FEEDFORWARD CONTROLLER DESIGN BASED ON H ANALYSIS

271 FEEDFORWARD CONTROLLER DESIGN BASED ON H ANALYSIS Eduardo J. Adam * and Jacinto L. Marchetti Instituto de Desarrollo Tecnológico para la Industria Química (Universidad Nacional del Litoral - CONICET) Güemes 3450-3000 Santa Fe Argentina. E-mails: eadam@intec.unl.edu.ar, jlmach@.ceride.gov.ar Abstract. This paper presents a model based design and adjustment procedure or eedorward controllers that accounts or model uncertainties. It also analyses relevant properties o the eedorward-eedback structure and proposes a combined design method using robust control concepts in the requency domain. This approach allows the inspection o the perormance limits achievable or each particular application. The beneits o the suggested approach are discussed and illustrated through an application example. Keywords: eedorward controller, model-based control, H design. 1. Introduction There are many examples rom dierent engineering areas where an input disturbance has a strong eect on the controlled variable. Classic methods or tuning eedback controllers like Ziegler and Nichols (1942) among others have been used per decades. However, the level o perormance obtained in set-point changes is not always reached in disturbance rejections. Moreover, the combined eedorward-eedback control scheme considers simultaneously both the set-point tracking and the regulation problems. The use o this combined control is usually very common in the process industry: it can be ound in distillation columns (Rix et al., 1997), power plants (Weng and Ray, 1997; Mcaburn and Hughes, 1997), continuous reactors, among other examples. In spite o this, the synthesis o eedorward controllers has not received much attention. Consequently, the eedorward controller design and adjustment still ollows classic approaches (Seborg et al., 1989; Stephanopoulos, 1984). In this work, a method based on H design concepts is proposed which complements the classic design by providing a rational procedure to achieving a realizable controller transer unction. In order to present systematically this work, Section 2 gives the preliminary concepts related to eedorward-eedback control, Section 3 presents main theoretical undaments and includes an application example that reveals the capabilities o the proposed design technique. Finally, in Section 4, the conclusions are presented. 2. Feedorward-Feedback Control Let us start by analyzing the classical eedorward-eedback structure where linear representations with uncertain parameters are assumed or the process system. * To whom all correspondence should be addressed

272 Figure 1 shows an sketch o the combined control system where c(s) is the eedback controller, c (s) is the eedorward controller, p d (s) stands or the transer unction between the output y(s) and the exogenous disturbance d(s) and, p(s) is the transer unction between the output and the manipulated variable u(s). From this representation it is clear that p d (s) and p(s) are lineal time invariants (LTI) models representing the real plant. d( s) pd( s) c s ( ) d2( s) rs ( ) w( s) - u( s) c( s) - p( s) d1( s) y( s) Plant Fig. 1. Schematic representation o a eedorward-eedback control scheme. When the eedorward controller is implemented alone, the disturbance eect on the output variable is written as ys ( ) = pd( s) c ( sps ) ( ) ds ( ) (1) but in case that a eedback loop is also used, pd( s) c ( s) p( s) ys ( ) = ds ( ) (2) 1 + pscs ( ) ( ) In both cases, the optimal eedorward controller that minimizes pd c p is, 2 pd() s c() s = (3) ps () and it would lead to preect disturbance compensation in both cases i the c (s) is realizable. However, the expression (3) might result in an improper or unstable transer unction; when this happens c (s) should be chosen such to minimize the eect o the disturbance on the controlled variable. 2.1. Properties o the Feedorward-Feedback Control System Let us assume inputs d(t) and r(t) are bounded signals, i.e., d(t), r(t) L 2 [0, ), where L 2 [0, ) stands or any continuous signals on [0,) that have inite 2-norm (Green and Limebeer, 1995). Deinition 1 (Perect Disturbance Rejection). Feedorward control allows perect disturbance rejection when the controlled variable does not change as consequence o any bounded input disturbance, i.e., yt () = 0 t 0or dt () L2[0, ) and r(t) = 0. (4)

273 Equation (4) assumes the output variable y(t) is deined as a deviation rom the steady-state value. Note that, eedorward control allows the possibility o perect disturbance rejection while; this is not possible using eedback control neither or regulation nor or tracking problems (Morari and Zairou, 1989). In order to analyze the internal stability o the combined eedorward-eedback control system the ollowing deinition is introduced: Deinition 2 (Internal Stability o the Combined Control System). The system sketched in Fig. 2 is internally stable i the elements o the transer matrix S( pd c p) Scp Sc( pd c p) Sc M = with S : = 1/(1 + cp) (5) pd 0 c 0 rom [d r] T to [y w d 1 d 2 ] T belong to RH, or all bounded d(t) and r(t). Note that requiring internal stability to the eedback loop is not suicient or the combined system. To make sure internal stability it is also necessary that c (s) and p d (s) being stable plants. Furthermore, i this condition is satisied the internal stability o the eedback loop is not aected since; (i) all the input signals to the eedback loop are bounded and (ii) the eedorward controller is not included in the characteristic equation. In order to generalize the analysis, let us assume that this plant can be described by two amilies o models which are deined using the classic multiplicative uncertainty, p p0 Π u : = p: lmu : = lmu( ω), (6) p0 pd pd0 Π d : = pd : lmd : = lmd( ω). (7) pd0 Π u and Π d stand or the amilies, lmu( ω ) and lmd( ω ) are the upper bounds or the multiplicative uncertainty modulus lmu( jω ) and lmd( jω ) respectively. Thus, any member o the amilies o plants Π u and Π d can be represented as ps () = p0 ()(1 s + lmu()) s and pd() s = pd0 ()(1 s + lmd()) s respectively and, p 0 (s) and p d0 (s) are the nominal plants according to the designations in Fig. 1. Then, using the sensitivity unction S = 1/(1 + cp) and substituting p(s) and p d (s) in the expression (2) gives, ys () Ss (){ pd0()1 s[ lmd() s] c sp0()1 s[ lmu() s] } d() s = + +. (8) Based on the right side o this expression, a unctional denoted I is deined as ollows: [ ] [ ] I (): s = p ()1 s + lm () s c () s p ()1 s + lm () s. (9) d 0 d 0 u Note that i, S I γ or all 0 ω < then, according to previous deinition and Eqn. (2), it ensures that, y γ d( jω). It is important to remark that, i) S(s) does not depend o eedorward controller parameters and, ii) an acceptable disturbance attenuation in the output variable is reached i γ << 1. That

274 is, the appropriate design o both eedorward and eedback controllers should be such that a γ << 1 be reached. Furthermore, according to the comments ollowing to deinition 2, c(s) should assure stability o the eedback loop and c (s) should be stable. I the nominal plant satisies the perormance objectives, the system reaches the nominal perormance. This idea can be expressed or the eedorward-eedback control loop based on the expression (2) as, WSI p 0 0 1, where denotes the H ininity norm based on the classical deinition, S 0 is the nominal sensitivity unction deined as S 0 : = 1/(1 + cp 0 ), I 0 : = pd 0 c p0 and W p is a weight o the input signal included or a more general ormulation. On the other hand, S I γ is a robust perormance condition or the eedorward-eedback control system i the parameter γ is chosen less than one and the closedloop system is robustly stable. This condition can be ormalized as ollows, WSI p 1. Lemma 1. Let p(s) and p d (s) be plants that belong to the amilies (6) and (7) respectively, then, the eedorward-eedback control system reaches robust perormance i the ollowing condition is satisied: W ( j ω) S ( j ω) I ( j ω) + T ( j ω) lm ( j ω) 1 0 ω <. (10) p 0 0 The proo, given in Adam and Marchetti (2001), starts rom the condition WSI p 1 and consequently each element o the set (6) and (7) satisies Eqn. (10). 3. Feedorward Controller Design 3.1. The Standard Design Problem The eedorward controller can be synthesized according to the ollowing equation: c0 () s = { pd0( s)/ p0( s) } *, (11) where p d0 (s) and p 0 (s) are nominal transer unctions and {} * denotes a polynomial ratio such that it does not include: i) the zeros o p 0 (s) that belong to let hand side plane, ii) the unstable poles o p d0 (s) and iii) positive time delay that might result rom the dierence θ d0 (s) - θ 0 (s) where, θ d0 (s) and θ 0 (s) are the nominal plant delays o p d0 (s) and p 0 (s) respectively. The operation indicated in (11) gives a stable transer unction, in spite o p d0 (s) could be unstable and p 0 (s) could have right hal zeros, but in many cases a non realizable transer unction could result. Hence, a ilter is included in order to obtain realizability. This gives c ( s) = c 0 ( s) ( s), (12) where the ilter is deined as ollows: Deinition 3 (Feedorward Filter). The eedorward ilter (s) is chosen as a rational transer unction, K (): s =, (13) n ( λ s+ 1) with n = δ(ro) pd 0( s)/ p0( s). The unction δ is deined as, δ = δ( RO) : = 1 i RO p d 0( s )/ p 0( s ) < 0 or δ = δ( RO) : = 0 i RO p d 0( s )/ p 0( s ) 0 and, RO[ ] is an operator that computes the relative order o the

275 polynomial ratio contained inside the brackets. In other words, it gives the dierence between the denominator polynomial order and the numerator polynomial order. 3.2. Proposed Design Technique Since, c (s) has the gain and the ilter time constant as adjusting parameters, the ollowing optimization problem in the requency domain is proposed: min S I 0 ω < (14) K, λ s.t., K > 0 and λ 0. 3.3. Example Consider the boiler with natural recirculation modeled by Adam and Marchetti (1999). Adam (1996) demonstrates that or the operative conditions the non-linear dynamic model between level drum and the eed-water can be satisactorily modeled by the linear expression 4 6.1s p0 ( s) = 1.9510 e / s(5.72s + 1). Similarly, the transer unction between the level drum 4 and the steam load is pd 0 ( s) = 1.5810 (75.49s 1) / s(6.88s + 1). According to the eedorward controller design proposed in this paper c ( s ) = 0.810 K (75.49 s 1) /( λ s + 1). (15) Furthermore, using the extreme plant concept, it is possible to deine the multiplicative uncertainties lm u (s) and lm d (s) according to Eqn. (6) and (7). Thus, any plant ps Π ( ) or u p d() s Π d is included in the Nyquist plane inside the extreme plants 4 12.17 s 5 ps ( ) = 2.1810 e / s (1.55 s + 1) or p s d() s = p d0 () s e respectively. Figure 2 shows the S I ( jω ) when a PI eedback controller tuned by the TL method (Tyreus and Luyben, 1992) is used (Adam, 1996). Applying the optimization problem proposed in Section 3, the optimal parameters or the eedorward ilter are 5 K ot = 2.22 and λ = 3.11 and, SI = 1.05710 at requency ω = 0.6136. Figure 3 compares the level control perormance reached when the eedback and the combined scheme are used. For the last case, two eedorward controller are studied, the irst one is tuned by the proposed method and the other one is tuned as a static eedorward controller calculated or s = 0, according to classic textbook. In every case, a PI eedback controller tuned by the TL method was adopted. Clearly, it is observed the improvement reached on the controlled variable when the proposed eedorward controller is used in comparison with the other cases.

276 10-4 10-5 Sj ( ω) I 10-6 10-7 10-8 10-3 10-2 10-1 10 0 10 1 10 2 10 3 Frequency (rad/sec.) Fig. 2. S I ( jω ) using the PI controller tuned by TL method. 4. Conclusions Fig. 3. Level control using the eedback and eedorward-eedback with static and proposed eedorward controllers. In this work, a study o the eedorward-eedback control coniguration is presented. Under the ramework o the robust control theory, a method is proposed to achieve the realizable eedorward controller by including o a low pass ilter. The adjustment o this ilter is done by a minimization o an objective unctional in the requency domain and based on H design concepts that take into account the model uncertainties. Reerences Adam E. J. (1996). Modelado, Simulación y Control de Generadores de Vapor. Doctoral Thesis, Facultad de Ingeniería Química Universidad Nacional del Litoral. Adam E. J. and Marchetti J. L. (1999). Dynamic Simulation o Large Boilers with Natural Recirculation. Computers & Chemical Engineering, 23, 1031-1040. Adam E. J. and Marchetti J. L. (2001). Diseño de Controladores Feedorward Basándose en Propiedades de Robustez. Submited to RPIC 2001. Green M. and Limebeer D. J. N. (1995). Linear Robust Control. Prentice Hall, Inc.. Mcaburn A. and Hughes F. M. (1997). Feedorward Control o Solar Thermal Power Plants. Journal o Solar Energy Engineering, 119, 52-60. Morari M. and Zairiou E. (1989). Robust Process Control, Prentice Hall, Inc.. Rix A., Löwe K. and Gelbe H. (1997). Feedorward Control o a Binary High Purity Distillation Column. Chem. Eng. Comum., 159, 105-118. Seborg D. E., Edgar T. F. and Mellichamp D. A. (1989). Process Dynamics and Control. John Wile & Sons New York. Stephanopoulos G. (1984). Chemical Process Control an Introduction to Theory and Practice. Prentice- Hall. Tyreus B. D. and Luyben W. L. (1992). Tuning PI Controllers or Integrator / Dead Time Processes. Chem. Res, 31, 2625-2628. Weng C-K. and Ray A. (1997). Robust Wide-Range Control o Stean-Electric Power Plants. IEEE Transaction on Control Systems Technology, 5, 1, 74-88. Ziegler J. B. and Nichols N. B. (1942 ). Optimum Settings or Automatic Controllers. Trans. ASME, 64, 11, 759-768.