Abstract
For a class of discrete weakly nonlinear state-dependent coefficient (SDC) control systems, a suboptimal synthesis is constructed over a finite interval with a large number of steps. A one-point matrix Padé approximation (PA) of the solution of the initial problem for the discrete matrix Riccati equation is constructed based on the state-dependent Riccati equation (SDRE) approach and the asymptotics by the small-step of the boundary layer functions method. The symmetric gain coefficients matrix for Padé control synthesis is constructed based on the one-point PA. As a result, the parametric closed-loop control is obtained. The results of numerical experiments illustrate, in particular, the improved extrapolation properties of the constructed regulator, which makes the algorithm applicable in control systems for a wider range of parameter variation.
Highlights
In the literature, much attention is paid to the construction of optimal control laws for nonlinear systems and the corresponding approximate methods for their calculation
This can be explained by several factors; on the one hand, the greater accuracy of the description of dynamic systems in applications leads to the increase of their mathematical models’ dimension, and on the other hand, the calculations often need to be carried out in real time. This is especially true for finding feedback laws in nonlinear control systems, where the consideration of even weak nonlinearity in constructing synthesis laws based on linear control laws can lead to a significant improvement in the value of the quality criterion
The application field of the Kalman algorithm has been expanded to nonlinear control systems by the so-called state-dependent Riccati equation (SDRE) approach for continuous and discrete cases, where the systems are formally represented as linear systems in terms of state and control, the coefficients of the matrices are the functions of the state vector (state-dependent coefficients (SDC) systems), and the quality criterion is quadratic, but the quadratic forms matrices in the criterion can be state-dependent
Summary
Much attention is paid to the construction of optimal control laws for nonlinear systems and the corresponding approximate methods for their calculation. The asymptotic solution of the corresponding initial singularly perturbed problem for the discrete matrix Riccati equation with coefficients weakly dependent on the state and the corresponding one-point PA regulator is constructed using the SDRE approach. Taking into account the dependency of matrices Q(x, ε), A(x, ε), B(x, ε) on the small parameter a uniform asymptotic approximation of the second order for the solution of the singularly perturbed initial problem (9) is constructed using the boundary layer functions method (BLFM) [14,15].
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