Control Engineering Group

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1 Control Enineerin Group Automatic Control & Computer Science Department Public University of Navarra 36 Pamplona, Spain

2 Control Enineerin Group Automatic Control & Computer Science Department Public University of Navarra, 36 Pamplona, Spain Europe Head of the Group: Prof.Dr. Mario GarcíaSanz Doctors: Dr. Ior Eaña Santamarina Dr. Montserrat Gil Martínez Dr. Juan Carlos Guillén Dr. Xabier Ostolaza Zamora PhD. Students: Marta Barreras Ana Huarte Pamplona (Spain) Alejandro Asenjo Juan Antonio Osés Javier Villanueva Manuel Motilva Javier Castillejo Pablo Vital Juan José Martín Irene Euinoa

3 Control Enineerin Group Automatic Control & Computer Science Department Public University of Navarra, 36 Pamplona, Spain Basic Research on QFT Robust Control. Applied Research Projects with Industry. With Prof. Isaac Horowitz, October In the last 5 years: More than 5 papers (Journals and Conf.) on QFT. 7 Patents. 5 Ph.D. Dissertations on QFT. 6 Research Projects with Industry 3 Research Projects supported by Spanish Gov. Ph.D Students workin on QFT. Guest Editor, Robust Frequency Domain Special Issue, Int. J. Robust Nonlinear Control, Wiley, 3. Oranizer of the 5 th International Symposium on Quantitative Feedback Theory and Robust Frequency Domain Methods, Pamplona, Auust. 5 th QFT Symp., Pamplona, Auust NATO Lecturer. (USA, Italy, Portual, etc) 3

4 Applied Research Projects Wastewater Treatment Plants Drainae and Sewae Water Networks CEIT Public University of Navarra Control Enineerin Group UR QFT Research Fundamentals MIMO Processes Timedelay Systems Distributed Plants Nonlinear Controllers M.Torres Company Applied Research Projects Wind Turbines Desin Aerospace Machinery Paper Machinery Tetra Pak Georia Pacific International Paper Paper Division Enery Division Aerospace Division Boein Airbus A38 Eurofihter Eurocopter Northrop Gr. British Aerosp

5 EXTERNAL DISTURBANCE REJECTION IN UNCERTAIN MIMO SYSTEMS WITH QFT NONDIAGONAL CONTROLLERS M. GarcíaSanz, M. Barreras, I. Eaña, C.H. Houpis Automatic Control and Computer Science Department, Public University of Navarra, 36 Pamplona, SPAIN. Air Force Institute of Technoloy, Air Force Research Laboratory, WrihtPatterson AFB, Ohio, U.S.A. Authors wish to ratefully appreciate the support iven by the Spanish Ministerio de Ciencia y Tecnoloía (MCyT) under rants CICYT DPI 785 and DPI 3858C. 5

6 . Introduction Disturbance attenuation in multivariable systems is often an important and difficult problem to deal with. This work circumvents rejection of external disturbances in MIMO systems with plant model uncertainties. The Quantitative Feedback Theory (QFT) is applied to desin fully populated matrix controllers to attenuate the effect of external disturbances at plant input and plant output. 6

7 The study starts with some previous ideas suested by GarciaSanz and Eaña () about the desin of nondiaonal QFT controllers to reduce the loop couplin in trackin problems, GarcíaSanz, M. and I. Eaña (). Quantitative Nondiaonal Controller Desin for Multivariable Systems with Uncertainty. International Journal of Robust and NonLinear Control. Nº, pp John Wiley & Sons. and considers the approach introduced by Houpis and Rassmussen (999) about MIMO systems with external disturbances. Houpis, C.H. and S. J. Rasmussen (999). Quantitative Feedback Theory Fundamentals and Applications. Marcel Dekker, New York, ISBN:

8 . Rejection of External Disturbances at Plant Input Consider the n x n linear multivariable system shown in Fi.. d i P di d P d r = d i d v G u y P Fi. General structure of a MIMO system with plant output and input external disturbances. P is the plant, P є P and P is a set of possible plants due to uncertainty, G is the full populated matrix controller, P = p p p n p p p n p n p n ; p nn G = n n n P di and P do are the plant input and output disturbance transfer functions. d i and d o are the plant input and output external disturbances respectively. n nn 8

9 d i P di d P d r = d i v G u P y d denotin the external disturbance at plant input by d = i P di d ' i then the output due to the external disturbance at plant input is, y = ( I P G) P d = T d = T P d ' i Y / di i Y / di di i () and the closed loop transfer matrix T Y/di that relates the disturbance at plant input to the output y becomes, T Y / di = ( I P G) P () 9

10 Multiplyin Eq.() by (I P G), premultiplyin by and denotin P = p = P = G = G P [ ] d G where Λ is the diaonal part and B the balance of P respectively and G d is the diaonal part and G b the balance of G respectively b Λ B [ p ] P = P = = Λ B = p p nn p n n p G = G d G b = nn n n and rearranin it, yields the next expression T Y/di = ( I Λ G d ) Λ ( I Λ G d ) Λ [( B G b ) T Y/di ] (7)

11 By inspectin Equation (7) two different terms can be found: T Y/di = ( I Λ G d ) Λ ( I Λ G d ) Λ [( B G b ) T Y/di ] (7) A diaonal term T Y/did = ( I Λ G d ) Λ the only part which has a nondiaonal structure. It will be call the couplin matrix C di (8) A nondiaonal term T Y/dib = ( I Λ G d ) Λ [(B G b ) () T Y/di ] n di i d j = c di i = di j ii u i p ii y i u i y i ii p ii i n

12 This couplin matrix is essential to analyse the reduction of the crosscouplin effects. C di = ( B G b ) T Y/di () Each element of C di obeys, c di m = k= (p ik ik ) t kj ( δ ik ) (5) where is the delta of Kronecker. δ ki = δ δ ki ki = k = k = i i The influence of the nondiaonal elements ik (i j) on the crosscouplin elements described in Eq.(5) is difficult to analyse directly because of the complexity of the expression. For this reason, one hypothesis and two simplifications are stated in order to make the quantification of couplin effects easier.

13 Hypothesis H: The diaonal elements t jj in Eq. (5) are assumed to be much larer than the nondiaonal ones t kj, t jj ( p ) >> t ( p ) for k j kj ik ik Simplification S: Applyin Hypothesis H, Eq. (5) can be rewritten as, c di = t jj (p ) ; i j Simplification S: The elements t jj can be replaced usin the expression obtained from the equivalent system in Eq. (8). t jj = p jj jj p jj 3

14 Due to the previous considerations the couplin effect c di can be defined as, ( p ) cdi = ; i j (9) ( p jj ) jj Note, that every uncertain plant is represented by the followin family, N { p } ( ) = p p for i,j=,, n N Where p is the nominal plant and the non parametric uncertainty radii. db () p p N p N p φ In order to find out the optimum nondiaonal controller, Eq. (9) is made equal to zero, and a nominal plant that minimises the maximum nonparametric uncertainty radii p in Eq. () is chosen, opt N = p () 4

15 The minimum achievable couplin effects can be computed substitutin the optimum controller, Eq. (), in the couplin expression of Eq. (9) and takin into account the uncertainty radii of Eq. (). c pn OPT = = N jj ) p jj ( jj Effect of optimum nondiaonal controller In the same manner, the maximum couplin effect without any nondiaonal controller pure diaonal controller cases can be computed substitutin = in Eq. (9), c pn = = ( N jj ) p jj jj ( ) p jj, jj p, for i, j =,, n 5

16 3. Rejection of External Disturbances at Plant Output Consider the n x n linear multivariable system shown in Fi.. d i P di d P d r = d i v G u y P d Fi. General structure of a MIMO system with plant output and input external disturbances. P is the plant, P є P and P is a set of possible plants due to uncertainty, G is the full populated matrix controller, P = p p p n p p p n p n p n ; p nn G = n n n P di and P do are the plant input and output disturbance transfer functions. d i and d o are the plant input and output external disturbances respectively. n nn 6

17 d i P di d P d r = d i v G u P y d denotin the external disturbance at plant output by d = o P d o d ' o then the output due to the external disturbance at plant output is, y = ( I P G) d ' o = T Y / do d o = T Y / do P do d o (4) and the closed loop transfer matrix T Y/do that relates the disturbance at plant output to the output y becomes, T Y / do ( I P ) = G (5) 7

18 Multiplyin Eq.(5) by (I P G), premultiplyin by and denotin P = p = P = G = G P [ ] d G where Λ is the diaonal part and B the balance of P respectively and G d is the diaonal part and G b the balance of G respectively b Λ B [ p ] P = P = = Λ B = p p nn p n n p G = G d G b = nn n n and rearranin it, yields the next expression T Y/do = ( I Λ G ) ( I Λ G d ) Λ [ B (B G b T Y/do ) ] (6) 8

19 By inspectin Equation (6) two different terms can be found: T Y/do = ( I Λ G d ) ( I Λ G d ) Λ [ B (B G b T Y/do ) ] (6) A diaonal term T Y/dod = ( I Λ G d ) the only part which has a nondiaonal structure. It will be call the couplin matrix C do (7) A nondiaonal term T Y/dob = ( I Λ G d ) Λ [ B (B G b T Y/do )] (9) u i y i ii p ii do i i n = c u i ii p ii y i d j n do i = do j 9

20 This couplin matrix is essential to analyse the reduction of the crosscouplin effects. C do = B (B G b T Y/do ) (3) Each element of C do obeys, c do m = p ( δ ) ( pik ik ) tkj ( δ ik ) k= (3) where is the delta of Kronecker. δ ki = δ δ ki ki = k = k = i i The influence of the nondiaonal elements ik (i j) on the crosscouplin elements described in Eq.(3) is difficult to analyse directly because of the complexity of the expression. For this reason, one hypothesis and two simplifications are stated in order to make the quantification of couplin effects easier.

21 Applyin aain Hypothesis H and Simplifications S and S, the final expression of the couplin effect can be written as, p ( jj p ) c do = p ; i j (33) ( p jj ) jj Note, that every uncertain plant is represented by the followin family, N { p } ( ) = p p for i,j=,, n N Where p is the nominal plant and the non parametric uncertainty radii. db () p p N p N p φ In order to find out the optimum nondiaonal controller, Eq. (33) is made equal to zero, and a nominal plant that minimises the maximum nonparametric uncertainty radii p in Eq. () is chosen, N opt p = jj N p jj (34)

22 The minimum achievable couplin effects can be computed substitutin the optimum controller, Eq. (34), in the couplin expression of Eq. (33) and takin into account the uncertainty radii of Eq. (). Effect of pn optimum c = ( jj ) nondiaonal = OPT ( ) N jj p jj jj controller In the same manner, the maximum couplin effect without any nondiaonal controller pure diaonal controller cases can be computed substitutin = in Eq. (33), c pn = = ( ) N jj p jj jj ( ) p jj, jj p, for i, j =,, n

23 4. Desin Methodoloy Stability: It is necessary and sufficient that the plant of each successive loop is stabilised (Chait and Yaniv, 99) Before startin the sequential procedure, analyse the effect of interactions in the system and identify inputoutput pairins usin the Relative Gain Array, (Bristol, 966). Afterwards, rearrane matrix P so that ( p ) has the smallest phase marin frequency, ( ) the next smallest phase marin frequency, and so on, (Houpis, Rasmussen, 999). p Composed of n staes, as many as loops, performs the followin steps for every column of the matrix controller G. G = k n k n k n k n k n k k kk nk n n kn nn Step Step Step n 3

24 Step A: Desin the diaonal element of the controller kk for the inverse of equivalent plant described in Eq.(37), usin a standard QFT loopshapin method, [ e ] p = [ p ] ii k ii k ( ) ([ p ] [ ( )] ) [ p i(i) k i i k (i)i ] k [ (i )i ] [ p (i)(i) ] k [ ( i)( i) ] k k [ P ] = P ( 37 ) i k; k= Step B: Desin the (n) nondiaonal elements ik (i k, i =,,n) of the k th controller column, minimisin the couplin c diik or c doik described in Eqs. (9), (33), and applyin the optimum nondiaonal controller equations Eq. () and Eq. (34) respectively. 4

25 5. Example Set of plants P due to uncertainty y y k τ = k τ k τ k τ u u Coefficients of parametric uncertainties k k k min K max τ k..8 τ 8 k 5 τ 3 8 k 7 5 τ τ τ min τ max r e p p d Y Desired performance specifications: Robust Stability: L L i. i = i, e d p r Y p Robust Output Disturbance Rejection yi ( s) s d ( s) s i ω < 5 rad / s, i =, Reduction of couplin effect as much as possible 5

26 Relative Gain Array Value s of landa Values of landa. Percentae. landa Number of samples % Values of landa plants enerated due to uncertainty: λ є [,.) for plants λ є [.,.) for plants λ є [.,.3) for plants λ є [.3,.4) for 355 plants λ є [.4,.5] for 67 plants 6

27 DESIGN PROCEDURE ( >λ >.5) Step A: Standard QFT loopshapin for p 3 Templates 77 points..5 Gain (db) Phase (de) 7

28 DESIGN PROCEDURE ( >λ >.5) Step A: Standard QFT loopshapin for p 3 Te m plate s 9 points..5 Gain (db) Phase (de)

29 d Desin of r e p Y p 755s 78 = s 3s r e p p d Y 6 4 OpenLoop Gain (db) OpenLoop Phase (de)

30 d Step B: Optimum Nondiaonal controller for the first column r r e e p p p p d Y Y 8875 s = 3 3s 57 97s s s Step A: Once the first column has been desined, the equivalent plant of the second channel is calculated: e [ p ] = [ p ] ([ p ] [ ] ) ([ p] [ ] ) [ p] [ ] 3

31 e p Standard QFT loopshapin for Te m plate s 3 84 points..5 Gain (db) P has e (de) 45 3

32 Standard QFT loopshapin for 3 p e Te m plate s 34 points..5 Gain (db) Phase (de) 3

33 d Desin of r e p Y p 6745 s = 3 s 7 77 s 3 s s r e p p d Y 4 OpenLoop Gain (db) OpenLoop Phase (de) 33

34 Step B: Optimum Nondiaonal controller for the second column s 3.4 s.8 = 4 3 s 77. s s s s d r e p p Y p d p r e Y p d 34

35 d RESULTS r e p p Y Disturbance d at plant output e p r e Y p Response y Response y DIAGONAL CONTROLLER.5 DIAGONAL DIAGONAL CONTROLLER CONTROLLER y.5. y Time (seconds) Time (seconds) 35

36 d RESULTS r e p p Y Disturbance d at plant output e p r e Y p Response y Response y NON NONDIAGONAL DIAGONAL CONTROLLER.5 NON NONDIAGONAL DIAGONAL CONTROLLER y.5. y Time (seconds) Time (seconds) 36

37 RESULTS Disturbance d at plant output r r e e p Y p p d r Y p Response y Response y. DIAGONAL CONTROLLER.3 DIAGONAL CONTROLLER y y Time (seconds) Time (seconds) 37

38 RESULTS Disturbance d at plant output r r e e p Y p p d r Y p Response y Response y. NON NON DIAGONAL DIAGONAL CONTROLLER CONTROLLER.3 NON NON DIAGONAL DIAGONAL CONTROLLER CONTROLLER y y Time (seconds) Time (seconds) 38

39 6. Conclusions A QFT methodoloy to desin fully populated matrix controllers to solve the MIMO external disturbance rejection problem at both, plant input and output, and in the presence of model plant uncertainty, was presented. The definition of a specific couplin matrix allows the statement of the controller desin methodoloy. An example has been also presented to improve the understandin of the methodoloy. 39

40 7. Current Work Applyin the new methodoloy to control an industrial furnace. A 7x7 MIMO System Power: 9 kw Lenth: 37.6 m For Wind Turbine Blades Polymerisation Industrial furnace for composite manufacturin located at M.Torres (Spain) Group Group Group 3 Group 4 Group 5 Group 6 Group m 5 m 5 m 5 m 5 m 5 m 7.8 m 37.6 m 4

41 EXTERNAL DISTURBANCE REJECTION IN UNCERTAIN MIMO SYSTEMS WITH QFT NONDIAGONAL CONTROLLERS M. GarcíaSanz, M. Barreras, I. Eaña, C.H. Houpis Automatic Control and Computer Science Department, Public University of Navarra, 36 Pamplona, SPAIN. Air Force Institute of Technoloy, Air Force Research Laboratory, WrihtPatterson AFB, Ohio, U.S.A. Authors wish to ratefully appreciate the support iven by the Spanish Ministerio de Ciencia y Tecnoloía (MCyT) under rants CICYT DPI 785 and DPI 3858C. 4