Robust Mixed-Sensitivity H Control For a Class of MIMO Uncertain Nonlinear IPM Synchronous Motor Via T-S Fuzzy Model

Robust Mixed-Sensitivity H Control For a Class of MIMO Uncertain Nonlinear IPM Synchronous Motor Via T-S Fuzzy Model Ahmad Fakharian Faculty of Electrical and Computer Engineering Islamic Azad University, Qazvin Branch Qazvin, Iran E-mail: Ahmad.Fakharian@qiau.ac.ir Vahid Azimi Electrical Engineering Department Islamic Azad University, South Tehran Branch Tehran, Iran E-mail: Vahid.Azimii@gmail.com Abstract- This article presents robust mixed-sensitivity H output feedback controller by using loop shaping with regional Pole Placement for a class of MIMO uncertain nonlinear system. In order to design of controller first via Takagi and Sugeno's (T-S) fuzzy approach the nonlinear dynamic is represented by several linear sub systems. After that, loopshaping methodology and Mixed-sensitivity problem are introduced to formulate frequency-domain specifications and a systematic design of weighting matrices is presented. Then a regional pole-placement output feedback H controller is employed by using linear matrix inequality (LMI) approach for each linear subsystem. Parallel Distributed Compensation (PDC) is used to design the controller for the overall system and the total linear system is obtained by using the weighted sum of the local linear system. Several results show that the proposed method can effectively meet the performance requirements like robustness, good load disturbance rejection responses, good tracking responses and fast transient responses for the 3-phase interior permanent magnet synchronous motor (IPMSM) system. In addition, the superiority of the proposed control scheme is indicated in comparison with the feedback linearization and the H2/H controller's methods. Keywords: LMI, Mixed-sensitivity problem, PDC, Robust control, T-S fuzzy model, 3-phase interior permanent magnet synchronous motor (IPMSM) 1 INTRODUCTION Amid the customarily used motors, IPMSM has been progressively replacing DC and induction motors in a wide range of drive many industrial applications. The reason why an IPMSM has become so well-liked is essentially due to its many pleasant characteristics such as high efficiency, exceptional power density, excellent torque generating and high torque-to-current ratio and else. Numerous nonlinear and linear controllers have been developed for the IPMSM. For example: C.-K. Lin et al. [1]used Nonlinear position controller design with input output feedback linearization technique for an IPMSM control system. S.S. Yang et al. [2]implemented a Robust speed tracking of PMSM servo systems by equivalent disturbance attenuation. Y. X. Su et al.[3] Studied the Automatic Disturbances Rejection Controller for Precise Motion Control for a PMSM drive. Ming-Chang Chou et al. [4]has proposed a Development of Robust Current 2-DOF Controllers for a PMSM Drive With Reaction Wheel Load. Cheng-Kai Lin et al. [5]investigated the Adaptive Back ping PI Sliding-Mode Control for Interior Permanent Magnet Synchronous Motor Drive Systems. The available papers studied the sliding mode technique to PMSM or IPMSM drive systems [6, 7]. The published papers used the T-S fuzzy model technique to different drive systems [8, 9].The main contribution of this research is robust mixed-sensitivity H control for a class of MIMO uncertain nonlinear PM synchronous motor via T-S fuzzy model. In this paper the problem of robust mixedsensitivity H control for an IPMSM system which possesses not only parameter uncertainties but also external disturbances are considered. Several robust H, loop shaping and mixed-sensitivity problem schemes based on the use of LMIs theory have also been proposed in [10-12]. In the proposed method first nonlinear plant is represented by Takagi Sugeno (T-S) fuzzy model. Then by used of loopshaping methodology and mixed-sensitivity problem and by designed of a regular flowchart is proposed optimal weighting functions design method. The objectives in this research are rotor angular position tracking and reluctance effects and torque ripple reduce. Afterward, for each fuzzy linear subsystem a robust mixed-sensitivity H output feedback controller with regional pole-placement is designed by LMI-based design techniques. To design linear feedback control PDC is utilized to design the controller for the overall system.so the overall fuzzy model of the system is achieved by fuzzy "blending" of the local linear system models. The main advantage of the proposed T-S fuzzy control design approach is to deal with stability and performance over the complete operational range of the nonlinear system.and finally several results show that the proposed method can effectively meet the performance requirements like robustness, good load disturbance rejection responses, good tracking responses and fast transient responses for the 3-phase interior permanent magnet synchronous motor (IPMSM) system. The paper is organized as follows: IPMSM model dynamic in Section II. In Section III, T-S fuzzy model of IPMSM and problem formulation is introduced. Section IV describes H loopshaping and the mixed-sensitivity problem. In section V is proposed design of robust tracking controller. Simulation result of the closed-loop system with the proposed technique is presented in Section VI and finally the paper is concluded in Section VII. 2 IPMSM MODEL DYNAMIC The nonlinear electrical and mechanical equations for the 3- phase Interior permanent magnet synchronous motor in the d-q reference frame can be written as follows [1] : dθ dt =ω dω dt = 3 P L 2 J L i + i B ω J C J di dt =R L i L +P ω L i + 1 v L di dt =P L ω L P ω L i R i L + 1 v L (1) Corresponding author email: ahmad.fakharian@qiau.ac.ir 978-1-4673-2124-2/12/$31.00 2012 IEEE 546

In equation (1) θ is the angular position of the motor shaft, ω is the angular velocity of the motor shaft, i d is the direct current and i q is the quadrature current. is the flux linkage of the permanent magnet, P is the number of pole pairs, R s is the stator windings resistance, L d and L q are the direct and quadrature stator inductances respectively. J is the rotor moment of inertia, B the viscous damping coefficient and C l is the load torque. v d is the direct voltage and v q is the quadrature voltage. The electromagnetic torque of the motor can be described as T = 3 2 P L L i i + i (2) The parameters R s and B are supposed to differ from their nominal values R and B.The following equation indicates a state space representation of the synchronous motor x =x x = (η x + η )x + η x C J x = η x + η x x + η u x = η x + η x x + η x + η u (3) x=x x x x T = θ ω i i T (4) u u T = v v T η = 3 P L 2 J L η = 3 P 2 J η = R L η =P L η = 1 L L η =P L L η =P L 3 T-S FUZZY MODEL OF IPMSM AND FORMULATION PROBLEM In this section the T--S fuzzy dynamic model is described by fuzzy IF--THEN rules, which represent local linear input-- output relations of nonlinear systems [13].The fuzzy dynamic model is proposed by Takagi and Sugeno. The ith rule of T-S fuzzy dynamic model with parametric uncertainties can be described as follows: IF v (t) is M and and v (t) is M THEN x(t) =A + A x(t) +B + B w(t) + B + B u(t), x(0) =0 z(t) =C + C x(t) +D + D w(t) + D + D12iut η = B J η = 1 L η = R L y(t) =C + C x(t) +D + D w(t) + D + (5) D22iut i = 1,2,, r (7) Where, M is the fuzzy set; r is the number of IF-THEN Rules and v (t) v (t) are the premise variables. The matrices: ΔAi, B, B, C, C, D, D, D, D represent the uncertainties in the system and satisfy the following assumption: A =G(x(t),t)H B =G(x(t),t)H B =G(x(t),t)H C =G(x(t),t)H D =G(x(t),t)H D =G( (x(t),t)h C =G(x(t),t)H D =G( (x(t),t)h D =G(x(t),t)H (8) (6) Where, H ; j = 1,,9 are known matrix functions, that characterize the structure of the uncertainties. Furthermore, the following inequality holds: G(x(t),t), 0 Considering the nonlinear state space model (1), the parameters R s and B and Load torque disturbance input C are supposed to vary. According to the above local linearization approach, the local linear model matrices for the system (3) with mentioned variations at the ith selected operating point are obtained as follows: A =A +B A B +R A R (9) 0 1 0 0 0 0 B B = J = η 0 0 0 η 0 (10) 0 1 0 0 0 0 η A = x η 0 0 0 0 0 1 0 0 0 η x 0 0 A B = J 0 η η x 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 A R = 0 0 1 0 L 0 0 0 1 L (11) Consequently, just x 2 and x 4 are nonlinear terms that exist ina. The overall fuzzy model is of the following form x(t) = µ v(t)a + A x(t) +B + B w(t) + B2i+ B2iut, x0=0 z(t) = µ v(t)c + C x(t) +D + D w(t) + D12i+ D12iut y(t) = µ D21iwtD22i D22iut v(t) = v (t) v (t) µ v(t) = v(t) v(t) v(t)c + C x(t) +D + v(t) = M v (t) v(t) 0, i = 1,2,, r ; v(t) 0 µ v(t) 0, i = 1,2,, r ; µ v(t) =1 (16) 4 H LOOP SHAPING AND THE MIXED SENSITIVITY PROBLEM Loop shaping is a design procedure to formulate frequencyconstraints problems[14]. To domain specifications as H get a feeling for the loop-shaping methodology, consider the general pattern loop of Fig. 1. Fig. 1. The Control structure. (12) (13) (14) (15) 547

The H norm of a transfer function F(s) is corresponds to the peak gain of the frequency response F(j ), and is denoted = (()) (17) Alternatively, we can specify some maximum value for the closed-loop RMS gain as (, ) (18) Where is guaranteed H, ratio between z and w. In this case, the closed-loop transfer function T zw (s) is as follows: T () =(, ) = ()() ()() (19) Where S(s) is the sensitivity transfer matrix (the transfer function from r to e) and T(s) is the complementary sensitivity transfer matrix (the transfer function from r to y): () =(I+G(s)K(s)) () =G(s)K(s)(I + G(s)K(s)) (20) The W S (s) and W T (s) are two frequency dependent weighting functions (shaping filters), sensitivity weighting function and the complementary sensitivity weighting function respectively. Accordingly in order to realize above requirement can be used to normalize the constraint: ()() 1 T ()() 1 (21) In this paper, because the number of system outputs and tracking errors on both cases is equal to 2, in result the size of weighting functions W S (s) and W T (s) are 2 2 matrices as such in this case S(s) and T(s) are: () = = = = () = = = = (22) Where 1 and 2 are angular position and d-axis current, 1 and 2 are position command and direct current reference input and 1 and 2 are the tracking errors. Thereby W T (s) consists of a 2 2 square diagonal matrix with all its diagonal elements with the same transfer function () and so W S (s) is proposed to be a square diagonal matrix with same diagonal elements Sii (): T () = T () I () = S () I (23) The transfer functions Sii () and () must be stable, minimum phase and additionally should be proper. As well as Sii () and () must be low-pass and high-pass filters respectively. A practical formula to determine the performance and robustness weights are as follows. () = + 1 + 2 S () = + + (24) Where the gain for high frequency disturbances is, is the gain for low frequency control signal, is a constant and is the crossover frequency. is the frequency which differentiates the high frequency disturbance signal and the low frequency control signal. Now we have to introduce an effective procedure that how to design each weighting matrix. Therefore a global method can be achieved for all performance and robustness specifications for the synthesis of the robust H controller. The global regular weight selection algorithm is described in the flowchart shown in Table 1. TABLE 1. REGULAR WEIGHTS DESIGN METHOD Answer: NO Go back to: Step 2 and/or Step 5 Go back to: Step 2 and/or Step 5 Go back to: Step 2 and/or Step 5 Step Number - 1 2 3 4 5 6 7 - - Step description Start Get local linear subsystem, G i and new input values: a, b, d, w c Design weighting matrices, Build up the plant P and Augmentation (Gi, W T(s), W S(s)) Specify LMI Pole Placement Region, D and Find local controller by LMI approach, K Find the close-loop responses: S(s): S1(s), S2(s) and T (s): T1(s), T2(s) Is true this two constraints? σ S(jw) γ σ W (jw) σ T(jw) γ σ W T (jw) Is the guaranteed H less than 1? Is close-loop system steady state and transient responses expected for tracking and disturbance attenuation K is optimal and WT(s), WS(s) are desirable Stop Answer: YES In this flowchart for selecting of two weighting functions in order to formulate performance and robustness specifications of close-loop system, a, b, c, w values should be decremented or incremented as far as whole of clauses in each is realized and finally a stable controller is returned. 5 DESIGN OF ROBUST TRACKING CONTROLLER In this section focus to design a local pole-placement output feedback controller for each linear subsystem: () () () = (), = 1,2,, (25) Where K i (i =1,2,,r) are the local controller gains to be determined. For the system (7), the concept of parallel distributed compensation (PDC) is employed[15]. According to PDC approach, the control law of the whole system is the weighted sum of the local feedback control of the various subsystems. That is: u(t) =µ K y(t) (26) Where, the local pole-placement output feedback gains K j are determined by LMI-based design techniques in order to achieve the design requirements[14]. The LMI formulation is applicable for designing the local controllers which are introduced in Theorem1. Theorem1. Main objective is to design an outputfeedback controller u = Ky such that: maintains the H norm of T (s) (RMS gain) below some prescribed value > 0 maintains the H 2 norm of T 2 (s) (LQG cost) below some prescribed value > 0 places the closed-loop poles in some prescribed LMI region D Minimizes a trade-off criterion of the form αt +βt. T (s) and T 2 (s) are the closed-loop transfer functions from w to z and z 2, 548

respectively. For the control structure shown in Fig. 1, the linear fuzzy sub plant P(s) is given in the following statespace form = + + = + + = + + = + (27) And related controller K(s) is introduced by = + = + (28) With regard to P(s), K(s) and u = K y the closed-loop system is given by = + = + = + (29) Our three design objectives can be expressed in LMI formulation as follows: H performance: The closed-loop RMS gain from w to z does not exceed γ>0 if and only if theree exists a symmetric matrix X such that + 0,X 0 (30) H 2 performance: The H 2 norm of the closed-loop transfer function from w to z 2 does not exceed ν if and only if D cl2 = 0 and there exist two symmetric matrices X 2 and Q such that Q C X T 0 A T X +X A B X C X T B I 0 Trace(Q) ν (31) Pole placement: The closed-loop poles lie in the LMI region =z C:L+Mz+M T z0 L=L T =λ M = µ,, (32) If and only if there exists a symmetric matrix Xpol satisfying λ X +µ (A+B K)X +µ X +µ X (A+ B2KT1 i,j m>0, Xpol<0 (33) For tractability in the LMI framework, we seek a single Lyapunov matrix: X:=X =X 2 =Xpol that enforces all three sets of constraints. Factorizing X as X= 0 0 And introducing the change of controller variables + + ++ + + (34) The inequality constraints on χ are readily turned into LMI constraints in the variables R, S, Q, A K, B K, C K, and D K. This leads to the following suboptimal LMI formulation of our multi-objective synthesis problem: Minimize αγ +β Trace (Q) over R, S, Q, A K, B K, C K, D K and γ satisfying: + + + + + ++ + + + + + 0 + + + 0 + + + + + + + + 0 1 i,j m Trace(Q) ν γ γ + =0 (35) Given optimal solutions γ*, Q* of this LMI problem, the closed-loop H and H 2 performances are bounded by T γ T Trace(Q ) (36) The purpose is to design a suitable control which guarantees robust performance in presencee of parameters variation and load torque disturbance. In this case there are two control objectives. First, the rotor angular position x 1 must track a reference trajectory r 1. Secondly, the direct current x 3 to track a constant reference r 2 = 0. This objective is equivalent to the nonlinear electromagnetic torque can be linearized to avoid reluctance effects and torque ripple. Therefore to achieve these objectives just both tracking errors is minimize. Mentioned goals are realized through Construct the objectives z in a appropriate control loop. Under the above considerations, the structure of the fuzzy robust control loop is proposed that shown schematically in Fig. 2. Fig. 2. The fuzzy robust control loop structure. In above structure first nonlinear dynamic model is approximated to many local linear models in each rule which represent by T--S fuzzy approach. Then, designed two shaping filters W S (s) and W T (s) and Build up the augmented plant P and controller is designed for each linear sub plant based on LMI approach.. After that the total linear system is obtained by used of the weighted sum of the local linear system and is utilized rather than original nonlinear system. So according to PDC approach, the control law of the whole system is the weighted sum of the local feedback control of the various subsystems. Finally by used of whole system and global controller is applied a tracking loop in order to achieve to desirable specifications such as tracking performance, bandwidth, disturbance rejection, and robustness for close-loop system. 6 SIMULATION RESULT The motor type used in this paper is the 130-750MS-ZK-L2. [1]. The parameters of the IPMSM are shown in Table 2. The stator windings resistance R and the viscous damping coefficient B are varied between 50% and load torque disturbance is unknown [1, 6]. According to IPMSM characteristic and system operating points we can assume that v (t) =x 0 2000, v (t) =x 0 12 549

Which v and v are fuzzy variables. As a result the membership functions can be calculated M = v, M 3000 =1 v as: 3000 M = v, M 6 =1 v 6 TABLE 2. THE PARAMETERS OF IPMSM Rs 1.9 Ω Bmo (without load) 0.03 N m s/rad Bmo (with load) 0.0341 N m s/rad Ld 0.0151 H Lq 0.031 H 0.31 V s/rad Jmo (without load) 0.0005 kg.m 2 Jmo (with load) 0.0227 kg.m 2 Po 2 First to derive a T-S fuzzy model from the nonlinear system (3) that is modelled by used of fuzzy rules (7) with r=4.referring to (9)-(10)-(11) whole of system matrices are constant except A that varying in various rules according to following matrices: A 0 1 0 0 0 1 0 0 0 0 1145 1860 0 0 0 1860 = = 0 49 0 0 0 0 0 0 0 20 82119 0 0 20 82119 0 A 0 1 0 0 0 1 0 0 0 0 0 1860 A = 0 0 1145 1860 0 0 0 0 = 0 49 0 0 0 20 0 0 0 20 0 0 Referring to section IV, the weighting matrices W T (s) and W S (s) has been designed as follows: T () = T () I = 0.001 + 1 0.001 + 2 I 0.5 + 10 () = S () I = + 0.01 I Then by used of purposed control loop (Fig. 2) and mentioned weighting matrices and the application of theorem 1 to calculate the local controller for each linear subsystem. In order to design output feedback gains (K i ) for each subsystem, below s are done: Specify the LMI region (32), in order to place the closed-loop poles in this region (pole placement) and also to guaranteee some minimum decay rate and closed-loop damping. The characteristic of appointed region is: the intersection of the half-plane is x<-8 and of the sector centered at the origin and with inner angle π/2. Choose a four-entry vector specifying the H2/H trade-off criterion in theorem1: [γ0 ν0 α β]= [0 0 1 0]. As a matter of fact, in this case, constraint and criterion of H2 is not used (β=0) and just two design objectives, H performance and pole placement are employed Minimize H 2 /H cost function based on theorem 1 subject to the mentioned pole placement constraint by using (30)-(31)- (34)-(35)-(36). Finally by using a weighted average defuzzifer, is obtained the overall fuzzy system and the control law of the whole system. Global proposed T-S fuzzy model can exactly represents the nonlinear system in the region [0, 2000] rpm [0, 12] A on the x 2 -x 4 space for various operating point. The external load is obtained by using the one of two type disturbance. first, another synchronous motor is coupled to the shaft of the main IPMSM motor in order to request a load torque[6]. The manners of the load torque C l applied to the synchronous motor are represented in Fig. 3. Second A the external load 1 N m that proposed with input. Fig. 3 shows real direct current response that track 0 A in order to elimination of reluctance effects and torque ripple for proposed and H 2 /H method. Fig. 4 and 4 demonstrates disturbance rejection on angular position in three different methods, with load torque and benchmark load torque (Fig. 3( ) respectively, though, first the IPMSM is controlled to reach a fixed position, 90 degree. Fig. 3. Load torque d-axis Current Fig. 4. Disturbance rejection on angular position with load torque (1 Nm) with benchmark load torque (Fig. 3) Mentioned figures show the comparison of the proposed method, the H 2 /H controller and the feedback linearization technique (represented in Ref. [1]) in presence disturbances. As you can view, the proposed method has better disturbance attenuation in both type of external load. And table 3 abridges disturbance rejection differences for load torque (1 Nm) in three methods. According to this Table, the proposed method has the smallest undershoot value of another method on disturbance rejection responses. Fig. 5 and 5 shows the responses of the certain position (180 degree) and rectangular position commands respectively for the proposed, H 2 /H and feedback linearization (represented in Ref. [1]) controllers. According to this figures the proposed controller has the best values of settling time and raise time into another mentioned method. Table 3 also summarizes the results of transient responses for three mentioned methods. Referring to this table, the proposed method has the smallest settling time value of another method on position tracking responses. Fig. 5. Comparison of transient responses certain position rectangular position command Fig. 6 illustrates the position responses when both the parameters of the stator windings resistance R s and the viscous damping coefficient B m are varied between ±50%. As you can see, the system has good robustness when the parameters in the systems dynamic are varied in a wide 550

range. Fig. 6 illustrates the position tracking responses at different position commands by using the proposed controller. According to this figure, the proposed system has satisfactory performance for various position commands. Fig.6. Angular position responses with varying Rs and B m position tracking responses at different position reference TABLE 3. THE COMPARISON OF DESIGN CHARACTERISTIC IN THREE METHODS Item Settling time(sec) Undershoot Method value (Degree) Proposed 0.15 0.06 H2/H 0.28 0.2 Feedback linearization (represented in Ref. [1]) 0.4 1.1 Fig. 7 and 7 demonstrates W and W T, that they are greater than S 1, S 2 and T 1, T 2 on frequency domain respectively.as well as this weights are greater than sensitivity functions and complementary sensitivity functions variety correspondingly, that cause of this variation on S and T is parameters uncertainty. Fig.7. W matrix as an upper bound of the S 1and S 2 uncertainty W T matrix as an upper bound of the T 1and T 2 uncertainty 7 CONCLUSIONS In this paper, a robust mixed-sensitivity H controller has been designed in order to tracking and disturbance attenuation of position and current, for a MIMO nonlinear uncertain IPMSM system. First to approximate uncertain nonlinear system, the T-S fuzzy technique is employed. Then in order to improve frequency-domain specifications is presented loop-shaping methodology and Mixed-sensitivity problem. After that based on each linear model a robust pole-placement output feedback H controller is determined by LMI-based design techniques in order to achieve the robustness design of nonlinear uncertain systems. Final PDC is used to design the controller for the overall system and the total linear system is obtained by used of the weighted sum of the local linear system. The simulation results on IPMSM show that the robust control system has small position and current tracking error and has desired robustness against load torque disturbance and parameter variations. Proposed position and current control system has good transient responses and load disturbance rejection and tracking responses. In addition, the superiority of the proposed control scheme is indicated in comparison with the feedback linearization (represented in Ref. [1]) and the H2/H Controller controller's methods. Tables III represented performance comparison to three different methods. Referring to this table the proposed method had smallest settling time value on position tracking response and the lowest undershoot value on disturbance rejection response. The major achievements of this research are delivered as follows: (i) The performance requirements like good load disturbance rejection, tracking and fast transient responses in proposed method that have been presented by table III, were better than other control methods (ii) The proposed method had satisfactory tracking for various angular position commands (iii) System has good robustness when parameter trajectories of Rs(t) and B m (t) were changed in real time (iv) Direct current of system has tracked 0 A for elimination of reluctance effects and torque ripple. REFERENCES [1] C.-K. Lin, T.-H. Liu and S.-H. Yang, "Nonlinear position controller design with input output linearization technique for an interior permanent magnet synchronous motor control system,". IET Trans. Power Electron., vol.1, No. 1, pp. 14 26, 2008. [2] S.S. Yang and Y.S. Zhong, "Robust speed tracking of permanent magnet synchronous motor servo systems by equivalent disturbance attenuation,". 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