IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 56, NO. 3, JUNE 2007
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Indirect Measurements via a Polynomial Chaos Observer Anton H. C. Smith, Antonello Monti, Senior Member, IEEE, and Ferdinanda Ponci
Abstract—This paper proposes an innovative approach to designing algorithms for indirect measurements based on a polynomial chaos observer (PCO). A PCO allows the introduction and management of uncertainty in a process. The structure of this algorithm is based on the standard closed-loop structure of the observer that is originally introduced by Luenberger. This structure is extended here to formally include uncertainty in the measurement and in the model parameters. Possible applications of this structure are discussed. Index Terms—Kalman filtering, observability, observers, statespace method, stochastic system, uncertainty.
I. I NTRODUCTION
M
EASUREMENTS are a critical part of any control system. Many modern control structures are based on the complete measurement of the state. However, it is usually not feasible to include a sensor for each state variable. One of the main reasons can be cost reduction. As result, it is a common practice to substitute a sensor with an indirect measurement based on an observer. In other situations, there can be a technical issue that is related with the measurement process itself. In the control of an induction motor, e.g., it can be really important to measure the magnetic flux in the rotor windings, but it is a quite complicated procedure to insert the proper sensor in an offthe-shelf machine. Again, an indirect measurement based on an observer is the most widely applied solution [1]. Observers can play other important measurement roles: In the presence of noisy sensors, a state-estimation process can be used to improve the quality of the measurement. An example of this application is given by the estimation of the speed starting from an encoder-based position measurement [2]. For all these reasons, using the concept of state observability, it is a common practice to estimate states starting from a limited set of measurements. Such a process introduces uncertainty in the measurement process due to uncertainties in the parameter values and in the model structure itself, which affects the state measurement estimation.
Manuscript received June 15, 2006; revised February 13, 2007. This work was supported by the U.S. Office of Naval Research under Grant N00014-021-0623. The authors are with the Department of Electrical Engineering, University of South Carolina, Columbia, SC 29208 USA (e-mail:
[email protected];
[email protected];
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIM.2007.894914
Last but not least, observers are also widely used for monitoring and diagnostic purposes. A real-time model of the plant is integrated in parallel with the evolution of the system. The evolution of the real quantities is compared with the predicted behavior. Deviations from the prediction can be interpreted as possible faulty operations [3], [4]. Even in this case, the procedure is highly sensitive to parameter uncertainty in the model definition. The most widely used solution to parameter uncertainty in observer-based indirect measurement is the use of an extended Kalman filter (EKF) [5], [6]. In this case, uncertain parameters are treated as additional state variables. The time derivative of the uncertain parameters is usually defined as white noise. This paper proposes a new observer structure that is able to reconstruct the state in the presence of uncertainty by utilizing a stochastic model based on the polynomial chaos theory (PCT). This approach improves classical solutions such as the aforementioned EKF, which represents parameter uncertainty as a noise problem. The concept was originally introduced in [16], presenting some introductory results, while this paper presents complete details on the mathematical derivation of the observer and a comprehensive comparison with the EKF approach. The main advantages that are given by this new approach can be summarized in three points. 1) The PCT model completely exploits the available information on the possible statistical variations of the parameters. Theoretically, there are no limitations on the shape of the probability density function (pdf) of the parameter. Given the shape of the pdf, an optimal polynomial base can always be selected. 2) It supports more complex statistical processes than the classical Kalman filters, where only the mean and variance of the quantities are considered. By using higher polynomial order, more complex pdfs can be reconstructed. 3) It allows a complete reconstruction in real time of the pdf of each variable of the system. In this respect, the PCT is the only approach that allows estimation of the statistic of a random process in real time. This is possible due to the transformation of a set of stochastic differential equations in a set of deterministic differential equations with an extended set of state variables. This property makes the PCT an excellent solution for every online process based on indirect measurements that are affected by parameter uncertainties.
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In the following, the authors introduce PCT and then focus on the mathematical formalization of the polynomial chaos observer (PCO). In addition, the methodologies for the design of the observer are presented, as well as a complete application example. II. P OLYNOMIAL C HAOS T HEORY Polynomial chaos is a technique that uses a polynomialbased stochastic space to represent and propagate uncertainty in the form of pdfs. This concept was first introduced by Wiener in 1938 as “homogeneous chaos” [7]. The theory evolved into the Wiener–Askey polynomial chaos, which extended the theory to the entire Askey scheme of orthogonal polynomials [8]. Polynomial chaos can be described through the relationship and description of a generic second-order random process as follows: X(θ) = a0 Ψ0 +
∞
ai1 Ψ1 (ξi1 (θ))
i1 =1
+
i1 ∞
ai1 i2 Ψ2 (ξi1 (θ), ξi2 (θ))
i1 =1 i2 =1
+
i1 i2 ∞
ai1 i2 i3 Ψ3 (ξi1 (θ), ξi2 (θ), ξi3 (θ)) + · · ·
nv . The maximum order for the polynomial base np also has to be defined. Given the two values nv and np , the total number of terms that are needed for the description of each variable in the system is defined by (nv + np )! P = − 1. (2) nv !np ! The spectral representation of an uncertain parameter on this limited space is given by X(θ) =
P
ai Ψi (ξ(θ)) .
(3)
i=0
Notes on the selection of parameter P and on the convergence of the PCT approach are reported in detail in [8]. Variables that are represented through a polynomial chaos expansion can then be substituted in a generic dynamic model. Due to a Galerkin projection, the dependence on random variable ξ can be removed, and a set of deterministic state equations can be obtained. An example of an application will be described in the following paragraph. Polynomial chaos has been applied to numerous fields of study, including fluid dynamics and circuit simulation [9]–[12]. A previous application involving measurement is reported in [13].
i1 =1 i2 =1 i3=1
(1) where X(θ) ai Ψi ξi1
random process or function under analysis in terms of θ, which represents the random event; coefficients of the expansion; polynomials of the selected base; random variables with a suitable pdf that is defined according to the polynomial base.
The second-order random process from (1) represents the infinite space of a complete multivariable orthogonal polynomial space. If X varies with time, then generic coefficient ak will vary with time. The shape of the represented pdf is created in this stochastic space. Because the space is complete, any pdf can be represented using any polynomial base. Certain polynomial bases can be chosen to represent a given pdf with the fewest number of terms. Considering the properties of the polynomial base and the definition of variable ξ, a variable with perfect Gaussian distribution can be described using only two terms of the Hermite polynomial base, while more terms are required to represent a variable with Gaussian distribution with the Legendre polynomials. The Legendre polynomials, however, can represent variables with uniform distribution by using only two terms. For practical applications, the stochastic space described by (1) must be limited to a finite number of dimensions. The selection of this number is based on the number of independent sources of uncertainty (i.e., the number of independent variables ξ that are used to describe the process) in the system, i.e.,
III. G ENERIC PCT M ODEL Consider a linear time-invariant dynamic system in statespace form, i.e., ˙ x(t) = Ax(t) + Bu(t) y(t) = Cx(t)
(4)
where x(t) ∈ n u(t) ∈ r y(t) ∈ m A, B, C
system state vector; input vector; output vector; system matrices with appropriate dimensions.
Given that the parameters in A and/or B are uncertain and that the uncertain parameters can be described by a known pdf, the system can be expanded using PCT. The PCT-expanded system constitutes a new set of state equations that describe how the parameter uncertainty propagates to the output. The expansion of the system through PCT employs orthogonal polynomials of the Askey scheme to represent uncertain variables in the governing equation. This expansion technique associates individual random events with independent random variables ξi . In the PCT-expanded model, the pdf of the nv uncertain variables is assumed to be known. These variables or parameters can be described, as in (3), as stochastic uncertain parameters with a PCT base of np order. Starting from (4), the intermediate step will be Σk x˙ k (t)ψ(ξ) = A(ξ)Σk xk (t)ψ(ξ) + B(ξ)Σk uk (t)ψ(ξ) Σk yk (t)ψ(ξ) = C(ξ)Σk xk (t)ψ(ξ).
(5)
SMITH et al.: INDIRECT MEASUREMENTS VIA A POLYNOMIAL CHAOS OBSERVER
A new set of deterministic state equations can then be obtained by projecting the stochastic uncertain equations onto the random space that is spanned by the orthogonal polynomial basis Ψi by taking the inner product of the equation of each basis. The inner product can be determined by taking the Galerkin projection, which is realized by integrating each component of the system with the polynomial basis Ψi Ψj Ψk =
Ψi Ψj Ψk w(ξ)dξ
(6)
Ω
where w(ξ) depends on the choice of basis in the Askey scheme, and Ω is the region for which the chosen basis is valid. The solution of this new set of state equations provides the coefficients of the polynomial chaos expansion of the state variables. The zeroth order of such expansion represents the most likely value. This expanded state-space PCT model can be described as
Gain K pct is a design parameter, and different strategies for setting this gain matrix can be introduced. Such strategies include pole placement and optimal/suboptimal Kalman filter design. In the pole-placement design, K pct is chosen such that the poles of the observer are at desired locations. Choosing poles on the left-hand plane ensures that the PCO is asymptotically stable and that the error will converge to zero [14]. K pct can also be found in terms of a gain that minimizes the mean square of the error as in the Kalman filter design. The Kalman filter design is an optimal strategy that would require K pct to be time varying. The time-varying gain in the optimal Kalman filter design reaches a steady-state value far from the final time. Thus, a steady-state or suboptimal K pct relaxes the requirement of K pct to be time varying. The optimal and suboptimal Kalman gain K pct can be found using the nonlinear matrix differential Riccati equation or the algebraic Riccati equation, respectively [14]. The nonlinear matrix differential Riccati equation is given as P˙ (t) = P (t)ATpct + Apct P (t)
x˙ pct (t) = Apct xpct (t) + B pct upct (t) y pct (t) = C pct xpct (t)
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(7)
+ B pct S w B Tpct + P (t)C Tpct S −1 v C pct P (t)
(10)
and the algebraic Riccati equation is given as
where xpct (t) ∈ npct upct (t) ∈ rpct y pct (t) ∈ mpct Apct , B pct , C pct
expanded system state vector; expanded input vector; expanded output vector; system matrices with appropriate dimensions.
If the original system was of order n and each variable is expanded up to the pth order of the PCT, the new state vector will have the dimensions npct = n × p. The other matrices will be expanded accordingly. The states of the PCT model are observable from the output of the original system if
C pct C pct Apct Wo = .. . n
pct C pct Apct
(8)
−1
has a rank equal to npct , i.e., equal to the number of state variables in the PCT domain. The value of npct is determined by the original number of state variables, the number of independent sources of uncertainty in the model, and the order of polynomial expansion that is chosen to represent the random process. If the system is observable, a closed-loop state observer can be designed. Assuming that the set of variables y in (4) defines the measurable variables, the observer can be defined as ˆ˙ pct (t) = Apct x ˆ pct (t) + B pct upct (t) x ˆ pct (t) + K pct y pct (t) − C pct x
(9)
ˆ pct (t) ∈ npct is the estimation of the system state where x vector.
P ATpct + Apct P + B pct S w B Tpct + P C Tpct S −1 v C pct P = 0 (11) where S w and S v are the spectral density on the plant and measurement noise, respectively, and P is the unique symmetrical positive semidefinite solution of the algebraic Riccati equation. K pct can then be found by K pct = P C Tpct S −1 v .
(12)
The formulation of the model that is detailed in (9) describes the uncertainty that is propagated by the model uncertainty itself. A more comprehensive uncertainty propagation process can be defined if we assume, as is reasonable, that the measurement itself is affected by uncertainty. This uncertainty will introduce a new dimension to the PCT space. Instead of an uncertain parameter, we now deal with an uncertain input. In this case, the order of complexity increases. IV. E XAMPLE OF A PCT-E XPANDED S YSTEM W ITH U NCERTAINTY ON A M EASURED P ARAMETER Consider a linear time-invariant system model, for example, the averaged model of a buck dc/dc converter (see Fig. 1). For the sake of simplicity, we assume that the resistances that are associated with the inductor and the capacitor are null, as in the case of an ideal lossless converter. The average model of the buck converter can be written as di(t) −v(t) Vcc = + d dt L L dv(t) i(t) Gv(t) = − dt C C
(13)
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TABLE I PARAMETER LIST USED IN THE SIMULATION
Fig. 1. Buck converter schematic.
Because the chosen basis is orthogonal, the Galerkin projection can be used, allowing for the PCT expansion of the system, i.e.,
where L C G i(t) v(t) Vcc d
value of the inductor; value of the capacitor; value of the load conductance; current through the inductor; voltage across the capacitor; supply voltage; applied duty cycle.
din (t) Ω = dt dvn (t) Ω = dt
Equation (13) can be written in state-space form [15] as follows:
di(t) dt dv(t) dt
=
0 1 C
− L1 G −C
i(t) + v(t)
Vcc L
0
d.
(14)
Assuming that a single parameter is uncertain and its pdf is given, (13) can be expanded using PCT with a convenient polynomial basis that is chosen based on that pdf. If the uncertain parameter in (14) is G, the PCT expansion of all the uncertain variables is given as G=
P
P
v(t) =
Ω
Φ2n (ξ)dξ
dvn (t) dt Φn (ξ)w(ξ)dξ
Φ2n (ξ)dξ
.
(17)
Any orthogonal polynomial of the Askey scheme can be used for the expansion with appropriate weight w(ξ) and domain of integration Ω [8]. From (16) and (17), the PCT-expanded model of a buck converter that is affected by an uncertain conductance with Gaussian distribution, under the hypothesis of truncating the expansion at the first order, will result in the following equation: di0 (t) −v0 (t) Vcc = + d dt L L di1 (t) −v1 (t) = dt L
Ga Φa (ξ)
dv0 (t) i0 (t) G0 v0 (t) G0 v0 (t) = − − dt C C C
ib (t)Φb (ξ)
b=0 P
Ω
a=0
i(t) =
din (t) dt Φn (ξ)w(ξ)dξ
vc (t)Φc (ξ).
dv1 (t) i1 (t) G1 v0 (t) G0 v1 (t) = − − dt C C C
(15)
(18)
c=0
where Substituting (15) into (13)
di(t) = dt
−
P c=0
G0 G1
vc (t)Φc (ξ) L
+
Vcc d L
i0 i1
P
ib (t)Φb (ξ)(t) dv(t) b=0 = dt C P P ib (t)Ga Φa (ξ)Φb (ξ) a=0 b=0 − . C
v0 v1 (16)
expected value of the load conductance; first term of the polynomial expansion of the conductance and numerical match of the variance of the load conductance; expected value of the inductor current; first PCT term of expansion of the uncertainty on the inductor current and numerical match of the variance if the process is perfectly Gaussian; expected value of the capacitor voltage; first PCT term of expansion of the uncertainty on the capacitor voltage and numerical match of the variance if the process is perfectly Gaussian.
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Fig. 2. Simulation results using variation of parameter R within the predicted boundary (current and voltage sensor available). (a) Current and the upper and lower bounds estimated by the PCO compared with the measured current, (b) pdf of R used in the test, (c) estimated current calculated by the EKF compared with the measured current, and (d) pdf of the estimation of R using the EKF.
The state-space model can be written as
di0 (t) dt di1 (t) dt dv0 (t) dt dv1 (t) dt
0 0 = 1
0 0 0
0
1 C
C
− L1 0
−G0 C −G1 C
The state-space model can be written as
Vcc
0 i0 L − L1 ii 0 + d. −G1 v0 0 C −G0 v1 0 C (19)
Following the same approach, if we now suppose that the conductance has an uncertainty with uniform distribution, from (14), the PCT-expanded model is given as di0 (t) −v0 (t) Vcc = + d dt L L
0 0 = 1
0 0 0
0
1 C
C
Vcc 0 i0 L 1 − L ii 0 + d. −G1 v0 0 3C −G0 v1 0
− L1 0 −G0 C −G1 C
C
(21) Note that changing the assumption on the pdf of the uncertain parameters affects the coefficient of the state matrix, as expected. In particular, the cross terms between v0 and v1 change to one third. From (19) and (21)
Apct_Hermite
di1 (t) −v1 (t) = dt L
0 0 = 1
0 0 0
0
1 C
0 0 = 1
0 0 0
0
1 C
C
dv0 (t) i0 (t) G0 v0 (t) G1 v1 (t) = − − dt C C 3C dv1 (t) i1 (t) G1 v0 (t) G0 v1 (t) = − − . dt C C C
di0 (t) dt di1 (t) dt dv0 (t) dt dv1 (t) dt
Apct_Legendre (20)
C
− L1 0 −G0 C −G1 C − L1
0
−G0 C −G1 3C
0 1 −L −G1 C −G0 C
0 − L1 −G1 3C −G0 C
(22)
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Fig. 3. Simulation results using variation of parameter R outside the predicted boundary (current and voltage sensor available). (a) Current and the upper and lower bounds estimated by the PCO compared with the measured current, (b) pdf of R used in the test, (c) estimated current produced by the EKF compared with the measured current, and (d) pdf of the estimation of R using the EKF. TABLE II FIRST- AND SECOND-ORDER MOMENTS OF R WITHIN THE DESIRED RANGE AND WITH BOTH THE VOLTAGE AND CURRENT MEASUREMENTS (mean = 2.08 AND std = 0.43) USING DATA FROM FIG. 2(b) AND (d)
TABLE III FIRST- AND SECOND-ORDER MOMENTS OF R NOT WITHIN THE DESIRED RANGE, USING BOTH THE CURRENT AND VOLTAGE MEASUREMENTS IN THE E STIMATION (mean = 2 AND std = 0.63) U SING D ATA FROM FIG. 3(b) AND (d)
and
C pct =
1 0
0 0
0 1
0 . 0
(23)
Constructing the observability matrix W o for both the Hermite and Legendre, it can be shown that rank(W 0 ) = 4 ∀G1 .
(24)
Therefore, assuming that the outputs of the measured system are the most likely value of the measured variable, the uncertain
states of the PCT model are observable from the measured system for all values of G1 . The terms of the PCT expansion can be used to evaluate different characteristics of the uncertain variables. Boundary calculations can be easily performed by calculating the polynomial of the bases in suitable points. For example, if the uncertain parameter has uniform distribution and all the coefficients of the polynomial expansion of the output variable have the same sign, the two boundaries can be simply calculated using the
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Fig. 4. Simulation results using variation of parameter R within the predicted boundary (only current and sensor available). (a) Current and the upper and lower bounds estimated by the PCO compared with the measured current, (b) pdf of R used in the test, (c) estimated current calculated by the EKF compared with the measured current, and (d) pdf of the estimation of R using the EKF.
values of −1 and 1 for the random variable ξ that appears in the polynomial base. Therefore, for the upper boundary, C pct for the current is given as C pct = [1
1
1
1].
The lower boundary C pct for the current is given as C pct = [1
−1
−1
− 1].
Now, supposing that the measurement introduces a new element of uncertainty, it can be shown that the total order of the state-space formulation of the observer grows to 8, while matrix Cpct is modified as follows:
C pct
1 0 = 0 0
0 1 0 0
0 0 0 0
0 0 0 0
0 0 1 0
0 0 0 1
0 0 0 0
0 0 0 0
(25)
V. S IMULATION R ESULT The objective of this simulation is to test the hypothesis that, given a set of measured output, it is possible to determine if the
parameters in a model are within or outside a given uncertainty range (Table I). One method that is proposed by this paper is to use a PCO and the combination of the expected and uncertainty states to construct upper and lower boundaries. These boundaries describe the limits of the measured outputs given a priori the shape of the pdf of the uncertain parameters. To demonstrate the validity of the approach, an average model of a buck converter was used as a case study to obtain the uncertainty of the outputs given an uncertain load conductance G0 . Conductance G0 is considered to be Gaussian distributed; therefore, the PCO was designed using (19). The states of the observer were combined through the C pct matrix to obtain the expected upper and lower boundaries. Parameter R, which is 1/G, was randomly generated with a Gaussian distribution [Figs. 2(b) and 3(b)]. The measured outputs are then checked to verify if the measured states cross the upper or lower boundary [see Figs. 2(a) and 3(a)]. From Fig. 2(a), it can be seen that if R stays within the boundary of the defined uncertainty, then the current output stays in the boundaries that are calculated by the observer. Therefore, the expected value that is calculated from the observer is valid. From Fig. 3(a), it can also be seen that if R does not stay within the boundary of the defined uncertainty, then
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Fig. 5. Simulation results using variation of parameter R outside the predicted boundary (only current sensor available). (a) Current and the upper and lower bounds estimated by the PCO compared with the measured current, (b) pdf of R used in the test, (c) estimated current calculated by the EKF compared with the measured current, and (d) pdf of the estimation of R using the EKF.
the current output also does not stay in the boundaries that are calculated by the observer. To compare the effectiveness of the PCO approach, an EKF has also been used to study the same system. In the EKF method, the uncertain parameter is included in the model description as an additional state. After the EKF has converged [see Figs. 2(c) and 3(c)] and a sufficiently large number of estimates of parameter R is obtained [see Figs. 2(d) and 3(d)], the first and second moments of estimated R are calculated (see Tables II and III). From Table II, the first and second moments that are estimated by the EKF show that the value of R stay within the uncertainty boundary, while in Table III, when R is not within the uncertainty boundary, the EKF method correctly predicts that the variation exceeds the expected range. For both the PCO and the EKF, it is then possible to detect when the variations of a parameter are within or outside an expected range. Let us now consider a second case where only a current sensor is available. The observer now is supposed to estimate voltage as a result of the model integration still under uncertain conditions for parameter R. Using the PCO method even with just the current sensor, it is still possible to determine if R stays within a given uncertainty
TABLE IV FIRST- AND SECOND-ORDER MOMENTS OF R WITHIN THE DESIRED RANGE AND WITH ONLY THE CURRENT MEASUREMENTS (mean = 2.08 AND std = 0.43) U SING D ATA F ROM F IG . 4(b) AND (d)
range, as shown in Fig. 4(a), and if it does not, as shown in Fig. 5(a). By using the EKF method, a similar estimation can be performed, as shown in Fig. 4(d) or Table IV, as well as in Fig. 5(d) or Table V. VI. C OMPREHENSIVE R ESULT C OMPARISON From the simulation results that are discussed in the previous paragraph, some important elements of comparison between the PCO and the EKF can be defined. Both observers work very well when one parameter is uncertain in a linear system. The approach is anyhow different: In the case of the Kalman filter, the uncertain parameter becomes a state variable, making the filter itself a nonlinear system; in
SMITH et al.: INDIRECT MEASUREMENTS VIA A POLYNOMIAL CHAOS OBSERVER
TABLE V FIRST- AND SECOND-ORDER MOMENTS OF R NOT WITHIN THE DESIRED RANGE, USING ONLY THE CURRENT MEASUREMENTS IN THE ESTIMATION (mean = 2 AND std = 0.63) USING DATA FROM FIG. 5(b) AND (d)
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This paper demonstrates that a closed-loop estimator can be designed to estimate the uncertainty on output states. In order to perform the estimation, a model of the system must be first expanded using PCT. Structuring the PCT model of the system in state-space format allows an estimator to be designed through pole placement or through an optimal technique, which is a characteristic of the Kalman filter. PCO can be used to validate sensor readings since it can account for parametric uncertainty in the model. A complete comparison with the EKF completed the discussion. R EFERENCES
the case of the PCO, the observer is still linear, even when a parameter is uncertain. The prize to pay for the PCO is a higher order for the system. For this specific case, the EKF is a thirdorder nonlinear system, while the PCT is a fourth-order linear system. Furthermore, while the Kalman filter assumes a completely random variation for the parameter, the PCO requires an a priori assumption on the pdf of the uncertain parameter. On one hand, this means that the user needs more information to properly set up the PCO, but on the other hand, the PCO can process the information more efficiently. In effect, all in all, the PCO is computationally less intensive mostly because it does not require the inversion of a matrix for the evaluation of the observer gain at every step. This is, conversely, an important step for the EKF to keep the suboptimal performance of the filter itself. Performances are comparable as long as the statistics of the variables can be easily represented by two coefficients (Gaussian assumption). The PCO has the degree of freedom to consider higher moments by adopting an expansion based on more terms of the polynomial base. This is a quite important feature mostly for situations where we have more than one uncertainty in the system. Furthermore, the PCO directly provides information on the full pdf of the uncertain parameter through the coefficients of the expansion. This condensed information is very useful in online monitoring applications. The possibility of analytically predicting the probability for a parameter to be outside a predefined range is additional information that is made available by the PCO, which is not easily obtainable from the EKF. It should also be noted that the PCO is designed only to compensate for the so-called modeling noise (parameter uncertainty, unmodeled dynamic, etc.), while the Kalman filter is also useful in compensating for the noise coming from the sensors. For this reason, as also reported in a previous paragraph, a Kalman approach can be used to tune the PCO in order to reduce the effects of the noise coming from the sensors in the observer calculations. VII. C ONCLUSION This paper introduced an original structure for indirect measurements accounting for uncertainty. The theory of the approach has been tested with reference to a second-order model of a dc/dc power converter.
[1] G. C. Verghese and S. R. Sanders, “Observers for flux estimation in induction machines,” IEEE Trans. Ind. Electron., vol. 35, no. 1, pp. 85–94, Feb. 1988. [2] J. M. Corres and P. M. Gil, “High-performance feedforward control of IM using speed and disturbance torque observer with noise reduction of shaft encoder,” in Proc. IEEE ISIE, Jul. 8–11, 2002, vol. 2, pp. 403–408. [3] C. Kral, F. Pirker, and G. Pascoli, “Model based detection of rotor faults without rotor position sensor—The sensorless Vienna monitoring method,” in Proc. 4th IEEE SDEMPED, Aug. 24–26, 2003, pp. 253–258. [4] J. D. Kozlowski, C. S. Byington, A. K. Garga, M. J. Watson, and T. A. Hay, “Model-based predictive diagnostics for electrochemical energy sources,” in Proc. IEEE Aerosp. Conf., Mar. 10–17, 2001, vol. 6, pp. 3149–3164. [5] M. S. N. Said, M. E. H. Benbouzid, and A. Benchaib, “Detection of broken bars in induction motors using an extended Kalman filter for rotor resistance sensorless estimation,” IEEE Trans. Energy Convers., vol. 15, no. 1, pp. 66–70, Mar. 2000. [6] S. Bolognani, L. Tubiana, and M. Zigliotto, “Extended Kalman filter tuning in sensorless PMSM drives,” IEEE Trans. Ind. Appl., vol. 39, no. 6, pp. 1741–1747, Nov.–Dec. 2003. [7] N. Wiener, “The homogeneous chaos,” Amer. J. Math., vol. 60, no. 4, pp. 897–936, Oct. 1938. [8] D. Xiu and G. E. Karniadakis, “The Wiener–Askey polynomial chaos for stochastic differential equations,” SIAM J. Sci. Comput., vol. 24, no. 2, pp. 619–644, 2002. [9] D. Xiu and G. E. Karniadakis, “Modeling uncertainty in flow simulations via generalized polynomial chaos,” J. Comput. Phys., vol. 187, no. 1, pp. 137–167, May 2003. [10] D. Xiu, D. Lucor, C.-H. Su, and G. E. Karniadakis, “Stochastic modeling of flow-structure interactions using generalized polynomial chaos,” J. Fluid Eng., vol. 124, no. 1, pp. 51–59, Mar. 2002. [11] T. Lovett, A. Monti, and F. Ponci, “A polynomial chaos theory approach to the control design of a power converter,” in Proc. IEEE Power Electron. Spec. Conf., Jun. 20–25, 2004, pp. 4809–4813. [12] A. Monti, F. Ponci, and T. Lovett, “A polynomial chaos theory approach to uncertainty in electrical engineering,” in Proc. IEEE Intell. Syst. Application Power Syst. (ISAP), Washington, DC, Nov. 7–10, 2005, pp. 534–539. [13] T. Lovett, A. Monti, and F. Ponci, , “A polynomial chaos approach to measurement uncertainty,” in Proc. IEEE AMUEM, Niagara Falls, ON, Canada, May 2005, pp. 33–38. [14] J. F. Burl, Linear Optimal Control. Reading, MA: Addison-Wesley. [15] J. G. Kassakian, M. F. Schlecht, and G. C. Verghese, Principles of Power Electronics. Reading, MA: Addison-Wesley, 1991. [16] A. Smith, A. Monti, and F. Ponci, “Indirect measurements via polynomial chaos observer,” in Proc. AMEUM, Trento, Italy, Apr. 20–21, 2006, pp. 27–32.
Anton H. C. Smith received the B.S. degree in electrical engineering and the M.E. degree from the University of South Carolina (USC), Columbia, in 2002 and 2004, respectively. He is currently working toward the Ph.D. degree at the USC. He is currently a Research Assistant with the Virtual Test Bed Group, Department of Electrical Engineering, USC. His research interests include digital control and robotics.
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Antonello Monti (S’88–M’89–SM’02) received the M.S. degree in electrical engineering and the Ph.D. degree from Politecnico di Milano, Milan, Italy, in 1989 and 1994, respectively. From 1990 to 1994, he was with the Research Laboratory, Ansaldo Industria, Milan, where he was responsible for the design of digital control of a large power cycloconverter drive. From 1995 to 2000, he was with the Department of Electrical Engineering, Politecnico di Milano, as an Assistant Professor. Since August 2000, he has been an Associate Professor with the Department of Electrical Engineering, University of South Carolina, Columbia. He is the author and coauthor of more than 200 papers in the field of power electronics and electrical drives. Prof. Monti served as Chair of the IEEE Power Electronics Committee on Simulation, Modeling, and Control and as Associate Editor of the IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING.
Ferdinanda Ponci received the M.S. and Ph.D. degrees in electrical engineering from Politecnico di Milano, Milan, Italy, in 1998 and 2002, respectively. In 2003, she joined the Department of Electrical Engineering, University of South Carolina (USC), Columbia, as an Assistant Professor. She works with the Power and Energy Research Group, USC, on the research and development of the electric ship: a project that is funded by the U.S. Navy. Her current research interests include integrated environments for distributed measurement and advanced simulation for monitoring and diagnostics of electrical systems.
