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Integral sliding mode control based approach and landing guidance law with finite-time convergence
Accepted Manuscript Title: Integral sliding mode control based approach and landing guidance law with finite-time convergence Author: Xuanping Liao Lei Chen PII: DOI: Reference:
S0030-4026(16)30633-7 http://dx.doi.org/doi:10.1016/j.ijleo.2016.06.012 IJLEO 57798
To appear in: Received date: Accepted date:
10-4-2016 1-6-2016
Please cite this article as: Xuanping Liao, Lei Chen, Integral sliding mode control based approach and landing guidance law with finite-time convergence, (2016), http://dx.doi.org/10.1016/j.ijleo.2016.06.012 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 proof before it is published in its final 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.
1
Integral sliding mode control based approach
an
Abstract
us
Xuanping Liao∗ , Lei Chen
cr
convergence
ip t
and landing guidance law with finite-time
In this study, a novel finite-time sliding mode control law is proposed and its application to the
M
approach and landing (A&L) guidance problem is detailed. The proposed control strategy is designed through a combination of integral sliding mode (ISM) concept and polynomial feedback control law. The advantages of this approach are that the robustness is ensured throughout the motion, the convergence time
d
can be chosen in advance and the system states can be expressed in an analytical way. Through the A&L
te
guidance problem, the potential of the developed approach is demonstrated. The guidance scheme can land the reusable launch vehicle at the designated touchdown point with a reasonable amount of vertical
Ac ce p
velocity. The associated law features in its closed-loop form and the on-line trajectory generation ability. The required information only includes the instantaneous flight conditions and the terminal constraints. Numerical simulations in different scenarios are provided to validate the effectiveness of the proposed method.
Index Terms
Finite-time control, approach and landing, guidance, integral sliding mode, reusable launch vehicle.
I. I NTRODUCTION As a subclass of variable structure control, sliding mode control (SMC) is a popular nonlinear deterministic control method tackling the uncertainty and external disturbance under matching Xuanping Liao and Lei Chen are with the college of Aerospace Science and Engineering, National University of Defense Technology, Changsha 410073, China. ∗ Corresponding author. Email: Xuanping [email protected]
Page 1 of 30
conditions [1–3]. In traditional SMC design, a linear hyperplane based sliding mode manifold is used to design the controller which leads to infinite-time stabilization of the system. Finitetime control usually possesses superior properties, such as higher precision, faster convergence rate and better robustness against parameter variation [4–6]. Therefore, finite-time SMC problem has
ip t
recently drawn a good amount of attention from researchers around the world.
Based on the concept of terminal attractors [7], terminal sliding mode control (TSMC) technique
cr
is developed to achieve finite-time convergence [8]. However, the TSMC laws suffers from the
us
problem of singularity. This is because terms with negative fractional powers may exist in the associated controller. To deal with this problem, some methods have been investigated in [9, 10].
an
These methods, however, belong to indirect approaches and cannot resolve the problem completely. In [11], a novel concept of nonsingular terminal sliding mode control (NTSMC) is proposed which solves the singular problem fundamentally. In terms of the convergence property, NTSMC still has
M
two main drawbacks. One is that the system is sensitive to external disturbance and parameter variation in the reaching phase. The other is that the convergence time of the system states is hard
d
to know. This is because the convergence time under NTSMC is calculated from the the occurrence
te
of the sliding mode, and the associated results in the reaching phase is hard to obtain.
Ac ce p
To achieve finite-time convergence and better convergence property, integral sliding mode control (ISMC) technique has also been investigated. The key feature of ISMC lies in that the robustness of the controlled system can be enhanced without sacrificing the performance [12]. Unlike TSMC and NTSMC, a nominal controller is first designed in ISMC to achieve a desired performance for the nominal system, and then the sliding surface is constructed on the basis of the nominal controller. In [13], an ISMC based higher order sliding mode control scheme is developed. The main advantage of this method is that the time convergence can be chosen in advance. But the associated controller design depends on an open-loop control which cannot ensure a desired system performance if the system states are perturbed off the predicted trajectory. This method is further extended in [14] by adopting a continuous closed-loop control law. But the system can only be stabilized to a small vicinity of the origin but not zero. In this study, we intend to propose a novel finite-time sliding mode control method based on ISMC concept. The controller consists of two parts. One part is polynomial feedback control 2
Page 2 of 30
law which solves a finite-horizon optimal control problem with free-final-state. The discontinuous control is developed as the other part for the rejection of the bounded disturbance and uncertainty throughout the entire response of the system. The main feature of this control law is that it ensures the system states to converge to equilibrium point at the desired finite-time and the system state
ip t
can be expressed in an analytical way. In addition, the controller is global robust against matched model uncertainty and external disturbance and it does not depend on any off-line information.
cr
In order to demonstrate the potential of the proposed control method, the terminal guidance
us
problem for the approach and landing (A&L) phase of a reusable launch vehicle (RLV) is also investigated in this paper. The A&L phase is responsible for bringing the RLV from the approach
an
and landing interface (ALI) to runway touchdown [15]. During the early stages of the entry phase, extreme conditions such as aerosurface failures, poor vehicle performance and aerodynamic perturbation may be encountered. So the RLV needs to calculate the current status on-line and to re-
M
target an alternate landing position if necessary [16]. In this way, the predesigned A&L trajectory
generation capability.
d
may be no longer suitable and the guidance laws are required to possess the on-line trajectory
te
A trajectory-planning based guidance law for the A&L phase is presented in [15]. Afterwards,
Ac ce p
this method is further analyzed to land a RLV with limited normal acceleration capabilities in [17]. The design goal is achieved by determining the minimum constant equilibrium-glide load factor during the A&L phase. An Optimum-Path-To-Go (OPTG) algorithm is investigated in [18–20] to achieve a successful A&L movement. The potential of this approach is further demonstrated by dealing with control surface errors [21] and retargeting to more suitable alternative landing sites [22]. Using the state-dependent Riccati equation (SDRE) method, a novel finite horizon suboptimal comptroller is proposed in [23]. The corresponding guidance scheme is robust to initial conditions. In [24], a new second-order SMC is proposed and applied to solve the A&L guidance problem. The associated guidance law just depends on the boundary conditions of the A&L phase and it is able to generate new trajectories on-line. The aim of the A&L guidance is to ensure a feasible landing movement, i.e. the RLV is landed at the predetermined touchdown point with the least possible flight path angle and vertical velocity. As detailed in [23], the A&L guidance problem belongs to the finite time control category. In this 3
Page 3 of 30
paper, we apply the proposed ISMC method to the A&L guidance problem. The associated guidance law is suitable for onboard implementation due to its simple format and less information demand. The required information only includes the terminal constraints and the instantaneous states of the RLV. Compared with the existing publications, the proposed method makes some contributions
ip t
in the following aspects. First, the approaches outlined in [15, 17–20] can only be used to solve the A&L problem. The proposed ISMC law, however, is a practical control method and can be
cr
employed to deal with other control problems with a goal to be fulfilled in a given time. Second, although a closed-form solution to the finite horizon optimal control problem is addressed in [23],
us
the control law can only be expressed in the feedback from and the associated computational burden is high. The proposed method is also a kind of optimal control and it can be expressed in
an
an analytical form. Hence, it is more more practical and ease of mechanization. Finally, the system states can be analytically obtained using the proposed controller, whereas other finite-time control
M
laws [8–11, 13, 14] do not possess this feature.
te
A. Equations of Motion
FORMULATION
d
II. S YSTEM MODEL AND PROBLEM
Ac ce p
In this section, the dynamics of an RLV during the approach and landing phase is presented. It is assumed that the velocity direction of the RLV stays in a vertical plane during the entire landing process which implies the cross range to the center line of the runway is zero. To make it explicit, a diagram of the A&L phase is presented in Fig. 1. The velocity and the flight path angle of the
Fig. 1.
Diagram of the A&L phase
RLV is represented by V and γ respectively and the touchdown point is denoted by T . 4
Page 4 of 30
The two-dimensional point-mass dynamics of a RLV during the A&L phase is given as follows [15] (1)
y˙ = V sinγ
(2)
ip t
x˙ = V cosγ
D V˙ = − − gsinγ m
cr
L g − cosγ mV V
(4)
us
γ˙ =
(3)
where g = 32.2f t/sec2 is the gravity acceleration. L and D are aerodynamic lift and drag forces
an
defined in the usual manner
(5)
D = qSCD
(6)
M
L = qSCL
where q = 12 ρV 2 is the dynamic pressure, S represents the reference area of the RLV, CL and CD
d
denote the lift and drag coefficients respectively. The atmospheric density ρ is obtained from the
Ac ce p
te
1976 U.S. Standard Atmosphere [25] and CL and CD are modeled as follows [24] CL = CL0 sin2 αcosα
(7)
CD = CD0 + KCL2
(8)
where K, CL0 and CD0 are constants and α represents the angle of attack. The normal acceleration u = V γ˙ is treated as the control effort of the guidance system.
B. Problem statement To guarantee a successful touchdown movement, the soft landing requirement must be fulfilled, i.e., the descent rate of the RLV should be negative with a small magnitude as the altitude goes to
5
Page 5 of 30
zero. Consequently, the objective is to design a guidance law such that lim y = 0
x→xf
(9)
lim y˙ = Vyf
where Vyf is the desired vertical velocity at the final instant.
ip t
x→xf
cr
III. F INITE - TIME INTEGRAL SLIDING MODE CONTROL
us
This section details the basic principle of the finite-time ISMC. A class of second-order uncertain nonlinear systems is used for illustration. The control law is composed of two parts [26], i.e.,
an
integral sliding mode control and polynomial feedback control. The former is used to compensate any possible uncertainties during the entire control action, while the latter is designed to achieve
M
the finite time stability of the system. The optimality of this method is analyzed and the analytic solutions of the system states as well as the control command are obtained. Furthermore, feasible
d
sets of the control coefficients are discussed.
te
A. Design of integral sliding mode control
Ac ce p
Consider a second-order uncertain nonlinear system x˙ 1 = x2 (10)
x˙ 2 = f (X) + g(X) + b(X)U
where X = [x1 , x2 ]T is the system state vector, U denotes the system input signal, g(X) represents the lumped disturbances satisfying ||g(X)|| ≤ δg with δg > 0, f (X) and b(X) ̸= 0 are known functions of X. The design goal is to find a controller U such that both x1 and x2 can be driven to zero in a given finite time in the presence of uncertainties. As discussed in [13], the system described by Eq. (10) can be trivially rewritten as follows x˙ 1 = x2 (11) x˙ 2 = φ + U
6
Page 6 of 30
where φ = f (X) + g(X) + (b(X) − 1)U is treated as a new form of lumped uncertainties which is also bounded. The controller U is composed of two parts, as U = u0 + u1
(12)
ip t
where u0 is the ideal control, which is continuous and specially designed to stabilize system (11) without uncertainties, while u1 is the discontinuous control used to reject the bounded disturbances.
cr
We assume the ideal control u0 is already known and design the discontinuous control u1 .
an
S1 = x2 + z
us
Defining the following integral sliding manifold
(13)
where z is the integral term that needs to be determined. Differentiating S1 with respect to t yields
M
S˙ 1 = x˙ 2 + z˙ = φ + u0 + u1 + z˙
(14)
d
˙ Assuming the equivalent control of u1 is ueq 1 , a sufficient condition for guaranteeing S1 = 0 during
te
the entire control action can be obtained as
Ac ce p
z˙ = −u0
(15)
ueq 1 = −φ
Besides, the sliding motion is established from the very beginning as long as the following equation holds.
z(0) = −x2 (0)
(16)
In order to ensure the attractability of the sliding surface in spite of uncertainties, the discontinuous control u1 is designed as follows
u1 = −ηsgn(S1 )
(17)
where the switching gain satisfies η > ||φ||∞ . The stability of the system (11) can be easily proved using Lyapunov Second Method by considering V1 = 12 S12 as Lyapunov candidate [26].
7
Page 7 of 30
B. Polynomial feedback control law design In this subsection, a new polynomial feedback control law is proposed. The optimality of this controller and the selection of the control coefficients are also discussed.
ip t
By employing the integral sliding mode control approach, the system model (11) in sliding mode can be derived as
cr
x˙ 1 = x2
us
x˙ 2 = u0
(18)
an
which is a linear time-invariant system. Rewritten Eq. (18) into state space representation, one gets x˙ 0 1 x 0 1 = 1 + u0 (19) x˙ 2 0 0 x2 1 | {z } |{z} B
M
A
Recall the design goal is to drive the system states x1 and x2 to zero at the desired finite time. It
∫
te
quadratic performance index
d
can be considered as a finite horizon optimal control problem in which we minimize the following
tf
(
) X T (t)Q(t)X(t) + R(t)u20 (t) dt
(20)
t0
Ac ce p
1 1 J1 = X T (tf )Ptf X(tf ) + 2 2
where t0 and tf denote the initial and final time respectively, Ptf ∈ R2×2 and Q(t) ∈ R2×2 are symmetric and positive semi-definite, R(t) ∈ R+ . It is well-known that the resulting optimal control takes the form of u0 (t) = −R−1 (t)B T P (t)X(t), where P (t) ∈ R2×2 is the unique non-negative definite solution of the following matrix differential Riccati equation −P˙ (t) = AT P (t) + P (t)A − P (t)BR−1 (t)B T P (t) + Q(t)
(21)
It can be seen from the above equation that P (t) can be obtained by integrating Eq. (21) backward from tf to t with P (tf ) = Ptf [27]. This process, however, can hardly be applied on-line and thus this method is restrictive in particular for practical applications. In the sequel, the polynomial feedback control law will be introduced which provides a new solution to the finite horizon optimal control problem.
8
Page 8 of 30
Before moving on, let us first consider the following infinite-horizon nonlinear regulator problem: find u0 , which minimizes J, defined by 1 J= 2
∫
∞
(X T (t)QX(t) + Ru20 )dt
(22)
t0
ip t
Here, Q ≥ 0 and R > 0 represent the state weighting matrix and the control weighting matrix,
(23)
us
cr
respectively. Without loss of generality, we assume R = 1 and Q takes the form of q12 0 Q= 2 0 q2 The state feedback solution of the proposed optimal control problem is given by
an
u0 = −R−1 B T P X(t)
(24)
M
The matrix P is positive definite and satisfies the following algebraic Riccati equation −P A − AT P + P BR−1 B T P − Q = 0
(25)
te
d
p1 p2 and substitute it into Eq. (25), yields Let P = p2 p3
Ac ce p
] p1 p2 p1 p2 0 0 p1 p2 0 [ q12 0 p1 p2 0 1 + 0 1 − − =0 − 2 1 0 p2 p3 1 p2 p3 p2 p3 0 q2 0 0 p2 p3 After some algebra, we have p1 = q1
√ √ 2q1 + q22 , p2 = q1 and p3 = 2q1 + q22 , which, on
substitution in Eq. (24), yields
√ u0 = −q1 x1 − 2q1 + q22 x2
(26)
Then, the polynomial feedback control law will be derived. Choose the coefficients q1 and q2 to be functions of time-to-go
N t2go √ √ (υ − 2)N q2 = (υ − 2)q1 = tgo q1 =
9
(27)
Page 9 of 30
where N > 0, υ ≥ 2 and tgo = tf − t. Then, substitute Eq. (27) into Eq. (26), one gets √ N υN u0 = − x1 − x2 2 (tf − t) tf − t
(28)
The controller in Eq. (28) is the polynomial feedback control law. It should be noted, however,
ip t
the proposed polynomial feedback control law is not an optimal solution to the performance index Eq. (22). This is because the coefficients q1 and q2 are selected to be time-varying functions. As
cr
a matter of fact, the controller in Eq. (26) with constant q1 and q2 is the optimal control to the
us
cost function (22). Fortunately, there is no need to find an optimal solution to the infinite horizon performance index (22) since the design goal is to enforce x1 and x2 both to zero at the a designated time which belongs to the finite horizon control problem. Then, it invites a question: Is there any
an
cost function in the finite horizon format as Eq. (20) that results in the polynomial feedback control law denoted by Eq. (28)?
k1 k2 x1 − x2 2 (tf − t) tf − t
(29)
d
u∗ = −
M
Theorem 1: Consider the following control law
te
where k1 and k2 are positive constants. If µ is a constant which satisfies µ >
k12 , k2
the control law
Ac ce p
described by Eq. (29) can optimize the performance index (20), with q11 q12 Q(t) = q21 q22 where q11 =
k22 −k2 −2k1 R(t) (tf −t)2
k12 −3µ R(t) (tf −t)4
−
−
k2 ˙ R(t). (tf −t)
µ ˙ R(t), (tf −t)3
q12 = q21 =
k1 (k2 −2)−µ R(t) (tf −t)3
(30)
−
k1 ˙ R(t) (tf −t)2
and q22 =
P (t) is given by
P (t) =
µ R(t) (tf −t)3 k1 R(t) (tf −t)2
(31)
k2 R(t) (tf −t)
Proof The proof can refer to [28]. From Eq. (28), one gets k1 = N and k2 =
k1 R(t) (tf −t)2
√ υN . If the control law (28) is able to optimize a
cost function (20), the matrix P (t) determined by Eqs. (23), (25) and (27) should satisfy Eq. (31).
10
Page 10 of 30
From Eqs. (23), (25) and (27), we have P (t) =
√ N υN (tf −t)3
N (tf −t)2 √ υN (tf −t)
N (tf −t)2
(32)
Q(t) =
√ N 2 −3N υN (tf −t)4 −2N (tf −t)3
−2N (tf −t)3 √ (υ−2)N − υN (tf −t)2
us
cr
ip t
Substitute R(t) = 1 into Eq. (31) and compare the result with Eq. (32), we can easily observed √ that Eq. (31) and Eq. (32) are equivalent if only µ = N υN . Consider N > 0 and υ ≥ 2, it is √ √ k2 obvious that µ = N υN > k12 = N√υN . Then, Eq. (30) can be simplified as
(33)
an
According to Theorem 1, the control law denoted by Eq. (28) can minimize the performance index (20) with Ptf = P (tf ) = ∞, R(t) = 1 and Q(t) satisfying Eq. (33). In other words, Eq. (28) is a
M
solution to the finite horizon optimal control problem.
Then, another question arises: Can the design goal be achieved with bounded control effort,
te
to this question will be presented.
d
i.e., driving x1 and x2 both to zero at t = tf with u0 < ∞? In the following, theoretical analysis
Substitute Eq. (28) into Eq. (18), yields the following equation
Ac ce p
√ υN N x¨1 + x˙ 1 + x1 = 0 tf − t (tf − t)2
(34)
The above equation is a second-order Cauchy equation whose characteristic equation can be obtained by letting x1 = trgo .
√ r2 − ( υN + 1)r + N = 0
The roots of Eq. (35) become r1 = λ +
√
d and r2 = λ −
√
(35) √
d, where λ =
υN +1 2
and d = λ2 − N .
With respect to different values of d, Eq. (35) may have three different kinds of roots: 1. a real double root, if d = 0. 2. two distinct real roots, if d > 0. 3. two complex conjugate roots, if d < 0. which would lead to three different kinds of solutions for Eq. (34). Assume the initial condition of 11
Page 11 of 30
Eq. (18) is [x10 , x20 ]T , the associated solutions can be derived as [29]
tgo , tf
β=
d>0
(36)
d<0
√ |d|.
cr
where τ =
d=0
ip t
] [ λ τ (1 − λlnτ )x − t lnτ x , 10 f 20 τ r2 x1 = [(r1 − r2 τ r1 −r2 )x10 + tf (1 − τ r1 −r2 )x20 ] , r1 −r2 − τ λ {[λsin(βlnτ ) − βcos(βlnτ )]x + t sin(βlnτ )x } , 10 f 20 β
an
us
The system response x2 as well as the control input u0 can be further obtained as follows [ 2 ] τ λ−1 λ lnτ x + t (1 + λlnτ )x , d=0 10 f 20 t f [ ( r −r ) ( r −r ) ] τ r2 −1 1 2 1 2 x2 = (37) r r τ − 1 x + t r τ − r x20 , d>0 1 2 10 f 1 2 (r1 −r2 )tf λ−1 { [ ] } τ (λ2 − d)sin(βlnτ )x10 + tf λsin(βlnτ ) + βcos(βlnτ ) x20 , d < 0 βtf
Ac ce p
te
d
M
[ ] [ ] } λ−2 { − τ t2 λ2 1 + (λ − 1)lnτ x10 + tf (2λ − 1) + λ(λ − 1)lnτ x20 , d=0 f { [ ] [ ] } τ r2 −2 2 r1 r2 (r2 − 1) − (r1 − 1)τ r1 −r2 x10 + tf r2 (r2 − 1) − r1 (r1 − 1)τ r1 −r2 x20 , d > 0 (r1 −r2 )tf u0 = [ ] 2 (λ − d) (λ − 1)sin(βlnτ ) + βcos(βlnτ ) x10 λ−2 τ , d<0 − βt2 f +t [(λ2 − λ + d)sin(βlnτ ) + β(2λ − 1)cos(βlnτ )]x f 20 (38)
Some useful discussions have been presented in [29], which can be summarized in Table I. TABLE I N ECESSARY CONDITIONS FOR TERMINAL REQUIREMENTS
Terminal requirements x1 (tf ) = 0 x2 (tf ) = 0 u0 (tf ) < ∞
d=0 λ>0 λ>1 λ>2
d>0 r2 > 0 r2 > 1 r2 ≥ 2 or r2 = 1
d<0 λ>0 λ>1 λ≥2
From Table I, it can be seen that the system states, i.e., x1 and x2 , would both go to zero at the final instant as long as λ > 1 (for d ≤ 0) or r2 > 1 (for d > 0). From a practical point of view, the control command u0 should be bounded throughout the control process which implies the conditions λ ≥ 2 (for d < 0), λ > 2 (for d = 0), or r2 ≥ 2 (for d > 0) must hold. Then, the 12
Page 12 of 30
feasible sets of υ and N will be analyzed. Case 1. d < 0. From λ ≥ 2, the following relationship can be easily derived
ip t
√ 3 N≥√ υ
cr
Further consider d < 0, one gets
(40)
us
√ √ (υ − 4)N + 2 υ N + 1 < 0
(39)
Since υ ≥ 2, two situations will be analyzed, i.e., 2 ≤ υ < 4 and υ ≥ 4.
1 √ 2− υ
M
√ N>
√ N > 0, one gets
an
If 2 ≤ υ < 4 holds, solving Eq. (40) and consider
(41)
Since the following results can be easily observed 1√ , 2− υ
√3 < υ
1√ , 2− υ
Ac ce p
te
d
√3 ≥ υ
if
2≤υ≤
if
9 4
9 4
(42)
<υ<4
we can conclude
√ N≥ √N >
√3 , υ 1√ , 2− υ
if
2≤υ≤
if
9 4
9 4
(43)
<υ<4
Consider the case of υ ≥ 4, the following results can be obtained by solving Eq. (40) −
√ 1 1 √ < N< √ <0 2+ υ 2− υ
(44)
It clearly contradicts the truth. As a consequence, for the case of d < 0, the values of N and υ should be selected in accordance with Eq. (43). Case 2. d = 0. 13
Page 13 of 30
From λ > 2 and d = 0, we can obtain √ N> √N =
√3 υ
(45)
1√ 2− υ
1 √ , 2− υ
if
9 <υ<4 4
us
Case 3. d > 0.
(46)
cr
√ N=
ip t
Combine Eq. (42) and Eq. (45), the following conclusion can be drawn
as
if
2≤υ<4
(47)
υ≥4
√ √ (υ − 4)N + 2 υN + 1 ≤ υN − 3
(48)
Ac ce p
te
√
if
d
Then, from r2 ≥ 2, one gets
1√ , 2− υ
M
√ 0 ≤ N < √N ≥ 0,
an
First, let us consider d > 0. Following similar analysis as in Case 1, the results can be obtained
Solving, yields
√ √ √ N ≤ υ− υ−2 √N ≥ √3 υ
Then, the position relationship among The potential position of
√3 υ
√3 , υ
or
√
√ √ √ N ≥ υ+ υ−2
√ √ √ υ − υ − 2 and υ + υ − 2 is depicted in Fig. 2.
is shown as rings in Fig. 2. First, consider
after some algebra, one gets 3 < υ < √ √ In other words, √3υ ≥ υ − υ − 2.
9 4
(49)
√3 υ
<
√ √ υ − υ − 2 and
which means υ does not exist under such circumstances.
√ √ Then, the position relationship between √3υ and υ + υ − 2 will be analyzed. Letting √3υ ≤ √ √ υ + υ − 2. After some algebra, we have υ ≥ 94 . Consequently, the following results can be 14
Page 14 of 30
Position relationship among
√3 , υ
√
υ−
√
υ − 2 and
√
υ+
√
ip t
Fig. 2.
υ−2
if
υ≥
if
2≤υ<
9 4
an
√3 > √υ + √υ − 2, υ
us
√ √ √3 ≤ υ + υ − 2, υ
cr
drawn
(50)
9 4
M
Further, according to Eqs. (49) and (50), one can conclude
te
d
√ √ √ N ≥ υ + υ − 2, if √N ≥ √3 , if υ
υ≥
9 4
2≤υ<
(51) 9 4
Ac ce p
From Eq. (47) and Eq. (51), it can be drawn that the value of υ is divided into three segments, i.e., 2 ≤ υ ≤ 49 ,
9 4
< υ < 4 and υ ≥ 4.
For the case of 2 ≤ υ ≤ 94 , we have √ N ≥ √3 υ √ 0 ≤ N <
Then, letting Consider
9 4
1√ 2− υ
>
√3 , υ
(52) 1√ 2− υ
one gets υ > 94 . Therefore, N belongs to the null set in this case.
< υ < 4, one gets √ √ √ N ≥ υ+ υ−2 0 ≤ √N < 1√ 2− υ
15
(53)
Page 15 of 30
Assuming √ √ 1 √ − ( υ + υ − 2) 2− υ
ψ=
Both sides of Eq. (54) is multiplied by
(54)
√ √ υ − υ − 2 and after some algebra, yields
cr
ip t
√ 2(2 υ − 3)2 ] >0 √ √ ψ= √ √ [ √ ( υ − υ − 2)(2 − υ) (3 υ − 4) + υ − 2
Combine this result with Eq. (53), we can easily obtain the following result υ+
√ √ υ−2≤ N <
1 √ , 2− υ
if
an
Finally, consider the case of υ ≥ 4, yields
9 <υ<4 4
us
√
From Eq. (56), one gets
N≥
√ √ υ + υ − 2,
Ac ce p
te
√
d
M
√ √ √ N ≥ υ+ υ−2 √N ≥ 0
υ≥4
if
(55)
(56)
(57)
Using Eqs. (55) and (57), we can conclude N and υ should satisfy the following requirements under the circumstance of d > 0. √ √ √ υ+ υ−2≤ N < √N ≥ √υ + √υ − 2,
1√ , 2− υ
if if
9 4
<υ<4 (58)
υ≥4
Till now, the feasible sets of N and υ have be completely derived. All the above analysis in this section can be summarized in the following theorem. Theorem 2: For the uncertain system described in Eq. (10), the finite-time ISMC law which is determined by Eqs. (12), (17) and (28) can enforce the system states x1 and x2 both to zero at a predetermined finite time and the control command is bounded throughout the motion as long as η > ||φ||max and Eqs. (43), (46) or (58) hold.
16
Page 16 of 30
IV. A PPLICATION
OF FINITE - TIME
ISMC
TO
A&L G UIDANCE
In this subsection, the steps in the application of finite-time ISMC to the A&L guidance problem are demonstrated. Consider the boundary constraints of the A&L phase, a new variable is designed
ip t
as follows (59)
cr
σ1 = y + Vyf (tf − t)
It can be drawn that the altitude of the RLV will be driven to zero if σ1 goes to zero at the final
(60)
an
σ2 = σ˙ 1 = y˙ − Vyf
us
time. Then, differentiating σ1 with respect to t yields
An interesting phenomenon can be drawn from Eq. (60): If σ2 is enforced to zero at t = tf , a
M
desired decent rate is guaranteed at touchdown. From the above analysis, we can conclude that a successful touchdown movement can be achieved as long as σ1 and σ2 are both driven to zero at
te
of σ1 is taken and one gets
d
the final instant. In order to make the first appearance of the control input, the second derivative
σ˙ 2 = V˙ sinγ + V cosγ γ˙
Ac ce p
= (V˙ sinγ + V cosγ γ˙ − V γ) ˙ + V γ˙
(61)
=δ+u
Assume the control input u = V γ˙ is bounded, we have ||δ|| ≤ δu where δu is a positive constant. It is obvious that Eqs. (60) and (61) take the same form as Eq. (11). Hence, the proposed control method can be employed to achieve the desired objective. According to Eqs. (12), (17) and (28), the associated control input is formulated as follows √ N υN u = − 2 σ1 − σ2 − ksgn(s) tgo tgo
(62)
where k > δu , υ and N are positive constants which satisfy Eqs. (43), (46) or (58), s = σ2 + Z is
17
Page 17 of 30
the sliding function. Using Eqs. (15) and (16), the integral term Z can be obtained as follows. Z˙
=
N σ t2go 1
√
+
υN σ2 tgo
(63)
Z(0) = −σ (0) 2
ip t
From the above analysis, it seems that the guidance law is just able to control the altitude and the decent rate to their desired values. The downrange x, however, is not constrained. As a matter
cr
of fact, tgo is concatenated to the range-to-go information.
us
The range over closing velocity, which is characterized by its simple format and less information demand, is the most widely used approach to estimate tgo . Considering this method could provide
this study. Then, the following equation is derived
R Vc
M
tgo =
an
satisfactory results when the RLV is close to the touchdown point, it is selected to estimate tgo in
(64)
te
velocity, respectively.
d
where R and Vc represent the range between the RLV and the touchdown point and the closing
Although the estimated value of tf varies with time due to the estimation error of tgo , it would
Ac ce p
converge to a constant as the RLV goes close to the runway touchdown point. As a consequence, the actual system states would first disperse from the associated analytic solutions, but finally converge as the estimation error of tgo decays to zero. This statement is verified in the simulation results in Section V.
Note, from Eq. (62), that the discontinuous control term in the controller would give rise to chattering problems, which is undesirable. To suppress the chattering phenomenon, the continuous approximation technique [30] is utilized in this paper. By employing this technique, the discontinuous function sgn(s) is replaced by the continuous saturation function sat(s), which takes the form of ε−1 s, sat(s) =
|s| ≤ ε
sgn(s), otherwise
18
(65)
Page 18 of 30
where ε is the boundary layer thickness. In such a case, the sliding function is just constrained inside the boundary layer but no longer on the sliding surface exactly. The errors could be reduced to negligible values as long as ε is set enough small. Till now, the final expression of the control effort u can be given as follows (66)
cr
ip t
√ [√ ] υN V sinγ − ( υN − N )Vyf Vc N Vc2 y u=− − − ksat(V sinγ − Vyf + Z) R R2
In practical application, the angle of attack α is usually treated as the guidance command, and
us
thus, its instantaneous value should be obtained for on-line implementation. For a given control input u, the angle of attack can be calculated via Eqs. (4), (5) and (7) as follows. From Eqs. (4)
m(u + gcosγ) qS
(67)
M
CLd =
an
and (5), the desired lift coefficient can be found as
Then, the angle of attack can be obtained using Eq. (7). Newton’s method can be utilized to solve
d
for α.
te
Remark 1: It should be noted that the control saturation problem is not taken into account in the finite-time ISMC law derivation. By a practical point of view, the physical limits of the control
Ac ce p
and the dynamics of the system should be considered. Therefore, not all the values of N and υ that satisfy Eqs. (43), (46) or (58) can be applied in the guidance law (66). These two coefficients should be selected to avoid control saturation. The statement in Remark 1 is validated by a simulation shown in Section V.
V. N UMERICAL S IMULATIONS
In this section, the numerical simulations are presented to verify the effectiveness of the proposed guidance method. Firstly, simulations with different sets of (N, υ) are performed to verify the analytic solutions. Subsequently, various boundary conditions are considered and the corresponding simulations are carried out to validate the robustness of the proposed guidance law. Finally, simulation results with first-order system lag are analyzed. The simulation parameters used in this section are listed in Table II, where (x0 , y0 ) and (xf , yf ) are the coordinates of the initial position 19
Page 19 of 30
TABLE II S IMULATION PARAMETERS
Parameter x0 y0 V0 γ0 ε k
Value 0f t 10000f t 500f t/sec -25deg 0.004 6
ip t
Value 0.9118f t2 /slug 2.3 0.0975 0.1819 20000f t 0f t -5f t/sec
cr
Parameter S/m CL0 CD0 K xf yf Vyf
us
of the RLV and the touchdown point, respectively; V0 is the initial velocity and γ0 is the initial
an
flight path angle. In all the simulations, the angle of attack is limited within (−30, +40) deg.
A. Simulations with different sets of (N, υ)
M
In this subsection, different sets of N and υ are considered. Without loss of generality, we choose (υ = 2.125, N = 4.2353), (υ = 2.5, N = 5.6998) and (υ = 4, N = 11.6569) in accordance
te
3(a) and Fig. 3(b).
d
with Eqs. (43), (46) and (58), respectively. The results for this set of simulations are shown in Fig.
From Fig. 3(a), it can be clearly observed that the guidance law can lead to suitable touchdown
Ac ce p
movements in the presence of different υ and N . Further, we can see that the larger N and υ are, the faster the RLV will tend to the flare maneuver. As detailed in Section III-B, the control effort would remain bounded at the final time by choosing N and υ appropriately. This phenomenon can be verified by Fig. 3(b). Besides, it can be drawn that the magnitude of α increases as the values of υ and N increase.
Fig. 4 shows the comparison results between the estimated tf and the actual tf . As detailed in Section IV, the estimated value of tf varies with time owing to the estimation error of tgo . As can be seen in Fig. 4, the initial estimation error of tf is about 10 sec. However, it will converge to zero at the final time instant. Next, the comparisons between the guided results and the analytic solutions with (υ = 2.5, N = 5.6998) are depicted in Fig. 5. It can be drawn from these results that the flight trajectory and the vertical velocity disperse from the associated analytic solutions for most of the time. However, they converge to the analytic solutions ultimately as the estimation
20
Page 20 of 30
10000
40 υ=2.125, N=4.2353 υ=2.5, N=5.6998 υ=4, N=11.6569
30
y (ft)
α (deg)
6000 4000 2000 0 0
10
υ=2.125, N=4.2353 υ=2.5, N=5.6998 υ=4, N=11.6569
0
0.5
1 x (ft)
1.5
−10 0
2 4
10
20
x 10
(b)
30 t (sec)
40
50
60
cr
(a)
Results with different values of υ and N : (a) A&L trajectories, (b) angle of attack histories
us
Fig. 3.
20
ip t
8000
error of tf decays to zero. Note that although the value of tf is underestimated for most of the
an
flight, it is of advantage to improve the A&L movement. This is because the altitude of the RLV will descent faster than the analytic solution at first. As the estimation error of tf converges, the
M
gradual decrease in the altitude is realized which will lead to better flare maneuver. 56
te
f
t (sec)
52
d
54
50
Actual t
f
Ac ce p
48
Estimated t
f
46
44 0
Fig. 4.
10
20
30 t (sec)
40
50
60
Comparisons between the estimated tf and the actual tf
Then, another set of υ and N which satisfies Eq. (43) is taken into consideration. The values are chosen as υ = 3 and N = 14.0282. Correspondingly, the simulation results are plotted in Fig. 6. From this set of simulations, we can easily see that the desired goal is not achieved since the downrange of the touchdown point is less than 20000f t. This phenomenon is due to the control saturation during the final time period. These results reveal that not all the values of υ and N that satisfy Eqs. (43), (46) or (58) would lead to desired touchdown movements. Physical limits should be taken into account in the parameter selection.
21
Page 21 of 30
10000
0 Guided solution Analytic solution
−50
8000
Guided solution Analytic solution
V (ft/sec)
−100 −150
y
y (ft)
6000 4000 2000 0 0
ip t
−200 −250
10
20
30 t (sec)
40
50
−300 0
60
(b)
20
30 t (sec)
40
50
60
cr
(a)
10
10000
40 30
an
8000
us
Fig. 5. Comparisons between the guided results and the analytic solutions with υ = 2.5 and N = 5.6998: (a) altitude, (b) vertical velocity
20
y (ft)
α (deg)
6000
0
M
4000
10
−10
2000
−20
0.5
1 x (ft)
10
20
30 t (sec)
40
50
60
(b)
Results with υ = 3 and N = 14.0282: (a) A&L trajectory, (b) angle of attack history
Ac ce p
Fig. 6.
−30 0
2 4
x 10
te
(a)
1.5
d
0 0
B. Simulations with different boundary conditions The robustness of the proposed A&L guidance law with respect to different boundary conditions is studied in this subsection. The values of (υ, N ) is chosen as (2.5, 5.6998). Fig. 7 contains the simulation results with different xf of 17000, 20000 and 23000f t. Inspecting the results in Figs. 7(a) and 7(c), it can be seen that the RLV can be landed at the desired touchdown point with the descent rate of −5f t/sec. Further, from Fig. 7(d), we can conclude that the magnitude of α increases as xf increases. This phenomenon can be explained as follows. As can be seen from Fig. 7(b), the larger xf is, the more speed it consumes. Thus, a larger magnitude of α is required to realize a pull-up maneuver. In order to test the dependence of the guidance law to the initial altitude, the simulations with different initial altitudes of 8000, 10000 and 12000f t are performed and the results are presented 22
Page 22 of 30
10000
500 x =17000ft f
450
x =20000ft
8000
f
400
x =23000ft f
y (ft)
V (ft/sec)
6000 4000
350 x =17000ft
300
f
x =20000ft f
250
x =23000ft
2000
ip t
f
200 0 0
0.5
1
1.5
2
150 0
2.5 4 x 10
x (ft)
20
0
40
−50
35 30
60
70
60
70
x =17000ft f
x =20000ft f
x =23000ft f
α (deg)
−150 −200
x =17000ft
−250
x =20000ft
f
f
10
20
30 40 t (sec)
10
50
60
(c)
5 0
M
−350 0
20 15
f
x =23000ft −300
25
an
y
V (ft/sec)
−100
50
cr
(b)
30 40 t (sec)
us
(a)
10
70
10
20
30 40 t (sec)
50
(d)
te
d
Fig. 7. Simulations with different xf : (a) A&L trajectory, (b) velocity histories, (c) vertical velocity histories, (d) angle of attack histories
in Fig. 8. It can be noted from Figs. 8(a) and 8(b) that the A&L movements are satisfactory for
Ac ce p
all of the different initial altitude cases. Also, for each case the vertical velocity at touchdown is 5f t/s. Fig. 8(c) reveals that greater speed loss will be experienced with a lower initial altitude, which in turn leads to a larger control magnitude as can be seen in Fig. 8(d). The next set of simulations involves analyzing the effects on the system performance by varying the initial flight path angle. Correspondingly, the simulation results with initial flight path angles of −15, −25 and −35deg are described in Fig. 9. From Figs. 9(a) - 9(d), we can see that feasible touchdown movements are achieved with smooth and bounded control efforts for all the three cases since Vy and γ converge to −5f t/sec and roughly 1.2deg respectively. Besides, the terminal velocity for the three cases range between 224f t/sec and 270f t/sec as can be seen from Fig. 9(e), which are all acceptable results at the touchdown moment. In the sequel, the simulations are focused on different initial velocities of 400, 500 and 600f t/sec and the results are shown in Fig. 10. It can be observed from Fig. 10(a) that the variation of initial 23
Page 23 of 30
12000
500 y =8000ft
450
0
10000
y =10000ft 0
400
y =12000ft 0
V (ft/sec)
6000
350 y =8000ft 0
300
y =10000ft
4000
0
250 2000
y =12000ft 0
200
0 0
0.5
1 x (ft)
1.5
150 0
2 4
10
20
x 10
0
40
−50
35
−100
30
40
50
60
50
60
cr
(b)
30 t (sec)
us
(a)
ip t
y (ft)
8000
y =8000ft
α (deg)
−150 −200 y =8000ft 0
−250
0
10
20
30 t (sec)
40
50
(c)
0
10
y0=12000ft
5 0
M
−350 0
20
0
y =12000ft
15
y =10000ft
−300
25
an
y
V (ft/sec)
0
y =10000ft
60
10
20
30 t (sec)
40
(d)
te
d
Fig. 8. Simulations with different initial altitudes: (a) A&L trajectory, (b) velocity histories, (c) vertical velocity histories, (d) angle of attack histories
velocity appears to have relatively small effects on the resulting altitude profiles. Note, from Fig.
Ac ce p
10(b), that the velocity profile for the 400f t/s initial velocity case performs an initial pull-up behavior, while the velocity profiles for the other two cases perform gradual decrease behaviors. Besides, the final velocities for the three cases are all about 250f t/sec. These phenomena reveal that the guidance law can adapt to different initial velocities automatically. From Fig. 10(c) and 10(d), we can see that the final vertical velocity for each case is fixed on −5f t/sec as desired and enough operational margin for the guidance command is ensured. From all the simulation results in this subsection, we can conclude that the guidance law is quite robust to boundary conditions and the associated guidance commands can be maintained within the acceptable boundaries.
24
Page 24 of 30
10000
0 γ =−15deg
γ =−15deg
0
0
γ =−25deg
8000
−50
γ =−25deg
−100
γ =−35deg
0
0
γ =−35deg
0
−150
y
y (ft)
V (ft/sec)
0
6000 4000 2000
−250 0.5
1 x (ft)
1.5
−300 0
2 4
(a)
20
(b) 0
35 γ =−15deg 0
γ =−25deg
30
γ =−35deg
25
0
−10
α (deg)
50
60
50
60
γ =−15deg 0
γ =−25deg
20
γ =−35deg 0
an
−20 −25
15
−30
10
10
20
30 t (sec)
40
5 0
10
20
40
50
60
M
−35 −40 0
40
0
0
−15
30 t (sec)
us
−5
γ (deg)
10
x 10
cr
0 0
ip t
−200
50
60
40
(d)
500
te
450
d
(c)
30 t (sec)
γ =−15deg
350
0
Ac ce p
V (ft/sec)
400
300
γ =−25deg 0
γ =−35deg 0
250
200 0
10
20
30 t (sec)
(e)
Fig. 9. Simulations with different initial flight path angles: (a) A&L trajectory, (b) vertical velocity histories, (c) flight path angle histories, (d) angle of attack histories, (e) velocity histories
C. Simulations with first-order system lag During practical implementation of the proposed guidance law, the guidance command cannot be achieved immediately after its generation. Such phenomenon is considered as the system lag. To evaluate the robustness of the proposed guidance law with respect to the system lag, simulations
25
Page 25 of 30
10000
600 V =400ft/sec 0
V =400ft/sec
550
V =500ft/sec
8000
0
V =500ft/sec
0
0
500
V =600ft/sec 0
y (ft)
V (ft/sec)
6000 4000
V =600ft/sec 0
450 400 350
ip t
300
2000
250 0.5
1 x (ft)
1.5
200 0
2 4
10
20
x 10
(a)
(b) 0
30 V =400ft/sec 25
0
V =600ft/sec
60
50
60
V =500ft/sec 0
V =600ft/sec
0
0
α (deg)
−150 −200
20
an
y
V (ft/sec)
50
us
0
V =500ft/sec
−100
40
V =400ft/sec
0
−50
30 t (sec)
cr
0 0
−250
15 10
−300 10
20
30 t (sec)
40
50
(c)
5 0
M
−350 0
60
10
20
30 t (sec)
40
(d)
te
d
Fig. 10. Simulations with different initial velocities: (a) A&L trajectory, (b) velocity histories, (c) vertical velocity histories , (d) angle of attack histories
Ac ce p
are carried out in this subsection. Consider the first-order system lag as follows αr 1 = αc Ts + 1
(68)
where αc and αr are the commanded angle of attack and the realized angle of attack respectively, T is the first-order time constant which varies from 0.1sec to 0.8sec. The values of υ and N are selected as 2.5 and 5.6998.
The A&L trajectories and the final vertical velocities with respect to different T is plotted in Fig. 11. It can be noted from Fig. 11(a) that the first-order system lag has a fairly negligible effect on the resulting A&L trajectories and the desired touchdown movements can be guaranteed. From Fig. 11(b), we can observe that although the vertical velocity errors at touchdown point increase as the time constant increases, they are still very small and the soft landing requirement can be achieved.
26
Page 26 of 30
−4.965
8000
−4.97
y (ft)
6000 4000 2000 0 0
−4.98 −4.985
0.5
1 x (ft)
1.5
−4.99 0
2 4
0.2
x 10
(b)
0.4 T (sec)
0.6
0.8
cr
(a)
Results with with first-order system lag: (a) A&L trajectory, (b) Vertical velocity at touchdown
an
VI. C ONCLUSION
us
Fig. 11.
−4.975
ip t
Vy (ft/sec)
10000
In this paper, a finite-time ISMC method is proposed to achieve finite time stabilization of a class of second-order uncertain nonlinear systems, and through a challenging A&L guidance problem,
M
the potential of the developed approach is demonstrated. By using the finite-time ISMC strategy, the system can be stabilized in a given finite time and the system states can be analytically expressed.
d
In addition, global robustness throughout the motion is ensured by the virtue of ISMC. After well
te
developed, the proposed control method is applied to the A&L guidance law design. The guidance law features in its simple form, less information demand as well as the on-line trajectory generation
Ac ce p
capability. Simulation results show that the guidance scheme is quite robust to boundary conditions as well as system lag. Therefore, the approach outlined in this work can be considered as a viable candidate for the A&L guidance. Moreover, since many control objectives need to be achieved in a given finite time, the finite-time ISMC algorithm shows a lot of potential in dealing with such problems.
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ip t
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cr
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te
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Ac ce p
[11] Feng Y, Yu XH, Man ZH, Non-singular adaptive terminal sliding mode control of rigid manipulators, Automatica, 38: 2159–2167, 2002. [12] Cong BL, Chen Z, Liu XD, On adaptive sliding mode control without switching gain overestimation, International Journal of Robust and Nonlinear Control, 24(3): 515–531, 2014. [13] Laghrouche S, Plestan F, Glumineau A, Higher order sliding mode control based on integral sliding mode, Automatica, 43: 531–537, 2007. [14] Wang L, Sheng YZ, Liu XD, A novel adaptive higher-order sliding mode control based on integral sliding mode, International Journal of Control, Automation, and Systems, 12(3): 459– 472, 2014. [15] C. A. Kluever, Unpowered Approach and Landing Guidance Using Trajectory Planning, Journal of Guidance, Control, and Dynamics, 27(6): 967–974, 2004. [16] J. M. Hanson, D. J. Coughlin, G. A. Dukeman, J. A. Mulqueen and J. W. McCarter, Ascent, Transition, Entry, and Abort Guidance Algorithm Design for the X-33 Vehicle, AIAA Paper 28
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ip t
Reusable Launch Vehicles, AIAA Paper 2001-4429, Aug. 2001.
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cr
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[22] J. Schierman and J. Hull, In-Flight Entry Trajectory Optimization for Reusable Launch Vehicles, AIAA Paper 2005-6434, Aug. 2005.
d
[23] A. Heydari and S. N. Balakrishnan, Path Planning Using a Novel Finite Horizon Suboptimal
te
Controller, Journal of Guidance, Control, and Dynamics, 36(4): 1210–1214, 2013. [24] N. Harl and S. N. Balakrishnan, Reentry Terminal Guidance Through Sliding Mode Control,
Ac ce p
Journal of Guidance, Control, and Dynamics, 33(1): 186–199, 2010. [25] US COESA, Committee on Extension to the Standard Atmosphere, US Standard Atmosphere 1976, NOAA-S/T 76-1562, US Gov. Printing Office, Washington, DC, Sup. DOCS, 1976. [26] V. Utkin and J. Shi, Integral Sliding Mode in Systems Operating under Uncertainty Conditions, Proceedings of the 35th Conference on Decision and Control, 1996: 4591–4596. [27] F. L. Lewis and V. L. Syroms, Optimal Control, Second Edition, John Wiley & Sons, 1995. [28] Y. I. Lee, S. H. Kim and M. J. Tahk, Optimality of Linear Time-Varying Guidance for Impact Angle Control, IEEE Transactions on Aerospace and Electronic Systems, 48(3): 2802–2817, 2012. [29] Y. I. Lee, S. H. Kim, J. I. Lee and M. J. Tahk, Analytic Solutions of Generalized ImpactAngle-Control Guidance Law for First-Order Lag System, Journal of Guidance, Control, and Dynamics, 36(1): 96–112, 2013. [30] J. Slotine, Sliding mode controller design for nonlinear system, International Journal of 29
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Ac ce p
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d
M
an
us
cr
ip t
Control, 40(2): 421–434, 1984.
30
Page 30 of 30
S0030-4026(16)30633-7 http://dx.doi.org/doi:10.1016/j.ijleo.2016.06.012 IJLEO 57798
To appear in: Received date: Accepted date:
10-4-2016 1-6-2016
Please cite this article as: Xuanping Liao, Lei Chen, Integral sliding mode control based approach and landing guidance law with finite-time convergence, (2016), http://dx.doi.org/10.1016/j.ijleo.2016.06.012 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 proof before it is published in its final 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.
1
Integral sliding mode control based approach
an
Abstract
us
Xuanping Liao∗ , Lei Chen
cr
convergence
ip t
and landing guidance law with finite-time
In this study, a novel finite-time sliding mode control law is proposed and its application to the
M
approach and landing (A&L) guidance problem is detailed. The proposed control strategy is designed through a combination of integral sliding mode (ISM) concept and polynomial feedback control law. The advantages of this approach are that the robustness is ensured throughout the motion, the convergence time
d
can be chosen in advance and the system states can be expressed in an analytical way. Through the A&L
te
guidance problem, the potential of the developed approach is demonstrated. The guidance scheme can land the reusable launch vehicle at the designated touchdown point with a reasonable amount of vertical
Ac ce p
velocity. The associated law features in its closed-loop form and the on-line trajectory generation ability. The required information only includes the instantaneous flight conditions and the terminal constraints. Numerical simulations in different scenarios are provided to validate the effectiveness of the proposed method.
Index Terms
Finite-time control, approach and landing, guidance, integral sliding mode, reusable launch vehicle.
I. I NTRODUCTION As a subclass of variable structure control, sliding mode control (SMC) is a popular nonlinear deterministic control method tackling the uncertainty and external disturbance under matching Xuanping Liao and Lei Chen are with the college of Aerospace Science and Engineering, National University of Defense Technology, Changsha 410073, China. ∗ Corresponding author. Email: Xuanping [email protected]
Page 1 of 30
conditions [1–3]. In traditional SMC design, a linear hyperplane based sliding mode manifold is used to design the controller which leads to infinite-time stabilization of the system. Finitetime control usually possesses superior properties, such as higher precision, faster convergence rate and better robustness against parameter variation [4–6]. Therefore, finite-time SMC problem has
ip t
recently drawn a good amount of attention from researchers around the world.
Based on the concept of terminal attractors [7], terminal sliding mode control (TSMC) technique
cr
is developed to achieve finite-time convergence [8]. However, the TSMC laws suffers from the
us
problem of singularity. This is because terms with negative fractional powers may exist in the associated controller. To deal with this problem, some methods have been investigated in [9, 10].
an
These methods, however, belong to indirect approaches and cannot resolve the problem completely. In [11], a novel concept of nonsingular terminal sliding mode control (NTSMC) is proposed which solves the singular problem fundamentally. In terms of the convergence property, NTSMC still has
M
two main drawbacks. One is that the system is sensitive to external disturbance and parameter variation in the reaching phase. The other is that the convergence time of the system states is hard
d
to know. This is because the convergence time under NTSMC is calculated from the the occurrence
te
of the sliding mode, and the associated results in the reaching phase is hard to obtain.
Ac ce p
To achieve finite-time convergence and better convergence property, integral sliding mode control (ISMC) technique has also been investigated. The key feature of ISMC lies in that the robustness of the controlled system can be enhanced without sacrificing the performance [12]. Unlike TSMC and NTSMC, a nominal controller is first designed in ISMC to achieve a desired performance for the nominal system, and then the sliding surface is constructed on the basis of the nominal controller. In [13], an ISMC based higher order sliding mode control scheme is developed. The main advantage of this method is that the time convergence can be chosen in advance. But the associated controller design depends on an open-loop control which cannot ensure a desired system performance if the system states are perturbed off the predicted trajectory. This method is further extended in [14] by adopting a continuous closed-loop control law. But the system can only be stabilized to a small vicinity of the origin but not zero. In this study, we intend to propose a novel finite-time sliding mode control method based on ISMC concept. The controller consists of two parts. One part is polynomial feedback control 2
Page 2 of 30
law which solves a finite-horizon optimal control problem with free-final-state. The discontinuous control is developed as the other part for the rejection of the bounded disturbance and uncertainty throughout the entire response of the system. The main feature of this control law is that it ensures the system states to converge to equilibrium point at the desired finite-time and the system state
ip t
can be expressed in an analytical way. In addition, the controller is global robust against matched model uncertainty and external disturbance and it does not depend on any off-line information.
cr
In order to demonstrate the potential of the proposed control method, the terminal guidance
us
problem for the approach and landing (A&L) phase of a reusable launch vehicle (RLV) is also investigated in this paper. The A&L phase is responsible for bringing the RLV from the approach
an
and landing interface (ALI) to runway touchdown [15]. During the early stages of the entry phase, extreme conditions such as aerosurface failures, poor vehicle performance and aerodynamic perturbation may be encountered. So the RLV needs to calculate the current status on-line and to re-
M
target an alternate landing position if necessary [16]. In this way, the predesigned A&L trajectory
generation capability.
d
may be no longer suitable and the guidance laws are required to possess the on-line trajectory
te
A trajectory-planning based guidance law for the A&L phase is presented in [15]. Afterwards,
Ac ce p
this method is further analyzed to land a RLV with limited normal acceleration capabilities in [17]. The design goal is achieved by determining the minimum constant equilibrium-glide load factor during the A&L phase. An Optimum-Path-To-Go (OPTG) algorithm is investigated in [18–20] to achieve a successful A&L movement. The potential of this approach is further demonstrated by dealing with control surface errors [21] and retargeting to more suitable alternative landing sites [22]. Using the state-dependent Riccati equation (SDRE) method, a novel finite horizon suboptimal comptroller is proposed in [23]. The corresponding guidance scheme is robust to initial conditions. In [24], a new second-order SMC is proposed and applied to solve the A&L guidance problem. The associated guidance law just depends on the boundary conditions of the A&L phase and it is able to generate new trajectories on-line. The aim of the A&L guidance is to ensure a feasible landing movement, i.e. the RLV is landed at the predetermined touchdown point with the least possible flight path angle and vertical velocity. As detailed in [23], the A&L guidance problem belongs to the finite time control category. In this 3
Page 3 of 30
paper, we apply the proposed ISMC method to the A&L guidance problem. The associated guidance law is suitable for onboard implementation due to its simple format and less information demand. The required information only includes the terminal constraints and the instantaneous states of the RLV. Compared with the existing publications, the proposed method makes some contributions
ip t
in the following aspects. First, the approaches outlined in [15, 17–20] can only be used to solve the A&L problem. The proposed ISMC law, however, is a practical control method and can be
cr
employed to deal with other control problems with a goal to be fulfilled in a given time. Second, although a closed-form solution to the finite horizon optimal control problem is addressed in [23],
us
the control law can only be expressed in the feedback from and the associated computational burden is high. The proposed method is also a kind of optimal control and it can be expressed in
an
an analytical form. Hence, it is more more practical and ease of mechanization. Finally, the system states can be analytically obtained using the proposed controller, whereas other finite-time control
M
laws [8–11, 13, 14] do not possess this feature.
te
A. Equations of Motion
FORMULATION
d
II. S YSTEM MODEL AND PROBLEM
Ac ce p
In this section, the dynamics of an RLV during the approach and landing phase is presented. It is assumed that the velocity direction of the RLV stays in a vertical plane during the entire landing process which implies the cross range to the center line of the runway is zero. To make it explicit, a diagram of the A&L phase is presented in Fig. 1. The velocity and the flight path angle of the
Fig. 1.
Diagram of the A&L phase
RLV is represented by V and γ respectively and the touchdown point is denoted by T . 4
Page 4 of 30
The two-dimensional point-mass dynamics of a RLV during the A&L phase is given as follows [15] (1)
y˙ = V sinγ
(2)
ip t
x˙ = V cosγ
D V˙ = − − gsinγ m
cr
L g − cosγ mV V
(4)
us
γ˙ =
(3)
where g = 32.2f t/sec2 is the gravity acceleration. L and D are aerodynamic lift and drag forces
an
defined in the usual manner
(5)
D = qSCD
(6)
M
L = qSCL
where q = 12 ρV 2 is the dynamic pressure, S represents the reference area of the RLV, CL and CD
d
denote the lift and drag coefficients respectively. The atmospheric density ρ is obtained from the
Ac ce p
te
1976 U.S. Standard Atmosphere [25] and CL and CD are modeled as follows [24] CL = CL0 sin2 αcosα
(7)
CD = CD0 + KCL2
(8)
where K, CL0 and CD0 are constants and α represents the angle of attack. The normal acceleration u = V γ˙ is treated as the control effort of the guidance system.
B. Problem statement To guarantee a successful touchdown movement, the soft landing requirement must be fulfilled, i.e., the descent rate of the RLV should be negative with a small magnitude as the altitude goes to
5
Page 5 of 30
zero. Consequently, the objective is to design a guidance law such that lim y = 0
x→xf
(9)
lim y˙ = Vyf
where Vyf is the desired vertical velocity at the final instant.
ip t
x→xf
cr
III. F INITE - TIME INTEGRAL SLIDING MODE CONTROL
us
This section details the basic principle of the finite-time ISMC. A class of second-order uncertain nonlinear systems is used for illustration. The control law is composed of two parts [26], i.e.,
an
integral sliding mode control and polynomial feedback control. The former is used to compensate any possible uncertainties during the entire control action, while the latter is designed to achieve
M
the finite time stability of the system. The optimality of this method is analyzed and the analytic solutions of the system states as well as the control command are obtained. Furthermore, feasible
d
sets of the control coefficients are discussed.
te
A. Design of integral sliding mode control
Ac ce p
Consider a second-order uncertain nonlinear system x˙ 1 = x2 (10)
x˙ 2 = f (X) + g(X) + b(X)U
where X = [x1 , x2 ]T is the system state vector, U denotes the system input signal, g(X) represents the lumped disturbances satisfying ||g(X)|| ≤ δg with δg > 0, f (X) and b(X) ̸= 0 are known functions of X. The design goal is to find a controller U such that both x1 and x2 can be driven to zero in a given finite time in the presence of uncertainties. As discussed in [13], the system described by Eq. (10) can be trivially rewritten as follows x˙ 1 = x2 (11) x˙ 2 = φ + U
6
Page 6 of 30
where φ = f (X) + g(X) + (b(X) − 1)U is treated as a new form of lumped uncertainties which is also bounded. The controller U is composed of two parts, as U = u0 + u1
(12)
ip t
where u0 is the ideal control, which is continuous and specially designed to stabilize system (11) without uncertainties, while u1 is the discontinuous control used to reject the bounded disturbances.
cr
We assume the ideal control u0 is already known and design the discontinuous control u1 .
an
S1 = x2 + z
us
Defining the following integral sliding manifold
(13)
where z is the integral term that needs to be determined. Differentiating S1 with respect to t yields
M
S˙ 1 = x˙ 2 + z˙ = φ + u0 + u1 + z˙
(14)
d
˙ Assuming the equivalent control of u1 is ueq 1 , a sufficient condition for guaranteeing S1 = 0 during
te
the entire control action can be obtained as
Ac ce p
z˙ = −u0
(15)
ueq 1 = −φ
Besides, the sliding motion is established from the very beginning as long as the following equation holds.
z(0) = −x2 (0)
(16)
In order to ensure the attractability of the sliding surface in spite of uncertainties, the discontinuous control u1 is designed as follows
u1 = −ηsgn(S1 )
(17)
where the switching gain satisfies η > ||φ||∞ . The stability of the system (11) can be easily proved using Lyapunov Second Method by considering V1 = 12 S12 as Lyapunov candidate [26].
7
Page 7 of 30
B. Polynomial feedback control law design In this subsection, a new polynomial feedback control law is proposed. The optimality of this controller and the selection of the control coefficients are also discussed.
ip t
By employing the integral sliding mode control approach, the system model (11) in sliding mode can be derived as
cr
x˙ 1 = x2
us
x˙ 2 = u0
(18)
an
which is a linear time-invariant system. Rewritten Eq. (18) into state space representation, one gets x˙ 0 1 x 0 1 = 1 + u0 (19) x˙ 2 0 0 x2 1 | {z } |{z} B
M
A
Recall the design goal is to drive the system states x1 and x2 to zero at the desired finite time. It
∫
te
quadratic performance index
d
can be considered as a finite horizon optimal control problem in which we minimize the following
tf
(
) X T (t)Q(t)X(t) + R(t)u20 (t) dt
(20)
t0
Ac ce p
1 1 J1 = X T (tf )Ptf X(tf ) + 2 2
where t0 and tf denote the initial and final time respectively, Ptf ∈ R2×2 and Q(t) ∈ R2×2 are symmetric and positive semi-definite, R(t) ∈ R+ . It is well-known that the resulting optimal control takes the form of u0 (t) = −R−1 (t)B T P (t)X(t), where P (t) ∈ R2×2 is the unique non-negative definite solution of the following matrix differential Riccati equation −P˙ (t) = AT P (t) + P (t)A − P (t)BR−1 (t)B T P (t) + Q(t)
(21)
It can be seen from the above equation that P (t) can be obtained by integrating Eq. (21) backward from tf to t with P (tf ) = Ptf [27]. This process, however, can hardly be applied on-line and thus this method is restrictive in particular for practical applications. In the sequel, the polynomial feedback control law will be introduced which provides a new solution to the finite horizon optimal control problem.
8
Page 8 of 30
Before moving on, let us first consider the following infinite-horizon nonlinear regulator problem: find u0 , which minimizes J, defined by 1 J= 2
∫
∞
(X T (t)QX(t) + Ru20 )dt
(22)
t0
ip t
Here, Q ≥ 0 and R > 0 represent the state weighting matrix and the control weighting matrix,
(23)
us
cr
respectively. Without loss of generality, we assume R = 1 and Q takes the form of q12 0 Q= 2 0 q2 The state feedback solution of the proposed optimal control problem is given by
an
u0 = −R−1 B T P X(t)
(24)
M
The matrix P is positive definite and satisfies the following algebraic Riccati equation −P A − AT P + P BR−1 B T P − Q = 0
(25)
te
d
p1 p2 and substitute it into Eq. (25), yields Let P = p2 p3
Ac ce p
] p1 p2 p1 p2 0 0 p1 p2 0 [ q12 0 p1 p2 0 1 + 0 1 − − =0 − 2 1 0 p2 p3 1 p2 p3 p2 p3 0 q2 0 0 p2 p3 After some algebra, we have p1 = q1
√ √ 2q1 + q22 , p2 = q1 and p3 = 2q1 + q22 , which, on
substitution in Eq. (24), yields
√ u0 = −q1 x1 − 2q1 + q22 x2
(26)
Then, the polynomial feedback control law will be derived. Choose the coefficients q1 and q2 to be functions of time-to-go
N t2go √ √ (υ − 2)N q2 = (υ − 2)q1 = tgo q1 =
9
(27)
Page 9 of 30
where N > 0, υ ≥ 2 and tgo = tf − t. Then, substitute Eq. (27) into Eq. (26), one gets √ N υN u0 = − x1 − x2 2 (tf − t) tf − t
(28)
The controller in Eq. (28) is the polynomial feedback control law. It should be noted, however,
ip t
the proposed polynomial feedback control law is not an optimal solution to the performance index Eq. (22). This is because the coefficients q1 and q2 are selected to be time-varying functions. As
cr
a matter of fact, the controller in Eq. (26) with constant q1 and q2 is the optimal control to the
us
cost function (22). Fortunately, there is no need to find an optimal solution to the infinite horizon performance index (22) since the design goal is to enforce x1 and x2 both to zero at the a designated time which belongs to the finite horizon control problem. Then, it invites a question: Is there any
an
cost function in the finite horizon format as Eq. (20) that results in the polynomial feedback control law denoted by Eq. (28)?
k1 k2 x1 − x2 2 (tf − t) tf − t
(29)
d
u∗ = −
M
Theorem 1: Consider the following control law
te
where k1 and k2 are positive constants. If µ is a constant which satisfies µ >
k12 , k2
the control law
Ac ce p
described by Eq. (29) can optimize the performance index (20), with q11 q12 Q(t) = q21 q22 where q11 =
k22 −k2 −2k1 R(t) (tf −t)2
k12 −3µ R(t) (tf −t)4
−
−
k2 ˙ R(t). (tf −t)
µ ˙ R(t), (tf −t)3
q12 = q21 =
k1 (k2 −2)−µ R(t) (tf −t)3
(30)
−
k1 ˙ R(t) (tf −t)2
and q22 =
P (t) is given by
P (t) =
µ R(t) (tf −t)3 k1 R(t) (tf −t)2
(31)
k2 R(t) (tf −t)
Proof The proof can refer to [28]. From Eq. (28), one gets k1 = N and k2 =
k1 R(t) (tf −t)2
√ υN . If the control law (28) is able to optimize a
cost function (20), the matrix P (t) determined by Eqs. (23), (25) and (27) should satisfy Eq. (31).
10
Page 10 of 30
From Eqs. (23), (25) and (27), we have P (t) =
√ N υN (tf −t)3
N (tf −t)2 √ υN (tf −t)
N (tf −t)2
(32)
Q(t) =
√ N 2 −3N υN (tf −t)4 −2N (tf −t)3
−2N (tf −t)3 √ (υ−2)N − υN (tf −t)2
us
cr
ip t
Substitute R(t) = 1 into Eq. (31) and compare the result with Eq. (32), we can easily observed √ that Eq. (31) and Eq. (32) are equivalent if only µ = N υN . Consider N > 0 and υ ≥ 2, it is √ √ k2 obvious that µ = N υN > k12 = N√υN . Then, Eq. (30) can be simplified as
(33)
an
According to Theorem 1, the control law denoted by Eq. (28) can minimize the performance index (20) with Ptf = P (tf ) = ∞, R(t) = 1 and Q(t) satisfying Eq. (33). In other words, Eq. (28) is a
M
solution to the finite horizon optimal control problem.
Then, another question arises: Can the design goal be achieved with bounded control effort,
te
to this question will be presented.
d
i.e., driving x1 and x2 both to zero at t = tf with u0 < ∞? In the following, theoretical analysis
Substitute Eq. (28) into Eq. (18), yields the following equation
Ac ce p
√ υN N x¨1 + x˙ 1 + x1 = 0 tf − t (tf − t)2
(34)
The above equation is a second-order Cauchy equation whose characteristic equation can be obtained by letting x1 = trgo .
√ r2 − ( υN + 1)r + N = 0
The roots of Eq. (35) become r1 = λ +
√
d and r2 = λ −
√
(35) √
d, where λ =
υN +1 2
and d = λ2 − N .
With respect to different values of d, Eq. (35) may have three different kinds of roots: 1. a real double root, if d = 0. 2. two distinct real roots, if d > 0. 3. two complex conjugate roots, if d < 0. which would lead to three different kinds of solutions for Eq. (34). Assume the initial condition of 11
Page 11 of 30
Eq. (18) is [x10 , x20 ]T , the associated solutions can be derived as [29]
tgo , tf
β=
d>0
(36)
d<0
√ |d|.
cr
where τ =
d=0
ip t
] [ λ τ (1 − λlnτ )x − t lnτ x , 10 f 20 τ r2 x1 = [(r1 − r2 τ r1 −r2 )x10 + tf (1 − τ r1 −r2 )x20 ] , r1 −r2 − τ λ {[λsin(βlnτ ) − βcos(βlnτ )]x + t sin(βlnτ )x } , 10 f 20 β
an
us
The system response x2 as well as the control input u0 can be further obtained as follows [ 2 ] τ λ−1 λ lnτ x + t (1 + λlnτ )x , d=0 10 f 20 t f [ ( r −r ) ( r −r ) ] τ r2 −1 1 2 1 2 x2 = (37) r r τ − 1 x + t r τ − r x20 , d>0 1 2 10 f 1 2 (r1 −r2 )tf λ−1 { [ ] } τ (λ2 − d)sin(βlnτ )x10 + tf λsin(βlnτ ) + βcos(βlnτ ) x20 , d < 0 βtf
Ac ce p
te
d
M
[ ] [ ] } λ−2 { − τ t2 λ2 1 + (λ − 1)lnτ x10 + tf (2λ − 1) + λ(λ − 1)lnτ x20 , d=0 f { [ ] [ ] } τ r2 −2 2 r1 r2 (r2 − 1) − (r1 − 1)τ r1 −r2 x10 + tf r2 (r2 − 1) − r1 (r1 − 1)τ r1 −r2 x20 , d > 0 (r1 −r2 )tf u0 = [ ] 2 (λ − d) (λ − 1)sin(βlnτ ) + βcos(βlnτ ) x10 λ−2 τ , d<0 − βt2 f +t [(λ2 − λ + d)sin(βlnτ ) + β(2λ − 1)cos(βlnτ )]x f 20 (38)
Some useful discussions have been presented in [29], which can be summarized in Table I. TABLE I N ECESSARY CONDITIONS FOR TERMINAL REQUIREMENTS
Terminal requirements x1 (tf ) = 0 x2 (tf ) = 0 u0 (tf ) < ∞
d=0 λ>0 λ>1 λ>2
d>0 r2 > 0 r2 > 1 r2 ≥ 2 or r2 = 1
d<0 λ>0 λ>1 λ≥2
From Table I, it can be seen that the system states, i.e., x1 and x2 , would both go to zero at the final instant as long as λ > 1 (for d ≤ 0) or r2 > 1 (for d > 0). From a practical point of view, the control command u0 should be bounded throughout the control process which implies the conditions λ ≥ 2 (for d < 0), λ > 2 (for d = 0), or r2 ≥ 2 (for d > 0) must hold. Then, the 12
Page 12 of 30
feasible sets of υ and N will be analyzed. Case 1. d < 0. From λ ≥ 2, the following relationship can be easily derived
ip t
√ 3 N≥√ υ
cr
Further consider d < 0, one gets
(40)
us
√ √ (υ − 4)N + 2 υ N + 1 < 0
(39)
Since υ ≥ 2, two situations will be analyzed, i.e., 2 ≤ υ < 4 and υ ≥ 4.
1 √ 2− υ
M
√ N>
√ N > 0, one gets
an
If 2 ≤ υ < 4 holds, solving Eq. (40) and consider
(41)
Since the following results can be easily observed 1√ , 2− υ
√3 < υ
1√ , 2− υ
Ac ce p
te
d
√3 ≥ υ
if
2≤υ≤
if
9 4
9 4
(42)
<υ<4
we can conclude
√ N≥ √N >
√3 , υ 1√ , 2− υ
if
2≤υ≤
if
9 4
9 4
(43)
<υ<4
Consider the case of υ ≥ 4, the following results can be obtained by solving Eq. (40) −
√ 1 1 √ < N< √ <0 2+ υ 2− υ
(44)
It clearly contradicts the truth. As a consequence, for the case of d < 0, the values of N and υ should be selected in accordance with Eq. (43). Case 2. d = 0. 13
Page 13 of 30
From λ > 2 and d = 0, we can obtain √ N> √N =
√3 υ
(45)
1√ 2− υ
1 √ , 2− υ
if
9 <υ<4 4
us
Case 3. d > 0.
(46)
cr
√ N=
ip t
Combine Eq. (42) and Eq. (45), the following conclusion can be drawn
as
if
2≤υ<4
(47)
υ≥4
√ √ (υ − 4)N + 2 υN + 1 ≤ υN − 3
(48)
Ac ce p
te
√
if
d
Then, from r2 ≥ 2, one gets
1√ , 2− υ
M
√ 0 ≤ N < √N ≥ 0,
an
First, let us consider d > 0. Following similar analysis as in Case 1, the results can be obtained
Solving, yields
√ √ √ N ≤ υ− υ−2 √N ≥ √3 υ
Then, the position relationship among The potential position of
√3 υ
√3 , υ
or
√
√ √ √ N ≥ υ+ υ−2
√ √ √ υ − υ − 2 and υ + υ − 2 is depicted in Fig. 2.
is shown as rings in Fig. 2. First, consider
after some algebra, one gets 3 < υ < √ √ In other words, √3υ ≥ υ − υ − 2.
9 4
(49)
√3 υ
<
√ √ υ − υ − 2 and
which means υ does not exist under such circumstances.
√ √ Then, the position relationship between √3υ and υ + υ − 2 will be analyzed. Letting √3υ ≤ √ √ υ + υ − 2. After some algebra, we have υ ≥ 94 . Consequently, the following results can be 14
Page 14 of 30
Position relationship among
√3 , υ
√
υ−
√
υ − 2 and
√
υ+
√
ip t
Fig. 2.
υ−2
if
υ≥
if
2≤υ<
9 4
an
√3 > √υ + √υ − 2, υ
us
√ √ √3 ≤ υ + υ − 2, υ
cr
drawn
(50)
9 4
M
Further, according to Eqs. (49) and (50), one can conclude
te
d
√ √ √ N ≥ υ + υ − 2, if √N ≥ √3 , if υ
υ≥
9 4
2≤υ<
(51) 9 4
Ac ce p
From Eq. (47) and Eq. (51), it can be drawn that the value of υ is divided into three segments, i.e., 2 ≤ υ ≤ 49 ,
9 4
< υ < 4 and υ ≥ 4.
For the case of 2 ≤ υ ≤ 94 , we have √ N ≥ √3 υ √ 0 ≤ N <
Then, letting Consider
9 4
1√ 2− υ
>
√3 , υ
(52) 1√ 2− υ
one gets υ > 94 . Therefore, N belongs to the null set in this case.
< υ < 4, one gets √ √ √ N ≥ υ+ υ−2 0 ≤ √N < 1√ 2− υ
15
(53)
Page 15 of 30
Assuming √ √ 1 √ − ( υ + υ − 2) 2− υ
ψ=
Both sides of Eq. (54) is multiplied by
(54)
√ √ υ − υ − 2 and after some algebra, yields
cr
ip t
√ 2(2 υ − 3)2 ] >0 √ √ ψ= √ √ [ √ ( υ − υ − 2)(2 − υ) (3 υ − 4) + υ − 2
Combine this result with Eq. (53), we can easily obtain the following result υ+
√ √ υ−2≤ N <
1 √ , 2− υ
if
an
Finally, consider the case of υ ≥ 4, yields
9 <υ<4 4
us
√
From Eq. (56), one gets
N≥
√ √ υ + υ − 2,
Ac ce p
te
√
d
M
√ √ √ N ≥ υ+ υ−2 √N ≥ 0
υ≥4
if
(55)
(56)
(57)
Using Eqs. (55) and (57), we can conclude N and υ should satisfy the following requirements under the circumstance of d > 0. √ √ √ υ+ υ−2≤ N < √N ≥ √υ + √υ − 2,
1√ , 2− υ
if if
9 4
<υ<4 (58)
υ≥4
Till now, the feasible sets of N and υ have be completely derived. All the above analysis in this section can be summarized in the following theorem. Theorem 2: For the uncertain system described in Eq. (10), the finite-time ISMC law which is determined by Eqs. (12), (17) and (28) can enforce the system states x1 and x2 both to zero at a predetermined finite time and the control command is bounded throughout the motion as long as η > ||φ||max and Eqs. (43), (46) or (58) hold.
16
Page 16 of 30
IV. A PPLICATION
OF FINITE - TIME
ISMC
TO
A&L G UIDANCE
In this subsection, the steps in the application of finite-time ISMC to the A&L guidance problem are demonstrated. Consider the boundary constraints of the A&L phase, a new variable is designed
ip t
as follows (59)
cr
σ1 = y + Vyf (tf − t)
It can be drawn that the altitude of the RLV will be driven to zero if σ1 goes to zero at the final
(60)
an
σ2 = σ˙ 1 = y˙ − Vyf
us
time. Then, differentiating σ1 with respect to t yields
An interesting phenomenon can be drawn from Eq. (60): If σ2 is enforced to zero at t = tf , a
M
desired decent rate is guaranteed at touchdown. From the above analysis, we can conclude that a successful touchdown movement can be achieved as long as σ1 and σ2 are both driven to zero at
te
of σ1 is taken and one gets
d
the final instant. In order to make the first appearance of the control input, the second derivative
σ˙ 2 = V˙ sinγ + V cosγ γ˙
Ac ce p
= (V˙ sinγ + V cosγ γ˙ − V γ) ˙ + V γ˙
(61)
=δ+u
Assume the control input u = V γ˙ is bounded, we have ||δ|| ≤ δu where δu is a positive constant. It is obvious that Eqs. (60) and (61) take the same form as Eq. (11). Hence, the proposed control method can be employed to achieve the desired objective. According to Eqs. (12), (17) and (28), the associated control input is formulated as follows √ N υN u = − 2 σ1 − σ2 − ksgn(s) tgo tgo
(62)
where k > δu , υ and N are positive constants which satisfy Eqs. (43), (46) or (58), s = σ2 + Z is
17
Page 17 of 30
the sliding function. Using Eqs. (15) and (16), the integral term Z can be obtained as follows. Z˙
=
N σ t2go 1
√
+
υN σ2 tgo
(63)
Z(0) = −σ (0) 2
ip t
From the above analysis, it seems that the guidance law is just able to control the altitude and the decent rate to their desired values. The downrange x, however, is not constrained. As a matter
cr
of fact, tgo is concatenated to the range-to-go information.
us
The range over closing velocity, which is characterized by its simple format and less information demand, is the most widely used approach to estimate tgo . Considering this method could provide
this study. Then, the following equation is derived
R Vc
M
tgo =
an
satisfactory results when the RLV is close to the touchdown point, it is selected to estimate tgo in
(64)
te
velocity, respectively.
d
where R and Vc represent the range between the RLV and the touchdown point and the closing
Although the estimated value of tf varies with time due to the estimation error of tgo , it would
Ac ce p
converge to a constant as the RLV goes close to the runway touchdown point. As a consequence, the actual system states would first disperse from the associated analytic solutions, but finally converge as the estimation error of tgo decays to zero. This statement is verified in the simulation results in Section V.
Note, from Eq. (62), that the discontinuous control term in the controller would give rise to chattering problems, which is undesirable. To suppress the chattering phenomenon, the continuous approximation technique [30] is utilized in this paper. By employing this technique, the discontinuous function sgn(s) is replaced by the continuous saturation function sat(s), which takes the form of ε−1 s, sat(s) =
|s| ≤ ε
sgn(s), otherwise
18
(65)
Page 18 of 30
where ε is the boundary layer thickness. In such a case, the sliding function is just constrained inside the boundary layer but no longer on the sliding surface exactly. The errors could be reduced to negligible values as long as ε is set enough small. Till now, the final expression of the control effort u can be given as follows (66)
cr
ip t
√ [√ ] υN V sinγ − ( υN − N )Vyf Vc N Vc2 y u=− − − ksat(V sinγ − Vyf + Z) R R2
In practical application, the angle of attack α is usually treated as the guidance command, and
us
thus, its instantaneous value should be obtained for on-line implementation. For a given control input u, the angle of attack can be calculated via Eqs. (4), (5) and (7) as follows. From Eqs. (4)
m(u + gcosγ) qS
(67)
M
CLd =
an
and (5), the desired lift coefficient can be found as
Then, the angle of attack can be obtained using Eq. (7). Newton’s method can be utilized to solve
d
for α.
te
Remark 1: It should be noted that the control saturation problem is not taken into account in the finite-time ISMC law derivation. By a practical point of view, the physical limits of the control
Ac ce p
and the dynamics of the system should be considered. Therefore, not all the values of N and υ that satisfy Eqs. (43), (46) or (58) can be applied in the guidance law (66). These two coefficients should be selected to avoid control saturation. The statement in Remark 1 is validated by a simulation shown in Section V.
V. N UMERICAL S IMULATIONS
In this section, the numerical simulations are presented to verify the effectiveness of the proposed guidance method. Firstly, simulations with different sets of (N, υ) are performed to verify the analytic solutions. Subsequently, various boundary conditions are considered and the corresponding simulations are carried out to validate the robustness of the proposed guidance law. Finally, simulation results with first-order system lag are analyzed. The simulation parameters used in this section are listed in Table II, where (x0 , y0 ) and (xf , yf ) are the coordinates of the initial position 19
Page 19 of 30
TABLE II S IMULATION PARAMETERS
Parameter x0 y0 V0 γ0 ε k
Value 0f t 10000f t 500f t/sec -25deg 0.004 6
ip t
Value 0.9118f t2 /slug 2.3 0.0975 0.1819 20000f t 0f t -5f t/sec
cr
Parameter S/m CL0 CD0 K xf yf Vyf
us
of the RLV and the touchdown point, respectively; V0 is the initial velocity and γ0 is the initial
an
flight path angle. In all the simulations, the angle of attack is limited within (−30, +40) deg.
A. Simulations with different sets of (N, υ)
M
In this subsection, different sets of N and υ are considered. Without loss of generality, we choose (υ = 2.125, N = 4.2353), (υ = 2.5, N = 5.6998) and (υ = 4, N = 11.6569) in accordance
te
3(a) and Fig. 3(b).
d
with Eqs. (43), (46) and (58), respectively. The results for this set of simulations are shown in Fig.
From Fig. 3(a), it can be clearly observed that the guidance law can lead to suitable touchdown
Ac ce p
movements in the presence of different υ and N . Further, we can see that the larger N and υ are, the faster the RLV will tend to the flare maneuver. As detailed in Section III-B, the control effort would remain bounded at the final time by choosing N and υ appropriately. This phenomenon can be verified by Fig. 3(b). Besides, it can be drawn that the magnitude of α increases as the values of υ and N increase.
Fig. 4 shows the comparison results between the estimated tf and the actual tf . As detailed in Section IV, the estimated value of tf varies with time owing to the estimation error of tgo . As can be seen in Fig. 4, the initial estimation error of tf is about 10 sec. However, it will converge to zero at the final time instant. Next, the comparisons between the guided results and the analytic solutions with (υ = 2.5, N = 5.6998) are depicted in Fig. 5. It can be drawn from these results that the flight trajectory and the vertical velocity disperse from the associated analytic solutions for most of the time. However, they converge to the analytic solutions ultimately as the estimation
20
Page 20 of 30
10000
40 υ=2.125, N=4.2353 υ=2.5, N=5.6998 υ=4, N=11.6569
30
y (ft)
α (deg)
6000 4000 2000 0 0
10
υ=2.125, N=4.2353 υ=2.5, N=5.6998 υ=4, N=11.6569
0
0.5
1 x (ft)
1.5
−10 0
2 4
10
20
x 10
(b)
30 t (sec)
40
50
60
cr
(a)
Results with different values of υ and N : (a) A&L trajectories, (b) angle of attack histories
us
Fig. 3.
20
ip t
8000
error of tf decays to zero. Note that although the value of tf is underestimated for most of the
an
flight, it is of advantage to improve the A&L movement. This is because the altitude of the RLV will descent faster than the analytic solution at first. As the estimation error of tf converges, the
M
gradual decrease in the altitude is realized which will lead to better flare maneuver. 56
te
f
t (sec)
52
d
54
50
Actual t
f
Ac ce p
48
Estimated t
f
46
44 0
Fig. 4.
10
20
30 t (sec)
40
50
60
Comparisons between the estimated tf and the actual tf
Then, another set of υ and N which satisfies Eq. (43) is taken into consideration. The values are chosen as υ = 3 and N = 14.0282. Correspondingly, the simulation results are plotted in Fig. 6. From this set of simulations, we can easily see that the desired goal is not achieved since the downrange of the touchdown point is less than 20000f t. This phenomenon is due to the control saturation during the final time period. These results reveal that not all the values of υ and N that satisfy Eqs. (43), (46) or (58) would lead to desired touchdown movements. Physical limits should be taken into account in the parameter selection.
21
Page 21 of 30
10000
0 Guided solution Analytic solution
−50
8000
Guided solution Analytic solution
V (ft/sec)
−100 −150
y
y (ft)
6000 4000 2000 0 0
ip t
−200 −250
10
20
30 t (sec)
40
50
−300 0
60
(b)
20
30 t (sec)
40
50
60
cr
(a)
10
10000
40 30
an
8000
us
Fig. 5. Comparisons between the guided results and the analytic solutions with υ = 2.5 and N = 5.6998: (a) altitude, (b) vertical velocity
20
y (ft)
α (deg)
6000
0
M
4000
10
−10
2000
−20
0.5
1 x (ft)
10
20
30 t (sec)
40
50
60
(b)
Results with υ = 3 and N = 14.0282: (a) A&L trajectory, (b) angle of attack history
Ac ce p
Fig. 6.
−30 0
2 4
x 10
te
(a)
1.5
d
0 0
B. Simulations with different boundary conditions The robustness of the proposed A&L guidance law with respect to different boundary conditions is studied in this subsection. The values of (υ, N ) is chosen as (2.5, 5.6998). Fig. 7 contains the simulation results with different xf of 17000, 20000 and 23000f t. Inspecting the results in Figs. 7(a) and 7(c), it can be seen that the RLV can be landed at the desired touchdown point with the descent rate of −5f t/sec. Further, from Fig. 7(d), we can conclude that the magnitude of α increases as xf increases. This phenomenon can be explained as follows. As can be seen from Fig. 7(b), the larger xf is, the more speed it consumes. Thus, a larger magnitude of α is required to realize a pull-up maneuver. In order to test the dependence of the guidance law to the initial altitude, the simulations with different initial altitudes of 8000, 10000 and 12000f t are performed and the results are presented 22
Page 22 of 30
10000
500 x =17000ft f
450
x =20000ft
8000
f
400
x =23000ft f
y (ft)
V (ft/sec)
6000 4000
350 x =17000ft
300
f
x =20000ft f
250
x =23000ft
2000
ip t
f
200 0 0
0.5
1
1.5
2
150 0
2.5 4 x 10
x (ft)
20
0
40
−50
35 30
60
70
60
70
x =17000ft f
x =20000ft f
x =23000ft f
α (deg)
−150 −200
x =17000ft
−250
x =20000ft
f
f
10
20
30 40 t (sec)
10
50
60
(c)
5 0
M
−350 0
20 15
f
x =23000ft −300
25
an
y
V (ft/sec)
−100
50
cr
(b)
30 40 t (sec)
us
(a)
10
70
10
20
30 40 t (sec)
50
(d)
te
d
Fig. 7. Simulations with different xf : (a) A&L trajectory, (b) velocity histories, (c) vertical velocity histories, (d) angle of attack histories
in Fig. 8. It can be noted from Figs. 8(a) and 8(b) that the A&L movements are satisfactory for
Ac ce p
all of the different initial altitude cases. Also, for each case the vertical velocity at touchdown is 5f t/s. Fig. 8(c) reveals that greater speed loss will be experienced with a lower initial altitude, which in turn leads to a larger control magnitude as can be seen in Fig. 8(d). The next set of simulations involves analyzing the effects on the system performance by varying the initial flight path angle. Correspondingly, the simulation results with initial flight path angles of −15, −25 and −35deg are described in Fig. 9. From Figs. 9(a) - 9(d), we can see that feasible touchdown movements are achieved with smooth and bounded control efforts for all the three cases since Vy and γ converge to −5f t/sec and roughly 1.2deg respectively. Besides, the terminal velocity for the three cases range between 224f t/sec and 270f t/sec as can be seen from Fig. 9(e), which are all acceptable results at the touchdown moment. In the sequel, the simulations are focused on different initial velocities of 400, 500 and 600f t/sec and the results are shown in Fig. 10. It can be observed from Fig. 10(a) that the variation of initial 23
Page 23 of 30
12000
500 y =8000ft
450
0
10000
y =10000ft 0
400
y =12000ft 0
V (ft/sec)
6000
350 y =8000ft 0
300
y =10000ft
4000
0
250 2000
y =12000ft 0
200
0 0
0.5
1 x (ft)
1.5
150 0
2 4
10
20
x 10
0
40
−50
35
−100
30
40
50
60
50
60
cr
(b)
30 t (sec)
us
(a)
ip t
y (ft)
8000
y =8000ft
α (deg)
−150 −200 y =8000ft 0
−250
0
10
20
30 t (sec)
40
50
(c)
0
10
y0=12000ft
5 0
M
−350 0
20
0
y =12000ft
15
y =10000ft
−300
25
an
y
V (ft/sec)
0
y =10000ft
60
10
20
30 t (sec)
40
(d)
te
d
Fig. 8. Simulations with different initial altitudes: (a) A&L trajectory, (b) velocity histories, (c) vertical velocity histories, (d) angle of attack histories
velocity appears to have relatively small effects on the resulting altitude profiles. Note, from Fig.
Ac ce p
10(b), that the velocity profile for the 400f t/s initial velocity case performs an initial pull-up behavior, while the velocity profiles for the other two cases perform gradual decrease behaviors. Besides, the final velocities for the three cases are all about 250f t/sec. These phenomena reveal that the guidance law can adapt to different initial velocities automatically. From Fig. 10(c) and 10(d), we can see that the final vertical velocity for each case is fixed on −5f t/sec as desired and enough operational margin for the guidance command is ensured. From all the simulation results in this subsection, we can conclude that the guidance law is quite robust to boundary conditions and the associated guidance commands can be maintained within the acceptable boundaries.
24
Page 24 of 30
10000
0 γ =−15deg
γ =−15deg
0
0
γ =−25deg
8000
−50
γ =−25deg
−100
γ =−35deg
0
0
γ =−35deg
0
−150
y
y (ft)
V (ft/sec)
0
6000 4000 2000
−250 0.5
1 x (ft)
1.5
−300 0
2 4
(a)
20
(b) 0
35 γ =−15deg 0
γ =−25deg
30
γ =−35deg
25
0
−10
α (deg)
50
60
50
60
γ =−15deg 0
γ =−25deg
20
γ =−35deg 0
an
−20 −25
15
−30
10
10
20
30 t (sec)
40
5 0
10
20
40
50
60
M
−35 −40 0
40
0
0
−15
30 t (sec)
us
−5
γ (deg)
10
x 10
cr
0 0
ip t
−200
50
60
40
(d)
500
te
450
d
(c)
30 t (sec)
γ =−15deg
350
0
Ac ce p
V (ft/sec)
400
300
γ =−25deg 0
γ =−35deg 0
250
200 0
10
20
30 t (sec)
(e)
Fig. 9. Simulations with different initial flight path angles: (a) A&L trajectory, (b) vertical velocity histories, (c) flight path angle histories, (d) angle of attack histories, (e) velocity histories
C. Simulations with first-order system lag During practical implementation of the proposed guidance law, the guidance command cannot be achieved immediately after its generation. Such phenomenon is considered as the system lag. To evaluate the robustness of the proposed guidance law with respect to the system lag, simulations
25
Page 25 of 30
10000
600 V =400ft/sec 0
V =400ft/sec
550
V =500ft/sec
8000
0
V =500ft/sec
0
0
500
V =600ft/sec 0
y (ft)
V (ft/sec)
6000 4000
V =600ft/sec 0
450 400 350
ip t
300
2000
250 0.5
1 x (ft)
1.5
200 0
2 4
10
20
x 10
(a)
(b) 0
30 V =400ft/sec 25
0
V =600ft/sec
60
50
60
V =500ft/sec 0
V =600ft/sec
0
0
α (deg)
−150 −200
20
an
y
V (ft/sec)
50
us
0
V =500ft/sec
−100
40
V =400ft/sec
0
−50
30 t (sec)
cr
0 0
−250
15 10
−300 10
20
30 t (sec)
40
50
(c)
5 0
M
−350 0
60
10
20
30 t (sec)
40
(d)
te
d
Fig. 10. Simulations with different initial velocities: (a) A&L trajectory, (b) velocity histories, (c) vertical velocity histories , (d) angle of attack histories
Ac ce p
are carried out in this subsection. Consider the first-order system lag as follows αr 1 = αc Ts + 1
(68)
where αc and αr are the commanded angle of attack and the realized angle of attack respectively, T is the first-order time constant which varies from 0.1sec to 0.8sec. The values of υ and N are selected as 2.5 and 5.6998.
The A&L trajectories and the final vertical velocities with respect to different T is plotted in Fig. 11. It can be noted from Fig. 11(a) that the first-order system lag has a fairly negligible effect on the resulting A&L trajectories and the desired touchdown movements can be guaranteed. From Fig. 11(b), we can observe that although the vertical velocity errors at touchdown point increase as the time constant increases, they are still very small and the soft landing requirement can be achieved.
26
Page 26 of 30
−4.965
8000
−4.97
y (ft)
6000 4000 2000 0 0
−4.98 −4.985
0.5
1 x (ft)
1.5
−4.99 0
2 4
0.2
x 10
(b)
0.4 T (sec)
0.6
0.8
cr
(a)
Results with with first-order system lag: (a) A&L trajectory, (b) Vertical velocity at touchdown
an
VI. C ONCLUSION
us
Fig. 11.
−4.975
ip t
Vy (ft/sec)
10000
In this paper, a finite-time ISMC method is proposed to achieve finite time stabilization of a class of second-order uncertain nonlinear systems, and through a challenging A&L guidance problem,
M
the potential of the developed approach is demonstrated. By using the finite-time ISMC strategy, the system can be stabilized in a given finite time and the system states can be analytically expressed.
d
In addition, global robustness throughout the motion is ensured by the virtue of ISMC. After well
te
developed, the proposed control method is applied to the A&L guidance law design. The guidance law features in its simple form, less information demand as well as the on-line trajectory generation
Ac ce p
capability. Simulation results show that the guidance scheme is quite robust to boundary conditions as well as system lag. Therefore, the approach outlined in this work can be considered as a viable candidate for the A&L guidance. Moreover, since many control objectives need to be achieved in a given finite time, the finite-time ISMC algorithm shows a lot of potential in dealing with such problems.
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ip t
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cr
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[11] Feng Y, Yu XH, Man ZH, Non-singular adaptive terminal sliding mode control of rigid manipulators, Automatica, 38: 2159–2167, 2002. [12] Cong BL, Chen Z, Liu XD, On adaptive sliding mode control without switching gain overestimation, International Journal of Robust and Nonlinear Control, 24(3): 515–531, 2014. [13] Laghrouche S, Plestan F, Glumineau A, Higher order sliding mode control based on integral sliding mode, Automatica, 43: 531–537, 2007. [14] Wang L, Sheng YZ, Liu XD, A novel adaptive higher-order sliding mode control based on integral sliding mode, International Journal of Control, Automation, and Systems, 12(3): 459– 472, 2014. [15] C. A. Kluever, Unpowered Approach and Landing Guidance Using Trajectory Planning, Journal of Guidance, Control, and Dynamics, 27(6): 967–974, 2004. [16] J. M. Hanson, D. J. Coughlin, G. A. Dukeman, J. A. Mulqueen and J. W. McCarter, Ascent, Transition, Entry, and Abort Guidance Algorithm Design for the X-33 Vehicle, AIAA Paper 28
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ip t
Reusable Launch Vehicles, AIAA Paper 2001-4429, Aug. 2001.
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cr
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[22] J. Schierman and J. Hull, In-Flight Entry Trajectory Optimization for Reusable Launch Vehicles, AIAA Paper 2005-6434, Aug. 2005.
d
[23] A. Heydari and S. N. Balakrishnan, Path Planning Using a Novel Finite Horizon Suboptimal
te
Controller, Journal of Guidance, Control, and Dynamics, 36(4): 1210–1214, 2013. [24] N. Harl and S. N. Balakrishnan, Reentry Terminal Guidance Through Sliding Mode Control,
Ac ce p
Journal of Guidance, Control, and Dynamics, 33(1): 186–199, 2010. [25] US COESA, Committee on Extension to the Standard Atmosphere, US Standard Atmosphere 1976, NOAA-S/T 76-1562, US Gov. Printing Office, Washington, DC, Sup. DOCS, 1976. [26] V. Utkin and J. Shi, Integral Sliding Mode in Systems Operating under Uncertainty Conditions, Proceedings of the 35th Conference on Decision and Control, 1996: 4591–4596. [27] F. L. Lewis and V. L. Syroms, Optimal Control, Second Edition, John Wiley & Sons, 1995. [28] Y. I. Lee, S. H. Kim and M. J. Tahk, Optimality of Linear Time-Varying Guidance for Impact Angle Control, IEEE Transactions on Aerospace and Electronic Systems, 48(3): 2802–2817, 2012. [29] Y. I. Lee, S. H. Kim, J. I. Lee and M. J. Tahk, Analytic Solutions of Generalized ImpactAngle-Control Guidance Law for First-Order Lag System, Journal of Guidance, Control, and Dynamics, 36(1): 96–112, 2013. [30] J. Slotine, Sliding mode controller design for nonlinear system, International Journal of 29
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Ac ce p
te
d
M
an
us
cr
ip t
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30
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