Design of non-parametric process-specific optimal tuning rules for PID control of flow loops

The problem of designing optimal process-specific rules for non-parametric tuning is undertaken in the paper. It is shown that producing non-parametric process-specific optimal tuning rules for PID controllers leads to the problem that can be characterized as optimization under uncertainty. This happens due to the fact that tuning rules, unlike tuning constants, are produced not for a particular process or plant model but for a set of models from a certain domain. The novelty of the proposed approach is that the problem of obtaining optimal tuning rules for a flow process is formulated and solved as a problem of optimization of an integral performance criterion parametrized through values that define the domain of available process models. The considered non-parametric tuning assumes the use of the modified relay feedback test (MRFT) recently proposed in the literature. It allows one to tune the PID controller satisfying the requirements to gain or phase margins that is achieved through coordinated selection of tuning rules and test parameters. This approach constitutes a holistic approach to tuning. In the present paper, optimal tuning rules coupled with MRFT, for flow loops, are proposed. Final results are presented in the form of tables containing coefficients of optimal tuning rules for the PI controller, obtained for a number of specified gain margins. The produced non-parametric tuning rules well agree with the practice of loop tuning.     

Dynamic harmonic balance principle and analysis of rocking block motions

The harmonic balance (HB) principle is a powerful and convenient tool for finding periodic solutions in nonlinear systems. In the present paper, this principle is extended to transient processes in systems with one single-valued odd-symmetric nonlinearity and linear plant not having zeros in the transfer function, and named the dynamic HB. Based on the dynamic HB, first the equations for the amplitude, frequency, and amplitude decay of an oscillatory process in the Lure system are derived. It is then applied to analysis of rocking block decaying motions. An example is provided.     

Stabilization of artificial gas-lift process using nonlinear predictive generalized minimum variance control

Artificial gas-lift (AGL) is one of the most widely used methods in oil production to maintain acceptable oil flow to the processing equipment and sales when the reservoir pressure is not high enough. In spite of its popularity, the AGL process is prone to casing-heading instability, which is revealed as significant flow oscillation. This is undesirable as it results in production losses and unstable behavior that has negative impact on the downstream equipment. Controller design for such a process is very challenging as it exhibits highly nonlinear dynamics. In this work, the predictive generalized minimum variance control (NPGMV) is employed to derive a robust controller based on the state estimation to stabilize AGL process when casing-heading phenomenon occurs. A closed-form optimal control law is obtained based on the Taylor series approximation. Further, a nonlinear state observer is produced and combined with the controller to ensure closed-loop control through variables that are most beneficial to the system performance, which are unmeasurable and can be obtained only via estimation. Through simulation studies, the effectiveness of the proposed controller is demonstrated.     

Analysis of orbital stability in lure system,based on dynamic harmonic balance

A criterion of orbital stability of a limit cycle in a Lure system is formulated using the dynamic harmonic balance (DHB) principle and the describing function (DF) method. It is demonstrated that a more precise formulation of orbital stability than that provided by the Loeb's criterion can be produced. This enhancement is achieved through elimination of the assumption that is used in Loeb's criterion derivation. It is demonstrated in the paper that this assumption does not hold. An example of analysis is given.     

On relative degree,chattering and fractal nature of parasitic dynamics in sliding mode control

With respect to relative degree and chattering in sliding mode (SM) control systems, the notion of fractal dynamics is introduced, and a conjecture is formulated that the character of parasitic dynamics of real control systems is fractal. A model of fractal dynamics is proposed. The characteristics of fractal dynamics are studied in the frequency and time domains. It is shown that with fractal parasitic dynamics SM control systems will always feature chattering and non-ideal closed-loop performance. An example of analysis is provided.     

Finding exact periodic solutions in Lure systems through periodic signal mapping

A solution of the periodic problem in a nonlinear system comprising a single-valued symmetric nonlinearity (Lure system) and linear dynamics is presented. The solution is designed as an iterative application of Periodic Signal Mapping to refine the approximate solution obtained through the describing function method. The algorithm is based upon the transformation of the original nonlinear system into an equivalent nonlinear system for which the filtering hypothesis is satisfied exactly, and for that reason the latter being suitable for application of the developed algorithm. The solution sought for is a fixed point of the periodic signal mapping. Conditions of asymptotic convergence of the proposed algorithm are given. The proposed approach is illustrated by two examples of analysis of periodic motions in a relay feedback system and chattering in a terminal sliding mode.     

Frequency domain precision analysis and design of sliding mode observers

Estimation precision and bandwidth of sliding mode (SM) observers are analyzed in the frequency domain for different settings of the observer design parameters. It was shown previously that the SM observer could be analyzed as a relay feedback-feedforward system. It is feedback with respect to the measured variable of the system being observed, and feedforward with respect to the control applied to the system being observed. This approach is now further extended to analysis of effects of design parameter change on observer performance. An example of SM observer design for estimation of DC motor speed from the measurements of armature current is considered in the paper. The input-output properties of observer dynamics are analyzed with the use of the locus of a perturbed relay system (LPRS) method.     

Design of rules for in-flight non-parametric tuning of PID controllers for unmanned aerial vehicles

Designing high performance controllers for multirotors is a rigorous task that is often solved by trial and error approach. Trial and error tuning usually results in non-optimal controller parameters. Tuning controllers based on the existing quadrotor models would result in poor performance of quadrotors due to simplifications and inaccuracies in the underlying models. In this paper optimal tuning rules for quadrotor attitude dynamics are designed, which guarantees near-optimal performance and robustness. A single in-flight run of the Modified Relay Feedback Test that takes only few seconds with guaranteed stability is enough to have near-optimal tuning of the controller. The designed tuning rule is tested experimentally in-flight on a custom-built quadrotor. The results showed significant advantages in performance and robustness due to the proposed approach.     

On asymmetric periodic solutions in relay feedback systems

Asymmetric self-excited periodic motions or periodic solutions which are produced by relay feedback systems that have symmetric characteristics are studied in the paper. Two different mechanisms of producing an asymmetric oscillation by a system with symmetric properties are noted and analyzed by the locus of a perturbed relay system (LPRS) method. Bifurcation between the ability to excite symmetric and asymmetric oscillation with variation of system parameters is analyzed. An algorithm of finding asymmetric solutions is proposed.     

Optimal non-parametric tuning of PID controllers based on classification of shapes of oscillations in modified relay feedback test

In this work, tuning rules of the PID controller have been developed by categorizing a system's response into distinct classes. The classes are formed using the shapes of the test oscillations induced by the system under the Modified Relay Feedback Test (MRFT) produced by specific system models. It is proposed that a physical system can be categorized into one of the proposed classes and thus the tuning rules for a particular class can apply to any kind of system from this class. The idea of producing tuning rules that are based on the shape of the oscillations induced in the loop containing the process comes from the observations that oscillatory responses of physical systems reveal just a few different shapes depending on system dynamics. For applying the developed optimal tuning rules for an arbitrary system, first, certain system characteristics are determined using a priori knowledge of the class model. Then the system's response with the application of the MRFT is examined to classify the oscillation waveform/shape. In this work, such classification is carried out using a cross-correlation algorithm. Finally, a class tuning rules are applied.