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Two numerical computational techniques are presented for improvement of the stable-manifold method, which is effective for nonlinear optimal control. The first technique is for generation of points on the stable manifold in a robust way against numerical errors. There, a special numerical method that preserves Hamiltonian is used to solve a differential equation sensitive to numerical errors. The second technique is a sort of shooting method to generate a point corresponding to the desired system state. Again, numerical robustness is an issue there. The two techniques are applied to an example system and shown to be effective.

This paper presents nonlinear model predictive control (NMPC) for nonlinear systems and apply to the inverted pendulum. There is distinction between the optimal control and NMPC. The optimal control aims to minimize the cost function over the finite horizon and applies the control input to the system. On the other hand, NMPC employs the state feedback and updates the control input at each time step. At each sampling time, the control design of NMPC is formulated as a two-point boundary-value (TPBV) problem. When the optimal control law is calculated, only the first control input is applied to the controlled system. We employ an iterative steepest descent algorithm to solve the TPBV problem. Numerical results show comparison between the optimal control and NMPC applied to the inverted pendulum.

A problem in the field of nonlinear dynamics is considered with a technique of control theory. In particular, for preventing a nonlinear dynamical system from making undesirable bifurcation, it is ...

Two conservative approaches are proposed to a semidefinite programming problem nonlinearly dependent on uncertain parameters. These approaches are applicable to general nonlinear parameter dependence not necessarily polynomial or rational. They are based on the mild assumption that the parameter dependence is expressed as the difference of two convex functions. The first approach uses constant bounds on the parameter dependence. Optimization of the bounds is reduced to convex nonsmooth minimization. The second approach uses parameter-dependent bounds for a less conservative result. Optimization of the bounds is immediate when the centroid is computable for the parameter set. Numerical examples are presented for illustration of the approaches.

Two conservative approaches are proposed to a semidefinite programming problem nonlinearly dependent on uncertain parameters. These approaches are applicable to general nonlinear parameter dependence not necessarily polynomial or rational. They are based on a mild assumption that the parameter dependence is expressed as the difference of two convex functions. The first approach uses constant bounds on the parameter dependence. Optimization of the bounds is reduced to convex nonsmooth minimization in general. The second approach uses parameter-dependent bounds for a less conservative result. Optimization of the bounds is immediate when the gravity center is computable for the parameter set. Numerical examples are presented for illustration of the approaches.

Stability analysis of an aperiodic sampled-data control system is considered for application to networked and embedded control. The stability condition is described in a linear matrix inequality to be satisfied for all possible sampling intervals. Although this condition is numerically intractable, a tractable sufficient condition can be constructed with the mean value theorem. Special attention is paid to tightness of the sufficient condition for less conservative stability analysis. A region-dividing technique for the reduction of conservatism and generalization to stabilization are also discussed. An example demonstrates the efficacy of the approach.

A new approach to model-set identification is proposed based on an agnostic learning theory. The squared prediction error is estimated together with its uncertainty uniformly in some parameter region. Based on this estimation, a model set is constructed so as to include the best model. The proposed approach does not require assumptions on the true dynamics or the noise, neither does it need infinite number of input-output data in order to justify its result. But it guarantees that the size of the identified model set converges to zero as the number of input-output data increases. Improvement of the precision is considered on the proposed identification method. Generalization of the approach is discussed and a numerical example is presented.

A guaranteed cost regulator design is presented for uncertain linear discrete-time systems, where its objective is to find a state feedback such that a quadratic performance index is less than a given value for any uncertainty. The design is recast as a feasibility problem described by a parametrized linear matrix inequality (LMI). A notion of probabilistic solution that satisfies the LMI for almost all uncertainties is introduced and an efficient randomized algorithm based on the gradient method with random samples of the uncertainty is developed. The proposed algorithm always stops in a finite number of iterations. Then, it can either find a probabilistic solution with a given probabilistic confidence or detect the infeasibility of the problem in the sense that the feasibility set is too small to contain a ball with a given radius.

In this paper, we consider robust semidefinite programs with functional variables. In the proposed approach, an approximate semidefinite program is constructed based on approximation with the sum-of-squares technique. Unlike the conventional use of the sum-of-squares technique, the quality of approximation is improved by dividing the parameter region into several subregions. The idea is generalized from that of Jennawasin and Oishi (in Proceedings of the 46th IEEE Conference on Decision and Control, New Orleans, USA, December 2007), where robust semidefinite programs without functional variables are considered. The present approach is asymptotically exact in the sense that the optimal value of the approximate problem converges to that of the original problem as the resolution of the division becomes finer. Our approach also gives an a priori upper bound on the discrepancy between the optimal values of the two problems in terms of the resolution of the division.

This paper presents a gradient-based randomized algorithm to design a guaranteed cost regulator for a plant with general parametric uncertainties. The algorithm either provides with high confidence a probabilistic solution that satisfies the design specification with high probability for a randomly sampled uncertainty or claims that the feasible set of the design parameters is too small to contain a ball with a given radius. In both cases, the number of iterations executed in the algorithm is of polynomial order of the problem size and is independent of the dimension of the uncertainty.

A nonlinear dynamical system is made robust against undesirable bifurcation with a matrix inequality technique, which is widely used in the field of control theory. The proposed method is based on the idea in Chap. 2. Namely, a stability index is defined for each parameter value of the system and the parameter is updated so as to optimize the index. Since this optimization problem is not differentiable in the original form, it is transformed to a differentiable optimization problem and here the technique of a matrix inequality is used. The proposed method is applied to a simple dynamical system and shown to be effective. Its generalization is considered for driving a system to avoid chaos.