Kiyotaka Shimizu

A solution method for the static constrained Stackelberg problem via penalty method

This note presents a new solution method for the static constrained Stackelberg problem. Through our approach, the Stackelberg problem is completely transformed into a one-level unconstrained problem such that the newly introduced overall augmented objective function is minimized with repect to the leaders and the followers variables jointly. It can be proved that a sequence of solutions to the transformed problems converges to the solution of the original problem, when the penalty parameters are updated.

Necessary conditions for min-max problems and algorithms by a relaxation procedure

For decision making under uncertainty, a rational optimality criterion is min-max. Min-max problems such that the minimizer makes an optimal decision against the worst case that might be chosen by the maximizer are studied. This paper presents necessary conditions and computational methods for a min-max solution (not a saddle point solution). Those conditions are stated in a form like Kuhn-Tucker theorem. The computational methods are based on the relaxation procedure. A min-max problem such that the minimizer and the maximizer are subject to separate constraints is primarily studied. But it is shown that the obtained results can be applied for the unseparate constraint case by use of duality theory.

A new computational method for Stackelberg and min-max problems by use of a penalty method

This paper is concerned with the Stackelberg problem and the min-max problem in competitive systems. The Stackelberg approach is applied to the optimization of two-level systems where the higher level determines the optimal value of its decision variables (parameters for the lower level) so as to minimize its objective, while the lower level minimizes its own objective with respect to the lower level decision variables under the given parameters. Meanwhile, the min-max problem is to determine a min-max solution such that a function maximized with respect to the maximizers variables is minimized with respect to the minimizers variables. This problem is also characterized by a parametric approach in a two-level scheme. New computational methods are proposed here; that is, a series of nonlinear programming problems approximating the original two-level problem by application of a penalty method to a constrained parametric problem in the lower level are solved iteratively. It is proved that a sequence of approximated solutions converges to the correct Stackelberg solution, or the min-max solution. Some numerical examples are presented to illustrate the algorithms.

Tracking Control of General Nonlinear Systems by Direct Gradient Descent Method

This paper is concerned with tracking control of nonlinear multi- variable systems whose relative degree is more than one. The control method is based on a direct steepest descent method using the gradient of a perfor- mance index. Simulation results demonstrate the usefulness of the proposed method.

Necessary and sufficient conditions for the efficient solutions of nondifferentiable multiobjective problems

The necessary conditions for the efficient solutions of multiobjective problems with locally Lipschitz objective and constraint functions are presented. Under the assumption of convexity, these conditions are also sufficient conditions. For its application, the optimality conditions for the multiobjective problems including extremal-value functions are obtained. The necessary conditions are derived by means of the generalized Motzkins lemma and are expressed in terms of Clarkes generalized gradients (1975).

Nonlinear state observers by gradient descent method

This paper concerns with a new nonlinear state observer of deterministic general nonlinear systems; which is designed based on the gradient descent method for decreasing squared estimation errors. The convergence of the proposed observer is proved by using the Bihari type inequality which generalizes the Gronwall inequality. We confirm from various simulations that our method works very well as a nonlinear state observer with superior convergence properties.

Performance improvement of direct gradient descent control for general nonlinear systems

This paper is concerned with direct gradient descent control (DGDC) of general nonlinear systems. DGDC manipulates control inputs directly so as to decrease a performance function like the squared error, based on the gradients of the performance function and the steepest descent method. We propose several methods to improve performances of DGDC. Convergence speed and asymptotical stability of DGDC are improved by two approaches. The first one is to consider time derivative of the performance function. Namely, to improve the convergence, we modify the DGDC by decreasing both the performance function and its time derivative. The second one is to introduce a new artificial control vector and to consider the extended controller for DGDC such that two kinds of performance function are decreased. Our simulation results with the modified DGDC showed very good performances for various plants.

Constrained optimization in Hilbert space and a generalized dual quasi-Newton algorithm for state-constrained optimal control problems

Studies a modified Newton method for constrained optimization in Hilbert space, and generalizes the dual quasi-Newton algorithm to Hilbert space. A new algorithm for solving state-constrained optimal control problems is proposed by applying the generalized dual quasi-Newton method. >

PID controller tuning via quasi pole assignment method

In this paper, a new tuning method for PID control is proposed. We set the controlled system to be a controllable and observable SISO system. First, we consider an optimal servo problem by two methods. For each method, the optimal state feedback control law is obtained by applying the optimal regulator theory, and optimal poles of the closed-loop system are calculated. Next, we consider the problem that poles of the closed-loop system by PID control are best approximated to the optimal poles as close as possible. This technique is called the quasi-pole assignment. The problem becomes a nonlinear programming problem with some equality constraints, and is solved by an exterior point penalty method. Finally, the simulation results is reported, demonstrating the effectiveness of the proposed methods.

Constrained optimization methods in Hilbert spaces and their applications to optimal control problems with functional inequality constraints

This note generalizes constrained optimization methods in a finite-dimensional space into Hilbert spaces and investigates computational methods for optimal control problems with functional inequality constraints. Two methods are proposed by applying the feasible direction method and the constrained quasi-Newton method. Subsidiary problems for direction-finding that are originally linear-quadratic programming in a Hilbert space can be transformed into linear-quadratic ones in Rn. Thus, the control problem can be solved by a series of finite-dimensional programming problems.