Sliding mode-based intercept guidance with uncertainty and disturbance compensation

Author's Accepted Manuscript

Sliding mode-based intercept guidance with uncertainty and disturbance Compensation Takeshi Yamasaki, S.N. Balakrishnan, Hiroyuki Takano, Isao Yamaguchi

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PII: DOI: Reference:

S0016-0032(15)00335-X http://dx.doi.org/10.1016/j.jfranklin.2015.08.014 FI2422

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Journal of the Franklin Institute

Received date: Revised date: Accepted date:

30 October 2013 8 July 2015 20 August 2015

Cite this article as: Takeshi Yamasaki, S.N. Balakrishnan, Hiroyuki Takano, Isao Yamaguchi, Sliding mode-based intercept guidance with uncertainty and disturbance Compensation, Journal of the Franklin Institute, http://dx.doi.org/10.1016/j.jfranklin.2015.08.014 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Sliding Mode-Based Intercept Guidance with Uncertainty and Disturbance Compensation

1

Takeshi Yamasaki2, National Defense Academy of Japan, Yokosuka, Kanagawa, 239-8686, Japan S. N. Balakrishnan3, Missouri University of Science and Technology, Rolla, MO, 65409 Hiroyuki Takano4 and Isao Yamaguchi5 National Defense Academy of Japan, Yokosuka, Kanagawa, 239-8686, Japan

Abstract: This paper presents a missile intercept guidance law that can account for uncertainty and disturbance. The proposed guidance law is developed based on a novel second order sliding surface, along with an uncertainty and disturbance estimator. The proposed sliding surface is invariant for the non-impact angle constraint case as well as the impact angle constraint case. The sliding surface guarantees finite time convergence of the line-of-sight(LOS) rate and (if specified) the LOS angle error, characterized by a simple function of time-to-go. Furthermore, the proposed strategy is also effective against a maneuvering target. The line-of-sight rate convergence property can be adjusted using only one parameter that makes the design process simple. Potential of the proposed guidance laws is demonstrated with some simulations.

1

A major part of this paper was presented at the AIAA Guidance, Navigation, and Control Conference, Boston, Aug. 2013, Paper No. is AIAA 2013-5115. 2 Research Associate, Department of Aerospace Engineering. ([email protected]) 3 Curators’ Professor, Department of Mechanical and Aerospace Engineering. ( [email protected]) 4 Associate Professor, Department of Aerospace Engineering. ([email protected]) 5 Professor, Department of Aerospace Engineering. ([email protected]) 1

Keywords: high-order sliding mode, uncertainty and disturbance estimator, missile guidance

I. Introduction Intercepting a maneuvering target is a challenging issue for a guided missile. Proportional navigation (PN) is a wellknown guidance law and is widely used because of its simplicity in implementation[1]. The PN guidance law uses only the angular rate of the line-of-sight (LOS), and if available, the closing velocity between the missile and the target, otherwise, the missile velocity. If there is no system time delay, acceleration limitation, or measurement noise in the missile system, and the target is moving with a constant velocity and bearing, the PN can guide a missile to achieve a perfect intercept, in other words, a ‘zero-miss’. The PN and its alternative representation as the zeroeffort-miss guidance have been the basis for many advanced guidance methodologies. For example, the augmented PN (APN) guides the missile with a compensation for the target’s the lateral constant acceleration[2]. Compensated PN guidance (CPNG)[3] (or velocity compensated PN guidance[4]) considers the missile’s axial acceleration component that affects the LOS rate which the basic PN tries to annul. Results from the optimal guidance (OG) law show that the PN is optimal in the sense that it is a minimum energy guidance law with a zero-miss distance against non-maneuvering targets for no-time-lag systems with the effective navigation constant set to three[2]. Both the APN and the OG[5,6] make allowances for the target maneuvers. If the target maneuver bounds are known, the differential game theory (DGT)[7] or the sliding mode guidance (SMG)[8,9,10] can capture a maneuvering target without the need for precise target acceleration information. SMG laws for missile intercept use nonlinear discontinuous terms with a high gain to compensate for external disturbance such as target maneuvers, which might cause unnecessarily large input accelerations. In order to avoid this, a target acceleration component normal to the LOS is first estimated in [11,12] with a disturbance observer under the assumption that the target acceleration bound is available. . Then, such disturbance can be cancelled by an intercept guidance system[13] instead of using high gains with the discontinuous term. This work uses a second order sliding mode (SOSM) approach[14,15] that can mitigate the so-called chattering phenomenon associated with the sliding mode approach. The sliding mode control can generally be separated into two phases; the reaching surface phase and the sliding surface phase[16]. The magnitude of the control command for the reaching surface phase tends to be large and it may cause control saturation. This study uses the concept proposed by Ackerman and Utkin[17, 18 that can eliminate the reaching 2

surface phase by adding a dummy variable to the original sliding surface for a missile engagement. However, a time-varying coefficient is introduced instead of the discontinuous term [17, 18] to drive the LOS rate and the LOS angle error converge to zero. Recently, an ‘inertial delay control’ (IDC) based-SMG[19] was proposed wherein the IDC-based SMG law does not need to know the target maneuver bounds or the missile’s flight path angle. The IDC-based SMG assures that the sensing system requirement equals to the one required for the PN. The IDC methodology was developed based on the uncertainty and disturbance estimator (UDE) proposed by Zhong[20], where the UDE was modified from a ’time delay control’ (TDC)[21,22,23] by considering the frequency domain instead of the time domain. The stability analysis of the TDC technique is relatively more difficult since it involves numerical differentiation as compared to that of the UDE which uses a low-pass filter. Thus, the design process of the UDE is relatively simple. This paper follows the idea of the IDC-based SMG zeroing the LOS rate, but extends it for the important case with an impact angle constraint and the same formulation can be used for a case with no impact angle constraint too. For this purpose, a sliding surface is developed with an SOSM[13,14,15] for the missile intercept engagement. The proposed sliding surface guarantees that the missile will follow the pre-specified LOS rate trajectory regardless of the target maneuvers. Furthermore, the LOS rate trajectories can be characterized with only one parameter, and the LOS rate shaping is derived in a closed form. Consequently, we can design the LOS rate trajectory ahead of time. Intercept angle constraint requirement has spawned another interesting research area recently. See [ e.g., 5, 24, 25, 26] and the references therein. In this study, the proposed sliding surface developed for a missile engagement without an impact angle constraint is extended and unified in a guidance law that includes an impact angle constraint. To unify the sliding surface, a time-varying coefficient, which is different from an integral sliding mode-based design [27, 28], is applied to the surface in order to drive the LOS rate and the LOS angle error to zero simultaneously at the end of the engagement. The unified sliding surface also guarantees that the missile will follow the pre-specified LOS angle and the LOS rate trajectories regardless of the target maneuvers. Expressions for the trajectories of the impact angle error, the LOS rate, and its derivative are derived in a closed form. These trajectories are characterized with only one additional parameter. In the unified guidance law, one can select a no-impact angle constraint mode by simply setting the additional parameter to zero without re-tuning the other parameter. This study therefore, provides a unified framework for the guidance law with or without impact angle constraint.

3

It is worth noting that the guidance law developed in this study by using time-varying coefficients on sliding surfaces where the resulting guidance law includes an equivalent input term that represents the widely-used PN for intercept without an LOS angle constraint whereas the equivalent input term represents the biased PN for intercept with an LOS angle constraint. These results have great significance on the design for intercept guidance laws because the PN and BPN guidance laws have come from optimal control theories in the sense of minimizing the integral of the squared lateral accelerations for interception with a lag-free system against a nonmaneuvering target. The resulting nonlinear input term compensates for target maneuvers as well as uncertain dynamics to provide excellent guidance system performance. The rest of the paper is organized as follows: section 2 presents a model of the engagement geometry and the UDE and SOSM-based guidance laws. First, a basic sliding variable is introduced and the basic sliding motion guarantees the convergence of the LOS rate and the LOS angle error to zero. Then, a sliding variable for the SOSMbased guidance is introduced to mitigate any large initial input. A unified guidance law with or without an impact angle constraint along with a UDE methodology is also developed. In section 3, the potential of the novel UDE and SOSM-based unified guidance law are demonstrated with some simulation results. Section 4 provides conclusions from this study.

II. Development of Guidance Laws A. Engagement Geometry Engagement geometry of the missile and the target is depicted in Fig. 1, where both vehicles are treated as point masses. As shown in Fig. 1, the LOS angle and the LOS range are respectively denoted by
and  whereas

variables associated with the two vehicles such as the velocity,  , the flight path angle,  , and the lateral

acceleration(latax), , are distinguished by the subscripts,  for the missile, and  for the target.

4

Vt at Vm

γt

R

γm λ

am

Fig. 1 Engagement Geometry.

In order to develop the basic guidance law, some assumptions are made as follows: (A1)The missile and the target are moving in a plane. (A2)The autopilot and the seeker dynamics are perfect. (A3)Required data such as LOS, LOS rate, range, and range rate are available. All variables in assumption A3 are available by using, for example, a Doppler radar-based seeker with respect to the missile fixed body reference. Under these assumptions, the engagement kinematics for the two vehicles can be represented by the following differential equations.  =  cos −
 −  cos  −


λ =

1  sin −
 −  sin −
 

 =  ⁄

 =  ⁄

(1)

(2) (3) (4)

where the dot denotes the time derivative. By taking the time derivative of Eq. (2) and using Eqs. (1), (3), and (4) , one can get the LOS rate dynamics as
 =

1 −2
 −  + ℎ 

where 5

(5)

 sin −
 ℎ =  1 − cos  −
 +  cos −
 +   sin −
 − 

(6)

Note that ‘h’ includes ‘disturbance’ due to the target maneuver (the second term in Eq. (6)) and the target velocity change (the third term in Eq. (6) ), and ‘uncertainty’ in modeling that appears in the first and the last terms in Eq. (6). The first term in Eq. (6) relates to the difference between the flight path angle and LOS angle, whereas the last term in Eq. (6) is due to the missile’s acceleration/deceleration. Therefore, the first and the last terms in Eq. (6) are treated as an uncertain dynamics. These uncertainty and disturbance effects on the LOS rate are estimated by an uncertainty and disturbance estimator which will be described later. The missile lateral acceleration (latax)  is treated as an input.

B. A basic sliding surface for intercept guidance with or without an impact angle constraint In this study, we develop a sliding mode approach similar to Phadke et al.[19], however, the new formulation can incorporate impact angle constraints. Furthermore, it is a unified formulation applicable to scenarios with or without an impact angle constraint. The variable to be annulled( for a basic sliding surface) is defined as

!

=
 + "#


−
$% &'

(7)

where
$% implies a desired terminal LOS angle to be pre-specified with a constant value, &' is a time-to-go, and "# is a design parameter. The variable

!

and the surface

!

= 0 are respectively referred to as a ‘basic sliding

variable’ and a ‘basic sliding surface’ in this study. Note that the parameter "# can be selected as 0 for the case with

no specific impact angle constraint, whereas "# can be set greater than 1 when an impact angle constraint exists. The time-to-go &' is approximated with the closing velocity, ) as

&' = $ −  ≅ /,− - = /.

(8)

where  indicates a current time while $ is the time when the two vehicles are closest. This approximated value tends to be smaller than the real one[29,30] especially for the case where the missile tries to intercept the target with an impact angle constraint. However, this fact does not cause any problem since the shorter time-to-go estimate 6

drives system to the siding mode faster than in case of using the real time-to-go. (In that case, the lateral acceleration initially required for the engagement is slightly larger than that of using the real time-to-go which is usually calculated in a recursive way [29,30].) Furthermore, the approximated value, which is usually calculated in every guidance-update period, typically converges to the real time-to-go towards the end of the engagement. The following Lemma, which is inspired by one of the authors’ previous work[31,32], is introduced to show a property of the basic sliding surface in Eq. (7). Lemma 1. Assuming the system states can be constrained such that

= 0 with "# = 0 or "# > 1 in Eq. (7) holds,

!

then, (i) the LOS rate
 converges to zero at  = $ for "# = 0 (ii) for "# > 1, the LOS rate
 and the LOS angle error
−
$% converge to zero at  = $ .

Proof. (i) The case of "# = 0 is clear since Eq. (7) with "# = 0 always holds, that is,

!

=
 = 0.

(ii) In case of "# ≠ 0, the differential equation of Eq. (7) can be solved with an initial LOS angle
# and the constant pre-specified impact angle
$% (see Appendix B for farther detailed derivation) as
−
$% = ,
# −
$% - 11 −

 2 $

34

(9)

Equation (9) implies that the LOS angle error goes to zero at  = $ with "# > 0. Taking the time derivative of Eq. (9) yields
 = −,
# −
$% -

"#  11 − 2 $ $

When "# > 1, the LOS rate
 converges to zero at  = $ . Thus,

!

34 56

(10)

= 0 with "# > 1 fulfills the requirement of

zeroing the LOS rate
 and the LOS error angle,
−
$% . This completes the proof. ∎

From Lemma 1, once the system states (the LOS angle and the LOS rate) are constrained to move on the basic sliding surface;

!

= 0, the LOS rate and the LOS error angle can be driven to zero. Therefore, the design objective

is to drive the system states to the basic sliding surface:

!

= 0.

C. A sliding surface for an SOSM-based intercept guidance law 7

Analogous to the work by Ackerman and Utkin[17,18], or Phadke and Talole[19], a dummy variable 9 is added to the basic sliding variable as :=

!

+9

(11)

where 90 = − ! 0 9 =

" &'

(12)

!

(13)

with design flexibility added with parameter ". The initial condition for the dummy variable 90 = − ! 0 of Eq. (12) suggests that the initial sliding surface variable of :0 =

! 0

+ 90 achieves zero already. This implies that

the system states lie on the sliding surface : = 0 at the beginning itself and therefore the system can skip the

reaching surface phase and can begin the sliding surface phase directly. Note that previous works[17,18,19] contain a discontinuous term; However, by using a time-varying constant in this study this is obviated and the LOS rate


and the LOS angle error
−
$% converge to zero at  = $ . The analytical proof will be discussed later.

Before discussion on the convergence properties of the LOS rate and the LOS angle error with time, a helpful lemma is introduced first. With the new sliding mode controller, the input is selected such that the system states are at the sliding surface at the beginning and are always constrained to be on the sliding surface: that is, : = : = 0

holds always. In this section, the sliding dynamics is discussed assuming : = : = 0 holds.

The following Lemma is inspired by the work by Phadke and Talole[19], but it is extended to a time-varying coefficient explicitly based on one of the authors’ previous work [31,32]. Lemma 2. Assuming the system states are always constrained to be on the surface : = 0 with " > 1 in Eq. (13), the basic sliding variable

!

and the time derivative ! converge to zero at  = $ .

Proof. Taking the time derivative of : = 0 with Eq. (11) yields : = ! + 9

8

(14)

= ! +

3

;< !

=0

The second equality is made using Eq. (13). Solving the differential equation of Eq. (14) with an initial condition of ! 0

≜

!#

yields

!

The basic sliding variable

!

 = !# 11 − 2 $

3

(15)

with " > 0 goes to zero at  = $ . The time derivative of Eq. (15) is evaluated as ! = −

Equations (15) and (16) imply that when " > 1, ! =

Lemma 2 suggests that the basic sliding surface

" $

!

!

!# 11 −

 2 $

356

(16)

= 0 at  = $ . This completes the proof∎

= 0 and its time derivative ! = 0 will be achieved at  = $

as long as the sliding surface : = 0 holds. Lemma 2, however, cannot guarantee the convergence of the LOS rate


and the LOS angle error,
−
$% , at  = $ . The following theorem guarantees the convergence property.

Theorem 1. Assuming that the system states are always constrained on the surface : = 0 with " > 1 in Eq. (13)

(i) in case of no impact angle constraint with "# = 0 in Eq. (7), then, the LOS rate
 and its derivative
 converge to zero at  = $ ,

(ii) in case of an impact angle constraint with "# > 2 in Eq. (7) and "# ≠ " + 1, then, the LOS rate
, its derivative
, and the LOS angle error,
−
$% converge to zero at  = $ .

Proof. (i) Substituting Eq. (7) with "# = 0 for Eqs. (15) and (16) in Lemma 2 yields
 =
# 11 −

9

 2 $

3

(17)


 = −

" 
# 11 − 2 $ $

356

(18)

Equations (17) and (18) imply that the LOS rate
 and its derivative
 converge to zero at  = $ when " > 1. The

value of " determines the order of the convergence. In other words, " = 1 implies that > = 0 at  = $ ; " = 2

implies that > = > = 0 at  = $ , whereas " = ?> 2 implies that > = > = ⋯ = > A = 0 at  = $ .

(ii) Next, the impact angle constraint case for Theorem 1 is proved. Substituting the basic sliding variable of Eq. (7) into Eq. (14) yields : =  ! + =

" &'

!


−
$%
−
$% B " 1
 + "# 2+ 1
 + "# 2 &' &' B &'

=
 +

" + 1"# " + "#
 + ,
−
$% - = 0 &' &' C

(19)

Solving the differential equation Eq. (19) for the LOS angle error,
−
$% with the initial conditions of the LOS

angle
# , and the LOS rate
# provide (see Appendix B for further detailed derivation)
−


%$ = −D6 11 − 2 $

3E6

 + DC 11 − 2 $

34

(20)

where D6 = DC =

"# ,
# −
$% - + $
# " − "# + 1 " + 1,
# −
$% - + $
# " − "# + 1

(21)

and "# ≠ " + 1. Taking the second and third time derivative of Eq. (20) leads to
 = D6
 = −D6

" + 1  "#  11 − 2 − DC 11 − 2 $ $ $ $

"" + 1  11 − 2 $ $ C

3

356

34 56

"# "# − 1  + DC 11 − 2 $ $ C

10

(22) 34 5C

(23)

Equations (20) through (23) show that the LOS angle error,
−
$% , the LOS rate
, and its derivative
 converge to zero at  = $ when "# > 2, " > 1, and "# ≠ " + 1. Note that the minimum of " + 1, or "# , that is, min [" + 1, "# ] determines the order of the convergence. Therefore, a value of more than or equal to two guarantees the angle and LOS rate convergence to zero. This completes the proof∎

D. SOSM and UDE-based guidance law In order to mitigate the usual chattering effect observed in sliding mode control, a second order sliding mode approach (SOSM) is used in this study. The compensated dynamics for the SOSM[14, 15] is dictated as : = −6 |:|KL signξ + O O = −C |:|KP sign:

(24) (25)

where 6 , C , Q6 , QC are design parameters and conditions, 6 , C > 0, 1 > Q6 > QC > 0 should hold for stability and finite time convergence of the system[15]. On the other hand, the sliding dynamics can be written from Eq. (19) as : =
 +

" + 1"# " + "#
 + ,
−
$% &' &' C

(26)

Substitution of the LOS rate dynamics Eq. (5) in Eq. (26) yields : =

" + 1"# 1 " + "# −2
 −  + ℎ +
 + ,
−
$%  &' &' C

(27)

To achieve the compensated dynamics of Eqs. (24) and (25), the right hand side of Eqs. (24) is set equal to (27) yielding an expression for the missile latax,  . Equating the right hand sides of Eqs. (24) and (27) yields −6 |:|KL sign: + O =

" + 1"# 1 " + "# −2
 −  + ℎ +
 + ,
−
$%  &' &' C

(28)

Solving for  with the relation in Eq. (8), the required latax  can be derived as  = RS + RT + ℎ where 11

(29)

RS = " + "# + 2)
 +

" + 1"# ) ,
−
$% &'

RT = −−6 |:|KL sign: + O

(30) (31)

The first term RS in Eq. (29) is a so-called 'equivalent input' which works such that the system states are constrained to lie on the sliding surface when the system states have reached the sliding surface and when there is no uncertainty and disturbance, that is, ℎ = 0. It is interesting to note that the equivalent input RS in Eq. (30) results in the PN

when "# = 0, whereas it represents the so-called ‘biased PN’ when the parameters are set, for example, as "# = 1, and " = 1.

Since ℎ indicates uncertain and/or disturbance term (UDT), an estimated value is required to calculate the

missile latax command. Defining the estimated value as ℎU, Eq. (29) can be represented as  = RS + RT + ℎU

(32)

In this study, the UDT is estimated with an uncertainty and disturbance estimator (UDE) [20]. In the work by Phadke and Talole[19], the derivation of the UDT estimator starts with the time derivative of the LOS rate. In this study, however, the derivation of the UDT estimator uses the time derivative of the sliding variable, :. The time

derivative of the sliding variable : of Eq. (27) can be reduced as : =

= =

" + 1"# 1 " + "# 2)
 −  + ℎ +
 + ,
−
$%  &' &' C

" + 1"# 1 V− + ℎ + " + "# + 2)
 + ) ,
−
$% -W  &' 1 − + ℎ + RS  

(33)

where Eq. (30) is used in the last equality. Solving for ℎ yields

ℎ = : − RS + 

(34)

Equation (34) includes the time derivative of the sliding variable which contains the time derivative of the LOS rate as seen in Eq. (26). It is not advisable to include a term in the guidance law that contains the differentiation of measurement data(noisy). Therefore, the estimate of the UDT, ℎU is evaluated using a low-pass filter with unity steady-state gain and broad enough bandwidth [19, 20] as

12

ℎU = X$  ,: − RS +  -

(35)

where

with the time constant denoted by Z and

G$   =

1 Z +1

(36)

is the complex constant in the Laplace transform. Equation (35) can be

represented in the time domain as ZℎU + ℎU = : − RS + 

(37)

Substituting Eq. (32) for Eq. (37) leads to ZℎU + ℎU = : − RS + RS + RT + ℎU =

B: − : + RT + ℎU B

(38)

Solving Eq. (38) for ℎU using the relation of Eq. (8) , one obtains

1 B: ℎU = V + ) : + RT W Z B

(39)

ℎU = : − # :# + OC /Z

(40)

O C = ) : + RT

(41)

ℎU = : + OC /Z

(42)

Integrating Eq. (39) with respect to time with the initial LOS range, # and the initial sliding variable, :# yields

where

Since the sliding variable : starts with :# = 0, Eq. (40) can be reduced to

Finally, the SOSM and UDE-based controller in Eq. (32) can be summarized as  = RS + RT + ℎU where by using Eqs. (30), (31), and (42), it can be shown that

13

RS = " + "# + 2)
 +

" + 1"# ) ,
−
$% &'

RT = −−6 |:|KL sign: + O ℎU = : + OC /Z

Referring to Eqs. (11), (12), (13), (7), (25), and (41), expressions for the remaining variables are :=

! !

+ 9, 90 = − ! 0

=
 + "# 9 =

" &'


−
$% &' !

O = −C |:|KP sign: O C = ) : + RT

where "# ("# = 0 or "# > 2) and " " > 1 are design parameters by which the LOS rate and the LOS angle shapes are characterized, 6 , C , Q6 , QC (6 , C > 0, 1 > Q6 > QC > 0) are design parameters for the SOSM control, and Z

is a design parameter which determines the bandwidth of the low pass filer (the time constant of the low-pass filter) for the UDE. It is worth noting that this guidance system requires only the same amount of information as that of the PN when no impact angle constraint is imposed and as that of the biased PN, when there is an impact angle constraint. The stability analysis of the whole guidance system is given in Appendix C.

III. SIMULATION A. Property of sliding dynamics Before investigating the performance of the SOSM and UDE-based intercept guidance laws, a property of the sliding dynamics is first demonstrated. Once the sliding mode holds, the convergence property of the system states such as the LOS angle or the LOS rate can be characterized by the sliding dynamics. Performance modulation with parameters, "# , and " are discussed in this subsection. First, the LOS rate characterization with time without impact angle constraint is discussed. Then, the LOS angle error change with time for the guidance law with an impact angle 14

constraint is explained. Analogous to the preceding sections, the sliding mode is always assumed to hold: : = : = 0 in this section. Before discussing the sliding dynamics with no impact angle constraint, the following variable for the LOS rate is defined for convenience. > ≜


(43)

By using Eq.(43), Eq. (17) can be represented as >  = 11 − 2 ># $

3

(44)

where ># indicates the initial LOS rate. Equation (44) suggests that the magnitude of the LOS rate monotonically decreases with time and it goes zero at $ if " > 0. The number " determines the order of convergence. In other

words, " = 1 suggests that > = 0 at  = $ , and, " = 2 implies that > = > = 0 at  = $ , whereas " = ?> 2

implies that > = > = ⋯ = > A = 0 at  = $ .

Figure 2 shows the LOS rate history based on Eq. (44) with various choices of ". As can be seen in Fig. 2, the

convergence property of the LOS rate can be characterized by the choice of ". Note that the convergence property with " = 1, 2, 3,∙∙∙ is similar to those of the PN where the corresponding effective navigation constants are ] ^ =

3, 4, 5,∙∙∙, respectively.

1 k=1 k=2 k=3 k=4 k=5

σσ

0.8 0

/

0.6 0.4 0.2 0

0

0.1

0.2

0.3

0.4

0.5 t/tf

0.6

Fig. 2 LOS rate vs time.

15

0.7

0.8

0.9

1

Next, the sliding dynamics with an impact angle constraint is discussed. The following variable for the LOS angle error
−
$% is defined for convenience. a ≜
−


%$(45)

Using Eq. (45), Eqs. (20) and (22) can be represented respectively as

a = −D6 11 − a =
 = σ = D6

 2 $

3E6

+ DC 11 −

 2 $

34

" + 1  "#  11 − 2 − DC 11 − 2 $ $ $ $ 3

(46) 34 56

(47)

Analogous to the previous case with no impact angle constraint, Eqs. (46) and (47) also suggest that the magnitudes of the LOS angle error, a =
−
$ , and the LOS rate, >, can be driven to zero at  = $ for " > 0 and "# > 1. Figure 3 shows the LOS angle error history with a various " against  "# = 2.5, c "# = 5.5, D "# = 10.5 for the

case of positive initial LOS angle error; a# = 1 with ># = −1 where a# , and ># indicate the initial values of a, and

>. As can be seen in Fig. 3, the convergence of the LOS angle error can be characterized by " and "# . Note that, for

the impact angle constraint case("# ≠ 0), LOS angle error is less sensitive to the parameter, " as shown in Fig. 3 if compared to Fig.2; the case without impact angle constraint. It can be seen that the LOS angle and the LOS rate tend to vary more rapidly and with a larger curvature as the parameter "# increases, a trend that may lead to input saturation.

16

1

1 k =1 k =2 k =3 k =4 k =5

0

/

0.6 0.4

0

/

0.2 0

0.1

0.2

0.3

0.4

0.5 t/tf

0.6

0.7

0.8

0.9

0

1

0

0.1

0.2

0.3

0.4

0.5 t/tf

0.6

0.7

0.8

0.9

1

0

0.1

0.2

0.3

0.4

0.5 t/tf

0.6

0.7

0.8

0.9

1

0 -1

-1

σ

σ

0

-0.5

-0.5

-1.5 -2

-1.5 -2

0.6 0.4

0.2 0

k =1 k =2 k =3 k =4 k =5

0.8 η η

η η

0.8

-2.5 0

0.1

0.2

0.3

0.4

0.5 t/tf

0.6

0.7

0.8

0.9

-3

1

a "# = 2.5

b "# = 5.5 1 k =1 k =2 k =3 k =4 k =5

η η

0.8 0

/

0.6 0.4 0.2 0

0

0.1

0.2

0.3

0.4

0.5 t/tf

0.6

0.7

0.8

0.9

1

0

0.1

0.2

0.3

0.4

0.5 t/tf

0.6

0.7

0.8

0.9

1

0

σ

-1 -2 -3 -4

c "# = 10.5 Fig. 3 LOS angle error and LOS rate vs time.

B. Simulation settings In order to investigate the performance of the SOSM and UDE-based intercept guidance laws, two sets of simulations with the same scenarios are made: One set with no impact angle constraint and the other set with a terminal impact angle constraint. Two scenarios are considered in each set: with and without a maneuvering target. Table 1 shows the initial kinematics for the simulations. Initial target velocity is assumed to be three times the initial missile velocity.

17

In each Case, two values for the LOS rate shaping parameter " are used with the parameter "# set to zero when

there is no impact angle constraint and "# = 2.5 for the scenarios with an impact angle constraint. The other parameters using in the simulation can be seen in Table 2. Table 1 Initial states’ values for Case 1 and 2 missile

Target

Case 1 & 2

Case 1

Max latax:

No maneuver

50g’s

Case 2 Weaving frequency: π/3 (rad) Max latax: 30g’s

velocity(m/s)

1,000

3,000

x(m)

0

50,000

y(m)

0

15,000

flight path angle(deg)

40

-175

Table 2 Parameter settings Parameters

Values

6 , C

1, 1

α6 , αC

0.7, 0.6

Z

0.01

For reference purposes, simulations made with a few other related guidance laws are compared with the proposed guidance law. For the engagement scenario with no impact angle constraint, the proportional navigation guidance (PN) and SOSM-based guidance with a nonlinear disturbance differentiator developed by Shtessel et al.[13] referred in this study as ‘SOSM-NDD’ is used. The PN guidance law is represented as[1, 2, 5]

18

,hi = ])


(48)

where ] is the effective navigation constant set as 3 in the simulations. As described before, the preceding equation is the same as with the equivalent input of Eq. (30) when setting " = 1 and "# = 0. The SOSM-NDD law is represented as[13] ,jkjl5imm = ,−]′
 + R%6 + o ,p -⁄cos 
−  

where o ,p is the estimated target acceleration component normal to the LOS and R%6 = Q6Tq |

C⁄r sign Tq  Tq | Tq

Tq |

+ QCTq s|

6⁄r

sign

Tq B

= 
 − D# √

−

(49)

D# 

2√

(50)

(51)

The parameters in the preceding guidance laws are set as D# = 0.1, ]′ = 4, Q6Tq = 0.5, QCTq = 1. For the engagement scenario with an impact angle constraint, the biased proportional navigation guidance (BPN) developed by Kim et al. [24] and integral sliding mode control-based guidance with a nonlinear disturbance observer developed by Zhang et al.[27] referred in this study as ‘ISMC-NDOB’ are also applied and compared with the proposed guidance law. The BPN guidance law[2, 5, 24] used in this simulation study is given by ,hi = ]6 )
 +

]C ) 
−
$%  &'

(52)

where the two constants are set as ]6 = 4, and ]C = 2[2,5]. As described before, the preceding equation reduces to the equivalent input in Eq. (30) when " = 1 and "# = 1. The ISMC-NDOB law is represented as[27] ,ujl.5imkv

 sin
−   + asign = ,−2
 + R%C + 

wq 

+ o ,p -⁄cos 
−  

(53)

where Kyz

R%C = Q6wq x
−
$% x wq

CKyz

sign,
−
$% - + QCwq x
xKyz E6 sign,
-

(54)



=
 −
0 + s R%C B #

19

(55)

Parameters in the preceding guidance laws are set as Q6wq = 0.5, QCwq = 1, Qwq = 0.3, a = 0.1. Zhang et al. have also developed an ISMC-based guidance for a time-lag system that provides excellent results[28]. However, their recent methodology is not used for this simulation study because it requires precise information of the second derivatives of the LOS range and the LOS angle: and
, that are impracticable to get. The maximum absolute value of the target acceleration component normal to the LOS required for the observers in SOSM-NDD and ISMCNDOB is assumed as xo ,p x ≤ 50g′s . Furthermore, for reference purposes, simulations using an HOSM differentiator[11,12] for UDT estimation are also conducted in each scenario.

C. Simulation with no impact angle constraint Figures 4 through 6 show the simulation results against a non-maneuvering target without an impact angle constraint. These results are obtained from using five guidance laws; the labels ‘PN’ and ‘SOSM-NDD’ means the results using the respective PN guidance and the SOSM-NDD guidance laws, the label ’SOSM-HOSM’ shows the results with " = 2 but it uses the HOSM differentiator for the UDT estimation for reference purpose, the labels

‘SOSM-UDE(" = 1)’and ‘SOSM-UDE(" = 2)’ means the results using the variable " set as 1 and 2, respectively.

In the case of " = 1, (although that is not recommended values in the proposed methodology) where Theorem 1

demands " > 1 for the convergence of the derivative of the LOS rate, it corresponds to the PN with ] = 3 and

therefore " = 1 is applied for reference purposes. Figure 4 shows the trajectories of the missile and the target. The resulting miss distances are shown in Table 3. As can be seen, the missile can intercept the target with a very small miss distance regardless of guidance methodologies. Figure 5 shows the latax histories of the missile and the target in the upper figure, the LOS rate > in the middle and the basic sliding surface

!

histories are plotted at the bottom.

For reference purpose, the target latax histories (that also appear in the later sections) are plotted with opposite sign for convenience. In this non-maneuvering target scenario, the results obtained using SOSM-HOSM and SOSMUDE(" = 2) are almost the same. Therefore, the results made with SOSM-HOSM related plot overlap with the

SOSM-UDE(" = 2) results. The latax histories made with PN and SOSM-UDE(" = 1) almost coincide with each

other. As expected in Eq. (17), the LOS rate changes linearly for the case using SOSM-UDE(" = 1), whereas the LOS rate trajectory changes with parabolic shape regardless of the UDT estimation methodologies when using " = 2(SOSM-HOSM and SOSM-UDE(" = 2)), the latax history changes corresponding to the LOS rate change as 20

expected. The SOSM-NDD guidance law can swiftly drive the LOS rate to around zero at the beginning while the corresponding latax requirement is higher than those of the other methodologies. The LOS rate histories with PN (] = 3) is supposed to coincide with that of SOSM-UDE(" = 1) if there is no uncertainty in the LOS rate dynamics. However, convergence characteristics of the LOS rate with PN is slightly different from that with SOSM-UDE(" = 1) because the PN guidance force is applied normal to the missile velocity vector and not the LOS. This difference

or uncertainty might be the cause for the difference between the PN and SOSM-UDE(" = 1) results. All the basic sliding variables except for SOSM-NDD converge to zero at the end. Note that the sliding surface for SOSM-NDD in the figure is divided by the LOS range to reduce the order of the magnitude, that is,

Tq /

is plotted for

comparison purpose. Therefore, the sliding surface plot with SOSM-NDD seems to be deviated from zero as the LOS range is supposed to converge to zero. Indeed, the original sliding surface for SOSM-NDD converges to zero. These smooth convergence histories with SOSM-UDE or SOSM-HOSM that can be made with Ackermann’s scheme can avoid the unnecessary fluctuating input (latax) as seen in Fig. 5. The plot is linear for SOSM-UDE(" = 1) and quadratic for SOSM-UDE(" = 2) as observed in Eq. (15). Figure 6 shows estimation plots in the upper

figures along with their real values and the corresponding estimation error plots in the lower figures. The figures on the left side in Fig. 6 relate to the UDT values estimated in the SOSM-HOSM and SOSM-UDE algorithms, whereas the figures on the right side are concerned with the target acceleration components normal to the LOS that are estimated in the SOSM-NDD algorithm. Since the UDT is affected by not just the target maneuvers but also by the missile maneuvers, the real values of such UDT are different in both guidance laws. However, it is found that the real UDT value is almost similar both for SOSM-HOSM and SOSM-UDE(" = 2). Thus, the real UDT values for

SOSM-UDE(" = 2) and SOSM-UDE(" = 1) are only plotted in red broken dotted and dotted lines, respectively.

Although the resulting latax and LOS rate histories made with SOSM-HOSM and SOSM-UDE(" = 2) can be observed to be similar, the UDT estimation results are slightly different from each other as shown in the left figures in Fig. 6. Fluctuation of the UDT estimation error with the HOSM differentiator (SOSM-HOSM) can be observed initially whereas the UDT estimation error is low amount even at the beginning. A large initial estimation error is also observed with the SOSM-NDD method.

21

15000

10000 y

PN SOSM-NDD SOSM-HOSM SOSM-UDE(k =1) SOSM-UDE(k =2) target

5000

0

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

]

5 4

x 10

x

Fig. 4. 2D trajectories of the two vehicles: Case 1.

Table 3 Miss distance: Case 1 without an impact angle constraint

Guidance law

PN

SOSM-NDD

SOSM-HOSM

SOSM-UDE (k=1)

SOSM-UDE (k=2)

Miss distance [m]

2.79e-8

9.98e-8

1.91e-12

1.27e-10

1.16e-12

400 PN SOSM-NDD SOSM-HOSM SOSM-UDE(k =1) SOSM-UDE(k =2)

200

s / m ( x a t a l

0 -200

-a t

0

2

4

6

8

10

12

14

16

18

-3

6 ) s / d a r (

σ

2

)

x 10

PN SOSM-NDD SOSM-HOSM SOSM-UDE(k =1) SOSM-UDE(k =2)

4 2 0 -2

0

2

4

6

8

10

12

14

16

18

-3

5

b

s

x 10

SOSM-NDD SOSM-HOSM SOSM-UDE(k =1) SOSM-UDE(k =2)

0

-5

0

2

4

6

8

10

22

12

14

time

Fig. 5. Latax, LOS rate and sliding surface vs time: Case 1.

16

18

8 h

300 n o tia re le 200 c c a te gr 100 at d e ta 0 im ts e -100

6

T D 4 U d e ta 2 im ts e 0 -2

SOSM-HOSM SOSM-UDE(k =1) SOSM-UDE(k =2) real value for UDE(k =1) real value for UDE(k =2) 0

2

4

6

8

10

12

14

SOSM-NDD real value

0

2

4

6

time 6 T 4 D U fo ro 2 rr e 0 n o tia m tis -2 e -4

100 axt la te 0 gr at fo ro -100 rr e n o tia -200 im ste -300

6 4 2 0 -2

h

0

0.1

0.2

0.3 SOSM-HOSM SOSM-UDE( k=1) SOSM-UDE( k=2)

0

2

4

6

8

10

12

14

time

8

10

12

14

SOSM-NDD

0

2

4

6

8

10

12

14

time

time

Fig. 6. UDT and estimation error vs time: Case 1.

Figures 7 through 9 show the results of Case 2 where the missile is engaging a weaving target. As can be seen in Fig. 7, the missile can capture the target in all cases. The resulting miss distance with PN, however, is much larger than all other methods as shown in Table 4. From the top figures in Fig. 8, the missile latax with PN tends to diverge with time in response to the target maneuver. On the other hand, the missile latax with the other laws seems to converge to the target latax. The latax histories with SOSM-UDE(" = 1) and SOSM-UDE(" = 2) are almost the same. Magnitudes of the LOS rates with SOSM-NDD decrease slower than that in Case 1 while the LOS rate trends for the SOSM-HOSM and the SOSM-UDE are almost the same as in Case 1 regardless of the target's weaving maneuvers. The basic sliding variables except for SOSM-NDD converge to zero at the end and the convergence characteristics are similar to the case against a non-maneuvering target. The contribution of the target maneuver to the UDT is dominant for Case 2 as shown in the left upper figure in Fig. 9 where the real UDT plots for SOSMUDE(" = 1) and SOSM-UDE(" = 2) overlap with each other. Similar to the previous case, the UDT and the target latax estimation errors with the HOSM differentiator in SOSM-HOSM and SOSM-NDD, respectively, are large at the beginning but swiftly converge to zero whereas the estimation errors with the proposed UDE are bounded to a 23

small amount as shown in Fig.9. Regardless of such large initial UDT estimation errors by the HOSM differentiator, the proposed SOSM-based guidance law along with Ackermann’s scheme can compensate for such errors without producing initially large and transient latax commands.

15000

PN SOSM-NDD SOSM-HOSM SOSM-UDE( k =1) SOSM-UDE( k =2) target

10000 y

5000

0

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5 x 10

x

4

Fig.7. 2D trajectories of the two vehicles : Case 2. Table 4 Miss distance: Case 2 without an impact angle constraint

Guidance law

PN

SOSM-NDD

SOSM-HOSM

SOSM-UDE (k=1)

SOSM-UDE (k=2)

Miss distance [m]

7.88

3.39e-7

2.49e-12

3.60e-10

2.41e-10

400 2

s /

0

m ( x a t a l

PN SOSM-NDD SOSM-HOSM SOSM-UDE( k =1) SOSM-UDE( k =2)

200

)

-200 -400 -600

-at

0

2

4

6

8

10

12

14

0.02

σ

-0.02

5

-0.04 -0.06

0

-0.08

-5

-0.1 5

b

s

0 -3 x 10

x 10

-3

0

2 2

4 4

6

8 6

10 8

12 10

14 12

14

16

18

SOSM-NDD SOSM-HOSM SOSM-UDE( k =1) SOSM-UDE( k =2)

0

-5

18

PN SOSM-NDD SOSM-HOSM SOSM-UDE( k =1) SOSM-UDE( k =2)

0 ) s / d a r (

16

0

2

4

6

8

10

12

14

time

24 Fig.8. Latax, LOS rate and sliding surface vs time : Case 2.

16

18

SOSM-HOSM SOSM-UDE(k =1) SOSM-UDE(k =2) real value for UDE( k =1) real value for UDE( k =2)

600 400 T D U 200 de t 0 a m it s -200 e h

-400 0

2

4

6

8

10

SOSM-NDD real value

no 600 it ar el 400 ec ca 200 te gr at 0 de ta -200 im ste -400

12

14

0

2

4

6

time 20 h

10

T D U 0 fo r o rre -10 n o it -20 a m it se -30 -40

SOSM-HOSM SOSM-UDE( k =1) SOSM-UDE( k =2)

0

2

4

6

8

10

12

14

time

8

10

12

14

200 xa ta l 100 te gr 0 at fo ro -100 rr e no -200 it a itm-300 se -400

SOSM-NDD

0

2

4

6

8

10

12

14

time

time

Fig.9. UDT and estimation error vs time: Case 2. D. Simulation with an impact angle constraint Figures 10 through 12 show the results from applying the proposed UDE and the SOSM-based guidance laws with an impact angle constraint against a non-maneuvering target along with other guidance laws. The specified impact angle is set as
$% = |/12 rad. Parameter values are set as " = 1 for label ‘SOSM-UDE( " = 1)’ and " = 2 for label ‘SOSM-UDE( " = 2)’ and ‘SOSM-HOSM’, and "# = 2.5. Analogous to the previous case, results labelled

‘BPN’ and ‘ISMC-NDOB’ were obtained using the BPN and ISMC-NDOB guidance laws, whereas label ‘SOSMHOSM’ indicates results from using the HOSM differentiator for the UDT estimation. The missile can intercept the target with small miss distances in all cases as shown in Table 5. Saturation of latax command with ISMC-NDOB is observed initially in the upper figure in Fig.11. This may be due to the fact that the required latax for the LOS angle error compensation, in general, increases with the LOS range even when the LOS error angle is very small. What happens here is that the ISMC-NDOB guidance law tries to reduce the LOS angle error swiftly regardless of the magnitude of the LOS range, which may cause initial saturation. Furthermore, fluctuations of the latax are observed 25

with ISMC-NDOB around 2.5 s through 3 s, which result from the zero-division in the latax command calculation in Eq. (53) when cos
−   becomes zero where the missile switches the latax direction between head-on and head

pursuit with high frequency. The latax histories made with SOSM-UDE(" = 2) and SOSM-HOSM almost coincide

with each other. The second figure from the bottom in Fig. 11 indicates the LOS angle error histories. It is observed that the LOS angle error and the LOS rate for ISMC-NDOB are driven to zero faster than with other methods whereas the LOS angle error and the LOS rate for BPN, SOSM-HOSM, SOSM-UDE(" = 1), and SOSM-UDE(" =

2) are driven to zero at the final moment in both cases as expected by substituting  = $ for Eqs. (20) and (22). All the basic sliding variables converge to zero at the end. The convergence characteristics of the LOS angle error and the LOS rate are different from the case of no-impact angle constraint; the plots of the basic sliding variables form linear for SOSM-UDE(" = 1) and quadratic for SOSM-UDE(" = 2) conform to the analysis of Eq. (15). Similar to

the results in previous sections, results made with SOSM-UDE(" = 2) and SOSM-HOSM in Fig. 11 show that they are less affected by the estimation methodologies when applying the SOSM-based guidance law with Ackermann’s scheme. However, the initial estimation error can be attenuated using the proposed UDE if compared with the results using the HOSM differentiator as shown in Fig. 12.

15000

10000 y

BPN ISMC-NDOB SOSM-HOSM SOSM-UDE(k=1) SOSM-UDE(k=2) target

5000

0

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5 4

x 10

x

Fig. 10. 2D trajectories of the two vehicles(Case 1 with an impact angle constraint). Table 5 Miss distance: Case 1 with an impact angle constraint Guidance law

BPN

ISMC-NDOB

SOSM-HOSM

SOSM-UDE (k=1)

SOSM-UDE (k=2)

Miss distance [m]

1.94e-12

1.46e-9

3.53e-11

5.34e-11

1.76e-12

26

500

BPN ISMC-NDOB SOSM-HOSM SOSM-UDE(k =1) SOSM-UDE(k =2)

)

2

s /

0

m ( x a t a l

-a t

-500

0

2

4

6

8

10

12

14

16

18

0.01 BPN ISMC-NDOB SOSM-HOSM SOSM-UDE(k =1) SOSM-UDE(k =2)

0.005 ) s / d a r (

0

σ

-0.005 -0.01 -0.015

0

2

4

6

8

10

12

14

16

18

0.04

λ λ

BPN ISMC-NDOB SOSM-HOSM SOSM-UDE(k =1) SOSM-UDE(k =2)

0.03

) d a r(

0.02 0.01

f

-

0 -0.01

0

2

4

6

0.4

s

10

12

14

16

0.01

0.3

b

8

0.2

0

0.1

-0.01

18

ISMC-NDOB SOSM-HOSM SOSM-UDE(k =1) SOSM-UDE(k =2) 0

5

10

15

0 -0.1 -0.2

0

2

4

6

8

10

12

14

16

18

time

Fig. 11. Latax, LOS rate, LOS angle error, and sliding surface vs time(Case 1 with an impact angle constraint). 30 SOSM-HOSM SOSM-UDE( k =1) SOSM-UDE( k =2) real value for UDE(k =1) real value for UDE(k =2)

20

h

T D 10 U de ta 0 m tis e -10 -20

0

2

4

6

8

10

12

14

40 n io ta re 20 le cc 0 a t eg -20 ra t de -40 ta -60 im ts e -80

ISMC-NDOB real value

0

5

10

time 20

15

h

T 15 D U f o ro 10 rre no 5 it a im ts 0 e -5

SOSM-HOSM SOSM-UDE( k =1) SOSM-UDE( k =2)

10 5 0

0

2

0

0.5

4

6

8 time

15

time

10

12

14

xa 80 atl te 60 gr at 40 f o r 20 o rre n 0 o it a -20 im t es -40

100

ISMC-NDOB

50 0 -50

0

0

0.01

0.02

0.03

5

10 time

Fig. 12 UDT and estimation error vs time27 (Case 1 with an impact angle constraint).

15

Figures 13 through 15 show the results obtained with an impact angle constraint against a maneuvering target. The parameters values are the same as before. In all cases, the missile can still capture the target. The resulting miss distances with the BPN and the ISMC-NDOB are slightly larger than those of the other three cases. The latax command with BPN tends to diverges with time. Saturation of the latax command with ISMC-NDOB is also observed at the beginning while the LOS angle error and the LOS rate with ISMC-NDOB converge to zero around 6.5 s and remain zero. On the other hand, regardless of the target maneuver or the estimation methodology for SOSM-UDE and SOSM-HOSM, the LOS angle error and the LOS rate convergence characteristics are almost the same as in the case against non-maneuvering target. Again, the plots of the basic sliding variables are linear for SOSM-UDE(" = 1) and quadratic for SOSM-UDE(" = 2) follow the trends predicted in Eq. (15).

15000

10000 y

BPN ISMC-NDOB SOSM-HOSM SOSM-UDE(k =1) SOSM-UDE(k =2) target

5000

0

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

x

Fig. 13. 2D trajectories of the two vehicles (Case 2 with an impact angle constraint).

28

5 4

x 10

Table 6 Miss distance: Case 2 with an impact angle constraint

Guidance law

BPN

ISMC-NDOB

SOSM-HOSM

SOSM-UDE (k=1)

SOSM-UDE (k=2)

Miss distance [m]

2.87e-1

4.47e-3

2.46e-10

7.27e-12

1.60e-10

500 BPN ISMC-NDOB SOSM-HOSM SOSM-UDE(k =1) SOSM-UDE(k =2)

)

2

s / m (

0

x a t a l

-a t

-500

0

2

4

6

8

10

12

14

16

18

0.02 BPN ISMC-NDOB SOSM-HOSM SOSM-UDE(k =1) SOSM-UDE(k =2)

0.01 ) s / d a r (

0

σ

-0.01 -0.02 -0.03

0

2

4

6

8

10

12

14

16

18

0.04

λ λ

BPN ISMC-NDOB SOSM-HOSM SOSM-UDE(k =1) SOSM-UDE(k =2)

0.02

) d a r (

f

-

0 -0.02

0

2

4

6

8

10

12

14

16

18

0.5

0.3

s

b

ISMC-NDOB SOSM-HOSM SOSM-UDE(k =1) SOSM-UDE(k =2)

0.01

0.4

0

0.2

-0.01

0.1

0

5

10

15

0 -0.1 -0.2

0

2

4

6

8

10

12

14

16

18

time

Fig. 14. Latax, LOS rate, LOS angle error, and sliding surface vs time(Case 2 with an impact angle constraint).

29

SOSM-HOSM SOSM-UDE(k=1) SOSM-UDE(k=2) real value for UDE(k =1) real value for UDE(k =2)

600 h

400 T D U 200 d e ta m 0 tis e -200 -400

0

2

4

6

8

10

12

14

n 600 o tia re le 400 c c a te 200 gr at 0 de ta -200 im ste -400

ISMC-NDOB real value

0

2

4

6

time 40 SOSM-HOSM SOSM-UDE(k=1) SOSM-UDE(k=2)

h

T D 20 U fo ro rr 0 e n o tia -20 im ste -40

0

2

4

6

8

10

12

14

time

8

10

12

14

xa 80 ta l 60 te gr at 40 fo r 20 o rr e 0 n o it a -20 m tis e -40

ISMC-NDOB

100 0 -100

0

0

2

0.005

0.01

4

0.015

6

0.02

8

10

12

14

time

time

Fig. 15. UDT and estimation error vs time (Case 2 with an impact angle constraint). Similar to the all previous scenarios, the UDT estimation error with the HOSM differentiator for SOSM-HOSM and ISMC-NDOB is initially large whereas the error with the proposed UDE is small even at the beginning. Regardless of such errors, the SOSM-based guidance law with Ackermann’s scheme can compensate the UDT estimation error. Except for the initial transient for the HOSM differentiator, the UDE as well as the HOSM differentiator can attenuate the estimation error to a small bound from the beginning. It is important to note that, in order to obtain the estimated UDT values described thus far, the HOSM differentiator requires six differential equations and seven tuning parameters for this study, whereas the proposed UDE demands only one differential equation and one tuning parameter; the time constant Z for the UDT estimation. From this point of view, the proposed UDE has good potential as a simple design process.

IV. Conclusions Missile-target intercept guidance with uncertainties and disturbances was considered in this study. A novel unified intercept guidance law which can work in cases with/without angle constraints and which can follow desired LOS rate and/or the LOS angle trajectories regardless of the target maneuver, was developed in this study. Simulation results show that the new guidance law has desirable performance characteristics. This unified guidance law which incorporates the PN and the biased PN strategies in its equivalent control can be a powerful tool and it is equipped with an uncertainty and disturbance compensation while its measurement requirements are moderate and 30

the same as the conventional PN and the biased PN. Such a low implementation requirement for sensing and the design simplicity equipped with robustness for uncertainty and disturbances make the proposed guidance system as practicable as the widely-used PN and/or BPN guidance laws for ease of implementation into a real system. Integration of the proposed guidance with an autopilot system taking account of a nonlinear rigid body six-degreesof-freedom model, and the implementation of the proposed guidance law in a real system under sensor noise circumstances should be considered in the future.

Appendix A. Nomenclature LOS

= line-of-sight



= lateral missile acceleration, m/s2



X$  

= lateral target acceleration, m/s2 = low-pass filter

ℎ

= term of uncertainty and disturbance, m/s2

"

= LOS rate shaping parameter

ℎU "#

]^

 

!

$ )



= estimated value of ℎ

= parameter for an impact angle constraint = effective navigation constant = relative distance or LOS range =

the basic sliding surface

= time = the time when the two vehicles locate in closest point. = closing velocity, m/s = missile velocity, m/s



= target velocity, m/s

x, y

= x-axis and y-axis components in the 2D plane, m 31

9

= variable for sliding surface




= LOS angle, rad




%$= desired terminal LOS angle, rad



= missile flight path angle, rad



= target flight path angle, rad

>

= LOS rate, rad/s

Z

= time constant for the low-pass filter

ξ

= sliding surface variable related to SOSM-based guidance law with an impact angle constraint

  

 #

= time derivative = initial value

B. Closed Form Solutions of Differential Equations in Sec. II. In this Appendix, solutions for first and second order differential equations are derived for reference purposes. In relation to the basic sliding mode, Eqs. (7) and (14) have the forms as } +

" }=0 &'

(B-1)

where &' = $ − , and ", and $ are assumed to be constant. Note that if x is set as } =
−
$% with a constant
$% , then, Eq. (7) take the form of Eq. (B-1). Rearranging Eq. (B-1) can be expressed as } −" = } &'

(B-2)

Integrating Eq. (B-2) over time with the integral constant D yields 3

} = D,&' - = D,$ − -

3

(B-3)

Supposing the initial value of } as }# , then, D = }# $ 53 , thus Eq. (B-3) can be rearranged as } = }# 1

&' 2 $

53

= }# 11 −

 2 $

53

This is the closed form solution of the first-order differential equation of Eq. (B-1).

32

(B-4)

Next a closed form solution of a second-order differential equation is derived. The differential equation used in this study has the following form: } +

" + 1"# " + "# } + }=0 &' &' C

(B-5)

Similar to the preceding first-order equation, if x is set as } =
−
$% with a constant value of
$% , then, Eq. (19) takes the form of Eq. (B-5). Assuming that a special solution of Eq. (B-5) is } = D,$ − -

~

(B-6)

where D and  are nonzero constants. The first and second time derivatives can be calculated as ~56

} = −D,$ − -

} = D − 1,$ − -

(B-7)

~5C

(B-8)

Substituting Eqs. (B-6)-(B-8) into Eq. (B-5) yields  − " − 1 − "# D,$ − -

~5C

=0

(B-9)

Thus,  = " + 1, or  = "# can satisfy the preceding equation. Therefore, the general solution of the second-order differential equation of Eq. (B-5) can be represented by the linear combination as } = D,$ − -

3E6

34

+ D# ,$ − -

(B-10)

Taking the time derivative of Eq. (B-10) yields 3

34 56

} = −D" + 1,$ − - − D# "# ,$ − -

(B-11)

Assuming the initial values of } and } as }# and } # , then, one can get the constant values D and D# as D=

−"# }# − } # $ 5356  , " − "# + 1 $

D# =

" + 1}# + } # $ 53 $ 4 " − "# + 1

(B-12)

Substituting Eq. (B-12) for Eq. (B-10) leads to "# }# + } # $  }=− 11 − 2 " − "# + 1 $

3E6

" + 1}# + } # $  + 11 − 2 " − "# + 1 $

34

Equation (B-13) represents the closed form solution of the second-order differential equation of Eq. (B-5)

33

(B-13)

C. Stability Analysis In this Appendix, the stability of the whole system is discussed. As discussed in Sec. II.D, the proposed guidance system is represented by Eqs. (33) , (25), and (41) rewritten here as : =

1 − + ℎ + RS  

O = −C |:|KP sign:

(C-1)

(C-2)

O C = ) : + RT

(C-3)

 = RS + RT + ℎU

(C-4)

where

RT = −−6 |:|KL sign: + O ℎU = : + OC /Z

(C-5)

(C-6)

from Eqs. (32), (31), and (42). As can be seen, the system of Eqs. (C-1), (C-2), and (C-3) has the equilibrium point at :, O, OC  = 0,0, Zℎ where Z is a constant as discussed in Sec. II.D. This system is said to be stable when the system states; :, O, OC converge to the neighborhood of the equilibrium point. Without loss of generality, assume

that the UDT ℎ is bounded, then the system stability can be proved if : and O converge to the neighborhood of zero and ℎU converges to the neighborhood of ℎ since the convergence of OC is equivalent to the convergence of ℎU and :

under the finite values of and Z. Thus, the convergent property of the variables :, O and ℎ − ℎU are discussed. Assume that the time derivative of the UDT is bounded as xℎx < 

(C-7)

ℎ‚ = ℎ − ℎU

(C-8)

Define an error UDT as

and its time derivative using Eq. (C-6) with Eqs. (C-3) and (C-4) is expressed as ℎ‚ ℎ‚ = ℎ − Z Substituting Eqs. (C-4), (C-5), and (C-8) into (C-1) yields 34

(C-9)

: = −6 |:|KL sign: + O +

ℎ‚ 

(C-10)

where the last term in the preceding equation denotes a UDT in sliding dynamics. To investigate the convergence property of :, O and ℎ‚, let a candidate Lyapunov function be |‡|

ƒ ,:, O, ℎ‚- = s C „ KP dζ + #

O C ℎ‚C + 2 2

(C-11)

where ƒ ,:, O, ℎ‚- > 0 except at : = O = ℎ‚ = 0. The time derivative of the preceding Lyapunov function results in ƒ ,:, O, ℎ‚- = C |:|KP sign:: + OO + ℎ‚ℎ‚

(C-12)

Substituting Eqs. (C-9) and (C-10) for Eq. (C-12) yields ƒ ,:, O, ℎ‚- = −C |:|KP V6 |:|KL −

ℎ‚ ℎ‚ sign:W − ℎ‚ 1 − ℎ2  Z

(C-13)

As long as the following two inequalities hold, that is, 6 |:|KL >

xℎ‚x and xℎ‚x > Z 

(C-14)

the time derivative of the Lyapunov function is always negative: ƒ ,:, O, ℎ‚- < 0. When the minimum range for the

missile engagement is assumed to be set as A~ known as ‘blind sight’, then the absolute values of the sliding variable |:| and the UDT error xℎ‚x are ultimately bounded in the neighborhood of the equilibrium as Ω = ‰:, ℎ‚ ∈ ℝŒ|:| < 

6

Z KL Ž , xℎ‚x < Z  6 A~

(C-15)

Since the sliding variable : is ultimately bounded, the integral of such value over a finite-time of interval is also

bounded, which deduces from Eq. (C-2) that the absolute value of the variable O is bounded as KP

Z KL |O| < C $ − #   Ž 6 A~

(C-16)

Thus, the proposed guidance system is said to be stable. Note that, if the UDT is unchanged, then ℎ = 0  ≈ 0 that implies the UDT can be estimated exactly and the sliding variable converges to zero as seen in Eq. (C-15). For the finite time convergence, please refer to the previous work [15].

35

In order for the stability discussion for all states to be complete, one needs to show the boundedness of the remaining variables; the LOS angle and LOS rate. The boundedness of Eq.(C-15) with Eq.(C-16) leads toanother bounded value from Eq.(C-10); that is, the absolute value of the time derivative of the sliding variable, : x:x = | ! + 9 | = ‘ ! +

KP

" &'

Z Z KL Ž + C $ − #   Ž ≜ ’6 > 0 !‘ < 2  A~ 6 A~

(C-17)

where the definitions of Eqs. (11) and (13) are used in the first and second equality. Since the square of the basic sliding variable,

!

C

decreases as long as

! !

< 0 holds, (this is satisfied when ’6 <

3

;<

| ! |). Therefore, the basic

sliding variable is ultimately bounded as | !| <

&' ’ " 6

(C-18)

Analogous to the preceding discussion, the condition 
−
$% 
 < 0 decreases the square of the LOS angle error,


−
$% C . On the other hand,
 = 
−
$% 
 < 0 is satisfied when

34

;<

!

− "#

p5p“”

;<

from Eq. (7), which implies that the decreasing condition

x
−
$% x > | ! |. Since | ! | is bounded within

LOS angle error, x
−
$% x can be ultimately bounded as x
−
$% x <

&' C ’ ""# 6

;< 3

’6 , the absolute value of the

(C-19)

The boundedness inequalities of the LOS angle error as in Eq. (C-19) and the basic sliding variable as in Eq. (C-18), and the definition of the basic sliding variable of Eq. (7) imply that the LOS rate ultimately comes to be bounded. Thus, all the states in the proposed guidance system is said to be stable.

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