This invention relates generally to controlling electric motors, and more particularly to a method and system for controlling the angular speed of a rotor of in a speed sensorless induction motor.
Adjustable speed induction motors are widely used due to low maintenance cost and acceptable performance. The speed control design, or equivalently, the motor drive design, of the induction motor is challenging due to highly nonlinear dynamics. Among various means, the vector (field oriented) control appears to be a good and most popular solution that has evolved as a mature technology.
This invention considers speed sensorless control technologies for the induction motor, i.e., an inductor motor where the rotor speed or position is not measured by encoders. Speed sensorless motor drives (motor drives) are desirable due to the elimination of speed sensors, lower cost, and improved reliability of the resultant system. Current speed sensorless motor drives suffer significant performance degradation due to the absence of the encoder to sense the rotor position and speed. Thus, their applications remain limited to fields requiring low or medium performance.
The performance of speed sensorless motor drives relies heavily on the performance of a flux and speed estimator 106-107 described below with reference to
A flux control block 101 determines a stator current in the d-axis, denoted by signal 114. A signal 115, as an estimate or true stator current in the d-axis, is produced by the flux estimator 106.
A difference signal 116 between signals 115 and 114 is used by a current control block 103 to determine a reference stator voltage in the d-axis, which is part of signal 123. Similarly, a signal 117 denotes a desired rotor speed reference for the induction motor. A signal 118 denotes an estimated rotor speed produced by a speed estimator block 107 based on an output signal 126 of the flux estimator 106.
A difference signal 119 between signals 117 and 118 is input to the speed control block 102 to determine a reference stator current 120 in the q-axis. An estimated or true stator current 121 in the q-axis is compared to the reference stator current 120 in the q-axis to produce a difference signal 122. The current control block 103 determines a desired stator voltage signal 123 in d- and q-axis, on the basis of the difference signals 116 and 122.
An inverse Clarke/Park transformation 104 convert the desired stator voltages signals in d- and q-axis into desired well-known three-phase voltages, and produces three-phase voltages, denoted as by 124, to drive the induction motor 105.
Note that the flux estimator 106 takes the three-phase voltages 124, sensed 131 phase currents 125, see
All these estimation methods fit an architecture shown in
This fact is elaborated by
Overall, most of existing speed sensorless motor drives produce limited speed tracking performance due to performing estimator design in appropriate state coordinates and under unnecessary assumption. Also, most speed-sensorless estimation methods are ad-hoc and the resultant estimation error dynamics are not guaranteed to be stable.
The embodiments of the invention provide a method and system for controlling rotor angular speed of an induction motor. The invention focuses on estimation methods that guarantee stability and practical effectiveness. The main contributions of the invention are as follows.
The invention uses an induction motor model that admits a non-triangular observable form by changing state coordinates. Then, high gain observer design can be performed in new coordinates.
However, the high gain observer in the new coordinates is not implementable due to failure to establish a closed form expression of an inverse state transformation. Therefore, the embodiments use several observers, which can be implemented in the original coordinates.
The implementable observers in the original coordinates involve a matrix inversion, which can be subject to numerical stabilities from time to time. Consequentially, the invention uses several techniques to address this issue, and makes the high gain observer practical.
The embodiments of our invention provide a method and system for controlling rotor angular speed of an induction motor.
We define the following notations. For a variable ζ of the induction motor, we denote
Clarke/Park Transformation
The Clarke transformation (α−β transformation), transforms quantities in three-phase circuits to a space vector representation, and thus simplifies the analysis of the three-phase circuits. In this disclosure, the Clarke transformation is limited to the case that transforms quantities in three-phase circuits such as three-phase stator voltages into a space vector in a fixed frame.
Similarly, the Park transformation (d−q transformation), projects quantities in the fixed frame onto a rotating two-axis reference frame. The Clarke/Park transformation and its inverse are well-known for those skilled in the art. Generally, applying the optional Park transformation after the Clarke transformation facilitates the control design.
Induction Motor Models
With the Clarke/Park transformation, a general model of the induction motor is
where y represents sensed stator current signals, ω1 is the rotational speed of a reference frame. If ω1=0, Eqn. (1) is reduced to
which represents the induction motor model without applying the Park transformation, and corresponds to block 201 in
Generally, the system in Eqn. (1) cannot be used to design convergence-guaranteed nonlinear observers. In order to design observers giving provably convergent estimation error dynamics, it is beneficial to transform the system (1) into certain structures by a change of coordinates, for simplicity in this invention, a change of state coordinates.
The invention uses the following method and system for state estimation based on a transformed induction motor model obtained by applying a change of state coordinates to the induction motor model (1) or (2).
As shown in
The block 322 outputs a high gain estimator 401 in the original state coordinates, which can be implemented in processor to produce estimates 402 of induction motor states including the stator currents, the rotor fluxes, and the rotor speed, also see
The dynamic gain estimator in the original coordinates 401 produces state estimates (îds,îqs,{circumflex over (Φ)}dr,{circumflex over (Φ)}qr,{circumflex over (ω)})T 402 for the controller 101, which in turn determines command voltages 123 in two-axis space for the motor. The command voltages 123 are transformed into command voltages in three-phase to the induction motor using the inverse Clarke/Park transformation 104.
Next described is an embodiment of new coordinates which simplifies state estimation in speed sensorless motor drives.
Non-Triangular Observable Form
A high gain observer design can be carried out when the system (1) is transformed into the following non-triangular observable form by a change of state coordinates x=ϕ(ζ)
{dot over (x)}=Ax+Φ(x,u)
y=Cx, (3)
where the state x∈n, and for 1≤k≤p,
where, xk denotes the state of the kth subsystem associated with the kth output yk, and λk, 1≤k≤p are the dimensions of all subsystems. For 1≤k≤p, we call λk subsystem indices, and Σk=1p λk=n. The vector field φ(x,u) is
Specifically, φik has the following structure: for 1≤i≤λk−1,
φik=φik(x1, . . . ,xk−1,x1k, . . . ,xik,x1k+1, . . . ,x1p,u) (4)
and for i=λk
φii=φλ
Motor Drives: Verifying that System (1) Admits the Form (3)
A concise representation of system (1) is
{dot over (ζ)}=f(ζ)+g1uds+g2uqs,
where
ζ=(dds,iqs,Φdr,Φqr,ω)T,
g1=(1/σ,0,0,0,0)T,
g2=(0,1/σ,0,0,0)T, and
f, g1, g2 are appropriately defined according to (1).
System (1) is transformable to the non-triangular observable form by the following change of state coordinates, for the kth subsystem
where λk are observability indices. Observability ensures that
x=ϕ(ζ)=((ϕ1(ζ))T, . . . ,(ϕp(ζ))T)T
defines new coordinates, i.e., ϕ(ζ) is a local diffeomorphism.
The observable form is not uniquely defined due to the non-uniqueness of observability definitions and observability indices.
In one embodiment, for system (1) with outputs h1=ids and h2=iqs, observability indices can be taken as (3, 2). For example, we take (λ1, λ2)=(3, 2) and verify that new state coordinates, given by
x=(h1,Lfh1,Lf2h1,h2,Lfh2)T (6)
transforms (1) into the non-triangular observable form (3). For a smooth vector field f(ζ): Rn→Rn and a smooth function h(ζ), we denote
the Lie derivative of h(ζ) along f(ζ). Repeated Lie derivative are defined as
with k>1.
Given Eqn. (6), the system in x-coordinates is
{dot over (x)}=Ax+φ(x,u)
y=Cx,
where A=diag{A1, A2} with A1∈R3×3 and A2∈2×2, C=diag{C1, C2} with C1∈R3 and C2∈2. One can verify that φ satisfies the triangular condition in Eqn. (4).
In another embodiment, for system (1) with outputs h1=iqs and h2=ids, observability indices can be taken as (λ1, λ2)=(3,2), and we can verify that new state coordinates x=(h1,Lfh1,Lf2h1,h2,Lfh2)T transforms (1) into the non-triangular observable form (3).
High Gain Estimator Design in New Coordinates x
For the case when the inverse of the state transformation ζ=ϕ−1(x) is unsolvable, the high gain estimator design in new coordinates x=ϕ(ζ) is not trivial.
Case When ζ=ϕ−1(x) is Solvable
Given the state transformation x=ϕ(ζ) and its inverse ζ=ϕ−1(x), one can obtain the system representation in the non-triangular observable form. Observer design is performed in the non-triangular observable coordinates to estimate x. Afterwards, estimates of ζ, denoted by {circumflex over (ζ)}, is reconstructed out of {circumflex over (x)}. For the kth subsystem, we use the following observer
and Sk is solved from
Sk+AkTSk+SkAk=CkTCk. (8)
Note that θ depends on the Lipschitz constant ω.
Case When ζ=ϕ−1(x) is Unsolvable
Without the closed form expression of the inverse state transformation (=ϕ−1(x), one cannot obtain the system representation in the non-triangular observable form because the Lipschitz constant φ(x,u) is unknown. This poses significant challenges to observer design approaches with state transformation involved because the observer (7) cannot be designed or implemented without knowing the expression of the Lipschitz constant φ(x,u). This section describes that one can still perform observer design in the non-triangular observable coordinates.
We describe how to derive the upper bound of the Lipschitz constant of φ. The design of observer (7) necessitates the expression of φ(x,u) in order to estimate the Lipschitz constant of φ(x,u) in x. We show how the Lipschitz constant of φ(x,u) in x can be estimated on the basis of ϕ(ζ). Assume that φ(x,u) is globally Lipschitz with respect to x uniformly in u, i.e., ∀x1, x2∈n, ∃Lx>0
|φ(x1,u)−φ(x2,u)≤Lx|x1−x2|. (9)
Note that Lx is unknown and to be estimated. From Eqn. (9), and considering that φ(x,u) is smooth about its arguments, we have for all u∈m
where
and
Because
is known, the upper bound of its infinity norm κ1 can be established. To derive the upper bound of
we take into account the fact that for a non-singular matrix M, its infinity norm (maximum absolute eigenvalue) is equal to the inverse of the minimum absolute eigenvalue of M−1,
The above equation produces an upper bound of
as
Therefore, we establish κ1κ2 as the upper bound of the Lipschitz constant Lx.
Motor Drives: High Gain Estimator Design in New Coordinates
Given the observer (7) for the kth subsystem and an estimated upper bound of the Lipschitz constant φ(x,u), an observer can be designed for each subsystem. For the induction motor case, we take λ1=3, λ2=2, p=2, and have design parameters
Note that θ is a tuning parameter of the observer. Substituting the above design parameters into the kth subsystem for k={1, 2}, we can obtain observer in x-coordinates as follows
{dot over (x)}1=A1{circumflex over (x)}1+1+G1C1{tilde over (x)}1
{dot over (x)}2=A2{circumflex over (x)}2+2+G2C2{tilde over (x)}2,
ŷ1=C1{circumflex over (x)}1
ŷ2=C2{circumflex over (x)}2, (10)
where
G1=θδ
G2=θδ
In practice, a less rigorous approach to design the high gain observer in new coordinates can select a sufficiently large θ, which balances performance and sensitivity to noises.
Dynamic Gain Estimator Design in Original Coordinates
One can estimate the state x using the observer designed in x-coordinates. In practice, it is more natural to have the original state estimate {circumflex over (ζ)}. For many cases, given the estimator in new coordinates, it is not easy to derive the corresponding estimator in the original coordinates.
As shown in
and then takes the output (estimated state in the original coordinates) of the dynamic gain estimator 401, finally evaluate the numerical value of the Jacobian matrix point-wisely at the estimated state, i.e.,
Qo(ζ)|ζ={circumflex over (ζ)}=Qo({circumflex over (ζ)})
with {circumflex over (ζ)}=(îds,îqs,{circumflex over (Φ)}qr,{circumflex over (Φ)}qr,{circumflex over (ω)}). We therefore have a point-wise matrix Qo({circumflex over (ζ)}).
It is likely the matrix Qo({circumflex over (ζ)}) is singular or close to singular, i.e., ill-conditionedness. Both facts lead to divergence of state estimation. Block 602 addresses the issues raising from the ill-conditionedness of Qo({circumflex over (ζ)}), and determines an approximate inverse of the point-wise matrix Qo({circumflex over (ζ)}) by replacing the ill-conditioned matrix Qo({circumflex over (ζ)}) with its non-singular and well-conditioned approximation. The dynamic gain estimator 401 is implemented based on the high gain estimator in the new coordinates and the approximate inverse of the point-wise matrix Qo({circumflex over (ζ)}).
Case When ζ=ϕ−1(x) is Solvable
Two approaches can be used to construct {circumflex over (ζ)}. The first solves for {circumflex over (ζ)} from {circumflex over (x)} based on the closed form expression of the inverse transformation ζ=ϕ−1(x). That is {circumflex over (ζ)}=ϕ−1({circumflex over (x)}). The second implements the observer (7) in ζ-coordinates as follows
where
Θ=diag{θδ
Δ−1=diag{Δ1−1(θ), . . . ,Δp−1(θ)}
S−1=diag{S1−1, . . . ,Sp−1}.
Notice that Eqn. (11) includes the term
which assumes the closed form formula of the inverse transformation ζ=ϕ−1(x), and thus may not be implementable with an explicit expression of ζ=ϕ−1(x).
Case When ζ=ϕ−1(x) is Unsolvable
With the expression of φ(x,u) unknown, the observer (7) cannot be realized unless φ is represented as a function of ϕ(ζ),u). Consequently, the following implementable x-observer can be obtained
{circumflex over ({dot over (x)})}k=Ak{circumflex over (x)}k+φk(ϕ({circumflex over (ζ)}),u)+θδ
ŷ=Ck{circumflex over (x)}k, (12)
where φk(ϕ({circumflex over (ζ)}),u) are the rows corresponding to the kth subsystem in φ(ϕ({circumflex over (ζ)}),u) given by
and {circumflex over (ζ)} is numerically solved from the following n nonlinear algebraic equations
{circumflex over (x)}=φ({circumflex over (ζ)}). (13)
As a result, the observer (12) is implemented as (13) and
Alternatively, considering the state transformation ϕ(ζ) is a local diffeomorphism, and thus its Jacobian
is non-singular. One can therefore implement the observer (11) as follows
Note that Eqn. (15) is different from Eqn. (11) where the Jacobian of the inverse state transformation
is used.
Motor Drives: High Gain Estimator Design in Original Coordinates
Two approaches can be used to estimate ζ on the basis of Eqn. (10). The estimated state {circumflex over (ζ)} can be constructed by implementing the observer in ζ-coordinates, i.e., as in Eqn. (15).
Point-Wise Matrix Inverse Approximation
To shorten the notation, we use
Implementation of the observer in Eqn. (15) requires one to solve the inverse of Qo in realtime, more accurately, to solve the observer gain Gζ in ζ-coordinates from the following linear system
Gx=QoGζ, (16)
where Gx=ΘΔ−1(θ)S−1CT. Numerical instability could be serious when Qo is ill-conditioned, and we cannot easily rely on the solutions coining out of an ill-conditioned Qo, for instance the least square solution
Gζ=(QoTQo)−1QoTGx
becomes useless. Let cond(Qo)=|Qo|28 |Qo−1|∞ be the condition number of Qo. One can expect a loss of roughly log10(cond(Qo)) decimal places in the accuracy of the solutions.
Next, we describe techniques to remedy the aforementioned numerical instability.
Preconditioning
A number of method are known for obtaining a good approximate inverse of an ill-conditioned Qo using techniques such as pre-conditioning, scaling, etc. An intuitive pre-conditioning, also called the Tikhonov regularization, uses a positive constant h into (16) and tries to solve
Gζ=(QoTQo+hI)−1QoTGx.
By using h, the Tikhonov regularization reduces the condition number of QoTQo and results in an improvement of the solution accuracy. Alternatively, one can regularize the problem by adding extra information, for instance the smoothness of Gζ. The basic idea is that the correct solution among all “near” solutions is characterized by requiring a “smoothness” of some function, curve or surface constructed from Gζ at a fixed time instant.
Dynamic Gain
When a smooth function of ζ(t) is continuous over time, Gζ is continuous over time as well. However, this contradicts the fact that an ill-conditioned Qo typically results in discontinuity of Gζ over time. Therefore, during implementation, one can force that Gζ is be continuous over time.
As shown in
One embodiment of the matrix smoothing mechanism can be
where k is the time step, and c is a positive constant. Another embodiment of the matrix smoothing mechanism uses a low pass filter.
Functional Matrix Inverse Approximation Qo
The above described preconditioning methods concentrate on Eqn. (16), and try to resolve the ill-conditionedness of the numerical matrix Qo at a fixed time instant (point-wisely). As shown in
where Q is a functional space, and |Qo−{circumflex over (Q)}o| is certain norm representing the distance between Qo and {circumflex over (Q)}o. The problem in Eqn. (17) is of infinite dimensions and very difficult to solve. Instead, approximation of Qo is more often accomplished intuitively.
An embodiment of the function approximation idea solves (17) by exploiting the structure of Qo. Notice that in the fixed frame, the matrix Qo is periodically time-varying, and thus makes the numerical stability issue more complicated.
Hence, we first rewrite the system (1) in a frame rotating at ω1 so that at a steady state, ids,iqs,Φdr,Φqr are constants. Then, we remove 804 some components in Qo to get {circumflex over (Q)}o and avoid ill-conditionednesstrim Qo, which is suitable for realtime implementation and give reasonable state estimation performance. For illustration purpose, we express
Notice that the elements in the last row of matrix Qo are large numbers, and cause the ill-conditionedness. Also, the last row uses large feedback gains in the ĩds and ĩds-dynamics, and thus make the entire error dynamics sensitive to measurement noises. Therefore, we approximate matrix {circumflex over (Q)}o by letting the first four elements of the last row in Qo be zero. We implement the observer (15) with
equivalently Qo, replaced by
Although the invention has been described by way of examples of preferred embodiments, it is to be understood that various other adaptations and modifications can be made within the spirit and scope of the invention. Therefore, it is the object of the appended claims to cover all such variations and modifications as come within the true spirit and scope of the invention.
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20170201198 A1 | Jul 2017 | US |