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Nonlinear control of permanent magnet stepper motors
Communications in Nonlinear Science and Numerical Simulation 9 (2004) 443–458 www.elsevier.com/locate/cnsns
Nonlinear control of permanent magnet stepper motors Ahmad M. Harb a
a,*
, Ashraf A. Zaher
b
Department of Electrical Engineering, Jordan University of Science and Technology, P.O. Box 3030, Irbid, Jordan b Department of Electrical and Systems Engineering, School of Enginnering and Computer Science, Oakland University, Rochester, MI 48309, USA Received 10 April 2002; received in revised form 17 July 2002; accepted 5 August 2002
Abstract The nonlinear dynamics of a permanent magnet stepper motor is studied by means of modern nonlinear theories such as bifurcation and chaos. A three-phase stepper motor is considered as a case study in this paper. The study shows that the system experiences a dynamic bifurcation (Hopf bifurcation) at high frequencies. Since this kind of motors is widely used in some important applications such as printers, disk drives, process control systems, X –Y records, and robotics, controlling such instabilities is the main concern of this paper. A nonlinear robust model-reference controller is introduced. The study shows how to stabilize the system, while having a satisfactory performance, even in the case when some of the motor parameters were uncertain. Ó 2003 Elsevier B.V. All rights reserved. PACS: 84.50.þd; 05.45.)a Keywords: Nonlinear systems; Robust control; Model-reference control; Bifurcation and chaos; Electrical machines; Electric power applications
1. Introduction It is well known that the stepper motor is an electromagnetic incremental actuator that converts digital pulse inputs to analog shaft motion outputs. It rotates by a specific number of degrees in response to an electrical pulse input; therefore the stepper motor is used in digital control systems. Stepper motors are widely used in our daily life. They are used in practical applications that require incremental motion such as printers, tape drives, hard drives in PCÕs, machine tools, process control systems, X –Y records, and robotics. *
Corresponding author. Tel.: +962-2-720-1000/22504; fax: +962-2-709-5018. E-mail address: [email protected] (A.M. Harb).
1007-5704/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/S1007-5704(02)00133-8
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Resolution of stepper motors, available in the market, ranges from several steps per revolution to hundreds of steps per revolution. Stepper motors have been built to follow signals as rapid as 1200 pulses per second with power ratings up to several horsepower. Two types of stepper motors are widely used, the variable reluctance type and the permanent magnet (PM) type. In this paper, the PM type is used. PM stepper motors have higher inertia and therefore slower acceleration than variable reluctance types. They also produce more torque per ampere stator current than the variable reluctance types [1]. Stepper motors suffer from an oscillation or unstable phenomenon that severely restricts their dynamic performance at high-speed operation. The oscillation usually occurs at stepping rates lower than 1000 pulses per second. Verghese [2] and Mellor [3] recognize these oscillations as local instability. In addition, there is another type of unstable phenomenon, or oscillation that is recognized as global instability, known as out of synchronism, and it occurs at higher stepping rates. Many researchers have studied the dynamic stability of this phenomenon. Traft and Gauthier [4], and Traft and Harned [5] used mathematical concepts such as limit cycles and separatics in their analysis to study the dynamics of loss of synchronism for the stepper motors. Verghese [2] used a linearized model in his analysis. Pickup and Russell [6,7] presented a detailed analysis of the so-called modulation method into a two-phase stepper motor. They used Jacobi series to solve an ordinary differential equation, and a set of nonlinear algebraic equations that were solved numerically. In this paper, modern nonlinear theories such as bifurcation and chaos are used to analyze a three-phase PM stepper motor. Modern nonlinear theories have been widely used in power systems. Abed and Fu [8], Nayfeh et al. [9], and Harb et al. [10] used bifurcation analysis applied to power systems. Lately, control of the instability in nonlinear systems is being the nowadays work. Harb et al. [11,12] used a new nonlinear recursive backstepping controller to stabilize the nonlinear dynamical systems. In this paper a robust nonlinear model-reference controller is used to control the instabilities in PM stepper motors. This paper is organized as follows: in Section 2, the mathematical model of a three-phase PM stepper motor is discussed. In Section 3, the nonlinear analysis and uncontrolled numerical simulation are discussed. Section 4 introduces the control of the instabilities in the PM stepper motor. Finally, conclusions are drawn in Section 5.
2. Mathematical model In this paper, we consider the stepper motor shown in Fig. 1. It is a three-phase salient stator and PM rotor. This kind of motors operates on the same principle as that of synchronous motors. The mathematical model for this kind of motors is derived based on q–d-axes and ParkÕs transformation from Krause [13]. 2.1. Voltage equations Eqs. (1)–(3) gives the dynamic model for the voltages: va ¼ Ria þ L
dia dib dic dkma M M þ dt dt dt dt
ð1Þ
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Fig. 1. Schematic diagram of the PM stepper motor.
vb ¼ Rib þ L
dib dia dic dkmb M M þ dt dt dt dt
ð2Þ
vc ¼ Ric þ L
dic dia dib dkmc M M þ dt dt dt dt
ð3Þ
where R and L are the resistance and inductance of the windings respectively, M is the mutual inductance between the windings, and kij are the flux linkages of the phases due to the PM and kma ¼ k1 sinðN hÞ
ð4Þ
kmb ¼ k1 sinðN h 2p=3Þ
ð5Þ
kmc ¼ k1 sinðNh þ 2p=3Þ
ð6Þ
and N is the number of rotor teeth and k1 is the maximum value of the flux linkages. Based on ParkÕs transformation [13], the transformation matrix from abc frame (three-phase system a–b–c as shown in Fig. 1 to q–d–o frame (the direct (d)-axis, centered magnetically in the center of the north pole and the quadrature (q)-axis, 90 electrical degrees ahead of the d-axis) is given by: 2 3 cosðN hÞ cosðN h 2p=3Þ cosðNh þ 2p=3Þ 24 sinðN hÞ sinðN h 2p=3Þ sinðN h þ 2p=3Þ 5 ð7Þ P¼ 3 3 3 3 2
2
2
vabc ¼ Pvdqo
ð8Þ
Substituting Eqs. (1)–(7) into Eq. (8), and using the symmetry (ia þ ib þ ic ¼ 0), we obtain: vd ¼ Rid þ L1
did NL1 iq x dt
vq ¼ Riq þ L1
diq þ NL1 id x þ N k1 x dt
ð9Þ ð10Þ
where L1 ¼ L þ M, and x is the rotor speed. The motion and the motorÕs torque equations are given by [6,7]: 3 T ¼ N k1 iq 2
ð11Þ
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J x_ þ Bf x ¼ T TL
ð12Þ
d_ ¼ x x0
ð13Þ
where Bf is the friction viscosity coefficient, x0 is the synchronous rotor speed (steady state), TL is the load torque, d is the load angle, J is the inertia constant, and x0 ¼ x1 =N . Assuming Vm to be the peak voltage, the input voltages take the forms: vd ¼ Vm sinðNdÞ
ð14Þ
vq ¼ Vm cosðN dÞ
ð15Þ
Finally, from Eqs. (1)–(15), the mathematical model can be represented in a state space form using four nonlinear ordinary differential equations as follows: x_ 1 ¼
Vm R Nk1 cosðNx4 Þ x1 Nx2 x3 x3 L1 L1 L1
ð16Þ
x_ 2 ¼
Vm R sinðNx4 Þ x2 þ Nx1 x3 L1 L1
ð17Þ
x_ 3 ¼
3N k1 Bf TL x1 x3 2J J J
ð18Þ
x_ 4 ¼ x3 x0
ð19Þ
where x1 ¼ iq , x2 ¼ id , x3 ¼ x, and x4 ¼ d.
3. Numerical simulation (uncontrolled case) This section discusses the nonlinear dynamics of the uncontrolled system.The analysis will show that the system has multiple equilibrium points, Xe ¼ ½ x1e x2e x3e x4e T . The stability of each equilibrium point is now investigated. 3.1. Equilibrium solutions and their stabilities At equilibrium, we have: x_ 1 ¼ x_ 2 ¼ x_ 3 ¼ x_ 4 ¼ 0 Thus, from Eq. (19) x1 x3e ¼ N and from Eqs. (18) and (21) x1e ¼
2 ðBf x1 þ TL NÞ 3N 2 k1
Using Eqs. (21) and (22), Eqs. (16) and (17) can be cast in the following form:
ð20Þ
ð21Þ
ð22Þ
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cosðNx4e Þ ¼ a þ bx2e
ð23Þ
sinðNx4e Þ ¼ c þ dx2e
ð24Þ
and, where a¼
1 ðRx1e þ N k1 x3e Þ; Vm
b¼
L1 Nx3e ; Vm
c¼
L1 Nx1e x3e Vm
and d ¼
R Vm
ð25Þ
Thus ðb2 þ d 2 Þx22e þ 2ðab þ cdÞx2e þ ða2 þ c2 1Þ ¼ 0
ð26Þ
From which x2e ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðab þ cdÞ ðab þ cdÞ2 ðb2 þ d 2 Þða2 þ c2 1Þ ðb2 þ d 2 Þ
ð27Þ
and, 1 1 ð28Þ cos1 ða þ bx2e Þ ¼ sin1 ðc þ dx2e Þ N N Using numerical analysis, it can be shown that all equilibrium points for the system are unstable and that the system exhibits chaotic behavior for the given range of 825:315 6 x1 6 1274:581. The nominal values of the motor parameters are given in Appendix A. Taking x1 as the only variable where 825:315 6 x1 6 1274:581, we can decide on an operating point for which stability analysis is further investigated. It is seen, from Fig. 2, where xei is the equilibrium value for state i, that each value of x1 results in two different equilibrium points provided that 0 6 Nx4e 6 2p Choosing x1 to be equal to 1100, the following two equilibrium points are generated: x4e ¼
T
X1e ¼ ½ 0:322411 0:951936 22 0:12502
ω1
ω1
ð29Þ
ω1
ω1
Fig. 2. Equilibrium points analysis of the stepper motor using nominal parameters.
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ω1 Fig. 3. Hopf bifurcation diagram.
with eigenvalues 298:6 1139:0i and þ5:0 647:3i, and T
X2e ¼ ½ 0:322411 4:129869 22 0:073153
ð30Þ
with eigenvalues 293:2 115:6i, þ660.4, and )661.1. The variation of the state-variable, x, with the control parameter x1 is shown in Fig. 3. Starting with x1 ¼ 100, the system will be stable. As we increase x1 , the system losses its stability at the Hopf bifurcation point H1 , x1 ¼ 825:315. As x1 is further increased, the system will remain unstable till the control parameter x1 ¼ 1274:581, where the system regain its stability at the second Hopf bifurcation point H2 . After that the system will remain stable. 3.2. Dynamical solutions and their stabilities As illustrated, there are two Hopf bifurcation points H1 and H2 . To study the stability of these bifurcation points, we used the Floquet theory of Nayfeh and Balachandran [14]. First we need to know the type of these Hopf bifurcation points, i.e. to reduce the dimension of the system near Hopf bifurcation point to its normal form, amplitude and phase. Hopf bifurcation points might be either supercritical or subcritical. A perturbation method such as the method of multiple scales is used to determine the type of the Hopf bifurcation point. A numerical algorithm encoded in MATHEMATICA [14] is used to find the normal form of both H1 and H2 . The normal form near the Hopf bifurcation is given by a_ ¼ a1 a þ a2 a3 and h_ ¼ a3 þ a4 a2 , where a and h are the amplitude and phase, and the ai are functions of the system parameters. When a2 < 0, the limit cycle is stable and hence the Hopf bifurcation is supercritical. On the other hand, when a2 > 0, the limit cycle is unstable and hence the Hopf bifurcation is subcritical. In Fig. 3, we found that a2 < 0 near H1 and H2 . So, both points are supercritical Hopf bifurcation points, i.e. a stable limit cycle will be born at the vicinity of both H1 and H2 . Starting from H1 , x1 ¼ 825:315, we see a small stable periodic solution (limit cycle) shown in Fig. 4. Further increase in the control parameter x1 results in an increase of the size of the limit cycle. Then, as x1 increases more, the limit cycle starts to decrease in size till the second Hopf bifurcation point H2 is reached, x1 ¼ 1274:581, where the system regains the equilibrium stable solutions. The stable periodic solution between the two Hopf bifurcation points is known as midfrequency oscillation. The undesirable oscillation needs to be controlled or eliminated. In Section 4, a nonlinear robust control is designed to eliminate these oscillations.
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(a)
(b)
(c)
(d)
Fig. 4. Uncontrolled numerical simulations, U ¼ 0: (a) x1 ¼ 826, (b) x1 ¼ 926, (c) x1 ¼ 1116, (d) x1 ¼ 1275.
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4. Controller design Designing and implementing systems that are capable of controlling unknown plants or adapting to unpredictable changes in the environment have been an active and rich area in control engineering for a long time. Many appealing concepts were proposed in which the notion of Lyapunov functions was often used [15]. Most conventional control techniques are based, either explicitly or implicitly, on a model of the process to be controlled. Problems with control arise when either the system to be controlled is in some way ill defined and/or it is not possible to gain access to the internal variables of the system. Both of these represent a lack of information about the underlying physical system. Usually deficient information about a model can be compensated for by having access to most of the variables and by designing a tight feedback control. This approach is most successful when applied to regulation problems, however for servomechanism the lack of information is much more crucial [16]. In this case, an explicit model can be used to generate estimates of the systemÕs behavior that can be used to modify the closed-loop time response and satisfy the performance specifications. Traditional adaptive schemes are classified as direct and indirect and as Lyapunov-based and estimation-based. They involve parameter identification with parameter estimators or identifiers. The vital part of the identifier is the parameter adaptation algorithm, commonly referred to as the parameter update law. The direct–indirect classification reflects the fact that the updated parameters are either those of the controller or the plant respectively. The distinction between Lyapunovbased and estimation-based schemes is more substantial and is indicated in part by the type of parameter update law and the corresponding proof of stability and convergence. When the true parameters of the systems are unknown, the controller parameters are either estimated directly (direct scheme) or computed by solving the same design equations with plant parameters estimates (indirect scheme). The resulting controller is called a certainty equivalence controller. A controller design is now proposed that is based on forcing the actual model to follow the response of a reference model. We will begin by assuming that the model parameters are known and that the controller structure will allow the decoupling of the motor dynamics. Four control signals are added to the original model given by Eqs. (16)–(19), so that the augmented system is given by: X_ ¼ f ðX ; UÞ ¼ gðX Þ þ U ¼ ½ g1 ðX Þ þ u1
g2 ðX Þ þ u2
g3 ðX Þ þ u3
T
g4 ðX Þ þ u4
ð31Þ
or x_ i
x_ i þ ui and gi is given by Eq: ð16 þ iÞ;
i ¼ 1; 2; 3; and 4
ð32Þ
Introducing the following linear reference model: X_ d ¼ Ad Xd þ Bd Xe
ð33Þ
where 2 6 Ad ¼ Bd ¼ 6 4
3
1=s1
7 7 5
1=s2 1=s3 1=s4
ð34Þ
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results in e ¼ Xd X
ð35Þ
where e is the racking error between the desired response (of the reference model) and the actual response (of the motor itself). Thus, re ¼ e_ ¼ X_ d X_ ¼ ðAd Xd þ Bd Xe Þ f ðX ; U Þ ¼ Ad e þ Ad X þ Bd Xe f ðX ; U Þ
ð36Þ
Hence, the control strategy reduces to forcing the error gradient to be negative definite, or forcing the error to eventually becoming zero. This will be achieved via a Lyapunov-based technique that has a major goal of assuring asymptotic stability of the system, while allowing additional flexibility to improve the transient response as well as resulting in a satisfactory performance [17,18]. This technique is illustrated now. Introducing the following quadratic Lyapunov function: V ðeÞ ¼ eT Pe;
P ¼ ½ P1
P2
P3
P4 ;
Pi ¼ ½ P1i
P2i
P3i
P4i T ;
i ¼ 1; 2; 3; and 4
ð37Þ
where P is a symmetric positive definite matrix yields: V_ ðeÞ ¼ e_ T Pe þ eT P e_ ¼ ðeT ATd þ X T ATd f T ðX ; UÞ þ XeT BTd ÞPe þ eT P ðAd e þ Ad X f ðX ; U Þ þ Bd Xe Þ ¼ eT ðATd P þ PAd Þe þ 2eT P ðAd X f ðX ; U Þ þ Bd Xe Þ ¼ eT Qe þ 2W
ð38Þ
Hence, conditions for asymptotic stability can be reduced to ATd P þ PAd ¼ Q;
Q is þ ve definite;
and W 6 0
ð39Þ
It is clear form Eq. (39) that there is no unique solution for the control signal, U . In fact the controller is very flexible as the only restriction imposed on the design is having a positive definite structure for the matrix P . A systematic approach is now introduced for the controller design for the two cases when the system parameters are completely known, and the uncertain case where no information about some or all of the parameters is provided except for an upper bound on their values. 4.1. Deterministic case If all the model parameters are known, the simplest way to force Eq. (38) to become negative definite is by using: W ¼ 0 ! f ðX ; U Þ ¼ Ad X þ Bd Xe
ð40Þ
U ¼ Ad X þ Bd Xe gðX Þ
ð41Þ
or Thus the system nonlinearities are cancelled, and the closed loop dynamics are now solely governed by the reference model. Some of the useful nonlinearities could have been retained in the design [19] to improve the performance of the system, however, the control law design will be
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more complicated, for which more versatile techniques such as backstepping or robust estimationbased control can be used [20,21]. Fig. 5 illustrates the response of the stepper motor for the deterministic case where si ¼ 0:05 s, i ¼ 1, 2, 3, and 4 for the nominal motor parameters, given in Appendix A, and compares it against the response of the same controller design when only one of the motor parameters, N , is uncertain. It is shown that, in the deterministic case, the stepper motor exhibits a smooth satisfactory performance that is identical to that of a typical first order system having a time constant of 0.05 s. The choice of si depends on the required speed of response, however, if the control signal, U , is bounded, an upper bound should be imposed on it to insure that the controller will not saturate. Analysis of the motor operating in the saturation region is beyond the scope of this paper. It is also clear that the states converged to their true values, which are the same as the expected equilibrium point for the used value of x1 . Although zero initial conditions were used in the simulation, it can be shown that the system will always reach its steady state form any initial condition without any offsets. The control signals shown in Fig. 5, where xdi is the desired value for state i which is the same as the equilibrium points of xi using the operating point, x1 ¼ 1100 as shown in Fig. 2, are smooth and causal, and can be easily implemented in the closed loop. These control signal, artificially introduced by the modelreference controller, eventually go to zero insuring stability of the closed loop system and compatibility with the original model. When N was assumed uncertain, and subject to a change of only 2%, such that the nominal value of N ¼ 50 is now 51 and 49 respectively, the response significantly deteriorated as depicted in Fig. 5 as a direct result of the nonexact cancellation of the unwanted nonlinear motor dynamics. This motivates the need for an improved version of the designed control law to accommodate inherent uncertainties in some or all of the motor parameters. The next section investigates the main contribution of this paper, i.e. the design of a robust nonlinear controller that only needs information about the upper bounds of the motor parameters as priori knowledge. 4.2. Adding robustness The controller design in the previous section depends mainly on having a deterministic model for the stepper motor. This might not be a realistic assumption, if one or more of the motor parameters are uncertain. Exact cancellations of the unwanted model nonlinearities will surely fail, if the wrong values of the parameters are used as depicted in Fig. 5. In this section, a robust design is adopted where it is assumed that the nonlinear model structure is correct, but one or more of the motor parameters are uncertain. It is also assumed that the uncertain parameters have a known structure, such that their nominal values are known along with upper bounds on their values. Zaher and coworkers in [11,12] used a backstepping technique to use some of the systemÕs states as virtual controls with implicit parameter estimators to account for uncertainty. The designed parameter update laws were designed such that the closed loop system is stable in the sense of Lyapunov. The innovation in this paper is to use a model reference technique that works fine for a given uncertainty domain without having to design either explicit or implicit parameter estimation algorithms.
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Fig. 5. Controlled response: (a) x1 and x2 , (b) x3 and x4 , (c) u1 and u2 , (d) u3 and u4 .
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If some of the model parameters were uncertain, the model can be put in the form: X_ ¼ f ðX ; U; pÞ ¼ gðX ; pÞ þ U ¼ gnom ðX Þ þ DgðX ; pÞ þ U
ð42Þ
where kDgðX ; pÞk 6 Dgmax
ð43Þ
and p represents the perturbed or uncertain parameters. In this case W , in Eq. (38), cannot be made exactly zero as f ðX ; U Þ is uncertain, instead W ¼ eT P fAd X þ Bd Xe gnom ðX Þ DgðX ; pÞ U g < 0
ð44Þ
The control strategy now is to use Eq. (44) to design the control law, U , that should be a function of the known parameters only along with the nominal and maximum values of the uncertain parameters. This new robust control law must guarantee stability for all values of the uncertain parameters confined within the uncertainty domain. The design must also allow for the capability of achieving a satisfactory performance. Thus the new robust design will have a dual function, insuring stability even in the worst case scenario of having only partial information about all the motors parameters, and maintaining a satisfactory response, i.e. a smooth transient response and a zero steady state offset. This can be achieved by choosing U to take the form: U ¼ Ad X þ Bd Xe gnom ðX Þ þ diag½ signðeT P1 Þ signðeT P2 Þ signðeT P3 Þ signðeT P4 Þ Dgmax
ð45Þ
resulting in
W ¼ eT P DgðX ; pÞ diag½ signðeT P1 Þ signðeT P2 Þ 4 X ¼ ðeT Pi Þ Dgi þ Dgimax signðeT P Þ
signðeT P3 Þ
signðeT P4 ÞDgmax
ð46Þ
i¼1
which is clear to be always negative. 4.3. A numerical example The control law, given by Eq. (46), is shown to be a strong function in P , which is the solution of Eq. (39). Thus a careful choice of the matrix Q must be made to guarantee having a satisfactory performance. A few off-line trails can be made to develop the best structure for Q. To exemplify the design technique, it is assumed that R is the only uncertain parameter, such that: R ¼ Rnom ð1 þ DRÞ;
jDRj 6 DRmax ;
Rmax ¼ Rnom ð1 þ DRmax Þ;
and
Rmin ¼ Rnom ð1 DRmax Þ
ð47Þ
which yields u1 ¼
1 1 Vm N k1 Rmax x1 þ x1e cosðNx4 Þ þ Nx2 x3 þ x3 þ x1 signfx1 ðeT P1 Þg s1 s1 L1 L1 L1
ð48Þ
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Fig. 6. Robust control: (a) x1 and x2 , (b) x3 and x4 , (c) u1 and u2 , (d) u3 and u4 .
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u2 ¼
1 1 Vm Rmax x2 þ x2e sinðNx4 Þ Nx1 x3 þ x2 signfx2 ðeT P2 Þg s2 s2 L1 L1
ð49Þ
u3 ¼
1 1 þ x3 g3 ðX Þ s3 s3
ð50Þ
u4 ¼
1 1 þ x3 g4 ðX Þ s4 s4
ð51Þ
and, from which ðx1 K1 Þ ðx2 K2 Þ V_ ðeÞ ¼ fRnom Rmax signðx1 K1 Þg þ fRnom Rmax signðx2 K2 Þg L1 L1
ð52Þ
where K is a row vector ¼ eT P , and Ki is its ith element. It is clear that the result of Eq. (52) is negative for all values of R and X . Fig. 6 shows the motor response for the case where Rnom ¼ 0:285 X and Rmax ¼ 0:3 X. The Q matrix was chosen to be the identity matrix. The rest of the motor parameters are the same as that for the deterministic case considered in the previous section. Eq. (53) gives the values of Ad and P matrices used in the simulation 2 3 2 3 20 0:025 6 7 6 7 20 0:025 7; 7 Ad ¼ 6 P ¼6 ð53Þ 4 5 4 5 20 0:025 20 0:025 It is clear from Fig. 6 that the same equilibrium points were used again, so that the robust performance could be easily compared against the original design. As discussed in Section 3, there are two equilibrium points for each value of the operating parameter, x1 . Using any of them will result in having zero steady state values for the introduced control signal, U , however, any value for the desired response could have been used as well. One final issue to be further investigated is the causality of the proposed controller. First, the controller, given by Eqs. (48)–(51) is a full-state nonlinear feedback system. For practical reasons, only some of these states will be available for measurements. This necessitates the need for a state observer to be added to the system. The structure of the state observer, along with its convergence rate, and biasness will increase the order of the controller, and strongly affects its performance. Also, it is luxury to have four controls directly acting on the four state variables, which is very often impossible. Detailed analysis of dealing with such issues is considered on [22].
5. Conclusion A bifurcation analysis has been applied to a PM stepper motor. Numerical simulations were done using MATHEMATICA for implementing the bifurcation multiple-scale algorithm for Chaos analysis, and MATLAB for investigating the control design. The results show that the system, without control, has a Hopf bifurcation point that lead to undesirable oscillations. For
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the parameter range 825:315 < x1 < 1274:581, there are two kinds of solutions, an unstable constant solution, and a stable periodic solution. The stable periodic solution between the two Hopf bifurcation points is known as mid-frequency oscillation. A nonlinear robust model-reference controller has been designed to control the bifurcation as well as incipient chaos via converting the region between the two Hopf bifurcation points, H1 and H2 , into a stable region. The control strategy was based on forcing the motor to follow the response of a reference model using a Lyapunov-based design that guarantees stability. Robustness of the controller design against parameters uncertainty was carefully studied, and a modified structure was used to implement the new design such that stability and good performance for the entire region of the uncertain parameters were insured. Finally useful comments regarding causality and extensibility of the controller were given. Appendix A The stepper motor used in this paper has the following parameters: Vm ¼ 3:14 V, L1 ¼ 0:00105 H, N ¼ 50 turns, R ¼ 0:285 X, k1 ¼ 0:00177 V s, Bf ¼ 0:0019 N m s/rad, J ¼ 0:00004 g cm2 , TL ¼ 0:001 N m, x1 ¼ 1100 rad/s.
References [1] Sen PC. Principles of electric machines and power electronics. John Wiley & Sons; 1997. [2] Verghese GC. Analysis of instability in electrical machines. IEEE Trans Ind Appl 1986;IA-22(5):853–64. [3] Mellor PH. Open loop stability characteristics of synchronous drive incorporating high field permanent motor. IEE Proc Power Appl 1991;138(4):175–84. [4] Traft CK, Gauthier RG. Stepping motor failure model. IEEE Trans Ind Electron Contr Instrum 1975;IEC22(3):375–85. [5] Traft CK, Harned TJ. Phase plane representation of the dynamics of two phase synchronous motors. In: Proceedings of the 12th Annual Symposium on Incremental Motion Control Systems and Devices. 1983. p. 139–48. [6] Pickup AE, Russell AP. Analysis of frequency and amplitude modulation in the stabilization of permanent magnet synchronous stepping motors. IEE Proc B, Elect Power Appl 1992;139(4):336–46. [7] Pickup E, Russell AP. Analysis of current waveforms in permanent magnet synchronous stepping motors and synthesis of a stabilizing signal. IEE Proc B, Elect Power Appl 1992;139(9):534–48. [8] Abed EH, Fu J-H. Local feedback stabilization and bifurcation control, I. Hopf bifurcation. Syst Contr Lett 1986;7:11–7. [9] Nayfeh AH, Harb A, Chin C. Bifurcations in a power system model. Int J Bifurcat Chaos 1996;6(3):497–512. [10] Harb AM, Nayfeh AH, Chin C, Mili L. On the effects of machine saturation on subsynchronous oscillations in power systems. Elect Mach Power Syst J 1999;28(11).
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[11] Harb A, Zaher A, Zohdy M. Nonlinear recursive chaos control. In: Proceedings of ACC, Anchorage, AL, USA, May 2002. p. 2251–4. [12] Harb A, Zaher A, Al-Qaisia A, Zohdy M. Estimation-based control of chaotic duffing oscillators. Int J Vibrat Contr, in press. [13] Krause PC. Analysis of electric machinery. McGraw-Hill; 1987. [14] Nayfeh AH, Balachandran B. Applied nonlinear dynamics. John Wiley & Sons; 1994. [15] Khalil H. Nonlinear systems. 3rd ed Prentice Hall; 2002. [16] Soltine JJE, Li W. Applied nonlinear control. Prentice Hall; 1991. [17] Zaher A, Zohdy M, Areed F, Soliman K. Real-time adaptive model-reference control of non-linear processes. In: Proceedings of the 2nd International ICCI, Manama, Bahrain, 13–15 November 2000. [18] Zaher A, Zohdy M, Areed F, Soliman K. Robust model-reference control for a class of non-linear and piece-wise linear systems. In: Proceedings of ACC, Arlington VA, USA, June 2001. p. 4514–9. [19] Zaher A, Zohdy M, Areed F, Soliman K. Convergent nonlinear control for uncertain process plants. In: ACC 2003, Denver, CO, USA. [20] Zaher A, Zohdy M, Areed F, Soliman K. Robust control of nonlinear processes with bounded inputs and hybrid structure. In: Proceedings of the 5th International SCI Conference, Orlando, FL, USA, 22–25 July 2001. [21] Zaher A, Zohdy M, Areed F, Soliman K. Robust estimation-based design for uncertain plants. In: Proceedings of AMSE conference, Dearborn, MI, USA, Oct. 25–28, 2001. [22] Zaher A. Novel nonlinear techniques for modeling, simulation, and control, with applications to biped robots. PhD Dissertation, Oakland University, Rochester, MI, USA, 2001.
Nonlinear control of permanent magnet stepper motors Ahmad M. Harb a
a,*
, Ashraf A. Zaher
b
Department of Electrical Engineering, Jordan University of Science and Technology, P.O. Box 3030, Irbid, Jordan b Department of Electrical and Systems Engineering, School of Enginnering and Computer Science, Oakland University, Rochester, MI 48309, USA Received 10 April 2002; received in revised form 17 July 2002; accepted 5 August 2002
Abstract The nonlinear dynamics of a permanent magnet stepper motor is studied by means of modern nonlinear theories such as bifurcation and chaos. A three-phase stepper motor is considered as a case study in this paper. The study shows that the system experiences a dynamic bifurcation (Hopf bifurcation) at high frequencies. Since this kind of motors is widely used in some important applications such as printers, disk drives, process control systems, X –Y records, and robotics, controlling such instabilities is the main concern of this paper. A nonlinear robust model-reference controller is introduced. The study shows how to stabilize the system, while having a satisfactory performance, even in the case when some of the motor parameters were uncertain. Ó 2003 Elsevier B.V. All rights reserved. PACS: 84.50.þd; 05.45.)a Keywords: Nonlinear systems; Robust control; Model-reference control; Bifurcation and chaos; Electrical machines; Electric power applications
1. Introduction It is well known that the stepper motor is an electromagnetic incremental actuator that converts digital pulse inputs to analog shaft motion outputs. It rotates by a specific number of degrees in response to an electrical pulse input; therefore the stepper motor is used in digital control systems. Stepper motors are widely used in our daily life. They are used in practical applications that require incremental motion such as printers, tape drives, hard drives in PCÕs, machine tools, process control systems, X –Y records, and robotics. *
Corresponding author. Tel.: +962-2-720-1000/22504; fax: +962-2-709-5018. E-mail address: [email protected] (A.M. Harb).
1007-5704/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/S1007-5704(02)00133-8
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Resolution of stepper motors, available in the market, ranges from several steps per revolution to hundreds of steps per revolution. Stepper motors have been built to follow signals as rapid as 1200 pulses per second with power ratings up to several horsepower. Two types of stepper motors are widely used, the variable reluctance type and the permanent magnet (PM) type. In this paper, the PM type is used. PM stepper motors have higher inertia and therefore slower acceleration than variable reluctance types. They also produce more torque per ampere stator current than the variable reluctance types [1]. Stepper motors suffer from an oscillation or unstable phenomenon that severely restricts their dynamic performance at high-speed operation. The oscillation usually occurs at stepping rates lower than 1000 pulses per second. Verghese [2] and Mellor [3] recognize these oscillations as local instability. In addition, there is another type of unstable phenomenon, or oscillation that is recognized as global instability, known as out of synchronism, and it occurs at higher stepping rates. Many researchers have studied the dynamic stability of this phenomenon. Traft and Gauthier [4], and Traft and Harned [5] used mathematical concepts such as limit cycles and separatics in their analysis to study the dynamics of loss of synchronism for the stepper motors. Verghese [2] used a linearized model in his analysis. Pickup and Russell [6,7] presented a detailed analysis of the so-called modulation method into a two-phase stepper motor. They used Jacobi series to solve an ordinary differential equation, and a set of nonlinear algebraic equations that were solved numerically. In this paper, modern nonlinear theories such as bifurcation and chaos are used to analyze a three-phase PM stepper motor. Modern nonlinear theories have been widely used in power systems. Abed and Fu [8], Nayfeh et al. [9], and Harb et al. [10] used bifurcation analysis applied to power systems. Lately, control of the instability in nonlinear systems is being the nowadays work. Harb et al. [11,12] used a new nonlinear recursive backstepping controller to stabilize the nonlinear dynamical systems. In this paper a robust nonlinear model-reference controller is used to control the instabilities in PM stepper motors. This paper is organized as follows: in Section 2, the mathematical model of a three-phase PM stepper motor is discussed. In Section 3, the nonlinear analysis and uncontrolled numerical simulation are discussed. Section 4 introduces the control of the instabilities in the PM stepper motor. Finally, conclusions are drawn in Section 5.
2. Mathematical model In this paper, we consider the stepper motor shown in Fig. 1. It is a three-phase salient stator and PM rotor. This kind of motors operates on the same principle as that of synchronous motors. The mathematical model for this kind of motors is derived based on q–d-axes and ParkÕs transformation from Krause [13]. 2.1. Voltage equations Eqs. (1)–(3) gives the dynamic model for the voltages: va ¼ Ria þ L
dia dib dic dkma M M þ dt dt dt dt
ð1Þ
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445
Fig. 1. Schematic diagram of the PM stepper motor.
vb ¼ Rib þ L
dib dia dic dkmb M M þ dt dt dt dt
ð2Þ
vc ¼ Ric þ L
dic dia dib dkmc M M þ dt dt dt dt
ð3Þ
where R and L are the resistance and inductance of the windings respectively, M is the mutual inductance between the windings, and kij are the flux linkages of the phases due to the PM and kma ¼ k1 sinðN hÞ
ð4Þ
kmb ¼ k1 sinðN h 2p=3Þ
ð5Þ
kmc ¼ k1 sinðNh þ 2p=3Þ
ð6Þ
and N is the number of rotor teeth and k1 is the maximum value of the flux linkages. Based on ParkÕs transformation [13], the transformation matrix from abc frame (three-phase system a–b–c as shown in Fig. 1 to q–d–o frame (the direct (d)-axis, centered magnetically in the center of the north pole and the quadrature (q)-axis, 90 electrical degrees ahead of the d-axis) is given by: 2 3 cosðN hÞ cosðN h 2p=3Þ cosðNh þ 2p=3Þ 24 sinðN hÞ sinðN h 2p=3Þ sinðN h þ 2p=3Þ 5 ð7Þ P¼ 3 3 3 3 2
2
2
vabc ¼ Pvdqo
ð8Þ
Substituting Eqs. (1)–(7) into Eq. (8), and using the symmetry (ia þ ib þ ic ¼ 0), we obtain: vd ¼ Rid þ L1
did NL1 iq x dt
vq ¼ Riq þ L1
diq þ NL1 id x þ N k1 x dt
ð9Þ ð10Þ
where L1 ¼ L þ M, and x is the rotor speed. The motion and the motorÕs torque equations are given by [6,7]: 3 T ¼ N k1 iq 2
ð11Þ
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J x_ þ Bf x ¼ T TL
ð12Þ
d_ ¼ x x0
ð13Þ
where Bf is the friction viscosity coefficient, x0 is the synchronous rotor speed (steady state), TL is the load torque, d is the load angle, J is the inertia constant, and x0 ¼ x1 =N . Assuming Vm to be the peak voltage, the input voltages take the forms: vd ¼ Vm sinðNdÞ
ð14Þ
vq ¼ Vm cosðN dÞ
ð15Þ
Finally, from Eqs. (1)–(15), the mathematical model can be represented in a state space form using four nonlinear ordinary differential equations as follows: x_ 1 ¼
Vm R Nk1 cosðNx4 Þ x1 Nx2 x3 x3 L1 L1 L1
ð16Þ
x_ 2 ¼
Vm R sinðNx4 Þ x2 þ Nx1 x3 L1 L1
ð17Þ
x_ 3 ¼
3N k1 Bf TL x1 x3 2J J J
ð18Þ
x_ 4 ¼ x3 x0
ð19Þ
where x1 ¼ iq , x2 ¼ id , x3 ¼ x, and x4 ¼ d.
3. Numerical simulation (uncontrolled case) This section discusses the nonlinear dynamics of the uncontrolled system.The analysis will show that the system has multiple equilibrium points, Xe ¼ ½ x1e x2e x3e x4e T . The stability of each equilibrium point is now investigated. 3.1. Equilibrium solutions and their stabilities At equilibrium, we have: x_ 1 ¼ x_ 2 ¼ x_ 3 ¼ x_ 4 ¼ 0 Thus, from Eq. (19) x1 x3e ¼ N and from Eqs. (18) and (21) x1e ¼
2 ðBf x1 þ TL NÞ 3N 2 k1
Using Eqs. (21) and (22), Eqs. (16) and (17) can be cast in the following form:
ð20Þ
ð21Þ
ð22Þ
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447
cosðNx4e Þ ¼ a þ bx2e
ð23Þ
sinðNx4e Þ ¼ c þ dx2e
ð24Þ
and, where a¼
1 ðRx1e þ N k1 x3e Þ; Vm
b¼
L1 Nx3e ; Vm
c¼
L1 Nx1e x3e Vm
and d ¼
R Vm
ð25Þ
Thus ðb2 þ d 2 Þx22e þ 2ðab þ cdÞx2e þ ða2 þ c2 1Þ ¼ 0
ð26Þ
From which x2e ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðab þ cdÞ ðab þ cdÞ2 ðb2 þ d 2 Þða2 þ c2 1Þ ðb2 þ d 2 Þ
ð27Þ
and, 1 1 ð28Þ cos1 ða þ bx2e Þ ¼ sin1 ðc þ dx2e Þ N N Using numerical analysis, it can be shown that all equilibrium points for the system are unstable and that the system exhibits chaotic behavior for the given range of 825:315 6 x1 6 1274:581. The nominal values of the motor parameters are given in Appendix A. Taking x1 as the only variable where 825:315 6 x1 6 1274:581, we can decide on an operating point for which stability analysis is further investigated. It is seen, from Fig. 2, where xei is the equilibrium value for state i, that each value of x1 results in two different equilibrium points provided that 0 6 Nx4e 6 2p Choosing x1 to be equal to 1100, the following two equilibrium points are generated: x4e ¼
T
X1e ¼ ½ 0:322411 0:951936 22 0:12502
ω1
ω1
ð29Þ
ω1
ω1
Fig. 2. Equilibrium points analysis of the stepper motor using nominal parameters.
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ω1 Fig. 3. Hopf bifurcation diagram.
with eigenvalues 298:6 1139:0i and þ5:0 647:3i, and T
X2e ¼ ½ 0:322411 4:129869 22 0:073153
ð30Þ
with eigenvalues 293:2 115:6i, þ660.4, and )661.1. The variation of the state-variable, x, with the control parameter x1 is shown in Fig. 3. Starting with x1 ¼ 100, the system will be stable. As we increase x1 , the system losses its stability at the Hopf bifurcation point H1 , x1 ¼ 825:315. As x1 is further increased, the system will remain unstable till the control parameter x1 ¼ 1274:581, where the system regain its stability at the second Hopf bifurcation point H2 . After that the system will remain stable. 3.2. Dynamical solutions and their stabilities As illustrated, there are two Hopf bifurcation points H1 and H2 . To study the stability of these bifurcation points, we used the Floquet theory of Nayfeh and Balachandran [14]. First we need to know the type of these Hopf bifurcation points, i.e. to reduce the dimension of the system near Hopf bifurcation point to its normal form, amplitude and phase. Hopf bifurcation points might be either supercritical or subcritical. A perturbation method such as the method of multiple scales is used to determine the type of the Hopf bifurcation point. A numerical algorithm encoded in MATHEMATICA [14] is used to find the normal form of both H1 and H2 . The normal form near the Hopf bifurcation is given by a_ ¼ a1 a þ a2 a3 and h_ ¼ a3 þ a4 a2 , where a and h are the amplitude and phase, and the ai are functions of the system parameters. When a2 < 0, the limit cycle is stable and hence the Hopf bifurcation is supercritical. On the other hand, when a2 > 0, the limit cycle is unstable and hence the Hopf bifurcation is subcritical. In Fig. 3, we found that a2 < 0 near H1 and H2 . So, both points are supercritical Hopf bifurcation points, i.e. a stable limit cycle will be born at the vicinity of both H1 and H2 . Starting from H1 , x1 ¼ 825:315, we see a small stable periodic solution (limit cycle) shown in Fig. 4. Further increase in the control parameter x1 results in an increase of the size of the limit cycle. Then, as x1 increases more, the limit cycle starts to decrease in size till the second Hopf bifurcation point H2 is reached, x1 ¼ 1274:581, where the system regains the equilibrium stable solutions. The stable periodic solution between the two Hopf bifurcation points is known as midfrequency oscillation. The undesirable oscillation needs to be controlled or eliminated. In Section 4, a nonlinear robust control is designed to eliminate these oscillations.
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(a)
(b)
(c)
(d)
Fig. 4. Uncontrolled numerical simulations, U ¼ 0: (a) x1 ¼ 826, (b) x1 ¼ 926, (c) x1 ¼ 1116, (d) x1 ¼ 1275.
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4. Controller design Designing and implementing systems that are capable of controlling unknown plants or adapting to unpredictable changes in the environment have been an active and rich area in control engineering for a long time. Many appealing concepts were proposed in which the notion of Lyapunov functions was often used [15]. Most conventional control techniques are based, either explicitly or implicitly, on a model of the process to be controlled. Problems with control arise when either the system to be controlled is in some way ill defined and/or it is not possible to gain access to the internal variables of the system. Both of these represent a lack of information about the underlying physical system. Usually deficient information about a model can be compensated for by having access to most of the variables and by designing a tight feedback control. This approach is most successful when applied to regulation problems, however for servomechanism the lack of information is much more crucial [16]. In this case, an explicit model can be used to generate estimates of the systemÕs behavior that can be used to modify the closed-loop time response and satisfy the performance specifications. Traditional adaptive schemes are classified as direct and indirect and as Lyapunov-based and estimation-based. They involve parameter identification with parameter estimators or identifiers. The vital part of the identifier is the parameter adaptation algorithm, commonly referred to as the parameter update law. The direct–indirect classification reflects the fact that the updated parameters are either those of the controller or the plant respectively. The distinction between Lyapunovbased and estimation-based schemes is more substantial and is indicated in part by the type of parameter update law and the corresponding proof of stability and convergence. When the true parameters of the systems are unknown, the controller parameters are either estimated directly (direct scheme) or computed by solving the same design equations with plant parameters estimates (indirect scheme). The resulting controller is called a certainty equivalence controller. A controller design is now proposed that is based on forcing the actual model to follow the response of a reference model. We will begin by assuming that the model parameters are known and that the controller structure will allow the decoupling of the motor dynamics. Four control signals are added to the original model given by Eqs. (16)–(19), so that the augmented system is given by: X_ ¼ f ðX ; UÞ ¼ gðX Þ þ U ¼ ½ g1 ðX Þ þ u1
g2 ðX Þ þ u2
g3 ðX Þ þ u3
T
g4 ðX Þ þ u4
ð31Þ
or x_ i
x_ i þ ui and gi is given by Eq: ð16 þ iÞ;
i ¼ 1; 2; 3; and 4
ð32Þ
Introducing the following linear reference model: X_ d ¼ Ad Xd þ Bd Xe
ð33Þ
where 2 6 Ad ¼ Bd ¼ 6 4
3
1=s1
7 7 5
1=s2 1=s3 1=s4
ð34Þ
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451
results in e ¼ Xd X
ð35Þ
where e is the racking error between the desired response (of the reference model) and the actual response (of the motor itself). Thus, re ¼ e_ ¼ X_ d X_ ¼ ðAd Xd þ Bd Xe Þ f ðX ; U Þ ¼ Ad e þ Ad X þ Bd Xe f ðX ; U Þ
ð36Þ
Hence, the control strategy reduces to forcing the error gradient to be negative definite, or forcing the error to eventually becoming zero. This will be achieved via a Lyapunov-based technique that has a major goal of assuring asymptotic stability of the system, while allowing additional flexibility to improve the transient response as well as resulting in a satisfactory performance [17,18]. This technique is illustrated now. Introducing the following quadratic Lyapunov function: V ðeÞ ¼ eT Pe;
P ¼ ½ P1
P2
P3
P4 ;
Pi ¼ ½ P1i
P2i
P3i
P4i T ;
i ¼ 1; 2; 3; and 4
ð37Þ
where P is a symmetric positive definite matrix yields: V_ ðeÞ ¼ e_ T Pe þ eT P e_ ¼ ðeT ATd þ X T ATd f T ðX ; UÞ þ XeT BTd ÞPe þ eT P ðAd e þ Ad X f ðX ; U Þ þ Bd Xe Þ ¼ eT ðATd P þ PAd Þe þ 2eT P ðAd X f ðX ; U Þ þ Bd Xe Þ ¼ eT Qe þ 2W
ð38Þ
Hence, conditions for asymptotic stability can be reduced to ATd P þ PAd ¼ Q;
Q is þ ve definite;
and W 6 0
ð39Þ
It is clear form Eq. (39) that there is no unique solution for the control signal, U . In fact the controller is very flexible as the only restriction imposed on the design is having a positive definite structure for the matrix P . A systematic approach is now introduced for the controller design for the two cases when the system parameters are completely known, and the uncertain case where no information about some or all of the parameters is provided except for an upper bound on their values. 4.1. Deterministic case If all the model parameters are known, the simplest way to force Eq. (38) to become negative definite is by using: W ¼ 0 ! f ðX ; U Þ ¼ Ad X þ Bd Xe
ð40Þ
U ¼ Ad X þ Bd Xe gðX Þ
ð41Þ
or Thus the system nonlinearities are cancelled, and the closed loop dynamics are now solely governed by the reference model. Some of the useful nonlinearities could have been retained in the design [19] to improve the performance of the system, however, the control law design will be
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more complicated, for which more versatile techniques such as backstepping or robust estimationbased control can be used [20,21]. Fig. 5 illustrates the response of the stepper motor for the deterministic case where si ¼ 0:05 s, i ¼ 1, 2, 3, and 4 for the nominal motor parameters, given in Appendix A, and compares it against the response of the same controller design when only one of the motor parameters, N , is uncertain. It is shown that, in the deterministic case, the stepper motor exhibits a smooth satisfactory performance that is identical to that of a typical first order system having a time constant of 0.05 s. The choice of si depends on the required speed of response, however, if the control signal, U , is bounded, an upper bound should be imposed on it to insure that the controller will not saturate. Analysis of the motor operating in the saturation region is beyond the scope of this paper. It is also clear that the states converged to their true values, which are the same as the expected equilibrium point for the used value of x1 . Although zero initial conditions were used in the simulation, it can be shown that the system will always reach its steady state form any initial condition without any offsets. The control signals shown in Fig. 5, where xdi is the desired value for state i which is the same as the equilibrium points of xi using the operating point, x1 ¼ 1100 as shown in Fig. 2, are smooth and causal, and can be easily implemented in the closed loop. These control signal, artificially introduced by the modelreference controller, eventually go to zero insuring stability of the closed loop system and compatibility with the original model. When N was assumed uncertain, and subject to a change of only 2%, such that the nominal value of N ¼ 50 is now 51 and 49 respectively, the response significantly deteriorated as depicted in Fig. 5 as a direct result of the nonexact cancellation of the unwanted nonlinear motor dynamics. This motivates the need for an improved version of the designed control law to accommodate inherent uncertainties in some or all of the motor parameters. The next section investigates the main contribution of this paper, i.e. the design of a robust nonlinear controller that only needs information about the upper bounds of the motor parameters as priori knowledge. 4.2. Adding robustness The controller design in the previous section depends mainly on having a deterministic model for the stepper motor. This might not be a realistic assumption, if one or more of the motor parameters are uncertain. Exact cancellations of the unwanted model nonlinearities will surely fail, if the wrong values of the parameters are used as depicted in Fig. 5. In this section, a robust design is adopted where it is assumed that the nonlinear model structure is correct, but one or more of the motor parameters are uncertain. It is also assumed that the uncertain parameters have a known structure, such that their nominal values are known along with upper bounds on their values. Zaher and coworkers in [11,12] used a backstepping technique to use some of the systemÕs states as virtual controls with implicit parameter estimators to account for uncertainty. The designed parameter update laws were designed such that the closed loop system is stable in the sense of Lyapunov. The innovation in this paper is to use a model reference technique that works fine for a given uncertainty domain without having to design either explicit or implicit parameter estimation algorithms.
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Fig. 5. Controlled response: (a) x1 and x2 , (b) x3 and x4 , (c) u1 and u2 , (d) u3 and u4 .
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If some of the model parameters were uncertain, the model can be put in the form: X_ ¼ f ðX ; U; pÞ ¼ gðX ; pÞ þ U ¼ gnom ðX Þ þ DgðX ; pÞ þ U
ð42Þ
where kDgðX ; pÞk 6 Dgmax
ð43Þ
and p represents the perturbed or uncertain parameters. In this case W , in Eq. (38), cannot be made exactly zero as f ðX ; U Þ is uncertain, instead W ¼ eT P fAd X þ Bd Xe gnom ðX Þ DgðX ; pÞ U g < 0
ð44Þ
The control strategy now is to use Eq. (44) to design the control law, U , that should be a function of the known parameters only along with the nominal and maximum values of the uncertain parameters. This new robust control law must guarantee stability for all values of the uncertain parameters confined within the uncertainty domain. The design must also allow for the capability of achieving a satisfactory performance. Thus the new robust design will have a dual function, insuring stability even in the worst case scenario of having only partial information about all the motors parameters, and maintaining a satisfactory response, i.e. a smooth transient response and a zero steady state offset. This can be achieved by choosing U to take the form: U ¼ Ad X þ Bd Xe gnom ðX Þ þ diag½ signðeT P1 Þ signðeT P2 Þ signðeT P3 Þ signðeT P4 Þ Dgmax
ð45Þ
resulting in
W ¼ eT P DgðX ; pÞ diag½ signðeT P1 Þ signðeT P2 Þ 4 X ¼ ðeT Pi Þ Dgi þ Dgimax signðeT P Þ
signðeT P3 Þ
signðeT P4 ÞDgmax
ð46Þ
i¼1
which is clear to be always negative. 4.3. A numerical example The control law, given by Eq. (46), is shown to be a strong function in P , which is the solution of Eq. (39). Thus a careful choice of the matrix Q must be made to guarantee having a satisfactory performance. A few off-line trails can be made to develop the best structure for Q. To exemplify the design technique, it is assumed that R is the only uncertain parameter, such that: R ¼ Rnom ð1 þ DRÞ;
jDRj 6 DRmax ;
Rmax ¼ Rnom ð1 þ DRmax Þ;
and
Rmin ¼ Rnom ð1 DRmax Þ
ð47Þ
which yields u1 ¼
1 1 Vm N k1 Rmax x1 þ x1e cosðNx4 Þ þ Nx2 x3 þ x3 þ x1 signfx1 ðeT P1 Þg s1 s1 L1 L1 L1
ð48Þ
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Fig. 6. Robust control: (a) x1 and x2 , (b) x3 and x4 , (c) u1 and u2 , (d) u3 and u4 .
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u2 ¼
1 1 Vm Rmax x2 þ x2e sinðNx4 Þ Nx1 x3 þ x2 signfx2 ðeT P2 Þg s2 s2 L1 L1
ð49Þ
u3 ¼
1 1 þ x3 g3 ðX Þ s3 s3
ð50Þ
u4 ¼
1 1 þ x3 g4 ðX Þ s4 s4
ð51Þ
and, from which ðx1 K1 Þ ðx2 K2 Þ V_ ðeÞ ¼ fRnom Rmax signðx1 K1 Þg þ fRnom Rmax signðx2 K2 Þg L1 L1
ð52Þ
where K is a row vector ¼ eT P , and Ki is its ith element. It is clear that the result of Eq. (52) is negative for all values of R and X . Fig. 6 shows the motor response for the case where Rnom ¼ 0:285 X and Rmax ¼ 0:3 X. The Q matrix was chosen to be the identity matrix. The rest of the motor parameters are the same as that for the deterministic case considered in the previous section. Eq. (53) gives the values of Ad and P matrices used in the simulation 2 3 2 3 20 0:025 6 7 6 7 20 0:025 7; 7 Ad ¼ 6 P ¼6 ð53Þ 4 5 4 5 20 0:025 20 0:025 It is clear from Fig. 6 that the same equilibrium points were used again, so that the robust performance could be easily compared against the original design. As discussed in Section 3, there are two equilibrium points for each value of the operating parameter, x1 . Using any of them will result in having zero steady state values for the introduced control signal, U , however, any value for the desired response could have been used as well. One final issue to be further investigated is the causality of the proposed controller. First, the controller, given by Eqs. (48)–(51) is a full-state nonlinear feedback system. For practical reasons, only some of these states will be available for measurements. This necessitates the need for a state observer to be added to the system. The structure of the state observer, along with its convergence rate, and biasness will increase the order of the controller, and strongly affects its performance. Also, it is luxury to have four controls directly acting on the four state variables, which is very often impossible. Detailed analysis of dealing with such issues is considered on [22].
5. Conclusion A bifurcation analysis has been applied to a PM stepper motor. Numerical simulations were done using MATHEMATICA for implementing the bifurcation multiple-scale algorithm for Chaos analysis, and MATLAB for investigating the control design. The results show that the system, without control, has a Hopf bifurcation point that lead to undesirable oscillations. For
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the parameter range 825:315 < x1 < 1274:581, there are two kinds of solutions, an unstable constant solution, and a stable periodic solution. The stable periodic solution between the two Hopf bifurcation points is known as mid-frequency oscillation. A nonlinear robust model-reference controller has been designed to control the bifurcation as well as incipient chaos via converting the region between the two Hopf bifurcation points, H1 and H2 , into a stable region. The control strategy was based on forcing the motor to follow the response of a reference model using a Lyapunov-based design that guarantees stability. Robustness of the controller design against parameters uncertainty was carefully studied, and a modified structure was used to implement the new design such that stability and good performance for the entire region of the uncertain parameters were insured. Finally useful comments regarding causality and extensibility of the controller were given. Appendix A The stepper motor used in this paper has the following parameters: Vm ¼ 3:14 V, L1 ¼ 0:00105 H, N ¼ 50 turns, R ¼ 0:285 X, k1 ¼ 0:00177 V s, Bf ¼ 0:0019 N m s/rad, J ¼ 0:00004 g cm2 , TL ¼ 0:001 N m, x1 ¼ 1100 rad/s.
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