Parameter Estimation of Polytopic Models for a Linear Parameter Varying Aircraft System

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(13) Submit to Transactions of JSASS. Parameter Estimation of Polytopic Models for a Linear Parameter Varying Aircraft System Atsushi FUJIMORI and Lennart LJUNG†. Department of Mechanical Engineering, Shizuoka University 3-5-1 Johoku, Hamamatsu 432-8561, Japan Phone & Fax: +81-53-478-1064, Email: tmafuji@ipc.shizuoka.ac.jp †. Department of Electrical Engineering, Link o¨ ping University SE-581 83, Linköping, Sweden. Abstract This paper presents a parameter estimation method of continuous-time polytopic models for a linear parameter varying (LPV) aircraft system. The prediction error method for linear time invariant (LTI) models is modified for polytopic models. The modified prediction error method is applied to the parameter estimation of an LPV aircraft system whose varying parameter is the flight velocity and estimation parameters are the stability and control derivatives (SCDs). In an identification simulation, the SCDs of the initial polytopic model are adjusted so as to fit the response of the model to the data obtained from the original LPV aircraft system. Moreover, the polytopic model is more suitable for expressing the behavior of the LPV system than the LTI model from the viewpoints of the time and the frequency responses.. Key Words: model. Polytopic model, System identification, Prediction error method, Aircraft. Mailing Address : Atsushi Fujimori, Dr. Department of Mechanical Engineering, Shizuoka University 3-5-1 Johoku, Hamamatsu 432-8561, Japan Phone & Fax :. +81-53-478-1064. E-mail :. tmafuji@ipc.shizuoka.ac.jp.

(14) 1. Introduction Linearized equations of aircraft are regarded as linear time invariant (LTI) systems if the altitude and the flight velocity are constant, but linear parameter varying (LPV) systems if they are varying. Recently, a number of flight control designs in which the aircraft is treated as an LPV . system have been proposed by gain scheduling techniques 1 2 . In those gain scheduling designs, the LPV system is expressed or approximated by a polytopic model which is constructed by a linear combination of multiple LTI models at the vertices of the operating region 1 . Then, the constraints in the gain scheduling control design are expressed by linear matrix inequalities (LMIs)3 . A gain scheduling controller is obtained by solving the LMIs numerically. Unfortunately, in general it is not always possible to exactly transform an LPV system into a polytopic model. It depends on the structure of the LPV system 4 . One of the simplest ways for constructing a polytopic model is that multiple operating points are first chosen on the range of the varying parameters, an LTI model is obtained at each operating point, and a polytopic model is then constructed by interpolating between the operating points 1 . However, the polytopic model may contain an error in the interpolated region, although it depends on the interpolating function. The error is one of causes of conservative controllers. Therefore, it is worthwhile to obtain a polytopic model which is suitable for the original LPV system. The aim of this paper is to obtain a desirable polytopic model for an LPV aircraft system. Since an polytopic model is constructed by interpolating between the operating points as mentioned above, it may not suitably express the behavior of the original LPV system over the entire operating region. In this paper, a desirable polytopic model is found by a system identification technique5 . The system identification technique used in this paper is the prediction error method based on the quadratic criterion error in the time domain 5 . This paper modifies the standard prediction error method for LTI models to the parameter estimation of polytopic models. The modified prediction error method is applied to the longitudinal linearized equation of aircraft where the flight velocity is the varying parameter. The estimated polytopic model is evaluated by the time response and the ν -gap metric, which is a criterion associated with the. 1.

(15) model uncertainty6 .. 2. Polytopic Model and Objective of Parameter Estimation Let us consider a continuous-time LPV system given by dx t dt y t . Ac τ ξ τ x t Bc τ ξ τ v t . (1). Cc τ ξ τ x t Dc τ ξ τ v t . where x t , v t and y t are respectively the nx -dimensional state, nv -dimensional input and ny -dimensional output vectors. τ t is a measurable varying parameter with respect to time t, but the argument t is usually suppressed. ξ τ is the p-dimensional estimation parameter vector which is varying with respect to τ . In particular, when the matrices of Eq. (1) are written as the following polytopic form: r. Ac τ ξ τ Cc τ ξ τ . . ∑ wi τ Ai τi ξi . i 1 r. ∑ wi. . i 1. τ Ci τi ξi

(16) . Bc τ ξ τ Dc τ ξ τ . Eq. (1) is called a polytopic model, where wi τ i. r. ∑ wi. . i 1 r. ∑ wi. . . i 1. τ Bi τi ξi (2). τ Di τi ξi

(17) . 1 r are the weighting functions. satisfying the following relations. wi τ 0 r. ∑ wi. i 1. τ . . i. (3a). 1. (3b). τi is a frozen varying parameter and is called the i-th operating point. A i τi ξi , Bi τi ξi , Ci τi ξi and Di τi ξi are constant matrices with including a constant estimation parameter vector ξi at the i-th operating point. The set of A i Bi Ci Di is called the i-th local LTI model in this paper. The polytopic form (2) is constructed by interpolating between r operating points with the weighting functions satisfying Eqs. (3a) and (3b). When r . 1, w 1 τ . 1; that is, the . polytopic model becomes an LTI model. This paper considers the case where r. . 2. A i τi ξi . preserves the same structure as Ac τ ξ with respect to the estimation parameters. The same. 2.

(18) holds for Bi τi ξi , Ci τi ξi and Di τi ξi . The element of ξi is denoted as. ξi. . . ξ1 i ξ p i T . (4) . The polytopic model is one of blended multiple models 7 8 in which the varying parameter depends on the input and/or the state. In the polytopic model considered in this paper, the varying parameter τ is independent of these, but depends on time t. As pointed out in Section 1, in general it is not always possible to exactly transform an LPV system into a polytopic model. It depends on the structure of the LPV system. One of methods for constructing a polytopic model is that r frozen τ i i . 1 r ; that is, the operating points. are chosen on the range of τ . A local LTI model is obtained at each operating point. A polytopic model is then constructed by interpolating between the operating points. However, a model error may be included in the interpolated region. There are options for constructing the polytopic form (2) associated with the operating points and the weighting functions. In the former, when the number of the operating points is increased, the model error is decreased9 , but the polytopic model is complicated. In the latter, there are a number of candidates for the weighting functions satisfying Eqs. (3a) and (3b). One of the simplest weight functions is a triangular function whose center is at the operating point as shown in Fig. 1. Other weighting functions are introduced in Ref. 10). This paper does not discuss the selection of the weighting functions, furthermore. Anyway, it is assumed that the number of the operating points is determined and the weighting functions w i τ . i . 1 r . are given in advance. The objective of this paper is summarized as follows; using the input- and the output-data measured from the original LPV system v t y t and the varying parameter τ t , estimate p times r parameters ξl i l . 1 p i. 1 r in the polytopic form (2) so that the output . of the model denoted by yˆ t is fitted to the output-data y t as close as possible. To avoid over-parameterized estimation11 , the size of ξ is restricted as . p. nx nv n˜ y nv n˜ y. Dc. nx nv n˜ y . Dc. . . 0. (5). 0. 3.

(19) where n˜ y is the number of the output elements which are independent in the sense of the linear operator. A simple method for estimating the parameters is that the parameter estimation is done at each local LTI model by using data which are obtained at each operating point. However, when τ t is varied with time t, there is no guarantee that the response of the constructed polytopic model is fitted to that of the LPV system, especially in the intermediate region because the interpolation of Eq. (2) may be an approximated expression of the original LPV system. For this situation, the model error contained in the polytopic model should be made as small as possible. This is a motivation why p times r estimation parameters in the polytopic form (2) have to be estimated at the same time. A similar parameter estimation for blended multiple models in the discrete-time is discussed in Ref. 12). This paper shows the estimation of which parameters are related to physical systems in the continuous-time.. 3. Prediction Error Method for Polytopic Model This section shows the estimation computation in which the prediction error method for LTI models is modified for the case of polytopic models. Compared to the case of LTI models, there are two novelties in the case of polytopic models: the first is that the number of the estimation parameters is proportional to the number of chosen operating points. The second is an assumption on the discretization of the predictor and the gradient. In fact, both of them are caused to increase the computational burden on the parameter estimation.. 3.1. Predictor A predictor of a polytopic model Eq. (1) with Eq. (2) is given by d xˆ t ξ dt yˆ t ξ . . Ac τ ξ xˆ t ξ Bc τ ξ v t Cc τ ξ xˆ t ξ . . Dc τ ξ v t

(20) . 4. Kc τ ξ y t yˆ t ξ . (6).

(21) where Kc τ ξ is a filter gain which is given so that A c τ ξ . Kc τ ξ Cc τ ξ becomes a. stable matrix. Using Eq. (2) and giving Kc τ ξ by the following polytopic form: r. Kc τ ξ . ∑ wi. . τ Ki τi ξi . i 1. . (7). Eq. (6) is then written as d xˆ t ξ dt yˆ t ξ . Fc τ ξ xˆ t ξ Gc τ ξ z t . Cc τ ξ xˆ t ξ . (8). Hc τ ξ z t

(22) . where z t. . T. yT t vT t . . Fc τ ξ . r. Gc τ ξ Hc τ ξ . r. r. ∑ wi τ Ki τi ξi . . . r. . ∑ wi. i 1. j 1. . τ C j τ j ξ j . ∑ wi τ Bi τi ξi . i 1. 0. ∑ wj. τ Ai τi ξi Ki τi ξi . i 1. . r. . ∑ wi. . . i 1. Ki τi ξi . r. ∑ wj. j 1. τ D j τ j ξ j . τ Di τi ξi . 3.2. Discretization When the data z t used for estimation are sampled by a constant time period T , a discrete representation of Eq. (8) is derived under an assumption that τ t is frozen for each sampling interval;. τ t. τk . const . kT. t kT . . T. . (9). Then, Fc τ ξ , Gc τ ξ , Cc τ ξ and Hc τ ξ are constant for a fixed ξ . Applying the zero order hold discretization to Eq. (8), the following piecewise discrete-time predictor is obtained. xˆ kT. T ξ . yˆ kT ξ kT . . . F τk ξ x kT ξ . C τk ξ x kT ξ t kT . . . G τk ξ z kT . H τk ξ z kT . T

(23) . 5. (10).

(24) where F τk ξ . eFc . T . G τk ξ C τk ξ . . . τk ξ T. eFc. . τk ξ s. 0. Cc τk ξ

(25) . dsGc τk ξ H τk ξ . Hc τk ξ . For convenience, hereafter, the notation ‘kT ’ in x, ˆ yˆ and z is replaced by ‘t’ and τk is written as. τt in the discrete-time representation.. 3.3. Quadratic function and Gauss-Newton method Let N be the number of the sampled data. The data set Z N is defined as ZN. . . zT 1 . . zT N . . (11). . The function to be minimized is given by the following a quadratic function JN ξ Z N . 1 N 1 2 e t ξ

(26) N t∑. 12 . (12). where e t ξ is the prediction error vector defined as e t ξ . y t. yˆ t ξ . (13). Then, the estimated parameter is obtained as. ξˆ. arg min JN ξ Z N . . (14). ξ. Since the estimation parameter ξ is implicitly included in the prediction error e t ξ , J N can not be explicitly expressed with respect to ξ . The minimization of J N is then done numerically by an iterative calculation. In this paper, the Gauss-Newton method is used to search a ξ minimizing JN . For convenience, let the estimation parameter vector of the polytopic model be collectively denoted as. ξ . . ξ1T. . ξrT T . (15). 6.

(27) Let the superscript k be the k-th iteration. The estimation parameter vector is updated by. ξˆ. ξˆ. k 1. . µ. k. k. . HNk. 1 . . JN ξˆ . k. . ZN . (16). where JN ξˆ. k. . HNk . ψ t ξˆ. ZN . 1 N ψ t ξˆ N t∑. 1 . 1 N ψ t ξˆ N t∑. 1 k. . ∂ yˆ t ξ ∂ ξˆ . k. . k. ψ T t ξˆ. e t ξˆ . k. k. . (17). . (18). k . (19). HNk is called the Hessian and is usually invertible. µ. k. is the step size.. 3.4. Gradient Differentiating the piecewise discrete-time predictor (10) with respect to ξ l i , the gradient of the prediction error ψ t ξˆ is obtained as the following piecewise discrete-time state-space representation.. ∂ xˆ t 1 ξ ∂ ξl i ψ t ξˆ . . F τt ξ . C τt ξ . ∂ xˆ t ξ ∂ ξl i. ∂ xˆ t ξ ∂ ξl i. . . . ∂ F τt ξ ∂ ξl i. ∂ G τt ξ ∂ ξl i. . xˆ t ξ z t ξ . . ∂ C τt ξ ∂ ξl i. ∂ H τt ξ ∂ ξl i. . xˆ t ξ . (20). z t ξ. The derivatives of F τt ξ , G τt ξ , C τt ξ and H τt ξ are numerically obtained as. ∂ F τt ξ ∂ ξl i . F τt ξ . δl i F τt ξ. δl i . 2δl i. (21). where δl i is a small positive value. Before closing this section, some comments on the estimation computation will be given. The procedure of the computation described in this paper is essentially the same as that for LTI models except the following two points: one is that the number of the estimation parameters is proportional to the number of the operating points. Another is that the discretization for. 7.

(28) the predictor and the gradient calculation; that is, Eqs. (10) and (20) has to be done at every sampling because they are piecewise discrete-time LTI representations. To estimate the parameters accurately, all local LTI models have to be excited by the input v t and the varying parameter τ t . In particular, τ t should be varied over the entire range. This means that all the weighting functions w i τ . i . 1 r may be not constant during. measuring the data. Otherwise, the Hessian HNk in Eq. (16) may be singular, so the iterative calculation is stopped. In this situation, some local LTI models are not needed in the polytopic form (2). Let us consider a simple scenario that w1 τ is given by a constant. If w1 τ . 0,. the 1st local LTI model A1 B1 C1 D1 is not needed because of no contribution to Eq. (2). If w1 τ . 1, the rest of local LTI models are not needed. Moreover, if w1 τ is a constant in the . range of 0 w1 τ 1, Eq. (3b) is written as r. ∑1. i 2. wi w1. 1 . Replacing wi by w˜ i is constructed by r. 4.. . (22) wi 1. w1 , Eqs. (3a) and (3b) are satisfied. Then, the polytopic model. 1 local LTI models.. Identification Simulation of LPV Aircraft System. This section presents an identification simulation of an LPV aircraft system. The estimated parameters in the linearized aircraft equations are the stability and control derivatives (SCDs) which express linear contributions of the perturbed velocities and the angular rates to the aerodynamic forces or moments. The SCDs are varied according to the flight conditions; that is, the flight velocity and the altitude. This section demonstrates the identification simulation of an LPV aircraft system when the flight condition is varying. An LPV aircraft system for the longitudinal motion and a polytopic model are first shown in the following subsection. The estimation of the SCDs in the LTI and the polytopic models is shown in the next. The estimated models are evaluated by the time response and the ν -gap metric6 which is one of measures of model error.. 8.

(29) 4.1. LPV aircraft system for longitudinal motion In steady flight, the dynamics of aircraft can be generally divided into two parts; the longitudinal and the lateral motions. This paper considers the identification of the longitudinal motion which is regarded as an LPV system. The longitudinal motion in the continuous-time is expressed as the following linearized equations: du dt. Xα α g cos Θ0 θ. Xu u. 0 . dα Zα α V Zq q g sin Θ0 θ Zδe δe dt dα dq Mα˙ Mα α Mq q Mδe δe dt dt. Zu u V Mu u dθ dt. (23). q . u is the x-axis velocity, α the angle of attack, θ the pitch angle, q the pitch rate and δ e the elevator angle. The notations used in Eq. (23) are based on the symbols which have been usually used in flight dynamics 13 . The variables denoted by small letters mean the perturbed values. Θ0 is the pitch angle in the steady-state. g is the acceleration of gravity. In Eq. (23), there are nine stability derivatives; Xu, Xα , Zu , Zα , Zq , Mu , Mα , Mα˙ and Mq , and two control derivatives; Zδe and Mδe , which are varied with the flight velocity V and the altitude H. Since V is more considerably influenced on the characteristics of Eq. (23) rather than H, the varying parameter considered in this paper is τ V1. V . V and its range is given by. . V1 V2 . V2 . . (24). Defining x t , y t and v t as . . u x t. . θ α. . . . u. . . . . . . . . y t. θ . α. q. . . . v t . δe . (25). a continuous-time LPV system (1) for the longitudinal motion of aircraft is then written as dx t dt y t . . Ac V ξ V x t Cc xc t Dc v t . . Bc V ξ V v t . . 9. (26).

(30) where matrices in Eq. (26) are given by E c 1 Qc B c. Ac. Ec 1 Rc. . . . . . . 1 0. 0. 0. 0 1. 0. 0. . g cos Θ0 Xα. Xu. 0. . Ec. . . . . . 0 0. V. 0 0. Mα˙ 1. . 0. Qc. . . . 0. 0. g sin Θ0. Zu Mu. 0. 1. . . . . Zα. V. Mα. . Zq . Mq. (27). . 0. . . Rc. 0. . . 0 . . . 0 . . . . Zδe. . . . . 1 0 0 0. Cc. . 0 1 0 0 . . . . Dc . 0 0 1 0. Mδe. . 0 . 0. Collecting the SCDs, the estimation parameter vector in the LPV system is given by . ξ V . Xu Xα Zu Zα Zq Mu Mα Mα˙ Mq Zδe Mδe T . (28). The elements of y t are selected to avoid the over-parameterized estimation11 . A polytopic model of the longitudinal motion of aircraft is constructed as follows: two operating points are chosen at the edges of Eq. (24); that is, V . V1 V2 and two local LTI. models are obtained. Using linear interpolation as shown in Fig. 1, A c and Bc of the polytopic model are then constructed as Ac V ξ . 2. ∑ wi V Ai Vi ξi

(31) . i 1. Bc V ξ . 2. . ∑ wi V Bi Vi ξi . (29). i 1. where w1 V . V2 V V2 V1. w2 V . V V1 V2 V1. (30). The number of the estimation parameters is then 11. 2 . 22.. 4.2. Data for parameter estimation In the identification simulation, the flight velocity was changed in the range of Eq. (24). As an example, it was considered a situation that the flight velocity V t in the continuous-time was constantly accelerated as V t . V1 avt. (31). 10.

(32) where av was the acceleration. The input was given by a random binary signal. Using the flight velocity and the random input, the output-data were generated by the LPV aircraft system (26) which was converted to the discrete-time representation at each sampling as shown in discretization of the predictor in Section 3.2. The SCDs of the LPV aircraft system at each sampling were obtained by an analytical method based on the quasi-steady aerodynamic theory. They were given by13. ρV S ρV S Cxu 2CL tan Θ0

(33) Zu Czu 2CL 2m 2m ρV Sc ρV 2S Cmu Xα Cxα Mu 2Iyy 2m ρV 2 S ρV 2 Sc ρV Sc2 Czα Mα Zα Cmα Mα˙ C (32) 2m 2Iyy 4Iyy mα˙ ρV Sc ρV Sc2 Czq Mq Cmq Zq 4m 4Iyy ρV 2S ρV 2 Sc Zδe Czδe Mδe C 2m 2m mδe where m was the mass of aircraft, S the main wing area, c the main wing chord, and b the Xu. main wing span. CL was the lift coefficient. Cxu , Cmα , etc. were the non-dimensional stability and control derivatives and were obtained from the structural parameters of the aircraft 13 . The numerical values were referred from Ref. 14). The change of the altitude of aircraft due to the response was taken into consideration in the calculation of the SCDs. The number of the data was N . 100. The sampling time was given by T. acceleration of the flight velocity was given by av xc 0 . . . 0 5 sec. The. 2 m/s2 . The initial state was given by. 0. Using the flight velocity and the data explained above, The estimation of the SCDs. was done in the cases of the LTI and the polytopic models.. 4.3. Parameter estimation results (1). LTI model case. Table 1 shows the initial and the estimated SCDs in the case of LTI. model, where the initial SCDs were given by Eq. (32) where the flight velocity was V . 110. m/s. The estimated SCDs were changed from their initial values. Figure 2 shows the outputs of the initial and the estimated LTI models. The solid- and the dashed-dotted-lines mean the. 11.

(34) output-data and the outputs of the models, respectively. The responses of both LTI models were not well fitted to the data. The root mean square (RMS) of the prediction error for each output channel is used to evaluate the outputs of the models quantitatively. The RMS indicates the averaged amplitude of the prediction error. Table 2 shows the RMS of the prediction error for each output channel of the initial and the estimated LTI models, where e u , eα and eθ are the prediction errors of x-axis velocity, the angle of attack and the pitch angle, respectively. Although the estimated LTI model showed smaller values of the RMS than the initial LTI model, it was not enough to be acceptable. The ν -gap metric is one of criteria measuring the model error in the frequency domain. It had been introduced in robust control theories associated with the stability margin 6 . Let Pl pv s V be the transfer function of the LPV system where the varying parameter is V . Let Plti s be that of the initial or the estimated LTI model. The ν -gap metric between Pl pv s V and Plti s is defined as. δν Pl pv Plti . sup κ Pl pv jω V

(35) Plti jω . (33). ω. where. κ X Y where σ. . . . σ. I YY. . 1 2 . Y. X I XX. . 1 2 . . means the maximum singular value. The range is δ ν. . 0 1 . A large δν means. that the model error is large. Figure 3 shows the plots of δ ν Pl pv Plti , where the solid- and the dashed-dotted-lines indicate that Plti s is the estimated and the initial LTI models, respectively. Since the initial SCDs were given at V LTI model was zero at V V . . 110 m/s, δν Pl pv Plti whose Plti s was the initial. 110 m/s. However, it was increased when V was shifted from . 110 m/s. On the other hand, the minimum of δ ν Pl pv Plti whose Plti s was the estimated. LTI model was moved to V. 93 m/s. Similar to the initial LTI model, it was increased in other . flight condition. It was seen that the LTI model was not enough to express the characteristics of the original LPV system in the time and the frequency domains.. 12.

(36) (2). Polytopic model case. The result of estimation of the polytopic model is shown in. next. Table 3 shows the initial and the estimated SCDs of the polytopic models, where the initial SCDs were given as the values at V . V1 , numbered by #1, and V2 , numbered by #2,. by using Eq. (32). The a polytopic model using the initial SCDs has been often used in gain . scheduling control design3 9 . The estimated SCDs were moved from the initial values to the inside of the range (24). Figure 4 shows the outputs of the initial and the estimated polytopic models. The response of the estimated polytopic model was better fitted to the output-data than that of the initial polytopic model. It was also seen in the RMS as shown in Table 4. Letting Ppoly s V be the transfer function of the initial or the estimated polytopic model, Fig. 5 shows the plots of δν Pl pv Ppoly , where the solid- and the dashed-dotted-lines indicate that Ppoly s V is the estimated and the initial LTI models, respectively. δ ν Pl pv Ppoly whose Ppoly s V was the estimated polytopic model was smaller than that whose Ppoly s V was the initial polytopic model except near both edges of the flight region. Summarizing the identification simulation which has been shown so far, the polytopic model was more suitable for expressing the behavior of the LPV aircraft system than the LTI model from the viewpoints of the time and the frequency responses. Applying the prediction error method to the polytopic model, the parameters of the polytopic model were adjusted so as to fit the response of the model to that of the original LPV system and make the model error small over the entire flight region. (3). Response for other input-data and varying parameter. The estimated polytopic. models was evaluated by using other input-data and the varying parameter. Table 5 shows the RMS of the prediction error in which different random binary signals were used as the inputdata and the varying parameter; that is, the flight velocity V t was given by the following three types: increase, decrease and sinusoidal as shown in Fig. 6. The RMS of the prediction error was not constant for each trial since the input-data were given by different random binary signals at each trial. Therefore, the values of the RMS in Table 5 are the ten-trial-average for each type of V t . Compared to the prediction error by using the estimation input-data, the errors were. 13.

(37) increased by using different input-data and different types of V t . In particular, the change of the varying parameter to “Decreased” and “Sinusoidal” was influenced on the prediction error. Table 6 shows the reduction ratio of the RMS of the prediction error between the estimated and the initial polytopic models for each output channel. The reduction ratio for the x-axis velocity, for example, is defined as rd eu . RMS eu estimated RMS eu initial. 100 % . . (34). “Original” means the reduction ratio when the estimation input-data was used. Except the RMS of eu in the sinusoidal case, the prediction error of the estimated polytopic model was reduced compared to the initial model. This means that the parameter estimation of polytopic models is effective in improving the quality of the model.. 5. Concluding Remarks This paper has the presented a parameter estimation method of continuous-time polytopic models for an LPV aircraft system. The prediction error method for LTI models was modified for polytopic models. The modified prediction error method was applied to the parameter estimation of an LPV aircraft system whose varying parameter was the flight velocity and estimation parameters were the stability and control derivatives (SCDs). In an identification simulation, the SCDs of the initial polytopic model were adjusted so as to fit the response of the model to the data obtained from the LPV aircraft system. The polytopic model was more suitable for expressing the behavior of the LPV system than the LTI model from the viewpoints of the time response (prediction error) and the frequency response (ν -gap metric). A polytopic model is used as an approximated representation for nonlinear systems in which . the reference trajectory is given in advance9 15 . The presented technique is applicable for adjusting the models of such nonlinear systems. In this paper, the weighting functions were assumed to be given in advance. Including the parameters of the weighting functions into the estimation parameters, the quality of the estimated polytopic model will be improved. This is. 14.

(38) also a future subject of research.. References 1) Apkarian, P. Gahinet, P. and Becker, G.: Self-Scheduled. ∞. Control of Linear Parameter-. varying Systems: a Design Example, Automatica, 31 (1995), pp. 1251-1261. 2) Apkarian, P. and Adams, R. J.: Advanced Gain-Scheduling Techniques for Uncertain Systems, Advances in Linear Matrix Inequality Methods in Control, SIAM (2000), pp. 209-248. 3) Boyd, S., Ghaoui, L. E., Feron, E. and Balakrishnan, V.: Linear Matrix Inequalities in System and Control Theory, SIAM, Vol. 15, 1994. 4) Fujimori, A.: Descriptor Polytopic Model of Aircraft and Gain Scheduling State Feedback Control, Transactions on the Japan Society for Aeronautical and Space Sciences, 47 (2004), pp. 138-145. 5) Ljung, L.: System Identification - Theory for the User, 2nd edn, Prentice Hall, Upper Saddle River, NJ., 1999. 6) Vinnicombe, G.: Uncertainty and Feedback (. ∞. Loop-Shaping and the ν -Gap Metric),. Imperial College Press, 2001. 7) Leith, D. J., and Leithead, W. E.: Analytic Framework for Blended Multiple Model Systems Using Linear Local Models, Int. J. Control, 75 (1999), pp. 605-619. 8) Shorten, R.,Murray-Smith, R. Bjorgan, R. and Gollee, H.: On the Interpretation of Local Models in Blended Multiple Model Structures, Int. J. Control, 75 (1999), pp. 620-628. 9) Fujimori, A., Terui, F. and Nikiforuk, P. N.: Flight Control Designs Using ν -Gap Metric and Local Multi-Objective Gain-Scheduling, AIAA Paper, 2003-5414-CP, Guidance, Navigation and Control Conference, 2003, pp. 1729-1745.. 15.

(39) 10) Boukhris, A.,Mourot, G. and Ragot, J.: Non-Linear Dynamic System Identification: a Multi-Model Approach, Int. J. Control, 75 (1999), pp. 591-604. 11) McKelvey, T. and Helmersson, A.: System Identification Using an Over-Parameterized Model Class - Improving the Optimization Algorithm, Proc. 35th IEEE Conference on Decision and Control, 1997, pp. 2984-2989 12) Verdult, V.,Ljung, L. and Verhaegen, M.: Identification of Composite Local Linear StateSpace Models Using a Projected Gradient Search, Int. J. Control, 75 (2002), pp. 13851398. 13) Schmidt, L. V.: Introduction to Aircraft Flight Dynamics, AIAA, Reston, 1998. 14) Isozaki, K., Masuda, K., Taniuchi, A. and Watari, M.: Flight test Evaluation of Variable Stability Airplane, K.H.I. Technical Review, 75 (1980), pp. 50-58 (in Japanese). 15) Fujimori, A., Gunnarsson, S. and Norrlöf, M.: A Gain Scheduling Control of Nonlinear Systems Along a Reference Trajectory, Proc. of 16th IFAC World Congress.. 16.

(40) List of Figures 1. Triangular interpolative function. . . . . . . . . . . . . . . . . . . . . . . . . .. 2. Comparison between output-data and outputs of initial and estimated LTI models. 19. 3. ν -gap metric between Pl pv s V and Plti s in flight region, 50 V t . 4. Comparison between output-data and outputs of initial and estimated polytopic. . 150 m/s. 20. models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. 6. ν -gap metric between Pl pv s V and Ppoly s V in flight region, 50 V t . 18. . 21. 150. m/s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 22. Three types of flight velocity. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 22. List of Tables 1. Initial and estimated SCDs in case of LTI model. . . . . . . . . . . . . . . . .. 2. RMS of prediction error for each output channel of initial and estimated LTI. 23. models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 23. 3. Initial and estimated SCDs in case of polytopic model. . . . . . . . . . . . . .. 23. 4. RMS of prediction error for each output channel of initial and estimated polytopic models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 5. Ten-trial-average of RMS of prediction error of estimated polytopic model for different input and varying parameter. . . . . . . . . . . . . . . . . . . . . . .. 6. 24. 24. Reduction ratio of RMS of prediction error between estimated and initial polytopic models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 17. 24.

(41) 1. 0. .... Fig. 1.. wi -1. wi. wi +1. τ i -1. τi. τ i+1. .... Triangular interpolative function.. 18. τ.

(42) Data vs Initial LTI model 20. [deg]. [m/s]. 10. 0. θ. u. 0. -10. -20. -20 0. 25. 50. t. 0. 25. [s]. 50. t. [s]. 5. [deg]. data initial model. α. 0. -5 0. 25. 50. t. [s]. (a) Initial LTI model.. [deg]. 20. 0. 0. θ. u. [m/s]. Data vs Estimated LTI model 10. -10. -20. -20 0. 25. 50. t. 0. 5. [deg]. 25. 50. t. [s]. [s]. data estimated model. α. 0. -5 0. 25. 50. t. [s]. (b) Estimated LTI model. Fig. 2. Comparison between output-data and outputs of initial and estimated LTI models. 19.

(43) ν-gap metric bwtween P P lpv and lti 0.8 initial lti P estimatedlti P. δv. 0.6. 0.4. 0.2. 0 50. 75. 100. 125. V. Fig. 3.. 150. [m/s]. ν -gap metric between Pl pv s V and Plti s in flight region, 50 V t . 20. . 150 m/s..

(44) Data vs Initial polytopic model 20. [deg]. [m/s]. 10. 0. θ. u. 0. -10. -20. -20 0. 25. 50. t. 0. 25. 50. [s]. t. [s]. 5. [deg]. data initial model. α. 0. -5 0. 25. 50. t. [s]. (a) Initial polytopic model.. [deg]. 20. 0. 0. θ. u. [m/s]. Data vs Estimated polytopic model 10. -10. -20. -20 0. 25. 50. t. 0. 25. [s]. 50. t. 5. [s]. [deg]. data estimated model. α. 0. -5 0. 25. 50. t. [s]. (b) Estimated polytopic model. Fig. 4. Comparison between output-data and outputs of initial and estimated polytopic models. 21.

(45) ν-gap metric bwtween P P lpv and poly 0.4 initial polyP estimatedpoly P. δv. 0.3. 0.2. 0.1. 0 50. 75. 100. 125. V. ν -gap metric between Pl pv s V and Ppoly s V in flight region, 50. [m/s]. 100. 150 100. V. [m/s] V. 200. 150. 50 0 0. 50. 25. 50. t [s] Sinusoidal. 0 0. 150 100 50 0 0. 25. 50. t. 25. 50. t. 200. [m/s]. . Decreased. Increased 200. V. Fig. 5. m/s.. 150. [m/s]. [s]. Fig. 6. Three types of flight velocity.. 22. [s]. V t . 150.

(46) Table 1.. Initial and estimated SCDs in case of LTI model.. Model. Xu. Xα. Initial. -0.0237. 7.8829. -0.2423 -88.877 -1.6869. Estimated. -0.0452. 4.4446. -0.2469 -85.799 -19.583. Mu. Mα. Mq. 0 -0.0743. Zu. Mα˙. Zα. Zδe. Zq. Mδe. -5.2892 -1.0613 -0.3274 -5.9923 -3.7699 -32.227. 27.776. -35.420 -21.065 -9.6394. Table 2. RMS of prediction error for each output channel of initial and estimated LTI models. Model. eu m/s. eθ deg. eα deg. Initial. 1.7100 1.7706 0.9801. Estimated 1.1029 1.3528 0.9319. Table 3. Initial and estimated SCDs in case of polytopic model. Model. Xu. Xα. Initial #1. -0.0108. 1.6287. -0.1101 -18.363 -0.7668. Initial #2. -0.0324. 14.658. -0.3303 -165.27 -2.3003. Estimated #1 -0.0148. 0.9756. -0.1201 -17.392 -0.9337. Estimated #2 -0.0298. 12.609. -0.3065 -161.54 -1.5464. Mu. Mα. Mq. Zu. Mα˙. Zα. Zδe. Zq. Mδe. 0. -1.0928 -0.4824 -0.1488 -1.2381 -77.890. 0. -9.8353 -1.4472 -0.4465 -11.143 -7.0101. 0.0047. 0.2142. -3.0443. 2.2265. -1.0250 -0.5593. 0.0013. -7.9295 -1.8485. 0.1167. -11.374 -5.9544. 23.

(47) Table 4. RMS of prediction error for each output channel of initial and estimated polytopic models. Model. eu m/s. eθ deg. eα deg. Initial. 0.4115 0.6847 0.4228. Estimated 0.0108 0.1255 0.1086. Table 5. Ten-trial-average of RMS of prediction error of estimated polytopic model for different input and varying parameter. Type of V t . eu m/s. eθ deg. eα deg. Increased. 0.1442 0.2276 0.1618. Decreased. 0.3457 0.5398 0.2971. Sinusoidal. 1.1852 0.9987 0.5009. Table 6. Reduction ratio of RMS of prediction error between estimated and initial polytopic models. Type of V t . rd(eu ) %. Original. 2.6315. 18.329. 25.697. Increased. 49.019. 36.827. 33.171. Decreased. 74.187. 66.591. 55.354. Sinusoidal. 107.808. 69.427. 72.059. 24. rd(eθ ) % rd(eα ) %.

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