Feedback-Invariant Approach to Time-Delay Estimation for

Article pubs.acs.org/IECR

Feedback-Invariant Approach to Time-Delay Estimation for Performance Monitoring Christopher A. Harrison*,† and S. Joe Qin*,‡ †

Marathon Petroleum Company LP, Louisiana Refining Division, P.O. Box AC, Garyville, Louisiana 70051-0849, United States Mork Family Department of Chemical Engineering and Materials Science and Ming Hsieh Department of Electrical Engineering, University of Southern California, Los Angeles, California 90089-1211, United States

‡

ABSTRACT: A simple time-delay estimation approach that requires temporarily retuning the existing PID controller is presented. Closed-loop output data are fitted with autoregressive moving average models at two different controller tunings. The time delay is determined by using the feedback-invariance principle that the closed-loop impulse response coefficients do not change inside the time-delay window. This approach requires no intervention to plant operation and does not estimate the openloop process model but the time delay only. The estimated time delay is particularly suitable for use in control performance monitoring.

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INTRODUCTION An estimate of time delay is necessary to estimate the minimum variance performance of the controller of a SISO process. If the time delay of the process is known, then the minimum output variance benchmark introduced by Harris can be computed from closed-loop process output data.1,2 A reliable technique which allows for estimation of the process time delay in the closed-loop with minimal disruption to routine process operation is needed to facilitate the use of this benchmark. Techniques for time-delay estimation from open-loop data of linear systems have been extensively studied. A comprehensive review of these techniques is given by Björklund.3 Ni et al. provide a summary of recent advances in time-delay estimation of univariate processes.4 These time-delay estimation approaches can be used in the closed-loop if sufficient process excitation is available. To obtain information about the excited process without disrupting routine operation, data gathered from setpoint changes can be used to identify the time delay.5 Another option for exciting the process is to add a reference signal to the output of the feedback controller.6 For the purpose of control performance monitoring, it is desirable to have a method which can be applied in the closed-loop, with minimal disruption to routine operation, and at a time of the practitioner's choosing. Time-delay estimation techniques are classified into four classes by Bjö rklund.3 The four classes are time-delay approximation methods, explicit time-delay parameter methods, area and moment methods, and higher-order statistical methods. With time-delay approximation methods, the time delay is extracted from a model of the process.3 The model does not contain the time delay as an explicit parameter. This category contains methods which determine time delay by finding the peak of the cross correlation between input and output. It also contains frequency domain approximation methods and methods which extract the time delay from a Laguerre model of the process. The next class consists of methods in which the time delay is an explicit parameter in the model. One approach in this class is © 2012 American Chemical Society

a one-step explicit method in which the model parameters and time-delay are estimated simultaneously.3 Such a method would include estimating several models such as an ARX model and choosing the best estimate among them. Also included in this class are two-step explicit methods which alternate between estimating the time delay and other parameters. A two-step explicit known as fixed model variable regressor estimation was developed by Elnaggar.7 Lynch and Dumont8 make use of this closed-loop time-delay estimation technique to assess the performance of SISO control loops. Sampling methods which make use of the sampling process to derive an expression for the time delay are also included in this class. The class of area and moment methods makes use of the relationship between the time delay and certain areas over or below the step response and certain moments of the impulse response.3 Higher-order statistics methods which take advantage of higher-order statistics to remove Gaussian noise comprise the fourth class.3 Ni et al. classify existing time-delay estimation method as either model-independent methods or model-dependent methods.4 Model-independent methods are data-based methods. The conventional method of generalized cross correlation would be considered a data-based method.4 Model-dependent methods require the use of the system model, by which the time delay is estimate is determined through minimizing a criterion. Many of the methods previously described estimate the time delay in the closed-loop by making use of setpoint changes to create an excitation signal for the process. Previous work has shown that a practical method for achieving excitation with the use of normal operating data is to use the input and output data resulting from a setpoint change to the process output.5,6 Swanda and Seborg9 have proposed a methodology to assess Received: Revised: Accepted: Published: 9094

March 24, 2012 May 28, 2012 June 4, 2012 June 4, 2012 dx.doi.org/10.1021/ie300786g | Ind. Eng. Chem. Res. 2012, 51, 9094−9100

Industrial & Engineering Chemistry Research

Article

the closed-loop transfer function. The result will have the following form:

the performance of PI controllers from closed-loop response data. This method requires a setpoint step change to estimate the apparent time delay of the process when it is modeled by a first-order plus time delay transfer function. Isaksson et al.5 propose a time-delay estimation which uses significant setpoint changes that have occurred in routine operation. They examine normal operating data and search for intervals sufficient excitation for identification. The time-delay estimate based on a Laguerre model of the process is then implemented. A time-delay estimation approach that requires temporarily changing the tuning of the PID controller is proposed in this work. This approach fits the definition of a time-delay approximation method defined by Bjö rklund3 and the definition of a model-dependent method defined by Ni et al.4 Closed-loop output data are fitted with autogressive moving average (ARMA) models at two different PID tunings. Only upper limits on the orders of the closed-loop transfer function need to be assumed. The time-delay estimate is determined by using the feedback-invariance principle that the closed loop impulse responses do not change inside the time-delay window. Specifically, the time delay is estimated by determining the time at which the Markov parameters of the two impulse responses become statistically different. The organization of the rest of this work is given as follows. In the second section, the theory for the feedback-invariant approach to time-delay estimation is addressed. In the next section, the method for estimating the impulse response of the closed-loop transfer function is discussed. In the fourth section, the procedure for obtaining the time-delay estimate is outlined. Lastly, time-delay estimation for three processes is simulated. These processes have various orders of plant and disturbance models.

where f is the number of integer periods of delay due to the process deadtime. The coefficients of the first f lags of the impulse response from the closed-loop transfer function, after the leading coefficient of one, are not affected by any feedback controller due to the feedback-invariant principle. They are identical to the first f+1 Markov parameters of the disturbance transfer function N(q−1),1 where the leading coefficient is one. The number of periods of time delay for the discrete transfer function is thus b = 1 + f. The feedback-invariance of the first few Markov parameters can be used to obtain an estimate of the time delay. No matter what feedback controller is used, the impulse response coefficients of the closed loop transfer function within the time delay do not change, while those outside the time-delay window do. Therefore, we propose to estimate the leading impulse response coefficients under two different controller tunings. One estimate is from the controller with the existing tuning and the other is from a controller detuned from the existing tuning. If the feedback controller detuning is significant, then the time delay of the process becomes discernible. This approach avoids the need of exciting the process through a setpoint change or through the addition of a reference signal. The plant transfer function and disturbance transfer function are not identifiable for a process under feedback control without input excitation. The data set will not be informative enough to identify an autoregressive moving average with exogenous inputs (ARMAX) model if there is a linear, timeinvariant, and noise-free relationship between y(t) and u(t).6 However, for a process under PID control, the closed-loop transfer function can be identified as an ARMA model by fitting the model to a time series of process output data. Two closedloop transfer functions under different PID tunings can be used to determine the time delay of the process. Once the time-delay estimate is obtained, the performance of the minimum variance controller can be calculated from the coefficients of the first f lags:1

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FEEDBACK-INVARIANT APPROACH TO TIME-DELAY ESTIMATION The closed-loop SISO system is described by the following equations: y(t ) = H(q−1)u(t ) + N (q−1)a(t )

(1)

u(t ) = −K (q−1)y(t )

(2)

where K is the transfer function for the PID controller: −1

K (q ) =

2 σmv = (1 + ψ12 + ... + ψ f2)σa2

κ0 + κ1q−1 + κ2q−2 1 − q −1

where is the variance of the white noise sequence that drives the disturbance N(q−1).

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(3)

It is assumed that H is stable, N can be either stable or with integrating elements, and a(t) is white noise. The a(t) has a variance of σ2a. The closed-loop transfer function, which is determined by applying feedback to the linear process, is given by the following expression: y(t ) =

N (q−1) 1 + K (q−1)H *(q−1)q−b

(6)

σ2a

ESTIMATE OF IMPULSE RESPONSE OF THE CLOSED-LOOP TRANSFER FUNCTION We desire to obtain an estimate of the two impulse responses of the closed-loop process under PID control from the available output history. The two impulse responses will be estimated under two different tunings of the controller. The following assumptions are made in this approach: 1 The plant dynamics and time delay are time-invariant over the time period in which the closed-loop data are collected. 2 The disturbance dynamics are time-invariant over the time period in which the closed-loop data are collected. In addition to estimating the two impulse responses, it is useful to determine their confidence limits so as to detect a significant change in the impulse responses. To obtain a

a(t ) = G(q−1)a(t ) (4)

where b is the number of periods of time delay in the discrete process and H* is the delay-free part of the process.10 As shown by Harris,1 the structure and parameters of the closed-loop transfer function are found by fitting a time-series model, typically a model of an ARMA process, to the closed-loop output data. The impulse response coefficients of the closedloop process can be determined by performing long division on 9095

dx.doi.org/10.1021/ie300786g | Ind. Eng. Chem. Res. 2012, 51, 9094−9100


Article

confidence interval for an impulse response estimate, we use multiple sample sets of the output historical data to obtain a distribution of the closed-loop transfer function of the process. A bootstrapping method is used to obtain an average impulse response estimate and the corresponding confidence intervals. This approach requires using windows of data from the output history. These windows of the time-series data can contain overlapping data.11 A closed-loop transfer function is fit to each window of data by finding the ARMA model, which minimizes the Akaike’s Final Prediction Error (FPE).12 The order of the numerator in the ARMA is searched from 1 to a specified upper limit and the denominator is searched linearly from 1 to a specified upper limit to find which pair of orders minimizes Akaike’s FPE. From the closed-loop transfer function computed using each window of data, a set of impulse responses is computed. A mean impulse response and its confidence limits are determined from the set of estimated impulse responses. As more estimates of the impulse response are included, the estimation error in the mean impulse response will decrease. As a result, the confidence limits for the mean impulse response will become tighter. It is well-known that the variance of the parameter estimates is inversely proportional the length N of the time series used in parameter identification for large N.6 By applying a bootstrap method known as block resampling which uses overlapping windows of the time series, it is possible to increase the rate of convergence of the parameter estimate. The bootstrap method assumes the parameter estimate obtained from the whole time series and the parameter estimate obtained from a subset of the time series come from approximately the same distribution.11 The confidence limits for the estimated impulse response are also calculated with the bootstrap method. In the field of econometrics, there is a wide body of literature for methods to determine impulse response confidence limits.13−18 Pesevanto and Rossi provide a review and comparison of these methods.19

done by overlaying the mean impulse response of G2 on the mean and 2σ confidence limits of the mean impulse response of G1. Assuming the impulse response estimate is from a normal distribution, the 2σ confidence limits correspond to the 95% confidence level and are used to determine if an impulse response coefficient for G2 is statistically similar to the corresponding coefficient in G1.

CLOSED-LOOP TRANSFER FUNCTIONS OF A PROCESS UNDER TWO PID CONTROLLER TUNINGS The first closed-loop transfer function is identified from the process under routine operation. A mean impulse response and confidence limits are calculated from the output history. The impulse response estimate for the closed-loop transfer function with historical data is represented by the following:

(11)

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SIMULATION EXAMPLES In the following section, time-delay estimation for three processes is simulated. These processes include a first-order plant and first-order disturbance, a first-order plant and a firstorder integrating disturbance, and a second-order plant and first-order disturbance. Simulation Example 1: First-Order Plant and FirstOrder Disturbance. We begin by illustrating the proposed time-delay estimation method for a process with a first-order plant and first-order disturbance. The process is described by the following continuous-time transfer function: 4 H (s ) = e − 5s (9) 4s + 1 The disturbance model is described by the following continuous-time transfer function:

N (s ) =

where a(t) is white noise with variance = 4. A sampling time of one is chosen for simulation, which is a fourth of the time constant for the first-order plant, and a fifth of the time constant for the first-order disturbance. The discrete-time plant model will have a time delay b = 6. The PID controller is represented by the following transfer function in continuous-time: ⎛ ⎞ 1 + τDs⎟ K (s ) = K c ⎜1 + τIs ⎝ ⎠

where Kc is the proportional gain, τI is the integral time and τD is the derivative time. Time-delay estimation is performed by comparing the impulse responses estimated from processes under two different PID tunings. The mean impulse response and its confidence limits are calculated by sampling for 3000 time steps, and using 21 windows of data with a 1000 time steps. These windows are found by shifted the window of 1000 time steps forward 100 time steps until the 21 windows are obtained. To identify the structure of the discrete-time closed-loop transfer function in each window, the numerator is searched from 1 to 6 and the denominator is searched from 1 to 12. The mean impulse and confidence limits are calculated from the impulse responses computed from the individual data windows. The proportional gain of the controller tuning is reduced and the mean impulse response is calculated again. The number of coefficients for the second impulse response that are within the 2σ confidence limits of the first impulse response determine the time delay. The process is sampled for 3000 time steps to estimate the second impulse response using 21 windows of data with each window containing 1000 time steps. The mean impulse and confidence limits are calculated from the impulse responses computed from the individual data windows. The estimated impulse responses of the closed-loop transfer function for the process controlled with proportional gain of

(7)

To obtain a contrast in the feedback-varying coefficients of the impulse response, a straightforward approach is to change the PID tuning of the controller significantly. Another closedloop transfer function is estimated from the output history for a specified processing length under the detuned PID controller. Because the controller has been modified from its routine operation, it is desirable to sample process for as little time as possible under the second controller tuning. The impulse response estimate from the closed-loop transfer function under the second PID tuning is given by the following: G2(q−1) = 1 + γ1q−1 + ... + γf q−f + ...

(10)

σ2a

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G1(q−1) = 1 + φ1q−1 + ... + φf q−f + ...

5 5s + 1

(8)

The time delay can be identified by determined how many coefficients of the mean impulse response of G2 are statistically the same as the mean impulse response of G1. This analysis is 9096



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Figure 1. Direct Comparison of Impulse Responses for Kc = 0.225 and Kc = 0.125.


0.225 and integral time of 10.0 and proportional gain of 0.125 and integral time of 10.0, respectively, are shown in Figure 1. Figure 1 indicates the first five impulse response coefficients (after the leading coefficient of 1) of the second impulse response significantly match those of the first impulse response. This result indicates the process has a time delay of six. As expected from theory, the disturbance impulse response matches the first impulse response through five coefficients, indicating a time delay of six. The proportional gain needs to be modified very little to obtain a time-delay estimate between five and seven. In most cases, an accurate assessment of the time delay can be obtained by reducing the proportional gain by 30%. A policy for obtaining the time delay could be to reduce the gain by 30% if the closed-loop response of the process is underdamped. This

occurs when the gain is about 0.25 or greater. If it is known that the closed-loop response is overdamped at routine operation, then the time delay could be estimated by increasing the gain. Simulation Example 2: Process with an Integrating Disturbance. Time-delay estimation is performed on another process which has a first-order plant and a first-order integrating disturbance. The plant is given by the following transfer function: H (s ) =

4 e − 5s 4s + 1

(12)

The disturbance model is described by the following transfer function: 9097



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N (s ) =

1 s

an output response that is initially overdamped, the time-delay estimate can be estimated by increasing the initial gain. Because of the presence of the integrating disturbance, this process is overdamped for all values of the proportional gain that stabilize the process with τI = 10. Therefore, increasing the gain would be an appropriate policy for most cases. Simulation Example 3: Second-Order Plus Time Delay Plant. To demonstrate the proposed time-delay estimation approach is applicable to a wide range of processes, it is also applied to a process with a second-order plus time delay plant. Processes with higher-order plant models present a unique challenge because of the lag time in the dynamic response that is defined as the time before the time constant of the plant is reached. In higher-order plants that are modeled with a firstorder plus time delay model, neglected dynamics can be modeled with the time-delay term.20 Time-delay estimation is performed on a process which has a second-order plant and a first-order disturbance. The plant is given by the following transfer function:

(13)

where a(t) is white noise with variance σ2a = 4. In simulation, the sampling time of the process is chosen to be one. When converted to a discrete transfer function, the time delay of the process b = 6. Time-delay estimation is performed by comparing the impulse responses estimated from processes under two different PID tunings. The mean impulse response and its confidence limits are calculated by sampling for 3000 time steps and using 21 windows of data with each window containing 1000 time steps. To identify the structure of the transfer function of the closed-loop in each window, the numerator is searched from 1 to 6 and the denominator is searched from 1 to 12. An ARMA model is identified by minimizing the Akaike’s FPE in each window. The mean impulse and confidence limits are calculated from the impulse responses computed from the individual data windows. Next, the proportional gain of the controller tuning is reduced and the mean impulse response is calculated again. The process is sampled for 3000 time steps to estimate the second impulse response using 21 windows of data with each window containing 1000 time steps. The mean impulse and confidence limits are calculated from the impulse responses computed from the individual data windows. The estimated impulse responses of the closed-loop transfer function for the process controlled with proportional gain of 0.225 and integral time of 10.0 and proportional gain of 0.125 and integral time of 10.0, respectively, are shown in Figure 2. Figure 2 indicates the first seven impulse response coefficients (after the leading coefficient of 1) of the second impulse response significantly match those of the first impulse response. This result implies that the process has a time delay of eight. As expected from theory, the disturbance impulse response matches the first impulse response through five coefficients, indicating the process has a time delay of six. A reduction in the gain by 30% gives a reasonably accurate estimate of the time delay for most initial controller gains. For

H (s ) =

26.67 e − 5s (6.67s + 1)(4s + 1)

(14)

The time constant for the second-order plant is approximately 5.17. The continuous-time plant transfer function is converted to a discrete transfer function by assuming a sampling time of one time step. When converted to a discrete transfer function, the time delay of the process is b = 6. The disturbance model is described by the following transfer function:

N (s ) =

5 5s + 1

(15)

where a(t) is white noise with variance σ2a = 4. A first-order model is fit to the response of the second-order plant model to a unit step input by the process reaction-curve method:20 H (s ) = 9098

26.7 e−6.3s 15.0s + 1

(16)



Article

The time delay θ is found to be 6.3 time units and the time constant τ is found to be 15.0 time units. The inflation of the delay from the delay of five time steps is due to the lag in the dynamic response of the second-order model. The actual delay and lag compose an apparent delay, which is essentially feedback-invariant to a PID controller. For this reason, the identified time delay b of the discrete closed-loop process is expected to be eight or greater. Time-delay estimation is performed by comparing the impulse responses estimated from processes under two different PID tunings. In the simulation, the sampling time of the process is chosen to be one time step. The mean impulse response and its confidence limits are calculated by sampling for 3000 time steps and using 21 windows of data with each window containing 1000 time steps. To identify the structure of the transfer function in each window of the time series, the numerator of an ARMA model is searched from 1 to 6 and the denominator is searched from 1 to 12. The mean impulse and confidence limits are calculated from the impulse responses computed from the individual data windows. The proportional gain of the controller tuning is reduced and the mean impulse response is calculated again. The process is sampled for 3000 time steps to estimate the second impulse response using 21 windows of data with each window containing 1000 time steps. The mean impulse and confidence limits are calculated from the impulse responses computed from the individual data windows. The estimated impulse responses of the closed-loop transfer function for the process controlled with proportional gain of 0.045 and integral time of 10.0 and proportional gain of 0.025 and integral time of 10.0, respectively, are shown in Figure 3. Figure 3 indicates the first nine impulse response coefficients (after the leading coefficient of 1) of the second impulse response significantly match those of the first impulse response. This result implies a time-delay estimate of 10 for the discrete process. A possible reason for the overestimation of the time delay is that the lag present in the second-order plant cannot be reduced by changing the proportional gain of the PID controller. This lag may be essentially feedback-invariant for a PID controller. The disturbance impulse response matches the first impulse response through seven coefficients. This indicates the time delay of the discrete process is eight, which differs from the actual delay of six given by the discrete transfer function model. This result could be due to error in the estimation of the impulse response for the closed-loop transfer function with routine historical data. Through many simulations, it is observed that the proposed method could overestimate the time delay, while no underestimate of the time delay was observed. This is appropriate for performance assessment as an underestimate of the time delay could lead to unrealistic performance potential for improvement.

excitation through setpoint changes or the introduction of a reference signal. Obtaining the time-delay estimate with the proposed approach requires a temporary modification of the proportional gain of the PID controller for a process in closed-loop operation. At each controller setting, the closed-loop transfer function is computed from output time-series data. A bootstrapping method is used in the identification of the impulse response and confidence limits of the closed-loop transfer function at each controller setting. Time-delay estimation for three processes was simulated. Typically, a 30 to 50% reduction in the proportional gain is necessary to obtain an accurate time-delay estimate. When retuning of the PID controller is not desirable, Harrison proposes the use of relay-feedback as the second controller under which to rebuild the closed-loop transfer function model.21 For the process with an integrating disturbance, the time delay is slightly overestimated. For the process with a second-order plant and first-order disturbance, the lag introduced by the higher-order plant results in an apparent time delay which is slightly greater than the actual time delay of the process. This lag may be essentially feedback-invariant for a PID controller. While an overestimate of the time delay leads to a more conservative benchmark, it is more appropriate than the alternative. Once the time delay is obtained, the minimum variance benchmark proposed by Harris can be computed to assess the controller performance. The data that are used for the time-delay estimates could also be used to estimate a confidence interval for the performance index. For multivariable processes where the performance assessment is based on delay information only,22,23 this approach can be applicable as well.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (C.A.H.), sqin@ usc.edu (S.J.Q.) Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This research was supported by a National Science Foundation Graduate Fellowship and a National Science Defense and Engineering Graduate Fellowship.

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REFERENCES

(1) Harris, T. Assessment of control loop performance. Can. J. Chem. Eng. 1989, 67, 856−861. (2) Qin, S. Control performance monitoringA review and assessment. Comput. Chem. Eng. 1998, 23, 173−186. (3) Björklund, S. A survey and comparison of time-delay estimation methods in linear systems. Ph.D. thesis. Department of Electrical Engineering, Linköping University: Sweden, 2003. (4) Ni, B.; Xiao, D.; Shah, S. Time delay estimation for MIMO dynamical systemsWith time-frequency domain analysis. J. Process Control 2010, 20, 83−94. (5) Isaksson, A.; Horch, A.; Dumont, G. Event-triggered deadtime estimation from closed-loop data. Proceedings of the American Control Conference; Arlington, VA, 2001; pp 3280−3285. (6) Ljung, L. System Identification: Theory for the User; Prentice-Hall, Inc: Englewood Cliffs, NJ, 1999. (7) Elnaggar, A.New method for delay estimation. Proceedings of the 29th Conference on Decision and Control; Honolulu, HI, 1990; pp 1629−1630.

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CONCLUSIONS The proposed feedback-invariant approach to time-delay estimation has been demonstrated to be successful when applied to processes under PID control. The proposed technique is an alternative to existing time-delay estimation approaches which would require obtaining a step response or other input excitation. The application of open-loop time-delay estimation techniques in the closed-loop requires process 9099



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(8) Lynch, C.; Dumont, G. Control loop performance monitoring. IEEE Trans. Cont. Sys. Tech. 1996, 4, 185−192. (9) Swanda, A.; Seborg, D.Controller performance monitoring based on setpoint response data. Proceedings of the American Control Conference; San Diego, CA, 1999; pp 3863−3867. (10) Ko, B. On Performance Assessment of Feedback Control Loops. Ph.D. thesis; Department of Chemical Engineering: The University of Texas at Austin, 2000. (11) Cherwick, M. Bootstrapping Methods; John Wiley and Sons, Inc.: New York, NY, 1999. (12) Ljung, L. System Identification Toolbox User’s Guide; The MathWorks, Inc.: Natick, MA, 1995. (13) Wright, J. Confidence intervals for univariate impulse responses with a near unit root. J. Bus. Econ. Stat. 2000, 18, 368−373. (14) Runkle, D. Vector autogression and reality. J. Bus. Econ. Stat. 1987, 5, 437−442. (15) Pesavento, E.; Rossi, B. Small sample confidence intervals for multivariate IRFs at long horizons. J. Appl. Econ. 2006, 21, 1135−1155. (16) Kilian, L. Finite-sample properties of percentile and percentile-t bootstrap confidence intervals for impulse responses. Rev. Econ. Stat. 1999, 81, 652−660. (17) Gospodinov, N. Asymptotic confidence intervals for impulse responses of near-integrated processes. Econ. J. 2004, 7, 505−527. (18) Andrews, D.; Chen, H. Approximately median-unbiased estimation of autoregressive models. J. Bus. Econ. Stat. 1994, 12, 187−204. (19) Pesavento, E.; Rossi, B. Impulse response confidence intervals for persistent data: What have we learned? J. Econ. Dyn. Control 2007, 31, 2398−2412. (20) Seborg, D.; Edgar, T.; Mellichamp, D. Process Dynamics and Control; John Wiley and Sons, Inc.: New York, NY, 1989. (21) Harrison, C. Towards the Peformance Monitoring of Constrained Control Systems. Ph.D. thesis; Department of Chemical Engineering: The University of Texas at Austin, 2006. (22) Xia, H.; Majecki, P.; Ordys, A.; Grimble, M. Performance assessment of MIMO systems based on I/O delay information. J. Process Control 2006, 16, 373−383. (23) Huang, B.; Ding, S.; Thornhill, N. Practical solutions to the multivariate feedback control performance assessment problem: reduced a priori knowledge of interactor matrices. J. Process Control 2005, 15, 573−583.

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Feedback-Invariant Approach to Time-Delay Estimation for

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