This paper proposes an optimized tracking control approach using neural network (NN)-based reinforcement learning (RL) for a class of nonlinear dynamic systems, which requires both tracking and optimizing to be performed simultaneously. Generally, for obtaining optimal control solution, HJB (Hamilton- Jacobi-Bellman) equation is expected to be solvable, but, owing to strong nonlinearity, the equation is solved by analytical methods to be very difficult or even impossible. Therefore, adaptive NN approximation based RL is usually considered. In the optimized control design, for driving output state following to the desired trajectory, an error term is split from optimal performance index function, then both actor and critic NNs are built to perform RL algorithm. Actor NN aims to execute control behaviors, and critic NN aims to appraise control performance and make feedback to actor. The proof of stability concludes that the desired control performances are obtained. A numerical simulation is designed and implemented, and the desired results are shown.

Practical stability of a fractional order discrete-time linear state-space systems was put up in recent years. It is usually checked by the eigenvalues of the state matrix, some methods have been given during these years. But if the size of the state matrix is large, the computations of eigenvalues can be very onerous. In this paper, some new conditions on practical stability for positive fractional discrete-time linear system are presented. Numerically checking method of practical stability is presented based on the new conditions given in this paper. It is illustrated by the numerical examples that our checking method is effective and true. Compared to the now existing methods, numerically checking method is an attractive method because it’s easily implemented.

Constrained regulation problem (CRP) for continuous-time stochastic systems is investigated in this article. New existence conditions of linear feedback control law for continuous-time stochastic systems under constraints are proposed. The computation method for solving constrained regulation problem of stochastic systems considered in this article is also presented. Continuous-time stochastic linear systems and stochastic nonlinear systems are focused on, respectively. First, the condition of polyhedral invariance for stochastic systems is established by using the theory of positive invariant set and the principle of comparison. Second, the asymptotic stability conditions in the sense of expectation for two types of stochastic systems are established. Finally, finding the linear feedback controller model and corresponding algorithm of constrained regulation problem for two types of stochastic systems are also proposed by using the obtained condition. The presented model of the stochastic constrained regulation problem in this article is formulated as a linear programming problem, which can be easily implemented from a computational point of view. Our approach establishes a connection between the stochastic constrained regulation problem and positively invariant set theory, as well as provides the possibility of using optimization methodology to find the solution of stochastic constrained regulation problem, which differs from other methods. Numerical examples illustrate the proposed method.

This paper investigates reachable set estimation and state-feedback controller design for linear time-delay control systems with bounded disturbances. By constructing an appropriate Lyapunov–Krasovskii functional, we obtain a delay-dependent condition, which determines the admissible bounding ellipsoid for the reachable set of the system we considered. Then, a sufficient condition in the form of liner matrix inequalities is given to solve the problem of controller design and reachable set estimation. Then, by minimizing the volume of the ellipsoid and solving the liner matrix inequality, we obtain the desired ellipsoid and controller gain. A comparative numerical example is given to show the effectiveness of our result.

Positive linear electrical circuits systems described by generalized fractional derivatives are studied in this paper. We mainly focus on the reachability and observability of linear electrical circuits systems. Firstly, generalized fractional derivatives and $\rho$-Laplace transform of f is presented and some preliminary results are provided. Secondly, the positivity of linear electrical circuits systems described by generalized fractional derivatives is investigated and conditions for checking positivity of the systems are derived. Thirdly, reachability and observability of the generalized fractional derivatives systems are studied, in which the $\rho$-Laplace transform of a Mittag-Leffler function plays an important role. At the end of the paper, illustrative electrical circuits systems are presented, and conclusions of the paper are presented.