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17th IFAC Symposium on System Identification 17th IFAC Symposium on Identification 17th IFAC Symposium on System SystemCenter Identification Beijing International Convention 17th IFAC IFAC Symposium on System SystemCenter Identification Beijing International Convention 17th Symposium on Identification Beijing International Convention Center Available online at www.sciencedirect.com October 19-21, 2015. Beijing, China Beijing International Convention Center October 19-21, 2015. Beijing, China Beijing International Center October 19-21, 2015. Convention Beijing, China October October 19-21, 19-21, 2015. 2015. Beijing, Beijing, China China

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IFAC-PapersOnLine 48-28 (2015) 739–744

Understanding closed-loop Understanding Understanding closed-loop closed-loop with ITCLI with with ITCLI ITCLI

identification identification identification

∗∗ ∗ ∗ ∗∗∗ D.E. a ∗∗ J.L. ∗∗∗ D.E. Rivera Rivera ∗∗ J.L. Guzm´ Guzm´ an n ∗∗∗ M. M. Berenguel Berenguel ∗∗∗ S. S. Dormido Dormido ∗∗∗ D.E. Rivera a n Berenguel Dormido ∗∗ J.L. Guzm´ ∗∗∗ ∗∗ ∗ M. ∗ S. D.E. Rivera J.L. Guzm´ a n M. Berenguel S. Dormido D.E. Rivera J.L. Guzm´ an M. Berenguel S. Dormido ∗∗∗ ∗ ∗ Dep. de Inform´ atica, tica, Universidad Universidad de de Almer´ Almer´ıa, ıa, 04120 04120 Almer´ Almer´ıa, ıa, Spain. Spain. ∗ Dep. de Inform´ a de Inform´ a tica, Universidad de Almer´ ıa, 04120 Almer´ ıa, Spain. ∗ ∗ Dep. Dep. a tica, Universidad de Almer´ ıa, 04120 Almer´ ıa, Spain. Email:{joseluis.guzman,beren}@ual.es Dep. de de Inform´ Inform´ a tica, Universidad de Almer´ ıa, 04120 Almer´ ıa, Spain. Email:{joseluis.guzman,beren}@ual.es Email:{joseluis.guzman,beren}@ual.es ∗∗ ∗∗ School for Email:{joseluis.guzman,beren}@ual.es the Engineering of Matter, Transport, and Energy, ∗∗ School for Email:{joseluis.guzman,beren}@ual.es the Engineering of Matter, Transport, and Energy, ∗∗ School for the Engineering of Matter, Transport, and Energy, ∗∗ School State for the theUniversity. Engineering of Matter, Matter, Transport,USA. and Energy, Energy, Arizona State University. Tempe AZ 85287-6106 85287-6106 USA. Email: School for Engineering of Transport, and Arizona Tempe AZ Email: Arizona State University. Tempe AZ 85287-6106 USA. Email: Arizona Tempe AZ 85287-6106 USA. Email: [email protected] Arizona State State University. University. Tempe AZ 85287-6106 USA. Email: [email protected] [email protected] ∗∗∗ ∗∗∗ [email protected] Dep. Inform´ a tica y Autom´ a tica, UNED. C/ Juan del Rosal, 16, [email protected] ∗∗∗ Dep. Inform´ a tica y Autom´ a tica, UNED. C/ Juan del Rosal, 16, Inform´ a tica yy Autom´ a tica, UNED. C/ Juan del Rosal, 16, ∗∗∗ ∗∗∗ Dep. Dep. Inform´ a tica Autom´ a tica, UNED. C/ Juan del Rosal, 16, 28040, Madrid, Spain. Email: [email protected] Dep.28040, Inform´ a tica y Autom´ a tica, UNED. C/ Juan del Rosal, 16, Madrid, Spain. Email: [email protected] 28040, Madrid, Spain. Email: [email protected] 28040, Madrid, Madrid, Spain. Spain. Email: Email: [email protected] [email protected] 28040,

Abstract: Abstract: Abstract: Abstract: Prior work by by the the authors authors (featured (featured in in software software sessions sessions held held during during SYSID SYSID 2009 2009 and and 2012) 2012) Abstract: Prior work Prior work by the authors (featured in software sessions held during SYSID 2009 and 2012) Prior work by the authors (featured in software sessions held during SYSID 2009 and 2012) described ITSIE and ITCRI, a series of interactive tools designed to aid students and Prior work by the authors (featured in software sessions held during SYSID 2009 and 2012) described ITSIE and ITCRI, a series of interactive tools designed to aid students and described ITSIE and ITCRI, a series of interactive tools designed to aid students and described ITSIE and ITCRI, a series of interactive tools designed to aid students and practitioners in better understanding the role of design variables associated with classical described ITSIE and ITCRI, a series of interactive tools designed to aid students and practitioners in better understanding the role of design variables associated with classical practitioners in better understanding the role of design variables associated with classical practitioners in in and better understandingidentification, the role role of of design design variables associated with classical classical prediction-error and control-relevant identification, respectively. This paper presents presents ITCLI practitioners better understanding the variables associated with prediction-error control-relevant respectively. This paper ITCLI prediction-error and control-relevant identification, respectively. This paper presents ITCLI prediction-error control-relevant identification, respectively. This ITCLI (Interactive Tool and for Closed-Loop Closed-Loop Identification), an interactive interactive software tool for forpresents understanding prediction-error and control-relevant identification, respectively. This paper paper presents ITCLI (Interactive Tool for Identification), an software tool understanding (Interactive Tool for Closed-Loop Identification), an interactive software tool for understanding (Interactive Tool for Closed-Loop Identification), an interactive software tool for understanding design variables in closed-loop identification of LTI SISO systems using prediction-error (Interactive Tool for Closed-Loop Identification), an interactive software tool for understanding design variables in closed-loop identification of LTI SISO systems using prediction-error design variables in closed-loop identification of LTI SISO systems using prediction-error design variables in identification of SISO using prediction-error techniques. The tool enables an interactive evaluation how bias and variance design variables in closed-loop closed-loop identification of LTI LTIregarding SISO systems systems using prediction-error techniques. The tool enables an interactive evaluation regarding how bias and variance effects effects techniques. The tool enables an interactive evaluation regarding how bias and effects techniques. Thewhen tool enables enables an interactive interactive evaluation regarding how bias and and variance variance effects are manifested when identification is performed performed underregarding closed-loop circumstances. The role role of techniques. The tool an evaluation how bias variance effects are manifested identification is under closed-loop circumstances. The of are manifested when identification is performed under closed-loop circumstances. The role of are manifested when identification is performed under closed-loop circumstances. The role of external signal design, design, choice of of model model structure, controller controller tuning during during identification testing, are manifested when identification is performed under closed-loop circumstances. The role of external signal choice structure, tuning identification testing, external signal design, choice of model structure, controller tuning during identification testing, external signal design, choice of model structure, controller tuning during identification testing, and signal injection points (at either the manipulated variable or the setpoint) while in the external signal design, choice of model structure, controller tuning during identification testing, and signal injection points (at either the manipulated variable or the setpoint) while in the and signal injection points (at either the manipulated variable or the setpoint) while in the and signal injection (at manipulated variable or while presence of autocorrelated can be evaluated in aa comprehensive and easy-to-operate and signal injection points pointsdisturbances (at either either the the manipulated variable or the the setpoint) setpoint) while in in the the presence of autocorrelated disturbances can be evaluated in comprehensive and easy-to-operate presence of autocorrelated disturbances can be evaluated in a comprehensive and easy-to-operate presence ofsoftware autocorrelated disturbances can be evaluated evaluated inprovided a comprehensive comprehensive and easy-to-operate easy-to-operate tool. Theof software is developed developed using can Sysquake and is is in provided as aa stand-alone stand-alone executable presence autocorrelated disturbances be a and tool. The is using Sysquake and as executable tool. The software is developed using Sysquake and is provided as a stand-alone executable tool. The software is Sysquake and version in the the Windows and Mac Mac using OS operating operating environments. tool. The software is developed developed using Sysquakeenvironments. and is is provided provided as as aa stand-alone stand-alone executable executable version in Windows and OS version in the Windows and Mac OS operating environments. version in the Windows and Mac OS operating environments. version in the Windows and Mac OS operating environments. © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Closed-loop identification, interactivity, direct prediction-error estimation, control Keywords: Closed-loop identification, interactivity, direct prediction-error estimation, control Keywords: Closed-loop identification, identification, interactivity, interactivity, direct direct prediction-error prediction-error estimation, estimation, control control Keywords: Closed-loop education Keywords: Closed-loop identification, interactivity, direct prediction-error estimation, control education education education education 1. INTRODUCTION aa MATLAB-like language with fast execution and excellent 1. INTRODUCTION language with fast execution and excellent 1. INTRODUCTION INTRODUCTION aa MATLAB-like MATLAB-like language with fast execution and excellent 1. MATLAB-like language with fast execution and facilities for interactive graphics (Piguet, 2004). 1. INTRODUCTION afacilities MATLAB-like language with fast execution and excellent excellent facilities for interactive interactive graphics (Piguet, 2004). for graphics (Piguet, 2004). facilities for for interactive interactive graphics graphics (Piguet, (Piguet, 2004). 2004). facilities In recent years, advances in information technologies have In recent years, advances in information technologies have Closed-loop identification is an important practical probidentification is an important practical probIn recent recent years, years, advances advances in in information information technologies technologies have have Closed-loop Closed-loop identification is an important practical probIn provided powerful software tools for training engineers In recent years, advances in information technologies have provided powerful software tools for training engineers Closed-loop identification is an important practical problem in system identification (Wellstead, 1977; Forssell and Closed-loop identification is an important practical problem in system identification (Wellstead, 1977; Forssell and provided powerful software tools for training engineers lem in system identification (Wellstead, 1977; Forssell provided powerful software tools for training engineers (Dormido, 2004; Guzm´ aan et al., 2009). Moreover, interprovided powerful software tools for training engineers (Dormido, 2004; Guzm´ n et al., 2009). Moreover, interlem in in system system identification (Wellstead, 1977;situations, Forssell and and Ljung, 1999; Ljung, Ljung, 1999). In In(Wellstead, many practical practical situations, it lem identification 1977; Forssell and Ljung, 1999; 1999). many it (Dormido, 2004; Guzm´ a n et al., 2009). Moreover, interLjung, 1999; Ljung, 1999). In many practical situations, it (Dormido, 2004; Guzm´ a n et al., 2009). Moreover, interactive software have proven as particularly (Dormido, 2004; tools Guzm´ an etbeen al., 2009). Moreover, inter- Ljung, active software tools have been proven as particularly 1999; Ljung, 1999). many situations, it is not not possible possible to identify identify a In system inpractical the open-loop; open-loop; hence Ljung, 1999; Ljung, 1999). In many practical situations, it is to a system in the hence active software tools have been proven as particularly is not possible to identify a system in the open-loop; hence active software tools have been proven as particularly useful techniques with high impact on control education active software tools have been proven as particularly useful techniques with high impact on control education is not possible to identify a system in the open-loop; hence closed-loop identification becomes a necessity. A fundais not possible to identify a system in the open-loop; hence closed-loop identification becomes a necessity. A fundauseful techniques with high impact on control education identification becomes aa necessity. A useful techniques with high on education (Guzm´ aan et al., 2005, Interactive tools provide a closed-loop useful techniques with 2008). high impact impact on control control (Guzm´ n et al., 2005, 2008). Interactive tools provide closed-loop identification becomes necessity. A fundafundamental presented by identification identification becomes a necessity. A fundamental challenge challenge presented by closed-loop closed-loop identification (Guzm´ n et et al., al., 2005, 2005, 2008). Interactive tools education provide aaa closed-loop mental challenge presented by closed-loop identification (Guzm´ aa n 2008). Interactive tools provide real-time connection between decisions made during the (Guzm´ a n et al., 2005, 2008). Interactive tools provide a real-time connection between decisions made during the mental challenge presented by closed-loop identification in contrast contrast to open-loop open-loop is that that there exists exists correlation mental challenge presented by closed-loop identification in to is there correlation real-time connection between decisions made during the contrast to open-loop is that there exists correlation real-time connection between decisions made during the design phase and results obtained in the analysis phase real-time connection between decisions made during the in design phase and results obtained in the analysis phase in contrast to open-loop is exists correlation between the manipulated and the disturbance as in contrast to open-loop variable is that that there there exists correlation between the manipulated variable and the disturbance as design phase phase and results results obtained in the the analysis phase between the manipulated variable and the disturbance as design and obtained in analysis phase of any control-related project. Prior work involving the design phase and results obtained in the analysis phase of any control-related project. Prior work involving the between the manipulated variable and the disturbance as a result of the action of a closed-loop system. Despite between the manipulated variable and the disturbance as a result of the action of a closed-loop system. Despite of any control-related project. Prior work involving the a result of the action of a closed-loop system. Despite of any control-related project. Prior work involving the authors has resulted in ITSIE, an Interactive software of any control-related project. Prior work involving the athe authors has resulted in ITSIE, an Interactive software a result result of the the between action of of these a closed-loop closed-loop system. Despite the correlation between these signals, it itsystem. is possible possible to of action a Despite correlation signals, is to authors has resulted in ITSIE, an Interactive software the correlation between these signals, it is possible to authors has resulted in an Tool for System Identification Education (Guzm´ aasoftware n et al., authors has resulted in ITSIE, ITSIE, an Interactive Interactive software Tool for System Identification Education (Guzm´ n et al., the correlation between signals, it is possible to consistently estimate boththese the plant plant and disturbance disturbance modthe correlation between these signals, it is possible to consistently estimate both the and modTool for System Identification Education (Guzm´ a n et al., consistently estimate both the plant and disturbance modTool for System Identification Education (Guzm´ a n et al., 2009, 2012b) and ITCRI an Interactive Tool for Control Tool for System Identification Education (Guzm´ a n et al., 2009, 2012b) and ITCRI an Interactive Tool for Control consistently estimate both the plant and disturbance models in the absence of any external excitation (outside of consistently estimate both the plant and disturbance models in the absence of any external excitation (outside of 2009, 2012b) and ITCRI an Interactive Tool for Control els in the absence of any external excitation (outside of 2009, 2012b) and Interactive Tool Control ´án ´´ for 2009, 2012b) and ITCRI ITCRI an Interactive Tool for Control Relevant Identification ((Alvarez et al., 2011; Alvarez et al., els in the absence of any external excitation (outside of what may be naturally present in the closed-loop system) Relevant Identification Alvarez et al., 2011; Alvarez et al., ´ ´ els in the absence of any external excitation (outside of what may be naturally present in the closed-loop system) Relevant Identification ( Alvarez et al., 2011; Alvarez et al., what may be naturally present in the closed-loop system) ´ ´ Relevant Identification ( Alvarez et al., 2011; Alvarez et al., ´ ´ 2013). ITSIE focuses exclusively on open-loop system idenwhat may maythere be naturally naturally present in knowledge the closed-loop closed-loop system) provided there is detailed a priori knowledge regarding the Relevant Identification ( Alvarez et al., 2011; Alvarez et al., 2013). ITSIE focuses exclusively on open-loop system idenwhat be present in the system) provided is detailed a priori regarding the 2013). ITSIE focuses exclusively on open-loop system idenprovided there is detailed a priori knowledge regarding the 2013). ITSIE focuses exclusively on system tification, while ITCRI deals with the control-relevant there detailed a regarding the model structure structure (Ljung, 1999). In knowledge practice this this is often often not not 2013). ITSIE focuses exclusively on open-loop open-loop system ideniden- provided tification, while ITCRI deals with the control-relevant provided there is is(Ljung, detailed1999). a priori priori knowledge regarding the model In practice is tification, while ITCRI deals with with the control-relevant control-relevant model structure (Ljung, 1999). In practice this is often not tification, while ITCRI deals the identification based on open-loop prefiltered predictionmodel structure (Ljung, 1999). In practice this is often not the case, so it is important to establish how appropriate tification, while ITCRI deals with the control-relevant identification based on open-loop prefiltered predictionmodel structure (Ljung, 1999). In practice this is often not the case, so it is important to establish how appropriate identification based on open-loop prefiltered predictionthe case, so it is important to establish how appropriate identification based on prefiltered predictionerror estimation procedures. Our team develthe case, is to how appropriate and selection of other design variables identification based on open-loop open-loop predictionerror estimation procedures. Our team has has also also develexperimental the case, so so it itdesign is important important to establish establish how appropriate experimental design and selection of other design variables error estimation estimation procedures. Our prefiltered team has also develexperimental design and selection of other design variables error procedures. Our team has also developed i-pIDtune, an interactive tool that integrates system experimental design and and selection of other other design design variables in the identification process can facilitate the closed-loop error estimation procedures. Our team has also developed i-pIDtune, an interactive tool that integrates system experimental design selection of variables in the identification process can facilitate the closed-loop oped i-pIDtune, an interactive tool that integrates system in the identification process can facilitate the closed-loop oped i-pIDtune, an interactive tool that integrates system identification and PID controller design (Guzm´ a n et al., in the identification process can facilitate the closed-loop identification problem. This includes the use, excitation, oped i-pIDtune, an interactive tool that integrates system identification and PID controller design (Guzm´ a n et al., in the identification process can facilitate the closed-loop identification problem. This includes the use, excitation, identification and and PID PID controller controller design design (Guzm´ (Guzm´ n et et al., identification problem. This includes the use, excitation, identification aa 2012a). All these tools are coded in Sysquake, identification problem. This the use, excitation, location of an experimental signal, as well as sensible identification andinteractive PID controller (Guzm´ an n et al., al., and 2012a). All these interactive tools are coded in Sysquake, identification problem. This includes includes the use, excitation, and location of an experimental signal, as well as sensible 2012a). All All these these interactive tools design are coded coded in Sysquake, Sysquake, and location of an experimental signal, as well as sensible 2012a). interactive tools are in and location of an experimental signal, as well as sensible of the closed-loop system. The purpose of ITCLI, 2012a). All these interactive tools are coded in Sysquake, tuning and location of an experimental signal, as well as sensible tuning of the closed-loop system. The purpose of ITCLI, tuning of the closed-loop system. The purpose of ITCLI, tuning of the the closed-loop closed-loop system. The purpose of System ITCLI, an Interactive Software Tool for Closed Loop System tuning of system. The purpose of ITCLI, an Interactive Software Tool for Closed Loop This work has been partially funded by the following projects: an Interactive Software Tool for Closed Loop System This work work has has been been partially partially funded funded by by the the following following projects: projects: This an Interactive Software Tool for Closed Loop System Identification is Software to examine examine these various design variables This work has been partially an Interactive Tool for Closed Loop System Identification is to these various design variables DPI2014-55932-C2-1-R, DPI2014-56364-C2-1-R and DPI2012-31303 funded by the following projects: Identification is to examine these various design variables DPI2014-55932-C2-1-R, DPI2014-56364-C2-1-R DPI2012-31303 This work has been partially funded by the and following projects: DPI2014-55932-C2-1-R, DPI2014-56364-C2-1-R and DPI2012-31303 Identification is to these design variables in closed-loop identification in an accessible and Identification issystem to examine examine these various various design variables in closed-loop system identification in an accessible and (financed by the the SpanishDPI2014-56364-C2-1-R Ministry of of Economy and and Competitiveness DPI2014-55932-C2-1-R, and DPI2012-31303 (financed Ministry DPI2014-55932-C2-1-R, andCompetitiveness DPI2012-31303 in closed-loop system identification in an accessible and (financed by by the Spanish SpanishDPI2014-56364-C2-1-R Ministry of Economy Economy and Competitiveness in closed-loop system identification in an accessible informative software environment. and EU-ERDF funds). (financed by the Spanish Ministry of Economy and Competitiveness in closed-loop systemenvironment. identification in an accessible and and informative software environment. and EU-ERDF funds). (financed by the Spanish Ministry of Economy and Competitiveness informative software and EU-ERDF funds). informative software software environment. environment. and informative and EU-ERDF EU-ERDF funds). funds). Copyright IFAC 2015 2015 739 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © 2015, IFAC (International Federation of Automatic Control) Copyright 739 Copyright © © IFAC IFAC 2015 739 Copyright IFAC 2015 739 Peer review© of International Federation of Automatic Copyright ©under IFAC responsibility 2015 739Control. 10.1016/j.ifacol.2015.12.218

2015 IFAC SYSID 740 October 19-21, 2015. Beijing, China

D.E. Rivera et al. / IFAC-PapersOnLine 48-28 (2015) 739–744

The paper is organized as follows. First, a brief description of issues in closed-loop identification is presented in Section 2. Section 3 describes the functionality of the tool, with illustrative examples discussed in Section 4. Section 5 provides the main conclusions and future research work. 2. BASIC ISSUES IN CLOSED-LOOP IDENTIFICATION Error in system identification is a consequence of bias and variance effects. This section is a summary of the presentation in Guzm´ an et al. (2014). 2.1 Problems caused by bias Bias expressions can be very helpful in terms of relating design variables in identification to the performance objective of the parameter estimation problem. Bias expressions for open-loop least-squares prediction-error identification are well known, based on the seminal work by Ljung (1999). ITCLI considers a linear, time-invariant plant with disturbance represented by the equation (1): y(t) = p(q)(u(t) + n1 (t)) + n2 (t) = p(q)u(t) + ν(t)

(1) (2)

where y(t) is the measured output signal, u(t) is a quasistationary time series with power spectra Φu that is designed by the user, p(q) is the zero-order-hold-equivalent transfer function for p(s), n1 (t) is a stationary white noise that allows to evaluate the effects of autocorrelated disturbances in the data and n2 (t) is a second stationary white noise that is introduced directly to the output signal. The total disturbance signal is represented as ν(t) = p(q)n1 (t) + n2 (t). n1 (t) and n2 (t) have variances σn2 1 and σn2 2 respectively. Consequently, u, n1 and n2 are all mutually uncorrelated. Furthermore, we will consider prefiltered input/output data yF (t) = L(q)y(t)

uF (t) = L(q)u(t)

(3)

The objective of the parameter estimation procedure is to approximate (1) to a model according to (4). y(t) = p˜(q)u(t) + p˜e (q)e(t)

(4)

where p˜(q) refers to the estimated plant model and p˜e (q) is the noise model, The general family of prediction-error models evaluated in ITCLI corresponds to: B(q) C(q) A(q)y(t) = u(t − nk) + e(t) (5) F (q) D(q) A(q) through F (q) are polynomials in q, while nk is the system delay. Estimation is accomplished by minimizing the prefiltered prediction error (eF (t) = L(q)e(t)) min p, ˜ p˜e

N

e2F (t)

(6)

t=1

Using Parseval’s theorem, it becomes possible to express the least-squares parameter estimation problem in the frequency domain. From open-loop analysis, we can infer that if u(t) is a persistently exciting input (i.e., Φu = 0 for all frequencies) and p˜ and has the correct model structure, 740

Fig. 1. Closed-loop feedback system considered in IT CLI, with signal injection points r and ud , and external disturbance ν(t) = p(q)n1 (t) + n2 (t). then an optimum is reached when p˜ = p; consistent estimation of the plant model p is possible, even with an incorrect structure for the noise model p˜e . When addressing closed-loop direct prediction-error estimation, similar expressions to open-loop prediction-error estimation can be obtained which relate the objective function to the estimated model, the prefilter, the manipulated and disturbance transfer functions, and the closed-loop transfer functions. The closed-loop structure considered is according to Figure 1, which consists of a classical feedback structure with possible signal injection points at r and ud . The derivation of the bias expression is not presented for reasons of brevity; the final result is shown below: π N 1 2 lim eF (t) = ΦeF (ω)dω (7) N →∞ 2π t=1 −π

where Φ eF =

and

|L|2 |p − p˜|2 |p−1 η|2 Φr + ||2 Φud |˜ pe |2

+ |1 + p˜c|2 ||

2

Φν (ω)

|p(ejω )|2 σn2 1 + σn2 2 )

(8)

η = pc(1 + pc)−1 Complementary sensitivity function = (1 + pc)−1 Sensitivity function The important fact to consider from (8) is that in closedloop identification, consistent estimation of p with p˜ is not guaranteed even if the external signals are white noise and uncorrelated to each other. In this case, the crosscorrelation between the input signal u and the disturbance signals n1 and n2 is nonzero because of the action of the controller. Plainly speaking, the controller “gets in the way” of the identification, affecting the quality of the parameter estimates. Hence, in addition to the effects discussed in the open-loop case, the feedback controller c also introduces an additional source of bias to the parameter estimation problem, which must be analyzed carefully if one is to obtain adequate models from closed-loop data. The effect of the feedback controller c is reflected in two ways, as seen in (8). We see it in its effect on the closed-loop transfer functions and p−1 η, which directly weight the additive error term p− p˜; we also see it in the term (1+ p˜c), which establishes a trade-off between the magnitudes of the input signal power spectral density Φu (ω) and the disturbance spectrum Φν (ω) = |p(ejω )|2 σn2 1 + σn2 2 .

2015 IFAC SYSID October 19-21, 2015. Beijing, China


2.2 Problems caused by variance Similarly, there exist expressions for variance in the closedloop that contrast those in the open-loop (Gevers et al., 2001). We shall focus primarily on the variance associated with p˜. For closed-loop identification under the feedback structure described in Fig. 1, the corresponding asymptotic covariance expression is n n Φν (ω) Φν (ω) = (9) Cov˜ p(ejω ) ∼ N Φext N |p−1 η|2 Φr + ||2 Φud u (ω) where n is the model order and N is the number of data. We see the continuing effects of controller tuning, as reflected in the closed-loop transfer functions and p−1 η, in influencing the variance of the plant estimate, as well as the important role of the power spectral densities in the external signals r and ud . 3. INTERACTIVE TOOL DESCRIPTION 3.1 Preliminary This section summarizes the main features of the interactive tool, which can be downloaded for free at http://aer.ual.es/ITCLI/. The plant to be identified consists of a fifth-order system according to 1 . (10) p(s) = (s + 1)5 The model shown in (10) is sampled at a user-specified value (default value T = 1 min) and is subject to noise and disturbances as described in Section 2. The input signals used in ITCLI are: (i) Pseudo-Random Binary Sequences (PRBS) and (ii) multisine signals. The input signal can be designed through direct parameter specification or by applying time constant-based guidelines. The input signal guidelines and parameters discussed in previous work; the interested reader is referred to Guzm´ an et al. (2012b) for a detailed description. Data preprocessing in ITCLI supports mean subtraction, differencing, and subtraction of baseline values. The interactive tool uses data from (1) and the userdefined input signal to estimate prediction-error models from a closed-loop system using the direct approach. The five most popular PEM models are evaluated in ITCLI, with FIR belonging as a subset of ARX models. The tool also includes PEM estimation of state-space models. Closed-loop control implemented in ITCLI stems from the application of the IMC design procedure to restricted complexity approximations for the plant according to (10), using the control-relevant identification procedure implemented in i-pIDtune. The resulting controllers conforming to PI, PID, and PID with filter structures (summarized in Table 1) have an adjustable parameter λ that corresponds to roughly the closed-loop speed-of-response. 3.2 Interactive Tool Features The graphical distribution of the tool has been developed according to the most important steps in a closedloop identification problem. It is described as follows (see Fig. 2): 741

741

Model

KKc

K(−βs+1) τ s+1 K(−βs+1) τ 2 s2 +2ζτ s+1 K(−βs+1) τ 2 s2 +2ζτ s+1

τ β+λ 2ζτ β+τ 2ζτ 2β+λ

τI τ

τD -

τF -

2ζτ

τ 2ζ τ 2ζ

βλ 2β+λ

2ζτ

-

Table 1. IMC-PID tuning rules for open-loop stable first and second-order plants and with β > 0. The PID controller form is represented 1 by c(s) = Kc (1 + τI1s + τD s) (τF s+1) . • Input signal definition. The input signal information in the tool is characterized by four different areas. A section called Input signal parameters is located at the top center zone of the tool. This section is devoted to choose the type of the input signal (PRBS or multisine) and by means of the checkbox called Guidelines to decide between specifying the input signal directly or following the guidelines given in Guzmán et al. (2012b). For instance, if the PRBS is selected without activating the checkbox Guidelines, a text edit and two sliders appear to modify the number of cycles (N Cycles), the number of registers (N Reg), and the switching time (Tsw). At the center right area and top right corner, there are two graphics namely Input signal and Power Spectrum. The graph in the top, Input signal, shows one cycle of the input signal, the graph below represents the input signal power spectrum. The input signal can be also modified dragging on both graphics. Once an input signal has been configured, the full signal with the total number of cycles is shown in Full input signal graph, located at the left-bottom of the central part of the main screen. • Model estimation. The different model structures can be selected from a set of checkboxes located on the top of the Step responses graphic, at the top left part of the tool. When a model structure is selected, estimation and validation results for that model are calculated and shown in the corresponding graphics of the tool. The model parameters can be modified from the section called Model parameters, which is available below the Input signal parameters section. Several radio buttons are available to choose between the different model structures. Once a model structure is selected, different sliders appear being possible to modify the associated orders interactively. Regarding the estimation process, once an input signal has been configured, the full input signal is applied to a high-order process model in order to obtain the simulated “real data” (shown in black in the Output signal graphic), which is used as real process data in the estimation process. In this tool, the input signal can be applied for open-loop or closed-loop identification purposes. On the top of the Full input signal graph, there are two sets of radio buttons allowing to switch between these two options. The first radio button group, located on the top right part of the Full input signal graph, allows to choose between open-loop or closed-loop identification. When the open-loop option is selected, the full input signal is applied directly in open loop to the high-order model like done in ITSIE (Guzm´ an et al., 2012b). This option has been kept here to compare the results between open-loop and closed-



Fig. 2. Main ITCLI screen displaying results for the first example in Section 4, using a PRBS input introduced at the setpoint. Controller tuning at an “intermediate” setting (i.e., not too fast and not too slow) enables closed-loop identification conditions without significant controller-induced bias. loop identification methodologies. On the other hand, when the closed-loop option is active, the full input signal is applied in closed loop on the high-order model. The input signal can be introduced in the loop at the reference or at the manipulated variable paths; this option can be selected from the second radio button group located at the top center part of the Full input signal graph. Based on the selected option (reference or manipulated variable), the full input signal will be shown at the Output signal or Full input signal graphic, respectively. The parameters for the closed-loop simulation are available at the Closed loop and simulation parameters section. In the Output signal graphic, there is an interactive pink vertical line defining the estimation and validation data. The area shown in yellow (at the left of the vertical line) defines the estimation data, whereas the white area represents the validation data (at the right side of the vertical line). Therefore, when a model structure is selected, the open-loop or closed-loop estimation data is used to estimate the model parameters and the validation data to test the resulting model. Then, for each selected model structure, the full input signal is applied to the obtained model, and the results are shown in the Output signal graphic together with the original data of the highorder system. Different colors are used to distinguish between the results of each model. • Model validation. Validation data is represented in white in the Output signal graphic, and is used for crossvalidation purposes. Model validation results are displayed in two different graphics: Step Responses and Correlation function of residuals (which can be 742

selected via the radio buttons located on the top of the Power Spectrum graphic). The Step Responses graph, located at the top left-hand side of the tool, shows the step responses for each selected model, with a legend representing its goodness of fit in %. • Closed-loop identification. The closed-loop parameters are located below the Model parameters section. PI, PID or PID with filter controller structures can be selected from three radio buttons. Moreover, a slider called Lambda allows to specify the parameter λ for the IMC filter time constant according to IMCPID tuning rules (Rivera et al., 1986) summarized in Table 1. Other two sliders called Noise 1 and Noise 2 determine the level of noise in the input (n1 ) and the output (n2 ) signals, respectively. When the closedloop option is selected from the radio button located at top right part of the Full input signal graph, the closed-loop simulation data is shown at the Output signal or Full input signal graphics for the output and control signals, respectively. At the lower right corner of the tool, there is a graph that shows the Bode magnitude of the additive error (|p − p˜|) for each selected model as well as the magnitude of the sensitivity function, ||, when the external signal is introduced at the manipulated variable or, |˜ p−1 η| = |c |, when the external signal is introduced at the reference. The plots of these frequency responses are very useful for studying bias shifts and variance effects as a result of changes in controller tuning (according to the analysis in Section 2 and Equation (8)).

2015 IFAC SYSID October 19-21, 2015. Beijing, China


4. ILLUSTRATIVE EXAMPLES We consider the simulated fifth-order system presented previously and described by the transfer function 1 (11) p(s) = (s + 1)5 with a default sample time of Ts = 1 min. This plant is identified in the closed-loop using built-in PI / PID / PID w/ filter controllers obtained from i-pIDtune. In all these controllers, the IMC filter λ can be adjusted as part of the tool. The magnitude-only Bode plots for the closed-loop transfer functions are evaluated (along with the additive error) and shown in the bottom right of the ITCLI screen. In the illustrative example depicted in Figure 2, a PRBS input signal is used for identification, with parameters: m = 2 (number of cycles), α = 2, (factor representing the closed-loop speed of response), β = 3 (factor representing L the settling time of the process), τdom = 3 (low estimate H of τdom ) and τdom = 7 (high estimate of τdom ). For more information about these parameters see Guzmán et al. (2012b). A signal injection point at the setpoint r is evaluated. Both ARX and OE model structures are estimated, as seen in Figure 2. Working with the tool shows that controller tuning during identification must be at some “intermediate” level; that is, the amplitude |p−1 η| must be as flat as possible. It can be seen that model fits obtained from both ARX and OE estimation agree well with the true plant; the simulated model output closely follows the real data. Because of the sensible tuning of the control loop and the presence of the external signal, direct closed-loop identification provides a good result. The second result, shown in Figure 3, shows that for the same external input, but now introducing the signal at the manipulated variable, the controller must be detuned substantially through a significant increase in λ. Despite the detuning, the low frequencies are still being attenuated by the action of the controller, and consequently the steady-state gain is not estimated as precisely compared to the set point case. The final result, shown in Figure 4, illustrates one of the main points stressed in Section 2 regarding the reasons why external excitation in closed-loop identification is so important. In this case, the excitation in the data is provided completely by the disturbance signal ν(t); the external input magnitude is lowered effectively to zero. Here we see how closed-loop identification displays a perfect fit to data from completely erroneous models. This is illustrated for both ARX and Output Error identification. 5. CONCLUSIONS This paper describes an interactive tool for evaluating important aspects of closed-loop identification. By using ITCLI it is possible to achieve this understanding interactively; this being the motivating philosophy behind the methodology described in this paper. The tool provides different functionality modes which make it possible for students and engineers to use its capabilities with a small learning curve. The tool is available free of charge from http://aer.ual.es/ITCLI/. Future work will be oriented 743

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to include additional illustrative model examples beyond the fifth-order system, and to incorporate the evaluation of additional design variables in the closed-loop identification problem, such as data prefiltering. REFERENCES ´ Alvarez, J.D., Guzmán, J.L., Rivera, D.E., Berenguel, M., and Dormido, S. (2011). ITCRI: An Interactive Software Tool for Control-Relevant Identification Education. In Proceedings of the 18th IFAC World Congress, Milan, Italy. ´ Alvarez, J., Guzmán, J.L., Rivera, D.E., Berenguel, M., and Dormido, S. (2013). Perspectives on controlrelevant identification through the use of interactive tools. Control Engineering Practice, 21(2), 171 – 183, http://aer.ual.es/ITCRI/. Dormido, S. (2004). Control learning: present and future. Annual Reviews in Control, 28(1), 115–136. Forssell, U. and Ljung, L. (1999). Closed-loop identification revisited. Automatica, 35, 1215–1241. Gevers, M., Ljung, L., and van den Hof, P. (2001). Asymptotic variance expressions for closed-loop identification. Automatica, 37, 781 – 786. Guzm´ an, J.L., ˚ Astrom, ¨ K.J., Dormido, S., Hägglund, T., Berenguel, M., and Piguet, Y. (2008). Interactive learning modules for PID control. IEEE Control System Magazine, 28(5), 118–134. Available: http://aer.ual.es/ilm/. Guzmán, J.L., Berenguel, M., and Dormido, S. (2005). Interactive teaching of constrained generalized predictive control. IEEE Control Systems Magazine, 25(2), 52–66. Available: http://aer.ual.es/siso-gpcit/. Guzmán, J.L., Rivera, D.E., Berenguel, M., and Dormido, S. (2012a). An Interactive Tool for Integrated System Identification and PID Control. In Proceedings of the IFAC Conference in PID Control, PID’12, Brescia, Italy, http://aer.ual.es/i--pidtune/. Guzmán, J.L., Rivera, D.E., Berenguel, M., and Dormido, S. (2014). ITCLI: An interactive tool for closed-loop identification. In 19th World Congress of the Federation of Automatic Control, 12249–12254. Cape Town, South Africa. Guzmán, J.L., Rivera, D.E., Dormido, S., and Berenguel, M. (2009). ITSIE: An interactive software tool for system identification education. In 15th IFAC Symposium on System Identification, http://aer.ual.es/ITSIE/. St. Malo, France. Guzmán, J.L., Rivera, D., Dormido, S., and Berenguel, M. (2012b). An interactive software tool for system identification. Advances in Engineering Software, 45(1), 115–123. Ljung, L. (1999). System Identification: Theory for the User. Prentice-Hall, New Jersey, 2nd edition. Piguet, Y. (2004). SysQuake 3 User Manual. Calerga S‘arl, Lausanne (Switzerland). Rivera, D., Morari, M., and Skogestad, S. (1986). Internal Model Control 4. PID Controller Design. Industrial & Engineering Chemistry Process Design and Development, 25, 252–265. Wellstead, P. (1977). Reference signals for closed-loop identification. International Journal of Control, 26, 945.



Fig. 3. Results for the second illustrative example explained in Section 4. Here the external excitation is introduced at the manipulated variable, with the closed-loop system needing to be detuned substantially to achieve identification with low estimation error. Despite this, precise estimation of the steady-state gain is difficult to achieve.

Fig. 4. Results for the third illustrative example explained in Section 4. If only excitation from disturbances is considered, it is possible in closed-loop identification to fit to data “perfectly” with highly erroneous models. This motivates the need for a sensibly designed external input signal.

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