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An L1 adaptive closed-loop guidance law for an orbital injection problem
J Roshanian, M Zareh, H H Afshari and M Rezaei Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering 2009 223: 753 DOI: 10.1243/09596518JSCE789 The online version of this article can be found at: http://pii.sagepub.com/content/223/6/753
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An L1 adaptive closed-loop guidance law for an orbital injection problem J Roshanian, M Zareh*, H H Afshari, and M Rezaei Department of Aerospace Engineering, K.N. Toosi University of Technology, Tehran, Iran The manuscript was received on 12 April 2009 and was accepted after revision for publication on 11 June 2009. DOI: 10.1243/09596518JSCE789
Abstract: The current paper presents the determination of a closed-loop guidance law for an orbital injection problem using two different approaches and, considering the existing timeoptimal open-loop trajectory as the nominal solution, compares the advantages of the two proposed strategies. In the first method, named neighbouring optimal control (NOC), the perturbation feedback method is utilized to determine the closed-loop trajectory in an analytical form for the non-linear system. This law, which produces feedback gains, is in general a function of small perturbations appearing in the states and constraints separately. The second method uses an L1 adaptive strategy in determination of the non-linear closedloop guidance law. The main advantages of this method include characteristics such as improvement of asymptotic tracking, guaranteed time-delay margin, and smooth control input. The accuracy of the two methods is compared by introducing a high-frequency sinusoidal noise. The simulation results indicate that the L1 adaptive strategy has a better performance than the NOC method to track the nominal trajectory when the noise amplitude is increased. On the other hand, the main advantage of the NOC method is its ability to solve a non-linear, two-point, boundary-value problem in the minimum time. Keywords:
1
L1 adaptive control, optimal control, optimal guidance
INTRODUCTION
Optimal formulations of non-linear dynamic systems, either through dynamic programming or variational approaches, lead to non-linear partial differential equations. Numerical solution of such equations dealing with complex non-linear systems is always difficult, especially for real-world physical problems. Obtaining closed-loop control laws intensifies the inherent difficulty involved and is only exceptionally determined in some rare cases. Besides, open-loop control laws, dependent on initial conditions, are highly sensitive to noise and external disturbances and therefore are not preferred for realworld applications. On the contrary, closed-loop control policies are desirable owing to their natural robustness to perturbations. In addition, there exist *Corresponding author: Department of Aerospace Engineering, K.N. Toosi University of Technology, East Vafadar Street, 4th Tehranpars Square, Tehran 1656983211, Iran. email:
[email protected] JSCE789
certain difficulties associated with the numerical determination of open-loop optimal control solutions for non-linear systems, such as slow convergence rate and high sensitivity to initial guesstimates. The technique of neighbouring optimal control (NOC) produces time-variant feedback control that minimizes a performance index to second order for perturbation from a nominal optimal path. Perturbations in the nominal optimal states are compensated for, but errors in the system dynamic model parameters, such as mass variation and environment disturbance, may substantially affect the performance. Also, the neighbouring extremals are given for orbital injection [1]. The time-optimal solution of a non-linear landing mission in polar coordinates was investigated utilizing a numerical technique named linear programming. This law is an exact solution to the two-point boundary-value problem associated with the necessary conditions for first variation [2]. The results of minimum-time feedback laws [3] are used to validate the results of an Proc. IMechE Vol. 223 Part I: J. Systems and Control Engineering
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analytical open-loop strategy proposed in the present paper. Jardin and Bryson used the technique of NOC to develop an algorithm for optimizing aircraft trajectories in general wind fields by computing timevarying linear feedback gains [4, 5]. Furthermore, Pourtakdoust et al. presented a time-optimal openloop strategy for a non-linear lunar landing mission using an analytical technique. To create closed-loop fuzzy guidance logic, a fuzzy algorithm was augmented to the variational function of the problem [6, 7]. Palma and Magni have worked on optimal predictive control by discretizing non-linear dynamic systems [8]. Recently, Afshari et al. employed analytical approaches to obtain non-linear optimal guidance policies of spacecraft missions [9, 10]. There are many reasons why researchers prefer adaptive laws to find control signals, especially for non-linear problems for which an analytical solution is barely possible. Uncertainties in the system, disturbances, and measurement noise are some of these reasons. The L1 adaptive strategy has recently been introduced by Cao and Hovakimyan [11–13]. This new approach presents much better results, especially during the transient phase. As proved previously, it can provide robust performance against external disturbances and measurement noises. The other advantages of L1 adaptive architecture, such as improved asymptotic tracking and guaranteed time-delay margin, achieved via smooth control input, have also been discussed previously. The L1 adaptive control approach replaces the conventional model reference adaptive control (MRAC) by first specifying an equivalent companion model architecture, which enables insertion of a low-pass filter in the closed loop. To ensure asymptotic stability of the closed-loop system, the L1 gain of the cascaded system involving the lowpass filter and the desired closed-loop reference system needs to be less than the inverse of the upper bound on the unknown parameters used in the projection-type adaptation law [14–18]. The present study demonstrates the first usage of L1 adaptive control in the determination of a closed-loop guidance law for spacecraft missions. For this purpose, the selected companion model should track the optimal trajectory with the optimal control signal considered as the reference input. To have a smooth input signal, a low-pass filter is utilized by satisfying the L1 small gain theorem [19–21]. 2
OPEN- VERSUS CLOSED-LOOP SOLUTION
Optimal control of non-linear dynamic systems will typically be of open-loop type, usually determined as
a function of time, u(t). An optimal control policy is designed to move the system from its initial state x(0) the specified terminal hypersurface, towards y xtf ,tf ~0, in a manner that minimizes the cost function. In this sense any point on the computed trajectory from [x(t0),t0] to the terminal hypersurface could be a possible initial point for which optimal control is already at hand, but for other initial points not lying on the predetermined trajectory, the current policy would no longer be optimal and the problem must be solved again. Bearing in mind the open-loop nature of the control history in non-linear optimal control problems, one can easily justify the need for many solutions linked to various initial conditions. To overcome this problem, a family of optimal trajectories is needed to envelop all of the feasible initial conditions, a task not easily accomplished. Generally speaking, for each initial condition [x(t0),t0] there exists an optimal trajectory to arrive at the specified target hypersurface. The corresponding control history u0(t) can be expressed as u0 ~u0 ðx,t Þ
ð1Þ
which is now in the so-called closed-loop form. This means that the optimal control action is a function of state x(t) and current time t. For a stationary system having performance measures and constraints that are not explicit functions of time, the optimal control will also be an implicit function of time, namely u0 ~u0 ðx Þ
ð2Þ
However, determination of closed-loop optimal control policies for non-linear systems is a formidable task. Moreover, if one desires the advantages associated with closed-loop policies, simplifying assumptions must be made to linearize the system around some working conditions [6].
3
OPEN-LOOP SOLUTION TO THE SPACECRAFT INJECTION PROBLEM
In this section, the results of an analytical timeoptimal solution obtained for a spacecraft injection mission are utilized. In this way, consider an idealized spacecraft at the origin of inertial frame (x,y) at t 5 0, moving under the action of a constant propulsive force making a control angle b with the horizon. Obviously, the position and velocity vector of the vehicle will change due to the forces acting
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An L1 adaptive closed-loop guidance law
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yðbÞ~{0:223 tan bz0:025 lnðsec bz tan bÞ z0:025 tan b sec b{0:008 sec2 bz0:689
ð9Þ
It should be noted that the above results are in the open-loop form and considered the nominal optimal solution. In section 4, this solution is utilized for a mass-varying spacecraft; in fact, the optimal parameters mentioned in equations (5) to (9) are assumed as reference parameters.
4 Fig. 1 Geometry of the injection mission
upon it. Based on Fig. 1, the governing non-dimensional equations are 8 du > > ~w1 cos b > > > dt > < dv {w2 ~w1 sin b > dt > > > > > : dy ~w3 v dt
ð3Þ
where w1, w2, and w3 are constant non-dimensional multipliers [6]. Also, the non-dimensional boundary conditions for soft landing on the moon surface are
APPLICATION OF NOC LAW IN DETERMINATION OF THE CLOSED-LOOP SOLUTION
The NOC law [1] allows development of the closedloop optimal guidance law. In order to determine the perturbation thrust angle, first the coefficient matrices A, B, and C are computed using the existing relationships in reference [1] in the following form 2
0
6 AðtÞ~4 0 0
0 0
3
7 0 05 1 0 2
sin2 b
{ sin b cos b
0
cos2 b 0
6 BðtÞ~4:523 cos b4 { sin b cos b
ðt~0Þ~0, vðt~0Þ~0, yðt~0Þ~0 u uðt~tf Þ~1, vðt~tf Þ~0, yðt~tf Þ~1
0
3
7 05 0
ð4Þ Next, by using the calculus of variation theory and an analytical strategy [1, 2, 6], the optimal thrust angle and state trajectories can be found in the following form tf ~2:2113,
~1:422 rad, b ~{1:2748 rad b 0 f ð5Þ
CðtÞ~03|3 ð10Þ The general form of NOC law is d uðt Þ~{ fuT S{RQ{1 RT zHux dx {1 zfuT RQ{1 dy Huu ? ~{L1 ðt Þdx{L2 ðt Þdy
~tan{1 ð6:671{4:5tÞ b
ð6Þ
ðbÞ~0:577{0:222lnðsecbztanbÞ u
ð7Þ
vðbÞ~0:0741|
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{13:57 cos b{ sin bz3 cos b
ð8Þ
ð11Þ
where R(t), S(t), and Q(t) are unknown matrices. Having obtained A(t), B(t), and C(t), it is possible to determine R(t) as discussed in reference [1] 2
0 dR3|3 6 ~4 {R31 dt 0
0 {R32 0
3 0 7 {R33 5 0
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J Roshanian, M Zareh, H H Afshari, and M Rezaei
Rðtf Þ~I3|3
ð12Þ
Each member of the matrix equation (12) forms a differential equation that must be integrated backwards from the terminal time to the current time. Due to the fact that the thrust angle b(t) is related to time, the matrix Q(t) can be easily determined by solving a set of independent differential equations with corresponding boundary conditions in the following form 2 dQ3|3 6 ~4:523 cos b|4 dt
Fig. 3
L1 adaptive scheme
sin2 b
{ sin b cos b
{ðtf {tÞ sin b cos b
{ sin b cos b
cos2 b
ðtf {tÞ cos2 b
{ðtf {tÞ sin b cos b
ðtf {tÞ cos2 b
ðtf {tÞ2 cos2 b
3 7 5
Qðtf Þ~03|3 ð13Þ Therefore, Q(t) is obtained analytically by backward integration from each member of matrix differential equation (13) with respect to time. In effect, the differential equations are swept backwards from the terminal condition to the current condition in the exact solution. Since the co-state equations are not functions of state variables, the matrix S(t) is equal to zero and solving the Riccati equation is not required. Note that because the spacecraft injection problem is in the class of a terminal guidance problem, the perturbation on terminal constraints will be equal to zero. As a result, the NOC law is a function of perturbation on state variables only. The NOC block diagram for determining the optimal closed-loop control law for the spacecraft injection problem is depicted in Fig. 2. Also, by computing the time
Fig. 2
Block diagram of the NOC law for design of the closed-loop trajectory
perturbation, it is found to be less than 1023, so that it is negligible. By regarding the previous relationships, the NOC law can be applied in the following formulation dbðtÞ~Ku ðtÞduðtÞzKv ðtÞdvðtÞzKy ðtÞdy ðtÞ
ð14Þ
where the perturbed terms du(t), dv(t), and dy(t) are small perturbations applied to the state variables of the spacecraft injection system, and Ku(t), Kv(t), and Ky(t) are neighbouring optimum gains for determination of the optimal closed-loop guidance law.
5
USING THE L1 ADAPTIVE APPROACH TO REACH A CLOSED-LOOP OPTIMAL GUIDANCE
MRAC is one of the main approaches for adaptive control. The reference model is chosen to generate the desired trajectory ym that the plant output yp has to follow. The closed-loop plant is made up of an ordinary feedback control law that contains the plant, a controller C(h), and an adjustment mechanism that generates the controller parameter estimates h(t) on-line. It is possible to ensure that the input and output of an uncertain linear system track the input and output of a desired linear system during the transient phase, in addition to the asymptotic tracking, by using the L1 adaptive approach. These features are established by first performing an equivalent re-parametrization of MRAC, the key difference of which with respect to MRAC is in the definition of the error signal for
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An L1 adaptive closed-loop guidance law
adaptive laws. This new architecture, called the companion model adaptive controller (CMAC), allows for incorporation of a low-pass filter into the feedback loop that enables one to enforce the desired transient performance by increasing the adaptation gain [12]. A block diagram of this method is shown in Fig. 3. To describe the idea, consider an imaginary spacecraft parallel to a real one that with an optimal control signal produces an optimal trajectory. So if the adaptation mechanism is able to reduce the tracking error e 5 yp 2 ym to zero, then the real spacecraft will behave like the imaginary one and so will track the optimal trajectory. By reformulating equation (3) we have X_ ~f ðX ,bÞ
X_ m ~f ðX m ,bc Þ
ð16Þ
where Xm 5 [um vm ym] and bc is not like that of conventional MRAC, being defined by a CMAC structure as follows bc ~b{h X
The aim of L1 adaptive control is to find b as a function of states such that the input and output of the system track the input and output of a desired system during the transient phase, in addition to the asymptotic tracking. To reach this end, a low-pass filter is added to the control signal b~FLP b zhT X
ð18Þ
where FLP{b* + hTX} denotes a low-pass filter. The vector h contains adaptation parameters. By defining deviation between the real output and the desired output as e~X m {X
ð19Þ
gives e_ ~f ðX m ,bc Þ{f ðX ,bÞ
ð20Þ
Let the adaptation parameters vector be such that f ~sf
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2
ð21Þ
h1
6 s~4 0 0
0 h2 0
0
3
7 05 h3
If the error goes to zero then it is clear that h 5 [1,1,1]T. So we define h ~½1,1,1T By this definition one can obtain from equation (20) that e_ ~ðI{sÞf in which I363 is the unique matrix. Now matrices A363 and B361 are defined so that A is Hurwitz, (A, B) is controllable, and e_ ~AezB X T h~
ð22Þ
where h˜ 5 h 2 h*. Consider the Lyapunov function as follows 1 h~T h~ ePez V~ 2 c
ð17Þ
T
where
ð15Þ
where X 5 [u v y]. Now consider the following reference model dynamics that satisfies the optimal trajectory
757
! ð23Þ
where P is a 363 matrix and c is a real constant scalar. To have a stable system dv/dt must be negative. Differentiating from equation (23), one obtains ATP + PA 5 2Q in which Q is positive definite and h_ ~{cProj e,X BT P
ð24Þ
Proj(?) denotes the projection operator [22].
6
RESULTS AND DISCUSSION
The time histories of neighbouring optimal feedback gains for the orbital injection problem are depicted in Fig. 4. Having obtained these gains, the timeoptimal closed-loop control solution can be computed using equation (14). By observing the plots in Fig. 4, it can be easily seen that the NOC law is able to satisfy the required boundary conditions. The perturbed thrust angle for protecting the launched spacecraft against environment disturbances is depicted in Fig. 5.
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J Roshanian, M Zareh, H H Afshari, and M Rezaei
where the noise frequency is v1 5 250. The design parameters of the L1 adaptive controller are chosen as follows 2
1 0
16 0 1 P~{ 6 24 0 0 2 3 1 6 7 7 B~6 415 1
Fig. 4
Neighbouring optimum gains for the spacecraft injection mission
c~3 and
0
3
2
{1
7 6 6 07 5, A~4 0 1 0
FLP ~
0 {1 0
0
3
7 0 7 5, {1
400 sz400
Limitation of the adaptation parameters is defined as jhj¡hmax ~10 To compare the robustness potentials of each method, two scenarios are designed. In the first, a sinusoidal noise with 0.01 amplitude is applied to the dynamic system; therefore e 5 0.01. Thereafter, the closed-loop solutions of the two methods are computed. In the other scenario, the amplitude of the sinusoidal noise is taken as 0.1, i.e. e 5 0.10. Comparisons between the NOC and L1 adaptive results are illustrated in Figs 6 to 13. As can be seen in Figs 6 to 9, when the noise amplitude is equal to 0.01, an excellent agreement
Fig. 5
Perturbed thrust angle, due to applying disturbance, obtained using the NOC law
There are several source of noise or disturbance in the operating environment that affect the performance of the determined optimal feedback laws. To investigate the robustness potentials of these two methods, i.e. NOC and L1 adaptive laws, the problem dynamic system is analysed under the influence of small disturbances exerted on the state feedback (measurement unit) of the system. These state disturbances, modelled similar to the control actuation noise, are taken as jðt Þ~e sinðvt Þ
ð25Þ
and zjðt Þ, vj ~vzjðt Þ j ~u u
ð26Þ
Fig. 6 Comparison between NOC and L1 closed-loop solutions for thrust angle (e 5 0.01)
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Fig. 7
Comparison between NOC and L1 closed-loop solutions for spacecraft horizontal velocity (e 5 0.01)
Fig. 9
Comparison between NOC and L1 closed-loop solutions for spacecraft altitude (e 5 0.01)
Fig. 8
Comparison between NOC and L1 closed-loop solutions for spacecraft vertical velocity (e 5 0.01)
Fig. 10
Comparison between NOC and L1 closed-loop solutions for thrust angle (e 5 0.10)
exists between the open-loop solution and the two proposed closed-loop solutions in the state-space trajectories. However, when the noise amplitude is equal to 0.10, Figs 10 to 13 show that only the L1 adaptive method is in good agreement with the open-loop solution. Of course, this behaviour is concluded from the adaptive nature of this methodology. One can easily verify from the results that the JSCE789
measurement disturbance has no effect on the performance of L1 adaptive methods, while the system behaves in an oscillatory fashion when it is not in the closed-loop form. Because of the observed agreement of L1 adaptive methods in real-time application, the presented methodology can be successfully utilized in real-world applications with good robustness to noise or disturbance in each of the state variables. Proc. IMechE Vol. 223 Part I: J. Systems and Control Engineering
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Fig. 11
Comparison between NOC and L1 closed-loop solutions for spacecraft horizontal velocity (e 5 0.10)
Fig. 13 Comparison between NOC and L1 closed-loop solutions for spacecraft altitude (e 5 0.10)
guidance law is the analytical solution process that has been used. Consequently, several complications related to the numerical determination of control strategies, such as slow convergence, unexpected singularities, and high sensitivity to the initial guess, have been resolved. The other advantage of the NOC method is the use of an optimal approach for optimizing the problem performance measure. However, if the noise amplitude is increased to 10 per cent, the only method with good accuracy is the L1 adaptive strategy. This behaviour results from the adaptive and robustness characteristics of the new proposed strategy. As expected, with the L1 adaptive controller a smooth control signal is obtained. The two proposed methodologies can be utilized in realtime aerospace applications. Fig. 12
7
Comparison between NOC and L1 closed-loop solutions for spacecraft vertical velocity (e 5 0.10)
CONCLUSION
A closed-loop control law for a non-linear orbital injection problem has been achieved in the present study using two different approaches. The simulation results indicate that the NOC and L1 adaptive strategies have good accuracy for tracking the nominal trajectories when the noise amplitude is equal to 1 per cent. The main advantage of the NOC
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