European Journal of Control (2009)6:599–612 # 2009 EUCA DOI:10.3166/EJC.15.599–612
Tracking of Partially Unknown Trajectories for Permanent Magnet Synchronous Motors1, 2 Luca Gentili1,, Andrea Paoli1, and Claudio Bonivento1, 1
Center for Research on Complex Automated Systems (CASY) ‘‘Giuseppe Evangelisti’’, DEIS – Department of Electronic, Computer Science and Systems, University of Bologna, Viale Risorgimento 2, 40136 Bologna, Italy
In this paper, an adaptive internal model-based control architecture is designed to deal with an exogenous trajectory tracking problem for a Permanent Magnet Synchronous Motor (PMSM). More in detail, we show how to design a controller able to guarantee the asymptotic tracking of partially unknown exogenous trajectories belonging to a given family, embedding in the regulator the internal model of this family; the control algorithm is able to attain the asymptotic tracking without the knowledge of reference derivatives and exploiting only instantaneous error feedbacks. The theoretical machinery exploited is the nonlinear regulation theory, specialized for the energy-based portHamiltonian formalism. The same methodology is proved to be able to deal even with the presence of exogenous voltage disturbances superimposed to the control voltages. Keywords: Disturbance suppression; Internal model control; Output regulation; Permanent magnet synchronous motor
Correspondence to: L. Gentili, E-mail:
[email protected] E-mail:
[email protected] E-mail:
[email protected] 1 2
The research is supported by MIUR. A preliminary version of this paper was presented at the European Control Conference 2007 ([12])
1. Introduction During last years permanent magnet synchronous motor (PMSM) has grown of importance in industrial automation community due to its high efficiency. In fact the absence of an externally supplied rotor (substituted by a magnetic core) avoid magnetic losses in rotor windings augmenting performances. Moreover, the introduction of new magnetic materials makes possible to generate high-density magnetic fields increasing motor performances. Finally the absence of rotor windings and brushes reduces motor wear and hence maintenance costs. For an exhaustive introduction to PMSMs the reader is referred to [16]. PMSMs became therefore one of the fundamental components of automated systems and the problem of controlling their behaviour has attracted the interests of most researchers in nonlinear control area. Several works presenting position and speed controllers for PMSMs can be found in literature. Among these, it is possible to find controllers based on passivity ([24]), controllers designed using feedback linearization ([14]), and backstepping ([9]). It is worth to mention the work described in [25]. In that paper the authors design an almost globally stable controller for PMSMs using the well-known energyshaping controller design technique. The controller is based on the interconnection and damping assignment passivity based control technique (IDA-PBC): energyshaping and passivity principles with focus on the Received 8 April 2008; Accepted 18 March 2009 Recommended by Z.P. Jiang, A.J. van der Schaft
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interconnection and damping structures of the system are deeply exploited. The final control scheme consists of a static-state feedback enriched with a nonlinear observer estimating the unknown load torque. Moreover, authors show how their controller is consistent with industrial practice, where a nonlinear observer is used instead of a simple PI speed loop. In [15], authors use an adaptive controller fitting the Hamiltonian framework to assure asymptotic tracking for currents and velocity. This result is proved to be robust with respect to load torque and stator resistance. A different typical problem to deal with when working with PMSM is torque ripple suppression. An interesting work in this field is [27]. In fact the main problem of PMSMs is that the generated torque is nonuniform and corrupted by a torque ripple due to high harmonics in magnetic interaction between stator windings and rotor magnet. Since PMSMs aim for high-performance applications, torque oscillations are not acceptable. So authors designed an adaptive control algorithm for minimization of torque ripple in PMSM drives, based on energy shaping and damping injection. The exploited adaptation law extracts, from the electrical subsystem, information about torque ripple harmonics and a current controller uses these information to achieve ripple minimization. Here, we are interested in solving a tracking problem for a PMSM: this is a simple but very significant issue as the tracking of a particular velocity profile is probably the main task to take into account when dealing with PMSM drives. The reader can find an exhaustive literature regarding tracking problems for PMSM in [10], [23], [29], [30], [31], [34] and references therein. More in detail, in this paper we show how to design a controller able to guarantee the asymptotic tracking of exogenous trajectories belonging to a known family. In [5], an analogous tracking problem complicated by the presence of unknown disturbances has been solved but for known trajectories. In this paper, the novelty is that the foreknowledge of reference is not assumed and hence time derivatives of reference are not available for ‘‘classic’’ feedback–feedforward control. Examples of this setup are trajectory tracking problems with respect to floating reference (e.g. Vertical Take-Off and Landing aircraft approaching a moving platform) or tracking problem with visual feedback (visual servoing). The main contribution of the paper with respect to other tracking problem solutions relies therefore in the definition of a control law able to attain the asymptotic tracking without the knowledge of reference derivatives and exploiting only instantaneous error feedback. In order to prove tracking properties of the exposed solution, we exploit the nonlinear regulation theory:
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the regulator embeds an adaptive internal model of the possible trajectory family. These concepts are specialized for the energy-based port-Hamiltonian formalism which is indeed really helpful to describe the problem, starting from an energetic description of the synchronous motor, and to find an elegant solution. For an introduction to the nonlinear regulation problem and its solution using an internal model controller the reader is referred to [18], [8]. Theoretically, the key point presented in this paper is the use of a suitably defined change of coordinates in order to define for the PMSM a tracking error system fitting in the port-Hamiltonian framework. With this setup it will be shown that the tracking problem can be cast into a regulation problem for the error system and an adaptive internal model based controller will be designed in order to asymptotically solve the tracking problem. Moreover, the same methodology is proved to be able to deal even with exogenous voltage disturbances added on the control voltages: the same asymptotic tracking problem is indeed solved in the presence of unknown voltage disturbance. The problem of input disturbance suppression is a very important topic in control theory as can be used to model the case in which malfunctions on the systems can be modelled as signals superimposed to the input channels; in real case it is possible to assume that the malfunction effect belongs to a known class of signals while their parameters (amplitude and even phase and frequencies in case of periodic signals) are unknown. For example malfunctioning on rotating systems driven by a power electronic part (e.g. electrical drives, magnetic levitation systems etc.) leads to asymmetries reflecting in spurious harmonics in the electrical variables (see [11], [6], [5], [1], [3]). The hypothesis of not perfect knowledge of the characteristic frequencies introduces a complex issue to deal with: in the last years this problem has been pointed out and addressed using different design techniques (see [28], [19], [21], [22], [2] and references therein). The paper is organized as follows: in Section 2, first the PMSM model is presented and the tracking problem is stated; then the change of coordinates to cast the tracking error system in the port-Hamiltonian framework is presented and the tracking problem is formalized as a regulation problem. In Section 3, an adaptive internal model-based controller is designed in order to asymptotically solve the problem. In Section 4, the same methodology exposed in the last section is extended to deal with exogenous disturbances acting superimposed to the input channels. In Section 5, some simulation results are presented to show the effectiveness of the proposed control techniques. Section 6 concludes the work with some final remarks.
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In Eq. (2) the Hamiltonian function is defined by
2. Problem Statement and Preliminary Positions
1 HðxÞ ¼ xT M1 x 2
Aim of this section is to introduce the model of a PMSM and to formally state the tracking problem. This problem can be described in words as follows: the motor should follow a desired velocity profile assuring, at the same time, zero flux current in order to obtain a perfect decoupling between flux and torque generation. This task should be asymptotically achieved despite of the presence of an unknown constant load torque. In our framework, the foreknowledge of the desired velocity trajectory is not considered; therefore time derivatives of the reference are not available for feedback–feedforward usage. The desired profile will be assumed only to belong to the class of signals generated by a linear, autonomous, and neutrally stable system, called exosystem. In this set up, for instance, any trajectory obtained by a combination of constant and sinusoidal signals can be modelled. This key assumption is used to cast the problem of trajectory tracking as a regulation problem as pointed out more precisely at the end of this section. In the following the system model is described. A PMSM (in a rotating reference, i.e. the dq frame) can be modelled as Ld i_d ¼ Lq !iq Rs id þ vd Lq i_q ¼ Ld !id q0 ! Rs iq þ vq j l !_ ¼ ðLd Lq Þid iq þ q0 iq ; np np
ð1Þ
where id and iq are the stator currents, ! is the angular velocity, Ld and Lq are the stator inductances, j is the inertia momentum, np is the number of pole pairs, Rs is the stator winding resistance, q0 is a constant term due to interaction of the permanent magnet, and the stator. The stator voltages vd and vq are the available control inputs while the constant load torque l is an unknown input. Model (1) can be rewritten as a port-Hamiltonian system with dissipation (see [26], [23], and [32]) considering the state vector 2 3 2 3 Ld 0 0 id 6 0 Lq 0 7 7 M¼6 x ¼ M4 iq 5 with 4 j 5: 0 0 ! np Hence the PMSM can be modelled as a portHamiltonian system of the form @HðxÞ þ g½ v d x_ ¼ ½JðxÞ R @x
vq
l T :
ð2Þ
while matrices JðxÞ, R and g are determined to be 3 2 0 0 L0 x3 7 6 0 q0 5; JðxÞ ¼ 4 L0 x3 0 q0 0 2 3 2 3 1 0 0 Rs 0 0 60 1 6 7 0 7 7; R ¼ 4 0 Rs 0 5; g ¼ 6 4 15 0 0 0 0 0 np where L0 ¼ Ld np =j ¼ Lq np =j under the assumption that we are using round-rotor machines (as opposed to salient-pole) for which the simplification Ld ¼ Lq holds. As stated previously, the control objective is to steer the motor to follow an exogenous, desired velocity des assuring, at the same time, trajectory xdes 3 ¼ ðj=np Þ ! zero flux current in order to obtain a perfect decoupling between flux and torque generation (i.e. des ides d ¼ 1=Ld x1 ¼ 0). Obviously this task should be asymptotically achieved despite of the presence of an unknown constant load torque l ; in that way the desired trajectory des for the iq current (ides q ¼ 1=Lq x2 ) results defined by an des des appropriate choice of ! ; id , and by l . As announced, let us assume that the desired trajectory is xdes ¼ hðÞ in which is generated by the linear, neutrally stable autonomous system (exosystem) _ ¼ Z 2 3 3 h1 ðÞ xdes 1 6 7 6 7 ¼ 4 xdes 2 5 ¼ hðÞ ¼ 4 h2 ðÞ 5 h3 ðÞ xdes 3 2 3 0 6 Lq @h3 ðÞ l 7 7; Z þ ¼6 4 q0 np 5 @ h3 ðÞ 2
xdes
ð3Þ
where 2 Rs and Z is defined by Z ¼ diagfZ0 ; Z1 ; . . . ; Z g; with Z0 ¼ 0, 0 Zi ¼ !i
!i 0
s ¼ 2 þ 1
!i > 0
i ¼ 1; . . . ;
and ð0Þ 2 Z Rs , bounded compact set. In this discussion, the dimension s of matrix Z will be considered known but all characteristic frequencies
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!i are unknown and ranging within known compact sets, i.e. !imin !i !imax . In this set up, the lack of knowledge of desired velocity profile reflects into the lack of knowledge of the initial state ð0Þ of the exosystem and of the characteristic frequencies. Moreover, the profile hðÞ (and h3 ðÞ in particular) will be assumed to be polynomial in : hence the class of signal considered is not restricted to a simple purely linear case, but will take into account a polynomial combination of constant and sinusoidal signals with unknown frequencies, amplitudes, and phases. Before introducing the main contribution of the paper, it is worth to deepen the polinomiality hypothesis made on the speed profile to be tracked h3 ðÞ. This is due to the main proposition in [17] which assures that, as h3 ðÞ is polynomial in , there exists some set of r real numbers a0 ; a1 ; . . . ; ar1 such that LrZ h3 ðÞ ¼ a0 h3 ðÞ þ a1 LZ h3 ðÞ þ r1 h3 ðÞ; þ ar1 LZ
ð4Þ
where Lf h denote the Lie derivative of function h along vector field f. Moreover, still in [17], the existnk 2 where r ¼ 2nk þ 1 0 ¼ 0 and !1 ; . . . ; ! ence of ! 1 þ . . . : þ lk !k 0; l1 ; . . . ; lk ¼ 0; 1; and ¼ fl1 ! 2; . . .g, such that
nk Y
2l Þ ¼ r a0 a1 ar1 r1 ð2 þ !
l¼1
ð5Þ is proved. From Eq. (5) it can be determined that ai ¼ 0 for i even. As known, condition (4) implies that the autonomous system _ ¼ Z u ¼ h3 ðÞ is immersed by a map ¼ ðÞ into the linear observable system (see [8], [18]) defined by _ ¼ ð6Þ u ¼ where matrices 2 0 1 60 0 6. . . . ¼6 6. . 40 0 0 a1
and are defined as 3 0 0 0 0 1 0 0 07 .. .. . . .. .. 7 . . . . .7 7; 0 0 0 15 0 a3 ar2 0
2 3 1 607 6 7 07 T ¼ 6 6 . 7: 4 .. 5 0
It is easy to realize that system (6) is equivalent to a linear system, and therefore immersed into: z_ ¼ Sz u ¼ z where matrices S and are defined as ¼ ½1
0
1 0
1
0
1;
S ¼ diagfS0 ; S1 ; . . . ; Sk g with S0 ¼ 0 and 0 Si ¼ i !
!i 0
!i > 0
i ¼ 1; . . . ; nk :
To this aim we use the linear transformation defined by z ¼ T with h i T1 ¼ T ST T ST r1 T : In the end we can write again the exosystem (3) and the desired exogenous trajectories as z_ ¼ Sz 2
xdes
3 0 6 Lq Sz þ l 7 ¼4 5: q0 np z
ð7Þ
The dimension of the characteristic matrix S (i.e. ¼ 2r ¼ 2ð2nk þ 1Þ) is still known, but, again, all i are unknown but ranging characteristic frequencies ! min max within known compact sets, i.e. ! !i ! . i i All the discussed assumptions allow us to cast the tracking problem as a regulation problem (see [7], [13]), complicated by the lack of knowledge of the matrix S (see [28], [4]), and suggest to look for a controller which embeds an internal model of the exogenous signals, augmented by an adaptive part in order to estimate the characteristic frequencies. To take into account the unknown load torque as a constant value generated by the exogenous system, let us define a new exogenous variable w such that Lq ¼ Þ; simple computations show w; ð xdes 2 ¼ q0 that its dynamics are governed by w_ ¼ Sw. Remark: It is worth to remark that in [5] a similar tracking problem was solved considering available for des feedback even the tracking errors id ides d and iq iq regarding the stator currents: from a technological point of view, this was a severe limitation as the derivative of the exosystem signals was assumed to be available for feedback.
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It is also important to stress that we consider as available for feedback the measure of stator currents id and iq , angular velocity !, and the measure of the angular velocity tracking error !des !. The novelty of the architecture that will be designed is based on the fact that it does not rely on ‘‘classic’’ feedforward actions as they are impossible in our framework due to the lack of knowledge of desired profile time derivatives.3
stabilizing asymptotically to the origin the new error variables (~ x1 ; x~2 ; x~3 ) is equivalent to accomplish the original tracking objective i.e. x~ ! 0 ) x ! xdes . Deriving error coordinates (8), it is possible to write the x~1 dynamics L0 Rs L0 L0 x~_ 1 ¼ x~2 x~3 x~1 x~2 x~3 þ x2 x3 þ vd ; Lq Ld Lq Lq the x~2 dynamics
3. Internal Model Unit Design In this section, we present the design procedure for the controller embedding the internal model of the exosystem generating the desired trajectory. It turns out to be useful to define a new change of coordinates defining error coordinates as: x~1 ¼ x1 x~2 ¼ x2
Lq ^ k3 x~3 q0
ð8Þ
x~3 ¼ x3 z;
ð9Þ
where N, F, and G will be suitably defined in the followings with the constraint that (F; G) is a controllable pair. It is important to stress that the new error coordinates are all available for feedback. Approaching to the problem with a standard design methodology (i.e. considering known the exosystem matrix S and hence the characteristic frequencies) the matrix should be computed solving in Y the Sylvester equation YS ¼ FY þ G
q0 np L0 ð~ x3 x3 Þ ð~ x1 x~3 x1 x3 Þ þ Ld j q0 q0 2 Rs þ ð~ x~2 x2 x2 Þ k3 k x~3 Lq Lq Lq 3 d Lq ^ dt q0 þ
and the x~3 dynamics
where k3 is a suitable gain determined later on in Proposition 1 and is the state of the internal model unit ^ þ Nð~ ^ _ ¼ ðF þ GÞ x1 ; x~2 ; x~3 ; Þ;
q0 np L0 Rs x~_ 2 ¼ x~1 x~3 x~3 x~2 Ld j Lq ^ þ k3 w þ vq k3
ð10Þ
1 ; in our framework the matrix and imposing ¼ Y S is unknown, hence the Sylvester equation is not solvable and, in our internal model unit, we are going ^ to use an estimation of the matrix called ðtÞ (later on we are going to define the adaptation law in order to obtain an asymptotic estimate). The main idea is to design the internal model unit and the voltage control inputs in order to stabilize to the origin these new error variables and to assure that, asymptotically, the internal model state tracks the exogenous state variable w (i.e. ! Yw): hence it is immediate, just inspecting equation (8), to state that
q0 q0 ^ w : x~_ 3 ¼ x~2 þ k3 x~3 þ Lq Lq Let us define two new error variables regarding the ^ internal model state and the estimate : ¼ Yw þ
1 G~ x2 k3
~ ¼ ^
ð11Þ
and consider the new Hamiltonian as ~ xÞ ¼ 1 x~T M1 x~ : Hð~ 2 On the basis of the x~i dynamics (i ¼ 1; 2; 3), the following preliminary control actions can be considered in order to obtain a simpler expression, able to enlighten a useful port-Hamiltonian structure: L0 L0 x~2 x~3 x2 x3 Lq Lq q0 np L0 ð~ x3 x3 Þ vq ¼ vq0 ð~ x1 x~3 x1 x3 Þ Ld j q0 q0 2 Rs ð~ x~2 þ x2 x2 Þ þ k3 k x~3 Lq Lq Lq 3 d Lq ^ : þ dt q0
vd ¼ vd0 þ
ð12Þ
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In previous expression vd0 and vq0 are new control inputs to be designed. By substituting (12) in x~ dynamics we obtain: ~ ~ L0 Rs @H @H Rs þ vd0 x~2 x~3 x~1 þ vd0 ¼ L0 x~3 x~_ 1 ¼ Lq Ld @ x~2 @ x~1 ð13Þ q0 np L0 Rs x~_ 2 ¼ x~1 x~3 x~3 x~2 þ vq0 Ld j Lq ^ k3 ð wÞ ~ ~ ~ @H @H @H q0 Rs þ vq0 @ x~1 @ x~3 @ x~2 ^ wÞ k3 ð ð14Þ
ð15Þ
^ w can be rewritten as: The exogenous term 1 ^ w ¼ ^ ð ^ Þð ~ þ G~ x2 Þ k3 ~ þ 1 G~ ^ 1 G~ x2 þ ð x2 Þ þ ¼ k3 k3 and additional control inputs vd0 and vq0 designed as vd0 ¼ vdst vq0 ¼
k1 x~1 Ld k2 ¼ x~2 ; Lq
vdst ¼ vqst
where k1 and k2 are gains whose values are determined by the later introduced Proposition 1. Let now focus the attention on the variable defined in (11). It is immediate to find that its dynamics can be written as
¼ L0 x~3
q0 q0 ^ w x~_ 3 ¼ x~2 þ k3 x~3 þ Lq Lq ~ q0 k3 j @ H ~ @H ^ w : þ þ ¼ q0 Lq np @ x~3 @ x~2
The stabilizing terms can be therefore suitably designed to inject two damping terms into error states x~1 and x~2
~ Lq ^ @ H ^ x2 þ vqst G~ G k3 @ x~3
where vdst and vqst are stabilizing terms. With these considerations in mind and with simple substitutions and computations, we obtain: ~ ~ @H @H Rs þ vdst x~_ 1 ¼ L0 x~3 @ x~2 @ x~1 ~ ~ @H _x~2 ¼ L0 x~3 @ H q0 Lq G ^ k3 @ x~1 @ x~3 ~ @H ~ þ 1 G~ Rs k3 ð x2 Þ k3 þ vqst @ x~2 k3 ~ ~ Lq ^ @ H q0 k3 j @ H x~_ 3 ¼ q0 G þ Lq np @ x~3 k3 @ x~2 1 ~ þ G~ þ ð x2 Þ þ : k3
^ þ N FYw Gw _ ¼ ðF þ GÞ q0 np 1 L0 x~3 þ G x~1 x~3 Ld j k3 ~ Lq ^ @ H Rs k2 ^ þ x~2 þ G k3 x~2 þ k3 w Lq k3 Lq @ x~3 1 1 x2 F G~ x2 þ N ¼ F Yw þ G~ k3 k3 q0 np 1 L0 Rs þ G x~1 x~3 x~3 þ x~2 Ld j Lq k3 ~ k2 Lq ^ @ H þ G x~2 ; k3 @ x~3 Lq where N is an additional term that can be suitably designed. If N is chosen as q0 np 1 1 L0 x~3 N ¼ F G~ x2 G x~1 x~3 Ld j k3 k3 ~ k2 Lq ^ @ H Rs þ x~2 þ G x~2 ; Lq k3 @ x~3 Lq then the dynamics become: _ ¼ F : Finally it is possible to design the following adaptation law in order to preserve a port-Hamiltonian structure: T np 1 1 ^_ ¼ k3 þ 1 G~ x~3 : x~2 þ G~ x2 x2 j k3 Lq k3 ð16Þ With this in mind we obtain the following portHamiltonian system: @H ð Þ ~ Rð Þ ~ þ ð Þ : _ ¼ ½Jð Þ @
ð17Þ
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In Eq. (17) the state vector is defined by T ~T ¼ x~1 x~2 x~3
Proof: Thanks to the control structure just described, the proof is standard and consists in choosing as Lyapunov function V ¼ H and decomposing matrix F as the sum of two matrices F1 and F2 (F ¼ F1 þ F2 ). With this in mind it turns out that
and the Hamiltonian function H ð Þ by ~ ~T : ~ xÞ þ 1 T þ 1 H ð Þ ¼ Hð~ 2 2 Moreover, the skew-symmetric interconnection matrix ~ turns out to be Jð Þ 0 1 0 L0 x~3 0 0 0 T B L0 x~3 0 J~1 ð Þ 0 J~2 ð Þ C B C ~ B ~ Jð Þ ¼ B 0 J1 ð Þ 0 0 J~3 ð ÞT C C; @ 0 0 0 0 0 A 0 0 J~2 ð Þ J~3 ð Þ 0 where Lq ^ J~1 ð Þ ¼ q0 G k3 1 x2 J~2 ð Þ ¼ k3 þ G~ k3 1 x2 : J~3 ð Þ ¼ þ G~ k3
x1 ; x~2 ; x~3 ; Þ ¼ ð0; 0; 0; 0Þ lim ð~
0
~ is bounded. Hence while adaptation error variable the proposition is proved and the tracking problem is solved.3 T
1 0 0C C C 0C C 0A 0
and the term ð Þ is
Now it is possible to apply twice the Young’s np inequality, first, fixed F1 , to overcome x~3 with k3 j k3 x~2 with k2 . and then, fixed F2 , to overcome Lq Note that a known bound on is required and can be easily derived (see (10)) from the known bounds on i once fixed F1 and F2 . frequencies ! Doing this it is immediate to prove that, thanks to Barbalat’s Lemma ([20]), system asymptotically converges: t!1
The semipositive-definite damping matrix R~ is 0 ðRs k1 Þ 0 0 0 B 0 0 0 ðRs k2 Þ B B
k j R~ ¼ B 0 0 Lq0q np3 0 B @ 0 0 0 F 0 0 0 0
ð Þ ¼ ð 0 k3
ðRs k1 Þ 2 ðRs k2 Þ 2 q0 k3 2 V_ x~1 þ x~2 þ x~ Lq 3 Ld Lq np k3 x~3 : x~2 þ þ T F1 þ T F2 j Lq
0 ÞT :
Remark: Let again remark that regulating (17) to the origin is equivalent to attain the original tracking objective: x~3 ! 0 ) x3 ! xdes 3 ; moreover ! 0 and x~2 ! 0 means that ! Yw and, from the definition of Lq x~2 ; x2 ! w ¼ xdes 2 .3 q0 Proposition 1: Consider the error system (17) obtained as interconnection between the PMSM (2) written in coordinates (8), the exosystem (3) generating the desired trajectory, the internal model controller (9), (12), and the adaptation law (16). It is possible to find suitable values for matrices F and G and gains k1 ; k2 , and k3 such that the tracking problem is asymptotically x1 ; x~2 ; x~3 Þ ¼ ð0; 0; 0Þ). solved (i.e. limt!1 ð~
Remark: Proposition 1 holds if the following explicit conditions are satisfied during the tuning of the control law. Matrices F and G can be chosen according to standard internal model tuning procedure: F ¼ F1 þ F2 ¼ diag ð1;1 ; . . . ; 1;2nk þ 1 Þ þ diag ð2;1 ; . . . ; 2;2nk þ1 Þ, G ¼ ½1; . . . ; 1T . The tuning procedure for gains k1 ; k2 ; k3 follows standard Young’s arguments3. – Choose k1 s.t. Rs k1 < 0. , choose " s.t. max ðF1 Þ – Once fixed F 1 1 k 3 1 < 0. þ L 2" q 1 Rs k2 "1 k3 < 0. þ – Choose k2 s.t. Lq 2 Lq max ðF2 Þ – Once fixed F2 , choose "2 s.t. np 1 þ j 2" < 0. 2 q0 k3 "2 np < 0: þ – Choose k3 s.t. j Lq 2 3
3
max ðAÞ is the maximum eigenvalue of matrix A.
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Remark: The presented design procedure assures that the tracking problem is achieved even if just a subset of the considered sinusoidal signals generated by the exosystem are present. The tracking errors are in fact proved to converge asymptotically to zero for any initial condition including the case of having some components of ð0Þ equal to zero (not all frequencies !i are excited). This fact reflects in assuring robustness of the approach with respect to exosystem order uncertainty if the number of harmonics are over-estimated: the dimension s of the matrix Z that is considered known is referred to an upper bound on the number of possible exogenous signals.3 Remark: The approach presented can be slightly modified in order to be robust even if the stator winding resistance is considered unknown but ranging in a known compact set. In this case the control input vq is modified deleting actions depending on Rs and adding a term k4 x2 where k4 is a design gain: q0 np L0 ð~ x3 x3 Þ ð~ x1 x~3 x1 x3 Þ Ld j q0 q0 2 d Lq ^ x~2 þ k3 x~3 þ þ k3 Lq Lq dt q0 ~ Lq ^ @ H ^ k2 G~ x2 x~2 þ k4 x2 : þ G k3 Lq @ x~3
vq ¼
ð18Þ Moreover, the internal model term N cannot use terms depending on Rs : q0 np 1 1 L0 x~3 x2 G x~1 x~3 N ¼ F G~ Ld j k3 k3 ~ k2 Lq ^ @ H þ G x~2 : k3 @ x~3 Lq The resulting derivative of the Lyapunov function V ¼ H is ðRs k1 Þ 2 ðRs k2 Þ 2 q0 k3 2 x~1 þ x~2 þ x~ V_ Lq 3 Ld Lq np k3 x~3 x~2 þ þ T F j Lq Rs 1 Rs x2 x~2 G x2 þ k4 x2 x~2 : Lq k3 Lq Following domination arguments as in Proposition 1, it Rs is possible to dominate terms ð þ k4 Þx2 x~2 Lq 1 Rs G x2 once fixed F suitably choosing gains k2 and k3 L q k4 . The proposed approach is hence robust with respect
to this parameter: this is a crucial feature as, physically, this electrical parameter is subject to large variations due to working temperature (see [33]).3
4. Disturbances Rejection In this section, the proposed methodology is specialized to take into account also external voltage disturbances superimposed to the input variables; this may represent the case in which malfunctions on the systems can be modelled as signals superimposed to the input channels. In real case it is possible to assume that the malfunction effect belongs to a known class of signals while their parameters (amplitude and even phase and frequencies in case of periodic signals) are unknown. In this case, the design procedure introduced is able to obtain a fault tolerant behavior: the asymptotic tracking is assured even in presence of a fault, and hence in presence of the resulting disturbances superimposed to the control inputs. The design procedure can be cast into the so-called implicit Fault Tolerant Control framework introduced in [6]. According to this approach the control reconfiguration does not rely on an explicit Fault Detection and Isolation design but is achieved by a proper design of a dynamical controller which is implicitly fault tolerant to all the possible faults whose model is embedded in the regulator by means of an internal model. Let us consider the PMSM (2) subject to two input disturbances dd ¼ 1 z1 and dq ¼ 2 z2 : x_ ¼ ½JðxÞ R
T @HðxÞ þ g ðvd dd Þ ðvq dq Þ l ; @x ð19Þ
with
z_1 ¼ S1 z1 ; z_2 ¼ S2 z2 dd ¼ 1 z 1 ; dq ¼ 2 z 2 ;
where matrices S1 ; S2 ; 1 and 2 are defined as 1 ¼ ½ 1 0
1
0
1
0
1
2 ¼ ½ 1 0
1
0
1
0
1
S1 ¼ diagfS10 ; S11 ; . . . ; S1n g S2 ¼ diagfS20 ; S21 ; . . . ; S2m g with S10 ¼ S20 ¼ 0 and ji 0 ! ji > 0 Sji ¼ ! ji 0 !
j ¼ 1 ; i ¼ 1; . . . ; n; j ¼ 2 ; i ¼ 1; . . . ; m:
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The dimensions of the characteristic matrices S1 and S2 (i.e. 2n þ 1 and 2m þ 1) are known, but, again, all characteristic frequencies !ji are unknown but ranging max !ji ! within known compact sets, i.e. !min ji ji . The design procedure is based on the change of coordinates (8) and on the definition of an extended internal model: the presence of three exogenous systems (the trajectory and the two disturbance generators) requires the presence of three internal models. Change of coordinates (8), internal model (9) and preliminary control actions defined as Rs k1 x~1 x~1 Ld Ld Rs k2 ^ vq0 ¼ vq00 þ x~2 x~2 þ k3 Lq Lq
vd0 ¼ vd00 þ
^ 2 Þ2 þ N2 ; _2 ¼ ðF2 þ G2
ð24Þ
where N1 ; N2 ; F1 ; F2 and G1 ; G2 will be suitably defined in the followings with ðF1 ; G1 Þ and ðF2 ; G2 Þ controllable pairs. Again matrices 1 and 2 should be computed solving in Y1 and Y2 Sylvester equations and Y2 S3 ¼ F2 Y2 þ G2 3 Y1 S1 ¼ F1 Y1 þ G1 1 1 and imposing 1 ¼ 1 Y1 and 2 ¼ 3 Y2 ; since in 1 our framework matrices S1 and S3 are unknown, Sylvester equations are not solvable and, in our internal model units, we are going to use estimates ^ 2 ðtÞ. ^ 1 ðtÞ and Firstly consider x~1 dynamics (20), it is possible to define new coordinates 1 ¼ 1 Y1 z1 G1 x~1 and ^ 1 1 . Simple computations show that ~1 ¼ ~ 1 ð1 G1 x~1 Þ þ 1 1 þ ^ 1 G1 x~1 : ^ 1 1 1 z 1 ¼
make it possible to write again Eqs. (13), (14), (15) as ~ ~ @H @H x~_ 1 ¼ L0 x~3 k1 þ vd00 1 z1 @ x~2 @ x~1
Designing the control action vd00 as ð20Þ
^ 1 1 ^ 1 G1 x~1 ; vd00 ¼ it is possible to write (20) as
~ ~ ~ @H @H @H x~_ 2 ¼ L0 x~3 q0 k2 þ v00q @ x~1 @ x~3 @ x~2 2 z 2 þ k3 w
L0 k1 ~ 1 ð1 G1 x~1 Þ þ 1 1 : x~_ 1 ¼ x~2 x~3 x~1 þ Lq Ld ð25Þ
ð21Þ ~ q0 k3 j @ H ~ @H ^ w : x~_ 3 ¼ q0 þ þ Lq np @ x~3 @ x~2
Let us focus on 1 dynamics: ð22Þ
Consider a new exogenous disturbance k3 w 2 z2 ¼ 3 z3 with z_3 ¼ S3 z3 ; again matrices S3 and 3 are defined as 3 ¼ ½ 1 0
1
0
1
0 1;
Designing N1 ¼ F1 G1 x~1 þ G1
S3 ¼ diagfS30 ; S31 ; . . . ; S3 g; with S30 ¼ 0 and 0 S3i ¼ 3i !
3i ! 0
L0 k1 ^ 1 G1 x~1 ; x~2 x~3 x~1 Lq Ld
we obtain
!3i > 0 i ¼ 1; . . . ; :
_ 1 ¼ F1 1 :
The dimension of matrix S3 is known and, again, 3i are unknown but all characteristic frequencies ! min ranging within known compact sets, i.e. ! 3i max 3i . 3i ! ! Now it is possible to define two internal model units: ^ 1 Þ1 þ N1 _1 ¼ ðF1 þ G1 and
^ 1 1 þ N1 Y1 S1 z1 G1 x~_ 1 _ 1 ¼ F1 1 þ G1 ¼ F1 1 þ F1 G1 x~1 þ N1 L0 k1 ^ x~2 x~3 x~1 1 G1 x~1 : G1 Lq Ld
ð23Þ
ð26Þ
Let us design the adaptation law as ^_ T ¼ ð1 G1 x~1 Þ x~1 ¼ ~_ T : 1 1 Ld
ð27Þ
Considering now x~2 dynamics (21), it is possible to define new coordinates 2 ¼ 2 Y2 z3 G2 x~2 and ^ 2 2 . Simple computations show that ~2 ¼ ~ 2 ð2 G2 x~2 Þ þ 2 2 þ ^ 2 G2 x~2 : ^ 2 2 3 z 3 ¼
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Designing the control action vq00 as
we obtain _ ¼ F :
^ 2 2 ^ 2 G2 x~2 ; vq00 ¼ it is possible to write (21) as q0 np L0 k2 x~3 x~2 x~_ 2 ¼ x~1 x~3 Ld j Lq ~ ~ þ 2 ð2 G2 x2 Þ þ 2 2 :
ð28Þ
Finally, the adaptation law for the last internal model unit is designed as: ^_ T ¼ np x~3 ¼ ~_ T : ð33Þ j From (25), (28), (31), (26), (29), (32), (27), (30), and (33) we obtain the overall port-Hamiltonian system
Let us focus now on 2 dynamics:
@H ð Þ ~ Rð Þ ~ þ ð Þ ; _ ¼ ½Jð Þ @
^ 2 2 þ N2 Y2 S3 z3 G2 x~_ 2 _ 2 ¼ F2 2 þ G2 L0 ¼ F2 2 þ F2 G2 x~2 þ N2 G2 x~1 x~3 Ld q0 np k2 ^ 2 G2 x~2 : x~3 x~2 j Lq
and the Hamiltonian function H ð Þ by ~ xÞ þ 1 T 1 þ 1 T 2 þ 1 T H ð Þ ¼ Hð~ 2 1 2 2 2 1 ~ ~T 1 ~ ~T 1 ~ ~T þ 1 1 þ 2 2 þ : 2 2 2
q0 np L0 N2 ¼ F2 G2 x~2 þ G2 x~1 x~3 x~3 Ld j k2 ^ 2 G2 x~2 ; x~2 Lq
Moreover, the skew-symmetric ~ turns out to be matrix Jð Þ
we obtain
0
ð29Þ
Again the adaptation law can be designed as ^_ T ¼ ð2 G2 x~2 Þ x~2 ¼ ~_ T : 2 2 Lq
ð30Þ
Let us finally consider x~3 dynamics (22); it is possible to define new coordinates ¼ Yz G~ x3 and ~ ¼ ^ . Simple computations show that q0 q0 ~ þ þ G~ x~2 þ k3 x~3 þ x3 x~_ 3 ¼ Lq Lq q0 q0 ~ þ : x~2 þ ¼ k3 þ G x~3 þ Lq Lq ð31Þ The dynamics can be derived as ^ þ N YSw Gx~_ 3 _ ¼ F þ G q0 q0 x~2 þ ¼ F þ FG~ x3 þ N G k3 x~3 ; Lq Lq choosing q0 q0 N ¼ FG~ x3 þ G x~2 þ k3 x~3 ; Lq Lq
ð34Þ
where the state vector is defined by T ~T ~ T 2 ~T ¼ x~1 x~2 x~3 1 1 2
Designing
_ 2 ¼ F2 2 :
ð32Þ
0 B L0 x~3 B B 0 B B 0 B ~ Jð Þ ¼ B B J~1 ð Þ B 0 B B 0 B @ 0 0
L0 x~3 0 q0 0 0 0 J~2 ð Þ 0 0
0 q0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
interconnection
J~1 ð ÞT 0 0 0 0 0 0 0 0
0 0 0 J~2 ð ÞT 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
1 0 0 C C T C C 0 C C 0 C C; 0 C C 0 C C 0 A 0
with J~1 ð Þ ¼ ð1 G1 x~1 Þ J~2 ð Þ ¼ ð2 G2 x~2 Þ ; and the semipositive-definite damping matrix R~ is 0 B B B B B B B R~ ¼ B B B B B B @
k1 0
0 k2
0
0
0 0 0 0 0 0
0 0 0 0 0 0
0 0
q0 k3 j þ G n Lq 0 0 0 0 0 0
0 0
0 0
0 0
0 0
0
0
0
0
F1 0 0 0 0 0
0 0 0 0 0 F2 0 0 0 0 0 0
0 0 0 0 0 0
1 0 0C C 0 0C C C 0 0C C 0 0C C 0 0C C 0 0C C A F 0 0 0 0 0
and the term ð Þ is ð Þ ¼ ð 1 1
2 2
0
0
0 0
0
0 ÞT :
Tracking of Partially Unknown Trajectories for PMSM
From the above, the asymptotic tracking of the exogenous trajectory despite the presence of voltage disturbances is equivalent to regulate (34) to the origin. Proposition 2: Consider the error system (34). It is possible to find suitable values for matrices F1 , F2 , F, G1 , G2 and G and gains k1 , k2 and k3 such that the tracking problem is asymptotically solved despite the presence of input disturbances (i.e. x1 ; x~2 ; x~3 Þ ¼ ð0; 0; 0Þ). limt!1 ð~ Proof: Again, thanks to the control structure, the proof follows the one exploited for Proposition 1: choosing as Lyapunov function V ¼ H , it turns out that q0 k3 k1 2 k2 2 _ þ G x~23 V x~1 x~2 þ Ld Lq Lq x~1 þ T F þ T1 F1 1 þ T2 F2 2 þ 1 1 Ld np x~2 þ 2 2 þ x~3 : Lq j Now it is possible to apply three times the Young’s x~1 inequality, first to overcome 1 1 with k1 once Ld x~2 fixed F1 , then to overcome 2 2 with k2 once fixed Lq np F2 and finally x~3 with k3 once fixed F. Note that j known bounds on , 1 and 2 are required and can be easily derived from the known bounds on freji once fixed F, F1 and F2 . i , ! quencies ! Doing this it is immediate to prove that, thanks to Barbalat’s Lemma ([20]), error variables asymptotically converge: x1 ; x~2 ; x~3 ; 1 ; 2 ; Þ ¼ ð0; 0; 0; 0; 0; 0Þ lim ð~
t!1
~ T, ~ T and ~ T are while adaptation error variables 1 2 bounded. Hence the proposition is proved and the tracking problem is solved.3
5. Simulation Results In order to check the performances of the regulator above presented, some simulation tests have been performed. A PMSM with the following parameters has been considered: Rs ¼ 2:7 , q0 ¼ 1 Wb, Ld ¼ Lq ¼ 12:23 103 H, np ¼ 4, j ¼ 0:003 kg m2 . The angular velocity desired trajectory has been with parameters defined as !des ¼ V0 þ V1 sinð!tÞ
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¼ 1 rad/sec V0 ¼ 133 rad/sec, V1 ¼ 93 rad/sec and ! ¼ 2 rad/sec from time from time 0 sec to time 10 sec, ! 10 sec to time 20 sec. The switching between two frequencies is made in order to enlighten the adaptation properties of the proposed solution, i.e. the ability of solving the tracking problem without requiring the knowledge of the characteristic frequencies of the exogenous trajectory. The motor is supposed to be affected by load torque varying during the simulation: in particular, from time t ¼ 0 to time t ¼ 2:5 sec, the load torque is defined as l ¼ 8 N m, from time t ¼ 2:5 sec to time t ¼ 12:5 sec l ¼ 2 N m and, finally, from time t ¼ 12:5 sec to t ¼ 20 sec l ¼ 8 N m. Controller gains have been chosen, according to Proposition 1, as: 3 2 0:4 0 0 7 6 F¼4 0 0:8 0 5; 0 0 1:2 2 3 1 6 7 G ¼ 4 1 5 ; k1 ¼ 10 ; k2 ¼ 100 ; k3 ¼ 0:06 : 1 ð35Þ The motor is initially considered stopped; the internal model is initialized to zero while, considering the fre ranging within [0.6; 2.1], the adaptation quency ! ^ variable is initially set to ð0Þ ¼ ½ 0:65 5 6:75 which corresponds to the lower bound of !. Fig. 1 shows simulation results: from upper to lower plot on the left column the desired trajectory xdes (as defined in equation (3)) to be tracked by the whole state vector x, load torque, tracking errors des des ex1 ¼ x1 xdes 1 , ex2 ¼ x2 x2 and ex3 ¼ x3 x3 are depicted. The reader can see their asymptotic convergence to zero which shows the effectiveness of the proposed control technique and the correctness of Proposition 1. Fig. 1 shows from upper to lower plot on the right column the zoom from 9.7 sec to 10.5 sec of tracking error variables to show transient behavior of the system around time t ¼ 10 sec (i.e. when trajectory frequency change: it is possible to note that the controller is able to adapt itself compensating the new frequency), stator currents id , iq and measured angular velocity !. Finally the lower plot of fig. 1 shows control inputs applied to the motor. The behavior of the same PMSM subject to an exogenous disturbance and controlled by the controller designed in Section 4 has been simulated: the tracking objective is the same of the above simulation but, from time t ¼ 5 sec a disturbance dq ¼ 2 t is acting on the system (d0 ¼ 3; d0 þ d1 sin !
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Fig. 1. From upper to lower plot on the left column: exogenous trajectories to be tracked and the load torque, tracking error variables des des ex1 ðtÞ ¼ x1 xdes 1 , ex2 ðtÞ ¼ x2 x2 and ex3 ðtÞ ¼ x3 x3 . From upper to lower plot on the right column: zoom from 9.7 sec to 10.5 sec of des , e ðtÞ ¼ x xdes tracking error variables ex1 ðtÞ ¼ x1 xdes x2 2 1 2 and ex3 ðtÞ ¼ x3 x3 , id , iq , and !. The lower plot shows vd and vq .
2 ¼ 1:5). The disturbance frequency is und1 ¼ 7; ! known to the controller; the knowledge is limited to the range [1.35; 1.65]; hence adaptation law started ^2 ¼ 1:35. Moreover, since it is a with initial value ! common assumption to consider Rs as piecewise
constant, to check the robustness of the proposed algorithm, at time t ¼ 8 sec stator resistance suddenly changes from 2.7 to 5 . To this aim the control law has been modified according to equation (18).
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Fig. 2. From upper to lower plot on the left column: exogenous trajectories to be tracked, the load torque, the stator resistance and des the disturbance dq ðtÞ, tracking error variables ex1 ðtÞ ¼ x1 xdes 1 , ex2 ðtÞ ¼ x2 x2 . From upper to lower plot on the right column: id ; iq ; !; vd and vq .
Controller gains have been suitably chosen as (35) with k4 ¼ 100; F1 ¼ F2 ¼ F; G1 ¼ G2 ¼ G. In Fig. 2 system variables trends are shown, enlightening the asymptotic achievement of objectives.
6. Conclusions In this paper, an adaptive internal model based control scheme is presented to deal with tracking of partially unknown trajectories problem for a PMSM. More in detail in Section 2 the port-Hamiltonian model of the motor is presented, the problem is stated and a suitable change of coordinates is introduced to define the desired error system. This step makes it possible to cast the tracking problem into a regulation problem. In Section 3 a controller able to guarantee the asymptotic tracking of partially unknown exogenous trajectories belonging to a certain family, embedding the internal model of this family, is presented; the theoretical machinery exploited is the
nonlinear regulation theory, specialized for the energy-based port-Hamiltonian formalism. The main contribution of the paper with respect to other tracking problem solutions relies in the definition of a control architecture able to attain the asymptotic tracking without the knowledge of reference derivatives and exploiting only instantaneous error feedback. The same methodology is proved to be able to deal even with exogenous voltage disturbances acting superimposed to the control voltages. The application of the presented techniques and the performances of the adopted solutions are shown in Section 5 with some simulation results.
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