference accelerates from 0 r/min to 300 r/min, and for thesecond step change, when the velocity reference deceleratesfrom 300 r/min to 100 r/min, respectively. As anticipated, theproposed LPV controller outperforms the PI controller withrespect to the overshoot/undershoot, rise time, and speed ofthe response in both acceleration and deceleration intervalsdue to its scheduling structure. Figures 4 and 5 show thecurrents and control input voltages of the proposed LPVcontroller, both in the α − β axis. Prepared using sagej.cls asoujian et al. Figure 3.
Closed-loop velocity tracking performance of the LPVcontroller and the fixed structure PI controller with nodisturbance during deceleration period.
Figure 4.
Currents of the LPV controller in the α − β axis. Figure 5.
Control input voltages of the LPV controller in the α − β axis. Next, we assume that the SPMSM is experiencingtemperature variation with a temperature profile shownin Figure 7 and an output disturbance. The temperaturevariation affects the model’s resistance and magnet flux asdescribed in (8). The disturbance under consideration is aconstant torque load disturbance as shown in Figure 6. Theclosed-loop performance of the proposed LPV controllerand the PI controller in tracking a given ramp-type velocityreference command with a step disturbance is shown inFigure 8. Additionally, Figures 9 and 10 demonstrate the
Figure 6.
Load torque disturbance.
Figure 7.
SPMSM operating temperature variation. currents and the input voltages of the LPV controller inthe α − β phases, respectively. In order to evaluate therobustness of the proposed design, the closed-loop responseof the proposed robust LPV gain-scheduling controller isinvestigated in the presence of model parameter variations.To this end, we select the stator inductance L s and themoment of inertia J m to be under-estimated by , andthe stator resistance R s and viscous friction coefficient B to be over-estimated by , which corresponds to aworst-case perturbation scenario. The closed-loop velocitytracking performance of the system with the proposedrobust LPV control design (obtained through condition (20)and Theorem 2) is compared to the response of the LPVcontroller designed without considering uncertainty obtainedusing the results of Theorem 1. As per Figure 11, the controlwithout considering uncertainty in the design demonstratessignificant oscillatory behavior, higher overshoots andsettling time, which are undesirable. Hence, as the resultsdemonstrate, the proposed robust LPV control designis capable of compensating for parameter uncertaintiesand modeling mismatches. Therefore, by investigating thepresented results, we conclude that the proposed LPV controlmethod demonstrates superior results in terms of velocitytracking, disturbance rejection and robustness under differentsimulated scenarios in the presence of parameter variations,disturbances and model uncertainty. Prepared using sagej.cls
Submitted to Transactions of the Institute of Measurement and Control (preprint)
Figure 8.
Closed-loop velocity tracking performance of the LPVcontroller and the fixed structure PI controller subject to loadtorque disturbance.
Conclusion
In the present paper, a linear parameter-varying (LPV) gain-scheduled output feedback controller has been proposedfor the speed control of the surface permanent magnetsynchronous motors (SPMSMs). The dynamic model ofthe motor has been developed in the α − β stationaryreference frame, and an LPV model representation hasbeen utilized to capture the nonlinear SPMSM dynamics.The effect of temperature on the variability of SPMSMsmodel parameters is taken into account in the model. Thelinear matrix inequality (LMI) framework has been used toformulate the controller synthesis conditions as numericallytractable convex optimization computational problem.Subsequently, the proposed controller was designed toguarantee the closed-loop stability and minimize thedisturbance amplification in terms of the induced L -norm performance specification of the closed-loop system.The effectiveness of the proposed controller was validatedvia comparisons with a conventional PI controller in theMATLAB/Simulink environment. The results demonstratedthe effectiveness and superiority of the proposed approachin improving the transient performances in terms of settlingtime, overshoot, disturbance attenuation, and parametricuncertainty compensation. Future research would focus ondesigning a disturbance observer to empower the controldesign to cope better with unknown disturbances. Hence, theestimated disturbance can be compensated in the controlleroutput to improve the stability of the motor and speedtracking performance. References
Apkarian P and Adams RJ (1998) Advanced gain-schedulingtechniques for uncertain systems.
IEEE Transactions onControl Systems Technology
IET Electric Power Applications
Vector Control of AC Drives . CRCpress.
Figure 9.
Currents of the LPV controller in the α − β axis. Figure 10.
Control input voltages of the LPV controller in the α − β axis. Figure 11.
Closed-loop velocity tracking performance of therobust LPV controller in the presence of model parameteruncertainty subject to torque load disturbance.
Boyd S, El Ghaoui L, Feron E and Balakrishnan V (1994)
LinearMatrix Inequalities in System and Control Theory , volume 15.Siam.Briat C (2014) Linear parameter-varying and time-delay systems.
Analysis, observation, filtering & control H ∞ -synthesis. Prepared using sagej.cls asoujian et al. In:
IEEE Industrial Electronics Conference . pp. 8602–8607.Chang S, Chen P, Ting Y and Hung S (2010) Robust currentcontrol-based sliding mode control with simple uncertaintiesestimation in permanent magnet synchronous motor drivesystems.
IET Electric Power Applications
IEEE Transactions on Industrial Electronics
AC Electric Motors Control: Advanced DesignTechniques and Applications . John Wiley & Sons.Hwang H, Lee Y, Shin D and Chung CC (2014) H control basedon LPV for speed control of permanent magnet synchronousmotors. International Conference on Control, Automation andSystems : 922–927.Jun BS, Park JS, Choi JH, Lee KD and Won CY (2018) Temperatureestimation of stator winding in permanent magnet synchronousmotors using d-axis current injection.
Energies
IEEE Transactionson Industrial Electronics
Electric Motor Control: DC, AC, and BLDCMotors . Elsevier.Kim SK, Lee JS and Lee KB (2016) Self-tuning adaptive speedcontroller for permanent magnet synchronous motor.
IEEETransactions on Power Electronics H ∞ control based onLPV for load torque compensation of PMSM. In: InternationalConference on Control, Automation and Systems (ICCAS) . pp.1013–1018.Lee Y, Lee SH and Chung CC (2017) LPV H ∞ Control withDisturbance Estimation for Permanent Magnet SynchronousMotors.
IEEE Transactions on Industrial Electronics
IEEE International Conferenceon Robotics and Automation . pp. 284–289.Pillay P and Krishnan R (1988) Modeling of permanent magnetmotor drives.
IEEE Transactions on Industrial Electronics
Automatica
IEEE Transactions on Energy Conversion
Control of Electric Machine Drive Systems ,volume 88. John Wiley & Sons.Tasoujian S, Grigoriadis K and Franchek M (2019a) Delay-dependent output-feedback control for blood pressure regula-tion using LPV techniques. In:
ASME Dynamic Systems andControl Conference (DSCC) .Tasoujian S, Salavati S, Franchek M and Grigoriadis K (2019b)Robust IMC-PID and parameter-varying control strategies forautomated blood pressure regulation.
International Journal ofControl, Automation and Systems
IETControl Theory & Applications .Wu F and Grigoriadis KM (2001) LPV systems with parameter-varying time delays: analysis and control.
Automatica H ∞ control of systems withparameter uncertainty. International Journal of Control
IEEE International Conference onElectrical Machines and Systems , volume 2. pp. 884–887.Yu JS, Kim SH, Lee BK, Won CY and Hur J (2007) Fuzzy-logic-based vector control scheme for permanent-magnetsynchronous motors in elevator drive applications.
IEEETransactions on Industrial Electronics
IEEE Transactions onPower Electronics
Transactions ofthe Institute of Measurement and Control
Control Theory and Technology
IEEE Transactions on Power Electronics
IEEEInternational Conference on Electrical Machines and Systems .pp. 1–5.