Convergence rates for direct transcription of optimal control problems using collocation at Radau points

Abstract

We present convergence rates for the error between the direct transcription solution and the true solution of an unconstrained optimal control problem. The problem is discretized using collocation at Radau points (aka Gauss-Radau or Legendre-Gauss-Radau quadrature). The precision of Radau quadrature is the highest after Gauss (aka Legendre-Gauss) quadrature, and it has the added advantage that the end point is one of the abscissas where the function, to be integrated, is evaluated. We analyze convergence from a Nonlinear Programming (NLP)/matrix algebra perspective. This enables us to predict the norms of various constituents of a matrix that is "close" to the KKT matrix of the discretized problem. We present the convergence rates for the various components, for a sufficiently small discretization size, as functions of the discretization size and the number of collocation points. We illustrate this using several test examples. This also leads to an adjoint estimation procedure, given the Lagrange multipliers for the large scale NLP.

Consistent approximation of a nonlinear optimal control problem with uncertain parameters

“…We contrast this condition to that of Polak (1997, Section 3.3), which in addition requires a condition on stationary points. We note that the consistency property in Theorem 1 differs from the consistency results in Gong et al (2006), Kameswaran and Biegler (2008), Kang (2010) and Polak (1997) because the discretization occurs in the parameter space instead of the time domain. This results in a sequence of standard optimal control problems which can be further approximated using existing time discretization schemes (Gong et al, 2006;Kameswaran & Biegler, 2008;Schwartz & Polak, 1996).…”

“…Based on the numerical approximation of the integral over the stochastic parameters in the objective functional, the considered uncertain optimal control problem can be approximated by a sequence of standard nonlinear optimal control problems, which can in turn be solved using existing computational methods such as Runge-Kutta (Kameswaran & Biegler, 2008;Schwartz & Polak, 1996) and pseudospectral (Gong et al, 2006) approaches. To ensure meaningful results in this computational framework, it is essential to guarantee that the discretization schemes provide valid approximations to the original non-standard optimal control Problem B.…”

“…That is, accumulation points of a sequence of optimal solutions to the approximate problem are optimal solutions to the original uncertain optimal control problem. We contrast this condition to consistency and convergence results on standard optimal control problems, for example results in Gong et al (2006), Kameswaran and Biegler (2008), Kang (2010) and Polak (1997), as the discretization in this work occurs in the parameter space rather than the time domain. In Section 4 we address the convergence of the corresponding adjoint states, and show that the accumulation points of a sequence of optimal state-adjoint pairs satisfy a necessary condition of Pontryagin Minimum Principle type, which facilities assessment of the optimality of numerical solutions.…”

“…4), Runge-Kutta (Kameswaran & Biegler, 2008;Schwartz & Polak, 1996), and Pseudospectral (Gong, Kang, & Ross, 2006;Kang, 2010;Ross & Karpenko, 2012). These computational optimal control methods have achieved great success in many areas of control applications (Bedrossian, Bhatt, Kang, & Ross, 2009;Bedrossian, Karpenko, & Bhatt, 2012;Chung, Polak, Royset, & Sastry, 2011;Li, Ruths, Yu, & Arthanari, 2011).…”

Direct Trajectory Optimization and Costate Estimation of State Inequality Path-Constrained Optimal Control Problems Using a Radau Pseudospectral Method

“…In even more recent years, a great deal of research has been done on the class of direct collocation pseudospectral methods. 3,12,16,[18][19][20][21] In a pseudospectral method, the state is approximated using a basis of either Lagrange of Chebyshev polynomials and the dynamics are collocated at points associated with a Gaussian quadrature. The most common collocation points, which are the roots of a linear combination of Legendre polynomials or derivatives of Legendre polynomials, are Legendre-Gauss (LG), Legendre-Gauss-Radau (LGR), and Legendre-Gauss-Lobatto (LGL) points.…”