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The combination of the time-scaling transformation and control parameterization has proven to be an effective approach in addressing optimal control problems involving switching systems with predefined subsystem sequences. However, this approach has certain limitations. First, the number of control switchings is required to be no less than the number of subsystem switchings. Second, the switching of the subsystem must be accompanied by the switching of the control. Third, this scheme introduces many hyperparameters, leading to combinatorial explosion. To address these drawbacks, we introduce a novel computational approach such that the control switching can be independent of subsystem switching. The superiority of this novel approach can be clearly observed from the solutions obtained using the proposed method for solving two illustrative examples. [ABSTRACT FROM AUTHOR]

Control parameterization technique is an effective method to solve optimal control problems. It works by approximating the control function by piece-wise constant (or linear) function. In this way, the optimal control problems are approximated by optimal parameter selection problems, which can be regarded as finite-dimensional optimization problems, where the control heights and switching times of the piece-wise constant function are taken as the decision variables. They can be solved by using gradient-based optimization methods. For this, it requires the gradient formulas of the objective and constraint functions with respect to the decision variables. There are two methods for deriving the required gradient formulas. The aim of this paper is to survey various numerical methods developed based on control parameterization for solving optimal control problems governed by time-delay systems. Particular focus is on those numerical methods for optimizing switching time points. This is because the process of optimizing the switching time points is much more complicated for optimal control problems with time delays. In addition, we shall also review the computational methods developed based on control parameterization technique for solving two optimal control problems involving more complicated time delays. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of time-delay optimal control problem (TDOCP) with canonical equality and inequality constraints. By applying control parameterization method together with time-scaling transformation, a TDOCP can be readily solved by gradient-based optimization methods. The partial derivative of the cost as well as the constraint functions with respect to the decision variables are obtained by variational approach, which is inefficient when the discretization for the control function is relatively dense. For general optimal control problem without time-delay, co-state approach is an effective way to compute the gradients, however, when time-delay is involved in the dynamic system, the co-state system is not known. In this paper, we derive the co-state system for TDOCP to compute the gradients of the cost and constraints. Numerical results show that the computational efficiency is much higher when compared with the traditional variational approach. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of optimal control problems with control constraints on a set of characteristic time instants. By applying the control parameterization technique, these constraints are imposed on the subdomains that contain the characteristic time points. The values of the control functions as well as the lengths for their corresponding subdomains become decision variables. Time-scaling transformation is an effective technique to optimize the length of each subdomain in a new time horizon. However, the characteristic time instants in the original time horizon become variable time instants in the new time horizon, and hence the control constraints imposed on these characteristic time points are difficult to be formulated in the new time horizon. We propose a surrogate condition and show that the characteristic time control constraints will be satisfied once the surrogate condition holds. Moreover, this surrogate condition is easy to formulate in the new time horizon. The resulting approximate problem can be readily solved by many existing computational methods for solving constrained optimal control problems. Finally, we conclude this paper by solving two examples. [ABSTRACT FROM AUTHOR]

In this paper, we consider an optimal control problem in which the control is almost smooth and the state and control are subject to terminal state constraints and continuous state and control inequality constraints. By introducing an extra set of differential equations for this almost smooth control, we transform this constrained optimal control problem into an equivalent problem involving both control function and system parameter vector as decision variables. Then, by the control parametrization technique and a time scaling transformation, the equivalent problem is approximated by a sequence of constrained optimal parameter selection problems, each of which is a finite dimensional optimization problem. For each of these constrained optimal parameter selection problems, a novel exact penalty function method is constructed by appending penalized constraint violations to the cost function. This gives rise to a sequence of unconstrained optimal parameter selection problems; and each of which can be solved by existing optimization algorithms or software packages. Finally, a practical container crane operation problem is solved, showing the effectiveness and applicability of the proposed approach. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of control theoretic spline model, which can be formulated as a linear quadratic optimal control problem. The unknown initial condition and the control are to be chosen optimally such that the output best fits a set of measurement data which are corrupted by noise with crucial knowledge of its distribution. We first transform the uncertain objective function into a deterministic objective function. The solution method is based on the control parameterization technique. We show that the approximate optimal controls obtained from the approximate finite dimensional problems converge to the optimal control of the original control problem in the weak ⋆ topology of L ∞ ( [ 0 , T ] , R r ) . Numerical results show that the proposed method is effective. [ABSTRACT FROM AUTHOR]

The control parameterization method used together with time-scaling transformation is an effective approach to transform optimal control problems into optimal parameter selection problems. The approximate problems can then be solved by gradient-based optimization methods. However, the conventional time-scaling transformation requires that all the control component functions switch simultaneously, which can be undesirable in practice. In this paper, we introduce a novel technique where the switching times for each of the control component functions can be adaptively selected. To demonstrate the effectiveness and flexibility of the proposed approach, two example problems with control input and terminal state constraints are solved using the method proposed. Numerical results show that the new method achieves much better objective values with some increase in computation time compared to the conventional time-scaling transformation technique. [ABSTRACT FROM AUTHOR]

Efficient and reliable integrators are indispensable for the design of sequential solvers for optimal control problems involving continuous dynamics, especially for real-time applications. In this paper, optimal control problems for systems represented by index-1 differential-algebraic equations are investigated. On the basis of a time-scaling transformation, the control is parameterized as a piecewise constant function with variable heights and switching time instants. Compared with control parameterization with fixed time grids, the flexibility of adjusting switching time instants increases the chance of finding the optimal solution. Furthermore, error constraints are introduced in the optimization procedure such that the optimal control obtained has a guarantee of integration accuracy. For the derived approximate nonlinear programming problem, a function evaluation and forward sensitivity propagation algorithm is proposed with an embedded implicit Runge–Kutta integrator, which executes one Newton iteration in the limit by employing a predictor-corrector strategy. This algorithm is combined with a nonlinear programming solver Ipopt to construct the optimal control solver. Numerical experiments for the solution of the optimal control problem for a Delta robot demonstrate that the computational speed of this solver is increased by a factor of 0.5–2 when compared with the same solver without the predictor-corrector strategy, and increased by a factor of 20–40 when compared with solver embedding IDAS , the Implicit Differential-Algebraic solver with Sensitivity capabilities developed by Lawrence Livermore National Laboratory. Meanwhile, the accuracy loss compared with the one using IDAS is small and admissible. [ABSTRACT FROM AUTHOR]