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Proportional navigation guidance using predictive and time delay control
ARTICLE IN PRESS
Control Engineering Practice 14 (2006) 1445–1453 www.elsevier.com/locate/conengprac
Proportional navigation guidance using predictive and time delay control S.E. Talole, A. Ghosh, S.B. Phadke Institute of Armament Technology, Girinagar, Pune 411 025, India Received 4 March 2005; accepted 18 November 2005 Available online 27 December 2005
Abstract A new formulation of the proportional navigation guidance law using the continuous time nonlinear predictive control approach is proposed. The guidance law needs information about the target acceleration for its implementation, which is generally not available. In this paper, this problem is addressed by estimating the target acceleration using the time delay control (TDC). The effectiveness of the guidance law and the estimation of the target acceleration is demonstrated by simulation in a realistic scenario against a highly maneuvering target. r 2005 Elsevier Ltd. All rights reserved. Keywords: Predictive control; Proportional navigation; Time delay control; Target acceleration estimation
1. Introduction Designing homing guidance laws for tactical missiles has been an area of intense research owing to the stringent performance requirements imposed upon them. Although the classical proportional navigation guidance (PNG) is one of the most widely proposed strategies in the homing phase, there exist situations such as the maneuvering target scenario where the simple PNG may not offer satisfactory performance (Lin, 1991). This fact has given rise to various variants of PNG such as the augmented proportional navigation (Nesline & Zarchan, 1981), modified proportional navigation (Ha, Hur, Ko, & Song, 1990), switched bias proportional navigation (Babu, Sarma, & Swamy, 1994) and proportional navigation based on predictive control (Talole & Banavar, 1998) as well as many optimal guidance laws (Asher & Matuszewski, 1974; Speyer, 1976; Salmond, 1996; Zarchan, 1997) which are derived by employing the optimal control theory. Formulation based on the well-known feedback linearization approach (Bezick, Rusnak, & Grey, 1995) is another example of
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E-mail address: [email protected] (S.E. Talole). 0967-0661/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.conengprac.2005.11.003
advanced guidance strategy proposed for tactical missiles engaging maneuvering targets. In this paper, an output tracking formulation of the continuous time nonlinear predictive control approach (Lu, 1994, 1995) is employed to derive a new proportional navigation guidance law. The resulting guidance law is optimal as it is obtained through minimization of a objective function based on the predicted errors. It is shown that under certain assumptions, this guidance law exhibits similarity with some of the well known guidance formulations. Closed loop stability of the system under the guidance law is studied and related results are presented. The predictive proportional navigation guidance (PPNG) law proposed here, like many other such formulations, needs information about target acceleration for its implementation. Since target acceleration cannot be measured directly by on-board sensors, it is necessary to estimate the same. To address this issue, we propose a novel approach based on the time delay control (TDC) (Youcef-Toumi & Ito, 1990) to estimate the target acceleration. The guidance law proposed here is similar to that proposed in Talole and Banavar (1998), in the sense that both are based on the predictive control approach. However in Talole and Banavar (1998) the target acceleration is assumed to be available. The present paper
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overcomes this limitation by explicitly estimating the acceleration of a maneuvering target in real time. Recently an expository application of TDC for estimating target acceleration in homing guidance where the objective is to minimize the relative range from missile to target is presented in (Talole, Phadke, & Singh, 2005) with the simulation results for an ideal scenario. In this paper, TDC is used under more realistic conditions like nonconstant velocities of missile and target, magnitude constraint on commanded acceleration and a practical autopilot. The paper is organized as follows. Section 2 briefly reviews the nonlinear predictive control theory while formulation of the guidance law based on this approach and related results are presented in Section 3. In Section 4, a brief introduction to time delay control along with its application to estimation of target acceleration is given. Simulation results showing the performance of the proposed formulation in a practical scenario are presented in Section 5 and finally Section 6 concludes this work.
yðtÞ, such that the component of control appears for the first time. Define zðxðtÞ; dÞ ¼ ½z1 ðxðtÞ; dÞ; . . . ; zm ðxðtÞ; dÞT , where zi ¼ dLf ðci Þ þ
d2 2 dl i l i Lf ðci Þ þ þ L ðci Þ; 2! li ! f
i ¼ 1; . . . ; m, (4)
where ci is the ith component of vector cðxÞ, d40 is a real number designated as the prediction horizon and Lkf ðci Þ denotes the kth order Lie derivative of ci with respect to f. In general the Lie derivatives of a scalar function hðxÞ with respect to f ðxÞ are defined as L0f hðxÞ ¼ hðxÞ, qhðxÞ f ðxÞ, qx L2f hðxÞ ¼ Lf ðLf hðxÞÞ
L1f hðxÞ ¼
.. . Lkf hðxÞ ¼ Lf ðLk1 hðxÞÞ, f
2. Predictive control The continuous time nonlinear predictive control (Lu, 1994, 1995) offers a systematic approach towards designing controllers for nonlinear systems. In this approach the state or output response of a nonlinear dynamic system is predicted by appropriate expansions and a quadratic cost function based on the predicted difference between the actual response and the desired response and current control expenditure is minimized point-wise to obtain an optimal, nonlinear feedback control law. As the guidance law formulated in this paper is based on the output tracking formulation, a brief outline of this formulation is presented here for the sake of completeness. The reader is referred to Lu (1994, 1995) for complete development. Consider a nonlinear system described by x_ ¼ f ðxÞ þ GðxÞu,
(1)
y ¼ cðxÞ,
(2)
where
where Lkf hðxÞ is the kth order Lie derivative of hðxÞ with respect to f ðxÞ. To expose fully the influence of uðtÞ on yi ðt þ dÞ for a small d40, one may approximate each yi ðt þ dÞ by an li th order Taylor series at t. In doing so, one can express yðt þ dÞ as a vector function of uðtÞ in a compact form as yðt þ dÞ yðtÞ þ zðxðtÞ; dÞ þ LðdÞW ðxðtÞÞuðtÞ,
(5)
mm
is a diagonal matrix with the elements where LðdÞ 2 R on the main diagonal being Lii ¼
dl i ; li !
i ¼ 1; . . . ; m
(6)
and W ðxÞ 2 Rmm has each of its rows in the form of wi ¼ ½Lg1 ðLlf i 1 ðci ÞÞ; . . . ; Lgm ðLlf i 1 ðci ÞÞ;
i ¼ 1; . . . ; m.
(7)
Similarly, expanding the ith component of y% ðt þ dÞ in the li th order Taylor’s series yields y% ðt þ dÞ y% ðtÞ þ dðt; dÞ,
(8) m
where the ith component of dðt; dÞ 2 R is
f 9½f 1 f 2 f n T ; n
G9½g1 g2 gm
(3) m
and xðtÞ 2 X R is the state, uðtÞ 2 U R represents the control and yðtÞ 2 Rm is the output vector, where X and U are compact sets in Rn and Rm spaces respectively. The functions f : Rn ! Rn ; c : Rn ! Rm and G : Rn ! Rnm are continuously differentiable nonlinear functions. Suppose that the desired output trajectory is specified by y% ðtÞ, 0ptptf , which is an outcome of Eqs. (1)–(2) for some feasible reference control u% ðtÞ 2 U for all t 2 ½0; tf . Given the present output of the system, yðtÞ, at any instant t 2 ½0; tf , the current control uðtÞ determines the output response in the immediate future. To predict the response of the system due to present control input, we follow the following procedure. Let li ; i ¼ 1; . . . ; m, be the lowest order of the derivative of yi , the ith component of
d i ðt; dÞ ¼ dy_ %i ðtÞ þ
d2 % dli ðli Þ% y ðtÞ; y€ i ðtÞ þ þ 2! li ! i
i ¼ 1; . . . ; m, (9)
ðli Þ% yi ðtÞ
where is the li th differentiation of yi with respect to time. In order to find the current control uðtÞ that improves the tracking accuracy at next instant, consider a point wise minimization of the performance index that penalizes the output tracking error at ðt þ dÞ and current control expenditure uðtÞ %
J½uðtÞ ¼ 12½yðt þ dÞ y% ðt þ dÞT Q½yðt þ dÞ y% ðt þ dÞ þ 12 uT ðtÞRuðtÞ, mm
ð10Þ mm
is positive definite and R 2 R positive where Q 2 R semi-definite weighting matrices. Replace yðt þ dÞ and
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y% ðt þ dÞ in Eq. (10) by predictions given by Eqs. (5) and (8), respectively. The control that minimizes the performance index is obtained by setting qJ=qu equal to zero as
output of system given by Eq. (12) i.e. _ y ¼ y.
ð11Þ
where eðtÞ9yðtÞ y ðtÞ is the current output tracking error. To use the control Eq. (11), one needs to choose the controller parameters i.e. the weightings Q and R and the prediction horizon d. Proper choice of these parameters is vital as their values affect the tracking performance of the controller. %
3. Predictive proportional navigation guidance In this section, we present a formulation of the PPNG law by employing the predictive control approach presented in Section 2. To this end, consider a two dimensional engagement geometry as shown in Fig. 1, where the missile and target are treated as point masses. We assume that the velocity of missile and target remains constant throughout the engagement with missile velocity being greater than that of target. Further we neglect autopilot and seeker loop dynamics. Under these assumptions, the engagement model can be represented by the following differential equations (Babu et al., 1994)
% % y_ ðtÞ ¼ y€ ðtÞ ¼ 0.
Following the procedure outlined in Section 2 to the present problem and using the definitions of f ðxÞ and gðxÞ, one obtains the various quantities appearing in Eq. (11) as LðdÞ ¼ d, cosðgm yÞ , W ðxÞ ¼ R % _ e ¼ y_ y_ ¼ y, zðx; dÞ ¼
1 y_ ¼ ½vt sinðgt yÞ vm sinðgm yÞ, R
dðt; dÞ ¼ 0.
at , vt
g_ m ¼
am ac ¼ , v m vm
(12)
where R is the relative range from missile to target, y the LOS angle, gm ; gt the flight path angles of missile and target, at the lateral acceleration of target, and ac , am the commanded and achieved lateral accelerations of missile.
(14)
Now one can re-write the single input system of Eq. (12) in the general form of Eq. (1) by defining the state x ¼ ½R y gm gt T , the input u ¼ am and the nonlinear functions f ðxÞ and GðxÞ ¼ gðxÞ as 2 3 2 3 0 vt cosðgt yÞ vm cosðgm yÞ 6 7 6 1 ½vt sinðg yÞ vm sinðg yÞ 7 607 t m 6 7 6 7 f ðxÞ ¼ 6 R ; gðxÞ ¼ 7 at 6 0 7. 4 5 4 5 vt 1 0 vm
R_ ¼ vt cosðgt yÞ vm cosðgm yÞ,
g_ t ¼
(13)
One can notice that with y_ as output, the relative degree is one and the desired reference output trajectory is
uðtÞ ¼ ½ðLðdÞW ðxÞÞT QLðdÞW ðxÞ þ R1 ½ðLðdÞW ðxÞÞT QðeðtÞ þ zðx; dÞ dðt; dÞÞ,
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d __ ð2Ry þ at cosðgt yÞÞ, R ð15Þ
Since the system represented by Eqs. (12) and (13) is single input single output, the weightings Q and R appearing in control Eq. (11) are scalar. Representing the weighting on predicted error by q and weighting on control expenditure by r and substituting Eq. (15) in Eq. (11) yields the PPNG law as ac (PPN) ¼
d2 q cosðgm yÞ ½d2 q cos2 ðgm yÞ þ rR2 ( ) Ry_ 2R_ y_ þ at cosðgt yÞ , d
ð16Þ
3.1. Formulation of guidance law The objective of proportional navigation is to drive the _ between missile and its target to zero while LOS rate, y, closing on the target. To ensure this, we choose y_ as the
Fig. 1. Two-dimensional engagement geometry.
where ac is the commanded lateral acceleration of the missile. The PPNG law (16), under certain assumptions, exhibits similarity with some of the known guidance laws appeared in literature. Bezick et al. (1995) have presented the design of input–output feedback linearizing homing guidance law based on the geometric control theory where the relative range R from missile to target, is chosen as output of the system (12). In the present case with LOS rate as output, one can show that the guidance law (16) is a special case of the input–output feedback linearizing control law. To show this, consider the system given by Eq. (12) with y_ as output. One can verify that with y_ as output the relative degree is one i.e. one needs to differentiate the output at least once for control to appear explicitly. The resulting output
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dynamics as obtained by differentiating y_ once is 1 y_ ¼ y€ ¼ ½2R_ y_ þ at cosðgt yÞ ac cosðgm yÞ. R For r ¼ 0 Eq. (16) reduces to ( ) 1 Ry_ 2R_ y_ þ at cosðgt yÞ . ac ¼ cosðgm yÞ d
(17)
(18)
_ Substituting Eq. (18) in Eq. (17) and defining v ¼ y=d yields y_ ¼ v
(19)
which is a linear relationship between output and input. Thus the guidance law (16) with r ¼ 0 achieves input–output linearization. Next, one can verify that the guidance law (18) closely resembles to the modified proportional navigation guidance (MPNG) law proposed by Ha et al. (1990). Also the switched biased proportional navigation (SBPN) guidance (Babu et al., 1994) shows similarity with (18) except that the target acceleration term is replaced by the switched bias term. In fact, the authors have shown that the switched bias term represents an estimate of the target acceleration. 3.2. Stability To establish stability of error dynamics, consider the system given by Eq. (12) with y_ as output. It is important to note that y_ itself represents output tracking error as the % desired output history y_ ¼ 0. Thus to obtain the tracking error dynamics differentiate y_ once to get 1 y€ ¼ ½2R_ y_ þ at cosðgt yÞ ac cosðgm yÞ. (20) R Now substituting the guidance law given by Eq. (16) with r ¼ 0 into Eq. (20) yields y_ (21) y€ þ ¼ 0. d Since d, the time prediction horizon is always positive, y_ ! _ 0 for any value of yð0Þ, implying asymptotic stability for the error dynamics. From (21), one can note that d represents a time constant of the error dynamics. When ra0, the error dynamics takes a form y€ þ K 1 y_ ¼ U,
input-bounded output stability for the time-varying error dynamics given by Eq. (22). From Eq. (16), it is clear that the knowledge of the component of target acceleration viz. at cosðgt yÞ is crucial for implementation of the guidance law. However, this component cannot be obtained by direct measurement by any on board sensors. In this paper we propose a method to estimate this component using the time delay control (TDC) approach, which is presented in the next section.
(22)
where _ 2Rð1 K 2 Þ K 2 K1 ¼ þ , R d d2 q cos2 ðgm yÞ , K2 ¼ 2 ½d q cos2 ðgm yÞ þ rR2 at cosðgt yÞð1 K 2 Þ . U¼ R Clearly, if one chooses controller parameters q; r, and d such that K 1 ðtÞ40, and noting the fact that the quantity at cosðgt yÞ is always bounded, one can ensure bounded
4. Time delay control Many real plants are affected by significant uncertainties and large unmeasurable external disturbances. Several techniques such as adaptive control, variable structure control, Liapunov function based control have been proposed to handle such systems. TDC is another method that can tackle uncertain systems. The idea underlying TDC is presented by Youcef-Toumi and Ito (1990) and Youcef-Toumi and Reddy (1992). In TDC, a function representing the effect of uncertainties and external disturbances is estimated directly by using information in the recent past and then a control is designed using this estimate in such a way as to cancel out the effect of the unknown dynamics and the external disturbances. Several diverse applications of TDC such as active vibration absorber (Olgac & Holm-Hansen, 1994), robot manipulators (Chang, Kim, & Park, 1995), brush-less DC motor (Song & Byun, 2000), telescopic handler (Park & Chang, 2004) are reported. However no application of TDC to missile guidance problems is seen in literature. In this work, we employ the TDC for estimation of target acceleration needed in the guidance law of Eq. (16). 4.1. The concept of Time Delay Control Consider the following system: _ ¼ f ðx; tÞ þ Bðx; tÞuðx; tÞ þ Bðx; tÞhðx; u; tÞ, xðtÞ
(23)
where xðtÞ 2 Rn is the state, uðx; tÞ 2 Rm is the control, f ðx; tÞ and Bðx; tÞ are known functions and hðx; u; tÞ is lumped uncertainty including disturbances. Assumption 1. The function hðx; u; tÞ does not change appreciably in a small interval of time L: hðx; u; tÞ hðx; u; t LÞ.
(24)
Now from the system given by (23), we have _ f ðx; tÞ Bðx; tÞuðx; tÞ Bðx; tÞhðx; u; tÞ ¼ xðtÞ
(25)
and _ LÞ f ðx; t LÞ Bðx; t LÞhðx; u; t LÞ ¼ xðt Bðx; t LÞuðx; t LÞ.
ð26Þ
ARTICLE IN PRESS S.E. Talole et al. / Control Engineering Practice 14 (2006) 1445–1453
Therefore
which is implementable. It may be noted that the estimation of target acceleration obtained in Eq. (31) can be used in other guidance laws requiring the target acceleration information.
hðx; u; t LÞ ¼ ½BT ðx; t LÞBðx; t LÞ1 BT ðx; t LÞ _ LÞ f ðx; t LÞ ðxðt Bðx; t LÞuðx; t LÞÞ.
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ð27Þ 5. Simulation and results
Let ^ u; tÞ ¼ hðx; u; t LÞ. hðx;
(28)
^ u; tÞ can act as an estimate of In view of Assumption 1, hðx; the lumped uncertainty and disturbance function hðx; u; tÞ. ^ u; tÞ can It can be seen from Eq. (27) that the estimate hðx; be obtained from past measurement at ðt LÞ and an ^ u; tÞ can be used in _ LÞ. The estimate hðx; estimate of xðt control to nullify the effect of uncertainty in the system. In the next subsection, we apply the TDC methodology outlined above to the problem of estimation of target acceleration. 4.2. Estimation of target acceleration using TDC Consider the PPNG given by Eq. (16) reproduced once again for the sake of continuity ac (PPN) ¼
d2 q cosðgm yÞ ½d2 q cos2 ðgm yÞ þ rR2 ( ) Ry_ 2R_ y_ þ at cosðgt yÞ . d
ð29Þ
In general, the range, range rate, LOS angle and LOS rate are obtained by processing on-board seeker measurement and hence are assumed to be available. Thus the guidance law given by Eq. (29) can be implemented if one can estimate the quantity at cosðgt yÞ which essentially represents a component of target acceleration. Differentiating the LOS rate i.e. y_ yields 1 y€ ¼ ½2R_ y_ ac cosðgm yÞ þ at cosðgt yÞ, (30) R wherein at cosðgt yÞ can be considered as an external disturbance to be estimated. Defining hðtÞ ¼ at cosðgt yÞ ^ as its estimate and following the TDC concept and hðtÞ presented in earlier section, we have
In this section, the performance of PPNG is compared with PNG by simulation under the same assumptions as used to derive the guidance law of Eq. (16) and consequently Eq. (32). We consider the target to be maneuvering with time-varying acceleration. It is important to note that for interception of the target, it is _ necessary to ensure Ro0 while driving the LOS rate to zero. In Ha et al. (1990) analytical conditions under which target interception is guaranteed for the conventional PNG and MPNG laws are presented. Here as we are comparing the performance of the PPNG with PNG and since the PPNG under certain assumptions resembles the MPNG law, we consider the initial conditions of the missile–target engagement such that they satisfy the same analytical conditions as derived in Ha et al. (1990). Also, it is assumed that the angle between missile velocity vector and LOS remains acute throughout the engagement i.e. jgm yjo90 . The target acceleration profile is taken as at ¼ 80 m=s2
for 0otp2 s,
at ¼ 0 m=s2
for 2otp4 s,
at ¼ 80 m=s2
for t44 s.
The other necessary simulation data, as taken from Babu et al. (1994), is Rð0Þ ¼ 4500 m, yð0Þ ¼ 20 , gt ð0Þ ¼ 140 , gm ð0Þ ¼ 60 , and vm ¼ 500 m=s, vt ¼ 300 m=s. The performance of the PPNG (32) is compared with the conventional proportional navigation guidance law. Now re-writing (32) as
^ ¼ ½at cosðg yÞjtL hðtÞ t € ¼ ½Ry þ 2R_ y_ þ ac cosðgm yÞjtL ½at cosðgt yÞjt .
5.1. Simulation under ideal conditions
^ ac (PPN) ¼ KfN 0 R_ y_ hg, ð31Þ
Here it is assumed that the quantity at cosðgt yÞ does not change appreciably in a small time L, i.e. ½at cosðgt yÞjt ½at cosðgt yÞjtL . Replacing the target acceleration term in Eq. (29) by its estimate (31) leads to the guidance law ( ) d2 q cosðgm yÞ Ry_ ^ 2R_ y_ þ hðtÞ , ac (PPN) ¼ 2 ½d q cos2 ðgm yÞ þ rR2 d (32)
(33)
where d2 q cosðgm yÞ , ½d2 q cos2 ðgm yÞ þ rR2 _ R 2Rd N0 ¼ . _ Rd K¼
ð34Þ
The weightings q and r in Eq. (34) are to be tuned to achieve satisfactory tracking performance by observing the simulated responses. The term N 0 represents the effective navigation ratio. For the purpose of valid comparison, the navigation ratio of the PN guidance law is modified by
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including the K term of (33). With this modification, the resulting PN guidance law is
4500
_ ac (PN) ¼ KN 0 R_ y.
3500
The navigation ratio, N 0 , is chosen as 4 in both the cases. However, as is evident from (34), to keep N 0 ¼ 4 requires certain adjustments. To this end, we modulate the value of the prediction horizon d in (34) such that N 0 ¼ 4 throughout the engagement. The resulting value of d is R d¼ 40. 2R_
3000 2500 2000 1500 1000 500
(36)
0
actual estimated
20
2
3
4
5
6
7
8
9
0.01 PPNG PNG
0 -0.01 -0.02 -0.03 -0.04 -0.05
0
2
4
6
8
10
time (s) Fig. 4. LOS rate profiles.
0
PPNG PNG
-20 -40 -60 -80 -100 -120 -140 -160
40
1
Fig. 3. Relative range profiles.
LOS rate (rad/s)
_ _ € LÞ yðtÞ yðt LÞ yðt (37) L and is used in Eq. (31) to compute the estimate of target acceleration as needed in guidance law of Eq. (33). It is important to note that use of numerical differentiation is undesirable when the signal is noisy. However, the TDC needs the system states and their derivatives for its implementation. In the literature on TDC, the issue of obtaining the states and the effect of noise on numerically obtained state derivatives are addressed (Xu & Cao, 2000; Morioka, Sabanovic, Uchibori, Wada, & Oka, 2001). Some solutions not requiring numerical differentiation are also suggested. For example, a model reference observer is proposed in Chang and Lee (1996) to reconstruct the states and their derivatives in a stable manner and is used in a practical application for a SCARA-type robot. In this paper we concentrate on showing the application of TDC to estimate the target acceleration and modification of this formulation to address the issue of noisy measurement could form a separate study.
0
time (s)
commanded acceleration (m/s2)
The purpose of obtaining d through (36) is only to ensure that the effective navigation ratio is same in both the cases for meaningful comparison. It is worth noting here that the performance of the PPNG may be better for some other values of d than the one obtained from (36). The weightings in (34) are chosen as q ¼ 10 000 000 and r ¼ 0:00001. From the LOS rate, the estimates of y€ are obtained by numerical differentiation as
at cos(γt−θ) (m/s2)
relative range (m)
(35)
PPNG PNG
4000
0
2
4
6
8
10
time (s) Fig. 5. Commanded acceleration profiles.
0 -20 -40 -60 -80
0
1
2
3
4
5
6
time (s) Fig. 2. Estimation of target acceleration.
7
8
9
Using the above data and initial conditions, simulations are carried out for PPNG and PNG and the results are presented in Figs. 2–5. Firstly, in Fig. 2, we compare the estimate of at cosðgt yÞ with the actual value and one can note that the TDC based approach gives highly satisfactory estimate of the target acceleration component. In Fig. 3, relative range R for PPNG as well as PN guidance is presented and one can observe that the relative range R ! 0 as expected for both the guidance laws. Fig. 4 shows the
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LOS rates for both laws from which it can be seen that the LOS rate with predictive guidance goes to zero near target interception, unlike with PN guidance law. Fig. 5 shows the corresponding commanded accelerations obtained from both guidance laws till Rp5 meters and one can observe that the acceleration profile due to predictive guidance is less demanding as compared to that of PN guidance.
where xm , hm and xt , ht denote the ground range and height for missile and target respectively. The quantity W ¼ mg is the weight of the missile, m is the mass and g represents the gravitational acceleration. The force of drag D is frequently represented by (Imado, Nagayama, & Tahk, 2001; Kim & Kim, 2004) D ¼ k1 v2m þ k2
5.2. Simulation in a practical scenario In this section we demonstrate the effectiveness of the guidance law (32) under more realistic conditions. Here we relax the assumptions of constant velocities of missile and target as well as that of ideal autopilot. Further a magnitude constraint is placed on the commanded acceleration. Therefore, we consider the missile–target engagement geometry incorporating the system dynamics as shown in Fig. 6, following which the equations of motion of the missile–target engagement model in the vertical plane assuming small angle of attack are obtained as T D W sin gm , m am g cos gm g_ m ¼ , vm x_ m ¼ vm cos gm , h_m ¼ vm sin g ,
(39)
1 1 (40) a_ m ¼ am þ ac , t t where t ¼ 0:1 represents the time constant of the autopilot dynamics. Thrust T and mass m are modelled as (Kim & Kim, 2004) ( T 0 ; tp15; T¼ (41) 0; t415; ( m¼
_ m0 mt; mf ;
tp15; t415;
(42)
_ ¼ 2:66 kg=s, and where T 0 ¼ 5880 N, m0 ¼ 165 kg, m mf ¼ 125 kg. It is assumed that the missile acceleration is constrained as 15p 15p vm if ac X vm , ac ¼ 180 180 15p 15p vm if ac p vm . ð43Þ ac ¼ 180 180
m
t
a2m , v2m
where the values of k1 ¼ 0:001 and k2 ¼ 1:0 are taken from (Kim & Kim, 2004). The autopilot is modelled as a first order system
v_m ¼
R_ ¼ vt cosðgt yÞ vm cosðgm yÞ, 1 y_ ¼ ½vt sinðgt yÞ vm sinðgm yÞ, R at g_ t ¼ , vt x_ t ¼ vt cos gt , h_t ¼ vt sin g ,
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ð38Þ
The above constraint on missile acceleration is arising from constraint on maximum possible missile flight path rate of 15 =s. The initial position of the missile and the target are chosen as follows: xm ð0Þ ¼ 0 m;
hm ð0Þ ¼ 0 m,
xt ð0Þ ¼ 3000 m;
ht ð0Þ ¼ 4000 m.
It is assumed that the target velocity is time-varying as vt ¼ 200 þ 100 sinð0:02ptÞ m=s.
(44)
The target acceleration profile is at ¼ 80 m=s2
Fig. 6. Missile–target engagement geometry.
for 0otp5 s,
at ¼ 0 m=s2
for 5otp10 s,
at ¼ 80 m=s2
for t410 s.
Other necessary data used in simulation are: gt ð0Þ ¼ 140 , gm ð0Þ ¼ 90 , and vm ð0Þ ¼ 1 m=s. The initial conditions for Rð0Þ and yð0Þ are decided by the initial coordinates
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of the target hm ð0ÞÞ2 1=2 and
Rð0Þ ¼ ½ðxt ð0Þ xm ð0ÞÞ2 þ ðht ð0Þ
8000 7000
ht ð0Þ hm ð0Þ . xt ð0Þ xm ð0Þ
6000
The values of the controller parameters q; r and the predictive horizon d are chosen as q ¼ 100, r ¼ 0:00001
height (m)
yð0Þ ¼ tan1
i.e.
5000 4000 3000 2000
missile target
1000 6000
0
0
1000
relative range (m)
4000
5000
3000 2000
0
0
5
10
15
20
25
20
25
time (s) Fig. 7. Relative range vs. time.
0.04 0.03
LOS rate (rad/s)
3000
Fig. 10. Target and missile trajectories.
4000
1000
0.02 0.01 0 -0.01 -0.02 -0.03 -0.04
0
5
10
15
time (s) Fig. 8. LOS rate vs. time.
100
missile acceleration (m/s2)
2000
downrange (m)
5000
commanded actual
80
and d ¼ 10 by trial and error procedure by observing the simulated responses. The value L required in Eq. (37) is taken as 0:01. It assumed that the guidance commences when the missile acquires a velocity of 200 m=s. Using the above data and initial conditions, simulation is carried out by using the guidance law (32) employing the target acceleration estimate as given by Eq. (31) and the results are presented in Figs. 7–10. Firstly in Fig. 7, the relative range profile for PPNG is presented from where we observe that the relative range R ! 0 as expected. The corresponding LOS rate history is given in Fig. 8 and one can note that y_ ! 0 as desired. Fig. 9 presents the corresponding commanded and achieved acceleration profiles. It is important to note that the overshoots for the accelerations at 10 and 15 s are arising due to the sudden change in the magnitude of target acceleration at t ¼ 10 s and sudden change in the magnitude of thrust at t ¼ 15 s. Lastly, Fig. 10 presents the planar trajectories of target and missile from where we observe that the missile has successfully engaged the target. The above results clearly demonstrate that the PPNG using the TDC based target acceleration estimation offers satisfactory results. Simulations were also carried out by using the classical PNG instead of PPNG in the practical scenario, however, as expected, the results (not included in the paper) are far from satisfactory.
60
6. Conclusion
40 20 0 -20 -40 -60 -80
0
5
10
15
20
time (s) Fig. 9. Commanded and achieved acceleration vs. time.
25
In this paper, a new formulation of the proportional navigation guidance law using the continuous time nonlinear predictive control approach is proposed. The incorporation of TDC based estimation of target acceleration, has made the proposed guidance law implementable. This approach can possibly be used in other guidance laws requiring target acceleration. The efficacy of the proposed formulation is verified by simulation in a realistic scenario for a highly maneuvering target.
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References Asher, R. B., & Matuszewski, J. P. (1974). Optimal guidance with maneuvering targets. Journal of Spacecraft and Rockets, 11(3), 204–206. Babu, K. R., Sarma, I. G., & Swamy, K. N. (1994). Switched bias proportional navigation for homing guidance against highly maneuvering targets. Journal of Guidance, Control, and Dynamics, 17(6), 1357–1363. Bezick, S., Rusnak, I., & Grey, W. S. (1995). Guidance of a homing missile via nonlinear geometric control methods. Journal of Guidance, Control, and Dynamics, 18(3), 441–448. Chang, P. H., Kim, D. S., & Park, K. C. (1995). Robust force/position control of a robot manipulator using time delay control. Control Engineering Practice, 3(9), 1255–1264. Chang, P. H., & Lee, J. W. (1996). A model reference observer for time delay control and its application to robot trajectory control. IEEE Transactions on Control Systems Technology, 4(1), 2–10. Ha, I. J., Hur, J. S., Ko, M. S., & Song, T. L. (1990). Performance analysis of PNG laws for randomly maneuvering targets. IEEE Transactions on Aerospace and Electronic Systems, 26(5), 713–720. Imado, F., Nagayama, M., & Tahk, M.-j. (2001). An extended study on a new missile guidance algorithm against a maneuvering target. AIAA2001-4342, AIAA guidance, navigation, and control conference and exhibit, Montreal, Canada. Kim, K. S., & Kim, Y. (2004). Design of generalized conceptual guidance law using aim angle. Control Engineering Practice, 12(3), 291–298. Lin, C. F. (1991). Modern navigation, guidance, and control processing. Englewood Cliffs, New Jeresy: Prentice-Hall. Lu, P. (1994). Nonlinear predictive controllers for continuous systems. Journal of Guidance, Control, and Dynamics, 17(3), 553–560. Lu, P. (1995). Optimal predictive control of continuous nonlinear systems. International Journal of Control, 62(3), 633–649. Morioka, H., Sabanovic, A., Uchibori, A., Wada, K., & Oka, M. (2001). Application of time-delay-control in variable structure motion control systems. ISIE, Pusan, Korea, 1313–1318.
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Nesline, F. W., & Zarchan, P. (1981). A new look at classical vs modern homing missile guidance. Journal of Guidance, Control, and Dynamics, 4(1), 78–85. Olgac, N., & Holm-Hansen, B. T. (1994). A novel active vibration absorption technique: Delayed resonator. Journal of Sound and Vibration, 176(1), 93–104. Park, J. Y., & Chang, P. H. (2004). Vibration control of a telescopic handler using time delay control and commandless input shaping technique. Control Engineering Practice, 12(6), 769–780. Salmond, D. J. (1996). Foundations of modern missile guidance. Journal of Defence Science, 1(2), 171–179. Song, J.-B., & Byun, K.-S. (2000). Design of time delay controller using variable reference model. Control Engineering Practice, 8(5), 581–588. Speyer, J. L. (1976). An adaptive terminal guidance scheme based on an exponential cost criterion with application to homing missile guidance. IEEE Transactions on Automatic Control, AC-26(5), 371–375. Talole, S. E., & Banavar, R. N. (1998). Proportional navigation through predictive control. Journal of Guidance, Control, and Dynamics, 21(6), 1004–1006. Talole, S. E., Phadke, S. B., & Singh, R. K. (2005). Predictive homing guidance using time delay control. AIAA-2005-6159, AIAA guidance, navigation, and control conference and exhibit, San Francisco, CA, USA. Xu, J., & Cao, W. (2000). Synthesized sliding mode and time-delay control for a class of uncertain systems. Automatica, 36, 1909–1914. Youcef-Toumi, K., & Ito, O. (1990). A time delay controller for systems with unknown dynamics. Journal of Dynamic Systems, Measurement, and Control, Transactions of ASME, 112, 133–142. Youcef-Toumi, K., & Reddy, S. (1992). Analysis of linear time invariant systems with time delay. Journal of Dynamic Systems, Measurement, and Control, Transactions of ASME, 114, 544–555. Zarchan, P. (1997). Tactical and strategic missile guidance. Progress in astronautics and aeronautics. Vol. 176, Reston, VA: AIAA.
Control Engineering Practice 14 (2006) 1445–1453 www.elsevier.com/locate/conengprac
Proportional navigation guidance using predictive and time delay control S.E. Talole, A. Ghosh, S.B. Phadke Institute of Armament Technology, Girinagar, Pune 411 025, India Received 4 March 2005; accepted 18 November 2005 Available online 27 December 2005
Abstract A new formulation of the proportional navigation guidance law using the continuous time nonlinear predictive control approach is proposed. The guidance law needs information about the target acceleration for its implementation, which is generally not available. In this paper, this problem is addressed by estimating the target acceleration using the time delay control (TDC). The effectiveness of the guidance law and the estimation of the target acceleration is demonstrated by simulation in a realistic scenario against a highly maneuvering target. r 2005 Elsevier Ltd. All rights reserved. Keywords: Predictive control; Proportional navigation; Time delay control; Target acceleration estimation
1. Introduction Designing homing guidance laws for tactical missiles has been an area of intense research owing to the stringent performance requirements imposed upon them. Although the classical proportional navigation guidance (PNG) is one of the most widely proposed strategies in the homing phase, there exist situations such as the maneuvering target scenario where the simple PNG may not offer satisfactory performance (Lin, 1991). This fact has given rise to various variants of PNG such as the augmented proportional navigation (Nesline & Zarchan, 1981), modified proportional navigation (Ha, Hur, Ko, & Song, 1990), switched bias proportional navigation (Babu, Sarma, & Swamy, 1994) and proportional navigation based on predictive control (Talole & Banavar, 1998) as well as many optimal guidance laws (Asher & Matuszewski, 1974; Speyer, 1976; Salmond, 1996; Zarchan, 1997) which are derived by employing the optimal control theory. Formulation based on the well-known feedback linearization approach (Bezick, Rusnak, & Grey, 1995) is another example of
Corresponding author. Tel.: +91 2024 389 550; fax: +91 2024 389 411.
E-mail address: [email protected] (S.E. Talole). 0967-0661/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.conengprac.2005.11.003
advanced guidance strategy proposed for tactical missiles engaging maneuvering targets. In this paper, an output tracking formulation of the continuous time nonlinear predictive control approach (Lu, 1994, 1995) is employed to derive a new proportional navigation guidance law. The resulting guidance law is optimal as it is obtained through minimization of a objective function based on the predicted errors. It is shown that under certain assumptions, this guidance law exhibits similarity with some of the well known guidance formulations. Closed loop stability of the system under the guidance law is studied and related results are presented. The predictive proportional navigation guidance (PPNG) law proposed here, like many other such formulations, needs information about target acceleration for its implementation. Since target acceleration cannot be measured directly by on-board sensors, it is necessary to estimate the same. To address this issue, we propose a novel approach based on the time delay control (TDC) (Youcef-Toumi & Ito, 1990) to estimate the target acceleration. The guidance law proposed here is similar to that proposed in Talole and Banavar (1998), in the sense that both are based on the predictive control approach. However in Talole and Banavar (1998) the target acceleration is assumed to be available. The present paper
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overcomes this limitation by explicitly estimating the acceleration of a maneuvering target in real time. Recently an expository application of TDC for estimating target acceleration in homing guidance where the objective is to minimize the relative range from missile to target is presented in (Talole, Phadke, & Singh, 2005) with the simulation results for an ideal scenario. In this paper, TDC is used under more realistic conditions like nonconstant velocities of missile and target, magnitude constraint on commanded acceleration and a practical autopilot. The paper is organized as follows. Section 2 briefly reviews the nonlinear predictive control theory while formulation of the guidance law based on this approach and related results are presented in Section 3. In Section 4, a brief introduction to time delay control along with its application to estimation of target acceleration is given. Simulation results showing the performance of the proposed formulation in a practical scenario are presented in Section 5 and finally Section 6 concludes this work.
yðtÞ, such that the component of control appears for the first time. Define zðxðtÞ; dÞ ¼ ½z1 ðxðtÞ; dÞ; . . . ; zm ðxðtÞ; dÞT , where zi ¼ dLf ðci Þ þ
d2 2 dl i l i Lf ðci Þ þ þ L ðci Þ; 2! li ! f
i ¼ 1; . . . ; m, (4)
where ci is the ith component of vector cðxÞ, d40 is a real number designated as the prediction horizon and Lkf ðci Þ denotes the kth order Lie derivative of ci with respect to f. In general the Lie derivatives of a scalar function hðxÞ with respect to f ðxÞ are defined as L0f hðxÞ ¼ hðxÞ, qhðxÞ f ðxÞ, qx L2f hðxÞ ¼ Lf ðLf hðxÞÞ
L1f hðxÞ ¼
.. . Lkf hðxÞ ¼ Lf ðLk1 hðxÞÞ, f
2. Predictive control The continuous time nonlinear predictive control (Lu, 1994, 1995) offers a systematic approach towards designing controllers for nonlinear systems. In this approach the state or output response of a nonlinear dynamic system is predicted by appropriate expansions and a quadratic cost function based on the predicted difference between the actual response and the desired response and current control expenditure is minimized point-wise to obtain an optimal, nonlinear feedback control law. As the guidance law formulated in this paper is based on the output tracking formulation, a brief outline of this formulation is presented here for the sake of completeness. The reader is referred to Lu (1994, 1995) for complete development. Consider a nonlinear system described by x_ ¼ f ðxÞ þ GðxÞu,
(1)
y ¼ cðxÞ,
(2)
where
where Lkf hðxÞ is the kth order Lie derivative of hðxÞ with respect to f ðxÞ. To expose fully the influence of uðtÞ on yi ðt þ dÞ for a small d40, one may approximate each yi ðt þ dÞ by an li th order Taylor series at t. In doing so, one can express yðt þ dÞ as a vector function of uðtÞ in a compact form as yðt þ dÞ yðtÞ þ zðxðtÞ; dÞ þ LðdÞW ðxðtÞÞuðtÞ,
(5)
mm
is a diagonal matrix with the elements where LðdÞ 2 R on the main diagonal being Lii ¼
dl i ; li !
i ¼ 1; . . . ; m
(6)
and W ðxÞ 2 Rmm has each of its rows in the form of wi ¼ ½Lg1 ðLlf i 1 ðci ÞÞ; . . . ; Lgm ðLlf i 1 ðci ÞÞ;
i ¼ 1; . . . ; m.
(7)
Similarly, expanding the ith component of y% ðt þ dÞ in the li th order Taylor’s series yields y% ðt þ dÞ y% ðtÞ þ dðt; dÞ,
(8) m
where the ith component of dðt; dÞ 2 R is
f 9½f 1 f 2 f n T ; n
G9½g1 g2 gm
(3) m
and xðtÞ 2 X R is the state, uðtÞ 2 U R represents the control and yðtÞ 2 Rm is the output vector, where X and U are compact sets in Rn and Rm spaces respectively. The functions f : Rn ! Rn ; c : Rn ! Rm and G : Rn ! Rnm are continuously differentiable nonlinear functions. Suppose that the desired output trajectory is specified by y% ðtÞ, 0ptptf , which is an outcome of Eqs. (1)–(2) for some feasible reference control u% ðtÞ 2 U for all t 2 ½0; tf . Given the present output of the system, yðtÞ, at any instant t 2 ½0; tf , the current control uðtÞ determines the output response in the immediate future. To predict the response of the system due to present control input, we follow the following procedure. Let li ; i ¼ 1; . . . ; m, be the lowest order of the derivative of yi , the ith component of
d i ðt; dÞ ¼ dy_ %i ðtÞ þ
d2 % dli ðli Þ% y ðtÞ; y€ i ðtÞ þ þ 2! li ! i
i ¼ 1; . . . ; m, (9)
ðli Þ% yi ðtÞ
where is the li th differentiation of yi with respect to time. In order to find the current control uðtÞ that improves the tracking accuracy at next instant, consider a point wise minimization of the performance index that penalizes the output tracking error at ðt þ dÞ and current control expenditure uðtÞ %
J½uðtÞ ¼ 12½yðt þ dÞ y% ðt þ dÞT Q½yðt þ dÞ y% ðt þ dÞ þ 12 uT ðtÞRuðtÞ, mm
ð10Þ mm
is positive definite and R 2 R positive where Q 2 R semi-definite weighting matrices. Replace yðt þ dÞ and
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y% ðt þ dÞ in Eq. (10) by predictions given by Eqs. (5) and (8), respectively. The control that minimizes the performance index is obtained by setting qJ=qu equal to zero as
output of system given by Eq. (12) i.e. _ y ¼ y.
ð11Þ
where eðtÞ9yðtÞ y ðtÞ is the current output tracking error. To use the control Eq. (11), one needs to choose the controller parameters i.e. the weightings Q and R and the prediction horizon d. Proper choice of these parameters is vital as their values affect the tracking performance of the controller. %
3. Predictive proportional navigation guidance In this section, we present a formulation of the PPNG law by employing the predictive control approach presented in Section 2. To this end, consider a two dimensional engagement geometry as shown in Fig. 1, where the missile and target are treated as point masses. We assume that the velocity of missile and target remains constant throughout the engagement with missile velocity being greater than that of target. Further we neglect autopilot and seeker loop dynamics. Under these assumptions, the engagement model can be represented by the following differential equations (Babu et al., 1994)
% % y_ ðtÞ ¼ y€ ðtÞ ¼ 0.
Following the procedure outlined in Section 2 to the present problem and using the definitions of f ðxÞ and gðxÞ, one obtains the various quantities appearing in Eq. (11) as LðdÞ ¼ d, cosðgm yÞ , W ðxÞ ¼ R % _ e ¼ y_ y_ ¼ y, zðx; dÞ ¼
1 y_ ¼ ½vt sinðgt yÞ vm sinðgm yÞ, R
dðt; dÞ ¼ 0.
at , vt
g_ m ¼
am ac ¼ , v m vm
(12)
where R is the relative range from missile to target, y the LOS angle, gm ; gt the flight path angles of missile and target, at the lateral acceleration of target, and ac , am the commanded and achieved lateral accelerations of missile.
(14)
Now one can re-write the single input system of Eq. (12) in the general form of Eq. (1) by defining the state x ¼ ½R y gm gt T , the input u ¼ am and the nonlinear functions f ðxÞ and GðxÞ ¼ gðxÞ as 2 3 2 3 0 vt cosðgt yÞ vm cosðgm yÞ 6 7 6 1 ½vt sinðg yÞ vm sinðg yÞ 7 607 t m 6 7 6 7 f ðxÞ ¼ 6 R ; gðxÞ ¼ 7 at 6 0 7. 4 5 4 5 vt 1 0 vm
R_ ¼ vt cosðgt yÞ vm cosðgm yÞ,
g_ t ¼
(13)
One can notice that with y_ as output, the relative degree is one and the desired reference output trajectory is
uðtÞ ¼ ½ðLðdÞW ðxÞÞT QLðdÞW ðxÞ þ R1 ½ðLðdÞW ðxÞÞT QðeðtÞ þ zðx; dÞ dðt; dÞÞ,
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d __ ð2Ry þ at cosðgt yÞÞ, R ð15Þ
Since the system represented by Eqs. (12) and (13) is single input single output, the weightings Q and R appearing in control Eq. (11) are scalar. Representing the weighting on predicted error by q and weighting on control expenditure by r and substituting Eq. (15) in Eq. (11) yields the PPNG law as ac (PPN) ¼
d2 q cosðgm yÞ ½d2 q cos2 ðgm yÞ þ rR2 ( ) Ry_ 2R_ y_ þ at cosðgt yÞ , d
ð16Þ
3.1. Formulation of guidance law The objective of proportional navigation is to drive the _ between missile and its target to zero while LOS rate, y, closing on the target. To ensure this, we choose y_ as the
Fig. 1. Two-dimensional engagement geometry.
where ac is the commanded lateral acceleration of the missile. The PPNG law (16), under certain assumptions, exhibits similarity with some of the known guidance laws appeared in literature. Bezick et al. (1995) have presented the design of input–output feedback linearizing homing guidance law based on the geometric control theory where the relative range R from missile to target, is chosen as output of the system (12). In the present case with LOS rate as output, one can show that the guidance law (16) is a special case of the input–output feedback linearizing control law. To show this, consider the system given by Eq. (12) with y_ as output. One can verify that with y_ as output the relative degree is one i.e. one needs to differentiate the output at least once for control to appear explicitly. The resulting output
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dynamics as obtained by differentiating y_ once is 1 y_ ¼ y€ ¼ ½2R_ y_ þ at cosðgt yÞ ac cosðgm yÞ. R For r ¼ 0 Eq. (16) reduces to ( ) 1 Ry_ 2R_ y_ þ at cosðgt yÞ . ac ¼ cosðgm yÞ d
(17)
(18)
_ Substituting Eq. (18) in Eq. (17) and defining v ¼ y=d yields y_ ¼ v
(19)
which is a linear relationship between output and input. Thus the guidance law (16) with r ¼ 0 achieves input–output linearization. Next, one can verify that the guidance law (18) closely resembles to the modified proportional navigation guidance (MPNG) law proposed by Ha et al. (1990). Also the switched biased proportional navigation (SBPN) guidance (Babu et al., 1994) shows similarity with (18) except that the target acceleration term is replaced by the switched bias term. In fact, the authors have shown that the switched bias term represents an estimate of the target acceleration. 3.2. Stability To establish stability of error dynamics, consider the system given by Eq. (12) with y_ as output. It is important to note that y_ itself represents output tracking error as the % desired output history y_ ¼ 0. Thus to obtain the tracking error dynamics differentiate y_ once to get 1 y€ ¼ ½2R_ y_ þ at cosðgt yÞ ac cosðgm yÞ. (20) R Now substituting the guidance law given by Eq. (16) with r ¼ 0 into Eq. (20) yields y_ (21) y€ þ ¼ 0. d Since d, the time prediction horizon is always positive, y_ ! _ 0 for any value of yð0Þ, implying asymptotic stability for the error dynamics. From (21), one can note that d represents a time constant of the error dynamics. When ra0, the error dynamics takes a form y€ þ K 1 y_ ¼ U,
input-bounded output stability for the time-varying error dynamics given by Eq. (22). From Eq. (16), it is clear that the knowledge of the component of target acceleration viz. at cosðgt yÞ is crucial for implementation of the guidance law. However, this component cannot be obtained by direct measurement by any on board sensors. In this paper we propose a method to estimate this component using the time delay control (TDC) approach, which is presented in the next section.
(22)
where _ 2Rð1 K 2 Þ K 2 K1 ¼ þ , R d d2 q cos2 ðgm yÞ , K2 ¼ 2 ½d q cos2 ðgm yÞ þ rR2 at cosðgt yÞð1 K 2 Þ . U¼ R Clearly, if one chooses controller parameters q; r, and d such that K 1 ðtÞ40, and noting the fact that the quantity at cosðgt yÞ is always bounded, one can ensure bounded
4. Time delay control Many real plants are affected by significant uncertainties and large unmeasurable external disturbances. Several techniques such as adaptive control, variable structure control, Liapunov function based control have been proposed to handle such systems. TDC is another method that can tackle uncertain systems. The idea underlying TDC is presented by Youcef-Toumi and Ito (1990) and Youcef-Toumi and Reddy (1992). In TDC, a function representing the effect of uncertainties and external disturbances is estimated directly by using information in the recent past and then a control is designed using this estimate in such a way as to cancel out the effect of the unknown dynamics and the external disturbances. Several diverse applications of TDC such as active vibration absorber (Olgac & Holm-Hansen, 1994), robot manipulators (Chang, Kim, & Park, 1995), brush-less DC motor (Song & Byun, 2000), telescopic handler (Park & Chang, 2004) are reported. However no application of TDC to missile guidance problems is seen in literature. In this work, we employ the TDC for estimation of target acceleration needed in the guidance law of Eq. (16). 4.1. The concept of Time Delay Control Consider the following system: _ ¼ f ðx; tÞ þ Bðx; tÞuðx; tÞ þ Bðx; tÞhðx; u; tÞ, xðtÞ
(23)
where xðtÞ 2 Rn is the state, uðx; tÞ 2 Rm is the control, f ðx; tÞ and Bðx; tÞ are known functions and hðx; u; tÞ is lumped uncertainty including disturbances. Assumption 1. The function hðx; u; tÞ does not change appreciably in a small interval of time L: hðx; u; tÞ hðx; u; t LÞ.
(24)
Now from the system given by (23), we have _ f ðx; tÞ Bðx; tÞuðx; tÞ Bðx; tÞhðx; u; tÞ ¼ xðtÞ
(25)
and _ LÞ f ðx; t LÞ Bðx; t LÞhðx; u; t LÞ ¼ xðt Bðx; t LÞuðx; t LÞ.
ð26Þ
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Therefore
which is implementable. It may be noted that the estimation of target acceleration obtained in Eq. (31) can be used in other guidance laws requiring the target acceleration information.
hðx; u; t LÞ ¼ ½BT ðx; t LÞBðx; t LÞ1 BT ðx; t LÞ _ LÞ f ðx; t LÞ ðxðt Bðx; t LÞuðx; t LÞÞ.
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ð27Þ 5. Simulation and results
Let ^ u; tÞ ¼ hðx; u; t LÞ. hðx;
(28)
^ u; tÞ can act as an estimate of In view of Assumption 1, hðx; the lumped uncertainty and disturbance function hðx; u; tÞ. ^ u; tÞ can It can be seen from Eq. (27) that the estimate hðx; be obtained from past measurement at ðt LÞ and an ^ u; tÞ can be used in _ LÞ. The estimate hðx; estimate of xðt control to nullify the effect of uncertainty in the system. In the next subsection, we apply the TDC methodology outlined above to the problem of estimation of target acceleration. 4.2. Estimation of target acceleration using TDC Consider the PPNG given by Eq. (16) reproduced once again for the sake of continuity ac (PPN) ¼
d2 q cosðgm yÞ ½d2 q cos2 ðgm yÞ þ rR2 ( ) Ry_ 2R_ y_ þ at cosðgt yÞ . d
ð29Þ
In general, the range, range rate, LOS angle and LOS rate are obtained by processing on-board seeker measurement and hence are assumed to be available. Thus the guidance law given by Eq. (29) can be implemented if one can estimate the quantity at cosðgt yÞ which essentially represents a component of target acceleration. Differentiating the LOS rate i.e. y_ yields 1 y€ ¼ ½2R_ y_ ac cosðgm yÞ þ at cosðgt yÞ, (30) R wherein at cosðgt yÞ can be considered as an external disturbance to be estimated. Defining hðtÞ ¼ at cosðgt yÞ ^ as its estimate and following the TDC concept and hðtÞ presented in earlier section, we have
In this section, the performance of PPNG is compared with PNG by simulation under the same assumptions as used to derive the guidance law of Eq. (16) and consequently Eq. (32). We consider the target to be maneuvering with time-varying acceleration. It is important to note that for interception of the target, it is _ necessary to ensure Ro0 while driving the LOS rate to zero. In Ha et al. (1990) analytical conditions under which target interception is guaranteed for the conventional PNG and MPNG laws are presented. Here as we are comparing the performance of the PPNG with PNG and since the PPNG under certain assumptions resembles the MPNG law, we consider the initial conditions of the missile–target engagement such that they satisfy the same analytical conditions as derived in Ha et al. (1990). Also, it is assumed that the angle between missile velocity vector and LOS remains acute throughout the engagement i.e. jgm yjo90 . The target acceleration profile is taken as at ¼ 80 m=s2
for 0otp2 s,
at ¼ 0 m=s2
for 2otp4 s,
at ¼ 80 m=s2
for t44 s.
The other necessary simulation data, as taken from Babu et al. (1994), is Rð0Þ ¼ 4500 m, yð0Þ ¼ 20 , gt ð0Þ ¼ 140 , gm ð0Þ ¼ 60 , and vm ¼ 500 m=s, vt ¼ 300 m=s. The performance of the PPNG (32) is compared with the conventional proportional navigation guidance law. Now re-writing (32) as
^ ¼ ½at cosðg yÞjtL hðtÞ t € ¼ ½Ry þ 2R_ y_ þ ac cosðgm yÞjtL ½at cosðgt yÞjt .
5.1. Simulation under ideal conditions
^ ac (PPN) ¼ KfN 0 R_ y_ hg, ð31Þ
Here it is assumed that the quantity at cosðgt yÞ does not change appreciably in a small time L, i.e. ½at cosðgt yÞjt ½at cosðgt yÞjtL . Replacing the target acceleration term in Eq. (29) by its estimate (31) leads to the guidance law ( ) d2 q cosðgm yÞ Ry_ ^ 2R_ y_ þ hðtÞ , ac (PPN) ¼ 2 ½d q cos2 ðgm yÞ þ rR2 d (32)
(33)
where d2 q cosðgm yÞ , ½d2 q cos2 ðgm yÞ þ rR2 _ R 2Rd N0 ¼ . _ Rd K¼
ð34Þ
The weightings q and r in Eq. (34) are to be tuned to achieve satisfactory tracking performance by observing the simulated responses. The term N 0 represents the effective navigation ratio. For the purpose of valid comparison, the navigation ratio of the PN guidance law is modified by
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including the K term of (33). With this modification, the resulting PN guidance law is
4500
_ ac (PN) ¼ KN 0 R_ y.
3500
The navigation ratio, N 0 , is chosen as 4 in both the cases. However, as is evident from (34), to keep N 0 ¼ 4 requires certain adjustments. To this end, we modulate the value of the prediction horizon d in (34) such that N 0 ¼ 4 throughout the engagement. The resulting value of d is R d¼ 40. 2R_
3000 2500 2000 1500 1000 500
(36)
0
actual estimated
20
2
3
4
5
6
7
8
9
0.01 PPNG PNG
0 -0.01 -0.02 -0.03 -0.04 -0.05
0
2
4
6
8
10
time (s) Fig. 4. LOS rate profiles.
0
PPNG PNG
-20 -40 -60 -80 -100 -120 -140 -160
40
1
Fig. 3. Relative range profiles.
LOS rate (rad/s)
_ _ € LÞ yðtÞ yðt LÞ yðt (37) L and is used in Eq. (31) to compute the estimate of target acceleration as needed in guidance law of Eq. (33). It is important to note that use of numerical differentiation is undesirable when the signal is noisy. However, the TDC needs the system states and their derivatives for its implementation. In the literature on TDC, the issue of obtaining the states and the effect of noise on numerically obtained state derivatives are addressed (Xu & Cao, 2000; Morioka, Sabanovic, Uchibori, Wada, & Oka, 2001). Some solutions not requiring numerical differentiation are also suggested. For example, a model reference observer is proposed in Chang and Lee (1996) to reconstruct the states and their derivatives in a stable manner and is used in a practical application for a SCARA-type robot. In this paper we concentrate on showing the application of TDC to estimate the target acceleration and modification of this formulation to address the issue of noisy measurement could form a separate study.
0
time (s)
commanded acceleration (m/s2)
The purpose of obtaining d through (36) is only to ensure that the effective navigation ratio is same in both the cases for meaningful comparison. It is worth noting here that the performance of the PPNG may be better for some other values of d than the one obtained from (36). The weightings in (34) are chosen as q ¼ 10 000 000 and r ¼ 0:00001. From the LOS rate, the estimates of y€ are obtained by numerical differentiation as
at cos(γt−θ) (m/s2)
relative range (m)
(35)
PPNG PNG
4000
0
2
4
6
8
10
time (s) Fig. 5. Commanded acceleration profiles.
0 -20 -40 -60 -80
0
1
2
3
4
5
6
time (s) Fig. 2. Estimation of target acceleration.
7
8
9
Using the above data and initial conditions, simulations are carried out for PPNG and PNG and the results are presented in Figs. 2–5. Firstly, in Fig. 2, we compare the estimate of at cosðgt yÞ with the actual value and one can note that the TDC based approach gives highly satisfactory estimate of the target acceleration component. In Fig. 3, relative range R for PPNG as well as PN guidance is presented and one can observe that the relative range R ! 0 as expected for both the guidance laws. Fig. 4 shows the
ARTICLE IN PRESS S.E. Talole et al. / Control Engineering Practice 14 (2006) 1445–1453
LOS rates for both laws from which it can be seen that the LOS rate with predictive guidance goes to zero near target interception, unlike with PN guidance law. Fig. 5 shows the corresponding commanded accelerations obtained from both guidance laws till Rp5 meters and one can observe that the acceleration profile due to predictive guidance is less demanding as compared to that of PN guidance.
where xm , hm and xt , ht denote the ground range and height for missile and target respectively. The quantity W ¼ mg is the weight of the missile, m is the mass and g represents the gravitational acceleration. The force of drag D is frequently represented by (Imado, Nagayama, & Tahk, 2001; Kim & Kim, 2004) D ¼ k1 v2m þ k2
5.2. Simulation in a practical scenario In this section we demonstrate the effectiveness of the guidance law (32) under more realistic conditions. Here we relax the assumptions of constant velocities of missile and target as well as that of ideal autopilot. Further a magnitude constraint is placed on the commanded acceleration. Therefore, we consider the missile–target engagement geometry incorporating the system dynamics as shown in Fig. 6, following which the equations of motion of the missile–target engagement model in the vertical plane assuming small angle of attack are obtained as T D W sin gm , m am g cos gm g_ m ¼ , vm x_ m ¼ vm cos gm , h_m ¼ vm sin g ,
(39)
1 1 (40) a_ m ¼ am þ ac , t t where t ¼ 0:1 represents the time constant of the autopilot dynamics. Thrust T and mass m are modelled as (Kim & Kim, 2004) ( T 0 ; tp15; T¼ (41) 0; t415; ( m¼
_ m0 mt; mf ;
tp15; t415;
(42)
_ ¼ 2:66 kg=s, and where T 0 ¼ 5880 N, m0 ¼ 165 kg, m mf ¼ 125 kg. It is assumed that the missile acceleration is constrained as 15p 15p vm if ac X vm , ac ¼ 180 180 15p 15p vm if ac p vm . ð43Þ ac ¼ 180 180
m
t
a2m , v2m
where the values of k1 ¼ 0:001 and k2 ¼ 1:0 are taken from (Kim & Kim, 2004). The autopilot is modelled as a first order system
v_m ¼
R_ ¼ vt cosðgt yÞ vm cosðgm yÞ, 1 y_ ¼ ½vt sinðgt yÞ vm sinðgm yÞ, R at g_ t ¼ , vt x_ t ¼ vt cos gt , h_t ¼ vt sin g ,
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ð38Þ
The above constraint on missile acceleration is arising from constraint on maximum possible missile flight path rate of 15 =s. The initial position of the missile and the target are chosen as follows: xm ð0Þ ¼ 0 m;
hm ð0Þ ¼ 0 m,
xt ð0Þ ¼ 3000 m;
ht ð0Þ ¼ 4000 m.
It is assumed that the target velocity is time-varying as vt ¼ 200 þ 100 sinð0:02ptÞ m=s.
(44)
The target acceleration profile is at ¼ 80 m=s2
Fig. 6. Missile–target engagement geometry.
for 0otp5 s,
at ¼ 0 m=s2
for 5otp10 s,
at ¼ 80 m=s2
for t410 s.
Other necessary data used in simulation are: gt ð0Þ ¼ 140 , gm ð0Þ ¼ 90 , and vm ð0Þ ¼ 1 m=s. The initial conditions for Rð0Þ and yð0Þ are decided by the initial coordinates
ARTICLE IN PRESS S.E. Talole et al. / Control Engineering Practice 14 (2006) 1445–1453
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of the target hm ð0ÞÞ2 1=2 and
Rð0Þ ¼ ½ðxt ð0Þ xm ð0ÞÞ2 þ ðht ð0Þ
8000 7000
ht ð0Þ hm ð0Þ . xt ð0Þ xm ð0Þ
6000
The values of the controller parameters q; r and the predictive horizon d are chosen as q ¼ 100, r ¼ 0:00001
height (m)
yð0Þ ¼ tan1
i.e.
5000 4000 3000 2000
missile target
1000 6000
0
0
1000
relative range (m)
4000
5000
3000 2000
0
0
5
10
15
20
25
20
25
time (s) Fig. 7. Relative range vs. time.
0.04 0.03
LOS rate (rad/s)
3000
Fig. 10. Target and missile trajectories.
4000
1000
0.02 0.01 0 -0.01 -0.02 -0.03 -0.04
0
5
10
15
time (s) Fig. 8. LOS rate vs. time.
100
missile acceleration (m/s2)
2000
downrange (m)
5000
commanded actual
80
and d ¼ 10 by trial and error procedure by observing the simulated responses. The value L required in Eq. (37) is taken as 0:01. It assumed that the guidance commences when the missile acquires a velocity of 200 m=s. Using the above data and initial conditions, simulation is carried out by using the guidance law (32) employing the target acceleration estimate as given by Eq. (31) and the results are presented in Figs. 7–10. Firstly in Fig. 7, the relative range profile for PPNG is presented from where we observe that the relative range R ! 0 as expected. The corresponding LOS rate history is given in Fig. 8 and one can note that y_ ! 0 as desired. Fig. 9 presents the corresponding commanded and achieved acceleration profiles. It is important to note that the overshoots for the accelerations at 10 and 15 s are arising due to the sudden change in the magnitude of target acceleration at t ¼ 10 s and sudden change in the magnitude of thrust at t ¼ 15 s. Lastly, Fig. 10 presents the planar trajectories of target and missile from where we observe that the missile has successfully engaged the target. The above results clearly demonstrate that the PPNG using the TDC based target acceleration estimation offers satisfactory results. Simulations were also carried out by using the classical PNG instead of PPNG in the practical scenario, however, as expected, the results (not included in the paper) are far from satisfactory.
60
6. Conclusion
40 20 0 -20 -40 -60 -80
0
5
10
15
20
time (s) Fig. 9. Commanded and achieved acceleration vs. time.
25
In this paper, a new formulation of the proportional navigation guidance law using the continuous time nonlinear predictive control approach is proposed. The incorporation of TDC based estimation of target acceleration, has made the proposed guidance law implementable. This approach can possibly be used in other guidance laws requiring target acceleration. The efficacy of the proposed formulation is verified by simulation in a realistic scenario for a highly maneuvering target.
ARTICLE IN PRESS S.E. Talole et al. / Control Engineering Practice 14 (2006) 1445–1453
References Asher, R. B., & Matuszewski, J. P. (1974). Optimal guidance with maneuvering targets. Journal of Spacecraft and Rockets, 11(3), 204–206. Babu, K. R., Sarma, I. G., & Swamy, K. N. (1994). Switched bias proportional navigation for homing guidance against highly maneuvering targets. Journal of Guidance, Control, and Dynamics, 17(6), 1357–1363. Bezick, S., Rusnak, I., & Grey, W. S. (1995). Guidance of a homing missile via nonlinear geometric control methods. Journal of Guidance, Control, and Dynamics, 18(3), 441–448. Chang, P. H., Kim, D. S., & Park, K. C. (1995). Robust force/position control of a robot manipulator using time delay control. Control Engineering Practice, 3(9), 1255–1264. Chang, P. H., & Lee, J. W. (1996). A model reference observer for time delay control and its application to robot trajectory control. IEEE Transactions on Control Systems Technology, 4(1), 2–10. Ha, I. J., Hur, J. S., Ko, M. S., & Song, T. L. (1990). Performance analysis of PNG laws for randomly maneuvering targets. IEEE Transactions on Aerospace and Electronic Systems, 26(5), 713–720. Imado, F., Nagayama, M., & Tahk, M.-j. (2001). An extended study on a new missile guidance algorithm against a maneuvering target. AIAA2001-4342, AIAA guidance, navigation, and control conference and exhibit, Montreal, Canada. Kim, K. S., & Kim, Y. (2004). Design of generalized conceptual guidance law using aim angle. Control Engineering Practice, 12(3), 291–298. Lin, C. F. (1991). Modern navigation, guidance, and control processing. Englewood Cliffs, New Jeresy: Prentice-Hall. Lu, P. (1994). Nonlinear predictive controllers for continuous systems. Journal of Guidance, Control, and Dynamics, 17(3), 553–560. Lu, P. (1995). Optimal predictive control of continuous nonlinear systems. International Journal of Control, 62(3), 633–649. Morioka, H., Sabanovic, A., Uchibori, A., Wada, K., & Oka, M. (2001). Application of time-delay-control in variable structure motion control systems. ISIE, Pusan, Korea, 1313–1318.
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Nesline, F. W., & Zarchan, P. (1981). A new look at classical vs modern homing missile guidance. Journal of Guidance, Control, and Dynamics, 4(1), 78–85. Olgac, N., & Holm-Hansen, B. T. (1994). A novel active vibration absorption technique: Delayed resonator. Journal of Sound and Vibration, 176(1), 93–104. Park, J. Y., & Chang, P. H. (2004). Vibration control of a telescopic handler using time delay control and commandless input shaping technique. Control Engineering Practice, 12(6), 769–780. Salmond, D. J. (1996). Foundations of modern missile guidance. Journal of Defence Science, 1(2), 171–179. Song, J.-B., & Byun, K.-S. (2000). Design of time delay controller using variable reference model. Control Engineering Practice, 8(5), 581–588. Speyer, J. L. (1976). An adaptive terminal guidance scheme based on an exponential cost criterion with application to homing missile guidance. IEEE Transactions on Automatic Control, AC-26(5), 371–375. Talole, S. E., & Banavar, R. N. (1998). Proportional navigation through predictive control. Journal of Guidance, Control, and Dynamics, 21(6), 1004–1006. Talole, S. E., Phadke, S. B., & Singh, R. K. (2005). Predictive homing guidance using time delay control. AIAA-2005-6159, AIAA guidance, navigation, and control conference and exhibit, San Francisco, CA, USA. Xu, J., & Cao, W. (2000). Synthesized sliding mode and time-delay control for a class of uncertain systems. Automatica, 36, 1909–1914. Youcef-Toumi, K., & Ito, O. (1990). A time delay controller for systems with unknown dynamics. Journal of Dynamic Systems, Measurement, and Control, Transactions of ASME, 112, 133–142. Youcef-Toumi, K., & Reddy, S. (1992). Analysis of linear time invariant systems with time delay. Journal of Dynamic Systems, Measurement, and Control, Transactions of ASME, 114, 544–555. Zarchan, P. (1997). Tactical and strategic missile guidance. Progress in astronautics and aeronautics. Vol. 176, Reston, VA: AIAA.