Impulsive control for permanent magnet synchronous motors with uncertainties: LMI approach

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1 Impulsive control for permanent magnet synchronous motors with uncertainties: LMI approach Li Dong( 李东 ) a)b) Wang Shi-Long( 王时龙 ) a) Zhang Xiao-Hong( 张小洪 ) c) and Yang Dan( 杨丹 ) c) a) State Key Laboratories of Mechanical Transmission Chongqing University Chongqing China b) College of Mathematics and Physics Science Chongqing University Chongqing China c) College of Software Engineering Chongqing University Chongqing China (Received 4 November 008; revised manuscript received 3 August 009) A permanent magnet synchronous motor (PMSM) may have chaotic behaviours under certain working conditions especially for uncertain values of parameters which threatens the security and stability of motor-driven operation. Hence it is important to study methods of controlling or suppressing chaos in PMSMs. In this paper the stability of a PMSM with parameter uncertainties is investigated. After uncertain matrices which represent the variable system parameters are formulated through matrix analysis a novel asymptotical stability criterion is established by employing the method of Lyapunov functions and linear matrix inequality technology. An example is also given to illustrate the effectiveness of our results. Keywords: permanent magnet synchronous motors impulsive control uncertainty linear matrix inequality PACC: Introduction Permanent magnet synchronous motors (PMSMs) are of great interest particularly for industrial applications in the low-medium power range since they have superior features such as compact size high torque/weight ratio high torque/inertia ratio and absence of rotor losses. 1] Moreover compared with induction motors PMSMs have the advantages of higher efficiency due to the absence of rotor losses and lower no-load current below the rated speed. ] The control technique analysis and systematic design for PMSMs are among the most important issues because the performance of the PMSMs is very sensitive to external load disturbances and parameter variations. Thus a considerable amount of control techniques have been developed to overcome these problems. Many approaches such as feed-forward control nonlinear control optimal control variable structure system control adaptive control hybrid control and robust control have been developed for the PMSMs to deal with uncertainties under various operating conditions. 1 1] Besides this we must point out that a PMSM is a nonlinear system. The control techniques for nonlinear systems have been widely studied such as impulsive control strategy 13 0] and fuzzy control strategy. 1] The impulsive control strategy has become an important control method for nonlinear systems recently. From the references cited above we find that they usually used a special Lyapunov function 1 xt x or 1 xt Γ x (where Γ is a special positive definite matrix). But the construction of the Lyapunov function itself is one of the important parts of the stability theory. The linear matrix inequality technique can be used for solving the problem. Moreover the PMSM is a special nonlinear system which is more rigorous than ordinary nonlinear systems. So a stability study on PMSMs is important. We cannot regard it as a simple application of the nonlinear system. The control schemes for PMSMs (or nonlinear systems) described above have been widely used in industry. However the values of parameters of the PMSM model are not constant but span some interval. The main reason for this is that the permanent magnetic-flux will vary with the following factors: the temperature rise in the permanent-magnet Project supported by the National Natural Science Foundation of China (Grant No ) the Chongqing Natural Science Foundation (Grant No. CSTC 008BB3308) and the Innovation Training Foundation of Chongqing University (Grant No. CDCX004). Corresponding author. lid@cqu.edu.cn (Li D.) c 010 Chinese Physical Society and IOP Publishing Ltd

2 material the variation of rotary inertia due to the attrition of the device the modeling errors and the measurement inaccuracy etc. However only a few studies on the nonlinear systems with variable parameters exist. 1 3] Therefore the control strategy of PMSMs with parameter variation is quite valuable. Impulsive control has the advantages of higher reliability and stronger anti-jamming ability. If applying analog voltage to control the motor the control voltage will rise to the limit rapidly when a blunder in wiring topology or some element breakage happens. But this kind of incident will not happen if applying impulse as control signal to control the motor. So impulsive control for PMSMs with parameter variation will be discussed in this paper. To the best of our knowledge few reports exist on stability results for uncertain impulsive dynamical systems and far fewer for PMSMs with parameter variation have been presented in the literature. The organization of the paper is as follows. In the next section the problems investigated in this paper are formulated and some preliminaries are presented. In Section 3 the asymptotic stability criteria for the equilibrium points of PMSMs with parameter variation are derived. A numerical example and simulations are presented in Section 4. Finally some conclusions are drawn in Section 5.. Problem formulation and preliminaries The machine model of a PMSM with parameter uncertainties can be described in the rotor rotating reference frame as follows: di d dt = ( Ri d + ωl q i q ) /L d di q dt = ( Ri q ωl d i d ωψ r ) /L q dω dt = n pψ r i q + n p (L d L q )i d i q T L βω] /J (1) where i d and i q are respectively d q axis stator currents ω is rotor speed L d and L q are d q axis stator inductance; R is stator resistance T L the external load torque ψ r the maximum flux induced by the rotor magnet n p the number of pole-pairs β the damping coefficient and J the moment of inertia of the rotor. Because there exist some uncertainties such as modeling errors and measurement inaccuracy in this uncertain system R R 1 R ] β β 1 β ] ψ r ψ 1 ψ ] and J J 1 J ] are unknown parameters which span their own intervals. Apply the affine transformation and time scale transformation to system (1) with x = λ x t = τ t where x = i d i q ω ] T x = ĩ d ĩ q ω ] T b = L q /L d k = β/(n p τψ r ) τ = L q /R with τ = L q R k = λ d 0 0 bk 0 0 λ = 0 λ q 0 = 0 k λ ω 0 0 1/τ Lq β n p τψ r R L q R 1 ] β1 n p τψ β n p τψ 1 ] R1 β 1 n p L q ψ R β n p L q ψ 1 The dimensionless mathematical model of PMSM can be described as follows: dĩ d dt = ĩ d + ωĩ q where dĩ q dt = ĩ q ωĩ d + γ ω dω dt = σ(ĩ q ω) + εĩ d ĩ q T L γ = ψ r np ψ1 n pψ ] kl q R β R 1 β 1 σ = βτ J β1 L q β ] L q J R J 1 R 1 ]. () ] T L = τ L J T q T 1 L R J L qt R1 J ε = n pbτ k (L d L q ) 1 J br 1 β1(l d L q ) n p R J ψ br β(l ] d L q ) n p R1 J 1ψ1. Considering a PMSM with a smooth air gap we have L d = L q = L and ε = n pbτ k (L d L q ) =0. J Let x = x 1 x x 3 ] T = ĩ d ĩ q ω ] T. The chaotic mathematical model of a smooth air gap PMSM with parameter variation and impulsive effects can be expressed by

3 ẋ 1 = x 1 + x 3 x ẋ = x 3 x 1 x + γx 3 t τ k k = 1... ẋ 3 = σ(x x 3 ) ( ) ( ) x t=tk = x τ + k x τ k = Bx k = 1... (3) where x = x 1 x x ] T are state variables without dimension B R 3 3 is a diagonal matrix representing the impulsive state control in place of the external load torque. B B 1 B ] represents the uncertain impulsive control where B 1 and B are also diagonal matrices. We denote and γ = ψ r kl q σ np ψ 1 R β n pψ R 1 β 1 ] = r 1 r ] βlq βl ] q = l 1 l ]. J R J 1 R 1 Obviously x = γ 1 γ 1 γ 1] x = γ 1 γ 1 γ 1] and x = 0 0 0] are the equilibrium points of system (3). We will discuss the stability of the origin of system (3). In the following in order to discuss the stability of system (3) we present some preliminary results with respect to impulsive differential equations initially. 3. The impulsive stability of a PMSM with uncertain parameters ] T. Let Φ(x) = x 3 x x 1 x 3 0 Under the initial condition x ( t + ) 0 = x0 0 system (3) becomes where ẋ = Ax + Φ(x) t τ k k N x t=τk = Bx k N x ( t + ) 0 = x A = 0 1 γ. 0 σ σ (4) The matrix A is an uncertain matrix due to γ r 1 r ] σ l 1 l ] and B B 1 B ]. In order to obtain the stability result for system (3) we shall establish some lemmas first. Lemma (i) Let A = 0 1 γ where γ r 1 r ] σ l 1 l ] and let P = 0 1 r 1 Q = 0 1 r. 0 σ σ 0 l 1 l 0 l l 1 Then A can be written as A = A 0 + EΣF (5) where A 0 = 1 (P + Q) H = (h ij) 3 3 = 1 (Q P ) ; Σ Σ = {Σ R 9 9 : Σ = diag(ε 11 ε 1... ε ε ε 33 ) ε ij 1 ε 3 = ε 33 i j = 1 3} E = ( h 11 e 1 h 1 e 1 h 13 e 1... h 31 e 3... h 33 e 3 ) R 3 9 F = ( h 11 e 1 h 1 e h 13 e 3... h 31 e 1... h 33 e 3 ) T R 9 3 e 1 = (1 0 0) T e = (0 1 0) T e 3 = (0 0 1) T. (ii) Let diagonal matrix B B 1 B ]; if B 0 = 1 B 1 + B ] G = 1 B B 1 ] = diag(g 1 g g 3 ) then B can be written as B = B 0 + Ẽ ΣẼ (6) where Σ Σ = { Σ R 3 3 : Σ = diag(ε 1 ε ε 3 ) ε i 1} Remark 3.1. Clearly for any Σ Σ and Σ Σ we have Ẽ = diag( g 1 g g 3 )

4 (i) ΣΣ T = Σ T Σ{ I Σ ΣT = Σ T Σ I (3 3 identity matrix); { (ii) EE T = diag h 1j h j h 3j } F T F = diag h i1 j=1 j=1 j=1 i=1 (iii) by Lemma 3.1 system (3) can be rewritten as 3 h i i=1 3 h i3 }. i=1 ẋ = A 0 x + EΣF x + Φ(x) t τ k k = 1... ( x t=τk = B 0 + Ẽ ΣẼ ) x k = 1... x ( τ 0 + ) = x0 0 (7) where A 0 B 0 E Σ F Ẽ and Σ are defined as in Lemma 3.1. Lemma 3. 7] If Σ Σ then for any positive scalar λ > 0 and for any ξ R n the following inequality holds ξ T η 1 λ ξt ξ + λη T η (8) especially the inequality holds for λ = 1. Lemma ] Let M E F and Σ be real matrices of appropriate dimensions with Σ satisfying Σ 1. Then we have (i) For any scalar λ > 0 EΣF + E T Σ T F T λ 1 EE T + λf F T. (9) (ii) For any matrix P > 0 and scalar ξ > 0 such that ξi F T P F > 0 we have (M + EΣF ) T P (M + EΣF ) M T P M + M T P F (ξi F T P F ) 1 F T P M + ξ 1 E T E. (10) Especially the above two inequalities hold for λ = 1 and ξ = 1. Lemma ] The following linear matrix inequality (LMI) is positive Q (x) S (x) > 0 S T (x) R (x) where Q(x) = Q T (x) R(x) = R T (x) and S(x) depend on x is equivalent to R (x) > 0 Q (x) S (x) R 1 (x) S T (x) > 0. (11) Theorem 3.1. Assume that there exist the positive constants λ > 0 β > 0 η > 1 and positive definite matrix Γ such that the following LMIs hold: (i) Γ A 0 + A T 0 Γ + F T F λγ E T Γ Γ E 0 (1) I (ii) (I + B 0) T Γ (I + B 0 ) + ẼT Ẽ βγ (I + B 0 ) T Γ Ẽ Ẽ T Γ (I + B 0 ) I (ẼT Γ Ẽ) 0. (13) If ln(ηβ) + λ(τ k+1 τ k ) 0 for all k then system (3) is robust and asymptotically stable. Proof Let the Lyapunov function V (x) = 1 xt Γ x. When t = τ k (k = 1... ) we have V (τ + k x) = 1 (x(τ k) + (B 0 + Ẽ ΣẼ)x(τ k)) T Γ (x(τ k ) + (B 0 + Ẽ ΣẼ)x(τ k)) = 1 x(τ k) T (I + B 0 + Ẽ ΣẼ)T Γ (I + B 0 + Ẽ ΣẼ)]x(τ k). (14) By Lemma 3.3 we have (I + B 0 + Ẽ ΣẼ)T Γ (I + B 0 + Ẽ ΣẼ) (I + B 0 ) T Γ (I + B 0 ) + (I + B 0 ) T Γ Ẽ(ξI ẼT Γ Ẽ) 1 Ẽ T Γ (I + B 0 ) + ξ 1 (ẼT Ẽ). (15)

5 Thus Eq. (14) turns into V (τ + k x) 1 x(τ k) T (I + B 0 ) T Γ (I + B 0 ) + (I + B 0 ) T Γ Ẽ(I ẼT Γ Ẽ) 1 Ẽ T Γ (I + B 0 ) + (ẼT Ẽ)]x(τ k ). (16) By the condition (ii) we have V (τ + k x) βv (τ + k x). (17) ] T When t τ k note that Φ(x) = x 3 x x 1 x 3 0 so we obtain Φ T (x)x+x T Φ(x) = 0. By the assumption and Lemmas and 3.4 for all (t x) (τ k 1 τ k ] R n (k = 1... ) we have D + V (t x) = 1 (ẋt Γ x + x T Γ ẋ ) = 1 (A 0x + EΣF x + Φ(x)) T Γ x + 1 xt Γ (A 0 x + EΣF x + Φ(x)) = 1 ( x T A T 0 + x T F T Σ T E T + Φ T (x) ) Γ x + 1 xt Γ (A 0 x + EΣF x + Φ(x)) = 1 xt A T 0 Γ + Γ A 0 + F T Σ T E T Γ + Γ EΣF ] x + 1 Φ T (x)γ x + x T Γ Φ(x) ]. (18) Note that 1 ΦT (x)γ x + x T Γ Φ(x)] 1 λ max (Γ ) ( Φ T (x)x + x T Φ(x) ) = 0 and F T Σ T E T Γ + Γ EΣF F T Σ T ΣF + Γ EE T Γ F T F + Γ EE T Γ. Thus by the condition (i) of the theorem we have D + V (t x) 1 xt A T 0 Γ + Γ A 0 + F T F + Γ EE T Γ ] x λv (t x). (19) Let k = 1 for all t (τ 0 τ 1 ] by Eq. (19) we have V (t x) V (τ 0 x) exp (λ(t τ 0 )). (0) Then it leads to V (τ 1 x) V (τ 0 x) exp (λ(τ 1 τ 0 )). (1) From the inequality (17) we have For all t (τ 1 τ ] we have V (τ + 1 x) βv (τ 1 x) βv (τ 0 x) exp (λ(τ 1 τ 0 )). () V (t x) V (τ + 1 x) exp (λ(α)(t τ 1)) βv (τ 0 x) exp (λ(t τ 0 )). (3) Similarly for all k and t (τ k τ k+1 ] we have By the condition ln(ηβ) + λ(α)(τ k+1 τ k ) 0 we have Hence for all t (τ k τ k+1 ] (k = 1... ) we have V (t x) β k V (τ 0 x) exp (λ(t τ 0 )) V (t x) β k V (τ 0 x) exp (λ(t τ 0 )). (4) β exp(λ(τ k+1 τ k )) 1 k = (5) η = V (τ 0 x) β exp (λ(τ k τ k 1 ))] β exp (λ(τ 1 τ 0 ))] exp (λ(t τ k )) V (τ 0 x) 1 η k exp (λ(t τ k)). (6) So if t then k and V (t x) 0. It follows that the conclusion holds. In the following we will present the design algorithm for the stabilization of system (3) based on Theorem

6 Algorithm In order to reduce the cost of control or implementation we hope that the impulsive distance is as large as possible. Hence we hope that λ > 0 and 0 < β < 1 are as small as possible such that the impulsive distance is larger. We have the following design algorithm for the robust stability of system (3). 1) Set a threshold T initialize λ > 0 β > 0. For example λ = 0.1 β = 0.1; ) Calculate P by (i) and (ii) of Theorem 3.1; 3) If P exists stop. Otherwise let β = β + β; 4) Repeat from step if β < 1/η. Otherwise θ = θ + θ and β is equal to the initial value; 5) Repeat from step if λ < T. Otherwise fail. Once the algorithm is successful we may determine the bound of the impulsive distance by (τ k+1 τ k ) ln(ηβ)/λ. Hence only if Algorithm 1 is slightly changed by satisfying 0 < β < 1/η may we obtain the asymptotical stability design algorithm of a chaotic system. Let B 1 = diag( ]) B = diag( ]). Set x 0 = ] T. Assume that the impulsive spacing is equal to say ε k ζ. Let η = 1.. We choose the initial iteration values of λ and β to be 0. and 0.1 respectively. We obtain Γ = λ = 7.4 and β = 0.5 Thus the impulsive distance for robust stability may be estimated as (τ k+1 τ k ) ln(β)/λ = By Theorem 3.1 and the Algorithm that we give the system is robust and asymptotically stable under the conditions of Theorem 3.1. Figures and 3 illustrate the impulsive control state diagram and the phase diagram of a PMSM with system parameter variation for impulsive spacing ζ = Example In system (1) let n p =1 L d = L q = mh R = ± Ω Ψ r = 0.057± N m/a J = ± K g m β = ± N/(rad s 1 ). So we have γ σ np ψ1 n pψ ] = ] R β R 1 β 1 β1 L q β ] L q = ]. J R J 1 R 1 During our experiment we set σ and γ in their own interval randomly. The PMSM will display chaos without impulsive control. The chaotic attractor is shown in Fig. 1. Fig.. The state diagram of a PMSM with impulsive control (ζ = 0.0). Fig. 3. The phase diagram of a PMSM with impulsive control (ζ = 0.0). Fig. 1. The chaotic attractor in a PMSM with system parameter variation. In summary according to the above simulation experiments we think that the proposed impulsive control for a PMSM with system parameter variation and the corresponding results can provide good effectiveness with PMSM control via the LMI approach

7 5. Conclusions We have investigated the impulsive control of PMSMs. The parameter variation in the system is considered and variable system parameters are formulated through matrix analysis. Sufficient conditions for robust asymptotic stability are established by employing the methods of Lyapunov functions and linear matrix inequality so as to obtain more flexibility in practical application. The effectiveness of our methods has been shown by computer simulation and the illustrated example shows that the system is robust and asymptotic stable under the conditions of Theorem 3.1. References 1] Elmas C and Ustun O 008 Control Engineering Practice ] Lee T S Lin C H and Lin F J 005 Control Engineering Practice ] Baik I C and Kim K H 000 Control Systems Technology ] Baik I C Kim K H and Youn M J 1998 IEE Proc. Electric Power Applications ] Li D Wang S L Zhang X H Yang D and Wang H 008 Chin. Phys. B ] Wei D Q Luo X S Fang J Q and Wang B H 006 Acta Phys. Sin (in Chinese) 7] Li D Wang S L Zhang X H Yang D and Wang H 008 Chin. Phys. Lett ] Karunadasa J P and Renfrew A C 1991 IEE Proc. Electric Power Applications ] Lin F J and Chiu S 1998 Control Theory and Applications ] Wai R J 001 Industrial Electronics ] Attaianese C Perfetto A and Tomasso G 1999 IEE Proc. Electric Power Applications ] Hsien T L Sun Y Y and Tsai M C 1997 IEE Proc. Electric Power Applications ] Yan J R and Shen J H 1999 Nonlinear Analysis ] Luo X S and Wang B H 001 Chin. Phys ] Zhao Y B Zhang D Y and Zhang C J 007 Chin. Phys ] Liu B Liu X Z and Liao X X 004 J. Math. Anal. Appl ] Sanchez E N and Perez J P 1999 IEEE Trans. on Circuits Systems ] Wu C L Ma S J Sun Z K and Fang T 006 Acta Phys. Sin (in Chinese) 19] Cao J D 003 Phys. Lett. A ] Liu J D and Yu Y M 007 Acta Phys. Sin (in Chinese) 1] Li Y Yang B J Yuan Y and Liu X H 007 Chin. Phys ] Liu X W Huang Q Z Gao X and Shao S Q 007 Chin. Phys ] Li D Wang S L Zhang X H and Yang D 009 Acta Phys. Sin (in Chinese)