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.

Further experimental results on modelling and algebraic control of a delayed looped heating-cooling process under uncertainties.

The aim of this research is to revise and substantially extend experimental modelling and control of a looped heating-cooling laboratory process with long input-output and internal delays under uncertainties. This research follows and extends the authors' recent results. As several significant improvements regarding robust modelling and control have been reached, the obtained results are provided with a link and comparison to the previous findings. First, an infinite-dimensional model based on mathematical-physical heat and mass transfer principles is developed. All important heat-fluid transport and control-signal delays are considered when assembling the model structure and relations of quantities. Model parameter values optimization based on the measurement data follows. When determining static model parameter values, all variations in steady-state measured data are taken into account simultaneously, which enhances previously obtained models. Values of dynamic model parameters and delays are further obtained by least mean square optimization. This innovative model is compared to two recently developed process models and to the best-fit model that ignores the measured variations. Controller structures are designed using algebraic tools for all four models. The designed controllers are robust in the sense of robust stability and performance. Both concepts are rigorously assessed, and the obtained conditions serve for controller parameter tuning. Two different control systems are assumed: the standard closed-loop feedback loop and the two-feedback-controllers control system. Numerous experimental measurements for nominal conditions and selected perturbations are performed. Obtained results are further analyzed via several criteria on manipulated input and controlled temperature. The designed controllers are compared to the Smith predictor structure that is well-established for time-delay systems control. An essential drawback of the predictor regarding disturbance rejection is highlighted.

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.

Parameter identification of a delayed infinite-dimensional heat-exchanger process based on relay feedback and root loci analysis.

The focus of this contribution is twofold. The first part aims at the rigorous and complete analysis of pole loci of a simple delayed model, the characteristic function of which is represented by a quasi-polynomial with a non-delay and a delay parameter. The derived spectrum constitutes an infinite set, making it a suitable and simple-enough representative of even high-order process dynamics. The second part intends to apply the simple infinite-dimensional model for relay-based parameter identification of a more complex model of a heating-cooling process with heat exchangers. Processes of this type and construction are widely used in industry. The identification procedure has two substantial steps. The first one adopts the simple model with a low computational effort using the saturated relay that provides a more accurate estimation than the standard on/off test. Then, this result is transformed to the estimation of the initial characteristic equation parameters of the complex infinite-dimensional heat-exchanger model using the exact dominant-pole-loci assignment. The benefit of this technique is that multiple model parameters can be estimated under a single relay test. The second step attempts to estimate the remaining model parameters by various numerical optimization techniques and also to enhance all model parameters via the Autotune Variation Plus relay experiment for comparison. Although the obtained unordinary time and frequency domain responses may yield satisfactory results for control tasks, the identified model parameters may not reflect the actual values of process physical quantities.

Experimental Investigation and Control of a Hot-Air Tunnel with Improved Performance and Energy Saving.

The paper is focused on the identification, control design, and experimental verification of a two-input two-output hot-air laboratory apparatus representing a small-scale version of appliances widely used in the industry. A decentralized multivariable controller design is proposed, satisfying control-loop decoupling and measurable disturbance rejection. The proposed inverted or equivalent noninverted decoupling controllers serve for the rejection of cross-interactions in controlled loops, whereas open-loop antidisturbance members satisfy the absolute invariance to the disturbances. Explicit controller-structure design formulae are derived, and their equivalence to other decoupling schemes is proven. Three tuning rules are used to set primary controller parameters, which are further discretized. All the control responses are simulated in the Matlab/Simulink environment. In the experimental part, two data-acquisition, communication, and control interfaces are set up. Namely, a programmable logic controller and a computer equipped with the peripheral component interconnect card commonly used in industrial practice are implemented. A simple supervisory control and data acquisition human-machine interface via the Control Web environment is developed. The laboratory experiments prove better temperature control performance measured by integral criteria by 35.3%, less energy consumption by up to 6%, and control effort of mechanical actuator parts by up to 17.1% for our method compared to the coupled or disturbance-ignoring design in practice. It was also observed that the use of a programmable logic controller gives better performance measures for both temperature and air-flow control.

Gridding discretization-based multiple stability switching delay search algorithm: The movement of a human being on a controlled swaying bow.

Delay represents a significant phenomenon in the dynamics of many human-related systems-including biological ones. It has i.a. a decisive impact on system stability, and the study of this influence is often mathematically demanding. This paper presents a computationally simple numerical gridding algorithm for the determination of stability margin delay values in multiple-delay linear systems. The characteristic quasi-polynomial-the roots of which decide about stability-is subjected to iterative discretization by means of pre-warped bilinear transformation. Then, a linear and a quadratic interpolation are applied to obtain the associated characteristic polynomial with integer powers. The roots of the associated characteristic polynomial are closely related to the estimation of roots of the original characteristic quasi-polynomial which agrees with the system's eigenvalues. Since the stability border is crossed by the leading one, the switching root locus is enhanced using the Regula Falsi interpolation method. Our methodology is implemented on-and verified by-a numerical bio-cybernetic example of the stabilization of a human-being's movement on a controlled swaying bow. The advantage of the proposed novel algorithm lies in the possibility of the rapid computation of polynomial zeros by means of standard programs for technical computing; in the low level of mathematical knowledge required; and, in the sufficiently high precision of the roots loci estimation. The relationship to the direct search QuasiPolynomial (mapping) Rootfinder algorithm and computational complexity are discussed as well. This algorithm is also applicable for systems with non-commensurate delays.

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.