Regions of robust relative stability for PI controllers and LTI plants with unstructured multiplicative uncertainty: A second-order-based example.

This example-oriented article addresses the computation of regions of all robustly relatively stabilizing Proportional-Integral (PI) controllers under various robust stability margins α for Linear Time-Invariant (LTI) plants with unstructured multiplicative uncertainty, where the plant model with multiplicative uncertainty is built on the basis of the second-order plant with three uncertain parameters. The applied graphical method, adopted from the authors' previous work, is grounded in finding the contour that is linked to the pairs of P-I coefficients marginally fulfilling the condition of robust relative stability expressed using the H∞ norm. The illustrative example in the current article emphasizes that the technique itself for plotting the boundary contour of robust relative stability needs to be combined with the precondition of the nominally stable feedback control system and with the line for which the integral parameter equals zero in order to get the final robust relative stability regions. The calculations of the robust relative stability regions for various robust stability margins α are followed by the demonstration of the control behavior for two selected controllers applied to a set of members from the family of plants.

Calculation of robustly relatively stabilizing PID controllers for linear time-invariant systems with unstructured uncertainty.

This article deals with the calculation of all robustly relatively stabilizing (or robustly stabilizing as a special case) Proportional-Integral-Derivative (PID) controllers for Linear Time-Invariant (LTI) systems with unstructured uncertainty. The presented method is based on plotting the envelope that corresponds to the trios of P-I-D parameters marginally complying with given robust stability or robust relative stability condition formulated by means of the H∞ norm. Thus, this approach enables obtaining the region of robustly stabilizing or robustly relatively stabilizing controllers in a P-I-D space. The applicability of the technique is demonstrated in the illustrative examples, in which the regions of robustly stabilizing and robustly relatively stabilizing PID controllers are obtained for a controlled plant model with unstructured multiplicative uncertainty and unstructured additive uncertainty. Moreover, the method is also verified on the real laboratory model of a hot-air tunnel, for which two representative controllers from the robust relative stability region are selected and implemented.

Frequency frame approach on tuning FOPI controller for TOPTD thermal processes.

The frequency frame is used to tune fractional order proportional-integral​ controllers for stability, performance and robustness of third order plus time delay plants. Such plants are frequently used in describing thermal processes such as an air heater or a fired boiler. The aim is to tune the controller to meet some frequency domain properties. As robustness is an indispensable issue for thermal processes, main inspiration of the paper comes from flattening the phase curve in the Bode plot to provide improved robustness for the system. In spite of some existing studies, flattening is not realized by equalizing the phase derivative to zero at a given frequency value. Firstly, gain and phase crossover frequency points are enclosed with a rectangular frame. Then, lengths of the edges of this frame are changed to tune phase and gain margins. Curves inside the frame can be flattened by proper tuning of the edges. This will enhance the robustness and also ensure the iso-damping property. Equations to obtain the controller are given with two theorems. Demonstrations are made on two different thermal plants which are a novel electrical air heater and a bagasse fired boiler and the results are given on detailed illustrations. The results proved that preferred gain and phase properties are successfully obtained and improved performance and robustness are provided for related systems.

Disturbance rejection FOPID controller design in v-domain.

Due to the adverse effects of unpredictable environmental disturbances on real control systems, robustness of control performance becomes a substantial asset for control system design. This study introduces a v-domain optimal design scheme for Fractional Order Proportional-Integral-Derivative (FOPID) controllers with adoption of Genetic Algorithm (GA) optimization. The proposed design scheme performs placement of system pole with minimum angle to the first Riemann sheet in order to obtain improved disturbance rejection control performance. In this manner, optimal placement of the minimum angle system pole is conducted by fulfilling a predefined reference to disturbance rate (RDR) design specification. For a computer-aided solution of this optimal design problem, a multi-objective controller design strategy is presented by adopting GA. Illustrative design examples are demonstrated to evaluate performance of designed FOPID controllers.

Frequency frame approach on loop shaping of first order plus time delay systems using fractional order PI controller.

This study proposes an analytical design method of fractional order proportional integral (FOPI) controllers for first order plus time delay (FOPTD) systems. Suggested technique obtains the general computation equations of controllers for such systems. These equations are used to tune controller parameters to meet specified frequency and phase properties to satisfy the stability of whole system. It is found that the designed controllers not only make the system stable, but also have positive effect on the performance and robustness of the system. Main contribution of the paper lays on this thought. There proposed a concept, "frequency frame" which encloses the curves between phase and gain crossover frequencies in Bode plot. Robustness of the control system can be improved by expanding or constricting the edges of this frame and flattening the curves inside the frame. Thus, any case that leads the system to instability can be avoided. Analytically derived equations are tested with proper examples and the results are shown illustratively. Advantages and disadvantages of the method are comparatively given.

Linear systems with unstructured multiplicative uncertainty: Modeling and robust stability analysis.

This article deals with continuous-time Linear Time-Invariant (LTI) Single-Input Single-Output (SISO) systems affected by unstructured multiplicative uncertainty. More specifically, its aim is to present an approach to the construction of uncertain models based on the appropriate selection of a nominal system and a weight function and to apply the fundamentals of robust stability investigation for considered sort of systems. The initial theoretical parts are followed by three extensive illustrative examples in which the first order time-delay, second order and third order plants with parametric uncertainty are modeled as systems with unstructured multiplicative uncertainty and subsequently, the robust stability of selected feedback loops containing constructed models and chosen controllers is analyzed and obtained results are discussed.

Robust stability of fractional order polynomials with complicated uncertainty structure.

The main aim of this article is to present a graphical approach to robust stability analysis for families of fractional order (quasi-)polynomials with complicated uncertainty structure. More specifically, the work emphasizes the multilinear, polynomial and general structures of uncertainty and, moreover, the retarded quasi-polynomials with parametric uncertainty are studied. Since the families with these complex uncertainty structures suffer from the lack of analytical tools, their robust stability is investigated by numerical calculation and depiction of the value sets and subsequent application of the zero exclusion condition.

Probabilistic robust stabilization of fractional order systems with interval uncertainty.

This study investigates effects of fractional order perturbation on the robust stability of linear time invariant systems with interval uncertainty. For this propose, a probabilistic stability analysis method based on characteristic root region accommodation in the first Riemann sheet is developed for interval systems. Stability probability distribution is calculated with respect to value of fractional order. Thus, we can figure out the fractional order interval, which makes the system robust stable. Moreover, the dependence of robust stability on the fractional order perturbation is analyzed by calculating the order sensitivity of characteristic polynomials. This probabilistic approach is also used to develop a robust stabilization algorithm based on parametric perturbation strategy. We present numerical examples demonstrating utilization of stability probability distribution in robust stabilization problems of interval uncertain systems.

A numerical investigation for robust stability of fractional-order uncertain systems.

This study presents numerical methods for robust stability analysis of closed loop control systems with parameter uncertainty. Methods are based on scan sampling of interval characteristic polynomials from the hypercube of parameter space. Exposed-edge polynomial sampling is used to reduce the computational complexity of robust stability analysis. Computer experiments are used for demonstration of the proposed robust stability test procedures.