Locus Of A Perturbed Relay System Research Articles

Analysis of chattering in homogenous sliding mode controllers using the locus of a perturbed relay system

AbstractThis work investigates the use of the locus of a perturbed relay system (LPRS) method for the analysis of chattering and other possible oscillations in discontinuous homogenous Sliding Mode Controllers (SMC). Particularly these homogeneous SMCs are considered as linear input‐to‐state and nonlinear separable state‐to‐output dynamics of Wiener type. The existence of non‐vanishing oscillations in these sliding mode control systems, when parasitic dynamics are present, is shown. This supports the claim that the finite‐time convergence for homogenous SMCs only occurs in the absence of any parasitic dynamics which is an ideal and non‐realizable condition. We investigate the chattering and other possible oscillations using the LPRS for the controllers proposed in the study by Ding et al. The undertaken analysis reveals multiple periodic solutions for the plants of relative degrees 2 through 4. The LPRS, originally developed for linear plants, is extended to linear input‐to‐state and nonlinear separable state‐to‐output dynamics. To investigate chattering, the parasitic dynamics are included in this analysis through a second‐order actuator model. The proposed approach to analysis provides exact values of the frequency and amplitude of chattering, as well as categorization of the obtained periodic solutions into orbitally stable and orbitally unstable. The analytical results are supported by simulations.

Analysis of Stability and Performance of a Cascaded PI Sliding-Mode Control DC–DC Boost Converter via LPRS

The analysis and design of a cascade proportional–integral (PI) sliding-mode control for a dc–dc boost converter using the locus of a perturbed relay system (LPRS) is presented in this article. The implemented system uses a sliding-mode controller for the inner loop and a PI controller for the outer loop to regulate the output voltage. An LPRS representation of the overall system is developed in this article. The latter is used to analyze the frequency and the amplitude of the chattering/ripple (self-excited oscillations) and relates them to the specifications. Besides, the LPRS-method-based analysis and design allows one to find the exact value of the equivalent gain, which is necessary for the analysis of disturbances propagation. Another important contribution is the detailed stability analysis of the two-loop controller through the LPRS method. System stability is analyzed for the two aspects, i.e., the chattering stability of the inner loop and the stability of the full-order two-loop averaged dynamics, through the concept of the equivalent gain. Furthermore, the LPRS is used to operate the converter at a specified switching frequency through defining the controller hysteresis value. Simulations and experimental results are provided to validate the theoretical analysis.

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.

Analysis of a Sliding Mode DC–DC Boost Converter Through LPRS of a Nonlinear Plant

An exact analysis of sliding mode boost converter dynamics based on the locus of a perturbed relay system (LPRS) method is presented in this article. An LPRS representation of the dc–dc boost converter based on a full nonlinear model is developed. This is a first known attempt to apply the LPRS concept to an industrial process having a nonlinear model. The frequency and amplitude of the self-excited oscillations (chattering/ripple) are analyzed through the LPRS method. The LPRS is also used to analyze the effect of the propagation of an external disturbance (fluctuations in the source voltage). It is found that the effect of source voltage fluctuations on the system dynamics can be attributed to a combination of the effects of amplitude modulation and input signal propagation. Further, LPRS is used in the design problem of the dc–dc boost converter prototype to determine the required controller hysteresis value which ensures the desired operating switching frequency. Simulation and experimental results are presented to validate and illustrate the theoretical analysis.

Analysis of a sliding mode boost converter under fluctuating input source voltage, using LPRS method

DC/DC converters require a controller that ensures high performance under the variations of load and source voltage, ensuring stability and a fast transient response at all operating points. This paper provides a detailed analysis of the dynamics of sliding mode boost converter using the locus of a perturbed relay system (LPRS) method. LPRS allows for an exact determination of the amplitude and the frequency of the output voltage oscillations as well as the equivalent gain. Furthermore, it describes the external signal propagation through the system. Analysis of the effect of the source voltage fluctuations on the output voltage waveform is presented in the paper as analysis of exogenous disturbance propagation. The mathematical analysis is verified through simulations and hardware experiments.

Analysis of oscillations in discontinuous Lurie systems via LPRS method

We discuss advantages and limitations of the harmonic balance method and the locus of a perturbed relay system (LPRS) method in the problem of finding periodic oscillations. In this paper we present the results of using harmonic balance method and LPRS method while investigating a 3rd order dynamic system in Lurie form. In this system a symmetric periodic oscillation is found, while other two asymmetric periodic motions are not found using both methods.

LPRS analysis of sliding mode buck converter

DC-DC buck (step-down) converters are widely used in various power systems. An exact analysis of buck converter dynamics based on the locus of a perturbed relay system (LPRS) is proposed. Beside conventional analysis tasks – determination of the frequency and amplitude of the self-excited oscillations (chattering), which characterize ripple of the converter output signal - analysis of the propagation of source voltage fluctuations is presented. The latter is conveniently done through the same LPRS method. The results are supported by simulations with the use of the conventional model of the converter.

Disturbance attenuation for systems with second‐order sliding modes via linear compensators

Non-ideal disturbance attenuation emerges in systems with sliding mode controllers when parasitic dynamics are present. In this work, the problem of non-ideal disturbance attenuation in systems controlled by second-order sliding modes is considered. Describing function and locus of a perturbed relay system (LPRS) methods are used for analysis of non-ideal disturbance attenuation. The proposed solution is to introduce a linear compensator (series or parallel) in the control loop to modify the system frequency response, that is, the Nyquist plot or LPRS. The design methodology for both parallel and series compensators is presented. The proposed solution allows one to improve disturbance attenuation and tracking quality, thereby reducing the effect of parasitic dynamics which is reflected in the system frequency response. Experiments supporting the proposed approach and simulations are provided.

Analysis and shaping of the self-sustained oscillations in relay controlled fractional-order systems

Abstract This work deals with Single-Input-Single-Output (SISO) fractional order systems with a discontinuous relay control element in the feedback loop. Stable self-sustained oscillations often occur in the closed loop relay system, and this work takes advantage of Describing Function (DF) analysis and of another more accurate approach, called Locus of a Perturbed Relay System (LPRS) method, for analyzing in the frequency domain the characteristics of the limit cycle oscillations. The use of fractional lead compensator is also suggested for the purpose of shaping the characteristics of the limit cycle. The proposed analysis and design procedures will be supported by thoroughly discussed simulation examples.

Frequency domain precision analysis and design of sliding mode observers

Abstract 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.

Oscillations and transfer properties of relay feedback systems with time-delay linear plants

Oscillations and transfer properties of relay feedback systems with linear plants having a dead time (time delay) are considered. Poincare maps are found explicitly in terms of the system parameters. The oscillatory and transfer properties are presented in the format of the locus of a perturbed relay system (LPRS), which represents the continuum of all possible periodic solutions and transfer properties of the system with a given linear plant. Conditions of local orbital asymptotic stability are found in terms of the system parameters. The proposed approach is illustrated by examples.

On the Transfer Properties of the “Generalized Sub-Optimal” Second-Order Sliding Mode Control Algorithm

A class of second order sliding mode control systems is studied in terms of its transfer properties, which allows for investigating the system response to an external signal representing either a disturbance, which the system is supposed to reject, or the reference input, which the system is supposed to track. An approach based on the locus of a perturbed relay system (LPRS) is developed making reference to linear plant/actuator/sensor dynamics and to the ldquogeneralized sub-optimalrdquo second-order sliding mode control algorithm in the closed loop. A thoroughly discussed example shows the feasibility of the presented procedure and confirms the theoretical results.

Oscillations and Transfer Properties of Relay Servo Systems With Integrating Plants

Asymmetric limit cycles and input-output properties of relay feedback systems with integrating plants are studied in the technical note. It is noted that an integrating plant introduces significantly different dynamics, so that a different than traditional system representation is needed to obtain a fixed point of Poincare map. Conditions of orbital asymptotic stability are obtained. The infinite set of possible periodic solutions and transfer characteristics is presented in the form of the locus of a perturbed relay system (LPRS). A formula of LPRS is derived. An example of analysis of oscillations and transfer properties is given.

Analysis of Closed-Loop Performance and Frequency-Domain Design of Compensating Filters for Sliding Mode Control Systems

For a sliding mode (SM) system containing parasitic dynamics, a model of chattering and a model of the averaged motions of the order equal to the original order of the system are obtained via the use of the locus of a perturbed relay system (LPRS) method. The latter model proves that the averaged motions in an SM system differ from those in the reduced-order model. In particular, such phenomena as the nonideal disturbance rejection and the nonideal input tracking are observed. It is shown that the non-reduced-order model transforms into the reduced one when the parasitic dynamics are removed. Yet, the chattering and the averaged motion - being the two components of the motion in the SM system - belong to two different frequency ranges (spectra). As a result, a frequency characteristic shaping aimed at achieving the desired frequency of chattering and enhancing the closed-loop performance becomes possible. This opportunity is explored in this paper. Examples that illustrate the proposed approach are provided.

Frequency Domain Input–Output Analysis of Sliding-Mode Observers

The sliding-mode (SM) observer dynamics are analyzed in the frequency domain. It is shown that the SM observer is essentially a relay feedback-feedforward system. It is feedback with respect to the measured variable of the system being observed, and the feedforward with respect to the control applied to the system being observed. Having noted this, the input-output properties of observer dynamics are analyzed with the use of the locus of a perturbed relay system (LPRS) method. The performance of the observer is assessed via the use of the frequency response techniques. An example of performance analysis of the observer dynamics in the frequency domain is given

SELF-TUNING PI CONTROLLER

A self-tuning algorithm comprising the steps of identification and tuning is proposed. The methodology of process parameters identification is based on the measurement of the locus of a perturbed relay system (LPRS) points from a single or multiple asymmetric relay feedback tests. An algorithm of identification of the first order plus dead time process model is developed in details. Also, a tuning algorithm using the criterion of minimal settling time subject to the overshoot constraints is developed and presented in the form of simple formulas for the first-order plus dead time process model. Examples of simulations and application are given.

SENSOR DYNAMICS COMPENSATOR FOR TEMPERATURE MEASUREMENT IN COMBUSTION CHAMBERS OF UTILITY BOILERS AND INCINERATORS

A sliding mode compensation scheme for the dynamics of the sensors used for temperature measurements in the combustion chambers of utility boilers and incinerators is proposed. The proposed compensator can be used as a part of the electronics of temperature transmitters. Analysis and design are carried out in the frequency domain with the use of the locus of a perturbed relay system (LPRS) method. Frequency-domain characteristics of the compensator dynamics are derived. An example of design is provided.

Analysis of sliding modes in the frequency domain

A frequency-domain approach to the analysis of real and ideal sliding modes (SM) in single input single output (SISO) systems with linear plants is proposed. The real SM is analysed as a superposition of fast oscillations and slow motions in a relay system. The approach is based on the notion of the locus of a perturbed relay system (LPRS). The proposed LPRS is defined as a characteristic of the response of a linear part to an unequally spaced pulse control of variable frequency in a closed-loop. Formulas for computing the LPRS are derived. It is demonstrated that beside the well know chattering phenomenon, real SM also exhibits a non-ideal disturbance rejection, which can be quantitatively assessed with the use of the proposed approach. An example of the LPRS analysis of a SM system is given.

Oscillations and transfer properties of relay servo systems—the locus of a perturbed relay system approach

A frequency-domain method of the input–output analysis of a class of relay feedback systems is proposed. The approach is based on the notion of the locus of a perturbed relay system (LPRS). The proposed LPRS is defined as a characteristic of the response of a linear part to an unequally spaced pulse control of variable frequency in a closed-loop system. Consequently, the LPRS comprises oscillatory and transfer properties of the relay system. Formulas for computing the LPRS are derived. An example of input–output analysis is given.