Pseudospectral optimal control

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Pseudospectral optimal control is a joint theoretical-computational method for solving optimal control problems. [1] [2] [3] [4] It combines pseudospectral (PS) theory with optimal control theory to produce a PS optimal control theory. PS optimal control theory has been used in ground and flight systems [1] in military and industrial applications. [5] The techniques have been extensively used to solve a wide range of problems such as those arising in UAV trajectory generation, missile guidance, control of robotic arms, vibration damping, lunar guidance, magnetic control, swing-up and stabilization of an inverted pendulum, orbit transfers, tether libration control, ascent guidance and quantum control. [5] [6]

Contents

Overview

There are a very large number of ideas that fall under the general banner of pseudospectral optimal control. [7] Examples of these are the Legendre pseudospectral method, the Chebyshev pseudospectral method, the Gauss pseudospectral method, the Ross-Fahroo pseudospectral method, the Bellman pseudospectral method, the flat pseudospectral method and many others. [1] [3] Solving an optimal control problem requires the approximation of three types of mathematical objects: the integration in the cost function, the differential equation of the control system, and the state-control constraints. [3] An ideal approximation method should be efficient for all three approximation tasks. A method that is efficient for one of them, for instance an efficient ODE solver, may not be an efficient method for the other two objects. These requirements make PS methods ideal because they are efficient for the approximation of all three mathematical objects. [8] [9] [10] In a pseudospectral method, the continuous functions are approximated at a set of carefully selected quadrature nodes. The quadrature nodes are determined by the corresponding orthogonal polynomial basis used for the approximation. In PS optimal control, Legendre and Chebyshev polynomials are commonly used. Mathematically, quadrature nodes are able to achieve high accuracy with a small number of points. For instance, the interpolating polynomial of any smooth function (C) at Legendre–Gauss–Lobatto nodes converges in L2 sense at the so-called spectral rate, faster than any polynomial rate. [9]

Details

A basic pseudospectral method for optimal control is based on the covector mapping principle. [2] Other pseudospectral optimal control techniques, such as the Bellman pseudospectral method, rely on node-clustering at the initial time to produce optimal controls. The node clusterings occur at all Gaussian points. [8] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]

Moreover, their structure can be highly exploited to make them more computationally efficient, as ad-hoc scaling [21] and Jacobian computation methods, involving dual number theory [22] have been developed. [19]

In pseudospectral methods, integration is approximated by quadrature rules, which provide the best numerical integration result. For example, with just N nodes, a Legendre-Gauss quadrature integration achieves zero error for any polynomial integrand of degree less than or equal to . In the PS discretization of the ODE involved in optimal control problems, a simple but highly accurate differentiation matrix is used for the derivatives. Because a PS method enforces the system at the selected nodes, the state-control constraints can be discretized straightforwardly. All these mathematical advantages make pseudospectral methods a straightforward discretization tool for continuous optimal control problems.[ citation needed ]

See also

Related Research Articles

<span class="mw-page-title-main">Optimal control</span> Mathematical way of attaining a desired output from a dynamic system

Optimal control theory is a branch of control theory that deals with finding a control for a dynamical system over a period of time such that an objective function is optimized. It has numerous applications in science, engineering and operations research. For example, the dynamical system might be a spacecraft with controls corresponding to rocket thrusters, and the objective might be to reach the Moon with minimum fuel expenditure. Or the dynamical system could be a nation's economy, with the objective to minimize unemployment; the controls in this case could be fiscal and monetary policy. A dynamical system may also be introduced to embed operations research problems within the framework of optimal control theory.

Trajectory optimization is the process of designing a trajectory that minimizes some measure of performance while satisfying a set of constraints. Generally speaking, trajectory optimization is a technique for computing an open-loop solution to an optimal control problem. It is often used for systems where computing the full closed-loop solution is not required, impractical or impossible. If a trajectory optimization problem can be solved at a rate given by the inverse of the Lipschitz constant, then it can be used iteratively to generate a closed-loop solution in the sense of Caratheodory. If only the first step of the trajectory is executed for an infinite-horizon problem, then this is known as Model Predictive Control (MPC).

In the numerical solution of partial differential equations, a topic in mathematics, the spectral element method (SEM) is a formulation of the finite element method (FEM) that uses high degree piecewise polynomials as basis functions. The spectral element method was introduced in a 1984 paper by A. T. Patera. Although Patera is credited with development of the method, his work was a rediscovery of an existing method

The Gauss pseudospectral method (GPM), one of many topics named after Carl Friedrich Gauss, is a direct transcription method for discretizing a continuous optimal control problem into a nonlinear program (NLP). The Gauss pseudospectral method differs from several other pseudospectral methods in that the dynamics are not collocated at either endpoint of the time interval. This collocation, in conjunction with the proper approximation to the costate, leads to a set of KKT conditions that are identical to the discretized form of the first-order optimality conditions. This equivalence between the KKT conditions and the discretized first-order optimality conditions leads to an accurate costate estimate using the KKT multipliers of the NLP.

DIDO is a MATLAB optimal control toolbox for solving general-purpose optimal control problems. It is widely used in academia, industry, and NASA. Hailed as a breakthrough software, DIDO is based on the pseudospectral optimal control theory of Ross and Fahroo. The latest enhancements to DIDO are described in Ross.

The Gauss–Kronrod quadrature formula is an adaptive method for numerical integration. It is a variant of Gaussian quadrature, in which the evaluation points are chosen so that an accurate approximation can be computed by re-using the information produced by the computation of a less accurate approximation. It is an example of what is called a nested quadrature rule: for the same set of function evaluation points, it has two quadrature rules, one higher order and one lower order. The difference between these two approximations is used to estimate the calculational error of the integration.

In numerical analysis, Gauss–Legendre quadrature is a form of Gaussian quadrature for approximating the definite integral of a function. For integrating over the interval [−1, 1], the rule takes the form:

In applied mathematics, the pseudospectral knotting method is a generalization and enhancement of a standard pseudospectral method for optimal control. The concept was introduced by I. Michael Ross and F. Fahroo in 2004, and forms part of the collection of the Ross–Fahroo pseudospectral methods.

The Legendre pseudospectral method for optimal control problems is based on Legendre polynomials. It is part of the larger theory of pseudospectral optimal control, a term coined by Ross. A basic version of the Legendre pseudospectral was originally proposed by Elnagar and his coworkers in 1995. Since then, Ross, Fahroo and their coworkers have extended, generalized and applied the method for a large range of problems. An application that has received wide publicity is the use of their method for generating real time trajectories for the International Space Station.

The Chebyshev pseudospectral method for optimal control problems is based on Chebyshev polynomials of the first kind. It is part of the larger theory of pseudospectral optimal control, a term coined by Ross. Unlike the Legendre pseudospectral method, the Chebyshev pseudospectral (PS) method does not immediately offer high-accuracy quadrature solutions. Consequently, two different versions of the method have been proposed: one by Elnagar et al., and another by Fahroo and Ross. The two versions differ in their quadrature techniques. The Fahroo–Ross method is more commonly used today due to the ease in implementation of the Clenshaw–Curtis quadrature technique. In 2008, Trefethen showed that the Clenshaw–Curtis method was nearly as accurate as Gauss quadrature. This breakthrough result opened the door for a covector mapping theorem for Chebyshev PS methods. A complete mathematical theory for Chebyshev PS methods was finally developed in 2009 by Gong, Ross and Fahroo.

Introduced by I. Michael Ross and F. Fahroo, the Ross–Fahroo pseudospectral methods are a broad collection of pseudospectral methods for optimal control. Examples of the Ross–Fahroo pseudospectral methods are the pseudospectral knotting method, the flat pseudospectral method, the Legendre-Gauss-Radau pseudospectral method and pseudospectral methods for infinite-horizon optimal control.

Named after I. Michael Ross and F. Fahroo, the Ross–Fahroo lemma is a fundamental result in optimal control theory.

The Bellman pseudospectral method is a pseudospectral method for optimal control based on Bellman's principle of optimality. It is part of the larger theory of pseudospectral optimal control, a term coined by Ross. The method is named after Richard E. Bellman. It was introduced by Ross et al. first as a means to solve multiscale optimal control problems, and later expanded to obtain suboptimal solutions for general optimal control problems.

The covector mapping principle is a special case of Riesz' representation theorem, which is a fundamental theorem in functional analysis. The name was coined by Ross and coauthors, It provides conditions under which dualization can be commuted with discretization in the case of computational optimal control.

Ross' π lemma, named after I. Michael Ross, is a result in computational optimal control. Based on generating Carathéodory-π solutions for feedback control, Ross' π-lemma states that there is fundamental time constant within which a control solution must be computed for controllability and stability. This time constant, known as Ross' time constant, is proportional to the inverse of the Lipschitz constant of the vector field that governs the dynamics of a nonlinear control system.

Isaac Michael Ross is a Distinguished Professor and Program Director of Control and Optimization at the Naval Postgraduate School in Monterey, CA. He has published a highly-regarded textbook on optimal control theory and seminal papers in pseudospectral optimal control theory, energy-sink theory, the optimization and deflection of near-Earth asteroids and comets, robotics, attitude dynamics and control, orbital mechanics, real-time optimal control and unscented optimal control. The Kang–Ross–Gong theorem, Ross' π lemma, Ross' time constant, the Ross–Fahroo lemma, and the Ross–Fahroo pseudospectral method are all named after him.

Fariba Fahroo is an American Persian mathematician, a program manager at the Air Force Office of Scientific Research, and a former program manager at the Defense Sciences Office. Along with I. M. Ross, she has published papers in pseudospectral optimal control theory. The Ross–Fahroo lemma and the Ross–Fahroo pseudospectral method are named after her. In 2010, she received, the AIAA Mechanics and Control of Flight Award for fundamental contributions to flight mechanics.

GPOPS-II is a general-purpose MATLAB software for solving continuous optimal control problems using hp-adaptive Gaussian quadrature collocation and sparse nonlinear programming. The acronym GPOPS stands for "General Purpose OPtimal Control Software", and the Roman numeral "II" refers to the fact that GPOPS-II is the second software of its type.

In mathematics, unscented optimal control combines the notion of the unscented transform with deterministic optimal control to address a class of uncertain optimal control problems. It is a specific application of Riemmann-Stieltjes optimal control theory, a concept introduced by Ross and his coworkers.

References

  1. 1 2 3 Ross, I. Michael; Karpenko, Mark (2012). "A review of pseudospectral optimal control: From theory to flight". Annual Reviews in Control. 36 (2): 182–97. doi:10.1016/j.arcontrol.2012.09.002.
  2. 1 2 Ross, I M. (2005). "A Roadmap for Optimal Control: The Right Way to Commute". Annals of the New York Academy of Sciences. 1065: 210–31. Bibcode:2005NYASA1065..210R. doi:10.1196/annals.1370.015. PMID   16510411. S2CID   7625851.
  3. 1 2 3 Fahroo, Fariba; Ross, I. Michael (2008). "Advances in Pseudospectral Methods for Optimal Control". AIAA Guidance, Navigation and Control Conference and Exhibit. pp. 18–21. doi:10.2514/6.2008-7309. ISBN   978-1-60086-999-0. S2CID   17819443.
  4. ↑ Ross, I.M.; Fahroo, F. (2003). "A unified computational framework for real-time optimal control". 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475). Vol. 3. pp. 2210–5. doi:10.1109/CDC.2003.1272946. ISBN   0-7803-7924-1. S2CID   122755607.
  5. 1 2 Qi Gong; Wei Kang; Bedrossian, Nazareth S.; Fahroo, Fariba; Pooya Sekhavat; Bollino, Kevin (2007). "Pseudospectral Optimal Control for Military and Industrial Applications". 2007 46th IEEE Conference on Decision and Control. pp. 4128–42. doi:10.1109/CDC.2007.4435052. hdl:10945/29677. ISBN   978-1-4244-1497-0. S2CID   2935682.
  6. ↑ Li, Jr-Shin; Ruths, Justin; Yu, Tsyr-Yan; Arthanari, Haribabu; Wagner, Gerhard (2011). "Optimal pulse design in quantum control: A unified computational method". Proceedings of the National Academy of Sciences. 108 (5): 1879–84. Bibcode:2011PNAS..108.1879L. doi: 10.1073/pnas.1009797108 . JSTOR   41001785. PMC   3033291 . PMID   21245345.
  7. ↑ Ross, I. M.; Proulx, R. J. (September 2019). "Further Results on Fast Birkhoff Pseudospectral Optimal Control Programming" (PDF). Journal of Guidance, Control, and Dynamics. 42 (9): 2086–2092. Bibcode:2019JGCD...42.2086R. doi:10.2514/1.g004297. ISSN   1533-3884. S2CID   191166808.
  8. 1 2 Gong, Q.; Kang, W.; Ross, I.M. (2006). "A Pseudospectral Method for the Optimal Control of Constrained Feedback Linearizable Systems". IEEE Transactions on Automatic Control. 51 (7): 1115–29. doi:10.1109/TAC.2006.878570. hdl:10945/29674. S2CID   16048034.
  9. 1 2 Hesthaven, J. S.; Gottlieb, S.; Gottlieb, D. (2007). Spectral methods for time-dependent problems. Cambridge University Press. ISBN   978-0-521-79211-0.[ page needed ]
  10. ↑ Gong, Qi; Ross, I. Michael; Kang, Wei; Fahroo, Fariba (2007). "Connections between the covector mapping theorem and convergence of pseudospectral methods for optimal control". Computational Optimization and Applications. 41 (3): 307–35. doi:10.1007/s10589-007-9102-4. hdl:10945/48182. S2CID   38196250.
  11. ↑ Elnagar, G.; Kazemi, M.A.; Razzaghi, M. (1995). "The pseudospectral Legendre method for discretizing optimal control problems". IEEE Transactions on Automatic Control. 40 (10): 1793–6. doi:10.1109/9.467672.
  12. ↑ Fahroo, Fariba; Ross, I. Michael (2001). "Costate Estimation by a Legendre Pseudospectral Method". Journal of Guidance, Control, and Dynamics. 24 (2): 270–7. Bibcode:2001JGCD...24..270F. doi:10.2514/2.4709. hdl:10945/29649. S2CID   122759455.
  13. ↑ Gong, Qi; Fahroo, Fariba; Ross, I. Michael (2008). "Spectral Algorithm for Pseudospectral Methods in Optimal Control". Journal of Guidance, Control, and Dynamics. 31 (3): 460–71. Bibcode:2008JGCD...31..460G. CiteSeerX   10.1.1.301.3354 . doi:10.2514/1.32908. hdl:10945/56995. S2CID   18145691.
  14. ↑ Elnagar, Gamal N.; Kazemi, Mohammad A. (1998). "Pseudospectral Chebyshev Optimal Control of Constrained Nonlinear Dynamical Systems". Computational Optimization and Applications. 11 (2): 195–217. doi:10.1023/A:1018694111831. S2CID   30241469.
  15. ↑ Fahroo, Fariba; Ross, I. Michael (2002). "Direct Trajectory Optimization by a Chebyshev Pseudospectral Method". Journal of Guidance, Control, and Dynamics. 25 (1): 160–6. Bibcode:2002JGCD...25..160F. doi:10.2514/2.4862.
  16. ↑ Benson, David A.; Huntington, Geoffrey T.; Thorvaldsen, Tom P.; Rao, Anil V. (2006). "Direct Trajectory Optimization and Costate Estimation via an Orthogonal Collocation Method". Journal of Guidance, Control, and Dynamics. 29 (6): 1435–40. Bibcode:2006JGCD...29.1435B. doi:10.2514/1.20478.
  17. ↑ Rao, Anil V.; Benson, David A.; Darby, Christopher; Patterson, Michael A.; Francolin, Camila; Sanders, Ilyssa; Huntington, Geoffrey T. (2010). "Algorithm 902: GPOPS, A MATLAB software for solving multiple-phase optimal control problems using the gauss pseudospectral method". ACM Transactions on Mathematical Software. 37 (2). doi: 10.1145/1731022.1731032 . S2CID   15375549.
  18. ↑ Garg, Divya; Patterson, Michael A.; Francolin, Camila; Darby, Christopher L.; Huntington, Geoffrey T.; Hager, William W.; Rao, Anil V. (2009). "Direct trajectory optimization and costate estimation of finite-horizon and infinite-horizon optimal control problems using a Radau pseudospectral method". Computational Optimization and Applications. 49 (2): 335–58. CiteSeerX   10.1.1.663.4215 . doi:10.1007/s10589-009-9291-0. S2CID   8817072.
  19. 1 2 Sagliano, Marco; Theil, Stephan (2013). "Hybrid Jacobian Computation for Fast Optimal Trajectories Generation". AIAA Guidance, Navigation, and Control (GNC) Conference. doi:10.2514/6.2013-4554. ISBN   978-1-62410-224-0.
  20. ↑ Huneker, Laurens; Sagliano, Marco; Arslantas, Yunus (2015). SPARTAN: An Improved Global Pseudospectral Algorithm for High-fidelity Entry-Descent-Landing Guidance Analysis (PDF). The 30th International Symposium on Space Science and Technology. Kobe, Japan.
  21. ↑ Sagliano, Marco (2014). "Performance analysis of linear and nonlinear techniques for automatic scaling of discretized control problems" (PDF). Operations Research Letters. 42 (3): 213–6. doi:10.1016/j.orl.2014.03.003.
  22. ↑ d'Onofrio, Vincenzo; Sagliano, Marco; Arslantas, Yunus E. (2016). "Exact Hybrid Jacobian Computation for Optimal Trajectories via Dual Number Theory" (PDF). AIAA Guidance, Navigation, and Control Conference. doi:10.2514/6.2016-0867. ISBN   978-1-62410-389-6.

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