An interactive approach to template generation in QFT methodology

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1 An interactive approach to template generation in QFT methodology J. M. Díaz, S. Dormido, J. Aranda Dpt. de Informática y Automática. ETSI Informática. UNED. Juan del Rosal nº Madrid. Spain Phone: Fax: josema@dia.uned.es Abstract The calculation of templates associated with plant uncertainty at some frequencies is one of the first steps in the design of robust controllers using the Quantitative Feedbac Theory (QFT) methodology. If the QFT designer does not correctly calculate the templates, the design will be unnecessarily conservative, or erroneous. This paper describes the main features of Template Interactive Generator (TIG), a new free software tool which calculates the templates of plants with special parameter dependencies, such as interval plants; and plants with affine parametric uncertainty. The paper also includes two examples which illustrate the ease of use and the strong interactivity of TIG. Keywords: Quantitative feedbac theory (QFT); Robust control; Control Systems Computer-Aided Design (CSCAD) 1.- INTRODUCTION A template Γ(ω i ) is the representation in the Nichols diagram of the plant uncertainty at a given frequency ω i. Templates are used in different robust control techniques, lie Quantitative Feedbac Theory (QFT) [1]-[3]. In QFT methodology, the templates calculation is one of the first steps. Templates are used to obtain the bounds associated with design specifications. In the loop-shaping stage, with these bounds it is possible to now where the open-loop transfer function must be placed in the Nichols diagram in order to fulfil the specifications. It is very important to obtain a good approximation of the templates in order to only add the necessary amount of conservatism in the controller designed. Typically, QFT designers wor with a finite set of template points, i.e., a discrete approximation of the template is used. The template boundary points are the most important, because they are the only ones used to calculate the bounds associated with the design specifications. Internal points only produce an unnecessary increase in the calculation time of the bounds associated with the specifications. It is Revised version (December 13 rd 26) J. M. Díaz, S. Dormido, J. Aranda 1

2 important to bear in mind that during a QFT design these bounds may have to be computed several times, because if the initial specifications are too demanding, there will be no solution to the problem considered and the specification will have to be relaxed. Therefore, the waste of computational time may be considerable. In general, QFT designers wor with all the template points, and not only with the template boundary, in spite of the waste of computational time. Besides, they usually use the grid method. This is a brute-force approach, which consists of selecting values inside each parameter uncertainty range, and evaluating the plant for these values at the desired frequency. While the procedure is direct and simple, it has several disadvantages [4]. Fortunately, in the last fifteen years other methods or algorithms for calculating templates have been developed. Each one uses different strategies: geometric considerations [4-5], Kharitonov segments [6], factorisation in simple elements and convolution [7], symbolic calculation [8], and interval analysis [9]. The most used and nown software tool to assist the designer in implementing the QFT methodology is the Matlab QFT toolbox [1]. However, this toolbox does not implement any algorithms to compute templates. This paper presents the software tool Template Interactive Generator (TIG) ( which assists the QFT designer in calculating template boundaries of interval plants, and plants with affine parametric uncertainty in their coefficients. These inds of plants are very usual in control problems. Besides, TIG can also find the template boundaries associated with other inds of plants whose templates have been previously computed in Matlab. The paper is organised as follows: Section 2 is devoted to describe the interactive tool TIG. Section 3 includes two examples to illustrate the use of TIG. Finally, conclusions are given in section TIG DESCRIPTION 2.1 Purpose of the interactive tool Let the following uncertain parameter vector be [ p p ] p =,..., (1) 1, 2 p L where p j min max [ p, ] j=1,2,..,l. With TIG it is possible to calculate the template boundaries of plants j p j whose transfer functions have the following structure: Revised version (December 13 rd 26) J. M. Díaz, S. Dormido, J. Aranda 2

3 m b ( p) s b( s; p) = P( s; p) = = (2) n a( s; p) a ( p) s = where the coefficients b (p) and a (p) are linear combinations of the uncertain parameters, i.e., L + j j= 1 b ( p) = β β p (3) L + j j= 1 j a ( p) = α α p (4) j In the previous expressions β j and α j j=,...,l are real constants. These inds of plants are nown as plants with affine parametric uncertainty. A particular case of (2) is obtained when the numerator uncertain parameters are independent of the denominator uncertain parameters: m b ( q) s b( s; q) = P( s; q, r) = = (5) n a( s; r) a ( r) s = where q h min max [ q, ] h=1,2,..,l n and r i min max [ r, ] i=1,2,..,l d, with L n +L d =L. The coefficients b (q) h q h i r i and a (r) also present an affine structure similar to (3) and (4). Another particular case of (2) is obtained when the plant transfer function coefficients are directly the uncertain parameters, such inds of plants are nown as interval plants: m q s b( s; q) = P( s; q, r) = = (6) n a( s; r) r s = min max min max where q [ q, ] and r [ r, ]. q r Revised version (December 13 rd 26) J. M. Díaz, S. Dormido, J. Aranda 3

4 For the plants described by (2), (5) and (6), TIG includes four algorithms to calculate the associated templates. These algorithms are the following: a) the Bailey & Hui algorithm [4], b) the Fu algorithm [5], c) the Kharitonov segment algorithm [6], and d) the grid algorithm. The Bailey & Hui algorithm directly generates template boundary points for plant types (5) and (6). If there is any dependency between the numerator uncertain parameters and the denominator uncertain parameters, then this algorithm generates a conservative template boundary. The number of template boundary points N c obtained with this algorithm can be determined using the expression N = 2 (7) c N b where N b is an algorithm parameter associated with the number of upper (or lower) template boundary points. The Fu algorithm considers L 2 1 L template edges (L is the number of uncertain plant parameters). A template edge can be external or internal. A template external edge forms part of the template boundary. Liewise, a template internal edge is contained inside the template boundary. Moreover, this algorithm generates N f points for each edge. N f is an algorithm parameter. Therefore, the number of generated template points is L N t 2 1 L = N (8) f The Kharitonov segment algorithm can be used for plant type (6). This algorithm considers 32 template edges (the Kharitonov segments), and it generates N points for each one. N is an algorithm parameter. Therefore, the total number of generated points is N = 32 (9) t N The grid algorithm generates N t points of a template, according to the following expression: N t = L j= 1 N p j (1) where min max N is the number of equispaced values for the uncertain parameters p j [ p, ] j=1,...,l p j j p j which are considered for obtaining the template points. The N are algorithm parameters. p j Revised version (December 13 rd 26) J. M. Díaz, S. Dormido, J. Aranda 4

5 The Bailey & Hui algorithm, Fu algorithm and Kharitonov segment algorithm can only be used if a(jω i ; p), for any combination of the uncertain parameter values. The plant template Γ(ω i ) is not bounded if a(jω i ; p) =. Besides, if the hyper-rectangle formed from q or r contains the origin of the complex plane, then the Bailey & Hui algorithm cannot be used. Moreover, the Bailey & Hui algorithm is the only one that directly generates the template boundaries. With the other two algorithms an additional algorithm is required to find the template boundaries. TIG uses the ε-algorithm proposed by F.M. Montoya [11]. It basically wors in the following way: it starts from a template boundary point, typically the template point with a greater real part. It traces a circle with radius ε, and it loos for the template point with the minimum phase contained inside the circle. This point also belongs to the template boundary. This procedure is repeated till all of the template boundary points are located. The ε-algorithm implemented in TIG has three parameters which can be tuned by users: Weight, Factor X and Factor Y. Weight is a positive real number that multiplies the diagonal D of the rectangle which encloses all the template points. The radius ε is related to the parameter Weight in the following expression: ε = Weight D (11) Moreover, Factor X and Factor Y are real positive numbers which divide the coordinate x or the coordinate y of all template points. 2.2 Main features The main features of TIG are its ease of use and strong interactivity. These are common characteristics of all the software tools developed with the program SysQuae [12]. All that users have to do is to place the mouse pointer over the different items which the tool displays on the screen. Any action carried out on the screen is immediately reflected on all the graphs generated and displayed by the tool. This allows users to visually perceive the effects of their actions. Thus, users do not have to write any script or command, unlie with Matlab. Some examples of TIG interactivity are: The wor template Γ w (ω w ) in the Nichols chart (or on the complex plane) is simultaneously modified if users change the configuration parameter of the template computation algorithm (N b for the Bailey & Hui algorithm, N f for the Fu algorithm or N for the Kharitonov segment algorithm) or the wor frequency ω. The boundary of the wor template Γ w (ω w ) in the Nichols chart (or on the complex plane) is simultaneously modified if users change the configuration parameter (Weight, Factor X or Factor Y) of the ε-algorithm used for computing the boundary template. The simultaneous visualization of the wor templates computed for each of the four template computation algorithms implemented in TIG. Revised version (December 13 rd 26) J. M. Díaz, S. Dormido, J. Aranda 5

6 The automatic selection and removing of all Γ w (ω w ) internal points. The manual selection and removing of any Γ w (ω w ) point. Another TIG interesting feature is its capability of cooperatively woring with other software tools, lie Matlab and Quantitative Feedbac Theory Interactive Tool (QFTIT) [13]. For example, TIG is able to import a template with interior points from Matlab, obtain its boundary, and export it to Matlab. Finally, TIG is a free distribution executable file available for Windows and Mac platforms. 2.3 Limitations The main limitation of this software tool is that its interactivity decreases if the computational load is heavy. In this case, there is a delay since users mae a change until it is represented in all the figures on the TIG window. TIG considers that the computational load begins to be heavy if the following relation is verified: T + T >. s (12) t c 6 where T t is the time (in seconds) that the wor template computation algorithm taes in computing Γ w (ω w ), and T c is the time (in seconds) that the ε-algorithm taes in finding the Γ w (ω w ) boundary. These times are function of the algorithm parameters and plant structure. In order to avoid a very great delay, the maximum value of the algorithm parameters and the feasible plant structure are limited. For example, in the present version of TIG, the limitations associated to the plant structure are: The maximum value of the plant denominator grade is n=7. Therefore, the maximum value of the plant numerator will be m=6. The maximum number of plant integrators or derivators is N ID =1. The maximum number of plant uncertain parameters is L=14. The maximum number of different uncertain parameters in the plant denominator is L d =7. The plant numerator has the same restriction. Remember that (L=L n +L d ). A complete enumeration of all the TIG limitations can be found in the User s Guide available at Revised version (December 13 rd 26) J. M. Díaz, S. Dormido, J. Aranda 6

7 2.4 Wor modes With TIG it is possible to wor in two different modes: configuration and analysis. The configuration mode consists of four sequential steps: Step 1. Configuration of the plant uncertain parameters. Step 2. Configuration of the plant transfer function denominator. Step 3. Configuration of the plant transfer function numerator. Step 4. Configuration of the wor frequencies set Ω. Furthermore, in the analysis mode, users can do, among other things, the actions previously described in section 2.2. In the following subsections, the TIG window items are briefly described for each wor mode. Readers who are interested can find a complete description in the User s Guide. The analysis mode is described first because the tool always starts in this mode on a predefined example Items available in the TIG window in the Analysis mode The main window of the tool in the Analysis mode is divided into several zones (see Figure 1), and each zone contains different items and figures. There are six zones on the left side of the window (Wor Mode, Status, Plant Uncertain Parameters, Analysis, Algorithm Information and Plant Transfer Function), and three zones on the right side of the window (Saved Templates, Frequencies and Wor Template). With the Wor Mode zone it is possible to select the wor mode. Users cannot force the transition from the configuration mode to the analysis mode. This transition is automatically done when they complete the last step of the configuration mode. However, users can go to the configuration mode when they wish. The Saved Templates zone contains a Nichols chart where the saved templates Γ(ω i ) ω i Ω are represented in different colours. A saved template is a valid template which can be exported to Matlab or QFTIT. Liewise, the template points are represented with o and their nominal value with x. Initially, the templates shown in this zone are computed by the Bailey & Hui algorithm (with N b =25), when the transition from the configuration mode to the analysis mode taes place. Users can save other templates in this zone throughout the wor session, clicing on the small red triangles located at both sides. With TIG it is only possible to save one template for each frequency ω i. Revised version (December 13 rd 26) J. M. Díaz, S. Dormido, J. Aranda 7

8 Figure 1. Example of the TIG window in the Analysis mode In the Frequencies zone, users can visualize and configure the wor frequencies set Ω, and select the wor frequency ω w. The zone contains a horizontal axis which is graduated in radians per seconds using a logarithmic scale. On this axis, a vertical segment for each frequency ω i Ω is drawn in different colours. The colour code is the same used in the Saved Templates zone. Besides, users can add and remove frequencies ω i Ω. The Wor Template zone contains a Nichols chart (or a complex plane) where the current wor template Γ w (ω w ) is drawn. Different colours are used to distinguish the algorithm which is being used to compute the templates. The Γ w (ω w ) points are represented in this zone with the symbol o. The Γ w (ω w ) boundary points are connected using a solid line. Other items located in this zone are the text field w and the button Save. With the text field it is possible to visualise and configure the current value of ω w. The button is used to save Γ w (ω w ) in the Saved Templates zone, i.e., if users clic on it, then Γ(ω w )= Γ w (ω w ). The Status zone contains three circular indicators: the upper one is associated with the Γ w (ω w ) calculation, the central one is associated with the Γ w (ω w ) boundary calculation, and the lower one is associated with the computational load state. Each indicator can tae three different colours (red, yellow or green) to warn users about certain events. Revised version (December 13 rd 26) J. M. Díaz, S. Dormido, J. Aranda 8

9 In the Plant Uncertain Parameter zone, users can visualise and configure the maximum, nominal and minimum values of the plant uncertain parameters. The Analysis zone allows users to do several actions: select the Γ w (ω w ) computation algorithm, configure the parameters of the selected algorithm (N b, N f or N ) and the parameters of the ε-algorithm (Weight, Factor X, and Factor Y), select and remove Γ w (ω w ) points, and correct the Γ w (ω w ) phase jumps. Finally, the Algorithm Information and Plant Transfer Function zones have merely an informative function. The Algorithm Information zone shows the following data: N t (number of Γ w (ω w ) points), N c (number of Γ w (ω w ) boundary points), R (algorithm yield (N c /N t )), T t (Γ w (ω w ) calculation time) and T c (Γ w (ω w ) boundary calculation time). Furthermore, the Plant Transfer Function zone shows the symbolic expression of the plant transfer function Items available in the TIG window in the Configuration mode In this wor mode, the TIG window s appearance changes. Users must wor principally in the Configuration Steps zone to configure the number of plant uncertain parameters, their values (minimum, maximum and nominal), and the wor frequencies set Ω. The upper part of this zone contains four text lines associated with the four sequential steps of this mode. The current step is always highlighted. In this wor mode, other useful zones are: Configure Plant Uncertain Parameters, Table of Plant Uncertain Parameters and Plant Transfer Function. The first zone offers an alternative way of configuring the plant uncertain parameters. The other two zones have an informative function. Thus, the Table of Plant Uncertain Parameters zone shows a table which contains the maximum, nominal and minimum values of the plant uncertain parameters. The Plant Transfer Function zone shows, apart from the denominator and numerator expressions, the field N ID with which it is possible to configure the number of plant integrators or derivators. 3. ILLUSTRATIVE EXAMPLES 3.1. Interval plant Let us assume a plant P( s, p) s + p s + p = (13) 3 2 s + p1 s + p2 s + p3 containing 5 uncertain parameters Revised version (December 13 rd 26) J. M. Díaz, S. Dormido, J. Aranda 9

10 p 1 [2.3, 19], p 2 [13.51, 147], p 3 [11.25, 49], p 4 [7, 9.5], p 5 [8.25, 9.5] (14) The nominal values are p = 1, p = 1, p = 25, p = 8, p 9 (15) = In order to eep the description as brief as possible, it is going to be assumed that the wor frequencies set consists of only one element: Ω={3} rad/s. This example is going to illustrate how it is possible to calculate Γ(3) and obtain its boundary using TIG. First of all, users have to go to the configuration mode and sequentially introduce step by step all the information which TIG needs to be able to wor: maximum, nominal and minimum values of the plant uncertain parameters, the numerator and denominator structures, and the wor frequencies set Ω. 1 5 Magnitude (db) Phase (º) Figure 2. Γ w (3) ( o and broen line) computed by the Bailey & Hui algorithm with N b =25, and Γ(3) ( * and solid line) computed by the same algorithm with N b =5 Once the configuration has finished, TIG automatically computes Γ(3) using by default the Bailey & Hui algorithm with N b =25, and goes to the Analysis mode. Since the numerator uncertain parameters are independent of the denominator uncertain parameters, another algorithm does not have to be used [16]. Remember that for this case, the exact template boundary may be directly calculated. The Wor Template zone shows Γ w (3) consisting of 5 points. It is advisable to study whether it is necessary to increase N b to improve the Γ w (3) boundary, or whether it is possible to decrease N b without Revised version (December 13 rd 26) J. M. Díaz, S. Dormido, J. Aranda 1

11 significantly changing the boundary shape. Users simply must drag the slider N b in the Analysis zone to do this study. It can be observed in the Wor Template that if users increase N b, the Γ w (3) boundary will have more points. However, the boundary shape will not significantly change. A good value is N b =5 to define better the boundary ends (see Figure 2). The wor template obtained with this value must be saved in the Saved Templates zone to export it later to Matlab or QFTIT. For this, users have to clic the Save button Plant with affine parametric uncertainty Let the plant with affine parametric uncertainty originally proposed in [5] be P( s, p) = s 3 2 s + (.4 p1 +.2 p2 + 4) s + ( p1 p3 + 2) 2 + (.5 p -.5 p +.5 p + 9.5) s + (2. p + p + 27) s - p + p (16) It has three uncertain parameters: { p, p } [ 3, 3] p (17) 1, 2 3 whose nominal values are p = 1.5, p = 1.5, p 1.5 (18) = In order to eep the description as brief as possible, it is going to be assumed that the wor frequencies set consists of only one element: Ω={1} rad/s. This example is going to illustrate how it is possible to calculate Γ(1) and obtain its boundary using TIG. Lie the previous example, first of all, the information associated with the plant must be introduced in TIG in the Configuration mode. Once the configuration has finished, TIG automatically computes Γ(1) using by default the Bailey & Hui algorithm with N b =25, and goes to the Analysis mode. The plant transfer function (16) has the same uncertain parameters in the numerator and the denominator. In this case, the Bailey & Hui algorithm generates conservative template boundaries. Therefore, another algorithm must be used. The best selection in this case is the Fu algorithm. Thus, this algorithm must be selected as the wor algorithm in the Analysis zone. Simultaneously, TIG calculates (using the Fu Revised version (December 13 rd 26) J. M. Díaz, S. Dormido, J. Aranda 11

12 algorithm with N f =5) and draws Γ w (1) in the Wor Template zone. However, TIG has not drawn the Γ w (1) boundary. This means that the ε-algorithm has not been able to find the boundary using the default values (Weight=1, Factor X=1, and Factor Y=1). Therefore, these parameters must be modified using the appropriate text fields or sliders. For example, if Weight=.37, a boundary of Γ w (1) is obtained. However, it is not correct because the Γ w (1) points placed at the lower right end do not belong to this boundary. Users can try to improve the boundary by decreasing Weight even more, but no improvement will be obtained. In fact, the boundary will disappear. Another option is to increase Factor X. Users can chec that a good boundary appears if Factor X is increased to 4 (see Figure 3). 1 2 Magnitude (db) Phase (º) Figure 3. Γ w (1) ( o ) calculated with the Fu algorithm, N f =5 and its boundary (solid line) found with the ε-algorithm, Weight=.37, Factor X=4 and Factor Y=1 Lie the previous example, users can analyse whether it is necessary to increase N f to improve the Γ w (1) boundary or whether it is possible to decrease N f without significantly changing the boundary shape. This study shows that N f =5 is a good value. Finally, before saving Γ w (1) in the Saved Templates zone, the non-boundary points must be removed. This action can be easily done in TIG, clicing on the Remove button located in the Analysis zone. 4. CONCLUSIONS The software tool TIG assists QFT designers in calculating template boundaries for interval plants, and plants with affine parametric uncertainty using different algorithms. Liewise, TIG can find template boundaries associated with other inds of plants if these templates have been previously calculated in Matlab. The main features of TIG are its ease of use and its strong interactivity. Any action carried out on the screen by users is immediately reflected on all the graphs generated and displayed by the tool. This Revised version (December 13 rd 26) J. M. Díaz, S. Dormido, J. Aranda 12

13 allows users to visually perceive the effects of their actions. Thus, QFT designers do not have to write any script or command, unlie with Matlab. The authors welcome any suggestions to improve the tool. ACKNOWLEDGMENTS This development was supported by CICYT of Spain under contracts DPI C4-1 and DPI Besides, we wish to express our sincere thans to Professor F. J. Montoya Dato for providing us the ε-algorithm code. REFERENCES [1] Horowitz, I. M., Survey of Quantitative Feedbac Theory (QFT), International Journal of Robust and Non-linear Control, Vol. 11, No. 1, pp (21). [2] Houpis, C. H., Rasmussen, S. J., and M. García-Sanz, Quantitative Feedbac Theory: fundamentals and applications, 2 nd Edition, CRC Taylor & Francis: Boca Ralen, (26). [3] Yaniv, O., Quantitative feedbac design of linear and nonlinear control systems, Kluwer Academic Publishers: Norwell, Massachusetts, (1999). [4] Bailey, F. N., and C. H. Hui, A fast algorithm for computing parametric rational functions, IEEE transactions on automatic control, Vol. 34, No. 11, pp (1989). [5] Fu, M., Computing the frequency response of linear systems with parametric perturbation, Systems & Control Letters, Vol. 15, pp (199). [6] Bartlett, A. C., Tesi, A., and A. Vicino,, Frequency response of uncertain systems with interval plants, IEEE transactions on automatic control, Vol. 38, No. 6, pp (1993). Revised version (December 13 rd 26) J. M. Díaz, S. Dormido, J. Aranda 13

14 [7] Gutman, P.O., Baril, C., and L. Neumann, An algorithm for computing value sets of uncertain transfer functions in factored real form. IEEE Transactions on Automatic Control, Vol. 29, No. 6, pp (1995). [8] Ballance D. J., and W. Chen, Symbolic computation in value sets of plants with uncertain parameters, Proc. UKACC International conference on control 98, pp (1998). [9] Nataraj, P. S. V., and G. Sardar, Template generation for continuous transfer functions using interval analysis. Automatica, Vol, 36, pp (2). [1] Borghesani C., Chait Y, and O. Yaniv, Quantitative Feedbac Theory Toolbox - for use with MATLAB. The MathWors Inc, Natic, MA, (1995). [11] Montoya, F. J., Diseño de sistemas de control no lineales mediante QFT: Análisis computacional y desarrollo de una herramienta CACSD. Ph.D. thesis. Universidad de Murcia (Spain), Departamento de Informática y Sistemas Informáticos, (1998). [12] Piguet, Y., Sysquae user manual, version 1., Calerga: Lausanne, Switzerland, (1999). [13] Díaz, J. M., Dormido, S., and J. Aranda, Interactive computer-aided control design using Quantitative Feedbac Theory: The problem of vertical movement stabilization on high-speed ferry, International Journal of Control, Vol. 78, No. 11, pp (25). Revised version (December 13 rd 26) J. M. Díaz, S. Dormido, J. Aranda 14