Integral Minimum Variance-Like Control for Pooled Nonlinear Representations with Application to an Aircraft System

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1 International Journal of Control Vol. 00, No. 00, DD Month 200x, 1 18 This is a preprint of an article whose final and definitive form has been published in the International Journal of Control 2007 copyright Taylor & Francis; International Journal of Control is available TM online at informaworld DOI: / Integral Minimum Variance-Like Control for Pooled Nonlinear Representations with Application to an Aircraft System Dimitrios G. Dimogianopoulos, John D. Hios and Spilios D. Fassois Stochastic Mechanical Systems and Automation (SMSA) Group, Department of Mechanical & Aeronautical Engineering, University of Patras, GR Patras, GREECE (Revised manuscript November 25, 2006) This paper presents an integral minimum variance-like controller design based upon a Constant Coefficient Pooled Nonlinear AutoRegressive Moving Average with exogenous excitation (CCP-NARMAX) representation. The use of pooling techniques significantly enhances the NARMAX representation s ability to accurately describe systems performing under various operating conditions such as aircraft systems, chemical processes, industrial systems and so on. The controller design introduces suitable modifications to account for the characteristics of the CCP-NARMAX representation. The control strategy is subsequently applied to a nonlinear aircraft system in order to obtain regulation of the pitch rate around a predetermined value. Comparisons with a conventional PID control design are also made under various operating conditions, including disturbances due to external input and turbulence. 1. Introduction 2. Problem Statement 3. The Aircraft System and its Modelling 3.1. CCP-NARMAX Modelling Procedure 3.2. CCP-NARMAX Modelling Results 4. Integral Minimum Variance Control Law Derivation 4.1. Standard Control Law Index to information contained in this document Keywords: Integral minimum variance, aircraft operations, nonlinear modelling, pooling techniques Modified Minimum Variance Control Law 5. Stability Analysis of the Controlled Aircraft System 6. Aircraft System Control Results 7. Conclusions Appendix A: Derivation of J Mod [t +1] and J MV [t +1] References 1 Introduction The Generalized Minimum Variance (GMV) control principle and its simplified MV version have long been used in linear time invariant industrial system applications (Aström and Wittenmark, 1971). They combine simplicity (at least for the univariate case), similar characteristics to Linear Quadratic Gaussian (LQG) designs in some cases, optimality under certain conditions (minimum phase plants) and acceptable performance at least for low frequency signals. The limitations are also quite well known: poor high frequency behavior, aggressive control actuator activity, and lack of integral action (if zero steady state tracking dynamics are needed) (Grimble, 2005). The minimum variance principle has certainly contributed to the development of the predictive control design techniques, which have quite successfully been used during the last two decades (Clarke et al., 1987). A recent effort in Grimble (2004) has concentrated on Corresponding author. Tel/Fax: (++ 30) , , dimogian@mech.upatras.gr, hiosj@mech.upatras.gr, fassois@mech.upatras.gr, Internet: sms. A short version of this manuscript was presented at the 14 th Mediterranean Conference on Control Automation MED 2006, June 28-30, 2006, Ancona, Italy.

2 2 Integral Minimum Variance-Like Control for Pooled Nonlinear Representations with Application to an Aircraft System the inclusion of the integral part for a MV controller based on a linear plant representation. Furthermore, a version of the GMV principle for a quite general nonlinear plant (referred to as the Nonlinear GMV or NGMV) has been proposed in Grimble (2005). These two works clearly motivate again research on this classical control solution, to the point where problems previously non treatable by this design could now be revisited. On the other hand, most of the plants treated to date are modelled by means of linear (time invariant or time varying) representations. Nonlinear discrete plants are usually controlled by means of PI(D) designs (Chen and Huang, 2004), PI(D) combined with inverse precompensation (Petridis and Shenton, 2003), or even loop shaping techniques (Glass and Franchek, 1999). Minimum variance techniques for nonlinear plants have mainly been connected with Neural Network (NN) representations (as in Gao et al. 2000, Zhu and Guo 2004) and, in a few cases, with plant modelled neuro-fuzzy techniques (Liu and Lara- Rosano, 2004). The more general case of plants modelled by means of Nonlinear AutoRegressive (Moving Average) with exogenous [NAR(MA)X] excitation representations and controlled by GMV based designs has received rather less attention (Goodwin and Sin 1984, Sales and Billings 1990, and a major part in Bittanti and Piroddi 1997). The last contribution makes several useful comments on the particularities of the NGMV designs and the difficulty of solving the control law equation even for affine plant models. The aim of this paper is to introduce a MV based controller which retains the straightforward operation of the original GMV design while attenuating its shortcomings when used with applications exhibiting fast dynamics, such as the considered aircraft systems. The NARMAX affine model class is considered, that is, a more general model class than the NARX in Bittanti and Piroddi (1997). This class approximates a great number of physical plants and systems (Zhu and Guo, 2004). In order to obtain a NARMAX representation globally valid for a flight envelope region, pooling techniques are utilized. Such techniques were originally proposed in the work of Billings and Chen (1989) for modelling nonlinear dynamics of (industrial) plants with multiple operating points by means of a single nonlinear representation. Recent experience with aircraft systems (Samara et al., 2003) shows that the use of pooled representations is equally well suited to aircraft (fast) system dynamics under various conditions: The Constant Coefficient Pooled NARMAX (CCP-NARMAX) representation identified in the present work, performs well for all points of the landing flight regime. An integral penalty is added (as in Grimble 2004) into the GMV criterion in order to compensate for the offset error, while the penalty on the input signal is retained. Thus, plants exhibiting non-minimum phase behavior may in principle be treated. Obviously, solving the control law equation for a NGMV design can be a tedious task (Bittanti and Piroddi, 1997), hence a proper modification of the NGMV control law expression is introduced in order to provide feasible (if suboptimal) control values in critical situations. Finally, this NGMV-like design is applied to the pitch rate control (regulation) of a 6 Degree-Of-Freedom (DOF) nonlinear aircraft simulator. An Input/Output

3 Integral Minimum Variance-Like Control for Pooled Nonlinear Representations with Application to an Aircraft System 3 (I/O) relation of this simulator is explicitly modelled as an affine SISO (Single Input Single Output) pooled NARMAX model, and the NGMV controller based upon this model is designed. Obviously, being a SISO controller this is not a global auto-pilot design. Advanced auto-pilots feature a more complex (generally not SISO) structure and may be based upon simple PI designs (Devaud et al., 2000), H (Clement et al., 2005), sliding mode control (Caferov and Tasaltin, 2001), or even receding horizon techniques (Kim et al., 2001). However, the 6 DOF simulator involves high frequency signals, significant disturbances and generally fast dynamics, thus being a thorough test for any controller design. 2 Problem Statement Consider the SISO nonlinear plant model, whose output y[t] should be regulated around a fixed reference value (zero without loss of generality): y[t] =A l (q 1 )y[t 1] + F nl (u[t k],..., u[t k n u +1]; y[t 1],...,y[t n y ]; e[t 1],..., e[t n e ]) + e[t] (1) where t stands for the normalized discrete time, y[t], u[t], e[t] are the system output, input and disturbance (assumed to be zero mean and uncorrelated) signals, respectively. The variables k, n u and n e designate the system delay and the maximum lags appearing in the model (model orders) for the corresponding signals y[t] u[t] and e[t]. Furthermore, A l (q 1 ) is a linear polynomial in the backward shift operator q 1. Finally, the operator F nl ( ;[t 1]) is a nonlinear polynomial mapping, admitting terms such as cross-products of lagged output, input and disturbance signals and their powers up to degree nl. The component signals of each cross product term may admit values of different lags (unlike Grimble 2005). The form of the nonlinear function F nl ( ;[t 1]) is crucial for the existence of an analytical solution of the control variable u[t]. In general F nl ( ;[t 1]) may be a high order nonlinear function of u[t], for which it is impossible to derive a general analytical solution. In this case, an assumption that the equation F nl ( ;[t]) = 0 has a unique solution with respect to u[t] (the invertibility condition in Bittanti and Piroddi 1994) is usually introduced. This condition is, however, difficult to satisfy especially when past values of e[t] are involved in F nl ( ). Furthermore, even if a unique solution exists, it is not always easy to be computed for real time operation. Hence, a less stringent class of F nl ( ;[t 1]) representations has been proposed in Bittanti and Piroddi (1997) and De Nicolao et al. (1997): Assumption 1: The plant model (1) is affine in the most recent control variable, that is the nonlinear function F nl ( ) may be rewritten as: F nl (y[t 1],...,y[t n y ]; u[t k],...,u[t k n u +1]; e[t 1],...,e[t n e ]) = G u [t 1]u[t k]+g F [t 1] (2)

4 4 Integral Minimum Variance-Like Control for Pooled Nonlinear Representations with Application to an Aircraft System The term G u [ ] in (2) includes all terms (cross products or other) which multiply the most recent control variable u[t k] (with u[t k] itself obviously not included), whereas the second one includes all remaining terms. The time argument [t 1] designates the respective time instant of the most recent signal involved in G u [t 1]; the same applies to G F [t 1]. Consequently, the process model of (1)-(2) may be rewritten as: y[t] =A l (q 1 )y[t 1] + G u [t 1]u[t k]+g F [t 1] + e[t] (3) Assumption 2: The plant model (3) does not have transmission (linear) zeros equal to one. This is a common assumption stated in De Nicolao et al. (1997), and implicitly mentioned in Grimble (2004) and Bittanti and Piroddi (1997). This assumption is useful for the design of an integral control law, as will be explained later. This model class may represent a considerable number of applications (see Bittanti and Piroddi, 1997, and references therein), while conceding little in terms of modelling accuracy. For this representation the commonly used assumption is that G u [ ] should be bounded away from zero. Part of this work is to relax this assumption, by the design of a suitable control law. 3 The Aircraft System and its Modelling 3.1 The CCP-NARMAX Modelling Procedure The aircraft system considered is simulated by means of a 6 DOF nonlinear model. Inputs are the pilot commands (stick, wheel and pedal), and outputs are the attitudes, the accelerations and the angular rates (see Fig. 1). The wind and turbulence effects can be simulated as system disturbances. Aircrafts are nonlinear systems, whose behavior varies with the flight regime (take off, clean flight or landing), their operational point in the flight envelope (with its boundaries causing major changes in the dynamics), and the external conditions (such as turbulence) during the flight. Hence, the system dynamics should be modelled using effective nonlinear representations. A stochastic CCP-NARMAX representation, which retains the property of being affine in the most recent control variable as in (3), is presently considered. This pooled representation features a more complete description of the system dynamics than a standard NARMAX one, as will be explained in the sequel, and has the form: y j [t] = L i=0 θ i p i,j [t]+e j [t] E{e j [t] e s [t τ]} = γ e [j, s]δ[τ] e j [t] NID (0,σe(j)) 2 with y j [t] and e j [t] being the model s output and prediction error or residual [assumed to be a zero-mean uncorrelated sequence with variance σ 2 e(j)] signals for the j-th flight, respectively. The input signal is (4)

5 Integral Minimum Variance-Like Control for Pooled Nonlinear Representations with Application to an Aircraft System 5 not presented explicitly in (4), but is involved in the regressors p i,j [t] (i=0,..., L and j=1,..., M). The regressors represent cross products between lagged values of the output, the input u j [t], and the residual e j [t], or powers of these signal values with p 0,j [t] = 1. The model coefficient corresponding to the i-th regressor is designated as θ i and is common for all flights. The maximum lags associated with the signals y j [t], u j [t] and e j [t] in (4) are n y, n u and n e (model orders), respectively. The argument E{ } designates statistical expectation, NID(.,.) stands for Normally Independently Distributed (with the indicated mean and variance), δ[τ] is the Kronecker delta (δ[τ] =1whenτ = 0 and δ[τ] =0whenτ 0) and γ e [j, s] the cross covariance of the error associated with j th and s th flights at zero lag. The first of equations (4) may be rewritten in a (pseudo)linear regression form as follows 1 : y j [t] =φ T j [t] θ + e j [t] (5) with φ j [t] =[p 0,j [t]... p L,j [t]] T the regressor vector and θ =[θ 0... θ L ] T the parameter vector, respectively. The form (5) is referred to as pseudolinear, as the terms p i,j [t] involved in φ j [t] may contain past values of e j [t], which obviously depend on the particular vector θ used. Hence, φ j [t] is indirectly related to θ. If M flights (each one consisting of N recorded data samples) are available, the following matrix equation is obtained: ȳ = Φ θ + ē (6) where ȳ =[y T 1 yt M ]T R [NM 1], Φ =[Φ T 1 Φ T M ]T R [NM (L+1)], ē =[e T 1 et M ]T R [NM 1].The vectors y j, e j R [N 1] designate data from the j-th flight, while Φ j =[φ j [1]...φ j [N]] T R N (L+1). Loosely speaking, ȳ contains all output data from all flights under all conditions considered, Φ contains all corresponding regressors, and θ is the parameter vector common to all flight conditions for which data are considered. The CCP-NARMAX form (6) is based upon a large number of different flights and, compared with a conventional NARMAX form identified using data from a single flight, is better suited to describing the aircraft system dynamics under the entire flight regime. Hence, the CCP-NARMAX is also more effective for control purposes with respect to the conventional NARMAX representation. Nevertheless, the CCP- NARMAX is reduced to a conventional NARMAX representation when data from a single flight are used. More details on pooled NARMAX representations may be found in Samara et al. (2003); pooled ARX models and their estimation are discussed in Kopsaftopoulos and Fassois (2006). Structure selection (that is, the choice of the appropriate regressors p i,j, i =0,..., L and j =1,..., M) and parameter estimation for the CCP-NARMAX model is carried out in two stages using (6), since the regressor matrix Φ contains past e j [t] values. 1 Lower case/capital bold face symbols designate column vector/matrix quantities, respectively.

6 6 Integral Minimum Variance-Like Control for Pooled Nonlinear Representations with Application to an Aircraft System In the first stage, an initial estimation of residuals e j [t] for j =1,..., M flights and t =1,..., N time instants (that is, the elements of ē) is performed by means of an auxiliary CCP-NARX model, with sufficiently high model orders of the input and output signals. This model has a structure similar to that in (4), except that the terms p i,j [t] do not include any (past or current) values of e j [t]. The identification of the CCP-NARX model structure (that is, the determination of the terms to be included in the model) as well as the estimation of the corresponding model parameter vector are based upon the orthogonal parameter estimation algorithm proposed in Korenberg et al. (1988). Briefly, for the available pooled data, a set of mutually orthogonal regressors is defined using an iterative forward search algorithm. Each regressor is evaluated from its contribution to the reduction of the Residual Sum of Squares to Signal Sum of Squares (RSS/SSS) ratio. At each iteration, the regressor leading to the most significant RSS/SSS reduction is selected. After a user-defined number of regressors is selected, the associated parameter vector is estimated (using an Ordinary Least Squares (OLS) or Weighted Least Squares (WLS) procedure as in Fassois 2001, p. 677) and the estimation of residuals e j [t] is also concluded. In the second stage, the CCP-NARMAX model in (4) of orders (n y,n u,n e ) is considered. The data set now involves both the output values and the previously estimated residual values from all flights. This data set is used along with the previously mentioned orthogonal algorithm to determine the CCP-NARMAX model structure. This procedure of term selection is carried out as in the first stage, except that now the selected regressors may involve past e j [t] values. As in the first stage, the procedure is terminated when the user-selected total number of CCP-NARMAX regressors is obtained and the parameter vector θ is estimated by OLS or WLS. 3.2 CCP-NARMAX Modelling Results In this study the integral minimum variance control objective is to regulate the aircraft pitch rate. For this reason an input/output relationship involving the pilot s stick movement and the pitch rate is considered. The system is represented by means of a CCP-NARMAX(15,15,15,1) model [that is, n y = 15, n u = 15, n e = 15 and k = 1, see (1) and (2)], with 45 regressor terms. The parameters are estimated using the pooling procedure previously mentioned, and are based upon 40 flights of 5001 samples each (sampling rate equal to 100 Hz), conducted inside the landing flight regime for a given aircraft configuration (constant weight-distribution and weather conditions). The choice of the CCP-NARMAX regressors [the matrix Φ] and the parameter estimation is based upon the orthogonal algorithm in Billings et al. (1989). The obtained regressors p i,j [t] are displayed in Table 1. In Fig. 2 the one-step-ahead prediction of the CCP-NARMAX model, as well as the prediction errors (residuals), are shown for a typical flight. The output prediction is quite accurate yielding quite small residual values. Furthermore, approximate uncorrelatedness of the residual sequence is obtained, and the

7 Integral Minimum Variance-Like Control for Pooled Nonlinear Representations with Application to an Aircraft System 7 standard cross correlation tests between the input signal and the residuals (see Billings and Woon 1986) are satisfied. Hence, the CCP-NARMAX model of the aircraft system admits the form (3) with: A l (q 1 )=θ 1 + θ 2 q 1 + θ 3 q 2 + θ 4 q 4 + θ 5 q 7 + θ 6 q 8 + θ 7 q 12 + θ 8 q 14 (7) G u [t 10] = θ 10 u[t 10] + θ 11 u[t 13] (8) and G F [t 1], the remaining terms of F nl (, [t 1]) which do not include the signal u[t 1] (see Table 1). Hence, the aircraft system model becomes: y[t] =A l y[t 1] + G u [t 10]u[t 1] + G F [t 1] + e[t] (9) In subsequent sections it will become clear, however, that this particular expression of G u [t 10] poses serious problems to the design of a minimum variance control law. 4 Integral Minimum Variance Control Law Derivation 4.1 Standard Control Law According to the minimum variance principle, the controller is designed to minimize the following performance index: {( ) 2 ( ) 2 } J[t + k] =E P (q 1 )y[t + k] + Q(q 1 t )f(u[t]) (10) with E{ t} designating the conditional expectation, P (q 1 ),Q(q 1 ) linear filters acting as dynamic weights, and f(u[t]) a function that includes all the terms in u[t] that are in the model (Sales and Billings, 1990). Since the objective is to design an integral minimum variance controller, P (q 1 ) assumes an integral form: P (q 1 )= k I(1 aq 1 ) 1 q 1 = k I + k I(1 a) 1 q 1 q 1 (11a) Moreover, f(u[t]) has the following form: f(u[t]) = F nl (u[t],...,u[t n u +1];e[t + k 1],...,e[t + k n e ]) F nl (u[t 1],..., u[t n u +1];e[t + k 1],...,e[t + k n e ]) (11b) Note that no cancellation of unit poles is possible since the plant (9) satisfies the Assumption 2 in section II. The values of k I, a in (11a) are tuned so that more weight is put on lower frequencies, thus placing a heavy penalty on the low frequency regulating errors, as in Grimble (2004).

8 8 Integral Minimum Variance-Like Control for Pooled Nonlinear Representations with Application to an Aircraft System Expanding (10) with the help of (1) and (11a)-(11b) gives: {( J[t + k] = E PA l y[t + k 1] + PF nl ( ;[t + k 1]) ) 2 } {( ) t 2 } t +(P k I )e[t + k]+qf(u[t]) + E k I e[t + k] (12) because the only element of Pe[t+k] that is uncorrelated with other terms in F nl ( ;[t+k 1]) is k I e[t+k]. Differentiating (12) with respect to u[t] yields: J[t + k] u[t] ( ) F =2E{ PA l y[t + k 1] + PF nl nl ( ;[t + k 1]) ( ;[t + k 1]) + (P k I )e[t + k]) u[t] +q 0 Qf(u[t]) f(u[t]) u[t] } t (13) However from (11b) we have: f(u[t]) u[t] = Fnl ( ;[t + k 1]) u[t] (14) Thus, inserting (14) into (13) and setting the result equal to zero, gives the following control law equation: { } E PA l y[t + k 1] + PF nl ( ;[t + k 1]) + (P k I )e[t + k]+q 0 Qf(u[t]) = 0 (15) Note that for k = 1, the conditional expectation operator E{ t} in (15) is dropped, because the expression contains known (present and past) values of the model s output, input and disturbance. From (2), and using (9) and (11b) it follows: f(u[t]) = G u [t 9]u[t] (16) Using (9), (11a), Q = q 0 in (15) (mainly for simplicity), the fact that k = 1 (see Table 1) and solving (15) with respect to the controlled variable u[t], the integral minimum variance control law is: u[t] = 1 (k I + q0 2)G H[t] (17a) u[t 9] ( ) H[t] =P A l y[t]+g F [t] + k I(1 a) ( ) 1 q 1 e[t]+g u [t 10]u[t 1] (17b) Note that, from (17a), it is quite obvious that when G u [ ] attains small values, u[t] becomes very large. It is easy to verify that even if the performance index J[t+1] is finite, the control law will not be computable.

9 Integral Minimum Variance-Like Control for Pooled Nonlinear Representations with Application to an Aircraft System Modified Minimum Variance Control Law In order to obtain a well defined control value when G u [ ] becomes very small, the design parameter k b is introduced, as follows: u Mod [t] = 1 (k I + q 2 0 )G u[t 9] + k b H[t] (18) Inserting (18) into the control law equation of (15) yields (see also the appendix): PA l y[t]+pf nl ( ;[t]) + (P k I )e[t]+q 0 Qf(u[t]) = Λ[t]H[t] (19a) Λ[t] = k b (k I + q 2 0 )G u[t 9] + k b (19b) The term k b permits shifting the threshold of (a rapidly growing) u[t], from G u [ ] =0toauser-chosen value, and is thus application-dependent. For instance, the choice of k b for the given aircraft system will be explained in section VI. Generally, k b may admit a constant user defined value or even be a function of G u [ ], as will be shown later. In this last case, the function k b is chosen such that the control law behaves as a pure minimum variance one when G u [ ] is of a certain magnitude and as a modified minimum variance one during the rest of the time. In the following section, the stability analysis of the closed loop system (9) and (17a)-(18) will be detailed for the case of a constant k b. 5 Stability Analysis of the Controlled Aircraft System As mentioned in the previous section, the switching strategy between the controllers (17a) and (18) depends upon the value of G u [t]. Consider a critical time instant t = t where for the first time G u [t ] <ɛ(but G u [t ] 0), with ɛ being a small user-defined number. Then, to avoid difficulties with the control law (17a), a switch to the controller (18) is performed. Replacing (18) into (12), and performing standard calculus using (13) and (14) for k = 1, the associated minimum variance performance index becomes (see Appendix A): [ ] J Mod [t q0 2 +1]= (q2 0 +1)G2 u[t 9] 2q0 2 ] 2 + [(k G u[t 9]k b ] 2 +Λ I + q0 2)G u[t 9] + k b [(k 2 [t ] H 2 [t ] I + q0 2)G u[t 9] + k b { } +p 2 0E e 2 Mod [t +1] (20) with Λ[t] defined in equation (19b), and the argument t in the conditional expectation operator dropped for simplicity reasons. At the same time, if the standard minimum variance controller (17a) had been used, the associated performance index [using equations (17a), (12), (13) and (14)- see also Appendix A] would

10 10 Integral Minimum Variance-Like Control for Pooled Nonlinear Representations with Application to an Aircraft System have been equal to: J MV [t +1]= q2 0 (q2 0 +1) { } (k I + q0 2)2 H2 [t ]+p 2 0E e 2 MV [t +1] (21) Since J MV [t +1] < (because it results from a minimization of (12) with J[t +1] < ), the signal H[t ] in (21) (composed from signals obtained for t t ) is a bounded quantity. Hence, J Mod [t +1]< in (20). However, due to J Mod [t + 1] resulting from a suboptimal control law, it should be expected that J MV [t +1] <J Mod [t + 1]. In fact, an approximate measure of the magnitude of J Mod [t +1] can be obtained. Considering (20), the first term inside the brackets may be rewritten as follows: q 2 0(q 2 0+1)G 2 u[t 9] [(k I+q 20)G u[t 9]+k b ] 2 = q 2 0(q 2 0+1)G 2 u[t 9] (k I+q0) 2 2 G 2 u[t 9]+ 2(k I+q0)k 2 bg u[t 9]+kb 2 1 = [ ] (k I +q 2 0 )2 q 2 0 (q2 0 +1) + [ 2(k I +q 2 0 )k b Gu[t 9]+k 2 b q 2 0 (q2 0 +1)G2 u [t 9] = 1 (k I +q 2 0 )2 q 2 0 (q2 0 +1) +M 2 ±M O( 1 M ) ] (22) with O( ) standing for order of magnitude and M a generic term for a finite large number depending on the choice of ɛ. Repeating the same procedure for the second term in (20), J Mod [t + 1] becomes: [ ] J Mod [t +1] O( 1 M ) ± O( 1 { } M )+Λ2 [t] H 2 [t ]+p 2 0E e 2 Mod [t +1] (23) with Λ[t ] 1 (from (19b) when G u [t 9] << k b ). Furthermore, in (21) q2 0(q0+1) 2 (k I+q < 1, or using standard 0) 2 2 calculus k I >> q q 0 q , which is always met in practice: k I admits substantial values due to P (q 1 ) in (11a) being a filter with integral behavior and q 0 >> 1 to account for non-minimum phase dynamics. Thus, assuming that the variances of the noise sequences e MV [t] and e Mod [t] are comparable to each other, the following inequality holds: J MV [t +1]<J Mod [t +1]< (24) Note, that proceeding in a same way, similar expressions may be obtained for J Mod [t +2]...J Mod [t + s], s>2. Consequently, the minimum variance-like controller results, at worst, to a slight increase of the performance index value in (12), but the latter is bounded and so are y[t] and u[t]. Then, as soon as G u [ ] becomes larger than ɛ, a switch to the minimum variance controller ensures that the performance index value is again minimal. The result is summarized as: Theorem 5.1 The system (9), satisfying Assumptions 1 and 2 in section II, under the control action (17a)- (17b) and (18) is stable and the I/O signals involved are, at worst, bounded.

11 Integral Minimum Variance-Like Control for Pooled Nonlinear Representations with Application to an Aircraft System 11 6 Aircraft System Control Results In this section the results of the control strategy (17a)-(17b) and (18) applied on the aircraft system will be detailed. For this particular system k b is chosen as: k b [t] = γ exp( G u [t 9] ) (25) with γ>0 a user defined positive number. Hence, the controller behaves as a pure MV one when G u [ ] is significant and as MV-like one when G u [ ] becomes small with respect to k b in (25) at that particular time instant. Thus, the transition between the (theoretical) control laws (17a)-(17b) and (18) is substantially simplified. As explained in the previous sections, the modelled aircraft I/O relation involves the pilot stick as control input and the aircraft pitch rate as output. Hence, all other pilot inputs (wheel and pedal, see Fig. 1) are considered as disturbances. For the present case, the pedal has been set to zero, since it does not really affect the pitch rate response for the considered flight scenarios. The wheel input, however, significantly affects the pitch rate response. During the test procedure, it is used as a structured disturbance (that is, of known form) to the modelled I/O relationship. Finally, the external turbulence is also considered as an additional source of (unstructured) disturbance. Hence, in the following tests the modelled I/O relationship is affected either by one (wheel) or two simultaneous disturbances (wheel and increased turbulence). A PID design is chosen as a benchmark controller, with parameters computed using a relevant nonlinear optimization procedure in the MATLAB TM Optimization Toolbox. The objective is to minimize the control error sum of squares over a given time period, resulting from the plant model (9) and the selected regulation set-point (equal to zero in this case). A Ziegler-Nichols gain selection procedure has also been used, but (as expected) the results have proven inferior. The discrete PID gains are thus computed as kp PID =5.6221, ki PID = and kd PID = On the other hand, the parameter γ =1.75 in (25), while in (18) k I = 1 and q 0 = 30. Since γ is application dependent, for the current case it is selected as follows: From (8) and Table I, G u O(10 6 ) u. However, the stick saturations for the given system are of O(10 2 ). Then max[ G u ] O(10 4 ), or [using (25)] γ O(10 2 ). The pitch rate response presented in Fig. 3 corresponds to an aircraft landing under low [(a),(b)] or increased [(c),(d)] turbulence with added disturbance from the wheel input [one pulse of +5 lbs and two of -6 lbs as seen in Figs. 3(b),(d)]. It is clearly seen that the MV-like controller features less overshoot (first pulse after t=10 s) and regulates the output faster than the PID (two successive pulses after t=30 s) in the low turbulence case. The PID controlled pitch rate signal is smoother than the MV-like one, but gets more slowly regulated around zero after a disturbance. This means that the aircraft stays more off-course. Similar results are obtained under increased turbulence external conditions, with a noisy behavior being

12 12 Integral Minimum Variance-Like Control for Pooled Nonlinear Representations with Application to an Aircraft System the main characteristic of the pitch rate response under both control schemes. 7 Conclusions In this work, an integral minimum variance like control design has been presented. The controller is designed for an affine (in the most recent input variable) CCP-NARMAX model, representing plants with fast dynamics such as aircraft systems. It has been suitably modified to account for possible singularities arising from the affine representation variable gain. Finally, the control strategy has been applied to a specific case of aircraft system: The regulation of the pitch rate signal around zero for a landing aircraft under low or increased turbulence and wheel input disturbances. Results have been quite satisfactory, notably featuring a restrained pitch rate amplitude (with respect to classical PID design) and low overall pitch rate variation. Acknowledgements Research supported by the European Commission [STREP project No on Innovative Future Air Transport System (IFATS) ]. The authors wish to thank the anonymous reviewers for their comments and suggestions to improve the paper. Appendix A: Derivation of J Mod [t + 1] and J MV [t +1] In this appendix the derivation of J Mod [t + 1] in (20) and J MV [t + 1] in (21) (that is, the performance indices obtained through the application at time instant t of the modified and the standard MV control laws, respectively, to the system) is presented in detail. A preliminary result concerns the derivation of (19a) for k = 1 and Q = q 0. From (2), (11a), the right hand part of (15), (11b) and the fact that p 0 = k I, it follows: PA l y[t]+pg u [t 9]u Mod [t]+pg F [t]+(p p 0 )e[t +1]+q0 2f(u Mod[t]) = PA l y[t]+pg u [t 9]u Mod [t]+pg F [t]+(p p 0 )e[t +1]+q 2 0 G u[t 9]u Mod [t] = (A1) PA l y[t]+ ki(1 a) 1 q 1 Then, from (17b), (18) and (A1), it results: G u [t 10]u Mod [t 1] + PG F [t]+ ki(1 a) 1 q e[t]+(k 1 I + q0 2)G u[t 9]u Mod [t] PA l y[t]+pg u [t 9]u Mod [t]+pg F [t]+(p p 0 )e[t +1]+q 2 0 G u[t 9]u Mod [t] = [ 1 (ki+q2 0)G u[t 9] ][ ki(1 a) (k I+q0)G 2 u[t 9]+k b 1 q 1 ] G u [t 10]u Mod [t 1] + PA l y[t]+pg F [t] ki(1 a) 1 q e[t] =Λ[t]H[t] 1 (A2) with Λ as in (19b). Applying the operator P (q 1 ) to (9) and adding and subtracting the term p 0 e[t +1]

13 Integral Minimum Variance-Like Control for Pooled Nonlinear Representations with Application to an Aircraft System 13 yields at instant [t +1]: Py[t +1]=PA l y[t]+pg u [t 9]u Mod [t]+pg F [t]+(p p 0 )e[t +1]+p 0 e[t + 1] (A3) From (11a), (A2) and (A3) the term Py[t + 1] is computed as follows: Py[t +1]=p 0 e[t +1] q 2 0G u [t 9]u Mod [t]+λ[t]h[t] (A4) Replacing (A3) into (10) and omitting (for simplicity reasons) the argument t from the conditional expectation operation, for k = 1 the associated modified minimum variance performance index becomes: J Mod [t +1]=E{(p 0 e[t +1] q 2 0 G u[t 9]u Mod [t]+λ[t]h[t]) 2 +(q 0 G u [t 9]u Mod [t]) 2 } = E{(p 0 e[t +1]) 2 } +(q 2 0 G u[t 9]u Mod [t]) 2 2q 2 0 G u[t 9]u Mod [t]λ[t]h[t] (A5) +(Λ[t]H[t]) 2 +(q 0 G u [t 9]u Mod [t]) 2 Inserting (18) into (A5) and using (19b) where appropriate, it follows: G 2 u[t 9] J Mod [t +1]=E{(p 0 e[t +1]) 2 } + q0 4 [(k I+q0)G 2 u[t 9]+k b] H 2 [t] 2 G u[t 9] +2q0 2 (k I+q0)G 2 u[t 9]+k b Λ[t]H 2 [t]+λ 2 [t]h 2 [t] (A6) +q 2 0 G 2 u [t 9] [(k I+q 2 0)G u[t 9]+k b] 2 H 2 [t] Grouping together the second and last term of (A6) and performing standard calculus leads to (20) for t = t. The derivation of J MV [t + 1] in (21) is carried out similarly. From (15), (11b) and the fact that p 0 = k I it follows: PA l y[t]+pg u [t 9]u[t]+PG F [t]+(p p 0 )e[t +1]+q 2 0G u [t 9]u[t] = 0 (A7) Note that, contrary to (A1), there is no Λ[t]H[t] term, as in the (suboptimal) case when u = u Mod. Hence: Py[t +1]=p 0 e[t +1] q 2 0G u [t 9]u[t] (A8) Replacing (A8) into (10), for k = 1 the associated minimum variance performance index becomes: J MV [t +1]=E{(p 0 e[t +1] q 2 0 G u[t 9]u[t]) 2 +(q 0 G u [t 9]u[t]) 2 } = E{p 0 e[t +1] 2 } +(q 2 0 G u[t 9]u[t]) 2 +(q 0 G u [t 9]u[t]) 2 (A9) = q 2 0 (q )(G u[t 9]u[t]) 2 + E{p 0 e[t +1] 2 } Replacing (17a) into (A9) and performing standard calculus leads to (21).

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16 16 Integral Minimum Variance-Like Control for Pooled Nonlinear Representations with Application to an Aircraft System Table 1. The CCP-NARMAX(15,15,15,1) Model Structure (landing flight regime, W= 31, 850 lbs, constant weight distribution, light turbulence). p 1 = y[t 1] p 24 = u[t 8] p 2 = y[t 2] p 25 = u[t 9] p 3 = y[t 3] p 26 = u[t 10] p 4 = y[t 5] p 27 = u[t 11] p 5 = y[t 8] p 28 = u[t 14] p 6 = y[t 9] p 29 = u[t 13] p 7 = y[t 13] p 30 = u[t 7] p 8 = y[t 15] p 31 = e[t 2]e[t 4] p 9 = y[t 1]e[t 2] p 32 = e[t 1]e[t 5] p 10 = u[t 1]u[t 10] p 33 = e[t 7]e[t 10] p 11 = u[t 1]u[t 13] p 34 = e[t 2]e[t 7] p 12 = u[t 5]u[t 14] p 35 = e[t 1]e[t 15] p 13 = u[t 3]u[t 12] p 36 = e[t 4]e[t 5] p 14 = u[t 8]u[t 15] p 37 = e[t 8]e[t 14] p 15 = u[t 11]u[t 14] p 38 = e[t 15] 2 p 16 = u[t 15] 2 p 39 = e[t 2] p 17 = u[t 15]e[t 2] p 40 = e[t 1] p 18 = u[t 14]e[t 1] p 41 = e[t 4] p 19 = u[t 14]e[t 15] p 42 = e[t 5] p 20 = u[t 3]e[t 15] p 43 = e[t 8] p 21 = u[t 14]e[t 2] p 44 = e[t 13] p 22 = u[t 6] p 45 = e[t 9] p 23 = u[t 15] u[t] : Pilot stick input (lbs) y[t] : aircraft pitch rate q Model orders: n y = 15, n u = 15, n e =15 Input delay : k = 1, Constant term: θ 0 =0 Flights used for parameter estimation: M = 40 Data record length for each flight: N = 5001 samples Data sampling frequency: f s = 100 Hz

17 Integral Minimum Variance-Like Control for Pooled Nonlinear Representations with Application to an Aircraft System 17 List of Figure Captions Figure 1. Aircraft model schematic. Figure 2. CCP-NARMAX(15,15,15,1) model performance: (a) actual pitch rate versus one-step-ahead prediction; (b) one-step-ahead prediction errors. Figure 3. PID versus MV-like control: (a) Pitch rate response (low turbulence); (b) Control input (low turbulence); (c) Pitch rate response (increased turbulence); (d) Control input (increased turbulence).

18 18 Integral Minimum Variance-Like Control for Pooled Nonlinear Representations with Application to an Aircraft System Wind Turbulence Stick Wheel Pedal AIRCRAFT Attitudes Angular rates Body Axes Accelerations Figure 1. Aircraft model schematic. pitch rate (deg/s) actual pitch rate one step ahead prediction (a) Figure 2. prediction error (deg/s) 5 x time (s) CCP-NARMAX(15,15,15,1) model performance: (a) actual pitch rate versus one-step-ahead prediction; (b) one-step-ahead prediction errors. (b) pitch rate (deg/s) 2 (a) 1 0 open loop system system with PID control system with MV control pitch rate (deg/s) 2 (c) 1 0 open loop system system with PID control system with MV control 1 1 Controlled input: Stick (lbs) 10 6 (b) PID control input 0 MV control input 4 10 disturbance time (s) Disturbance: Wheel (lbs) Controlled input: Stick (lbs) 10 6 (d) PID control input 0 MV control input 4 10 disturbance time (s) Disturbance: Wheel (lbs) Figure 3. PID versus MV-like control: (a) Pitch rate response (low turbulence); (b) Control input (low turbulence); (c) Pitch rate response (increased turbulence); (d) Control input (increased turbulence).