SDRE Control Stability Criteria and Convergence Issues: Where Are We Today Addressing Practitioners’ Concerns? Quang M. Lam1 Orbital Sciences Corporation Dulles, VA 20166 Ming Xin2 Mississippi State University, Mississippi State, MS 39762
James R. Cloutier3 Network Sensing Technologies, Bldg 13, Suite 212, Eglin AFB, FL 32542-6810
This paper has three purposes: 1) to provide a survey on the State-Dependent Riccati Equation (SDRE) stability analysis methodologies developed to date; these stability analysis techniques produce either a guarantee or a high degree of confidence that the closed-loop system is asymptotically stable over a domain of interest, 2) to present an argument that practical rules of thumb can be just as important as theoretical stability proofs with regard to real world implementation, and 3) to justify support of the above argument using some forms of actual implementation. The paper neither favors any particular stability analysis technique nor introduces a new stability analysis framework. Rather it presents a view of stability reasoned judgment and practical justification for actual implementation. The reasoned judgment and justification are mainly based on the space access vehicle control and the satellite attitude control examples whose performance becomes unstable in the presence of high gain magnitude conditions. The fundamental argument of this new view (i.e., reasoned judgment and justification) is that for even linear and static gain controllers such as the Linear Quadratic Regulator (LQR), stability of the system depends on the operational domain under practical implementation with nonlinear limiters in the loop. Therefore, for a variable gain or nonlinear controller like the SDRE method, the task of determining a region of stability either practically or theoretically is much more difficult as compared to a linear LQRbased approach. The SDRE designs of a space access vehicle control, a helicopter flight control, and a satellite attitude control are presented with a set of practical rules defined and applied to it as a practical design benchmark problem on how to pass the SDRE stability concern gate. The paper concludes with the recommendation of some practical design rules of thumb for practitioners
I. INTRODUCTION
It is no doubt that control designers always have their own respective preference in selecting a class of modern, ease of implementation, and robust controllers for their own control problem. There are problems that simply need the traditional LQR based static gain controller and they will perform just fine even though the actual system is nonlinear and non-stationary in some cases. How that can be accomplished is to perform the linearization process 1
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per operating condition and repeat that process throughout the entire performance envelope of the system to obtain multiple sets of LQR based controllers. These multiple sets of LQR based controllers are then implemented using the gain scheduling approach to address the performance of the vehicle at the system level. The drawback of the gain scheduling approach is that if the system behaves totally out of the predicted envelope (used during the linearization process), the vehicle will exhibit some performance degradations. Furthermore, the gain scheduling approach will not be suitable for the system or vehicle that reflects drastic variations and large parameter uncertainties which are practically unaccountable during the design process. It is also obvious that LQR based design is not capable of addressing adaptive reconfigurable control systems subject to sensors/actuators failures. There is no such thing called one size fits all, especially in solving challenging control problems across all industrial sectors (e.g., aircraft, missile, satellite, robotics, etc) and addressing various performance characteristics like nonlinear system control, robustness, optimality, adaptation to accommodate system/parameter variations, to name a few. However, the emerging SDRE class of controllers tends to approximate the one-size-fit-all category. Generally, the SDRE controller can eliminate the gain scheduling requirement and can offer many other performance characteristics that other controllers won’t be able to. Those characteristics include but are not limited to (i) optimality; (ii) nonlinear control; (iii) adaptive control; (iv) robustness; and (v) reconfigurability (subject to sensor/actuator failures). The SDRE based control design techniques have recently received a lot of attention within the Guidance, Navigation, and Control (GN&C) community due to its sophisticated capabilities. The extensive survey conducted by Cimen in 2009 definitely provides a good historical review of SDRE design strategies along with its applications to a variety of platforms across all industries. The survey also provides optimality conditions for the State Dependent Coefficient (SDC) along with the existence of the SDRE solution and its stability. This paper is not meant to repeat the enormous mission that Cimen has achieved in his SDRE control survey mentioned above. Rather, it focuses more on the stability analysis side of the technique, brings out some independent views judging SDRE’s stability behaviors, and provides practitioners some rules of thumb to get the SDRE controller into a practical implementation form so it can be confidently employed. The rest of the paper is organized as follows. Section II provides a survey on existing SDRE stability analyses with the focus on capturing imminent approaches. Section III discusses the validity of these stability analyses and presents some shortcomings of those approaches. Section IV introduces a different view on how to cope with those shortcomings described in Section III and provides guidelines on how to circumvent those situations to get the SDRE controller into the implementation stage. Section V provides new performance characteristics of the SDRE in coping with control actuator failures and an introduction of its ability to accommodate for the under-actuated control performance situation. Finally, Section VI wraps up the survey and offers some concluding remarks as going forward directions for actual design and implementation of the SDRE controller.
II. STABILITY ANALYSIS METHODOLOGIES A. The Star Lyapunov Approach Shamma and Cloutier (2001) presented an approach using the S-procedure for quadratic forms to arrive at a conclusion that if there exists any stabilizing feedback controller leading to a Lyapunov function with star shaped level sets, then there always exists a state dependent representation of the dynamics such that the SDRE approach is stabilizing. One important fact that Shammar and Cloutier pointed out in their preliminary SDRE stability investigation and we would like to emphasize here and repeat it throughout the entire paper, is that SDRE stability existence can be achieved by choice if designers select their state dependent coefficient (SDC) structure properly (due to its non-uniqueness) by “placing” their state dependent elements of x(t) (e.g., see example 1.1 in Shammar and Cloutier, 2001 of two equivalent representations by choice) in the appropriate matrix-indexed elements of the SDC pair [A(x(t)), B(x(t))]. The proposed approach is then followed up in 2003 by the same authors to indicate that there always exists a state dependent representation of the open loop state dynamics such that, (i) the closed-loop dynamics matrix in a state-dependent representation is pointwise Hurwitz; (ii) an alternative stabilizing feedback can be found via solving a state-dependent Riccati equation.
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The above approach requires the knowledge of a star Lyapunov function which can be difficult to find. Thus the approach in most cases is not practical. However, it supplies theoretical support to explain the observation that if a nonlinear technique can stabilize a system over a domain of interest, an SDRE controller can be designed to stabilize that system over the same domain. Specifically, there has yet to be a system presented that could be satisfactorily controlled by some nonlinear technique and not satisfactorily controlled by the SDRE method, even a problem containing an unstabilizable but bounded state. See the “Toy Nonlinear Control Problem” discussed in Hull et al (1998).
B. Vector Norm Based Approach or Region of Attraction As noted in Cloutier (1997), the number of successful applications have outpaced the available theoretical results and very little is known about the stability properties associated with SDRE controllers. While global stability has not yet been shown, local asymptotic stability can be shown in Cloutier, D’Souza, and Mracek (1996). However, a method for estimating a Region of Attraction (ROA) did not appear in the literature until Erdem and Alleyne (2002) began to establish some formalism to define the Region of Attraction (ROA). ROA can be found by computing three vector norm sets determined by the eigenvalues and the eigenvectors of the overvaluing matrix. Bracci, Innocenti, and Pollini (2006) extended this work by using a scalar Lyapunov function to determine the ROA, which was shown to be less conservative. Both methods require a grid of the state space and thus can be computationally intensive for large systems. In general, it is very difficult to claim Global Asymptotic Stability (GAS) for any class of controllers (and more specific arguments about this point will be provided in Section 3 below) let alone the SDRE controller whose control performance characteristics are classified as equivalent to indirect adaptive control (e.g., see Lam, Xin, and Cloutier, 2011). Erdem and Alleyne (2004) showed the GAS of the SDRE method for a class of second-order nonlinear systems by a convenient parameterization of the A(x) matrix. Analytical optimal control law can be also obtained. It is worth pointing out here that at least the ROA approach, if well investigated per specific application, can be embraced as the right technique to claim GAS victory. For instance, the work done by Shankar, Yedavalli, and Doman (2003) when applied to several aerospace control examples, establishes a good rule of thumb for getting the ROA methodology combined with the Constant Overvaluing Matrix to produce a stability assessment. One notable research by McCaffrey and Banks (2001) proposed a stability test for determining the size of the region on which large-scale asymptotic stability holds for the SDRE algorithm by using the geometrical construction of a viscosity-type Lyapunov function. The stability region estimates for the SDRE feedback are much closer to the true domain of attraction than the conservative estimates in the existing literature.
C. Control Lyapunov Function (CLF) Based Approach CLF approach has been known and recognized as a powerful method to analyze the stability of feedback systems in general. CLF coupled with SDRE has been employed by Pukdeboon and Zenobar (2009) to analyze the stability of the SDRE control system. Sontag [11] has shown that if a CLF is known for a nonlinear system that is affine in the control, then the CLF and the system equations can be used to find controllers that make the system asymptotically stable.
D. Exponential Stability by Contraction Region Chang and Chung (2009) have recognized that SDRE stability usually only exists in a local region and at its best will be the ROA which is the expanded local region of stability. They have come up with a new strategy called contraction analysis to estimate the exponential stability region for the SDRE controlled system. Their proposed method tends to outperform others because (i) it is based on exponential stability rather than asymptotic stability and (ii) it is numerical based approach with a fast convergent rate. The approach mainly relies on the contraction theory, which is a relatively new and very powerful tool for proving exponential stability of the nonlinear systems. By applying contraction analysis to the SDRE controlled systems, the exponential stability of the system is guaranteed whereas most of the aforementioned approaches guarantee only asymptotic stability.
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III. VALIDITY OF EXISTING SDRE STABILITY The main argument of this section is that we follow the rule and principle to design our controller and evaluate the controller subject to all practical conditions using highest possible fidelity of all applicable elements existing in our applied system. We then implement it at our best confidence knowing that all worst case conditions were examined and employed during the design and evaluation. “Knowing your worst case conditions” rule mentioned above is strongly recommended because stability theory vs state of practice ought to be balanced and judiciously judged. For instance, the SDRE stability theory developed by Banks (1992), there are flaws in the stability proof (as pointed out by Erdem (2001)). Erdem in 2001 also brings out some good discussions such as having the eigenvalues of the closed loop A(x)-B(x)K(x) matrix in the left-hand plane for all x is sufficient to deduce globally asymptotic stability but that condition provides no stability guarantee to the overall system performance at all times. The stability proof was also shown to be incorrect by a counterexample by Tsitoras, et al (1996). The example is re-used here for illustration purpose to state that the system has a finite escape time even though the system has negative eigenvalues everywhere, and thus the system is not globally asymptotically stable. Example 1: Consider the following nonlinear system
x 1 x1 x12 x 2 x 2 x 2
(1)
The above equation is rewritten in the SDC format as follows,
x A(x)x where
(2)
- 1 x12 A(x) 0 1 Where the eigenvalues of A(x) are all negative (i.e., -1) With the initial state of (x1(0),x2(0)=(2,2)), a routine calculation shows that for t belonging to [0, T c], with Tc=ln√2, the solutions are,
x1 (t )
2 x2 (t ) t and x2 (t ) 2e 2 x2 (t ) 2
(3)
With the above solution, at t=T c, x1(t) is undetermined or grows out of bound. Other key point of similar argument for this section is to provide practitioners an argument about the control system architecture reality wherein nonlinear elements exist due to either safety logic requirement or physical characteristics of an actuator (e.g., see Lam and Barkana, 2005). Because of these nonlinear elements (e.g., rate limiter, attitude limiter, actuator saturation, e.g., see Fig. 1) inherently residing in the control system, even LQR based controller performance and its proven stability become questionable. The results analyzed in Lam and Barkana, 2005 is re-used here to illustrate the LQR based (with PID feedback structure) performance degradation due to the input saturation when the state vector amplitude becomes large under degraded flight conditions. Fig. 2 (from Lam and Barkana, 2005) illustrates that the fixed gain, static, non-adaptive LQR based controller is unable to handle the operational flight condition variation when the state vector amplitude becomes increasingly large and at the 5th curve with large rate (i.e., 5 times of the normal rate), the LQR based controller loses its control stabilization. The above reality discussion implies that in practical implementation, stability of even the proven LQR based controller may not even hold.
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Fig. 1: Flight Control System Architecture With Nonlinear Element Existing in the Loop
The degraded performance of the LQR under the nonlinearity effect presented above can be ameliorated and accommodated by using either direct adaptive control or an SDRE/Theta-D controller (see Lam, Drake, and Ridgely 2007). Fig. 3 below illustrates that the SDRE/Theta-D controller can in fact act as an adaptive controller coping with various input torque magnitudes applied to the spacecraft attitude control system.
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Fig. 2: LQR Based Controller Degradation Due to Large Input State Vector
Fig. 3: SDRE Adaptation Capability for Handling Input Saturation 6 American Institute of Aeronautics and Astronautics
IV. SDRE IMPLMENTATION RULES AND REASONED STABILITY ASSESSMENT SDRE stability will be primarily dictated based on three related conditions: (1) point-wise controllability of the pair [A(x(t)), B(x(t))]; (2) bounded behavior of the state x(t) within its operational domain (which needs to be examined and verified by designers during the design stage regardless the nature of their specific application or problem); (3) selection of the state dependent factorization form of the pair [A(x(t)), B(x(t))]. The third condition is very important because it dictates the first two conditions. Due to the non-uniqueness of the state dependent factorization, designers need to select a certain structure of the State Dependent Coefficient (SDC) to ensure that the pair [A(x(t)), B(x(t))] will meet the first condition point-wise. Otherwise, the state-dependent Riccati equation will not have a solution. In the case of weak controllability, poor performance would occur due to numerical instability of the solution of the Riccati equation. Likewise, if the boundedness of the state vector x(t) is not established under the worst case condition via the Lyapunov Stability Criteria (i.e., simulation based stable eigen-structure of the system verified throughout), then potentially, the other two conditions can be violated at any time. The recommended rules of thumb for actual implementation and employment of the SDRE control system are: Define your mission/vehicle operational performance envelope against customers’ specifications Expand that performance envelope by adding more margin for SDRE control to be tuned to Define multiple gates of testing and evaluation wherein each gate will have modelling fidelity. growth and maturation complexity milestones. Perform a thorough evaluation at the system level using the Monte Carlo simulation approach There will always be a leap of faith in going from the paper design to implementation, thus test the designs with safety procedures handy and validate the SDRE technology insertion before putting it into real service The aforementioned items are suggested guidelines only. Practitioners should check with customers’ rules and regulations to derive actual test procedures at various gates of the process.
V. SDRE APPLICABILITY TO ACTUATOR FAILURES AND UNDER-ACTUATED CONTROL SDRE control methodology has been applied as an adaptive controller to accommodate the vehicle performance under extreme and drastic operating conditions like actuator failures. Lam and Oppenheimer (2010) have investigated the SDRE design as a combined Guidance and Control subsystem to address loss of various control surfaces (at various combinations) for a space access vehicle during the re-entry phase. The lesson learned from this investigation concerning stability is that we test and evaluate the SDRE design and performance essentially on a case by case basis. Stability evaluation to safeguard the pass-fail criteria was achieved using the simulation based approach which is considered the best path due to the complexity and high dimensionality of the system. The performance results of that investigation are re-used here to encourage practitioners to delve into SDRE control as a potential and effective design method for adaptive reconfigurable flight control. Fig. 4 presents the SDRE roll-axis controller performance of the Space Access Vehicle (SAV) under the loss of both left and right elevons (control surfaces). Note that a Dynamic Inversion (DI) based controller design was employed for comparative evaluation purpose. The DI-based controller was not able to handle such a degraded flight condition. Note that the dual-loop SDRE controller structure design employed by the SAV (Lam and Oppenheimer, 2010, Fig. 6a) was later used by Bogdanov (2010) to perform a control decoupling for a helicopter flight control system (FCS), shown in Fig 6b for helicopter state vector definition and configuration. The SDRE based dual-loop design configuration was again shown to outperform the single loop design configuration in the helicopter FCS application. Bogdanov in that same study also illustrated the SDRE adaptive controller’s inherent ability to provide adaptive treatment to an under-actuated control problem.
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Fig. 4: SAV Roll Axis Performance Subject to Stuck Left and Right Elevons Flight Condition. Similar great performance of the SDRE based controller is also observed for both pitch and yaw axes as presented in Figs. 5 and 6.
Fig. 5: SAV Pitch Axis Performance Subject to Stuck Left and Right Elevons Flight Condition. 8 American Institute of Aeronautics and Astronautics
Fig. 6: SAV Yaw Axis Performance Subject to Stuck Left and Right Elevons Flight Condition.
Fig. 6a: Dual-Loop SDRE Guidance and Control for SAV
The outer loop SDRE SDC of the SAV is described as follows, XOL=[ V I I]T d X OL (t) A OL (X OL (t))XOL (t) BOL (X OL (t)) u OL (t) dt Where aOL12 0 aOL14 0 aOL11 a aOL 22 0 aOL 24 0 OL 21 AOL 0 1 0 0 0 aOL 42 0 aOL 44 aOL 45 0 0 0 0 1 0 9 American Institute of Aeronautics and Astronautics
(4)
(5)
1 Fx ; aOL12 Fz ( tan( ) ) mV V m Fz 1 V aOL14 tan( ) Fy ; aOL 21 mV 2 mV FxV sin(2 ) ; Fz sin 2 ( ) aOL 22 aOL 24 mV 2 mV aOL11
aOL 42 0.5q aOL 45 0.5r
BOL
1
sin( ) tan( ) ; aOL 44 0.5 p
1
(6)
1
cos( ) tan( )
I
1 0 p g sin( ) c 2 t 1 0 0 0.5 0.5 s t 0 0
s c t 0 0.5 c t 0 0
(7) where is angle of attack, is roll angle, is pitch angle, V is airspeed, c=cosine, s=sine, t=tan, m is the vehicle mass, and the subscript I represents the integral of that variable. The inner SDRE SDC Model is described as follows, XIL=[p q r | s | p q r]T (8) The first 3 elements represent the angular body rate of the SAV while the last four elements represent actuator dynamics. d X IL (t) A IL (X IL (t))XIL (t) B IL (X IL (t)) u IL (t) dt
AIL (1 : 3,1 : 3) I 1S(I b (t))
(9)
where b=[p q r]T (angular body rate vector) I and I-1 are the inertia matrix and its inverse, respectively. S is the skew symmetric matrix defined as follows:
0 S (u ) u (3) u (2)
s 0 AIL (4 : 7,4 : 7) 0 0
u (3) 0 u (1)
s -s s u (2) p -p ( p - u p ) u (1) ; -q ( q - u q ) 0 q r -r ( r - u r )
0
0
s
0
0
q
0
0
1 0 0 0 0 B 0 0 IL 0 r 0 0
0 1 0 0 0 0 0
(10) 0 0 1 0 0 0 0
(11)
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With sp, q,r are design parameters to be selected to meet the dual loop interaction performance.
p des
uthrust
Outer Loop SDRE
v des
u attitude
q x
Inner Loop SDRE
des
Outer loop comp.
Vehicle dynamics
x
Inner loop comp.
x
Fig 6b: Dual-Loop SDRE for Helicopter FCS Bogdanov (2010) compared the helicopter FCS performance under the SDRE based dual loop configuration to the single loop SDRE controller performance. We use his work here to illustrate the dual loop performance effectiveness for both the nominal operating condition and the under-actuated operating condition. Figs 7a, 7b, and 7c present the performance comparison of single loop vs dual-loop for position, velocity, and altitude command tracking. 5 dual-loop single
4.5 4
Position error, m
3.5 3 2.5 2 1.5 1 0.5 0 15
20
25
30
35
40 45 Time, s
50
55
Fig. 7a: Position Command Tracking Error
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60
65
70
Fig. 7b: Velocity Command Tracking Error 1
0.5
Altitude error, m
0
-0.5
-1
-1.5
-2 10
dual-loop single 20
30
40
50
60
Time, s
Fig. 7c: Altitude Command Tracking Error
Table 1: Performance Comparison of Single Vs Dual 12 American Institute of Aeronautics and Astronautics
70
80
The numerical performance metric of single loop SDRE vs dual loop SDRE is summarized in Table 1 above. The main point for citing Bogdanov’s work here is to bring out one important fact that the SDRE design methodology along with its varying modelling structure (via the choice of state dependency selection, e.g., see Shamma and Cloutier (2001) for non-uniqueness of SDC model) does offer a “hidden” robustness and an inherent adaptive reconfigurable control capability dealing with both internal and external dynamic changes to the vehicle’s operating conditions. The under-actuated control application of SDRE design was also explored by Lam and Xin (2010) for a commercial GEO satellite pointing control evaluation. The study is not yet completed; however, the early results are very encouraging. The SDRE/Theta-D controller, shown in Figs. 8, 9, and 10, is able to maintain the satellite pointing accuracy subject to loss of two reaction wheels out of four mounted in a pyramid configuration (again for various failures combinations).
Fig. 8: Roll Axis Performance Subject to Loss of RWs 1 and 3 and Under Various Initial Conditions of Yaw Attitude Error
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Fig. 9: Pitch Axis Performance Subject to Loss of RWs 1 and 3 and Under Various Initial Conditions of Yaw Attitude Error
Fig. 10: Yaw Axis Performance Subject to Loss of RWs 1 and 3 and Under Various Initial Conditions of Yaw Attitude Error.
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The adaptation characteristics of the SDRE controller (under the loss of two RWs 1 and 2) is illustrated in Figs. 11a, 11b, and 11c for roll, pitch, and yaw axes, respectively (row 1 for roll axis, row 2 for pitch axis, and row 3 for yaw axis). The entire SDRE gain matrix (i.e., 3x9) are dynamically recomputed in real time to maintain the stability of a closed-loop system in the Lyapunov sense, i.e., all eigenvalues of respective states are in the left half plane throughout the entire period as shown in Figs. 12a, 12b, and 12c). Note that the SDRE gain adaptation is required nominally for the first 230 seconds to fight the new dynamics variation of the new situation. After that, they all reach their own respective steady state values and behave just like a LQR based controller. SDRE Adaptive Gain (Row 1), Failed RWs 1 and 2x 10-3 1.44 10
K13
1.4
0
1.38 1.36
0
5 K12
K11
1.42
0.02
0
200
400
-5
600
200
-0.02 -0.04
0
200
400
600
-0.06
2
K16
K14
K15
200
400
-1
600
K18
K17
1360.5
1360
0
200
400
600
0
200
400
600
-60
6
4
4
3
2
-2
0
200
400
600
0
200 400 seconds
600
-40
0 1359.5
600
-20
K19
0
400
0
0 190
200
20
1 195
0
2 1
0
200
400
600
0
Fig. 11a: SDRE Adaptive Gain History, Row 1
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SDRE Adaptive Gain (Row 2), Failed RWs 1 and 2 0.4 0.8
0.2 0.1 0
200
400
0.4
600
0 0
-0.05
500
140
30
0
200
400
80
600
8
0
-50
500
1465
0
200
400
600
0
200 400 seconds
600
6
4
K29
4 K28
K27
600
0
6 1460
2
2 0
400
50
100
10
200
100
120
20
0
150
K25
K24
0.05
0.5
40
0
0.6
K26
0
0.1 K23
0.7 K22
K21
0.3
0.15
0
200
400
600
1455
0
500
Fig. 11b: SDRE Adaptive Gain History, Row 2
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0
SDRE Adaptive Gain (Row 3), Failed RWs 1 and 2 0.3 0.2
0.1
0
-0.6
500
40
K35
K34
20 0 -20
0.1
500
50
200
0
180
-100 0
-150
500
0
0
0
500
0
500
160
120
500
1177
0
-5
500
0
140
5
K38
K37
0
-50
5
-5
0.15
-0.4
K39
-0.1
-0.2
K36
0
K33
0
K32
K31
0.2
0.2
0
1176.5
1176
500
0
500 seconds
Fig. 11c: SDRE Adaptive Gain History, Row 3 SDRE/LQR EIGENVALUES, Failed RWs 1 and 2
lamda1
-5
-10
-15
0
50
100
150
200
250 seconds
300
350
400
450
500
0
50
100
150
200
250 seconds
300
350
400
450
500
0
50
100
150
200
250 seconds
300
350
400
450
500
lamda2
-7 -7.5 -8 -8.5
lamda3
-8 -10 -12 -14
Fig. 12a: Closed-Loop Stable Eigenvalues (States 1 to 3)
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SDRE/LQR EIGENVALUES, Failed RWs 1 and 2
lamda4
0 -0.1 -0.2 -0.3
0
50
100
150
200
250 seconds
300
350
400
450
500
0
50
100
150
200
250 seconds
300
350
400
450
500
0
50
100
150
200
250 seconds
300
350
400
450
500
lamda5
-0.1
-0.15
-0.2
lamda6
0 -0.05 -0.1 -0.15
Fig. 12b: Closed-Loop Stable Eigenvalues (States 4 to 6)
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SDRE/LQR EIGENVALUES, Failed RWs 1 and 2
lamda7
0
-0.005
-0.01
0
50
100
150
200
250 seconds
300
350
400
450
500
0
50
100
150
200
250 seconds
300
350
400
450
500
0
50
100
150
200
250 seconds
300
350
400
450
500
lamda8
0
-0.005
-0.01
lamda9
0
-0.05
-0.1
Fig. 12c: Closed-Loop Stable Eigenvalues (States 7 to 9)
VI. Conclusion The paper provides self-contained information on the assessment of SDRE stability and how practitioners should view future SDRE control applications. Highlights of imminent stability analysis approaches were also provided to discuss their pros and cons. Independent viewpoints and practical arguments for stability concerns were stated to guide practitioners to step into the implementation phase along with rules to secure the safe employment and operations of the SDRE controller. The SDRE controller has produced consistent performance across all applications compiled in this paper (i.e., from helicopter to SAV to satellite pointing control). Thus, if a practitioner desires good adaptation performance in the face of degraded operating conditions, such as the loss of actuators and/or under-actuated control, there is substantial evidence that suggests that the SDRE controller should be the controller of choice.
References 1
Alleyne, A.G. and Erdem, E.B. (2002), “Estimation of Stability Regions of SDRE Controlled Systems Using Vector Norms” Proceedings of the American Control Conference 2 Bank, S.P. and Manha, K. J. (1992), Optimal Control and Stabilization for Nonlinear Systems, IMA Journal of Mathematical Control and Information, Vol. 9 pp. 179-196. 3 Bogdanov, A., (2010), “Dual-Loop Augmented State Dependent Riccati Equation Control for a Helicopter Model,” AIAA InfoTech Conference, Atlanta, GA.
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