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A Pseudospectral Method for the Optimal Control of Constrained Feedback Linearizable Systems

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A Pseudospectral Method for the Optimal Control of Constrained Feedback Linearizable Systems View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Calhoun, Institutional Archive of the Naval Postgraduate School Calhoun: The NPS Institutional Archive Faculty and Researcher Publications Faculty and Researcher Publications 2006-06-07 A Pseudospectral Method for the Optimal Control of Constrained Feedback Linearizable Systems Gong, Qi IEEE http://hdl.handle.net/10945/29674 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 7, JULY 2006 1115 A Pseudospectral Method for the Optimal Control of Constrained Feedback Linearizable Systems Qi Gong, Member, IEEE, Wei Kang, Member, IEEE, and I. Michael Ross Abstract—We consider the optimal control of feedback lin- of necessary optimality conditions resulting from Pontryagin’s earizable dynamical systems subject to mixed state and control Maximum Principle [3][35]. These methods are collectively constraints. In general, a linearizing feedback control does not called indirect methods. There are many successful implemen- minimize the cost function. Such problems arise frequently in astronautical applications where stringent performance require- tations of indirect methods including launch vehicle trajectory ments demand optimality over feedback linearizing controls. In design, low-thrust orbit transfer, etc. [3], [6], [9]. Although this paper, we consider a pseudospectral (PS) method to compute indirect methods enjoy some nice properties, they also suffer optimal controls. We prove that a sequence of solutions to the from many drawbacks [2]. For instance, the boundary value PS-discretized constrained problem converges to the optimal solu- problem resulting from the necessary conditions are extremely tion of the continuous-time optimal control problem under mild and numerically verifiable conditions. The spectral coefficients sensitive to initial guesses [8][2]. In addition, these neces- of the state trajectories provide a practical method to verify the sary conditions must be explicitly derived—a labor-intensive convergence of the computed solution. The proposed ideas are process for complicated problems that requires an in-depth illustrated by several numerical examples. knowledge of optimal control theory. Index Terms—Constrained optimal control, pseudospectral, Over the last decade, an alternative approach based on nonlinear systems. discrete approximations has gained wide popularity [27], [2], [13], [15], [26], [46] as a result of significant progress in large-scale computation and robustness of the approach. The I. INTRODUCTION essential idea of this method is to discretize the optimal control T IS well known [7], [8], [47] that it is extremely difficult to problem and solve the resulting large-scale finite-dimensional I analytically solve a state- and control-constrained nonlinear optimization problem. These types of methods are known as optimal control problem. The main difficulty arises in seeking direct methods whose roots can be traced back to the works a closed-form solution to the Hamilton–Jacobi equations, or in of Bernoulli and Euler [39]. The simplicity of direct methods solving the canonical Hamiltonian equations resulting from an belies a wide range of deeply theoretical issues that lie at application of the Minimum Principle. Over past decades, many the intersection of approximation theory, control theory and computational methods have been developed for solving non- optimization. Regardless, a wide variety of industrial-strength linear optimal control problems. For instance, in [8], various optimal control problems have been solved by this approach numerical methods such as neighboring extremal methods, gra- [2], [30], [33], [37]. dient methods and quasilinearization methods are discussed in Considering the widespread use of discrete approxima- detail. An update on these methods, along with extensive numer- tions, it might appear to a novice that theoretical questions ical results is presented in [7]. In [28], a generalized gradient regarding the existence of a solution and convergence of the method is proposed for constrained optimal control problems. approximations have been answered satisfactorily. While this In [11], the feasibility and convergence of a modified Euler dis- is somewhat true of Eulerian methods [29], [12], [13], [35], cretization method is proved. A unified approach based on a [11] corresponding results for higher-order methods are not piecewise constant approximation of the control is proposed in only absent, but a number of interesting results are reported [23]. in the literature. For example, Hager [26] has shown that a Numerical methods for solving nonlinear optimal control “convergent” Runge–Kutta method does not converge to the problems are typically described under two categories: direct continuous optimal solution despite the fact that it satisfies methods and indirect methods [3]. Historically, many early nu- the standard conditions in the Butcher tableau. On the other merical methods were based on finding solutions to satisfy a set hand, Betts et al. [1] show that a nonconvergent Runge–Kutta method converges for optimal control problems. Thus, it is not surprising that even for Eulerian methods, significant restric- Manuscript received July 6, 2005; revised November 30, 2005. Rec- ommended by Associate Editor A. Astolfi. This work was supported by tions and assumptions are necessary for proofs of convergence, the Secretary of the Air Force, the Naval Postgraduate School, and the Jet particularly for state-constrained problems [13]. Note that these Propulsion Laboratory. The work of Q. Gong was supported by the National issues are quite different from those that were raised in the Academies’ National Research Council postdoctoral program. Q. Gong and I. M. Ross are with the Department of Mechanical and Astronau- early days of optimal control as summarized in [35]. While tical Engineering, Naval Postgraduate School, Monterey, CA, 93943 (e-mail: Eulerian methods are widely studied and useful for a theoretical [email protected]; [email protected]). understanding of discrete approximations, they are not prac- W. Kang is with the Department of Applied Mathematics, Naval Postgraduate School, Monterey, CA, 93943 (e-mail: [email protected]). tical for solving industrial strength problems, especially the Digital Object Identifier 10.1109/TAC.2006.878570 so-called multiagent problems that require real-time solutions 0018-9286/$20.00 © 2006 IEEE 1116 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 7, JULY 2006 TABLE I COMPARISON OF DIFFERENT DISCRETIZATION METHODS Fig. 1. Demonstrating the exponential convergence rate of a PS method. [43]. Among many, one of the reasons for their limitation is that Example 1: It is straightforward to show that the exact they generate a much larger-scale optimization problem than optimal control is given by .Now, say, a higher order scheme like a Runge–Kutta method [14]. consider three different discretizations of the problem: Euler, In this paper we focus on direct pseudospectral (PS) methods. Hermite–Simpson [27] and PS. Part of the reason for choosing PS methods were largely developed in the 1970s for solving the Hermite–Simpson discretization scheme for comparison partial differential equations arising in fluid dynamics and me- is because it is a widely used method and forms the basis of teorology [10], and quickly became “one of the big three tech- well-known commercial optimal control software packages nologies for the numerical solution of PDEs” [45]. During the such as OTIS [34] and SOCS [2]. Since different imple- 1990s, PS methods were introduced for solving optimal control mentations, in terms of software and hardware, of the same problems [15]–[18]; and since then, have gained considerable discretization method may cause variations in the results, the attention [19], [30], [33], [37], [44], [48]. One of the main rea- simulations reported in Table I are to be considered as illustra- sons for the popularity of PS methods is that they demonstrably tive. The column labeled “Nodes” denotes the number of nodes offer an exponential convergence rate for the approximation of used in the discretization; the “Error” column denotes the max- analytic functions [45] while providing Eulerian-like simplicity. imum error in control between the discrete and exact solutions; Thus, for a given error bound, PS methods generate a signif- and, the “Time” column represents the computer run-time icantly smaller scale optimization problem when compared to for the calculation of the solution. All of the discretization other methods on problems with highly smooth solutions. This methods are implemented on the same computational hardware property is particularly attractive for control applications as it (Pentium 4, 2.4-GHz with 256 MB of RAM) and software places real-time computation within easy reach of modern com- (MATLAB 6.5 under Windows XP). The resulting nonlinear putational power [43]. To illustrate these points, consider the programming problems were solved by the sequential quadratic nonlinear optimal control problem shown in the equation at the programming (SQP) method of SNOPT [22]. A quick glance at bottom of the page. the table illustrates that the PS method achieves a much higher Minimize Subject to GONG et al.: A PSEUDOSPECTRAL METHOD FOR THE OPTIMAL CONTROL OF CONSTRAINED FEEDBACK LINEARIZABLE SYSTEMS 1117 Fig. 2. Discrete optimal solution for Example 2 by a PS method with 10 nodes. accuracy with a significantly smaller number of nodes and nodes. That this is not true is also demonstrated in
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