1 Algebraic Methods for Nonlinear Systems

1 Algebraic Methods for Nonlinear Systems: Parameter Identification and State Estimation John Chiasson1 , Kaiyu Wang1 , Mengwei Li1 , Marc Bodson2 and Leon M. Tolbert1,3 1

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ECE Dept, The University of Tennessee, Knoxville, TN 37996-2100, USA {chiasson,wkaiyu,mwl,tolbert}@utk.edu ECE Dept, The University of Utah, Salt Lake City, UT 84112 USA [email protected] Oak Ridge National Laboratory, NTRC, 2360 Cherahala Boulevard, Knoxville, TN 37932 [email protected]

Summary. Algebraic methods are presented for the problem of solving nonlinear least-squares type problems which arise in the parameter identification of nonlinear systems. The tracking of the induction motor rotor time constant is solved in detail. Also, an approach to estimating state variables using algebraic relationships (in contrast to dynamic observers) is discussed in the context of speed estimation for induction motors.

1.1 Introduction Algebraic methods have long been used as tools for problems in linear systems theory. In this work, some algebraic tools are explored as methods for nonlinear parameter identification and for constructing observers for nonlinear systems. Parameter and state estimation continue to be an important areas of research precisely because they are used in many practical engineering problems. For example, the parameters characterizing the internal working of a physiological system are almost never available for direct measurement and therefore must be approached indirectly as a parameter estimation problem. The recent Bode lecture by Professor Lennart Ljung outlined many of the challenges of nonlinear system identification as well as its particular importance to biological systems [18]. In these types of problems, the model developed for analysis is typically a nonlinear state space model with unknown parameter values. For example, in [24] a 15th order nonlinear differential equation model based on first principals is developed to model blood glucose uptake in humans, but only a few of the parameter values are directly measurable. In addition to the parameters not being directly measured, very few of the state variables are

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measurable so that the model must be reformulated as input-output model which invariably results in an overparameterized model. Though a standard least squares (regressor) approach can “theoretically” be applied, the overparameterized model typically results in an ill-conditioned problem and unreliable results. Here a method is proposed that can be used for parameter identification of a significant class of such systems of the type just described. The approach proposed here is related to the differential algebra tools for analysis of nonlinear systems developed by Michel Fliess [9][26] which has led to a clearer understanding of the nonlinear identification problem. Ollivier [22] as well as Ljung and Glad [19] have developed the use of the characteristic set of an ideal as a tool for identification problems. In particular, Ollivier developed the notion of the exhaustive summary of the model [22]. The use of these differential algebraic methods for system identification have also been considered in [20][25]. The focus of that research has been the determination of a priori identifiability of a given system model. However, as stated in [25], the development of an efficient algorithm using these differential algebraic techniques is still unknown. Though related, the approach proposed here is different from that in [20][25]. The approach presented in [25] typically leads to an overparameterized system which is well known to be ill conditioned. In [20] the emphasis is on a priori global identifiability conditions. Here it is shown, using the nonlinear techniques of elimination theory, that a significant class of nonlinear identification problems can be formulated as a nonlinear least-squares problem whose minimum value can be guaranteed to be found in a finite number of steps. The proposed methodology starts with an input-output linearly overparameterized model whose parameters are rationally related. (This is not atypical in many engineering examples as well be shown below.) After making appropriate substitutions, the problem is transformed into a nonlinear least-squares problem that is not overparameterized. It is then shown how the nonlinear least-squares problem can be solved using elimination theory. The computational issues are the symbolic computation of the Sylvester matrices to compute resultant polynomials based on elimination theory and the numerical computation of the roots of polynomials of high degree in a single variable. These issues are addressed below. Another important distinction of the work proposed here is that our concern is in actually computing the parameter values needed to characterize the input-output response rather than determining whether or not a system is a priori identifiable (cf. [20]). Even if a nonlinear state space model is not a priori identifiable in the sense of [20][25] (i.e., all of the parameters of the state space model being determined from input-output data), the model can be still be entirely adequate for the application at hand. For example, the induction motor is not a priori identifiable, yet the parameters that can be identified are complete enough to characterize the system for control purposes [27][36]. The technique described in the work below has potential to be an efficient and practical method for use in a variety of real world systems, especially in biological systems (See [2][3][4][15]). Unlike man-made systems such as electric

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machines, airplanes, space vehicles, etc., where the use of sensors is often a cost issue, in biological systems the technology is often not available to fully sense (measure) all the variables in a living organism. Consequently, one can only characterize such systems using input-output data.

1.2 Examples of Nonlinear System Models In this section, a set of mathematical models is given that characterize various biological systems. This is done to show that the class of problems to be considered here is a significant class. Example 1. The model for a glucose uptake system is (from [18]) dx1 x1 /θ2 − x2 /θ3 u − x1 + θ4 = −θ1 dt 1 + x1 /θ2 + x2 /θ3 1 + u/θ5 + x1 /θ5 + x1 u/θ5 dx2 x1 /θ2 − x2 /θ3 x2 /θ7 − θ8 − θ6 = θ1 dt 1 + x1 /θ2 + x2 /θ3 1 + x2 /θ2 + θ8 y = x2 The state space model is nonlinear in the unknown parameters θi . Example 2. A two compartment model describing the kinetics of a drug in the human body is given by (from [25]) dx1 k21 + VM x1 + k12 x2 + b1 u =− dt Km + x1 dx2 = k21 x1 − (k02 + k12 )x2 dt y = c1 x1

(1.1)

If Km were known a priori, then this model could be written using a linear regressor formulation if both state variables were measurable. On the other hand, as shown in [25], this can be rewritten as the input-output model d2 y 2 dy y + k21 k02 y 3 − c1 b1 (k21 + k02 ) y 2 u + (k21 + k12 + k02 ) y 2 2 dt dt 2 du 3 du 2 − c1 b1 y − Km c1 b1 + (2k21 Km k02 + k12 VM + k02 VM ) c1 y dt dt dy d2 y − 2 (k12 + k02 ) c21 b1 Km yu + 2Km c1 (k21 + k12 + k02 ) y + 2Km c1 y 2 dt dt du 2 − 2Km c21 b1 y b1 c31 (k12 + k02 ) u + Km c21 (k21 k02 + k12 VM + k02 VM ) y − Km dt 2 dy 2 2d y 2 2 c1 2 + Km c1 (k21 + k12 + k02 + VM ) + Km dt dt

0=

where it is now nonlinear in the parameters even if Km is known.

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Example 3. A model describing tumor targeting by antibodies is given by (from [30] where α, β, δ are known) dx1 dt dx2 dt dx3 dt dx4 dt dx5 dt y

= −k4 x2 − (k3 + k7 ) x1 + u = k3 x1 − k4 x2 + k5 x2 (α − x3 ) + k6 x3 − k5 x2 β (δ − x4 ) + k6 x4 = k5 x2 (α − x3 ) − k6 x4 = k5 x2 β (δ − x4 ) − k6 x4 = k7 x1 = c1 x1

This state space model is linear in the unknown parameters so that if all the state variables were measurable, linear least-squares techniques could be used to estimate the parameters. However, this is typically not the case and, as in the previous example, an input-output model results in a system that is nonlinear in the unknown parameters. Example 4. A model to describe microbial growth in a batch reactor is (from [10]) dx1 p1 x2 x1 − p3 x1 = dt p2 + x1 dx2 p1 x2 x1 = −p4 dt p2 + x1 x1 (0) = a x2 (0) = b Though these systems are all rational functions of the state variables and the unknown parameter values, this is not an a priori condition to be able to apply the methodology proposed below. The core requirements are that the coefficients of the input-output model are rational functions of the unknown parameters with no requirement that the model itself being a rational function of the inputs and outputs. The methodology provides a way to compute the identifiable parameters, that is, those that characterize the input-output model.

1.3 Nonlinear Least Squares Parameter Identification To introduce the methodology, an explicit example is considered, namely the induction motor. Though biological systems represent a rich new area for the methodology, the electric machine model of an induction motor will serve to

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illustrate the power of the proposed approach. Experimental results are also presented. Standard models of induction machines are available in the literature. For the development here, a two-phase equivalent state space model in the rotor coordinate system is the most convenient as given by [27] diSx dt diSy dt dψRx dt dψRy dt dω dt

= = = = =

β 1 uSx − γiSx + ψRx + np βωψRy + np ωiSy σLS TR β 1 uSy − γiSy + ψRy − np βωψRx − np ωiSx σLS TR 1 M iSx − ψRx TR TR 1 M iSy − ψRy TR TR τL M np (iSy ψRx − iSx ψRy ) − JLR J

(1.2) (1.3) (1.4) (1.5) (1.6)

where ω = dθ/dt with θ the position of the rotor, np is the number of pole pairs, and iSx , iSy are the (two-phase equivalent) stator currents in the rotor coordinate system and ψRx , ψRy are the (two-phase equivalent) rotor flux linkages also in the rotor coordinate system. The parameters of the model are the five electrical parameters, RS and RR (the stator and rotor resistances), M (the mutual inductance), LS and LR (the stator and rotor inductances), and the two mechanical parameters, J (the inertia of the rotor) and τL (the load 2 torque). The symbols TR = LR /R ¡ R , β2=¢ M/ (σLS LR ), σ = 1 − M / (LS LR ) 2 and γ = RS / (σLS ) + M RR / σLS LR have been used to simplify the expressions. TR is referred to as the rotor time constant while σ is called the total leakage factor. To simplify the presentation, the (important) special case in which only the two parameters RS , TR need be identified is considered. These two parameters will vary due to Ohmic heating and a method for online tracking of the values of TR and RS as they change due to Ohmic heating is presented. The electrical parameters M, LS , σ are now assumed to be known and not varying. Measurements of the stator currents iSa , iSb and voltages uSa , uSb as well as the position θ of the rotor are assumed to be available with velocity reconstructed from the position measurements. However, the rotor flux linkages ψRx , ψRy are not assumed to be measured. 1.3.1 Linear Overparameterized Model Standard linear least-squares methods for parameter estimation are based on equalities where known signals depend linearly on unknown parameters. The induction motor model does not fit in this category unless the rotor flux linkages are measured. However, as the rotor flux linkages are not usually measured, the first step is to develop an input-output model in which the

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fluxes ψRx , ψRy and their derivatives dψRx /dt, dψRy /dt are eliminated. The four equations (1.2), (1.3), (1.4), (1.5) can be used to solve for ψRx , ψRy , dψRx /dt, dψRy /dt, but one is left without another independent equation to set up a regressor system for the identification algorithm. A new set of independent equations are found by differentiating equations (1.2) and (1.3) to obtain diSx dψRy diSy 1 duSx d2 iSx β dψRx dω +γ = − − np βω − np βψRy − np ω σLs dt dt2 dt TR dt dt dt dt dω − np iSy (1.7) dt and diSy dψRx diSx d2 iSy β dψRy dω 1 duSy +γ = − + np βω + np βψRx + np ω σLs dt dt2 dt TR dt dt dt dt dω + np iSx . (1.8) dt Next, equations (1.2), (1.3), (1.4), (1.5) are solved for ψRx , ψRy , dψRx /dt, dψRy /dt and substituted into equations (1.7) and (1.8) to obtain ¶ µ ¶ µ diSx 1 duSx γ βM 1 d2 iSx diSy np ω + − γ+ − iSx − 2 + 0=− 2 + dt dt σLS dt TR dt TR TR 1 βM uSx 1 dω dω + iSy np ω( + )+ + np iSy − np × TR TR σLS TR dt dt σLS (1 + n2p ω 2 TR2 ) µ diSx diSy − γiSy σLS TR − iSx np ωσLS TR − np ωσLS TR2 −σLS TR dt dt ´ −γiSx np ωσLS TR2 + iSy n2p ω 2 σLS TR2 + np ωTR2 uSx + TR uSy (1.9) µ ¶ diSx 1 duSy γ βM d2 iSy diSy − ω + + − n − ( − i p Sy dt2 dt σLS dt dt TR2 TR 1 βM uSy 1 dω dω − iSx np ω( + )+ − np iSx + np × TR TR σLS TR dt dt σLS (1 + n2p ω 2 TR2 ) µ diSx diSy −σLS TR − γiSx σLS TR + iSy np ωσLS TR + np ωσLS TR2 dt dt ´ (1.10) +γiSy np ωσLS TR2 + iSx n2p ω 2 σLS TR2 − np ωTR2 uSy + TR uSx .

0=−

This set of equations may be rewritten in regressor form as y(t) = W (t)K where W (t) ∈ 0 and the solution pair that minimized E 2 (K1 , K2 ) was then chosen resulting in K1 = 241.1 and K2 = 7.6. The estimated machine parameters using (1.15) are 20

i

i sa

sa- sim

15

Current in Amperes

10

5

0

-5

-10

-15

-20 5.56

5.58

5.6

5.62

5.64 5.66 5.68 Time in seconds

5.7

5.72

5.74

Fig. 1.2. Phase a current iSa and its simulated response iSa_ sim .

TR = 0.132 s and RS = 5.1 Ω.

(1.24)

For comparison, the stator resistance was measured using an ohmmeter giving a value of 4.9 ohms. The sampled two phase equivalent current iSa and its simulated response iSa_sim are shown in Figure 1.2. The current iSa_sim is from a simulation using the measured input voltages from the experiment and the identified parameters from (1.24) - the other parameters are known. Note that both TR and RS are in the stator current equations (1.2) and (1.3). Further, the Hessian matrix was calculated at the minimum point according to (1.23) and was positive definite with a condition number of 295. It turns out that for this system to be sufficiently excited (i.e., the Hessian is positive definite so that TR and RS are identifiable), it is enough that the motor operates at constant speed under load (non zero rotor currents).

1.4 Computational Issues As seen from the development of the proposed estimation method, the main computational issues are the symbolic computation of the Sylvester matrices

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200

ω

Speed in radians/sec

150

ω

sim

100

50

0 5.4

5.5

5.6

5.7

5.8 5.9 6 Time in seconds

6.1

6.2

6.3

6.4

Fig. 1.3. Calculated speed ω and simulated speed ωsim .

to compute resultant polynomials based on elimination theory and the numerical computation of the roots of polynomials of high degree in a single variable. To increase the capability of the method proposed, these issues must be addressed. The results in [11][12] for the symbolic computation of the determinant of a matrix shows the potential for speeding up this computation by orders of magnitude over existing methods. The idea of the algorithm in [11][12] is based on polynomial methods in control and the discrete Fourier transform. To summarize the approach, rewrite the resultant polynomial (1.22) in the form N X r(K1 ) = pi K1i (1.25) i=0

where the unknowns pi and N are to be found. Any upper bound of the actual degree of r(K1 ) can be used for N . Such an upper bound is easily computed by finding the minimum of the sum of either the row or the column degrees 2πk of the Sylvester matrix [14]. Let K1k = e−j N+1 , n = 0, 1, ..., N be N + 1 different values of K1 . Then the Discrete Fourier Transform (DFT) of the set of numbers {p0 , p1 , ..., pN } is yk =

N X i=0

with inverse DFT given by

2πk

pi e−j N +1 i =

N X i=0

³ ´k 2π pi e−j N+1

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N

pi ,

2πi 1 X yk ej N+1 k . N +1

k=0

2πk

Here yk is just (1.25) evaluated at K1k = e−j N +1 . That is, one computes the numerical determinant of Sylvester matrix of the two polynomials (see (1.21) and (1.22)) at the N + 1 points K1k (this is fast) and obtains the DFT of the coefficients of (1.25). Then the pi are computed using the inverse DFT. That is, the symbolic calculation of the determinant is reduced to a finite number of fast numerical calculations. Such an approach has been shown to be as much as 500 times faster than existing methods [11]. The computation of the roots of a polynomial in one variable has been the object of research by numerical analysts for many years and is well documented in the literature (e.g., see [16]). In the research [36] on identifying the four (identifiable) parameters of the induction motor, it was necessary to use rational arithmetic to obtain accurate roots of a polynomial of degree 104. Care must be taken to ensure that the computed roots are indeed roots to the polynomial. Typically in an identification problem, one is looking for real roots which are restricted to a finite range simplfying the numerical search for the roots. (In particular, Sturm’s theorem (see [29], p. 5) can be used to find the precise number of real roots in a given interval).

1.5 Algebraic State Estimators In this section, the idea of using purely algebraic methods to estimate the state variables of a system are discussed. The method involves computing derivatives of the output variables and solving for the unknown state variable. However, as the equations are nonlinear, the goal is not a dynamic observer whose stability must be ascertained, but rather an algebraic observer with no stability issue. The issue of numerically differentiating the output signals is of course an important practical concern, and methods to address this problem are considered in [7][8]. As in the previous section, the induction motor is used as an example. If the speed of an induction motor is measured (along with stator currents), then the field-oriented control methodology provides the capability to precisely control the torque of the machine. The speed (along with the rotor time constant and the stator currents) is required for constructing a stable observer to estimate the rotor flux linkages in an induction motor. However, the speed/position sensor adds significant cost to the system and is one of the least reliable parts of the system. As a consequence, it is of interest to develop a speed sensorless field-oriented controller. Many different techniques have been proposed to estimate speed of an induction motor without a shaft sensor, but none has emerged as being completely satisfactory. Much literature exists in this area and the reader is re-

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ferred to [1][17][23][28][31][34][33][32][35] for an exposition of many of the existing approaches. Currently, induction motors are used throughout industry in pumps, fans, manufacturing machinery, conveyor belt drives, etc., where they are run open-loop. However, in many of these situations a speed sensorless field-oriented controller would be a distinct advantage. That is, it is desirable to have a field-oriented controller for its performance without the cost and reliability issues of having a speed/position sensor. For example, a conveyor belt in a mine that brings out coal in buckets is typically powered by induction motors running in open-loop from a 60 Hz voltage source. When power is lost, the coal buckets must all be emptied to reduce the load so that the motors can bring the conveyor belt system back up to speed. After the system is up to speed, the buckets may be refilled so that the motor is again fully loaded. The availability of a sensorless field-oriented controller capable of performing such a start up under full load would greatly reduce the downtime. Such reliable speed sensorless control algorithms would find application in a huge number of different industrial applications. Space Vector Model of the Induction Motor In the space vector model of the induction motor [17], one lets iS = iSa +jiSb , ψ R = ψRa + jψRb , and uS = uSa + juSb and the induction motor model may be written as β 1 d (1 − jnP ωTR ) ψ R − γiS + u i = dt S TR σLS S d 1 M (1 − jnP ωTR ) ψ R + i ψR = − dt TR TR S dω τL np M (iSb ψRa − iSa ψRb ) − . = dt JLR J

(1.26) (1.27) (1.28)

To replace the speed sensor, the estimation of rotor speed is based on measured stator voltages and currents at the motor terminals. The first step is to eliminate the flux linkage ψ R and its derivative dψ R /dt as they are not available measurements. The two equations (1.26) and (1.27) can be used to solve for ψ R , dψ R /dt, but one is left without another independent equations to set up a speed estimator. A new independent equation is found by differentiating the equation (1.26) which gives d2 dω β d 1 d d iS = (1 − jnP ωTR ) ψ R − jnP βψ R − γ iS + u . (1.29) 2 dt TR dt dt dt σLS dt S d Using these three (complex valued) equations, one can eliminate ψ R and ψ R dt to obtain

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µ ¶ 1 d 1 βM d2 i = − (1 − jn ωT ) + γi − u + 2 (1 − jnP ωTR ) iS i P R S dt2 S TR dt S σLS S TR µ ¶ d dω d 1 d jnP TR 1 − γ iS + uS uS − iS + γiS − . dt σLS dt 1 − jnP ωTR dt σLS dt The equation for dω/dt can be written as dω 1 − jnP ωTR (1 − jnP ωTR )2 + × =− 2 dt jnP TR jnP TR d 1 d d2 βM (1 − jnP ωTR ) iS − γ iS + uS − 2 iS 2 TR dt σLS dt dt . 1 d uS . iS + γiS − dt σLS

(1.30)

(These calculations are equivalent to those carried out for the parameter identification work in the previous section.) Equation (1.30) actually leads to two possibilities for a speed observer. If the signals are measured exactly and the motor satisfies its dynamic model, the right-hand side must be real. Breaking down the right side of (1.30) into its real and imaginary parts, the real part has the form dω = a2 (uSa , uSb , iSa , iSb ) ω 2 +a1 (uSa , uSb , iSa , iSb ) ω+a0 (uSa , uSb , iSa , iSb ) . dt (1.31) The expressions for a2 , a1 , a0 are lengthy and therefore not explicitly presented here. (The notational dependence is a little misleading as they depend on the derivatives of the currents and voltages as well.). Note that equation¯ (1.31) is singular, i.e., the denominator of (1.31) is zero, if and ¯ ¯ ¯ only if ¯ψ R ¯ ≡ 0. The equation (1.31) could be used as a dynamic speed observer. Specifically, one computes coefficients a2, a1 , a0 at each time step and then integrates the nonlinear time-varying differential equation (1.31) in real-time for an estimate ω ˆ . However, its stability must be ascertained, that is, if ω ˆ (t0 ) 6= ω (t0 ) , will ω ˆ (t) −→ ω (t)? For example, if uSa = constant and uSb = 0 (during initial flux build up in the machine), it turns out that a2 (uSa , uSb , iSa , iSb ) = a1 (uSa , uSb , iSa , iSb ) = a0 (uSa , uSb , iSa , iSb ) ≡ 0 and therefore dω/dt ≡ 0 which is an unstable observer under these conditions4 . On the other hand, the imaginary part of (1.30) must be zero leading to an algebraic equation satisfied by ω of the form q(t, ω) , q2 (uSa , uSb , iSa , iSb )ω 2 + q1 (uSa , uSb , iSa , iSb )ω + q0 (uSa , uSb , iSa , iSb ) ≡ 0. (1.32) If q2 6= 0, equation (1.32) can be solved to obtain 4

However, under normal operating conditions of an induction motor, the ai are not identically zero.

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p q12 − 4q2 q0 ω1 , 2q2 p −q1 − q12 − 4q2 q0 ω2 , . 2q2 −q1 +

(1.33) (1.34)

At least one of these two solutions must track the motor speed. This speed estimator does not have any stability issue, but a procedure to determine whether ω1 or ω2 is the actual motor speed is required. Further, there are circumstances for which (1.32) is singular, that is, q2 (uSa , uSb , iSa , iSb ) = q1 (uSa , uSb , iSa , iSb ) = q0 (uSa , uSb , iSa , iSb ) = 0. For example, during flux build up where uSa = constant and uSb = 0, the qi are all identically zero and the value of ω is not determined by (1.32). In order to determine the correct solution ω1 or ω2 , one can proceed as follows. Noting that q(t, ω) ≡ 0, its derivative is also identically zero, that is, (2q2 ω + q1 )

dω + q˙2 ω 2 + q˙1 ω + q˙0 ≡ 0. dt

(1.35)

Next, substitute the right-hand side of equation (1.31) for dω/dt to obtain r(t, ω) , 2q2 a2 ω 3 +(2q2 a1 + q1 a2 + q˙2 ) ω 2 +(2q2 a0 + q1 a1 + q˙1 ) ω+q1 a0 +q˙0 ≡ 0. (1.36) r(t, ω) in equation (1.36) is a third degree polynomial in ω. Dividing it by q (t, ω) which is a second degree polynomial in ω, the division algorithm gives r(t, ω) = (2a2 ω + 2a1 − q1 a2 /q2 + q˙2 /q2 ) q (t, ω) ¢ ¡ + 2q2 a0 − q1 a1 + q˙1 − 2q0 a2 + q12 a2 /q2 − q1 q˙2 /q2 ω + (q1 a0 + q˙0 − 2q0 a1 + q0 q1 a2 /q2 − q0 q˙2 /q2 ) ≡ 0. As q (t, ω) ≡ 0, it follows that ¡ ¢ 2q2 a0 − q1 a1 + q˙1 − 2q0 a2 + q12 a2 /q2 − q1 q˙2 /q2 ω + (q1 a0 + q˙0 − 2q0 a1 + q0 q1 a2 /q2 − q0 q˙2 /q2 ) ≡ 0

(1.37)

(1.38)

Equation (1.38) is now a first order polynomial in ω with a unique solution provided the coefficient of ω in (1.38) is nonzero. This last equation involves 3rd derivatives of the stator currents and 2nd derivatives of the stator voltages and therefore will be noisier due to the differentiations compared to the solutions of (1.32) whose coefficients have 2nd derivatives of the stator currents and only one derivative of the stator voltages. To avoid this additional noise, one could just use the solution of (1.38) to determine which of ω1 , ω2 is the correct motor speed. Specifically, choose either ω1 or ω2 as the speed estimate ω ˆ depending on which is closest to the solution of (1.38). As the simulations below show, one of the two solutions ω1 , ω2 is typically far from the actual motor speed.

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Simulation of the Algebraic Estimator As a first look at the viability of an algebraic observer, simulations are carried out. Here, a three-phase (two phase equivalent) induction motor model was simulated with machine parameter values given by np = 2, RS = 5.12 ohms, LS = 0.2919 H, J = 0.0021 kgm2 , τL_rated = 2.0337 Nm, Imax = 2.77 A, Vmax = 230 V. Figure 1.4 shows the block diagram of the system. In this

Induction Motor Load

Trajectory Generation

ωref

iSa iSb

CurrentConmmand FieldOriented Controller

ω

iSa iSb uSa uSb

uSa uSb

ωˆ

Algebraic Speed Esitimator

Fig. 1.4. Algebraic speed estimator.

system, a current command field-oriented controller is used [17]. The induction motor model is based on equations (1.26), (1.27) and (1.28). Figure 1.5 shows the simulation results of the motor speed and the two estimated speeds ω1 , ω2 for a low speed trajectory with full rated load on the motor. From

6

ω and ω

1

speed rad/s

4 2 0 ω

-2

2

-4 -6

0

1

2

time seconds

3

Fig. 1.5. ω, ω1 and ω2 for a low speed trajectory (ωmax = 5) with full rated load.

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ω and ω

speed rad/s

2

2

ω and ω

1

0 ω and ω

-2

1

-4 -6

ω and ω

ω

2

1

-8 0.35

0.4

0.45

time seconds

0.5

0.55

0.6

Fig. 1.6. An enlarged view of Figure 1.5.

t = 0 to t = 0.4 seconds, a constant uSa is applied to the motor to build up the flux with ω ≡ 0. At t = 0.4 seconds, the machine starts on the low speed trajectory (ωmax = 5 rad/s) with full load at the start. Figure 1.6 is an enlarged view of Figure 1.5 between 0.3 seconds and 0.6 seconds. Note that the correct solution for the speed does alternate between ω1 and ω2 .

1.6 Summary In this work, the problem of parameter identification and state estimation for nonlinear systems has been considered. It was shown that for a class of systems whose input-output models are nonlinear in the parameters, but rationally related, the parameters could be identified by solving a nonlinear least-squares problem using elimination theory. The problem of state estimation was considered from the point of view of finding an algebraic relationship between the unknown state variable and the known outputs. Such an approach entails computing higher order derivatives of the measured outputs and eliminating the derivatives of the state variable to be estimated.

References 1. Bodson M, Chiasson J (2002) A comparison of sensorless speed estimation methods for induction motor control. In: Proceedings of the 2002 American Control Conference Anchorage AK 2. Chauvet G. A (1995a) Theoretical Systems in Biology: Hierarchical and Functional Integration: Volume I - Molecules and Cells. Pergamon

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Index

Algebraic speed estimator, 17 Algebraic state estimators, 13

Nonlinear least-squares, 8

Biological system models, 3

Parameter identification, 4 Computational issues, 11 Experimental results, 10

Elimination theory, 9

Resultant, 9

Induction motor model, 4, 14

Sylvester matrix, 9