Quantitative Robust Control Engineering: Theory and Applications

Transcription

1 Quantitative Robust Control Enineerin: Theory and Applications Mario Garcia-Sanz Automatic Control and Computer Science Department Public University of Navarra. Campus Arrosadia 36 Pamplona SPAIN ABSTRACT This paper presents a summary of the main concepts and references of the Quantitative Feedback Theory QFT. It is a frequency domain enineerin method to desin robust controllers. It explicitly emphasises the use of feedback to simultaneously reduce the effects of model plant uncertainty and to satisfy performance specifications. QFT hihlihts the trade-off quantification amon the simplicity of the controller structure, the minimization of the cost of feedback, the existin model uncertainty and the achievement of the desired performance specifications at every frequency of interest. The technique has been successfully applied to control a wide variety of physical systems. After a brief introduction about the essential aspects of the QFT desin methodoloy, includin a wide set of QFT references, this paper presents a new method to extend the classical diaonal QFT controller desin method for MIMO plants with model uncertainty to a fully populated matrix controller desin method. The paper simultaneously studies three cases: the reference trackin, the external disturbance reection at plant input and the external disturbance reection at plant output. The work ends showin several real-world examples where the controllers have been desined usin QFT techniques: an industrial SCARA robot manipulator, a wastewater treatment plant, a variable speed wind turbine of.65 MW and an industrial furnace of MW.. INTRODUCTION Much of the current interest in frequency domain robust stability and robust performance dates from the oriinal works of H.W. Bode 945 [] and I. Horowitz 963 []. Since then, and durin the entire second half of the twentieth century, there has been a tremendous advance in the state-of-the-art of robust frequency domain methods. One of the main techniques, introduced by Prof. Isaac Horowitz in 959 [4], which characterises closed loop performance specifications aainst parametric plant uncertainty, mapped into open loop desin constraints, became known as Quantitative Feedback Theory QFT in the seventies [5-7]. This paper presents a summary of the main ideas and references of the QFT methodoloy. The method searches for a controller that uarantees the achievement of the desired performance specifications for every plant within the existin model uncertainty. QFT hihlihts the trade-off quantification amon the simplicity of the controller structure, the minimization of the cost of feedback bandwidth, the model uncertainty parametric and non-parametric and the achievement of the desired performance specifications at every frequency of interest. Followin this introduction, Section presents a brief description of the essential aspects of the QFT methodoloy. Section 3 introduces a method to desin non-diaonal QFT controllers for MIMO systems. Afterwards the paper describes some real-world applications of the technique, carried out by the author: an Garcia-Sanz, M. 6 Quantitative Robust Control Enineerin: Theory and Applications. In Achievin Successful Robust Interated Control System Desins for st Century Military Applications Part II pp Educational Notes RTO-EN-SCI-66, Paper. Neuilly-sur-Seine, France: RTO. Available from: RTO-EN-SCI-66 -

2 industrial SCARA robot manipulator in Section 4, a wastewater treatment plant of 5 m 3 /hour in Section 5, a variable speed wind turbine of.65 MW in Section 6 and an industrial furnace of 4 metres and MW in Section 7. The paper ends with a wide References Section that includes a representative collection of books and papers related with the theory and applications of QFT.. QUANTITATIVE FEEDBACK THEORY The Quantitative Feedback Theory QFT, first introduced by Prof. Isaac Horowitz in 959 [4], is an enineerin method, which explicitly emphasises the use of feedback to simultaneously reduce the effects of plant uncertainty and satisfy performance specifications. Horowitz s work is deeply rooted in classical frequency response analysis involvin Bode diarams, template manipulations and Nichols Charts NC. It relies on the observation that the feedback is needed principally when the plant presents model uncertainty or when there are uncertain disturbances actin on the plant. Frequency domain specifications and desired time-domain responses translated into frequency domain tolerances, lead to the so-called Horowitz-Sidi bounds or constraints. These bounds serve as a uide for shapin the nominal loop transfer function Ls Gs Ps, which involves the manipulation of ain, poles and zeros on the controller Gs. On the whole, the QFT main obective is to synthesize loop-shape a simple, low-order controller with minimum bandwidth, which satisfies the desired specifications and tackles feedback control problems with robust performance obectives. In the last few decades QFT has been successfully applied to many control problems. A wide collection of books and papers about the main aspects of the QFT methodoloy, theory and applications, is included in the references section: controller loop-shapin [4-44], existence conditions for controllers [45-47], multiinput multi-output MIMO systems [48-63], time-delay systems [64], diital QFT [65-66], distributed parameter systems [67-74], non-minimum phase systems [75-8], multi-loop systems [8-83], non-linear systems [84-9], linear time variant systems LTV [93-94], QFT software packaes [95-], real-world applications [3,7]. A detailed study about the history of QFT can be found in the papers written by Horowitz [9-], Houpis [] and Garcia-Sanz [3]. In 99, Houpis and Chandler oranized in Wriht-Patterson Dayton, Ohio the first International QFT Symposium []. Since then, and with the continuous support of Prof. Houpis, the Symposia have been oranized every two years: Indiana-USA-995 [], Glasow-UK-997 [], Durban - South Africa [3], Pamplona-Spain- [4], and Cape Town - South Africa -3 [5]. The next one will be in Kansas-USA-5. To o into the QFT theoretical aspects in depth, check the excellent tutorials written by Horowitz [6-7] and Houpis [8]. In addition, a maor analysis can be found in the books written by Horowitz [3], Houpis, Rasmussen and Garcia-Sanz [4], Yaniv [5] and Sidi [6]. Finally, three special issues of the International Journal of Robust and Nonlinear Control Wiley describe some of the more sinificant advances of QFT: Houpis 997 [7], Eitelber - [8] and Garcia-Sanz 3 [9]. The QFT desin methodoloy is quite transparent, allowin the desiner to see the necessary trade-offs to achieve the closed-loop system specifications. The basic steps of the procedure see also Fi. 4 are presented in the followin sub-sections. They are: Plant model with uncertainty, Templates eneration and nominal plant selection P o. Performance Specifications. QFT Bounds B. Loop-shapin the controller G. Pre-filter synthesis F. Simulation and Desin Validation. - RTO-EN-SCI-66

3 . Plant Model and Templates Generation The plant dynamics to be controlled may be described by frequency response data, or by linear or nonlinear transfer functions with mixed parametric and non-parametric uncertainty models. It can be defined takin into account the parameter uncertainty of the process at every frequency of interest i, that is to say the plant uncertainty templates, so that IP i {P i, i Ω k }. The templates are sets of complex numbers representin the frequency response of the family of uncertain plants at a fixed frequency IP i, i.e. a template is a proection of the n-dimensional parameter space onto the Nichols Chart. Fi. represents the QFT-template of the plant P s exp τ s/ s s ζ with three parameters, two with uncertainty ζ., n [.7,.], τ [, ], at rad/s. For more information about the QFT templates see [8-34]. n n τ τ puntos Proection P Manitude db n 5 rad/s Fiure : Template of the plant at rad/s Phase derees. Performance specifications The standard two deree of freedom system which best exemplifies the feedback problem considered in QFT is shown in Fi.. It includes the set of uncertain plants, -IP i {P i, i Ω k }-, the loop controller -G- and the pre-filter -F-, both to be desin, and the sensor dynamics -H-. On the other hand, R, E, U, Y and N are vectors representin respectively: the reference input, the error sinal, the controller output, the plant output and the sensor noise input. W, D and D are the external disturbance inputs. From the structure we can define the Eqs. to 3, W D D R F Pre-filter E - G Controller U P Plant Y H Sensor N Fiure : Standard two-deree-of-freedom feedback structure RTO-EN-SCI-66-3

4 N PGH PGH F R W PGH PG D PGH P D PGH Y D P D N PGH GH F R W PGH G U N PGH H F R PGH W PGH PGH D PGH PH D PGH H E 3 To achieve reliability and robustness, QFT deals with robust stability marins and robust performance specifications disturbance reection, reference trackin, etc as obectives in terms of the transfer functions of Eqs. to 3 over the frequencies of interest Table I..3 QFT-bounds For a nominal plant P, member of the family of plants within the uncertainty IP, the QFT methodoloy converts closed-loop system specifications and model plant uncertainty in a set of constrains or bounds Horowitz-Sidi Bounds for every frequency of interest that will have to be fulfilled by the nominal open-loop transfer function. They are represented on a Nichols chart. Such a reat interation of information in a set of simple curves the bounds will allow desinin the controller usin only a sinle plant, the nominal plant P. The i plant template, IP i {P i }, is approximated by a finite set of boundary plants {P r i, r,,m}. Each plant can be expressed in its polar form as P r i p i e θ i p θ, and likewise the Table I. Transfer functions and specification models H No.Eq. Ω δ, G P G P N Y D U F R Y T 4 Ω δ, G P D Y T Ω δ, G P P D Y T Ω δ, G P G F R U N U D U T 7 5 5sup 5 5inf Ω δ < δ, G P G P F R Y T Ω δ δ δ, P G P G P G P G inf sup d e e d 9-4 RTO-EN-SCI-66

5 controller polar form is G i i e φ φ. The controller phase φ varies from -π to. Therefore, for every frequency i, the feedback specifications { T k i δ k i, k,,5} in Table I Eqs. 4 to 9- are translated into the quadratic inequalities in Table II Eqs. to 4-, see [6]. k Table II. Quadratic inequality on Eq. δ p p cos φ θ δ p p cos φ θ 3 p p cos δ 4 p φ θ 4 p p cos φ θ 5 p p e d pe p d pe cos δ5 p δ5 δ 3 p δ 5 φ θ d cos φ θ p d d e e 3 4 The format of these quadratic expressions is: k i I p, θ, δk, φ a b c 5 Chait and Yaniv [35] developed an alorithm to compute the bounds based on quadratic inequalities see Table II, simplifyin much of the work on traditional manual bound computation. Takin these inequalities into account, it is possible to compute them at the NC. Once the bounds have been calculated for the performance specifications, they have to be rouped into a sinle variable. Then, the worst case bound, i.e. the most restrictive one for every phase, is computed for each frequency of the work array see Fiure 3. For more information about the QFT bounds see [35-4]..4 Controller desin In the desin stae loop-shapin, the controller Gs is synthesized on the NC by addin poles and zeros until the nominal loop, defined as L P G, lies near its bounds. Loop-shapin considers bounds on the NC to express the plant model with uncertainty and the performance specifications at every frequency. An optimal controller will be obtained if it meets its bounds over the continuous lines and under the dashed lines at every frequency and it has the minimum hih frequency ain see Fi.3. Althouh current CAD tools for QFT controller desin are very helpful [95-], the loop-shapin step must be still done manually usin desiner skills and experience. Even keepin the controller structure fixed, automatic tunin of parameters represents a reat challene [43-44]. RTO-EN-SCI-66-5

6 Open-Loop Gain L Open-Loop Phase Fiure 3: Loop shapin The eneral formulation for the controller structure is expressed by the followin transfer function: nrz s ncz / s Re z Π i Π s i zi i zi zi G s k G m rp m / s cp r s Re p s Π Π s p p p 6 where, k G is the ain, z i is a zero that may be complex n cz, number of complex zeros or real n rz, number of real zeros, and p is a pole real or complex with m rp the number of real poles and m cp the number of complex poles. Note that the amount of complex zeros or poles must be even, to have pairs of complex conuate numbers and obtain a polynomial with real coefficients. Controller may have some poles in the oriin and desiner can check the parameter r usually, or to set them. For more information about QFT loop-shapin see [4-44], and about existence conditions for controllers see [45-47]..5 Pre-filter synthesis If the feedback system involves trackin sinals, then the best choice is to use a pre-filter F. While controller G reduces the uncertainty and deals with stability, disturbance reection, etc, pre-filter F is desined to fulfil trackin requirements [4]..6 Simulation and desin validation Once the controller desin is finished, it is necessary to analyse the behaviour of the system with the controller previously obtained. Closed-loop response at several frequencies and time domain responses must be checked. The analysis will be carried out with the most unfavourable cases due to uncertainty [4]. - 6 RTO-EN-SCI-66

7 Nonlinear models uncertainty Start Linearization Mathematical model Templates In the Frequency Domain In the Frequency Domain Control Specifications MATHEMATICAL MODEL REVISION Bounds Computation Does a Solution exist? No SPECIFICATIONS DISCUSSION Yes Loop-shapin Gs Readustment Verification In the frequency domain? Yes Readustment Pre-filter Desin Fs No Verification In the freq. & time domains? No Readustment No Yes Satisfactory Simulation? Yes Experimental Verification? Yes Industrial Implementation End Fiure 4: QFT methodoloy RTO-EN-SCI-66-7

8 3. QUANTITATIVE NON-DIAGONAL COMPENSATOR DESIGN FOR MIMO SYSTEMS [6],[6]. A fully populated matrix controller allows the desiner much more desin flexibility to overn MIMO processes than the classical diaonal controller structure. This section introduces a new methodoloy to extend the classical diaonal QFT controllers desin for MIMO plants with model uncertainty to a fully populated matrix controller desin. The section simultaneously studies three cases: the reference trackin, the external disturbance reection at plant input and the external disturbance reection at plant output. Therefore, the role played by the non-diaonal controller elements i i is analysed in order to state a fully populated matrix controller desin methodoloy for QFT. The definition of three couplin matrices c i, c i, c 3i and a quality function q i of the non-diaonal elements come in useful to quantify the amount of loop interaction and to desin the non-diaonal controllers respectively. This yields a criterion to propose a sequential desin methodoloy of the fully populated matrix controller, in the QFT robust control frame. As a consequence the diaonal elements kk of the new non-diaonal method need less bandwidth than the diaonal elements of the previous diaonal methods. The work ends showin a realworld example section 4, where an industrial SCARA robot manipulator is controlled usin the new nondiaonal MIMO QFT methodoloy. 3. Introduction Control of multivariable systems multiple-input-multiple-output, MIMO with model uncertainty are still one of the hardest problems that the control enineer has to face in real-world applications. Two of the main characteristics that define a MIMO system are the input and output directionality -different vectors to actuate U and to measure Y-; and the couplin amon control loops -some outputs y i can be influenced by several inputs u i, and some inputs u i can influence several outputs y i. In the last few decades a very sinificant amount of work in MIMO systems, too numerous to list, has been done. Useful techniques for desinin multivariable feedback systems have been compiled in excellent references written by Rosenbrock [8], O Reilly [9], Macieowski [3], Skoestad and Postlethwaite [3], Houpis, Rasmussen and Garcia-Sanz [4], Marlin [3], Leithead and O Reilly [33], etc. Some of them collect the frequency domain approach, firstly introduced and adopted by Rosenbrock and MacFarlane in the UK and by Horowitz in Israel, in the sixties, when most of the academic community rearded the frequency domain as obsolete for MIMO processes. On the other hand, the oriinal works in MIMO systems were only made for fixed plants, without any uncertainty in the model. The first technique that made a quantitative synthesis and took into account quantitative bounds on the plant uncertainty, and quantitative tolerances on the acceptable closed-loop system response, was introduced by Horowitz in the fifties [4] and subsequently reinforced and thouhtfully studied as it has been introduced in Section. That technique is the Quantitative Feedback Theory QFT. Usin MIMO QFT, Horowitz [48-5] proposed to translate the oriinal nxn MIMO problem into n separate quantitative multiple-input-sinle-output MISO problems, each with plant uncertainty, external disturbances and closed-loop tolerances derived from the oriinal problem. The first improvement of the oriinal MIMO QFT method was also introduced by Horowitz [5]. It obtains a sinificant reduction of over desin in comparison with the previous method. It considers, in the successive steps of the iterative method, an equivalent plant that takes also into account the controllers desined in the previous steps. The book by Houpis, Rasmussen and Garcia-Sanz [4] presents a detailed compilation of both methods. - 8 RTO-EN-SCI-66

9 However, althouh such oriinal MIMO QFT methods take the couplin amon loops into account, they only propose the use of a diaonal controller G to overn the MIMO plant. This structure can be improved usin non-diaonal controllers. In fact a fully populated matrix controller allows the desiner much more desin flexibility to control MIMO plants than the classical diaonal controller structure. The use of the non-diaonal components can also ease the diaonal controller desin problem. The difficulty is how to desin the off-diaonal elements, especially if one must consider enineerin factors such as cost-benefit trade-offs of usin cross-feeds, stron structure in the plant uncertainty, system interity and plant input sinal levels. In the last few years some new methods for non-diaonal multivariable QFT robust control system desin have been introduced. Aain Horowitz desined and applied a procedure for non-diaonal G controllers [5]. The idea was to insert a non-diaonal matrix pre-compensator H before the plant, to have a new effective plant P e P H. Later, Franchek and Nwokah presented a sequential loop frequency approach that utilizes a fully populated matrix controller to meet performance specifications, which may include system interity requirements [56, 58]. Boe utilized the Perron-Frobenius root interaction measure to desin a pre-compensator that reduces the level of couplin between loops, before a diaonal QFT controller matrix is attempted [59]. Yaniv introduced an approach that emphasizes the bandwidth of a non-diaonal pre-controller multiplied by the classical diaonal controller [57]. Kerr and Jayasuriya presented a nonsequential MIMO QFT methodoloy [63]. In this context, this section oes on with a previous work [6, 6] and introduces a new methodoloy, based on QFT, to extend the classical QFT diaonal controller desin for MIMO plants with uncertainty to the fully populated matrix controller desin. The work simultaneously studies three cases: the reference trackin, the external disturbance reection at plant input and the external disturbance reection at plant output. It presents the definition of three specific couplin matrices c i, c i, c 3i, one for each case. They come in useful to quantify the amount of the loop interaction of the system. Furthermore, a quality function q i of the non-diaonal elements i i for the three problems is utilized to aid the desin of the fully populated matrix controllers. Based on the above ideas, the work introduces a sequential desin methodoloy for non-diaonal QFT controllers. It contemplates the desin of quantitative controllers able to achieve reference trackin and disturbance reection specifications, takin also into account the reduction of interaction amon loops. The diaonal elements kk of the new non-diaonal method need less bandwidth than the diaonal elements of the previous diaonal methods. The work beins with two sub-sections that formulate the couplin matrices and the couplin elements of the control system. Then the fourth sub-section introduces the expressions of the optimum non-diaonal controller. Sub-section five analyses and compares the couplin effects amon loops of both, the classical diaonal method and the new non-diaonal methodoloy. Sub-section six introduces a quality function to quantify the amount of loop interaction and to desin the non-diaonal controllers. Sub-section seven presents the sequential procedure to desin the fully populated matrix controller. Sub-section eiht deals with some practical issues for usin the method. Section 4 applies the new MIMO QFT methodoloy to control a real-world problem: a SCARA robot manipulator [5]. 3. The Couplin Matrix The obective of this section is to define a measurement index the couplin matrix that allows one to quantify the loop interaction in MIMO control systems. Consider a nxn linear multivariable system -see Fi. 5-, composed of a plant P, a fully populated matrix controller G, a pre-filter F, a plant input disturbance transfer function P di, and a plant output disturbance transfer function P do, where P IP, IP is the set of possible plants due to uncertainty, and, RTO-EN-SCI-66-9

10 p p pn n f f fn p p pn n f f f n P ; G ; F 7 p p p f f f n n nn n The reference vector r and the external disturbance vectors at plant input d i and plant output d o are the inputs of the system. The output vector y is the variable to be controlled. n nn n n nn d i P di s d o P do s r r u y Fs Gs Ps - T Y/R s d i d o Fi. 5 Structure of a Deree of Freedom MIMO System It is denoted P as the plant inverse so that, p p P - P [ p i] Λ B 8 p p nn n n G G d G b 9 nn n where Λ is the diaonal part and B is the balance of P ; and G d is the diaonal part and G b is the balance of G. The next pararaphs introduce a measurement index to quantify the loop interaction in the three classical cases: reference trackin, external disturbances at plant input, and external disturbances at plant output. That index is called the couplin matrix and, dependin on the case, shows three different expressions: C, C, C 3 respectively. n 3.. Trackin The transfer function matrix of the controlled system for the reference trackin problem, without any external disturbance, can be written as shown in Eq., y I P G P G r T r T F r' y / r y / r - RTO-EN-SCI-66

11 Usin Eq. 8 and 9, Eq. can be rewritten as, T y/r r I Λ- G Λ- G r I Λ- G Λ- G r B G T r d d d b b y/r In the expression of the closed-loop transfer function matrix of Eq., it is possible to find two different terms: i. A diaonal term T y/r_d, I Λ G - d Λ Gd T - y/r_d that presents a diaonal structure. Note that it does not depend on the non-diaonal part of the plant inverse B nor on the non-diaonal part of the controller G b. It is equivalent to n reference trackin SISO systems formed by plants equal to the elements of Λ - when the n correspondin parts of a diaonal G d control them, as shown in Fi. 6a. ii. A non-diaonal term T y/r_b, T y/r_b I Λ - G Λ- [ G B G T ] I Λ- G Λ- y/r C 3 d b b d that presents a non-diaonal structure. It is equivalent to the same n previous systems with internal disturbances c at plant input Fi. 6b. i r r i - ii u i p ii y i a n c r i u i y i - b ii p ii Fi. 6 i-th equivalent SISO and MISO systems In Eq. 3, the matrix C is the only part that depends on the non-diaonal parts of both the plant inverse B and the controller G. Hence, it comprises the couplin, and from now on C will be the couplin b matrix of the equivalent system for reference trackin problems, C Gb B Gb Ty/r 4 RTO-EN-SCI-66 -

12 Each element c i of this matrix obeys, c i i n i ik ik k ik 5 k δ p t δ where δ ki is the delta of Kronecker that is defined as, δki k i δ ki 6 δ k i ki 3.. Disturbance reection at plant input The transfer matrix from the external disturbance at plant input in Eq. 7, d ' i to the output y can be written as shown y I P G P d T d T P d ' i y / di i y / di di i 7 and then, T y/di d i I Λ Gd Λ di I Λ Gd Λ B Gb Ty /di di 8 In that expression -Eq. 8- it is possible to find two different terms: i. A diaonal term T y/di_d, y/di_d - - I Λ Λ T G 9 d Aain, Eq. 9 is equivalent to n reulator MISO systems, as shown in Fi. 7a. ii. Non diaonal term T y/di_b T y/di_b I Λ G Λ B Gb Ty/di I Λ G Λ C 3 d that presents a non-diaonal structure which is equivalent to the same n previous systems with external disturbances c di at plant input, as shown in Fi. 7b. i In Eq. 3, the matrix C comprises the couplin, and from now on C will be the couplin matrix of the equivalent system for external disturbance reection at plant input problems, C B G b Ty/di 3 d - RTO-EN-SCI-66

13 di i y i - ii u i p ii a n c di i u i y i ii - p ii b Fi. 7 i-th equivalent MISO systems Each element c i of this matrix obeys, n c i pik ik tk δ ik 3 k where δ ki is the delta of Kronecker defined in Equation Disturbance reection at plant output The transfer matrix from the external disturbance at plant output shown in Eq. 33, y I P G d T d T P d' o y / do o y / do do o d ' o to the output y can be written as 33 and then, T y/do d o I Λ- G d - o I Λ G Λ- B B Gb Ty/do do d d 34 In that expression -Eq. 34- it is possible to find two different terms: i. A diaonal term T y/do_d, I Λ G T - y/do_d d 35 Once more, Eq. 35 is equivalent to the n reulator MISO systems showed in Fi. 8a, RTO-EN-SCI-66-3

14 ii. Non diaonal term T y/do_b T y/do_b I Λ G Λ [ B B G Ty/do ] I Λ G Λ C3 d b d 36 that presents a non-diaonal structure. It is equivalent to the same n previous systems with external disturbances c 3i do at plant input, as shown Fi. 8b. do i y i - ii u i p ii a n c 3 do i u i y i ii - p ii b Fi. 8 i-th equivalent MISO systems In Eq. 36, the matrix C 3 comprises the couplin, and from now on it will be the couplin matrix of the equivalent system for external disturbance reection at plant output problems, B Gb Ty/do C3 B 37 Each element of the couplin matrix, c 3i obeys, c 3i n pi δ i pik ik tk δ ik k 38 where δ ki is the delta of Kronecker as defined in Equation The Couplin Elements In order to desin a MIMO controller with a low couplin level, it is necessary to study the influence of every non-diaonal element i on the couplin elements c i, c i and c 3i, defined in Eq. 5, 3 and 38. These elements can be simplified to quantify the couplin effects. Then it will be possible to analyse the loop decouplin and to state some conditions and limitations usin fully populated matrix controllers. - 4 RTO-EN-SCI-66

15 To analyse the couplin elements, one Hypothesis is stated. Hypothesis H: suppose that in Eq. 5, 3 and 38, i i t >> pik ik tk, for k, and in the bandwidth of t p 39 Note that the above expression is scale invariant and is typically fulfilled once the MIMO system has been ordered accordin to appropriate methods like the Relative Gain Analysis [34], etc. Then the diaonal elements t will be much larer that the non-diaonal ones t k, t t k, for k, and in the bandwidth of t >> 4 Now, two simplifications are applied to facilitate the quantification of the couplin effects c i, c i, c 3i. Simplification S: Usin the Hypothesis H, Eqs. 5, 3 and 38, which describe the couplin elements in the trackin problem, disturbance reection at plant input and disturbance reection at plant output respectively, are rewritten as shown Table III. Simplification S: The elements t are computed for each case from the equivalent system derived from Eqs., 9 and 35. The results are shown in Table III. Table III: Simplifications to quantify the couplin effects Reference trackin External disturbances at plant input External disturbances at plant output Simplification S c i t i p ; i i i 4 c i p ; i t i i 4 c 3i p t i p ; i i i 43 Simplification p t 44 S p p p t 45 t 46 p Due to Simplifications S and S, the couplin effects c i, c i, c 3i can be computed as, Trackin pi i p c ; i 47 i i Disturbance reection at plant input p i i p c ; i 48 i Disturbance reection at plant output RTO-EN-SCI-66-5

16 pi i p p c p ; i 49 3i i 3.4 The Optimum Non-diaonal Controller Consider non-diaonal controllers to reduce the couplin effect, as well as diaonal controllers that help to achieve the loop performance specifications. The optimum non-diaonal controllers for the three cases trackin and disturbance reection at plant input and output can be obtained makin the loop interaction of Eqs. 47, 48 and 49 equal to zero. Note that both elements, plant p i and p i can be any plant represented by the family, p, of these equations are uncertain elements of P. Every uncertain N { p } p, p, for i,,, n i i i i i 5 where N p i is the nominal plant, and p i the maximum of the non-parametric uncertainty radii i. N The nominal plants p i and follow the next rules: N p that will be chosen for the optimum non-diaonal controller will a If the uncertain parameters of the plants show a uniform Probability Distribution Fi. 9a which is typical in the QFT methodoloy-, then the elements p i and p for the optimum non-diaonal N controller will be the nominal plants N p i and p, which minimise the maximum of the nonparametric uncertainty radii p i and p that comprise the plant templates Fi. 9b. b If the uncertain parameters of the plants show a non-uniform Probability Distribution Fi. 9c, then N the elements p i and p for the optimum non-diaonal controller will be the nominal plants p i N and p, whose set of parameters maximize the area of the Probability Distribution in the reions [ a ε, a ε ] and [ a ε, a ε ] parameter a i, b i,, a, b respectively. i i These rules of selection will be analysed aain in Section 4.5, when we compute the couplin effects with the optimum non-diaonal controller. Now, makin Eqs. 47, 48 and 49 equal to zero and usin Eq. 5, the optimum non-diaonal controller for each case is obtained. - 6 RTO-EN-SCI-66

17 Uniform PD Parameter a i a a i_min a i_max Ima i p i p i i N p i N p i b Re Non-uniform PD Parameter a i A max c N a a N i i -ε a N i Fi. 9 Probability Distribution of the parameter a i, and Non-parametric uncertainty radii p i that comprise the plant templates 3.4. Trackin opt i p N i Fpd, for i 5 N p 3.4. Disturbance reection at plant input opt i F pd p i, for i 5 N RTO-EN-SCI-66-7

18 3.4.3 Disturbance reection at plant output N opt pi i Fpd for i N, 53 p where the function F pd A means in every case a proper function made from the dominant poles and zeros of the expression A. 3.5 The Couplin Effects The minimum achievable couplin effects -Eqs. 54, 56, 58- can be computed substitutin the optimum controller of Eqs. 5, 5 and 53 in the couplin expressions of Eqs. 47, 48 and 49 respectively, and takin into account the uncertainty radii of Eq. 5. Analoously, the maximum couplin effect without any non-diaonal controller -pure diaonal controller cases- can be computed substitutin i in the Eqs. 47, 48 and 49 respectively -Eqs. 55, 57, 59-. That is to say, 3.5. Trackin c i opt i i i i ψ c ψ i i i i Disturbance reection at plant input c ψ i opt i i i i i i i c ψ 57 i Disturbance reection at plant output c 3i opt i i 3i i ψ c ψ i i i i where, ψ i N pi 6 N p and the uncertainty is: pi, p, for i, i,, n - 8 RTO-EN-SCI-66

19 i The couplin effects, calculated in the pure diaonal controller cases, result in three expressions 55, 57 and 59 that still present a non-zero value when the nominal-actual plant mismatchin due to the uncertainty disappears: and. However, the couplin effects obtained with the optimum nondiaonal controllers -Eqs. 54, 56 and 58- tends to zero when that mismatchin disappears. 3.6 Quality of the Desined Controller Fi. shows the appearance of three different couplin bands for a common process. The maximum c i i -computed from Eqs. 55, 57 or 59- and the minimum couplin effects without any nondiaonal controller i limit the first one, as well as the second one is bounded by the maximum and the minimum couplin effects with a non-optimum decouplin element i. Finally, the minimum couplin effect c i opt opt ii with the optimum decouplin element i presents a maximum value, computed from Eqs. 54, 56 or 58. db Max of c i i Eqs. 55, 57, 59. Min of c i i Eqs. 55, 57, 59. rad/sec Max of c i ii opt Eqs. 54, 56, 58. Fi. Couplin effect bands with different non-diaonal controllers From those ideas a quality function q i is defined for a non-diaonal controller i i so that, max{ c i } i lo max{ c i } i i q i % 6 max{ c } i lo i max{ c } i opt i i The quality function becomes a proximity measure of the couplin effect c i to the minimum achievable couplin effect. Thus, the quality function is useful to quantify the amount of loop interaction and to desin the non-diaonal controllers. A suitable non-diaonal controller will maximise the quality function of Eq Desin Methodoloy The proposed controller desin method is a sequential procedure closin loops [58]. It uses a fully populated matrix controller that does not assin any special role to the upper and lower trianular elements of the controller G, and in addition, it can be used to desin the feedback controller of a DOF structure see Fi. 5. First it is necessary to fulfil the Hypothesis H and two new Hypotheses H and H3. RTO-EN-SCI-66-9

20 Hypothesis H: The plant P and its inverse P have to be stable and do not have any hidden unstable mode. Despite lookin very restrictive, note that it is only a sufficient condition to uarantee the stability of the system. Consequently, the desiner must pay close attention to systems with non-minimum phase or unstable elements [58, 6]. In the last few years, several works have studied the stability problem of MIMO systems with uncertainty, usin inverse plants in the QFT methodoloy. A deep analysis of the subect can be found in the next four references: [6, 6, 4, 5]. The second paper proofs that it is necessary and sufficient that the plant of each successive loop is stabilised. The third and fourth references expand the analysis. Hypothesis H3: The plant P is not 'ill-conditioned' for any of the possible plants in the whole set IP. This will uarantee the robustness of the desin. It is known [35] that 'ill-conditioned' plants, with lare elements of the RGA -Relative Gain Analysis [34]- matrix are difficult to control. Methodoloy Step A. First, the methodoloy beins parin inputs and outputs with the RGA technique [34], and arranin the matrix P so that p has the smallest phase marin frequency, p the next smallest phase marin frequency, and so on [4]. Later, the sequential controller desin technique Fi., composed of n staes -n loops-, will follow the next steps B and C for every column k to n. G k n k n k n k n k n k k kk nk n n kn nn Fi. Steps for controllers desin Step B. Desin the diaonal controller kk. This desin stae of kk is calculated throuh standard QFT e loop-shapin for the inverse of the equivalent plant p kk in order to achieve robust stability and robust performance specifications. The equivalent plant satisfies the next recursive relationship [58], e [ ] [ p ] [ ] [ i i ] [ pi i- ] [ ] i i- p k k i- i k [ pi- i- ] [ i- i- ] [ P ] P k k p ; i k; 6 k ii k ii k which is an extension for the non-diaonal case of the recursive expression proposed by Horowitz [5] as the Improved desin technique, also called Second method by Houpis et al. [4]. y/r If the control system requires trackin specifications as aii tii bii then, because y/r t t t -Eq.-, the trackin bounds b ii and a ii will have to be corrected with the couplin ii Step B and C a to C n Step B and C a to C n Step B n and C na to C nn rii cii k - RTO-EN-SCI-66

21 specification τ cii, so that: c c b ii b ii - τ cii, a ii a ii τ cii 63 t w c τ 64 cii ii ii cii c c ii rii ii a t b 65 These are the same corrections proposed by Horowitz [48, 5] and Houpis, Rasmussen and Garcia-Sanz [4] for the oriinal MIMO QFT methods and. However, with the new non-diaonal method these corrections will be less demandin. The couplin expression t cii w ii c ii is now minor than in the previous diaonal methods compare Eqs. 54 and 55-. The off-diaonal elements i i of the matrix controller will attenuate or cancel that cross couplin. Then the diaonal elements kk of the non-diaonal method will need less bandwidth than the diaonal elements of the previous diaonal methods. Step C. Desin the n- non-diaonal elements ik i k, i,,n of the k-th controller column, minimisin the couplin c ik -computed from Eqs. 47, 48 and 49-. To achieve this oal, the nominal optimum controller -Eqs. 5, 5 and 53- must be taken into account. Step D. Finally, the desin of the pre-filter F does not present any difficulty, if the complementary sensitivity function shows a low level of loop interaction. Therefore, the pre-filter F can be diaonal. 3.8 Some Practical Issues To use the above controllers there is a relevant condition to take into account: the plant P and its inverse P have to be stable and do not have any hidden unstable mode The H Hypothesis- [58]. This is only a sufficient condition. Hence, in some cases of not fulfillin this condition it is even possible to use a nondiaonal controller element like the above-mentioned [63]. In these situations some difficulties miht appear durin the sequential procedure such as the introduction of additional non-minimum phase nmp zeros due to hidden unstable modes of the plant or due to loop closures. This typically occurs in hihly coupled systems. e To avoid this problem, the elements of the resultin plants p kk must be checked in every step of the desin methodoloy to ensure that riht half plane transmission zeros or unstable modes have not been introduced by the new controller elements kk or ik, which would obviously cause an unnecessary loss of control performance [58]. If these nmp zeros appear due to the desined controller elements, supplementary constraints in the determinant of PG should be imposed to re-calculate the controller. On the other hand, if the plant elements, p kk or p ik, are the cause of the introduction of non-minimum phase e elements in the equivalent plant p kk, the theory proposed for nmp MISO feedback systems by Horowitz et al. [3, 75-79] and modified by Chen and Balance [8] can be applied to properly desin the controllers in the loop shapin step. At the same time, arbitrarily pickin the wron order of the loops to be desined can result in the nonexistence of a solution. This may occur if the solution process is based on satisfyin an upper limit of the phase marin frequency φ, for each loop. Hence, Loop i havin the smallest phase marin frequency will have to be chosen as the first loop to be desined. The loop that has the next smallest phase marin frequency will be next, and so on [4]. RTO-EN-SCI-66 -

22 e It is important to notice that the calculation of the equivalent plant p kk usually introduces some exact pole-zero cancellations. That operation can be precisely done usin symbolic mathematic tools, but could be erroneously done when usin numerical calculus due to the typical round errors. Finally, every step of the proposed methodoloy can be aided by existin QFT software packaes [95- ]. 4. NON-DIAGONAL MIMO QFT CONTROLLER FOR A SCARA ROBOT [6] The non-diaonal MIMO QFT controller desin technique is applied to control a real-world problem: a SCARA robot [5]. Fi. shows the AdeptOne robot manipulator, and the two oint anles δ and δ - that are considered in this example. In order to present the control of two oints of the SCARA robot, the plant model and the desired performance specifications are introduced in the next sections. δ 3 link link δ δ δ δ δ 4 δ 5 Fi. Adept One SCARA Robot 4. Plant Model The Larane equations' method is used to find Eqs. 66 and 67, which describe the non-linear dynamic behaviour of the two-link system [5]. The real inputs are the torques τ and τ -applied throuh power amplifier as u and u - commanded by electrical motors on oints and, and the outputs are anles δ and δ. [ α α cos δ ]&& δ [ α α cos δ ] && 3 3 δ ν δ & µ sn δ & τ u k 66 - RTO-EN-SCI-66

23 [ α α cos δ ] && 3 δ α && δ ν δ & µ sn δ & τ u k 67 where k is the power amplifiers' ain, ν i are coefficients of viscous friction, µ i Coulomb friction parameters associated with link i, and, α α α 3 I I m I m l m x x x m l x denotin I i, m i and x i as the moment of inertia, mass and position of the i-th link respectively, and l as the lenth of link. Input sinals u and u will be computed in counts [ct] and will be commanded to the robot motors by the amplifiers. After a robust parameter identification process [5], the coefficients of the robot model, with a uniform Probability Distribution, were found Table IV. Table IV: Coefficients of uncertain plant. Uniform Probability Distribution. 68 Minimum Maximum Nominal α k [ct s /rd] α k [ct s /rd] α 3 k [ct s /rd] ν k [ct s/rd] ν k [ct s/rd] µ k [ct] µ k [ct] Now it is possible to consider the Coulomb frictions as disturbances and the cosine value of δ as an uncertain parameter h between - and. Takin into account Eqs. 66 and 67, it is easy to find the followin linear transfer functions, which are the elements of the plant P defined as, δ u p p u P δ u p p u 69 α s ν p s s s k 7 - α α3 h p s s k 7 - α α3 h p s s k 7 α α 3 h s ν p s s s 73 k RTO-EN-SCI-66-3

24 where, η s η s η 74 with the followin coefficients, η η η α α ν α α3 h α α3 h ν ν α α h ν Performance Specifications The desired performance specifications for the SCARA robot manipulator are the followin, i. Robust Stability. t ii., for i,, involves a lower phase marin of at least 5º and a lower ain marin of at least db. ii. Control effort constraint. Control sinals have to be lower than 3767 [ct] for disturbance reection at plant output of about º. iii. Disturbance reection at plant input. The maximum allowed error has to be of 3º for torque disturbances of [ct]. iv. Loop Couplin. Reduction of couplin effect as much as possible. v. Trackin specifications. T has to achieve trackin tolerances defined by, y/r y/r a t b for i, 76 ii ii ii where,.5 [ /3 ] b ii a ii [ / ] [ ] The above specifications are limited by the achieved samplin time for the practical implementation, which is actually ms. 4.3 Controller Desin Step A: Couplin analysis and pairin The first step is the RGA. This analysis yields a very obvious result: anle δ will be controlled by motor u, and anle δ by motor u. The first element λ plotted in Fi. 3 also shows that the robot arm presents a very coupled behaviour. At low frequencies -below.6 rad/s- the coupled behaviour is very low, but as far as the frequency increases the system presents a more coupled dynamics. The required bandwidth of the system derived from trackin specifications lies between approximately and 3.5 rad/s. - 4 RTO-EN-SCI-66

25 In those frequencies the maximum value of λ is reater than λ Frequency rd/s Fi. 3 Element λ of the Relative Gain Analysis Matrix Step B.: Desin of the first loop controller, s. Takin into account the model uncertainty of p and the desired specifications for the Loop, the controller of Eq. 6 is found, satisfyin all the performance specifications -see Fi. 4a-..65 s s.69 9 s 79 s 5 s 89. s.545 Step C.: Desin of the decouplin element of control effort u on anle δ. Takin into account the optimum controller for reference trackin problems of Eq. 5, the controller of Eq. 8 is desined minimisin the couplin effect c. Fi 4c shows the frequency plot of the obtained couplin reduction. 386 s 3 s 63 s s.73 s.3 s.346 Step B.: Desin of the second loop controller, s. e The followin equivalent plant p e [ p ] [ p ] derived from Eq. 6 is calculated, [ p] [ ] [ p ] [ ] [ p] [ ] 8 The controller of Eq. 8 is found, satisfyin all the performance specifications -see Fi. 4d s 5 s s s.37 s.344s Step C.: Desin of the decouplin element of control effort u on anle δ. The controller of Eq. 83 is desined minimisin the couplin effect c. Fi 4b shows the frequency plot of the obtained couplin reduction. RTO-EN-SCI-66-5

26 6 s 56 s 57 s s.77 s 6.66 s.78 B L B L L 39-3 Manitude db 5 B. B 39-3 B 4-3 L 4-3 L. L 3.59 L 6.7 B 6.7 B 9.6 B 3.59 L 9.6 Manitude db c opt c - 4 Frequency rad /s Phase de p s L s s B.59 - L L Manitude db - c c opt 4 Frequency rad /s Manitude db B B.53 B.69 B.474 B 43.6 L.47 B.47 B 4.55 B 8. B 4. L.53 L.69 L.474 L 4.55 L 8. L 4. L Phase de e p s L s s Fi. 4 QFT MIMO controller desin. Step D: Pre-Filters. Open loop pre-filters of Eq. 84 and Eq. 85 are included in order to satisfy time domain specifications for reference trackin. 4.3 f s 84 s 7.56 s RTO-EN-SCI-66

27 3.6 f s 85 s s Experimental Results To control the AdeptOne SCARA robot, a diital form of the desined non-diaonal QFT compensator is implemented in a Motorola 684 microprocessor 5 MHz, with a 8 Mbyte DRAM memory and a VME bus card, and usin a samplin time of ms. The two plant outputs, anles δ and δ, are measured by encoders, and the two plant inputs, sinals u and u, are applied to a power amplifier that commands two direct drives that move the arms see Fi. 5. r f - u p δ p p r f - u p δ Fi. 5 Block diaram of the control system. The obtained results with the non-diaonal QFT controller, when the reference r for anle δ is commanded from up to 45 derees and while the reference r for anle δ is kept constant zero derees, are shown in Fi. 6. The same experiment with only the pure diaonal controller is shown in Fi. 7. Similarly, the results obtained with the non-diaonal QFT controller, when the reference r for anle δ is commanded from up to 45 derees and while the reference r for anle δ is kept constant derees, are shown in Fi. 8. The same experiment with only the pure diaonal controller is shown in Fi. 9. Both real experimental results show how the non-diaonal controller see Fi. 6 and 8 reach a sinificative reduction of the couplin effect with respect to the performance of the pure diaonal controller see Fi. 7 and 9. RTO-EN-SCI-66-7

28 q δ [de] q δ [de] u [cts] t [s] u [cts] t [s] Fi. 6 Step input at r with a fully populated non-diaonal matrix controller. δ q [de] δ q [de] u [cts] t [s] u [cts] t [s] Fi. 7 Step input at r with a pure diaonal matrix controller. - 8 RTO-EN-SCI-66

29 δ q [de] δ q [de] u [cts] t [s] u [cts] t [s] Fi. 8 Step input at r with a fully populated non-diaonal matrix controller. δ q [de] δ q [de] u [cts] t [s] u [cts] t [s] Fi. 9 Step input at r with a pure diaonal matrix controller. RTO-EN-SCI-66-9

30 5. QFT CONTROL OF A WASTEWATER TREATMENT PLANT [4] One of the main obectives of a Waste Water Treatment Plant WWTP is to protect the water environment from neative effects produced by residual pernicious substances. Fiures and show the new activated slude WWTP of Crispiana, Spain, which is able to reulate both the ammonia and nitrate concentration in the effluent, dealin with water influent of about 5 m 3 /hour. The control strateies desined to reulate that WWTP were based on a hierarchical structure where a hih-level or supervisor selects the set-point of the low-level or conventional controllers. The desin of the controllers was carried out usin the Quantitative Feedback Theory QFT. Nitrate control aims at the optimal use of the de-nitrification potential at any moment. For this purpose, the control alorithm continuously adapts an internal recycle flow in order to maintain a desired nitrate setpoint in the anoxic zone second loop. Ammonia control aims at maintainin the required averae concentration of ammonia in the effluent by manipulatin the Dissolved Oxyen set-point that commands several air flow turbines first loop. Mobile averae values of some variables were also introduced in order to eliminate the perturbations associated with the daily 4-hours profiles. The controllers were desined and verified usin lon-time dynamic simulations based on a multivariable and nonlinear mathematical model IWA nº previously calibrated with real data measured in the fullscale WWTP durin months. The results obtained in the reulation of the pilot plant show a tihter control of the effluent nitroen compounds and a sinificant reduction -enery savin- of the dissolved oxyen demand, reectin the plant disturbances and insurin robust stability. Fi. Wastewater Treatment Plant of Crispiana, Spain. Courtesy of AMVISA. The main control obective of the first loop is to uarantee the standard requirements of ammonia concentration in the plant effluent, fixed on a daily averae SNH lower than m/l. Accordinly, the set point is fixed to.3 m/l. The maximum allowed value saturation limit of the DO variable is m/l. In Fi. the daily averae of the effluent ammonia concentration SNH t is shown as a dashed line and the control input DOt is shown as a solid line, both in m/l, which are obtained with the G z controller, under a typical influent ammonia load and the temperature conditions. - 3 RTO-EN-SCI-66

31 4 NO CH O 4 H N 5 CO 7 H O NH 4 O NO - 3 H O H Influent Internal recycle IR Effluent D SNO Elimination N SNH Elimination SNH DO SST Settler SNO Facultative volumes Aeration System Slude recycle Wastae rate Slude blancket level Fi. Wastewater plant diaram D-N confiuration. Simultaneously, the main control obective of the second loop is to uarantee the standard requirements of nitrates concentration in the plant effluent, fixed on a daily averae SNO lower than m/l. For this reason, the controller tries to strenthen the denitrification process in the D tank to minimize the nitrates concentration in the plant effluent. Thus, in the present experiment the set-point of the nitrates concentration in the D tank is fixed to.5 m/l. The maximum allowed value of the IR variable is % of the desin value of the influent flow rate. In Fi. 3 the daily averae of the nitrates concentration SNO t is shown in the D tank as a dashed line [m/l] and the control input IRt is shown as a solid line [per unit of the influent flow rate] with the G z controller, under a typical influent ammonia load and the temperature conditions. Fiures and 3 show how the system is able to reulate within the required specifications for both loops, except in the intervals when external disturbances saturate the control inputs: First Loop. DO saturated: [ 7, 3 4, 6 7, ] days. Second Loop. IR saturated: [ 9 4, 9 34 ] days. 3 Performance of the First Loop. DO Level solid, Effluent Ammonia SNH dashed Time days Fi. First loop performance with G z. RTO-EN-SCI-66-3

32 Performance of the Second Loop. Internal Recycle solid, Nitrates SNO dashed Time days Fi. 3 Second loop performance with G z. In these cases, the system runs in the best possible conditions, obtainin a ood performance when the saturation disappears. The control system achieves satisfactory performance of the effluent ammonia and nitrates concentrations over the whole rane of operational conditions. It also obtains a notable reduction of the runnin costs, minimizin the oxyen supplied by the aeration system. 6. QFT CONTROL OF A LARGE WIND TURBINE [4, 5] Lare Wind Turbines WT present a very complex multi-obective control problem that combine critical reliability issues with non-linear optimization matters. Advanced QFT robust control strateies have been thorouhly applied in the desin, development and control of the new real Wind Turbines of 5 and 65 kw, made by M.Torres company Fi. 4. The WTs are a variable speed, pitch controlled, multipole synchronous enerator with two controlled IGBT s electrical power converters connected to the stator. The main dimensions are about 7 m of rotor diameter blades of 36 m and 65 m of tower. Fi. 4 Multipole Variable Speed Wind Turbine, 65 kw. Courtesy of M.Torres - 3 RTO-EN-SCI-66

33 Rotor speed Ref. Pitch anle Ref. Pitch speed Ref. I Pitch speed Pitch anle Rotor speed C Rotor - speed controller - C C 3 P 3 Pitch anle controller - Pitch speed controller P P Control System Fi. 5 Rotor speed control system block diaram Wind Speed -m/s- Rotor Speed -rpm Time sec Time sec Control Pitch Anle: refk,r,b,3 de blade Control Pitch Speed: r,b,3 de/sec blade.5 Pitch Anle Reference Pitch Anles Time sec Time sec Fi. 6. QFT Control under very hih wind speed conditions The principal tarets of the more than loops of the control system cover aspects such as the improvement of the maximum power efficiency for every wind speed, the attenuation of the transient RTO-EN-SCI-66-33

34 mechanical loads and fatiue stresses, the reduction of the electrical harmonics and flicker, and the robustness aainst parameters variation with a redundant fault tolerance system. In addition some critical problems arise in the desin of the WT control system, such as the difficulty to work safely with random and extreme usts, the complexity introduced by the stronly nonlinear, multivariable and time variable mathematical model and the impossibility to have a direct measurement of the wind speed experienced by the turbine, because of the hih uncertainty in the anemometer measurement and the stron influence of the blades movement. These set of motivations oblied the control enineer to et involved in the desin of every dynamic element of the wind turbine from the very beinnin of the proect, and to combine advanced control strateies such as QFT robust control techniques, adaptive schemes, multivariable methodoloy and predictive elements. The actual tests that were carried out in several Wind Turbines, for more than three years, with the proposed QFT control methodoloies showed very ood behavior of the WT, either in low, medium or hih winds and even with the extreme 3 m/s case. Fiure 5 shows a simplified block diaram of the rotor speed, pitch anle and pitch speed controllers. Fiure 6 shows some experimental results of the TWT65 with the QFT controllers under very hih wind speed conditions. 7. NON-DIAGONAL QFT CONTROLLER FOR A 3X3 INDUSTRIAL FURNACE [7] This section addresses the temperature control of a 3-input power supplies 3-output temperature sensors industrial furnace used to manufacture lare composite pieces see Fi. 7. Due to the multivariable condition of the process, the stron interaction between the three control loops and the presence of model uncertainties, a sequential desin methodoloy based on Quantitative Feedback Theory QFT is proposed to desin the controllers. The methodoloy derives a full matrix compensator that improves reliability, stability and control. It not only copes with furnace model uncertainties but also enhances the reference trackin and the homoeneousness of the composite piece temperature while minimizin the couplin effects amon the furnace zones and the operatin costs see Fi. 8. Fi. 7. Industrial furnace and piece to be manufactured inside. Siflexa, M.Torres-Spain - 34 RTO-EN-SCI-66