Bartels---Stewart algorithm is an effective and widely used method with an O(n 3) time complexity for solving a static Sylvester equation. When applied to time-varying Sylvester equation, the computation burden increases intensively with the decrease of sampling period and cannot satisfy continuous realtime calculation requirements. Gradient-based recurrent neural network are able to solve the time-varying Sylvester equation in real time but there always exists an estimation error. In contrast, the recently proposed Zhang neural network has been proven to converge to the solution of the Sylvester equation ideally when time goes to infinity. However, this neural network with the suggested activation functions never converges to the desired value in finite time, which may limit its applications in realtime processing. To tackle this problem, a sign-bi-power activation function is proposed in this paper to accelerate Zhang neural network to finite-time convergence. The global convergence and finite-time convergence property are proven in theory. The upper bound of the convergence time is derived analytically. Simulations are performed to evaluate the performance of the neural network with the proposed activation function. In addition, the proposed strategy is applied to online calculating the pseudo-inverse of a matrix and nonlinear control of an inverted pendulum system. Both theoretical analysis and numerical simulations validate the effectiveness of proposed activation function.

Accelerating a Recurrent Neural Network to Finite-Time Convergence for Solving Time-Varying Sylvester Equation by Using a Sign-Bi-power Activation Function

The Lyapunov equation is widely employed in the engineering field to analyze stability of dynamic systems. In this paper, based on a new evolution formula, a novel finite-time recurrent neural network (termed finite-time Zhang neural network, FTZNN) is proposed and studied for solving a nonstationary Lyapunov equation. In comparison with the original Zhang neural network (ZNN) model for a nonstationary Lyapunov equation, the convergence performance has a remarkable improvement for the proposed FTZNN model and can be accelerated to finite time. Besides, by solving the differential inequality, the time upper bound of the FTZNN model is computed theoretically and analytically. Simulations are conducted and compared to validate the superiority of the FTZNN model to the original ZNN model for solving the nonstationary Lyapunov equation. At last, the FTZNN model is successfully applied to online tracking control of a wheeled mobile manipulator.

Design and Analysis of FTZNN Applied to the Real-Time Solution of a Nonstationary Lyapunov Equation and Tracking Control of a Wheeled Mobile Manipulator

The Sylvester equation is often encountered in mathematics and control theory. For the general time-invariant Sylvester equation problem, which is defined in the domain of complex numbers, the Bartels-Stewart algorithm and its extensions are effective and widely used with an O(n³) time complexity. When applied to solving the time-varying Sylvester equation, the computation burden increases intensively with the decrease of sampling period and cannot satisfy continuous realtime calculation requirements. For the special case of the general Sylvester equation problem defined in the domain of real numbers, gradient-based recurrent neural networks are able to solve the time-varying Sylvester equation in real time, but there always exists an estimation error while a recently proposed recurrent neural network by Zhang et al [this type of neural network is called Zhang neural network (ZNN)] converges to the solution ideally. The advancements in complex-valued neural networks cast light to extend the existing real-valued ZNN for solving the time-varying real-valued Sylvester equation to its counterpart in the domain of complex numbers. In this paper, a complex-valued ZNN for solving the complex-valued Sylvester equation problem is investigated and the global convergence of the neural network is proven with the proposed nonlinear complex-valued activation functions. Moreover, a special type of activation function with a core function, called sign-bi-power function, is proven to enable the ZNN to converge in finite time, which further enhances its advantage in online processing. In this case, the upper bound of the convergence time is also derived analytically. Simulations are performed to evaluate and compare the performance of the neural network with different parameters and activation functions. Both theoretical analysis and numerical simulations validate the effectiveness of the proposed method.

Nonlinearly Activated Neural Network for Solving Time-Varying Complex Sylvester Equation.

The general coupled matrix equations over generalized bisymmetric matrices

Yong Wang

Papers

Weighted least squares solutions to general coupled Sylvester matrix equations

Brief paper: Detectability and observability of discrete-time stochastic systems and their applications

Least squares solution with the minimum-norm to general matrix equations via iteration

Stability analysis of linear stochastic neutral-type time-delay systems with two delays

Solutions to a family of matrix equations by using the Kronecker matrix polynomials