RESEARCH INTO COORDINATING SENSING, CONTROL AND ...
Transcript of RESEARCH INTO COORDINATING SENSING, CONTROL AND ...
AFRL-RV-PS- TR-2019-0102
AFRL-RV-PS- TR-2019-0102
RESEARCH INTO COORDINATING SENSING, CONTROL AND COMPUTATIONS FOR AUTONOMOUS VEHICLES
Ilya Kolmanovsky and Anouck Girard
Regents of the University of Michigan Office of Research and Sponsored Projects 503 Thompson Street Ann Arbor, MI 48109-1340
30 August 2019
Final Report
APPROVED FOR PUBLIC RELEASE; DISTRIBUTION IS UNLIMITED.
AIR FORCE RESEARCH LABORATORY Space Vehicles Directorate 3550 Aberdeen Ave SE AIR FORCE MATERIEL COMMAND KIRTLAND AIR FORCE BASE, NM 87117-5776
NOTICE AND SIGNATURE PAGE
Using Government drawings, specifications, or other data included in this document for any purpose other than Government procurement does not in any way obligate the U.S. Government. The fact that the Government formulated or supplied the drawings, specifications, or other data does not license the holder or any other person or corporation; or convey any rights or permission to manufacture, use, or sell any patented invention that may relate to them.
Qualified requestors may obtain copies of this report from the Defense Technical Information Center (DTIC) (http://www.dtic.mil).
AFRL-RV-PS-TR-2019-0102 HAS BEEN REVIEWED AND IS APPROVED FOR PUBLICATION IN ACCORDANCE WITH ASSIGNED DISTRIBUTION STATEMENT.
//SIGNED// DAVID WILT, Ph.D
//SIGNED// CHRISTOPHER PETERSEN Program Manager Technical Advisor, Space Based Advanced Sensing
And Protection
//SIGNED//JOHN P. BEAUCHEMIN Chief Engineer, Spacecraft Technology Division Space Vehicles Directorate
Approved for public release; distribution is unlimited
REPORT DOCUMENTATION PAGE Form Approved OMB No. 0704-0188
Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing this collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden to Department of Defense, Washington Headquarters Services, Directorate for Information Operations and Reports (0704-0188), 1215 Jefferson Davis Highway, Suite 1204, Arlington, VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to any penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number. PLEASE DO NOT RETURN YOUR FORM TO THE ABOVE ADDRESS. 1. REPORT DATE (DD-MM-YYYY)30-08-2019
2. REPORT TYPEFinal Report
3. DATES COVERED (From - To)14-Feb-2018 β 30-Aug-2019
4. TITLE AND SUBTITLEResearch Into Coordinating Sensing, Control and Computations for AutonomousVehicles
5a. CONTRACT NUMBER
5b. GRANT NUMBER FA9453-18-1-0010 5c. PROGRAM ELEMENT NUMBER 62601F
6. AUTHOR(S)Ilya Kolmanovsky and Anouck Girard
5d. PROJECT NUMBER 8809 5e. TASK NUMBER PPM00036957
5f. WORK UNIT NUMBER EF130350 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)
AND ADDRESS(ES)8. PERFORMING ORGANIZATION REPORT
NUMBERRegents of the University of Michigan Office of Research and Sponsored Projects 503 Thompson Street Ann Arbor, MI 48109-1340
9. SPONSORING / MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITORβS ACRONYM(S)Air Force Research LaboratorySpace Vehicles Directorate AFRL/RVSV 3550 Aberdeen Ave, SE 11. SPONSOR/MONITORβS REPORTKirtland AFB, NM 87117-5776 NUMBER(S)
AFRL-RV-PS-TR-2019-0102 12. DISTRIBUTION / AVAILABILITY STATEMENTApproved for public release; distribution is unlimited.13. SUPPLEMENTARY NOTES 14. ABSTRACTThe report addresses several strategies developed for coordinating sensing, estimation, control and computations in systemsthat must operate autonomously and satisfy constraints. The potential of the proposed algorithms is demonstrated inspacecraft simulation case studies. The developed algorithms and strategies contribute to advancing capabilities ofautonomous vehicles, in particular, in the areas of handling constraints and non-standard sensing-estimation-actuationrequirements. The proposed strategies also facilitate incorporating optimization-based control technology for onboardoperation in autonomous vehicles.15. SUBJECT TERMSAutonomous Vehicles, Constrained Control Algorithms, Model Predictive (MPC), Spacecraft Control
16. SECURITY CLASSIFICATION OF: 17. LIMITATIONOF ABSTRACT
18. NUMBEROF PAGES
19a. NAME OF RESPONSIBLE PERSON
Christopher Petersen a. REPORTUnclassified
b. ABSTRACTUnclassified
c. THIS PAGEUnclassified Unlimited 25
19b. TELEPHONE NUMBER (include areacode)
Standard Form 298 (Rev. 8-98) Prescribed by ANSI Std. 239.18
Approved for public release; distribution is unlimited i
TABLE OF CONTENTS Section Page
List of Figures .................................................................................................................................ii ACKNOWLEDGEMENTS AND DISCLAIMER ........................................................................ iii 1. SUMMARY ................................................................................................................................1
2. INTRODUCTION.......................................................................................................................2
3. METHODS, ASSUMPTIONS, AND PROCEDURES ..............................................................2
3.1 Optimal Strategies for Disjunctive Sensing and Control ...................................................... 2 3.1.1 Disjunctive Sensing and Actuation for Linear Systems. ............................................................... 2 3.1.2 Admissible Periodic Switching Sequences. .................................................................................. 3 3.1.3 Chance Constraints. ..................................................................................................................... 6 3.1.4 Selecting an Optimal Admissible Sequence. ................................................................................ 7
3.2 Computational Reference Governors for Safe Online Optimization .................................... 7 3.2.1 Assumptions.................................................................................................................................. 8 3.2.2 Methods........................................................................................................................................ 9 3.2.2.1 Nonlinear Model Predictive Control........................................................................................ 9
3.2.2.2 A Computational Reference Governor for Suboptimal MPC. .................................................. 11 3.3 Constrained control framework using a virtual net of chance constraint admissible sets and output measurements only ...........................................................................................................13
4. RESULTS AND DISCUSSION ................................................................................................14 4.1 Disjunctive-Based Constrained Spacecraft Relative Motion Planning ............................... 14
4.1.2 Theoretical Results. .................................................................................................................... 14 4.2 A Computational Reference Governor for Safe Online Optimization ................................ 18 4.3 Constrained Control Framework Using a Virtual Net of Chance Constraint Admissible Sets and Output Measurements Only.................................................................................................20
5. CONCLUSION ...........................................................................................................................21
6. REFERENCES............................................................................................................................22
LIST OF SYMBOLS, ABBREVIATIONS, AND ACRONYMS..................................................24
Approved for public release; distribution is unlimited ii
LIST OF FIGURES
Figure Page
Figure 1 A Comparison of Suboptimal MPC and Ideal MPC ............................................................... 11 Figure 2 Architecture of the Computational Reference Governor ......................................................... 12 Figure 3 Disjunctive Sensing and Control Trajectory Under a Length 8 Periodic Sequence ................ 16 Figure 4 Comparison of Length 4 and Length 7 Optimal Switching Sequences................................... 17 Figure 5 Demonstration of Satisfaction of Both Box and Chance Constraints ..................................... 18 Figure 6 The Time Histories of Control Torques, Angular Velocities and Euler Angles in Closed-loop
Simulation ....................................................................................................................................... 19 Figure 7 Closed-loop Traces of the Residuals and References with and without the CRG. ................. 20 Figure 8 Closed-loop Path of Spacecraft in CW Frame. Red Cone is the Obstacle and Black Dots are
the Reference Point Nodes in the Graph. ....................................................................................... 21
Approved for public release; distribution is unlimited iii
ACKNOWLEDGEMENTS AND DISCLAIMER
Acknowledgement This material is based on research sponsored by Air Force Research Laboratory under agreement number FA9453-18-1-0010. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon.
Disclaimer The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of Air Force Research Laboratory or the U.S. Government.
Approved for public release; distribution is unlimited 1
1. SUMMARY
The objective of the research was to advance algorithms for coordinating sensing, control and computations in systems that must operate autonomously and satisfy constraints. The focus was on situations where sensing, estimation, control and computations are inherently coupled and on developing strategies that are effective and could be straightforward to implement in real-world applications.
Specific progress has been made in the following areas:
1. Developing optimal periodic sequences of sensing and actuation actions in situations wheresensing and actuation cannot be performed simultaneously,
2. Establishing a controller that guarantees the boundedness of state and state estimationerrors and the eventual satisfaction of the imposed probabilistic chance constraints,
3. Demonstrating the effectiveness of this controller in simulations of an autonomousspacecraft application,
4. Developing a computational reference governor for safe online optimization.Demonstrating the potential of the computational reference governor in simulations whenpaired with a nonlinear model predictive controller,
5. Extending the virtual net constrained motion planning framework to the case when onlyoutput measurements are available, there is random measurement and process noise, andthe states of the system are estimated. Demonstrating the potential of the proposedapproach for spacecraft maneuvering while avoiding obstacle collision.
The following are publications which have appeared, been accepted or are in preparation related to the subject matter of this report:
β’ Sutherland, R., Kolmanovsky, I., Girard, A., Leve, F., Petersen, C., βOptimal strategies fordisjunctive sensing and controlβ [1],
β’ Sutherland, R., βAdvances in disjunctive and time-optimal predictive control methods,β[15],
β’ Berning, A., Kolmanovsky, I., Girard, A., Leve, F., and Petersen, C., βManeuverplanning using chained chance constrained admissible sets,β [14],
β’ Sutherland, R., Kolmanovsky, I., Girard, A., Leve, F., Petersen, C., βMinimum-time modelpredictive spacecraft attitude control for waypoint following and exclusion zoneavoidanceβ [2],
β’ Sutherland, R., Kolmanovsky, I., Girard, A., Leve, F., and Petersen, C., βOn closed-loopLyapunov stability with minimum-time MPC feedback laws for discrete-time systems,β[13]
2. INTRODUCTION
A key consideration in the design of control schemes for autonomous vehicles in general, and for spacecraft in particular, is the enforcement of constraints. Control actuation limits and pointwise in time and terminal state constraints must be satisfied. In this report, we summarize several research developments and outcomes in developing strategies for control of systems with constraints in situations when there is an interplay between sensing, actuation, estimation, control and computations. In particular, a non-standard setting of disjunctive sensing and control, when simultaneous actuation and sensing is not possible will be addressed. Furthermore, the notion of computational reference governor (CRG) to monitor and supervise the computational optimization within the nonlinear model predictive controller (NMPC) will be introduced and illustrated with simulation results. Finally, the previously proposed framework for motion planning/maneuvering based on chained invariant sets will be extended to the setting of output measurements and state estimation.
3. METHODS, ASSUMPTIONS, AND PROCEDURES
3.1 Optimal Strategies for Disjunctive Sensing and Control
3.1.1 Disjunctive Sensing and Actuation for Linear Systems.
Consider a system represented by a discrete-time linear model with stochastic state disturbance and measurement noise inputs, given by
π₯π₯ππ+1 = π΄π΄π₯π₯ππ + π΅π΅π’π’ππ + π€π€ππ, π¦π¦ππ = πΆπΆπ₯π₯ππ + π£π£ππ,
(1)
where π₯π₯ππ is a vector state and π’π’ππ is a vector control. The variables π€π€ππ and π£π£ππ are, respectively, the state disturbance and the measurement noise inputs, each assumed to be a sequence of zero-mean, independent (and jointly independent) identically distributed (i.i.d.) random variables with πΌπΌ[π€π€πππ€π€ππ
ππ] = Ξ£π€π€ = Ξ£π€π€ππ , πΌπΌ[π£π£πππ£π£ππππ] = Ξ£π£π£ = Ξ£π£π£ππ.
We consider a fixed-gain state feedback,
π’π’ππ = π’π’ππ + πΎπΎ(π₯π₯οΏ½ππ β π₯π₯ππ), (2)
where π₯π₯οΏ½ππ is the state estimate generated by a fixed-gain observer of the form
π₯π₯οΏ½ππ+1 = π΄π΄π₯π₯οΏ½ππ + πππππ΅π΅π’π’ππ + (1 β ππππ)πΏπΏ(π¦π¦ππ β πΆπΆπ₯π₯οΏ½ππ). (3)
The binary variable ππππ β {0, 1} represents the system operating mode. When ππππ = 1, the control is applied to the system. When ππππ = 0, the control is deactivated (π’π’ ππ = 0), and the sensed output π¦π¦ππ is obtained. The target equilibrium is denoted by π₯π₯ππ, and it is assumed that the feedforward control input, π’π’ , supports it in steady-s tate in the absence of π€π€ ππ and with ππ ππ = 1, i.e., π₯π₯ ππ = ππ + .
Approved for public release; distribution is unlimited 2
In a typical control design process, due to the Separation Principle, the gains πΎπΎ and πΏπΏ would be determined without consideration of the interference between actuation and sensing (possibly by different engineers) and then a coordination mechanism introduced by specifying ππππ, ππ β₯ 0. In this setting, offline-generated ππ-periodic switching sequences, {ππππ}, with ππππ = ππππ+ππ for all ππ ββ€β₯0, ππ β β€β₯0, are of particular interest, so that their repeated application leads to the attainment of the control objectives, including eventual satisfaction of state constraints. This approach, based on the application of the offline generated periodic sequence, has low computational footprint and is appealing in view of limited computing power and restrictive electrical power consumption budget onboard of small spacecraft.
3.1.2 Admissible Periodic Switching Sequences.
Suppose that {ππππ} is fixed and define οΏ½Μ οΏ½π΄ππ = π΄π΄ + πππππ΅π΅πΎπΎ, οΏ½ΜοΏ½π΄ππ = π΄π΄ + (1 β ππππ)πΏπΏπΆπΆ. The evolution of the state, π₯π₯ππ, and of the state estimation error, ππππ = π₯π₯ππ β π₯π₯οΏ½ππ, are driven by
π₯π₯ππ+1 = π΄π΄π₯π₯ππ + πππππ΅π΅πΎπΎπ₯π₯οΏ½ππ + π€π€ππ = οΏ½Μ οΏ½π΄πππ₯π₯ππ β πππππ΅π΅ππππ + πππππ΅π΅π’π’ππ + π€π€ππ,
(4)
and
ππππ+1 = [π΄π΄ + (1 β ππππ)πΏπΏπΆπΆ]ππππ + π€π€ππ + (1 β ππππ)πΏπΏπ£π£ππ = οΏ½ΜοΏ½π΄ππππππ + π€π€ππ + (1 β ππππ)πΏπΏπ£π£ππ.
(5)
In simulations, we will assume that π₯π₯ππ = 0 (and so π’π’ππ = 0). Based on the error dynamics and assumed independence and zero mean properties of the stochastic disturbance, π€π€ππ, and measurement noise, π£π£ππ, the error covariance matrix ππππ = πΌπΌ[ππππππππππ] satisfies
(6)
Definition: Let π΄π΄, π΅π΅, πΎπΎ, πΏπΏ, οΏ½Μ οΏ½π΄ππ, and οΏ½ΜοΏ½π΄ππ be defined as above, with none of οΏ½Μ οΏ½π΄ππ, οΏ½ΜοΏ½π΄ππ nilpotent. An ππ-periodic sequence of binary integers {ππ0, ππ1, β¦ , ππππβ1}, where, for all ππ β β€β₯0, ππππ β {0, 1} and ππππ+ππ = ππππ, is called admissible if the following contractivity conditions hold:
(7)
(8)
where ππ(β ) denotes the spectral radius operator. A non-periodic sequence of binary integers {ππ 0, ππ 1, β¦ } is admissible if
Approved for public release; distribution is unlimited 3
Approved for public release; distribution is unlimited 4
(9)
(10)
The constraint against nilpotent matrices ensures that the above limits approach zero rather than βjumpβ to zero. These definitions are consistent with the properties of discrete-time state transition matrices found in, for example, Chen [4]. Note that the existence of a non-periodic admissible sequence implies the existence of periodic admissible sequences.
In the existence of a periodic admissible sequence, the state and error dynamics coupled with πΌπΌ[π£π£ππ] = 0, πΌπΌ[π€π€ππ] = 0 imply that the state and estimation error mean, πππ₯π₯,ππ = πΌπΌ[π₯π₯ππ] and ππππ,ππ =πΌπΌ[ππππ], respectively, satisfy,
(11)
where οΏ½Μ οΏ½π΄ππ and οΏ½ΜοΏ½π΄ππ are periodic with the same period, ππ, as ππππ. The following results can be proven (see [1] for details):
Proposition 1: Suppose that the contractivity conditions hold. Then, as ππ β β, ππππ,ππ β 0 and πππ₯π₯,ππ β πππ₯π₯,ππ
π π exponentially, where {πππ₯π₯,πππ π } is the unique ππ-periodic solution of πππ₯π₯,ππ with ππππ,ππ β‘ 0.
Furthermore, if π’π’ππ = 0, then πππ₯π₯,πππ π = 0.
Proposition 2: Suppose that the second contractivity condition holds. Then, the error covariance matrix, ππππ, is bounded and, as ππ β β, converges to the unique ππ-periodic solution, {πππππ π }, with ππππ+πππ π = πππππ π . In addition, for any ππ β β€β₯0,
(12)
where
(13)
Remarks: 1) The steady-state periodic solution, {πππππ π }, can be computed by solving the conventional
discrete-time Lyapunov equation for the evolution of the lifted system error covariancematrix, i.e., of ππππ = diag{ππ0π π ,β― ,ππππβ1π π }.
2) limsupkββ βππππππβ β€ πΎπΎ / (1 β πποΏ½π΄π΄2).
3) The results in Propositions 1 and 2 generalize to non-constant ππ-periodic feedback andobserver gains. However, the analysis results benefit from both οΏ½Μ οΏ½π΄ππ and οΏ½ΜοΏ½π΄ππ having onlytwo possible values each, which is the case when the gains are constant.
We can take advantage of the dwell time conditions for stability analysis of hybrid systems to develop simpler sufficient conditions that can inform procedures for faster determination of admissible switching sequences. The discussion of the dwell time conditions follows Theorem 4.1 in Yu [5] and its proof.
Let Ξ©οΏ½0 = π΄π΄ and Ξ©οΏ½1 = π΄π΄ + π΅π΅πΎπΎ, and consider the first contractivity condition. By Gelfandβs theorem [6], limππββ βΞ©ππβ1\ππ = ππ(Ξ©), for any matrix Ξ© and norm ||β ||. This implies that there exist constants ππ0, ππ1 that do not depend on ππ such that for any ππ β₯ 1,
(14)
For any consistent matrix norm, there exists a finite ππβ such that οΏ½π΄π΄ππβοΏ½1/ππβ
β₯ βπ΄π΄ππβ1/ππ for all ππ β₯1. Then,
(15)
Let ππ = max {ππ0, ππ1}, ππ0 < ππ be the total time spent in the mode ππ = 0, ππ1 < ππ be the total time spent in the mode ππ = 1, and πππ π be the number of switches plus one. Then, the first contractivity condition holds if
(16)
Taking the logarithm of each side, it follows that this contractivity condition holds if
(17)
where οΏ½Μ οΏ½π0 = ππ(Ξ©οΏ½0) and οΏ½Μ οΏ½π1 = ππ(Ξ©οΏ½1). Similar analysis can be done in the case of the second contractivity condition to yield the complementary inequality
(18)
When seeking admissible sequences, it is possible to restrict the search to sequences for which ππ0, ππ1, and πππ π satisfy both logarithmic conditions, which together are sufficient (but not necessary) to ensure admissibility.
Approved for public release; distribution is unlimited 5
The search for admissible sequences can be made faster still by discarding sequences that replicate a known inadmissible sequence.
Definition: A sequence {π π ππ} of length ππ β β€ is called reducible if there exists a subsequence {π π ππ} of length ππ β β€>0 such that ππ < ππ, ππ is a multiple of ππ, and {π π ππ} = {π π ππ} β {π π ππ} ββ―β {π π ππ}, ππ/ππ times, where β is used to denote sequence concatenation. A sequence which is not reducible is called irreducible.
Proposition 3: Every (non-empty) sequence {π π ππ} contains a unique irreducible subsequence {π π ππ}.
Proposition 4: A binary sequence {π π ππ} is admissible if and only if the associated irreducible subsequence {π π ππ} is admissible.
Any reducible admissible sequence can be formed by propagating an irreducible admissible sequence forward in time. Thus, when investigating sequences of length ππ for admissibility, all sequences for which we have already evaluated the associated irreducible sub-sequence can be discarded.
3.1.3 Chance Constraints.
Consider the chance constraint,
(19)
where ππ is the set prescribed by the constraints of the problem, 0 β€ πΏπΏ < 1, and ππβ is sufficiently large. Define π§π§ = [π₯π₯ππ ππππ]ππ and ππ = [π£π£ππ π€π€ππ]ππ. Then, π§π§ππ+1 = οΏ½ΜοΏ½π΄πππ§π§ππ + ΞοΏ½ππππππ + πΊπΊοΏ½πππ’π’ππ, where
(20)
and πΊπΊοΏ½ππ = [ππππ πππππ΅π΅ππ]ππ. Under the contractivity conditions, we repeat the analysis in Propositions 1 and 2 for π§π§ππ+1. Let πποΏ½ππ = πΌπΌ οΏ½οΏ½π§π§ππ β πππ§π§,πποΏ½οΏ½π§π§ππ β πππ§π§,πποΏ½
πποΏ½, where πππ§π§,ππ = πΌπΌ[π§π§ππ]. Then,
(21)
ππ
and πποΏ½ππ β πποΏ½πππ π as ππ β β, where πποΏ½πππ π is the unique periodic solution to the above system. Then, the steady-state covariance matrix satisfies πποΏ½π₯π₯π π ,ππ = [ ππ ππ ]πποΏ½πππ π [ ππ ππ ]T. Thus, π₯π₯ππ converges to π₯π₯π π , which is a cyclostationary process with the ππ-periodic mean πππ₯π₯π π ,ππ and ππ-periodic covariance matrix πππ₯π₯π π ,ππ.
We have made no assumptions on the probability density functions of π€π€ππ and π£π£ππ, so to treat the chance constraint we resort to the multivariate Chebyshevβs inequality [7][8], which states
Approved for public release; distribution is unlimited 6
(22)
where πππ₯π₯ is the dimension of π₯π₯ and 0 < πΌπΌπ₯π₯ β€ 1. Given πΏπΏ > 0, choose πΌπΌπ₯π₯ = οΏ½nx/πΏπΏ. Suppose that for ππ = 0, β¦ ,ππ β 1, the following condition is verified:
(23)
where β(ππ, ππ) = {π₯π₯: π₯π₯πππππ₯π₯ β€ 1} is an ellipsoidal set and ~ denotes the Pontryaginβs set difference. Then, in steady state, the chance constraint holds.
3.1.4 Selecting an Optimal Admissible Sequence.
Since multiple admissible sequences may exist, we can select one by minimizing a cost functional. Consider the blended cost π½π½ that penalizes the estimation, the control objective, and the control effort:
(24)
where π π ππ = π π ππππ and π π π₯π₯ = π π π₯π₯ππ are positive semi-definite weighting matrices and ππππ β₯ 0 is a scalar weight. The ππβ1 factor in the cost functional normalizes for sequence length, ensuring that a given reducible/irreducible sequence pair will yield the same cost. To find the optimal sequence, fix ππ, check sequences for admissibility, then select the admissible sequence that yields the lowest cost. If no admissible sequence exists, then extend the sequence length and search again. The equivalence of admissibility between a sequence and its associated irreducible subsequence is invoked after incrementing the sequence length; if the subsequent search produces a reducible sequence for which the irreducible subsequence has already been evaluated, then the sequence can be skipped.
3.2 Computational Reference Governors for Safe Online Optimization
As spacecraft become more autonomous they will need increasingly complex online decision-making capabilities. One of the dominant paradigms for autonomous decision making is online optimization where actions are computed by solving a suitably formulated optimization problem. Optimization based frameworks have the major advantage that constraints can be naturally incorporated into the decision making process. In the context of control, the most common type of online optimization method is Model Predictive Control (MPC) wherein a control action is computed by solving a constrained Optimal Control Problem (OCP) at each sampling instance. If the dynamics are nonlinear, the OCP is necessarily non-convex and solving it in real-time is a significant challenge. In addition, the question of optimizer reliability is paramount.
Approved for public release; distribution is unlimited 7
Approved for public release; distribution is unlimited 8
Our research into safe online optimization has led to the development of a computational reference governor (CRG). By treating the optimizer as a dynamic system, we can impose constraints on it, e.g., that its optimality error be small, using the reference governor framework [9]. Our CRGessentially extends the continuous time computational reference governor developed in [3].
to the discrete time setting common in MPC applications. We have demonstrated the efficacy of the CRG for spacecraft orientation control benchmark problem and are in the process of developing the underlying mathematical theory.
3.2.1 Assumptions.
We consider a rigid spacecraft, actuated by three orthogonal thrusters. The orientation of the spacecraft is parametrized by 3-2-1 Euler angles. It is assumed that the attitude representation does not approach singularity; this is consistent with the limitations of the target physical testbed.
The equations of motion are given by
πΌπΌοΏ½ΜοΏ½π€+ π€π€ΓπΌπΌπ€π€ = π’π’ + ππ , οΏ½ΜοΏ½π = ππ(ππ)π€π€
(25)
where π€π€ is the vector of angular velocities, ππ are the Euler angles, ππ(ππ) is the inverse kinematic matrix, πΌπΌ is the moment of inertia matrix in a body-fixed frame, π’π’ are the control torques and ππ are unknown, possibly state dependent, disturbances. Collected into a compact form, the model is written as
οΏ½ΜοΏ½π₯ = ππππ(π₯π₯,π’π’,ππ), (26)
where π₯π₯ = [π€π€ππ ππππ]ππ .
The prediction model for NMPC is a discretized approximation of the continuous time model (using the Forward Euler integrator and zero order hold assumption) and is given by
π₯π₯ππ+1 = π₯π₯ππ + π‘π‘π π ππππ(π₯π₯ππ,π’π’ππ, 0), (27)
where π‘π‘π π is the discretization time step. The discrete-time prediction model is then considered in the following form,
π₯π₯ππ+1 = ππ(π₯π₯ππ,π’π’ππ). (28)
Approved for public release; distribution is unlimited 9
Note that since the disturbances are unknown, they cannot be incorporated into the prediction model and are assumed to be zero. At time π‘π‘ = πππ‘π‘π π , the state estimate is denoted by π₯π₯0 = π₯π₯οΏ½(π‘π‘).
3.2.2 Methods.
3.2.2.1 Nonlinear Model Predictive Control.
The NMPC optimal control problem (OCP) to be solved for the control to be applied, π’π’(π‘π‘) = π’π’0 has the following form,
minπ§π§
ππππππππππππππ + οΏ½οΏ½|ππππ|οΏ½ππ2
+ οΏ½|π’π’ππ|οΏ½π π 2
+ π π ππ+1
ππβ1
ππ=0π π . π‘π‘. π₯π₯ππ+1 = ππ(π₯π₯ππ ,π’π’ππ) + π₯π₯ππ β ππ(π₯π₯ππβ1,π’π’ππβ1), ππ = 0 β¦ππ β 1,
ππ(π₯π₯ππ) β€ π π ππ, ππ = 1 β¦ππ, ππ(π’π’ππ) β€ 0, ππ = 0 β¦ππ β 1,
π π ππ β₯ 0, ππ = 1 β¦ππ
(29)
where ππππ = π₯π₯ππ β ππ is the deviation from the target, π§π§ = [π’π’0ππ π₯π₯1ππ π’π’1ππ β¦ π₯π₯ππππ]ππ is the primal optimization variable, ππ is the length of the prediction horizon, ππ = ππππ β½ 0 is the terminal penalty, ππ = ππππ β½ 0 is the state penalty weight, π π = π π ππ β» 0 is the control penalty weight, and ππ,ππ are the stage state and control constraints respectively.
Note that the state constraints are softened using slack variables in order to ensure the problem is always feasible. The prediction model is written in rate-based form and will automatically compensate for low frequency unmodeled disturbances.
To solve the above OCP, we first write it in compact form as
minπ§π§
πΉπΉ(π§π§, ππ), π π . π‘π‘. ππ(π§π§, ππ) = 0,
ππ(π§π§) β€ 0, (30)
and introduce the Lagrangian,
πΏπΏ(π§π§, ππ, π£π£, ππ) = πΉπΉ(π§π§, ππ) + ππππππ(π§π§, ππ) + π£π£ππππ(π§π§), (31)
where the tuple ππ = (π₯π₯οΏ½(π‘π‘), ππ(π‘π‘)) collects the time varying parameters of the OCP and ππ and π£π£ are Lagrange multipliers. The following quadratic programming problem (QPP) is then solved at the ππth time-step
minβπ§π§
12βπ§π§πππ»π» βπ§π§ + ππππβπ§π§,
π π . π‘π‘. πΊπΊ π§π§ = β, π΄π΄ π§π§ β€ ππ,
(32)
Approved for public release; distribution is unlimited 10
where π»π» β βπ§π§2πΏπΏ(π§π§ππβ1, ππππβ1, π£π£ππβ1), ππ = βπ§π§πΏπΏ(π§π§ππβ1, ππππβ1, π£π£ππβ1), πΊπΊ = βπ§π§ππ(π§π§ππβ1), β = ππ(π§π§ππβ1),π΄π΄ = βπ§π§ππ(π§π§ππβ1), and ππ = βππ(π§π§ππβ1). The solution estimate is then updated as
π§π§ππ = βπ§π§β + π§π§ππβ1, (33)
and the multipliers are updated as
π£π£ππ = π£π£β, ππππ = ππβ, (34)
where (βπ§π§β, ππβ, π£π£β) is the solution of the QP. Under certain assumptions, this strategy generates a stabilizing feedback law and will converge to the optimum solution of the original nonlinear program [9]. The convergence of the SQP algorithm can be monitored using the natural residual of the KKT conditions
ππ(π₯π₯, ππ) = οΏ½οΏ½οΏ½βπ₯π₯πΏπΏ(π§π§, ππ, π£π£, ππ)
ππ(π₯π₯, ππ)min(βππ(π₯π₯),π£π£)
οΏ½οΏ½οΏ½. (35)
Written out explicitly, the feedback control law is computed by repeatedly solving QPs of the form:
minπ₯π₯,π’π’,π π
οΏ½βπ’π’βπ₯π₯οΏ½πποΏ½π»π»π’π’ 0
0 π»π»π₯π₯οΏ½ οΏ½βπ’π’βπ₯π₯οΏ½ + οΏ½πππ’π’πππ₯π₯
οΏ½πποΏ½βπ’π’βπ₯π₯οΏ½ + πΎπΎ11πππ π
π π . π‘π‘. π΄π΄βπ₯π₯ + π΅π΅βπ’π’ + ππ(οΏ½Μ οΏ½π₯,π’π’οΏ½) = 0 πΆπΆβπ₯π₯ + ππ(οΏ½Μ οΏ½π₯) β€ Ξπ π ππβπ’π’ + ππ(π’π’οΏ½) β€ 0
π π β₯ 0
(36)
where Ξ = πΌπΌππβ¨ 1ππ , πΆπΆ = βπ₯π₯ππ(οΏ½Μ οΏ½π₯), ππ = βπ’π’ππ(π’π’οΏ½), π΄π΄ = βπ₯π₯ππ(οΏ½Μ οΏ½π₯,π’π’οΏ½), π΅π΅ = βπ’π’ππ(οΏ½Μ οΏ½π₯,π’π’οΏ½), π»π»π₯π₯ =πππππππποΏ½πΌπΌππβ1 β¨ππ ,πππποΏ½, π»π»π’π’ = πΌπΌππ β¨π π .
To speed up computations we can condense the problem. Since for any sequence π’π’ the state sequence x is uniquely determined π΄π΄ must be invertible, allowing us to write
(37) βπ₯π₯ = βπ΄π΄ β1π΅π΅ βπ’π’ β π΄π΄ β1ππ (π₯π₯ Μ , οΏ½) = πΊπΊ βπ’π’ + π€π€
minβπ’π’,π π
οΏ½βπ’π’π π οΏ½πποΏ½π»π» 00 0οΏ½ οΏ½
βπ’π’π π οΏ½ + οΏ½ πππΎπΎ1 οΏ½
πποΏ½βπ’π’π π οΏ½
π π . π‘π‘. οΏ½ππ 0πΆπΆπΊπΊ βΞ0 βπΌπΌ
οΏ½ οΏ½βπ’π’π π οΏ½ + οΏ½οΏ½Μ οΏ½π
ππΜ + πΆπΆπ€π€0
οΏ½ β€ 0, (38)
where π»π» = π»π»π’π’ + πΊπΊπππ»π»π₯π₯πΊπΊ, and ππ = πππ’π’ + πΊπΊππ(π»π»π₯π₯π€π€ + πππ₯π₯). The iterates can then be updated as
π₯π₯ β π₯π₯Μ + πΊπΊβπ’π’ + π€π€, π’π’ β π’π’οΏ½ + βπ’π’. (39)
3.2.2.2 A Computational Reference Governor for Suboptimal MPC.
Suboptimal MPC (SOMPC), where optimization algorithm iterations are distributed over time, have the potential to reduce computation times and improve the reliability of numerical algorithms for MPC controller. In SOMPC an estimate of the OCP solution is maintained within the controller and a finite number of optimization algorithm iterations are performed at each sampling instance. This practice is commonly known as warmstarting. The solution updating process can be seen Figure 1, the result is a pair of coupled dynamic systems representing the plant and optimizer respectively.
FIGURE 1 A COMPARISON OF SUBOPTIMAL MPC AND IDEAL MPC
Model predictive control imposes constraints on the plant, the objective of the CRG is to impose constraints on the optimizer. Specifically, weβre interested in imposing the constraint
ππ(π§π§, π₯π₯) β€ πππ’π’π’π’ , (40)
Approved for public release; distribution is unlimited 11
We can then write the condensed problem as
Approved for public release; distribution is unlimited 12
οΏ½|π§π§ππ β π§π§β(ππππ)|οΏ½ β€ πποΏ½|π§π§ππβ1 β π§π§β(ππππβ1)|οΏ½ + π½π½οΏ½|βπ₯π₯ππ|οΏ½ + π½π½||βππππ||, (41)
where π§π§ is the primal dual solution estimate, π§π§β(ππ) is the parameter to solution mapping for the OCP and ππ = (π₯π₯, ππ) is the OCP parameter which consists of the measured state π₯π₯ and the reference command ππ.
The idea of the computational RG is to limit the residual ππ by controlling βππππ. To accomplish this, we use an add-on unit. The CRG is attached to the NMPC controller as shown in Figure 2 below:
FIGURE 2 ARCHITECTURE OF THE COMPUTATIONAL REFERENCE GOVERNOR
The CRG takes in the user commanded reference ππ, and the current OCP residual and outputs a modified auxiliary reference π£π£. Internally, the CRG solves the following optimization problem
minβπ£π£
οΏ½|ππππ β π£π£ππβ1 + βπ£π£|οΏ½22
π π . π‘π‘. |βπ£π£ππ| β€ βπ£π£ub, ππ = 1 β¦ 3,πππππ’π’ β€ π£π£ππβ1 + βπ£π£ β€ πππ’π’π’π’ ,
(42)
where ππππ is the reference, π£π£ππβ1 is the virtual reference at the previous step, πππππ’π’,π’π’π’π’ are the upper and lower bounds on the Euler angles. This is a quadratic program with only 3 decision variables; its computational cost is negligible compared to the MPC controller. The maximum step length is calculated as
βπ£π£π’π’π’π’ = maxοΏ½0, πΏπΏπ£π£(πππ’π’π’π’ β ππππβ1)
πππ’π’π’π’οΏ½ (43)
where ππππβ1 is the residual of the OCP at the previous sampling instance and the reference is updated as
π£π£ππ = π£π£ππβ1 + βπ£π£. (44)
where is the residual function for the OCP defined in (35). Enforcing this constraint ensures the quality of the numerical solution. The solver obeys the following convergence bounds
This forms a kind of nonlinear filter; the underlying intuition is to reduce the step size proportionally to the distance from the constraint ππ(π§π§, π₯π₯) β€ πππ’π’π’π’. The maximum reference step length πΏπΏπ£π£ and the maximum allowable residual πππ’π’π’π’ are user chosen parameters.
3.3 Constrained control framework using a virtual net of chance constraint admissible sets and output measurements only
Consider a system represented by a discrete-time linear model with stochastic state disturbance and measurement noise inputs, given by
π₯π₯ππ+1 = π΄π΄π₯π₯ππ + π΅π΅π’π’ππ + π€π€π€π€ππ, π¦π¦ππ = πΆπΆπ₯π₯ππ + πΉπΉπ£π£ππ,
(45)
where π₯π₯ππ is a vector state and π’π’ππ is a vector control. The variables π€π€ππ and π£π£ππ are, respectively, the state disturbance and the measurement noise inputs, each assumed to be a sequence of zero-mean, independent (and jointly independent) identically distributed (i.i.d.) random variables with πΌπΌ[π€π€πππ€π€ππ
ππ] = Ξ£π€π€ = Ξ£π€π€ππ , πΌπΌ[π£π£πππ£π£ππππ] = Ξ£π£π£ = Ξ£π£π£ππ.
We consider a fixed-gain state feedback,
π’π’ππ = πΎπΎπ₯π₯οΏ½ππ + πΊπΊππ, (46)
where ππ is the reference and π₯π₯οΏ½ππ is the state estimate generated by a fixed-gain, L, observer. Then the error, ππ = π₯π₯ππ β π₯π₯οΏ½ππ is normally distributed with covariance
ππβ = π΄π΄ππππβπ΄π΄ππππ + π΅π΅πππ΅π΅ππππ π΄π΄ππ = π΄π΄ + πΏπΏπΆπΆ,π΅π΅ππ = [π€π€, πΏπΏπΉπΉ]
(47)
Consider now enforcing a state constraint of the form,
π»π»π₯π₯ππ β€ β,βππ (48)
in a probabilistic manner. The resulting chance constraint can be shown to have the form (see [14] for details):
ππππππππ οΏ½[π»π» π»π»] οΏ½π₯π₯οΏ½πππππποΏ½ β€ β β π»π»(πΌπΌ β π΄π΄ππ)β1π΅π΅πππποΏ½ β₯ 1 β πΌπΌ, (49)
where πΌπΌ is the probability threshold and ππππ is the state estimation error. For this chance constraint, a chance constrained admissible set, ππβ(ππ), can be defined which is the set of initial state estimates such that the subsequent trajectories for the given reference, ππ, satisfy the chance constraint. In defining ππβ(ππ) certain technical assumptions need to be made such that the observed dynamics have reached the steady-state [16].
Approved for public release; distribution is unlimited 13
Following the same approach as in [10], now a virtual net of reference points ππ may be built to ensure safe transitions between reference points by declaring a connection between two nodes on the graph only exists if the a priori node lies within the chance constrained admissible set of a posteriori node. The maneuver planning problem on this graph is then solved using any standard graph search algorithm.
4. RESULTS AND DISCUSSION
4.1 Disjunctive-Based Constrained Spacecraft Relative Motion Planning
We consider a case study of spacecraft three-dimensional relative motion control. The relative motion dynamics are modeled with the linearized Clohessy-Wiltshire [11] equations,
(50)
where ππ is the chaser vehicleβs mass and ππ is the mean motion of the target vehicleβs orbit. We form a system of first-order equations with state vector π₯π₯ππ = οΏ½π₯π₯1,ππ,π₯π₯2,ππ,π₯π₯3,ππ, οΏ½ΜοΏ½π₯1,ππ, οΏ½ΜοΏ½π₯2,ππ, οΏ½ΜοΏ½π₯3,πποΏ½
ππ and
discretize using a Zero-Order Hold [12] with sampling period of 30 seconds, chaser vehicle mass of 140 kg, and target vehicle mean motion of 0.0010 radians per second. We choose πΆπΆ =[π°π°3 ππ3Γ3], which corresponds to relative position measurements. The goal in this scenario is to rendezvous the chaser vehicle with the target vehicle, i.e., to bring the chaser's state to the origin. For this controller and observer, the feedback gain matrix πΎπΎ is computed using LQR by solving the Discrete Time Algebraic Riccati Equation (DARE) and the observer gain matrix πΏπΏ computed by solving the dual DARE, with ππ = π°π°6 and π π = π°π°3 in each case.
4.1.2 Theoretical Results.
We consider a case study of spacecraft three-dimensional relative motion control. The relative motion dynamics are again modeled with the linearized CW equations,
(51)
where ππ represents the chaser vehicleβs mass and ππ is the mean motion of the target vehicleβs orbit. We form an equivalent system of first-order equations and discretize using Zero-Order Hold [12] with sampling period 30 sec, chaser vehicle mass of 140 kg, and target vehicle mean motion of 0.0010 rad/sec. We choose πΆπΆ = [ ππ ππ ππ 3Γππ ], which corresponds to
Approved for public release; destribution is unlimited14
Approved for public release; distribution is unlimited 15
relative position measurements. The goal in this scenario is to rendezvous the chaser vehicle with the target vehicle, i.e., to bring the chaserβs state to the origin.
For this controller and observer, the feedback gain matrix πΎπΎ is computed using LQR by solving the Discrete Time Algebraic Riccati Equation (DARE) and the observer gain matrix πΏπΏ computed by solving the dual DARE, with ππ = ππ6 and π π = ππ3 in each case . The dist ributions of the measurement noise, π£π£ππ, an d of th e st ate disturbance, π€π€ππ, are assu med to b e G aus sian with covariance matrices Ξ£π£π£ = 10β2 β ππ3 and Ξ£π€π€ = 10β4 β ππ6. The cost function is defined with ππππ = 0, π π ππ = ππ, and π π π₯π₯ = ππ. These choices ensure accurate estimates of the relative position states.
We will now construct a sequence that satisfies the sufficient conditions, and demonstrate that it leads to stable behavior. For this example, οΏ½Μ οΏ½π0 = 1.0063, ππ 1Μ = 0.2016, πποΏ½ 0 = 0.0332, and πποΏ½1 = 1.0063. Working with the Frobenius norm, ππβ = 1 and
(52)
We begin with the (inadmissible) sequence ππ2 = {0, 1}, with πππ π = 2, ππ0 = 1, and ππ1 = 1. To modify the sequence such that it satisfies the sufficient conditions, increase ππ1 by 1, and iterate until the first logarithmic condition is satisfied, then iterate on ππ0 until the second logarithmic condition is satisfied, and repeat as necessary until both inequalities are true. For this example, the iterative process yields ππ0 = 3 and ππ1 = 5. The resulting sequence, π π 8 = {0, 0, 0, 1, 1, 1, 1, 1}, will converge in relative position and state error estimates when propagated forward periodically, and a representative simulation is shown in Figure 3.
FIGURE 3 DISJUNCTIVE SENSING AND CONTROL TRAJECTORY UNDER A LENGTH 8 PERIODIC SEQUENCE
The constructed sequence satisfies the contractivity conditions but it not the optimum sequence under the chosen cost functional. To determine the optimum such sequence, we apply the algorithm discussed in Section 3.2.4. There are no admissible sequences of length 2 or length 3, thus we extend the search to sequences of length 4. The optimum such sequence is ππ4 = {0, 0, 1, 1}, with πποΏ½π΄π΄ = 0.5879 and πποΏ½ π΄π΄ = 0.0130. This sequence, however, is optimal only over sequences of length 4; longer sequences may exist that exhibit faster convergence. When the algorithm treats sequences of length 7, the optimum such sequence is ππ7 = {0, 0, 1, 1, 1, 0, 0}, with πποΏ½π΄π΄ = 0.07594 and πποΏ½π΄π΄ = 3.796 Γ 10β5. Si mulated re sults of propagating ea ch op timal sw itching se quence forward periodically are shown in Figure 4; the sequence ππ7 converges more quickly than the sequence ππ 4, which goes through several additional oscillations before settling to the origin.
Approved for public release; distribution is unlimited 16
FIGURE 4 COMPARISON OF LENGTH 4 AND LENGTH 7 OPTIMAL SWITCHING SEQUENCES
Of note, when treating sequences of length 8, the optimum such sequence is {0, 0, 1, 1, 0, 0, 1, 1}, a reducible sequence that has the previous length 4 optimal sequence ππ4 a s the corresponding irreducible subsequence.
The remaining objective is the treatment of chance constraints. Suppose that we wish to establish a constraint on the steady-state of the form ππ = {π₯π₯: βπ₯π₯ ββ β€ ππ}, so t hat the chas er spac ecraft remains within a box centered at the origin, of side lengths 2ππ, with Prob(ππ) β₯ 0.95. Consider the length 4 sequence ππ4; invoking the Chebyshev inequality with πΌπΌ π₯π₯ = οΏ½3/0.05, for each of πππ₯π₯π π ,0, β¦, πππ₯π₯π π ,3 yields four spheres, with minimum radius ππ = 2.79 and maximum radius ππ = 9.54. Only the maximum radius, the loosest of the four bounds, is guaranteed by the Chebyshev inequality, but even the tightest bound remains somewhat conservative. When setting ππ = 2.5, t he c hance constraints are violated in no more than 4% of trajectories at any given time step. Figure 5 illustrates the satisfaction of the chance constraints over the course of 200 simulation runs.
Approved for public release; distribution is unlimited 17
Approved for public release; distribution is unlimited 18
FIGURE 5 DEMONSTRATION OF SATISFACTION OF BOTH BOX AND CHANCE CONSTRAINTS
4.2 A Computational Reference Governor for Safe Online Optimization
We placed CRG and NMPC combination in closed-loop with the model described in Section 3.2.1 and commanded a pair of reorientation maneuvers. Figure 6 shows the closed-loop traces, the constraints are satisfied, and the system successfully completes the maneuver.
FIGURE 6 THE TIME HISTORIES OF CONTROL TORQUES, ANGULAR VELOCITIES AND EULERANGLES IN CLOSED-LOOP SIMULATION
Figure 7 compares the angle references and OCP residual with and without the CRG. Without the CRG the OCP residual violates the maximum residual constraint (ππππ β€ 10β4). With the CRG active the residual constraint is satisfied; the CRG filters the reference commands to limit the change in the reference. While these simulations provide a proof of concept, further development of the theory of computational reference governors will be pursued in future research.
Approved for public release; distribution is unlimited 19
FIGURE 7 CLOSED-LOOP TRACES OF THE RESIDUALS AND REFERENCES WITH AND WITHOUT THE CRG.
4.3 Constrained Control Framework Using a Virtual Net of Chance Constraint Admissible Sets and Output Measurements Only
For illustration, a spacecraft constrained relative motion maneuver is considered. The equations of motion are based on the Clohessy-Wiltshire equations introduced in Section 4.1 modified with random disturbance terms and measurement noise. Only the spacecraft relative position coordinates are measured and an observer has been constructed to estimate the state. A virtual net of stationary reference points in the relative frame and of the corresponding chance constrained admissible sets has then been constructed. The graph search is performed with fuel usage as the cost. Figure 8 illustrates the resulting maneuver, where the constraints correspond to avoiding a cone emanating from the origin and staying with the specified box.
Approved for public release; distribution is unlimited 20
Approved for public release; distribution is unlimited 21
FIGURE 8 CLOSED-LOOP PATH OF SPACECRAFT IN CW FRAME. RED CONE IS THE OBSTACLEAND BLACK DOTS ARE THE REFERENCE POINT NODES IN THE GRAPH.
5. CONCLUSION
The outcomes of research into coordinating sensing, control and computations for autonomous vehicles are as follows:
β’ Strategies to ensure stability and eventual chance constraint enforcement have been defined forsystems where simultaneous sensing and actuation cannot be performed and thus switchingbetween sensing and actuation is necessary.
β’ A computational reference governor has been defined to supervise and modify if necessaryreference commands to a nonlinear model predictive controller and ensure constraints on theresiduals of the optimization problems to ensure acceptable accuracy of the solution within limitedcomputational time.
β’ For the case of systems operating with output measurements only and a state observer used toreconstruct the full state, a methodology for constructing and chaining chance constraint admissiblesets has been defined which reduces the problem of constructing constraint admissible maneuversto a standard graph search.
β’ Simulations of spacecraft relative motion and attitude control have been used to illustrate thepotential of the proposed strategies.
Approved for public release; distribution is unlimited 22
6. REFERENCES
[1] Sutherland, R., Kolmanovsky, I., Girard, A., Leve, F., Petersen, C., βOptimal strategies for disjunctive sensing and control,β 2018 American Control Conference, AIAA, Milwaukee, WI, June 2018.
[2] Sutherland, R., Kolmanovsky, I., Girard, A., Leve, F., Petersen, C., βMinimum-time model predictive spacecraft attitude control for waypoint following and exclusion zone avoidance,β 2019 AIAA SciTech Forum, AIAA, San Diego, CA, January 2019.
[3] Liao-McPherson, D., Nicotra, M., and Kolmanovsky, I., βEmbedding constrained model predictive control in a continuous-time dynamic feedback,β IEEE Transactions on Automatic Control, 64, 5, pp. 1932-1946, 2019.
[4] Chen, C. Linear System Theory and Design, Oxford University Press, 3e, 1999.
[5] Yu, Qiang and Zhao, Xudong, βStability analysis of discrete-time switched linear systems with unstable subsystems,β Applied Mathematics and Computation, 273, pp. 718-725, 2016.
[6] Lax, Peter D., Functional Analysis, John Wiley and Sons, pp. 195-197, 2002.
[7] Navarro, Jorge, βA simple proof for the multivariate Chebyshev inequality,β arXiv preprint, arXiv:1305.5646, 2013.
[8] Chen, Xinjia, βA new generalization of Chebyshev inequality for random vectors,β arXiv preprint, arXiv:0707.0805, 2007.
[9] Garone, E., Di Cairano, S., and Kolmanovsky, I., βReference and command governors for systems with constraints: A survey on theory and applications,β Automatica, 75, pp. 306β328, 2017.
[10] Weiss, A., Kolmanovsky, I., Baldwin, M., Erwin, R.S., βSafe positively invariant sets for spacecraft obstacle avoidance,β AIAA Journal of Guidance, Control, and Dynamics, 36, 5, pp. 720-732, 2014.
[11] Clohessy, W. H., and Wiltshire, R. S., βTerminal guidance system for satellite rendezvous,β Journal of the Aerospace Sciences, 27, 9, pp. 653β658, 1960.
[12] Toth, R., Heuberger, P.S.C., and Van den Hof, P.M.J., βDiscretisation of linear parameter-varying state-space representations,β IET Control Theory and Applications, 4, pp. 2082β2096, 2010.
[13]Sutherland, R., Kolmanovsky, I., Girard, A., Leve, F., and Petersen, C., βOn closed-loop Lyapunov stability with minimum-time MPC feedback laws for discrete-time systems,β IEEE Conference on Decision and Control, Nice, France, 2019.
Approved for public release; distribution is unlimited 23
[14] Berning, A., Kolmanovsky, I., Girard, A., Leve, F., and Petersen, C., βManeuver planningusing chained chance constrained admissible sets,β American Control Conference, Denver,Colorado, in preparation, 2020.
[15] Sutherland, R., βAdvances in disjunctive and time-optimal predictive control methods,βPh.D. Dissertation, The University of Michigan, 2019.
[16] Gilbert, Elmer G., Ilya Kolmanovsky, and Kok Tin Tan. "Discreteβtime referencegovernors and the nonlinear control of systems with state and control constraints."International Journal of robust and nonlinear control 5.5, pp. 487-504, 1995.
Approved for public release; distribution is unlimited 24
LIST OF SYMBOLS, ABBREVIATIONS, AND ACRONYMS
CRG Computational Reference Governor
DARE Discrete-Time Algebraic Riccati Equation
DSC Disjunctive Sensing and Control
IID Independent and Identically Distributed
MILP Mixed-Integer Linear Program
MPC Model Predictive Control
NMPC Nonlinear Model Predictive Control
OCP Optimal Control Problem
DISTRIBUTION LIST
1 cy
1 cy
DTIC/OCP 8725 John J. Kingman Rd, Suite 0944 Ft Belvoir, VA 22060-6218
AFRL/RVIL Kirtland AFB, NM 87117-5776
Official Record Copy AFRL/RVS/Christopher Petersen 1 cy
25Approved for public release; distribution is unlimited