Faulttolerant control allocation for Mars entry vehicle using adaptive
INTERNATIONAL JOURNAL OF ADAPTIVE CONTROL AND SIGNAL PROCESSING Int. J. Adapt. Control Signal Process. 2011; 25:95–113 Published online 1 June 2010 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/acs.1183
Fault-tolerant control allocation for Mars entry vehicle using adaptive control Monika Marwaha¶ and John Valasek∗, †,‡,§ Vehicle Systems and Control Laboratory, Aerospace Engineering Department, Texas A&M University, College Station, TX, U.S.A.
SUMMARY Accurate and reliable control of planetary entry is a major challenge for planetary exploration vehicles. For Mars entry, uncertainties in atmospheric properties such as winds aloft and density pose a major problem for meeting precision landing requirements. Anticipated manned missions to Mars will also require levels of safety and fault tolerance not required during earlier robotic missions. This paper develops a nonlinear fault-tolerant controller specifically tailored for addressing the unique environmental and mission demands of future Mars entry vehicles. The controller tracks a desired trajectory from entry interface to parachute deployment, and has an adaptation mechanism that reduces tracking errors in the presence of uncertain parameters such as atmospheric density, and vehicle properties such as aerodynamic coefficients and inertias. This nonlinear control law generates the commanded moments for a discrete control allocation algorithm, which then generates the optimal controls required to follow the desired trajectory. The reaction control system acts as a non-uniform quantizer, which generates applied moments that approximate the desired moments generated by a continuous adaptive control law. If a fault is detected in the control jets, it reconfigures the controls and minimizes the impact of control failures or damage on trajectory tracking. It is assumed that a fault identification and isolation scheme already exists to identify failures. A stability analysis is presented, and fault tolerance performance is evaluated with non real-time simulation for a complete Mars entry trajectory tracking scenario using various scenarios of control effector failures. The results presented in the paper demonstrate that the control algorithm has a satisfactory performance for tracking a pre-defined trajectory in the presence of control failures, in addition to plant and environment uncertainties. Copyright 䉷 2010 John Wiley & Sons, Ltd. Received 16 September 2008; Revised 7 April 2010; Accepted 13 April 2010 KEY WORDS:
nonlinear adaptive control; fault tolerant; discrete control allocation; mixed integer linear programming; entry vehicle
1. INTRODUCTION One of the challenges in the design of a guidance law or a controller is handling uncertainties that the system will encounter in operation. Uncertainty can be present not only in the plant parameters, but also in the operating environment. Adaptive control and guidance are one of the options available to the engineer, and structured adaptive model inversion control (SAMI) [1–4] is one of the forms that has been successfully applied to various spacecraft problems. SAMI is based on ∗ Correspondence
to: John Valasek, Vehicle Systems and Control Laboratory, Aerospace Engineering Department, Texas A&M University, 3141 TAMU, College Station, TX 77843-3141, U.S.A. † E-mail: [email protected] ‡ Associate Professor. § Director. ¶ Graduate Research Assistant. Copyright 䉷 2010 John Wiley & Sons, Ltd.
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the concepts of feedback linearization, dynamic inversion, and structured model reference adaptive control. Because dynamic inversion assumes perfect knowledge of all the system parameters which are used to solve for the control, but is approximate in the actual implementation due to inaccurate modeling of system parameters, in SAMI an adaptive controller is wrapped around the dynamic inversion to handle the uncertainties in system parameters. Specific to the Mars entry problem, Restrepo and Valasek in [5] conducted the modeling and SAMI adaptive control formulation for a potential planetary entry vehicle that has 18 discrete reaction control system (RCS) jets. Another challenge during operation is recovery or tolerance in the case of actuator faults. Actuator faults can be due to any malfunction in the physical system or subsystem of the controller which results in its failure to perform as designed. The fault-tolerant control problem belongs to the domain of complex control systems in which inter-control-disciplinary information and expertise are required, with most application studies based upon aerospace systems [6]. Patton provides a thorough review of fault-tolerant control systems in [6]. When combined with adaptive control, control allocation algorithms provide the capability to recover from actuator faults. In their basic form control allocation algorithms are useful for finding solutions to meet the desired control objectives by delivering the desired moments [7]. Bodson provides a comprehensive survey of constrained, numerical-based optimization methods for control allocation in [8], and Page and Steinberg in [9] compare the closed and open-loop performance for 16 different control allocation approaches. Numerous linear control allocation algorithms are currently available. In [10] a faulttolerant control allocation scheme is introduced to handle actuator failures in discrete systems and is validated with a numerical example of a disturbance-free case. Tjonnas and Johansen [11] considers a dynamic approach by constructing actuator reference adaptive laws for over-actuated nonlinear time-varying systems. Shertzer et al. [12] discusses control allocation algorithms specifically in the context of next generation entry vehicles. Recently, Bolender and Doman have used a concept in which control allocation is used for aerodynamic surfaces, and dynamic inversion-based adaptive control is used for system identification, which helps in identifying any failure [13]. The control variable rates or moments are nonlinear functions of control positions. These schemes are applied to different aerovehicles and are shown to have good performance for continuous controls. However, this approach has not been applied to discrete controls, since they require a different allocation algorithm from aerodynamic surfaces [14]. This paper introduces a novel use of fault-tolerant control allocation for discrete controls to nonlinear, time-varying systems. Specifically, we address the problem of discrete control allocation coupled with adaptive control, which has not been addressed in the literature to date, and make three contributions. First, this paper integrates fault-tolerant control allocation for discrete controls with the SAMI adaptive control algorithm to produce a system which not only handles discrete control failures, but also accounts for uncertainties in the plant and in the operating environment. The discrete control allocation algorithm treats the discrete controls as quantized elements, and mixed integer linear programming is used to solve the optimization problem. The SAMI adaptive control algorithm provides the tracking of velocity level and kinematic level states and handles the uncertainties. This combined approach can achieve objectives which can be difficult to achieve individually. The structure of the scheme is shown in Figure 1. The tracking error between the reference trajectory and plant states is used to update the dynamic inversion-based control law. The controls (u cal ) calculated by the adaptive control module becomes the commanded control. The control signal feeds a control allocation module, and then the optimal control (u app ) calculated in this module become the control which is applied to the plant. It is assumed that all the faults can be detected and isolated and there exists a scheme to identify these faults. The reference trajectory, plant states, and tracking error are all fed back to the dynamic inversion controller. Second, faulttolerant control allocation for redundant discrete controls is used for the first time on a nonlinear plant, and a stability analysis is provided for quantized control of nonlinear, time-varying systems. The third contribution is the demonstration of a fault-tolerant control algorithm to a Mars entry vehicle with redundant discrete controls [5]. The paper is organized as follows. In Section 2 equations for the parameter update in adaptive control are derived, along with a proof of stability using Lyapunov analysis. Section 3 develops the control allocation scheme used for discrete controllers, and a stability proof for quantized Copyright 䉷 2010 John Wiley & Sons, Ltd.
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Reference Trajectory Reference States
97
Reference States
Feedback Loop Tracking Error
Plant States ucal
Fault Tolerant Control Allocation
uapp
Plant Plant States
SAMI Update Loop
Figure 1. Schematic figure of the fault-tolerant control allocation.
control. Modeling of the vehicle and the discrete controllers are presented in Section 4, and Section 5 defines the reference bank angle trajectory. Section 6 demonstrates the fault tolerance and uncertainty handling performance of the controller with several test cases. Conclusions are presented in Section 7.
2. STRUCTURED ADAPTIVE MODEL INVERSION (SAMI) The nonlinear plant can be modeled as ˙ = J ()
(1)
˙ = −I −1 (I ˜ )+ I −1 (u + Maero )
(2)
where ⎡
cl
⎤
1 ⎥ ⎢ Maero = Aerodynamic Moments = v 2 Sreflref ⎣ cm ⎦ 2 cn and represents modified Rodrigues parameters (MRPs) and represents angular velocity. J () is the nonlinear transformation relating ˙ and . This matrix exhibits orientation singularity at ±360◦ . Consider a reference model having a structure similar to that of the nonlinear plant with states r and r . The control objective is to calculate the commanded moments which track the reference trajectory in terms of r and r , where r represent the reference MRPs and r are the reference angular velocities and Equations (1) and (2) can be rearranged to obtain the following form [2]: ∗ Ia∗ ()+C ¨ ˙ ˙ = PaT ()(u + Maero ) a (, )
(3)
where Pa () = J −1 () Ia∗ () = PaT I Pa Ca∗ (, ) ˙ = Ia∗ J˙a Pa + PaT [ P ˙ Pa a ]I Copyright 䉷 2010 John Wiley & Sons, Ltd.
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The aerodynamic moments are modeled by separating the known and unknown terms. In the expression for the aerodynamic moments everything is known except the Mars atmospheric density and the aerodynamic coefficients of the vehicle. These uncertain terms are separated in a column denoted by d ∗ , and are multiplied to make sure that uncertain terms appear linearly in the equation. ⎤⎡ ⎡ ⎤ cl 0 0 ⎥⎢ ⎢ ⎥ 1 ⎥ ⎢ cm ⎥ 0 0 Maero = Sreflref v 2 ⎢ (4) ⎦⎣ ⎣ ⎦ 2 0 0
cn d∗
Let Dest be the guess for
d∗
and d be the vector which adapts itself so that d ∗ = Dest d
(5)
Equation (4) can now be written as ⎡
0 0
⎤
⎥ ⎢ 1 0 0⎥ Maero = Sreflref v 2 ⎢ ⎦ Dest d ⎣ 2 0 0
(6)
The product of the inertia matrix and any vector a can be written as I a = (a)h ∀a ∈ R3
(7)
with a minimal parameterization of the inertia matrix given by
⎡
I11
⎢ ⎢ I12 ⎣ I13
⎡
0
0
a2
a3
I22
⎤⎡ ⎤ ⎡ a1 a1 ⎥⎢ ⎥ ⎢ ⎢ ⎥ ⎢ I23 ⎥ ⎦ ⎣ a2 ⎦ = ⎣ 0
a2
0
a1
0
I23
I33
0
a3
0
a1
I12
I13
a3
0
I11
⎤
⎢ ⎥ ⎢I ⎥ 22 ⎥ ⎤⎢ ⎥ 0 ⎢ ⎢ ⎥ ⎥ ⎥⎢ ⎢ I33 ⎥ a3 ⎥ ⎢ ⎦⎢ ⎥ ⎥ ⎢ I12 ⎥ ⎢ a2 ⎢ ⎥ ⎥ ⎢ I13 ⎥ ⎣ ⎦ I23
(8)
The left-hand side of Equation (3) can now be linearly parameterized as ∗ ¨ ˙ r˙ = Ya (r, r, ˙ r)h ¨ Ia∗ (r)r+C a (r, r)
(9)
where h is the constant inertia parameter vector defined as h[I11 I22 I33 I12 I13 I23 ]T and ˙ r) ¨ is a regression matrix. The terms on the left-hand side of Equation (9) can be written Ya (r, r, as Ia∗ r¨ = PaT I Pa r¨ ¨ = PaT (Pa r)h
(10)
˙ PaT [ P ˙ Pa r˙ Ca∗ r˙ = −PaT I Pa J˙a Pa r+ a r]I ˙ ˙ ˙ P = PaT {−(Pa J˙a Pa r)+[ a r](P a r)}h
(11)
Combining Equations (10) and (11) we have the linear minimal parameterization for the inertia matrix [2]. ∗ ¨ ˙ r˙ = PaT {(Pa r)−(P ¨ ˙ ˙ ˙ Ia∗ (r)r+C P a J˙a Pa r)+[ a r](P a r)}h a (r, r)
˙ r)h ¨ = Ya (r, r, Copyright 䉷 2010 John Wiley & Sons, Ltd.
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Define the tracking error by ε = −r
(13)
Differentiating and multiplying by Ia∗ on both sides Ia∗ ¨ = Ia∗ − ¨ Ia∗ ¨ r
(14)
and letting Cd , K d and K i be the design matrices, adding (Ca∗ +Cd )˙ + K d + K i dt to both sides of Equation (14) produces ∗ ∗ ∗ ∗ Ia ¨+(Cd +Ca (, ))˙ ˙ +K d +K i dt = (Ia∗ +C ¨ ˙ )−(I ˙ ¨ r +Ca∗ (r , ˙r )˙r ) a (, ) a +Cd ˙+K d +K i
dt
(15)
Re-arranging and using Equation (3), the right-hand side of Equation (15) can be expressed in the form T RHS = Pa (u + Maero )−Ya (, , (16) ˙ ˙r , ¨r )+Cd ˙ + K d + K i dt Dynamic inversion is used to calculate the control law which is given by the following equation. ⎤ ⎡ ⎡ ⎤ 0 0 d1 1 ⎥ ⎢ ⎥ −T 2⎢ u = Pa Ya (, , ˙ ˙ r , ¨ r )−Cd ˙ − K d ε − K i dt − Sreflref v ⎣ 0 0⎦ Dest ⎣ d2 ⎦ (17) 2 0 0 d3
L
d
2.1. Update laws The control law of Equation (17) is not implementable due to uncertainties present in and d, hence estimated parameters are used instead of unknown parameters. This results in ⎡ ⎤ ⎤ ⎡ dˆ1 0 0 ⎢ ⎥ 1 ⎥ ⎢ ˆ u = Pa−T Ya (, , dt − Sreflref v 2 ⎣ 0 0⎦ Dest ⎢ ˙ ˙ r , ¨ r )−C dˆ2 ⎥ d ˙ − K d ε − K i ⎣ ⎦ (18) 2 0 0 dˆ3
L
Using Equation (18) the closed-loop dynamics becomes ∗ ∗ Ia ¨ +(Cd +Ca (, ))˙ ˙ + K d + K i dt = −PaT L dˆ + PaT Ld +Ya (, , ˙ ˙r , ¨r )˜ ∗ ∗ Ia ¨ +(Cd +Ca (, ))˙ ˙ + K d + K i dt = −PaT L d˜ +Ya (, , ˙ ˙r , ¨r )˜
dˆ
(19)
where d˜ = dˆ −d ˆ ˜ = − Equation (19) can be written as Ia∗ ¨ +(Cd +Ca∗ (, ))˙ ˙ + K d + K i Copyright 䉷 2010 John Wiley & Sons, Ltd.
˜ dt =
(20)
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where ˜ = [˜ T d˜ T ] = [Ya − PaT L] If vector
y=
and
⎡ ⎢ A=⎢ ⎣
ε dt T ε T ε˙ T
0
I3×3
0
0
0
I3×3
−(Ia∗ )−1 K i I3×3
−(Ia∗ )−1 K d I3×3
−(Ia∗ )−1 Cd I3×3
⎤ ⎥ ⎥ ⎦
Then Equation (20) can be written as ˜ y˙ = Ay +
(21)
To obtain the update laws for the estimated parameters, the following candidate Lyapunov function is selected ˜ T −1 ˜ V = y T P y + 12
(22)
where P is a positive-definite matrix and is a symmetric positive-definite gain matrix. Vector y ˜ consists of the consists of tracking error, its derivative and the integral of the tracking error. errors between the true and estimated inertia vector and the true and estimated vector d, where vector d includes uncertainties in aerodynamic coefficients and Mars atmospheric density. Taking the time derivative of the Lyapunov function ˙˜ T −1 ˜ V˙ = y T P y˙ + y˙ T P y +
(23)
and substituting the expression for y˙ in Equation (23) results in ˙˜ −1 ˜ ˜ V˙ = y T (AT P +PA)y + y T P+ T
(24)
As matrix A is Hurwitz, for any positive-definite matrix Q there exists a corresponding positivedefinite matrix P such that PA+ AT P = −Q
(25)
Equation (24) becomes ˙˜ T −1 ˜ ˜ V˙ = −y T Qy + y T P+ ˙˜ −1 ˜ ) = −y T Qy +(y T P+ T
(26)
and the update law selected for this system is ˙ˆ = T P y
(27)
2.2. Stability analysis This choice of update law in (27) can be used to show that V˙ = −y T Qy0 Copyright 䉷 2010 John Wiley & Sons, Ltd.
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˜ ∈ L ∞ and hence are bounded. This in turn proves that , ˙, and dt As V > 0 and V˙ 0, y and ∈ L ∞ as y is a vector consisting of all these terms. The reference trajectory is bounded, and hence r , ˙r are bounded. Having proved that and r are bounded, from the definition of tracking error ˙ ˙r , ¨r ) is bounded. Note that given by Equation (13), is also bounded. This implies that Ya (, , ˜ and hence V˙ is semi negative-definite. Using these V˙ = 0 only if y = 0 regardless of the value of , ˜ ∈ L ∞. properties of V and V˙ and Barbalat’s Lemma [15], it is concluded that y ∈ L 2 ∩ L ∞ and Thus we can conclude that and ˙ go to zero as t approaches ∞. Hence, it is concluded that → r and → r as t → ∞.
3. CONTROL ALLOCATION Control allocation here follows [16], where mixed integer linear programming (MILP) [17] is used to implement the quantization strategy, such that the controller closes the loop using a control allocator which minimizes the difference between the commanded and the actual moments delivered. The problem of control allocation can be posed as p p 3 min Ti,k u k + wk u k (29) i des − u i=1
k=1
k=1
subject to the constraints 0
p
Ti,k u k i
des
∀ i
des 0
(30)
Ti,k u k i
des
∀ i
des < 0
(31)
k=1
0
p k=1
Here u k is a binary number that can be either 0 or 1. It represents the discrete (on/off) state of the kth control effector. i des is the desired moment in the ith axis, where i=1(roll), 2(pitch) and 3(yaw). Ti,k is the torque produced by the kth jet on the ith axis, p is the number of jets, and wk 0 represents the penalty imposed on firing the kth jet. The inequality constraints given by Equations (30) and (31) are imposed to ensure that the effective torque will not exceed the magnitude of the torque commanded by the control laws. This constraint represents the quantization strategy, hence the control allocator transforms the continuous commanded torque vectors in R3 in to quantized torque vectors in R3 . These quantized torque vectors represent the optimal solution of the problem, implemented using the MILP formulation. The control allocation can be posed as a linear minimization problem stated as uk min[w1 w2 . . . w p |wroll wpitch wyaw ] (32) u,u s us where u is a vector of binary variables and represents the state (on/off) of each control effector. Here u s ∈ R3 is the vector of slack variables which are defined as u s i
des − T u k
(33)
where i des ∈ R3 is the desired torque vector and T ∈ R3× p is the matrix whose elements represent the torque that can be provided by each control effector. The constraints are −u s 0 T u k −u s i
(34) des
−T u k −u s − i Copyright 䉷 2010 John Wiley & Sons, Ltd.
des
(35) (36)
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and 0
p
Ti,k u k i
∀ i
des
des 0
(37)
k=1
0−
p
Ti,k u k < − i
des
∀ i
des < 0
(38)
k=1
3.1. Stability analysis of quantized control Glad shows in [18] that for a wide class of optimal regulators, the gain margin is infinite with respect to increase in gain, and that decreases down to 0.5 can be tolerated. This is proven for systems that are closed-loop asymptotically stable and on linearization the plant has eigenvalues strictly on left half of the plane. The system considered here is not asymptotically stable, but the same approach can be extended to prove the stability of the system with the quantized control. Quantized control can be treated as some fraction of the calculated control, and if we prove that the system has some gain margin, we can use the same approach as in [18] and the stability proofs of [19, 20]. The nonlinear system considered is of the form x˙ = f (x)+ G(x)u
(39)
where f (x) is the unforced dynamics and G(x) is the full rank control effectiveness matrix of the system. Let us assume that there exists a feedback control law u = k(x) with k(0) = 0 which makes the system globally asymptotically stable. It also ensures that for some class K∞ functions 1 , 2 , 3 , 1 there exists a C1 function V : n −→ which satisfies the inequalities 1 (|x|)V (x)2 (|x|)
(40)
and |x|1 (|e|) ⇒
*V f (x, k(x)+e)−3 (|x|) *x
(41)
for all x, e ∈ n . This implies that the perturbed closed-loop system x˙ = f (x)+ G(x)(k(x)+e)
(42)
is input to state stable (ISS) [21] with respect to the actuator disturbance input e. Assuming to be some class K ∞ function with the following property:
(r ) max |k(x)| ∀r 0 |x|r
we get |k(x)|| (x)| ∀x Let z be the variable being quantized and if q is defined as z q (z) = q
(43)
(44)
the closed-loop system with quantized feedback control law u = q (k(x))
(45)
x˙ = f (x)+ G(x)(q k(x))
(46)
becomes
This takes the form of Equation (42) if e = q k(x)−k(x) Copyright 䉷 2010 John Wiley & Sons, Ltd.
(47)
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The behavior of the trajectories of Equation (46) for fixed is characterized by the following Lemma. Lemma 3.1 Assume that we have 1 ◦ −1 (M ) > 2 ◦1 ( )
(48)
R1 ( ) := {x : V (x)1 ◦ −1 (M )}
(49)
R2 ( ) := {x : V (x)2 ◦1 ( )}
(50)
Then the sets
and
are invariant regions for the system of Equation (42). Moreover, all solutions of Equation (42) that start in the set R1 ( ) enter the smaller set R2 ( ) in finite time, given by the formula T =
1 ◦ −1 (M )−2 ◦1 ( ) 3 ◦1 ( )
(51)
The unforced system x˙ = f (x, 0)
(52)
is called forward complete if for every initial state x(0) the solution (x(0), .) is defined for all t0. Theorem 3.1 Assume that the system in Equation (52) is forward complete and we have −1 −1 2 ◦1 ◦ (M ) > 1 ( )
∀ > 0
(53)
then there exists a hybrid quantized feedback control policy that makes the system x˙ = f (x, u) globally asymptotically stable. Proof 1 The zooming-out stage:Set the control to zero, and let (0) = 1. Increase fast enough to dominate the rate of growth of |x(t)|. Then there will be a time t0 0 such that −1 |x(t0 )|1 ( (t0 )) < −1 2 ◦1 ◦ (M (t0 ))
(54)
hence x(t0 ) belongs to the set R1 ( (t0 )) given by Equation (49). The Zooming-in stage: For tt0 apply the control law in Equation (45). Let (t) = (t0 ) for t ∈ [t0 , t0 + T (t0 ) ], where T (t0 ) is given by Equation (51). Then x(t + T (t0 ) ) belongs to the set R2 . Using Equation (51) to calculate T( (t0 )) , where is the function defined by (r ) :=
1
◦−1 ◦2 ◦1 (r ), M
r 0
(55)
For t ∈ [t0 + T (t0 ) , t0 + T (t0 ) + T( (t0 )) ], let (t) = ( (t0 ))
(56)
We have (t0 + T (t0 ) ) < (t0 ) by Equation (48), and R2 ( (t0 )) = R1 ( (t0 + T (t0 ) ). Thus we can conclude that x(t0 + T (t0 ) + T( (t0 ) ) belongs to R2 ( (t0 + T (t0 ) )). Repeating the procedure now let (t) = ( (t0 + T (t0 ) )) for the next time interval of length given by Equation (51). Lyapunov stability of equilibrium x = 0 of the continuous dynamics follows from the adjustment policy for . Also the above analysis implies that x(t) → 0 as t → ∞. Copyright 䉷 2010 John Wiley & Sons, Ltd.
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Thus if the closed-loop system is asymptotically stable, it can be deduced to be asymptotically stable using quantized control. The system considered here satisfies all the assumptions above except for asymptotic stability, hence the quantized control does not guarantee the asymptotic stability. However, the weaker condition of boundedness can be deduced.
4. MODELING OF VEHICLE AND RCS SYSTEM 4.1. Vehicle model The Mars Ellipsled entry vehicle is a cylinder with a diameter of 3.75 m and a length of 6.323 m. The external physical characteristics are shown in Figure 2. The Mars Ellipsled has 18 RCS thrusters or jets, which can be characterized as force producing devices with only two states: on and off. Each jet is capable of producing some force F = f nˆ when it is on, where f is the magnitude of thrust and nˆ is the unit vector. Therefore, the torque produced by a particular jet is calculated as =r × f , where r is the distance between the center of gravity and the location of a particular jet. There are 2 p combinations of torques that can be achieved using p number of jets, and all the jets produce significant coupling in more than one axis (i.e roll, pitch and yaw). The small time lag for the discrete controls to reach a steady-state value and ramp down to zero is not modeled, and it is assumed that the jet performance will not decrease with time as the propellant decreases. Finally, all jets have the same efficiency, and they are constant. There are nine jets on each side, located in three clusters of three jets each: three side jets, three up jets and three down jets, and the torque matrix is assumed to be constant at the values specified in the Appendix. Table I provides the thrust characteristics for all 18 jets. The nonlinear, non real-time simulation model of the Mars Ellipsled is developed using the mass properties, e.g. location, and aerodynamics provided in [5], as are the standard guidance equations and rotational equations of motion.
5. REFERENCE TRAJECTORY Bank angle is modulated to provide a steering control during entry, and the reference trajectory is given in terms of non-smooth, discrete bank angle commands consisting of a series of step inputs (Figure 3), from which a polynomial fit is used to generate a smooth continuous trajectory. Although not an optimal trajectory, it does serve the purpose and scope of this work since it
Figure 2. Mars Ellipsled. Table I. Reaction control system jets. Location of jets
Identification numbers
Moment arm (m)
Single jet thrust (N )
Up jets Side jets Down jets
1, 2, 3, 4, 5, 6 7, 8, 9, 10, 11, 12 13, 14, 15, 16, 17, 18
1.88 2.98 2.98
14 14 95
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Figure 3. Reference trajectory. Source: Obtained from Reference [5].
keeps the vehicle ‘in the right part of the sky’ and permits the controller to adapt to the realistic atmospheric density.
6. NUMERICAL EXAMPLES The objective is to test the controller performance under plant uncertainties, with and without various combinations of jet failures. The performance is evaluated in terms of tracking error in different states of the vehicle, and the control effort is used to track these states. Each test case has uncertainties of 10–30% in all moments of inertia. Uncertainty in the atmospheric density is lumped along with the aerodynamic coefficients so that uncertainty is introduced in both terms together. The initial conditions for all cases are a speed of 7.3 km/s, altitude of 125 km and bank angle of 85◦ . An initial condition error of 5◦ is introduced into the bank angle. Control failures in all cases are introduced at 2.5 s, and consist of a fault in the on/off firing capability of certain jets, and a corresponding decrease in their efficiency. Three cases are evaluated. In Case 1 jets 1, 2, 17 and 18 are not producing any thrust, such that the control allocation algorithm cannot use them at all. In Case 2 jets 2 , 3, 4, 5, 8, 9 and 10 are failed, i.e. while reconfiguring the controller it is assumed that there is a certain amount of thrust which is always produced by these jets. In Case 3 it is assumed that some of the jets produce less than their full thrust due to leakage or a valve malfunction. For all of the test cases evaluated here, it is assumed that a fault identification and the isolation scheme already exists to identify failures. 6.1. Case 1 In Case 1 jets 1, 2, 17 and 18 do not provide thrust after 2.5 s. These discrete jets can only take the numerical value of 0 or 1, with 0 indicating that a jet is off. By comparing the failed jet response to the nominal case in Figure 4, the tracking performance degrades when the failure is introduced and the settling time increases. However, the system is able to regulate the tracking error to near-zero values within the first few seconds after the failure is introduced. Note that whenever a bank angle reversal occurs the error is greater in the failed jet case than in the nominal case. Figure 5 shows that the initial condition error makes tracking difficult, but the controller is able to compensate. This figure also shows the bank angle trajectory when fault tolerance is off, showing that fault-tolerant control allocation is effective. The adaptive controller performs as expected and updates both the inertia vector and d vector such that all of the adaptive parameters converge to constant values as defined by the update laws. How fast the adaptive parameters converge to constant values depends upon the gain matrix. The translational states , and velocity are shown in Figure 6. As the equations for velocity and do not depend on the rotational states and control, they are exactly the same for both the nominal and failed jet cases. Sideslip angle changes with a jet failure and settles down at a slightly different value. Velocity is reduced as soon as the dynamic pressure peaks, at which time the altitude rate also decreases. This effect is seen in Figure 6 at 250 s when the flight path angle changes sign from negative to positive. This sign change occurs when the aerodynamic forces have a greater Copyright 䉷 2010 John Wiley & Sons, Ltd.
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200 100 0
Nominal Case Ref Φ Case of Jet failure
Bank Angle Error (degrees)
0
50
100
150
200
250
300
350
150
200
250
300
350
5 0
Nominal Failed Jets 0
50
100
Time (sec)
Figure 4. Time histories of bank angle, Case 1.
4
Bank Angle Error (degrees)
3 2 1 0
Nominal Failed Jets Fault Tolerance off
Failure Introduced
0
2
4
6 Time (sec)
8
10
12
Figure 5. Time history of bank angle error, Case 1.
magnitude than the gravity force. In the plot of altitude versus downrange trajectory, the vehicle is seen to lose altitude at the same rate in both the simulations. The time histories of the commanded control and the applied control are shown in Figures 7 and 8 respectively, and the applied control in the failed case is clearly greater than the nominal case as expected. As the commanded and applied control are greater in the failed jet case, the control effort defined as u T u is correspondingly greater in the failed jet case (Figure 9). In summary, the results of Case 1 demonstrate that failed jets can be handled by the combined adaptive control and fault-tolerant control allocation algorithm. 6.2. Case 2 For Case 2 a moment is continuously generated in a particular direction since jets 2, 3, 4, 5, 8, 9 and 10 produce thrust continuously after 2.5 s. Figure 10 shows the effect of the initial condition error in the bank angle, and the introduction of the failure. The error on increases as expected due Copyright 䉷 2010 John Wiley & Sons, Ltd.
Int. J. Adapt. Control Signal Process. 2011; 25:95–113 DOI: 10.1002/acs
20
10
0
0
γ (deg)
β (deg)
FAULT-TOLERANT CONTROL ALLOCATION FOR MARS ENTRY VEHICLE
200 Time (sec)
0
10
5
0
200 Time (sec)
150 ALt (Km)
velocity (km/sec)
0
0
Nominal Failed Jets
100 50 0
200 Time (sec)
107
0
500 1000 Downrange(Km)
Figure 6. Time histories of translational states, Case 1. 6
Ucroll
1
x 10
0 0
50
100
150
200
250
300
350
50
100
150
200
250
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5
Ucpitch
5
x 10
0 0 6
Ucyaw
1
Nominal Failed Jets
x 10
0 0
50
100
150 200 Time (sec)
250
300
350
Figure 7. Commanded control, Case 1.
to the error in the initial condition, yet the algorithm efficiently handles both the initial condition error and the failure at the same time. Figure 11 shows that the error in the bank angle for the complete trajectory is greater than in the nominal case, but the controller is able to handle the error not only during the constant command, but also during the bank angle reversals. The error in both the dynamic and kinematic level states is also greater in the failed jet case, and Figure 12 shows that the control allocation algorithm attempts to reduce this error over time by supplying more control according to the given constraints. 6.3. Case 3 This case assumes that the thrust producing capacity of certain jets is diminished. Instead of producing torque equal to the constants given in matrix T in the Appendix, some of the jets produce less thrust. Figure 13 shows that the error due to the bank angle initial condition gets added to the error due to the jet failures at 2.5 s, resulting in a larger error during the first few seconds of the trajectory. Copyright 䉷 2010 John Wiley & Sons, Ltd.
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500 Uaroll
0 0
50
100
150
200
250
0
50
100
150
200
250
300
350
0
50
100
150 200 Time (sec)
250
300
350
2000
300 350 Nominal Failed Jets
Uapitch
0
2000 Uayaw
0
Figure 8. Applied control, Case 1. 5
14
x 10
Nominal Failed Jets
12
Control Effort
10 8 6 4 2 0
0
50
100
150
200
250
300
350
Time (sec)
Figure 9. Control effort, Case 1. 3 Nominal Failed Jets
Bank Angle Error (degrees)
2 1 0
Failure Introduced
0
2
4
6 Time (sec)
8
10
12
Figure 10. Time history of bank angle error, Case 2. Copyright 䉷 2010 John Wiley & Sons, Ltd.
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200 100 0 Nominal Case Ref Φ Case of Jet failure 50 100 150
Bank Angle Error (degrees)
0
200
250
300
350
150 200 Time (sec)
250
300
350
5 0
Nominal Failed Jets 0
50
100
Figure 11. Time histories of bank angle, Case 2. 5
14
x 10
Nominal Failed Jets
12
Control Effort
10 8 6 4 2 0
0
50
100
150 200 Time (sec)
250
300
350
Figure 12. Control effort, Case 2.
6.4. Case 4 fault detection delay Figure 14 shows the bank angle error for a time delay in the failure detection. A delay of 1.5 s is introduced to represent the time required to identify the fault, after which the correction is applied at time equal to 4 s instead of 2.5 s. Once the fault is identified and correction is applied, it takes approximately one additional second for the error to reduce to zero, as compared with Cases 1–3 where correction is applied without any delay.
7. CONCLUSIONS A fault-tolerant control allocation scheme has been developed using SAMI Control to handle parametric uncertainties, and MILP for the allocation of discrete control effectors. A stability analysis was performed on the adaptive control laws and on the control allocation. Numerical simulation examples using the Mars Ellipsled vehicle were used to demonstrate the recovery and performance in the presence of RCS jet failures, and uncertainties in aerodynamic coefficients density, and inertias. Based upon the results presented in the paper, it is concluded that the fault-tolerant control scheme successfully handles failures if one or several jets fail, or if the Copyright 䉷 2010 John Wiley & Sons, Ltd.
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3 Nominal Case Failed Jets
Bank Angle Error (degrees)
2 1 0
Failure Introduced
0
2
4
6 Time (sec)
8
10
12
Figure 13. Time history of bank angle error, Case 3.
3
Fault Tolerance Introduced Bank Angle Error (degrees)
2 1 0
Failure Introduced Nominal Failed Jets with no delay Failed Jets with 1.5 seconds delay
0
2
4
6 Time (sec)
8
10
12
Figure 14. Time history of bank angle error, Case 4.
efficiency of one or several jets diminishes. Recovery and re-tracking of reference states and angular velocities was achieved after all failures, and maximum bank angle error due to failures was ±5◦ . The controller was also able to track the kinematic states successfully during and after failures. Control effort required to track the desired trajectory was compared for nominal and failed jet cases, and demonstrated that while being greater for the failed jets case, it was still within acceptable limits. Finally, while this adaptive control scheme places no restriction on the magnitude of the desired control, it was observed that large control magnitudes and the non-existence of optimal solutions for a particular case may result.
APPENDIX A The torque matrix is T ∈ R3x18 and consists of the constant torques provided by each jet in three axes. T = [T1 T2 T3 ] Copyright 䉷 2010 John Wiley & Sons, Ltd.
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where ⎡
−26.3200
⎢ T1 = ⎣ 41.7200
−41.7200 ⎡
−26.3200
⎢ T2 = ⎣ 41.7200
−41.7200
⎤ −26.3200 ⎥ −41.7200 −41.7200 41.7200 −41.7200 −41.7200⎦ 0
26.3200
41.7200
26.3200
0
−41.7200 41.7200 −41.7200
41.7200
⎤ −26.3200 ⎥ −41.7200 −41.7200 41.7200 −41.7200 −41.7200⎦ 0
26.3200
41.7200
26.3200
(A1)
0
−41.7200 41.7200 −41.7200
(A2)
41.7200
and ⎡
−17.6000
⎢ T3 = ⎣ 283.1000
−283.1000
0
17.6000
17.6000
0
−17.6000
⎤
⎥ −283.1000 −283.1000 283.1000 −283.1000 −283.1000⎦ 283.1000
−283.1000 283.1000 −283.1000
(A3)
283.1000
NOMENCLATURE Variable
Definition
Cd cL cl cm cn Dest d q I Kd Ki lref M Maero p P Sref T uk u us V v ˙ r) ¨ Ya (r, r,
design matrix coefficient of lift rolling moment stability derivative pitching moment stability derivative yawing moment stability derivative best guess for d ∗ vector of adaptive parameters quantizer function inertia matrix design matrix design matrix reference length range of quantizer aerodynamic moments number of jets positive-definite matrix reference area torque produced by jets binary number control input vector slack variable Lyapunov function velocity vector regression matrix density of Mars atmosphere modified Rodrigues parameter Vector angle-of-attack sideslip angle flight path angle angular velocity
Copyright 䉷 2010 John Wiley & Sons, Ltd.
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P Q R ◦ ˜
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learning rate desired moment K ∞ function K ∞ function K ∞ function K ∞ function K ∞ function zoom variable quantization error constant inertia parameter vector tracking error error in angular velocity in roll axis error in angular velocity in pitch axis error in angular velocity in yaw axis function composition skew symmetric matrix
ACKNOWLEDGEMENTS
This material is based upon work supported in part by the U.S. Air Force Office of Scientific Research under contract FA9550-08-1-0038, with technical monitor Dr William H. McEneaney, and by the NASA Johnson Space Center under contract C08-00884, with technical monitor Steve M. Fitzgerald. This support is gratefully acknowledged by the authors. The authors thank Dr David B. Doman and Dr Michael A. Bolender of the U.S. Air Force Research Laboratory for their insightful comments and suggestions on this work. Any opinions, findings, conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the U.S. Air Force or NASA. REFERENCES 1. Schaub H, Akella MR, Junkins JL. Adaptive realization of linear closed-loop tracking dynamics in the presence of large system model errors. Journal of the Astronautical Sciences 2000; 48(4):537–551. 2. Subbarao K. Structured adaptive model inversion: theory and applications to trajectory tracking for non-linear dynamical systems Ph.D. Thesis, Aerospace Engineering Department, Texas A&M University, College Station, TX, 2001. 3. Tandale MD, Valasek J. Solutions for handling control position bounds in adaptive dynamic inversion controlled satellites. Journal of the Astronautical Sciences 2007; 55(2):171–194. 4. Tandale MD, Valasek J. Fault tolerant structured adaptive model inversion control. Journal of Guidance, Control, and Dynamics 2006; 29(3):635–642. 5. Restrepo C, Valasek J. Structured adaptive model inversion controller for mars atmospheric flight. Journal of Guidance, Control, and Dynamics 2008; 31(4):937–953. 6. Patton RJ. Fault tolerant control systems: The 1997 situation (survey). IFAC Symposium SAFEPROCESS’97, vol. 2. The University of Hull: England, 26–28 August 1997; 1033–1055. 7. Durham WC. Constrained control allocation. Journal of Guidance, Control, and Dynamics 1993; 16(4):717–725. 8. Bodson M. Evaluation of optimization methods for control allocation. Journal of Guidance, Control, and Dynamics 2002; 25(4): 703–711. 9. Page AB, Steinberg ML. A closed-loop comparison of control allocation methods. Guidance, Navigation and Control Conference, Denver, CO, 14–17 August 2000. No. AIAA-2000-4538. 10. Casavola A, Garone E. Adaptive fault tolerant actuator allocation for overactuated plants. American Control Conference, New York, 11–13 July 2007; 3985–3990. 11. Tjonnas J, Johansen TA. Optimizing adaptive control allocation with actuator dynamics. IEEE Conference on Decision and Control, New Orleans, LA, 12–14 December 2007; 3780–3785. 12. Shertzer RH, Zimpfer DJ, Brown PD. Control allocation for the next generation of entry vehicles. Guidance, Navigation and Control Conference, Monterey, CA, 5–8 August 2002. 13. Doman DB, Ngo AD. Dynamic inversion based adaptive/reconfigurable control of the X-33 on ascent. Journal of Guidance, Control, and Dynamics 2002; 25(2):275–284. 14. Bolender MA, Doman DB. Non-linear control allocation using piecewise linear functions: a linear programming approach. Journal of Guidance, Control, and Dynamics 2005; 28(3):558–562. 15. Ioannou PA, Sun J. Robust Adaptive Control. Prentice-Hall, Upper Saddle River: NJ, 07458, 1996. 16. Doman D, Gamble B, Ngo A. Quantized control allocation of reaction control jets and aerodynamic control surfaces. Journal of Guidance, Control, and Dynamics 2009; 32(1):13–24. Copyright 䉷 2010 John Wiley & Sons, Ltd.
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17. Reklaitis GV, Ravindran A, Ragsdell KM. Engineering Optimization-Methods and Applications. Wiley: New York, 1983. 18. Glad ST. On the gain margin of nonlinear and optimal regulators. IEEE Transactions on Automatic Control 1984; 29(7):615–620. 19. DePersis C. On feedback stabilisation of nonlinear systems under quantization. Conference on Decision and Control and the European Control Conference, Seville, Spain, 12–15 December 2005. 20. Liberzon D. Quantization, time delays and nonlinear stabilization. IEEE Transactions on Automatic Control 2006; 51(7):1190–1195. 21. Sontag ED. Smooth stabilization implies coprime factorization. IEEE Transactions on Automatic Control 1989; 34(4):435–443.
Copyright 䉷 2010 John Wiley & Sons, Ltd.
Int. J. Adapt. Control Signal Process. 2011; 25:95–113 DOI: 10.1002/acs
Fault-tolerant control allocation for Mars entry vehicle using adaptive control Monika Marwaha¶ and John Valasek∗, †,‡,§ Vehicle Systems and Control Laboratory, Aerospace Engineering Department, Texas A&M University, College Station, TX, U.S.A.
SUMMARY Accurate and reliable control of planetary entry is a major challenge for planetary exploration vehicles. For Mars entry, uncertainties in atmospheric properties such as winds aloft and density pose a major problem for meeting precision landing requirements. Anticipated manned missions to Mars will also require levels of safety and fault tolerance not required during earlier robotic missions. This paper develops a nonlinear fault-tolerant controller specifically tailored for addressing the unique environmental and mission demands of future Mars entry vehicles. The controller tracks a desired trajectory from entry interface to parachute deployment, and has an adaptation mechanism that reduces tracking errors in the presence of uncertain parameters such as atmospheric density, and vehicle properties such as aerodynamic coefficients and inertias. This nonlinear control law generates the commanded moments for a discrete control allocation algorithm, which then generates the optimal controls required to follow the desired trajectory. The reaction control system acts as a non-uniform quantizer, which generates applied moments that approximate the desired moments generated by a continuous adaptive control law. If a fault is detected in the control jets, it reconfigures the controls and minimizes the impact of control failures or damage on trajectory tracking. It is assumed that a fault identification and isolation scheme already exists to identify failures. A stability analysis is presented, and fault tolerance performance is evaluated with non real-time simulation for a complete Mars entry trajectory tracking scenario using various scenarios of control effector failures. The results presented in the paper demonstrate that the control algorithm has a satisfactory performance for tracking a pre-defined trajectory in the presence of control failures, in addition to plant and environment uncertainties. Copyright 䉷 2010 John Wiley & Sons, Ltd. Received 16 September 2008; Revised 7 April 2010; Accepted 13 April 2010 KEY WORDS:
nonlinear adaptive control; fault tolerant; discrete control allocation; mixed integer linear programming; entry vehicle
1. INTRODUCTION One of the challenges in the design of a guidance law or a controller is handling uncertainties that the system will encounter in operation. Uncertainty can be present not only in the plant parameters, but also in the operating environment. Adaptive control and guidance are one of the options available to the engineer, and structured adaptive model inversion control (SAMI) [1–4] is one of the forms that has been successfully applied to various spacecraft problems. SAMI is based on ∗ Correspondence
to: John Valasek, Vehicle Systems and Control Laboratory, Aerospace Engineering Department, Texas A&M University, 3141 TAMU, College Station, TX 77843-3141, U.S.A. † E-mail: [email protected] ‡ Associate Professor. § Director. ¶ Graduate Research Assistant. Copyright 䉷 2010 John Wiley & Sons, Ltd.
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the concepts of feedback linearization, dynamic inversion, and structured model reference adaptive control. Because dynamic inversion assumes perfect knowledge of all the system parameters which are used to solve for the control, but is approximate in the actual implementation due to inaccurate modeling of system parameters, in SAMI an adaptive controller is wrapped around the dynamic inversion to handle the uncertainties in system parameters. Specific to the Mars entry problem, Restrepo and Valasek in [5] conducted the modeling and SAMI adaptive control formulation for a potential planetary entry vehicle that has 18 discrete reaction control system (RCS) jets. Another challenge during operation is recovery or tolerance in the case of actuator faults. Actuator faults can be due to any malfunction in the physical system or subsystem of the controller which results in its failure to perform as designed. The fault-tolerant control problem belongs to the domain of complex control systems in which inter-control-disciplinary information and expertise are required, with most application studies based upon aerospace systems [6]. Patton provides a thorough review of fault-tolerant control systems in [6]. When combined with adaptive control, control allocation algorithms provide the capability to recover from actuator faults. In their basic form control allocation algorithms are useful for finding solutions to meet the desired control objectives by delivering the desired moments [7]. Bodson provides a comprehensive survey of constrained, numerical-based optimization methods for control allocation in [8], and Page and Steinberg in [9] compare the closed and open-loop performance for 16 different control allocation approaches. Numerous linear control allocation algorithms are currently available. In [10] a faulttolerant control allocation scheme is introduced to handle actuator failures in discrete systems and is validated with a numerical example of a disturbance-free case. Tjonnas and Johansen [11] considers a dynamic approach by constructing actuator reference adaptive laws for over-actuated nonlinear time-varying systems. Shertzer et al. [12] discusses control allocation algorithms specifically in the context of next generation entry vehicles. Recently, Bolender and Doman have used a concept in which control allocation is used for aerodynamic surfaces, and dynamic inversion-based adaptive control is used for system identification, which helps in identifying any failure [13]. The control variable rates or moments are nonlinear functions of control positions. These schemes are applied to different aerovehicles and are shown to have good performance for continuous controls. However, this approach has not been applied to discrete controls, since they require a different allocation algorithm from aerodynamic surfaces [14]. This paper introduces a novel use of fault-tolerant control allocation for discrete controls to nonlinear, time-varying systems. Specifically, we address the problem of discrete control allocation coupled with adaptive control, which has not been addressed in the literature to date, and make three contributions. First, this paper integrates fault-tolerant control allocation for discrete controls with the SAMI adaptive control algorithm to produce a system which not only handles discrete control failures, but also accounts for uncertainties in the plant and in the operating environment. The discrete control allocation algorithm treats the discrete controls as quantized elements, and mixed integer linear programming is used to solve the optimization problem. The SAMI adaptive control algorithm provides the tracking of velocity level and kinematic level states and handles the uncertainties. This combined approach can achieve objectives which can be difficult to achieve individually. The structure of the scheme is shown in Figure 1. The tracking error between the reference trajectory and plant states is used to update the dynamic inversion-based control law. The controls (u cal ) calculated by the adaptive control module becomes the commanded control. The control signal feeds a control allocation module, and then the optimal control (u app ) calculated in this module become the control which is applied to the plant. It is assumed that all the faults can be detected and isolated and there exists a scheme to identify these faults. The reference trajectory, plant states, and tracking error are all fed back to the dynamic inversion controller. Second, faulttolerant control allocation for redundant discrete controls is used for the first time on a nonlinear plant, and a stability analysis is provided for quantized control of nonlinear, time-varying systems. The third contribution is the demonstration of a fault-tolerant control algorithm to a Mars entry vehicle with redundant discrete controls [5]. The paper is organized as follows. In Section 2 equations for the parameter update in adaptive control are derived, along with a proof of stability using Lyapunov analysis. Section 3 develops the control allocation scheme used for discrete controllers, and a stability proof for quantized Copyright 䉷 2010 John Wiley & Sons, Ltd.
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Reference Trajectory Reference States
97
Reference States
Feedback Loop Tracking Error
Plant States ucal
Fault Tolerant Control Allocation
uapp
Plant Plant States
SAMI Update Loop
Figure 1. Schematic figure of the fault-tolerant control allocation.
control. Modeling of the vehicle and the discrete controllers are presented in Section 4, and Section 5 defines the reference bank angle trajectory. Section 6 demonstrates the fault tolerance and uncertainty handling performance of the controller with several test cases. Conclusions are presented in Section 7.
2. STRUCTURED ADAPTIVE MODEL INVERSION (SAMI) The nonlinear plant can be modeled as ˙ = J ()
(1)
˙ = −I −1 (I ˜ )+ I −1 (u + Maero )
(2)
where ⎡
cl
⎤
1 ⎥ ⎢ Maero = Aerodynamic Moments = v 2 Sreflref ⎣ cm ⎦ 2 cn and represents modified Rodrigues parameters (MRPs) and represents angular velocity. J () is the nonlinear transformation relating ˙ and . This matrix exhibits orientation singularity at ±360◦ . Consider a reference model having a structure similar to that of the nonlinear plant with states r and r . The control objective is to calculate the commanded moments which track the reference trajectory in terms of r and r , where r represent the reference MRPs and r are the reference angular velocities and Equations (1) and (2) can be rearranged to obtain the following form [2]: ∗ Ia∗ ()+C ¨ ˙ ˙ = PaT ()(u + Maero ) a (, )
(3)
where Pa () = J −1 () Ia∗ () = PaT I Pa Ca∗ (, ) ˙ = Ia∗ J˙a Pa + PaT [ P ˙ Pa a ]I Copyright 䉷 2010 John Wiley & Sons, Ltd.
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The aerodynamic moments are modeled by separating the known and unknown terms. In the expression for the aerodynamic moments everything is known except the Mars atmospheric density and the aerodynamic coefficients of the vehicle. These uncertain terms are separated in a column denoted by d ∗ , and are multiplied to make sure that uncertain terms appear linearly in the equation. ⎤⎡ ⎡ ⎤ cl 0 0 ⎥⎢ ⎢ ⎥ 1 ⎥ ⎢ cm ⎥ 0 0 Maero = Sreflref v 2 ⎢ (4) ⎦⎣ ⎣ ⎦ 2 0 0
cn d∗
Let Dest be the guess for
d∗
and d be the vector which adapts itself so that d ∗ = Dest d
(5)
Equation (4) can now be written as ⎡
0 0
⎤
⎥ ⎢ 1 0 0⎥ Maero = Sreflref v 2 ⎢ ⎦ Dest d ⎣ 2 0 0
(6)
The product of the inertia matrix and any vector a can be written as I a = (a)h ∀a ∈ R3
(7)
with a minimal parameterization of the inertia matrix given by
⎡
I11
⎢ ⎢ I12 ⎣ I13
⎡
0
0
a2
a3
I22
⎤⎡ ⎤ ⎡ a1 a1 ⎥⎢ ⎥ ⎢ ⎢ ⎥ ⎢ I23 ⎥ ⎦ ⎣ a2 ⎦ = ⎣ 0
a2
0
a1
0
I23
I33
0
a3
0
a1
I12
I13
a3
0
I11
⎤
⎢ ⎥ ⎢I ⎥ 22 ⎥ ⎤⎢ ⎥ 0 ⎢ ⎢ ⎥ ⎥ ⎥⎢ ⎢ I33 ⎥ a3 ⎥ ⎢ ⎦⎢ ⎥ ⎥ ⎢ I12 ⎥ ⎢ a2 ⎢ ⎥ ⎥ ⎢ I13 ⎥ ⎣ ⎦ I23
(8)
The left-hand side of Equation (3) can now be linearly parameterized as ∗ ¨ ˙ r˙ = Ya (r, r, ˙ r)h ¨ Ia∗ (r)r+C a (r, r)
(9)
where h is the constant inertia parameter vector defined as h[I11 I22 I33 I12 I13 I23 ]T and ˙ r) ¨ is a regression matrix. The terms on the left-hand side of Equation (9) can be written Ya (r, r, as Ia∗ r¨ = PaT I Pa r¨ ¨ = PaT (Pa r)h
(10)
˙ PaT [ P ˙ Pa r˙ Ca∗ r˙ = −PaT I Pa J˙a Pa r+ a r]I ˙ ˙ ˙ P = PaT {−(Pa J˙a Pa r)+[ a r](P a r)}h
(11)
Combining Equations (10) and (11) we have the linear minimal parameterization for the inertia matrix [2]. ∗ ¨ ˙ r˙ = PaT {(Pa r)−(P ¨ ˙ ˙ ˙ Ia∗ (r)r+C P a J˙a Pa r)+[ a r](P a r)}h a (r, r)
˙ r)h ¨ = Ya (r, r, Copyright 䉷 2010 John Wiley & Sons, Ltd.
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Define the tracking error by ε = −r
(13)
Differentiating and multiplying by Ia∗ on both sides Ia∗ ¨ = Ia∗ − ¨ Ia∗ ¨ r
(14)
and letting Cd , K d and K i be the design matrices, adding (Ca∗ +Cd )˙ + K d + K i dt to both sides of Equation (14) produces ∗ ∗ ∗ ∗ Ia ¨+(Cd +Ca (, ))˙ ˙ +K d +K i dt = (Ia∗ +C ¨ ˙ )−(I ˙ ¨ r +Ca∗ (r , ˙r )˙r ) a (, ) a +Cd ˙+K d +K i
dt
(15)
Re-arranging and using Equation (3), the right-hand side of Equation (15) can be expressed in the form T RHS = Pa (u + Maero )−Ya (, , (16) ˙ ˙r , ¨r )+Cd ˙ + K d + K i dt Dynamic inversion is used to calculate the control law which is given by the following equation. ⎤ ⎡ ⎡ ⎤ 0 0 d1 1 ⎥ ⎢ ⎥ −T 2⎢ u = Pa Ya (, , ˙ ˙ r , ¨ r )−Cd ˙ − K d ε − K i dt − Sreflref v ⎣ 0 0⎦ Dest ⎣ d2 ⎦ (17) 2 0 0 d3
L
d
2.1. Update laws The control law of Equation (17) is not implementable due to uncertainties present in and d, hence estimated parameters are used instead of unknown parameters. This results in ⎡ ⎤ ⎤ ⎡ dˆ1 0 0 ⎢ ⎥ 1 ⎥ ⎢ ˆ u = Pa−T Ya (, , dt − Sreflref v 2 ⎣ 0 0⎦ Dest ⎢ ˙ ˙ r , ¨ r )−C dˆ2 ⎥ d ˙ − K d ε − K i ⎣ ⎦ (18) 2 0 0 dˆ3
L
Using Equation (18) the closed-loop dynamics becomes ∗ ∗ Ia ¨ +(Cd +Ca (, ))˙ ˙ + K d + K i dt = −PaT L dˆ + PaT Ld +Ya (, , ˙ ˙r , ¨r )˜ ∗ ∗ Ia ¨ +(Cd +Ca (, ))˙ ˙ + K d + K i dt = −PaT L d˜ +Ya (, , ˙ ˙r , ¨r )˜
dˆ
(19)
where d˜ = dˆ −d ˆ ˜ = − Equation (19) can be written as Ia∗ ¨ +(Cd +Ca∗ (, ))˙ ˙ + K d + K i Copyright 䉷 2010 John Wiley & Sons, Ltd.
˜ dt =
(20)
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where ˜ = [˜ T d˜ T ] = [Ya − PaT L] If vector
y=
and
⎡ ⎢ A=⎢ ⎣
ε dt T ε T ε˙ T
0
I3×3
0
0
0
I3×3
−(Ia∗ )−1 K i I3×3
−(Ia∗ )−1 K d I3×3
−(Ia∗ )−1 Cd I3×3
⎤ ⎥ ⎥ ⎦
Then Equation (20) can be written as ˜ y˙ = Ay +
(21)
To obtain the update laws for the estimated parameters, the following candidate Lyapunov function is selected ˜ T −1 ˜ V = y T P y + 12
(22)
where P is a positive-definite matrix and is a symmetric positive-definite gain matrix. Vector y ˜ consists of the consists of tracking error, its derivative and the integral of the tracking error. errors between the true and estimated inertia vector and the true and estimated vector d, where vector d includes uncertainties in aerodynamic coefficients and Mars atmospheric density. Taking the time derivative of the Lyapunov function ˙˜ T −1 ˜ V˙ = y T P y˙ + y˙ T P y +
(23)
and substituting the expression for y˙ in Equation (23) results in ˙˜ −1 ˜ ˜ V˙ = y T (AT P +PA)y + y T P+ T
(24)
As matrix A is Hurwitz, for any positive-definite matrix Q there exists a corresponding positivedefinite matrix P such that PA+ AT P = −Q
(25)
Equation (24) becomes ˙˜ T −1 ˜ ˜ V˙ = −y T Qy + y T P+ ˙˜ −1 ˜ ) = −y T Qy +(y T P+ T
(26)
and the update law selected for this system is ˙ˆ = T P y
(27)
2.2. Stability analysis This choice of update law in (27) can be used to show that V˙ = −y T Qy0 Copyright 䉷 2010 John Wiley & Sons, Ltd.
(28)
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˜ ∈ L ∞ and hence are bounded. This in turn proves that , ˙, and dt As V > 0 and V˙ 0, y and ∈ L ∞ as y is a vector consisting of all these terms. The reference trajectory is bounded, and hence r , ˙r are bounded. Having proved that and r are bounded, from the definition of tracking error ˙ ˙r , ¨r ) is bounded. Note that given by Equation (13), is also bounded. This implies that Ya (, , ˜ and hence V˙ is semi negative-definite. Using these V˙ = 0 only if y = 0 regardless of the value of , ˜ ∈ L ∞. properties of V and V˙ and Barbalat’s Lemma [15], it is concluded that y ∈ L 2 ∩ L ∞ and Thus we can conclude that and ˙ go to zero as t approaches ∞. Hence, it is concluded that → r and → r as t → ∞.
3. CONTROL ALLOCATION Control allocation here follows [16], where mixed integer linear programming (MILP) [17] is used to implement the quantization strategy, such that the controller closes the loop using a control allocator which minimizes the difference between the commanded and the actual moments delivered. The problem of control allocation can be posed as p p 3 min Ti,k u k + wk u k (29) i des − u i=1
k=1
k=1
subject to the constraints 0
p
Ti,k u k i
des
∀ i
des 0
(30)
Ti,k u k i
des
∀ i
des < 0
(31)
k=1
0
p k=1
Here u k is a binary number that can be either 0 or 1. It represents the discrete (on/off) state of the kth control effector. i des is the desired moment in the ith axis, where i=1(roll), 2(pitch) and 3(yaw). Ti,k is the torque produced by the kth jet on the ith axis, p is the number of jets, and wk 0 represents the penalty imposed on firing the kth jet. The inequality constraints given by Equations (30) and (31) are imposed to ensure that the effective torque will not exceed the magnitude of the torque commanded by the control laws. This constraint represents the quantization strategy, hence the control allocator transforms the continuous commanded torque vectors in R3 in to quantized torque vectors in R3 . These quantized torque vectors represent the optimal solution of the problem, implemented using the MILP formulation. The control allocation can be posed as a linear minimization problem stated as uk min[w1 w2 . . . w p |wroll wpitch wyaw ] (32) u,u s us where u is a vector of binary variables and represents the state (on/off) of each control effector. Here u s ∈ R3 is the vector of slack variables which are defined as u s i
des − T u k
(33)
where i des ∈ R3 is the desired torque vector and T ∈ R3× p is the matrix whose elements represent the torque that can be provided by each control effector. The constraints are −u s 0 T u k −u s i
(34) des
−T u k −u s − i Copyright 䉷 2010 John Wiley & Sons, Ltd.
des
(35) (36)
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M. MARWAHA AND J. VALASEK
and 0
p
Ti,k u k i
∀ i
des
des 0
(37)
k=1
0−
p
Ti,k u k < − i
des
∀ i
des < 0
(38)
k=1
3.1. Stability analysis of quantized control Glad shows in [18] that for a wide class of optimal regulators, the gain margin is infinite with respect to increase in gain, and that decreases down to 0.5 can be tolerated. This is proven for systems that are closed-loop asymptotically stable and on linearization the plant has eigenvalues strictly on left half of the plane. The system considered here is not asymptotically stable, but the same approach can be extended to prove the stability of the system with the quantized control. Quantized control can be treated as some fraction of the calculated control, and if we prove that the system has some gain margin, we can use the same approach as in [18] and the stability proofs of [19, 20]. The nonlinear system considered is of the form x˙ = f (x)+ G(x)u
(39)
where f (x) is the unforced dynamics and G(x) is the full rank control effectiveness matrix of the system. Let us assume that there exists a feedback control law u = k(x) with k(0) = 0 which makes the system globally asymptotically stable. It also ensures that for some class K∞ functions 1 , 2 , 3 , 1 there exists a C1 function V : n −→ which satisfies the inequalities 1 (|x|)V (x)2 (|x|)
(40)
and |x|1 (|e|) ⇒
*V f (x, k(x)+e)−3 (|x|) *x
(41)
for all x, e ∈ n . This implies that the perturbed closed-loop system x˙ = f (x)+ G(x)(k(x)+e)
(42)
is input to state stable (ISS) [21] with respect to the actuator disturbance input e. Assuming to be some class K ∞ function with the following property:
(r ) max |k(x)| ∀r 0 |x|r
we get |k(x)|| (x)| ∀x Let z be the variable being quantized and if q is defined as z q (z) = q
(43)
(44)
the closed-loop system with quantized feedback control law u = q (k(x))
(45)
x˙ = f (x)+ G(x)(q k(x))
(46)
becomes
This takes the form of Equation (42) if e = q k(x)−k(x) Copyright 䉷 2010 John Wiley & Sons, Ltd.
(47)
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The behavior of the trajectories of Equation (46) for fixed is characterized by the following Lemma. Lemma 3.1 Assume that we have 1 ◦ −1 (M ) > 2 ◦1 ( )
(48)
R1 ( ) := {x : V (x)1 ◦ −1 (M )}
(49)
R2 ( ) := {x : V (x)2 ◦1 ( )}
(50)
Then the sets
and
are invariant regions for the system of Equation (42). Moreover, all solutions of Equation (42) that start in the set R1 ( ) enter the smaller set R2 ( ) in finite time, given by the formula T =
1 ◦ −1 (M )−2 ◦1 ( ) 3 ◦1 ( )
(51)
The unforced system x˙ = f (x, 0)
(52)
is called forward complete if for every initial state x(0) the solution (x(0), .) is defined for all t0. Theorem 3.1 Assume that the system in Equation (52) is forward complete and we have −1 −1 2 ◦1 ◦ (M ) > 1 ( )
∀ > 0
(53)
then there exists a hybrid quantized feedback control policy that makes the system x˙ = f (x, u) globally asymptotically stable. Proof 1 The zooming-out stage:Set the control to zero, and let (0) = 1. Increase fast enough to dominate the rate of growth of |x(t)|. Then there will be a time t0 0 such that −1 |x(t0 )|1 ( (t0 )) < −1 2 ◦1 ◦ (M (t0 ))
(54)
hence x(t0 ) belongs to the set R1 ( (t0 )) given by Equation (49). The Zooming-in stage: For tt0 apply the control law in Equation (45). Let (t) = (t0 ) for t ∈ [t0 , t0 + T (t0 ) ], where T (t0 ) is given by Equation (51). Then x(t + T (t0 ) ) belongs to the set R2 . Using Equation (51) to calculate T( (t0 )) , where is the function defined by (r ) :=
1
◦−1 ◦2 ◦1 (r ), M
r 0
(55)
For t ∈ [t0 + T (t0 ) , t0 + T (t0 ) + T( (t0 )) ], let (t) = ( (t0 ))
(56)
We have (t0 + T (t0 ) ) < (t0 ) by Equation (48), and R2 ( (t0 )) = R1 ( (t0 + T (t0 ) ). Thus we can conclude that x(t0 + T (t0 ) + T( (t0 ) ) belongs to R2 ( (t0 + T (t0 ) )). Repeating the procedure now let (t) = ( (t0 + T (t0 ) )) for the next time interval of length given by Equation (51). Lyapunov stability of equilibrium x = 0 of the continuous dynamics follows from the adjustment policy for . Also the above analysis implies that x(t) → 0 as t → ∞. Copyright 䉷 2010 John Wiley & Sons, Ltd.
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M. MARWAHA AND J. VALASEK
Thus if the closed-loop system is asymptotically stable, it can be deduced to be asymptotically stable using quantized control. The system considered here satisfies all the assumptions above except for asymptotic stability, hence the quantized control does not guarantee the asymptotic stability. However, the weaker condition of boundedness can be deduced.
4. MODELING OF VEHICLE AND RCS SYSTEM 4.1. Vehicle model The Mars Ellipsled entry vehicle is a cylinder with a diameter of 3.75 m and a length of 6.323 m. The external physical characteristics are shown in Figure 2. The Mars Ellipsled has 18 RCS thrusters or jets, which can be characterized as force producing devices with only two states: on and off. Each jet is capable of producing some force F = f nˆ when it is on, where f is the magnitude of thrust and nˆ is the unit vector. Therefore, the torque produced by a particular jet is calculated as =r × f , where r is the distance between the center of gravity and the location of a particular jet. There are 2 p combinations of torques that can be achieved using p number of jets, and all the jets produce significant coupling in more than one axis (i.e roll, pitch and yaw). The small time lag for the discrete controls to reach a steady-state value and ramp down to zero is not modeled, and it is assumed that the jet performance will not decrease with time as the propellant decreases. Finally, all jets have the same efficiency, and they are constant. There are nine jets on each side, located in three clusters of three jets each: three side jets, three up jets and three down jets, and the torque matrix is assumed to be constant at the values specified in the Appendix. Table I provides the thrust characteristics for all 18 jets. The nonlinear, non real-time simulation model of the Mars Ellipsled is developed using the mass properties, e.g. location, and aerodynamics provided in [5], as are the standard guidance equations and rotational equations of motion.
5. REFERENCE TRAJECTORY Bank angle is modulated to provide a steering control during entry, and the reference trajectory is given in terms of non-smooth, discrete bank angle commands consisting of a series of step inputs (Figure 3), from which a polynomial fit is used to generate a smooth continuous trajectory. Although not an optimal trajectory, it does serve the purpose and scope of this work since it
Figure 2. Mars Ellipsled. Table I. Reaction control system jets. Location of jets
Identification numbers
Moment arm (m)
Single jet thrust (N )
Up jets Side jets Down jets
1, 2, 3, 4, 5, 6 7, 8, 9, 10, 11, 12 13, 14, 15, 16, 17, 18
1.88 2.98 2.98
14 14 95
Copyright 䉷 2010 John Wiley & Sons, Ltd.
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105
Figure 3. Reference trajectory. Source: Obtained from Reference [5].
keeps the vehicle ‘in the right part of the sky’ and permits the controller to adapt to the realistic atmospheric density.
6. NUMERICAL EXAMPLES The objective is to test the controller performance under plant uncertainties, with and without various combinations of jet failures. The performance is evaluated in terms of tracking error in different states of the vehicle, and the control effort is used to track these states. Each test case has uncertainties of 10–30% in all moments of inertia. Uncertainty in the atmospheric density is lumped along with the aerodynamic coefficients so that uncertainty is introduced in both terms together. The initial conditions for all cases are a speed of 7.3 km/s, altitude of 125 km and bank angle of 85◦ . An initial condition error of 5◦ is introduced into the bank angle. Control failures in all cases are introduced at 2.5 s, and consist of a fault in the on/off firing capability of certain jets, and a corresponding decrease in their efficiency. Three cases are evaluated. In Case 1 jets 1, 2, 17 and 18 are not producing any thrust, such that the control allocation algorithm cannot use them at all. In Case 2 jets 2 , 3, 4, 5, 8, 9 and 10 are failed, i.e. while reconfiguring the controller it is assumed that there is a certain amount of thrust which is always produced by these jets. In Case 3 it is assumed that some of the jets produce less than their full thrust due to leakage or a valve malfunction. For all of the test cases evaluated here, it is assumed that a fault identification and the isolation scheme already exists to identify failures. 6.1. Case 1 In Case 1 jets 1, 2, 17 and 18 do not provide thrust after 2.5 s. These discrete jets can only take the numerical value of 0 or 1, with 0 indicating that a jet is off. By comparing the failed jet response to the nominal case in Figure 4, the tracking performance degrades when the failure is introduced and the settling time increases. However, the system is able to regulate the tracking error to near-zero values within the first few seconds after the failure is introduced. Note that whenever a bank angle reversal occurs the error is greater in the failed jet case than in the nominal case. Figure 5 shows that the initial condition error makes tracking difficult, but the controller is able to compensate. This figure also shows the bank angle trajectory when fault tolerance is off, showing that fault-tolerant control allocation is effective. The adaptive controller performs as expected and updates both the inertia vector and d vector such that all of the adaptive parameters converge to constant values as defined by the update laws. How fast the adaptive parameters converge to constant values depends upon the gain matrix. The translational states , and velocity are shown in Figure 6. As the equations for velocity and do not depend on the rotational states and control, they are exactly the same for both the nominal and failed jet cases. Sideslip angle changes with a jet failure and settles down at a slightly different value. Velocity is reduced as soon as the dynamic pressure peaks, at which time the altitude rate also decreases. This effect is seen in Figure 6 at 250 s when the flight path angle changes sign from negative to positive. This sign change occurs when the aerodynamic forces have a greater Copyright 䉷 2010 John Wiley & Sons, Ltd.
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106 Bank Angle (degrees)
M. MARWAHA AND J. VALASEK
200 100 0
Nominal Case Ref Φ Case of Jet failure
Bank Angle Error (degrees)
0
50
100
150
200
250
300
350
150
200
250
300
350
5 0
Nominal Failed Jets 0
50
100
Time (sec)
Figure 4. Time histories of bank angle, Case 1.
4
Bank Angle Error (degrees)
3 2 1 0
Nominal Failed Jets Fault Tolerance off
Failure Introduced
0
2
4
6 Time (sec)
8
10
12
Figure 5. Time history of bank angle error, Case 1.
magnitude than the gravity force. In the plot of altitude versus downrange trajectory, the vehicle is seen to lose altitude at the same rate in both the simulations. The time histories of the commanded control and the applied control are shown in Figures 7 and 8 respectively, and the applied control in the failed case is clearly greater than the nominal case as expected. As the commanded and applied control are greater in the failed jet case, the control effort defined as u T u is correspondingly greater in the failed jet case (Figure 9). In summary, the results of Case 1 demonstrate that failed jets can be handled by the combined adaptive control and fault-tolerant control allocation algorithm. 6.2. Case 2 For Case 2 a moment is continuously generated in a particular direction since jets 2, 3, 4, 5, 8, 9 and 10 produce thrust continuously after 2.5 s. Figure 10 shows the effect of the initial condition error in the bank angle, and the introduction of the failure. The error on increases as expected due Copyright 䉷 2010 John Wiley & Sons, Ltd.
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20
10
0
0
γ (deg)
β (deg)
FAULT-TOLERANT CONTROL ALLOCATION FOR MARS ENTRY VEHICLE
200 Time (sec)
0
10
5
0
200 Time (sec)
150 ALt (Km)
velocity (km/sec)
0
0
Nominal Failed Jets
100 50 0
200 Time (sec)
107
0
500 1000 Downrange(Km)
Figure 6. Time histories of translational states, Case 1. 6
Ucroll
1
x 10
0 0
50
100
150
200
250
300
350
50
100
150
200
250
300
350
5
Ucpitch
5
x 10
0 0 6
Ucyaw
1
Nominal Failed Jets
x 10
0 0
50
100
150 200 Time (sec)
250
300
350
Figure 7. Commanded control, Case 1.
to the error in the initial condition, yet the algorithm efficiently handles both the initial condition error and the failure at the same time. Figure 11 shows that the error in the bank angle for the complete trajectory is greater than in the nominal case, but the controller is able to handle the error not only during the constant command, but also during the bank angle reversals. The error in both the dynamic and kinematic level states is also greater in the failed jet case, and Figure 12 shows that the control allocation algorithm attempts to reduce this error over time by supplying more control according to the given constraints. 6.3. Case 3 This case assumes that the thrust producing capacity of certain jets is diminished. Instead of producing torque equal to the constants given in matrix T in the Appendix, some of the jets produce less thrust. Figure 13 shows that the error due to the bank angle initial condition gets added to the error due to the jet failures at 2.5 s, resulting in a larger error during the first few seconds of the trajectory. Copyright 䉷 2010 John Wiley & Sons, Ltd.
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M. MARWAHA AND J. VALASEK
500 Uaroll
0 0
50
100
150
200
250
0
50
100
150
200
250
300
350
0
50
100
150 200 Time (sec)
250
300
350
2000
300 350 Nominal Failed Jets
Uapitch
0
2000 Uayaw
0
Figure 8. Applied control, Case 1. 5
14
x 10
Nominal Failed Jets
12
Control Effort
10 8 6 4 2 0
0
50
100
150
200
250
300
350
Time (sec)
Figure 9. Control effort, Case 1. 3 Nominal Failed Jets
Bank Angle Error (degrees)
2 1 0
Failure Introduced
0
2
4
6 Time (sec)
8
10
12
Figure 10. Time history of bank angle error, Case 2. Copyright 䉷 2010 John Wiley & Sons, Ltd.
Int. J. Adapt. Control Signal Process. 2011; 25:95–113 DOI: 10.1002/acs
Bank Angle (degrees)
FAULT-TOLERANT CONTROL ALLOCATION FOR MARS ENTRY VEHICLE
109
200 100 0 Nominal Case Ref Φ Case of Jet failure 50 100 150
Bank Angle Error (degrees)
0
200
250
300
350
150 200 Time (sec)
250
300
350
5 0
Nominal Failed Jets 0
50
100
Figure 11. Time histories of bank angle, Case 2. 5
14
x 10
Nominal Failed Jets
12
Control Effort
10 8 6 4 2 0
0
50
100
150 200 Time (sec)
250
300
350
Figure 12. Control effort, Case 2.
6.4. Case 4 fault detection delay Figure 14 shows the bank angle error for a time delay in the failure detection. A delay of 1.5 s is introduced to represent the time required to identify the fault, after which the correction is applied at time equal to 4 s instead of 2.5 s. Once the fault is identified and correction is applied, it takes approximately one additional second for the error to reduce to zero, as compared with Cases 1–3 where correction is applied without any delay.
7. CONCLUSIONS A fault-tolerant control allocation scheme has been developed using SAMI Control to handle parametric uncertainties, and MILP for the allocation of discrete control effectors. A stability analysis was performed on the adaptive control laws and on the control allocation. Numerical simulation examples using the Mars Ellipsled vehicle were used to demonstrate the recovery and performance in the presence of RCS jet failures, and uncertainties in aerodynamic coefficients density, and inertias. Based upon the results presented in the paper, it is concluded that the fault-tolerant control scheme successfully handles failures if one or several jets fail, or if the Copyright 䉷 2010 John Wiley & Sons, Ltd.
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M. MARWAHA AND J. VALASEK
3 Nominal Case Failed Jets
Bank Angle Error (degrees)
2 1 0
Failure Introduced
0
2
4
6 Time (sec)
8
10
12
Figure 13. Time history of bank angle error, Case 3.
3
Fault Tolerance Introduced Bank Angle Error (degrees)
2 1 0
Failure Introduced Nominal Failed Jets with no delay Failed Jets with 1.5 seconds delay
0
2
4
6 Time (sec)
8
10
12
Figure 14. Time history of bank angle error, Case 4.
efficiency of one or several jets diminishes. Recovery and re-tracking of reference states and angular velocities was achieved after all failures, and maximum bank angle error due to failures was ±5◦ . The controller was also able to track the kinematic states successfully during and after failures. Control effort required to track the desired trajectory was compared for nominal and failed jet cases, and demonstrated that while being greater for the failed jets case, it was still within acceptable limits. Finally, while this adaptive control scheme places no restriction on the magnitude of the desired control, it was observed that large control magnitudes and the non-existence of optimal solutions for a particular case may result.
APPENDIX A The torque matrix is T ∈ R3x18 and consists of the constant torques provided by each jet in three axes. T = [T1 T2 T3 ] Copyright 䉷 2010 John Wiley & Sons, Ltd.
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111
FAULT-TOLERANT CONTROL ALLOCATION FOR MARS ENTRY VEHICLE
where ⎡
−26.3200
⎢ T1 = ⎣ 41.7200
−41.7200 ⎡
−26.3200
⎢ T2 = ⎣ 41.7200
−41.7200
⎤ −26.3200 ⎥ −41.7200 −41.7200 41.7200 −41.7200 −41.7200⎦ 0
26.3200
41.7200
26.3200
0
−41.7200 41.7200 −41.7200
41.7200
⎤ −26.3200 ⎥ −41.7200 −41.7200 41.7200 −41.7200 −41.7200⎦ 0
26.3200
41.7200
26.3200
(A1)
0
−41.7200 41.7200 −41.7200
(A2)
41.7200
and ⎡
−17.6000
⎢ T3 = ⎣ 283.1000
−283.1000
0
17.6000
17.6000
0
−17.6000
⎤
⎥ −283.1000 −283.1000 283.1000 −283.1000 −283.1000⎦ 283.1000
−283.1000 283.1000 −283.1000
(A3)
283.1000
NOMENCLATURE Variable
Definition
Cd cL cl cm cn Dest d q I Kd Ki lref M Maero p P Sref T uk u us V v ˙ r) ¨ Ya (r, r,
design matrix coefficient of lift rolling moment stability derivative pitching moment stability derivative yawing moment stability derivative best guess for d ∗ vector of adaptive parameters quantizer function inertia matrix design matrix design matrix reference length range of quantizer aerodynamic moments number of jets positive-definite matrix reference area torque produced by jets binary number control input vector slack variable Lyapunov function velocity vector regression matrix density of Mars atmosphere modified Rodrigues parameter Vector angle-of-attack sideslip angle flight path angle angular velocity
Copyright 䉷 2010 John Wiley & Sons, Ltd.
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112 i des 1 2 3 1
P Q R ◦ ˜
M. MARWAHA AND J. VALASEK
learning rate desired moment K ∞ function K ∞ function K ∞ function K ∞ function K ∞ function zoom variable quantization error constant inertia parameter vector tracking error error in angular velocity in roll axis error in angular velocity in pitch axis error in angular velocity in yaw axis function composition skew symmetric matrix
ACKNOWLEDGEMENTS
This material is based upon work supported in part by the U.S. Air Force Office of Scientific Research under contract FA9550-08-1-0038, with technical monitor Dr William H. McEneaney, and by the NASA Johnson Space Center under contract C08-00884, with technical monitor Steve M. Fitzgerald. This support is gratefully acknowledged by the authors. The authors thank Dr David B. Doman and Dr Michael A. Bolender of the U.S. Air Force Research Laboratory for their insightful comments and suggestions on this work. Any opinions, findings, conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the U.S. Air Force or NASA. REFERENCES 1. Schaub H, Akella MR, Junkins JL. Adaptive realization of linear closed-loop tracking dynamics in the presence of large system model errors. Journal of the Astronautical Sciences 2000; 48(4):537–551. 2. Subbarao K. Structured adaptive model inversion: theory and applications to trajectory tracking for non-linear dynamical systems Ph.D. Thesis, Aerospace Engineering Department, Texas A&M University, College Station, TX, 2001. 3. Tandale MD, Valasek J. Solutions for handling control position bounds in adaptive dynamic inversion controlled satellites. Journal of the Astronautical Sciences 2007; 55(2):171–194. 4. Tandale MD, Valasek J. Fault tolerant structured adaptive model inversion control. Journal of Guidance, Control, and Dynamics 2006; 29(3):635–642. 5. Restrepo C, Valasek J. Structured adaptive model inversion controller for mars atmospheric flight. Journal of Guidance, Control, and Dynamics 2008; 31(4):937–953. 6. Patton RJ. Fault tolerant control systems: The 1997 situation (survey). IFAC Symposium SAFEPROCESS’97, vol. 2. The University of Hull: England, 26–28 August 1997; 1033–1055. 7. Durham WC. Constrained control allocation. Journal of Guidance, Control, and Dynamics 1993; 16(4):717–725. 8. Bodson M. Evaluation of optimization methods for control allocation. Journal of Guidance, Control, and Dynamics 2002; 25(4): 703–711. 9. Page AB, Steinberg ML. A closed-loop comparison of control allocation methods. Guidance, Navigation and Control Conference, Denver, CO, 14–17 August 2000. No. AIAA-2000-4538. 10. Casavola A, Garone E. Adaptive fault tolerant actuator allocation for overactuated plants. American Control Conference, New York, 11–13 July 2007; 3985–3990. 11. Tjonnas J, Johansen TA. Optimizing adaptive control allocation with actuator dynamics. IEEE Conference on Decision and Control, New Orleans, LA, 12–14 December 2007; 3780–3785. 12. Shertzer RH, Zimpfer DJ, Brown PD. Control allocation for the next generation of entry vehicles. Guidance, Navigation and Control Conference, Monterey, CA, 5–8 August 2002. 13. Doman DB, Ngo AD. Dynamic inversion based adaptive/reconfigurable control of the X-33 on ascent. Journal of Guidance, Control, and Dynamics 2002; 25(2):275–284. 14. Bolender MA, Doman DB. Non-linear control allocation using piecewise linear functions: a linear programming approach. Journal of Guidance, Control, and Dynamics 2005; 28(3):558–562. 15. Ioannou PA, Sun J. Robust Adaptive Control. Prentice-Hall, Upper Saddle River: NJ, 07458, 1996. 16. Doman D, Gamble B, Ngo A. Quantized control allocation of reaction control jets and aerodynamic control surfaces. Journal of Guidance, Control, and Dynamics 2009; 32(1):13–24. Copyright 䉷 2010 John Wiley & Sons, Ltd.
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