In this paper, we propose a new method for the synthesis of robust proportional, integral, and derivative (PID) control systems. The proposed method is based on the idea of partial model matching. The stability feeler, a tool for robust stability analysis, is used to stabilize closed-loop systems with uncertainties. An advantage of the proposed method is that it guarantees robust stability of control systems. To clarify the effectiveness of the proposed method, two types of simulations are given. One is the case where the nominal plant is not stable. The other is the case where the plant is not robustly stable although the nominal plant is stable. The simulation results when the nominal plant is not stable show that the controller derived by the proposed method robustly stabilizes the closed-loop system, whereas that derived by conventional partial model matching method cannot stabilize it. Moreover, when the plant is not robustly stable, it can be seen that the controller by the proposed method can robustly stabilize the system. Therefore, it is shown that the proposed method enables one to derive PID controllers robustly stabilizing closed-loop systems.

Robust D-stability analyses of uncertain characteristic polynomials are considered. This paper shows that the stability feeler can be applied to robust D-stability analysis of linear time-invariant systems with disconnected stability region in the complex plane. We also show an application of the stability feeler to stability analysis for a lateral dynamics of an aircraft.

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SUMMARYStability analysis of a linear uncertain polynomial system in parameter space is one of the central themes of recent control theory. It often requires an enormous number of stability checks with conventional stability test methods. A delta operator polytopic polynomial, a convex combination of some delta operator vertex polynomials, is one of such cases. In this paper, robust stability is addressed for polytopic delta operator polynomials and several necessary and sufficient stability conditions are derived. The delta operator, an operator used to express discrete time systems, is known to have significant features: numerical advantage in implementation and ability to smoothly connect the z-operator with the Laplace operator. Based on this last feature and on the celebrated Edge theorem, we first derive three kinds of exact stability conditions for the uncertain delta operator polynomials. We then extend the directional stability radius method, which was developed for diamond polynomials, so that it can also be applied to polytopic polynomials. This extension gives rise to the fourth stability test. Furthermore, it is shown from the result of the numerical experiments with these stability analysis methods that one of these four methods, which uses eigenvalues of matrices, turns out to be most efficient for stability analysis of the polytope.

This paper gives a novel method to design less-conservative robust PID control systems. The partial model matching method is a design method for PID control systems. That is a method to design a control system that matches a reference model as closely as possible. This method does not guarantee robust stability of a PID control system because it is assumed that there are no uncertainties on the coefficients of the lower-order terms of the plant. Therefore, in our previous work, a robust design method based on the partial model matching method was proposed. However, the drawback of this conventional method is that the results of stability analysis are conservative. In this paper, we aim to reduce the conservativeness caused by the conventional method. To achieve this goal, we use a robust stability analysis method based on the Cremer-Leonhard-Mikhailov criterion and the mapping theorem. Moreover, we also propose a new method to update the value of a design parameter. A numerical example shown in this paper implies that the conservativeness is reduced by the proposed method.