Multiplicative Uncertainty Research Articles

A new analytical method for designing centralised PI controllers for unstable systems using a direct synthesis approach

This paper introduces a new analytical technique using a direct synthesis strategy to develop centralised proportional–integral (PI) controllers for multivariable processes. The current method design controller is focused on attaining the desired closed-loop response for multi input multi output (MIMO) processes that involve multiple time delays. The conventional multivariable PI controller is obtained by approximating the ideal multivariable controller by the Maclaurin series expansion. This is accomplished by choosing the desired closed-loop response order as the system order plus two. Subsequent analysis investigates the proposed method’s efficacy in designing multivariable PI controllers. Servo and regulatory problems studies were conducted to find the effectiveness of the proposed method and compared it with other methods recently reported in the literature, as well as assessment of performance indices like integral absolute error (IAE) and integral square error (ISE). The proposed controller's robustness is assessed by plotting the inverse maximum singular value against frequency, both input and output multiplicative uncertainties.

Robust Fractional-Order PI/PD Controllers for a Cascade Control Structure of Servo Systems

In this paper, a cascade control structure is suggested to control servo systems that normally include a servo motor in coupling with two kinds of mechanism elements, a translational or rotational movement. These kinds of systems have high demands for performance in terms of fastest response and no overshoot/oscillation to a ramp function input. The fractional-order proportional integral (FOPI) and proportional derivative (FOPD) controllers are addressed to deal with those control problems due to their flexibility in tuning rules and robustness. The tuning rules are designed in the frequency domain based on the concept of the direct synthesis method and also ensure the robust stability of controlled systems by using the maximum sensitivity function. The M-Δ structure, using multiplicative output uncertainties for both control loops simultaneously, is addressed to justify the robustness of the controlled systems. Simulation studies are considered for two kinds of plants that prove the effectiveness of the proposed method, with good tracking of the ramp function input under the effects of the disturbances. In addition, the robustness of the controlled system is illustrated by a structured singular value (µ) plot in which its value is less than 1 over the frequency range.

A dynamic optimal decoupling controller design for a multi-variable system with stability analysis: an algebraic approach

ABSTRACT This study describes the design and implementation of an algebraic approach for a dynamic optimal decoupling controller in a multi-variable liquid-level system. This method aligns with model-matching criteria and the frequency-matching technique. The prime objective of the proposed approach is to create a closed-loop feedback system using a PID controller that aligns with a user-defined linear system model in terms of both system dynamics and static behavioral response. To illustrate robust stability, the study incorporates multiplicative input and output uncertainty. Simulation results verify the efficacy of the proposed decentralized controller, presenting its effectiveness in achieving both set-point accuracy and disturbance attenuation. Furthermore, the study employs disk margin analysis to ascertain safe ranges for gain and phase margin.

A Novel Failure-Distribution-Dependent Non-Fragile $H_\infty$ Fault-Tolerant Load Frequency Control for Faulty Multi-Area Power Systems

This paper studies the non-fragile <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H_\infty$</tex-math></inline-formula> fault-tolerant load frequency control problem for multi-area power systems with actuator failures. At first, considering the distribution characteristics of partial failures, a stochastic distribution model is established for actuator partial failures. Next, for the perturbations of controller gains, not only the additive uncertainties but also the multiplicative uncertainties are incorporated into the fault-tolerant controller design. The coupling items which hardly directly solved have turned into a W-problem in fault-tolerant load frequency control. Furthermore, a novel failure-distribution-dependent non-fragile <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H_\infty$</tex-math></inline-formula> fault-tolerant load frequency control algorithm is given by using linear matrix inequalities technology and Lyapunov-Krasovskii functional method. Finally, a numerical example with a three-area power system are given to show the effectiveness of the proposed methods.

μ-Analysis and μ-Synthesis Control Methods in Smart Structure Disturbance Suppression with Reduced Order Control

In this study, we created an accurate model for a homogenous smart structure. After modeling multiplicative uncertainty, an ideal robust controller was designed using μ-synthesis and a reduced-order H-infinity Feedback Optimal Output (Hifoo) controller, leading to the creation of an improved uncertain plant. A powerful controller was built using a larger plant that included the nominal model and corresponding uncertainty. The designed controllers demonstrated robust and nominal performance when handling agitated plants. A comparison of the results was conducted. As an example of a general smart structure, the vibration of a collocated piezoelectric actuator and sensor was controlled using two different approaches with strong controller designs. This study presents a comprehensive simulation of the oscillation suppression problem for smart beams. They provide an analytical demonstration of how uncertainty is introduced into the model. The desired outcomes were achieved by utilizing Simulink and MATLAB (v. 8.0) programming tools.

ADP‐based robust consensus for multi‐agent systems with unknown dynamics and random uncertain channels

AbstractThis article pursues the robust consensus for linear discrete‐time multi‐agent systems with unknown dynamics and random uncertain channels. Stochastic multiplicative uncertainties depending on relative states of agents inevitably appear in the signal transmission channels. To handle these uncertainties, the distributed robust controllers are designed. Before designing this robust controller, the optimal control problem of a single‐agent system is considered first. Based on the technology of adaptive dynamic programming, a model‐free off‐policy algorithm is proposed to solve the optimal controller for every agent with fully unknown system dynamics. According to the theory of mean square stability, it is revealed that the distributed robust controller can be obtained by using the optimal controller of the single‐agent system and other related parameters without relying on the dynamic information of the systems. Besides, the relationship between uncertainty and the robust controller is described by a sufficient condition. Finally, to demonstrate the effectiveness of the theoretical analysis, a numerical example is presented.

Grid-Forming Inverter Primary Control Using Robust-Residual-Observer-Based Digital-Twin Model

Grid-forming (GFM) inverters with inductor-capacitor (LC) filters are used to interface distributed generation with the grid. However, the GFM inverter systems are prone to incipient multiplicative faults due to the inherent parameter drift phenomenon of the LC filter. Motivated by this scenario, this paper presents a primary active fault-tolerant control (PAFTC) strategy based on the robust-residual-observer-based-digital-twin model (RRODTM) to achieve the fault-tolerant control for the incipient multiplicative faults. The RRODTM is composed of a robust residual observer and a residual-driven compensation controller. Specially, the residual-driven compensation controller is designed by the data-driven method based on the multiplicative uncertainty caused by the incipient multiplicative faults. At the same time, this paper constructs a data-driven-event-trigger mechanism based on stability margin to achieve the plug-and-play (PnP) function of the residual-driven compensation controller. Based on these, the PAFTC strategy improves the control performance online according to the degree of the incipient multiplicative fault. Comprehensive experiments show that compared to well-known control schemes in the literature, the PAFTC strategy improves the power distribution accuracy and transient response speed as well as enhances the robustness of the system.

Two-Step Input Excitation Disturbance Estimation-Based Self-Adjustment Control for Spacecrafts With Multiplicative Uncertainties

Two-Step Input Excitation Disturbance Estimation-Based Self-Adjustment Control for Spacecrafts With Multiplicative Uncertainties

STABILITY ROBUSTNESS OF CONTROL SYSTEMS OF MULTIROTOR UNMANNED AERIAL VEHICLES

Multirotor unmanned aerial vehicles (UAVs) are broadly used in various military and civilian areas and tasks. In real flights of the multirotor UAVs, certain unexpected situations may occur, which can lead to improper functionality and even failures of elements or devices of the UAV’s control system. In some cases, this can even lead to the complete collapse of the UAV. Therefore, the safety of UAV flights is nowadays of prime importance. A method of analysis of the stability robustness of the UAV’s control systems with respect to the parameters multiplicative uncertainties (or perturbations), including possible efficiency degradation of DC motors, is proposed in the paper. Stability robustness is determined by a simple graphical procedure on the complex plane of the Nyquist or Nichols hodographs of the control system’s separate channels. That procedure is quite similar to the well-known classical feedback control procedure of determining the peak gain of the single-input, single-output closed-loop control system by the Nyquist hodograph of the open-loop system. A numerical example illustrating the proposed method is given.

Research on the Current Control Strategy of a Brushless DC Motor Utilizing Infinite Mixed Sensitivity Norm

During the brushless DC (BLDC) motor working process, the system encounters inevitable uncertainties. These ambiguities stem from potential fluctuations, random occurrences, measurement inaccuracies, varying operational conditions, environmental shifts such as temperature alterations, among other factors. Uncertainties, an inherent aspect of any real control system, can be broadly classified into two categories: sensor signal uncertainties and discrepancies between the mathematical and actual models due to parameter perturbations. To mitigate the impact of sensor noise and parameter perturbations on the BLDC motor, a robust control strategy utilizing infinite norm mixed sensitivity based on PI control strategy (PI-H∞-MIX) is proposed in this paper. Firstly, the closed-loop control structure and transfer function model of the BLDC motor control system current loop are analyzed based on the current loop circuit topology, and then, the model parameters perturbation is analyzed, and the multiplicative uncertainty bound is given. In addition, the appropriate weighting function is selected to ensure the robustness of the system. In this case, the controller design problem is transformed into the H∞ standard control problem, and then, the system augmentation matrix is established, and the controller is solved by Matlab/Simulink. Finally, the performances of the traditional PI control strategy and the PI-H∞-MIX are compared and analyzed. The results show that (1) the proposed PI-H∞-MIX strategy can improve the control system robustness under the parameter perturbation condition effectively, and (2) the proposed PI-H∞-MIX strategy can suppress the noise signal of the sensor.

A Particle Swarm Optimization tuned nonlinear PID controller with improved performance and robustness for First Order Plus Time Delay systems

The Proportional, Integral, and Derivative (PID) controller is a ubiquitous controller within industry. The conventional PID controller can struggle to provide a satisfactory response for the nonlinear systems faced by industry. In addition, conventional PID controllers have a trade-off between performance and robustness, where they cannot compensate for both without compromising stability or speed. In this paper, a novel Nonlinear gains Proportional, Integral, and Derivative (NLPID) control algorithm is proposed as a practical control strategy that shows improvements in the simultaneous set-point tracking and disturbance rejection, to control nonlinear systems. The paper shows the performance and robustness of the proposed controller for the case of a First Order Plus Time Delay (FOPTD) system, which heavily exists in industry. The Particle Swarm Optimization (PSO) algorithm is used to tune the proposed NLPID controller. The performance of the proposed NLPID controller is simulated and compared against established controllers in literature such as conventional PID, two degree of freedom PID, and Smith Predictor PID controllers in MATLAB/Simulink for an FOPTD system, with various uncertainties and disturbances. This study shows that the proposed NLPID controller maintains faster settling and rise time, with no overshoot and excellent disturbance rejection, without compromising stability or speed, and is robust against parametric, additive, and multiplicative uncertainties.

Adaptive Event-Triggered Fuzzy Positioning Control for Unmanned Marine Vehicles With Actuator Saturation and Hybrid Attacks

In this paper, we focus on the event-triggered observer-based fuzzy positioning control and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$L_{2}$</tex-math></inline-formula> -gain analysis of nonlinear unmanned marine vehicle (UMV) systems in network environments, which is subject to a hybrid attack consisting of DoS attack and deception attack. Firstly, based on the T-S fuzzy modeling method, a unified switched system model is established for the positioning control of nonlinear UMV systems with actuator saturation, multiplicative gain uncertainties, external disturbance and hybrid attacks. Then, by resorting to introducing the integral-type state error, a novel adaptive memory-based event-triggered mechanism (ETM) is proposed to lighten the communication load in UMV systems. Furthermore, a sufficient criterion is proposed to make the mean square exponential stability of UMV systems with a non-weighted <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$L_{2}$</tex-math></inline-formula> -gain level be guaranteed, and gain parameters of the designed controller and observer are also solved efficiently. Finally, the simulation example demonstrates the effectiveness of the proposed nonfragile control strategy.

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.

Explicit model predictive control through robust optimization

AbstractA strategy that calculates an explicit state feedback policy to regulate constrained uncertain discrete‐time uncertain linear systems is presented. We consider uncertain processes, affected by box‐bounded multiplicative uncertainty as well as bounded additive uncertainty with linear state and inputs constraints. The proposed method includes (i) the calculation of a terminal set constraint and (ii) the robust reformulation of state constraints in the prediction horizon. These features allow the derivation of the desired policy by solving a single multiparametric quadratic programming problem that guarantees feasible operation in the presence of uncertainty. Additionally, we employ variable and constraint elimination approaches to enhance the computational performance of the strategy. We demonstrate the steps and benefits of these developments with a numerical example and a chemical engineering case study.

Robust Consensus With Edge-Based Multiplicative Uncertainties via Recursive Channel Filters

This paper considers the robust consensus issue subject to channel uncertainties by adopting the first-order channel recursive filters for continuous-as well as discrete-time multi-agent systems (MASs). The considered uncertainties are edge-based multiplicative ones that are bounded in H∞ norm. The edge sensitivity function matrix (ESFM) and the edge complementary sensitivity function matrix (ECSFM) are defined for the continuous-and discrete-time MASs. Then, an edge sensitivity scheme is used for seeking the filter parameters, by which we show that the consensusability margins of MASs are associated with the edge complementary sensitivity function matrices of the MASs. The parameter conditions for the recursive filters are also be provided, under which the robust consensus with multiplicative uncertainties is achieved with maximal consensusability margins.

Robust consensus of multi‐agent systems with multiplicative uncertainties and its application in time synchronization

AbstractThis article first studies the robust consensus problem of discrete‐time single‐integrator multi‐agent systems (MASs) with multiplicative uncertainties in switching networks. Under the assumption that the uncertainty is polynomially decaying, we prove that the consensus of the MAS can be achieved by adopting the decaying gain protocol if the switching network is uniformly strongly connected (USC). The convergence rate is quantified in terms of the decay rate of uncertainties. Then, we use the consensus conclusion studying the convergence analysis problem of a typical time synchronization algorithm (the LSTS algorithm proposed in Tian [IEEE Trans Automat Contr. 2017;62(10):5445–5450]) in wireless sensor networks. We prove that the LSTS algorithm can still achieve the bounded time synchronization even if the switching network is USC, a more general topology condition compared with existing convergence results in Tian (IEEE Trans Automat Contr. 2017;62(10):5445–5450). Simulation results are given to verify the correctness of both the consensus conclusion for MASs and convergence conclusion for LSTS algorithm.

Negative imaginary feedback control of satellite orbit dynamical model

This article presents observer-based negative imaginary positive feedback control of satellite orbit dynamical model. In this article, satellite orbital model is transformed into a strongly strict negative imaginary system using observer-based transformation scheme, and then, negative imaginary control theory is applied to achieve stability and robustness in orbit control system in the presence of multiplicative uncertainty. The solutions of orbit control problem are investigated with two thrusters input as well as one thruster input. According to the numerical simulation findings reported in this study, the investigated control approach guarantees the robust stabilization of the satellite orbital system.

Stabilization of Triangular Nonlinear Systems With Multiplicative Stochastic State Sensing Noise

We present new state-feedback control designs for lower-triangular/upper-triangular nonlinear systems with multiplicative stochastic sensor uncertainty. For lower-triangular nonlinear systems with small sensor noise, we develop a novel control design where the control gains are suitably constructed to simultaneously dominate the nonlinear functions and sensor noise of sufficiently small multiplicative gain. For upper-triangular nonlinear systems, we propose a new low-gain domination design, the advantage of which is that it can effectively deal with the sensor noise with arbitrarily large intensities. These two designs can both ensure that closed-loop system has an almost surely unique global solution, the origin of the closed-loop system is mean-square stable, and the states can be regulated to zero almost surely. Finally, two simulation examples are given to illustrate the designs.

Supply chain price variability under the buyback contract

ABSTRACT Supply chain price variability, also known as the “Bullwhip effect in Pricing (BP),” refers to the absorption or amplification of the variability of prices from one stage to another in a supply chain. This article derives analytical conditions that result in BP considering a buyback contract and conducts numerical simulations to gain further insights. For this, a joint price and replenishment setting newsvendor model with a wholesale-Stackelberg game is considered. Two demand types (linear and isoelastic) are analyzed along with uniformly and normally distributed additive and multiplicative uncertainties. The outcome of this research reveals that the main influential factors that affect BP are the structure and error type of the demand functions. Absorption (amplification) in price fluctuations occurs for linear (isoelastic) demand cases. Moreover, the price variances and BP ratios differ under the buyback and wholesale-price-only cases. The overall results help understand the fluctuation of market prices under various conditions.

An Operator-Theoretic Approach to Robust Event-Triggered Control of Network Systems With Frequency-Domain Uncertainties

In this paper, we study the robustness of the event-triggered consensus algorithms against frequency- domain uncertainties. It is revealed that the sampling errors resulted by event triggering are essentially images of linear finite-gain L2-stable operators acting on the consensus errors of the sampled states and the event-triggered mechanism is equivalent to a negative feedback loop introduced additionally to the feedback system. In virtue of this, the robust consensus problem of the event-triggered network systems subject to additive dynamic uncertainties and network multiplicative uncertainties are considered, respectively. In both cases, quantitative relationships among the parameters of the controllers, the Laplacian matrix of the network topology, and the robustness against aperiodic event triggering and frequency-domain uncertainties are unveiled. Furthermore, the event-triggered dynamic average consensus (DAC) problem is also investigated, wherein the sampling errors are shown to be images of nonlinear finite-gain operators. The robust performance of the proposed DAC algorithm is analyzed, which indicates that the robustness and the performance are negative