Multiplicative Uncertainty Research Articles

Data-Driven Computation of Robust Control Invariant Sets With Concurrent Model Selection

Set invariance in the presence of uncertainty and disturbance is of central importance for the safety of control systems. This article proposes a data-driven method to compute an approximation of a minimal robust control invariant set (mRCI) from experimental data. For a given dynamical model with additive and multiplicative uncertainty, the proposed method is able to compute a polytopic mRCI with fixed complexity via linear programming (LP). Moreover, the method can be combined with model selection to enable mRCI computation directly from experiment data when the system dynamics are unknown. Specifically, given a model structure, our algorithm begins by identifying the set of admissible models with constraints extracted from the experimental data. Each model in the set of admissible models contains information about the nominal model and the characterization of the model uncertainties. Then, two iterative algorithms based on robust optimization are proposed to compute an mRCI while simultaneously searching for a model “optimal” with regard to the mRCI computation and the corresponding invariance-inducing controller. Finally, the method is demonstrated in an experiment with an autonomous vehicle lane-keeping control example.

Distributed stochastic model predictive control for systems with stochastic multiplicative uncertainty and chance constraints

The rapid development of technology and economy has led to the development of chemical processes, large-scale manufacturing equipment, and transportation networks, with their increasing complexity. These large systems are usually composed of many interacting and coupling subsystems. Moreover, the propagation and perturbation of uncertainty make the control design of such systems to be a thorny problem. In this study, for a complex system composed of multiple subsystems suffering from multiplicative uncertainty, not only the individual constraints of each subsystem but also the coupling constraints among them are considered. All the constraints with the probabilistic form are used to characterize the stochastic natures of uncertainty. This paper first establishes a centralized model predictive control scheme by integrating overall system dynamics and chance constraints as a whole. To deal with the chance constraint, based on the concept of multi-step probabilistic invariant set, a condition formulated by a series of linear matrix inequality is designed to guarantee the chance constraint. Stochastic stability can also be guaranteed by the virtue of nonnegative supermartingale property. In this way, instead of solving a non-convex and intractable chance-constrained optimization problem at each moment, a semidefinite programming problem is established so as to be realized online in a rolling manner. Furthermore, to reduce the computational burdens and amount of communication under the centralized framework, a distributed stochastic model predictive control based on a sequential update scheme is designed, where only one subsystem is required to update its plan by executing optimization problem at each time instant. The closed-loop stability in stochastic sense and recursive feasibility are ensured. A numerical example is employed to illustrate the efficacy and validity of the presented algorithm in this study.

Consensus of discrete-time multi-agent systems with multiplicative uncertainties and delays

This paper studies the consensus problem of discrete-time single-integrator multi-agent systems with multiplicative uncertainties and bounded communication delays in switching networks. Some sufficient conditions for non-stochastic consensus are obtained by adopting the constant-gain protocol. For the case that the uncertainty is exponentially decaying (i.e. is with ), it is proved for any bounded delays, the consensus can be achieved if the switching topology is uniformly rooted. For the case that the uncertainty is polynomially decaying (i.e. is with ), under a precondition that the system state is bounded, a similar conclusion is also obtained. The convergence rate of the consensus protocol is quantified in terms of the decay rate of uncertainties. Simulation results are given to verify the correctness of the conclusion.

A new robust tilt-PID controller based upon an automatic selection of adjustable fractional weights for permanent magnet synchronous motor drive control

This paper focuses on achieving a good trade-off between performance and robustness for a class of uncertainty models including unstructured multiplicative uncertainties. In robust control, the simultaneous improvement of the two secure margins for nominal performances and robust stability using a standard controller structure represents two contradictory objectives and guaranteeing simultaneously of these goals represents therefore a major challenge for most researchers. In this context, a robust tilt-proportional integral derivative (T-PID) controller synthesized with an automatic selection of adjustable fractional weights (AFWs) is discussed in our work. Their parameters are optimized through solving a weighted-mixed sensitivity problem using an optimization tool which is based on the genetic algorithm. This problem is formulated from performance and robustness requirements where a fitness function is accordingly determined. Furthermore, thus its search space is built according to some guidelines for ensuring an automatic selection of adequate AFWs. The proposed constrained optimization problem is initialized by using arbitrary T-PID speed controller as well as through initial fixed integer weights (FIWs) which were chosen previously by the designer. To highlight the proposed control strategy, the synthesized robust T-PID speed controller is applied on the permanent magnet synchronous motor. Their performance and robustness are compared to those provided by an integer-order PID (IO-PID) and two conventional fractional-order PID (FO-PID) controllers. This comparison reveals superiority of the proposed robust T-PID controller over the remaining controllers in terms of robustness with reduced control energy.

Moving Horizon H∞ Estimation of Constrained Multisensor Systems With Uncertainties and Fading Channels

This article considers the robust estimation problem of linear multisensor systems subject to constraints and additive and multiplicative uncertainties where the decaying of measurements is regarded as the multiplicative uncertainty when taking into account the fading channels between the estimator and sensors. Based on the H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> performance criterion, a moving horizon H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> estimation algorithm is proposed from an approximation of the H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> full information estimation, which describes the estimation problem as a minimax optimization problem. Sufficient conditions for the feasibility and stability of the proposed algorithm are discussed, under which the estimation errors are ultimately bounded. To reduce the difficulty of solving minimax problem, an approximate solution is provided. The simulation comparison results show the effectiveness of the proposed algorithm.

Robust Control for a Class of Nonlinearly Coupled Hierarchical Systems with Actuator Faults

This paper proposes an approach to addresses the control challenges posed by a fault-induced uncertainty in both the dynamics and control input effectiveness of a class of hierarchical nonlinear systems in which the high-level dynamics is nonlinearly coupled with a multi-agent low-level dynamics. The high-level dynamics has a multiplicative uncertainty in the control input effectiveness and is subjected to an exogenous disturbance input. On the other hand, the low-level system is subjected to actuator faults causing a time-varying multiplicative uncertainty in the dynamical model and associated control effectiveness. Moreover, the nonlinear coupling between the high-level and the low-level dynamics makes the problem even more challenging. To address this problem, an online parameter estimation algorithm is designed, coupled with an adaptive splitting mechanism which automatically distributes the control action among low level multi-agent systems. A nonlinear $\mathcal{L}_2$-gain-based controller, and then a state-feedback controller are designed in the high-level, and the low-level, respectively, to recover the system from faults with high performance in the transient response, and reject the exogenous disturbance. The resulting analysis guarantees a robust tracking of the high-level reference command signal.

Safe Adaptation with Multiplicative Uncertainties Using Robust Safe Set Algorithm

Maintaining safety under adaptation has long been considered to be an important capability for autonomous systems. As these systems estimate and change the ego-model of the system dynamics, questions regarding how to develop safety guarantees for such systems continue to be of interest. We propose a novel robust safe control methodology that uses set-based safety constraints to make a robotic system with dynamical uncertainties safely adapt and operate in its environment. The method consists of designing a scalar energy function (safety index) for an adaptive system with parametric uncertainty and an optimization-based approach for control synthesis. Simulation studies on a two-link manipulator are conducted and the results demonstrate the effectiveness of our proposed method in terms of generating provably safe control for adaptive systems with parametric uncertainty.

Sequential Multi-Stage and Tube-Based Robust MPC for Constrained Linear Systems With Multiplicative Uncertainty

Robust MPC algorithms using a multi-stage approach exhibit low levels of conservatism but suffer from an exponential growth of complexity with the prediction horizon. This may be an issue if long prediction horizons are required to achieve performance goals. In contrast, tube-based algorithms have a linear growth of computational complexity with the prediction horizon but show small domains of attraction. This letter provides a method for a trade-off between the two approaches. A multi-stage approach is used during the first part of the prediction horizon, while a tube-based approach is employed for the remaining part. Stability and recursive feasibility are guaranteed for the algorithm and the performance of the approach is validated with numerical examples.

A new control design and robustness analysis of a variable speed hydrostatic transmission used to control the velocity of a hydraulic cylinder

Controlling the velocity of a hydraulic cylinder is a common objective in fluid power industry. This objective is achieved by controlling the hydraulic fluid quantity that enters the cylinder. In this work, a control system that is used to control the velocity of a hydraulic cylinder was designed considering uncertainty in some of the design parameters. A fixed displacement pump was used to reduce the cost and complexity of the system. Furthermore, using variable speed drive eliminates the energy losses associated with valve controlled systems since no throttling or flow recirculation is needed. First, the system dynamics was modelled. Then, the stability and performance of the open loop system was studied using MATLAB/SIMULINK®. Next, controllers (PID and H-infinity) were designed and the stability and the performance of the closed loop system were studied and compared with those of the open loop system. Finally, the robustness of the system was studied considering multiplicative parametric uncertainty. Three parameters were considered as uncertain parameters which are the fluid bulk modulus, the viscous friction coefficient, and the leakage coefficient with a variation of ± 5% of their nominal values. The results showed that the open loop system is stable with a poor response in terms of input tracking and disturbance rejection. Using PID controller improved the system response. The system with the PID controller does not meet the robustness requirements. The system with H-infinity controller has better performance and satisfies the robustness requirements.

The optimal detuning approach based centralized control design for MIMO processes

This manuscript introduces an optimal detuning approach for designing a centralized PI control system for multi-input multi-output (MIMO) processes. Based on the approach, two multivariable PI controller designs are proposed in this manuscript. The proposed approach formulates the centralized PI controller design problem in an optimization framework and transforms it into an equivalent detuning parameter(s) design problem. While the centralized controller design with the proposed 1-Optimal Detuning Parameter (ODP) method is carried out as a multi-stage problem, the proposed 2-ODP method formulates the controller design as a single-stage optimization problem. An effective transfer function (ETF) parameterization followed by controller synthesis using the IMC theory and further the optimal detuning comprises the various stages in 1-ODP design. In this regard, a novel ETF parameterization procedure for large scale processes or systems with higher-order elements is presented in this manuscript. Contrary to the centralized controller design in the kp,ki space, where the dimensionality of the optimization problem blows at O(2n2), the proposed 1-ODP and 2-ODP design methods always aim at solving a uni- and bi-dimensional optimization problem, respectively, irrespective of the dimensionality of the MIMO process. Hence, the proposed method is easily applicable even to high-dimensional MIMO processes. Several illustrative industrial MIMO systems are considered to demonstrate the applicability of the proposed methods to highly interacting and large scale processes. The simulation studies show that the proposed methods performs better as compared to the other centralized PI controller designs. Even with parameter variations, the proposed methods give satisfactory closed-loop performance. Further, robust stability analysis for the proposed designs is performed by considering multiplicative input and output uncertainties.

Structured -Synthesis of Robust Attitude Control Laws for Quad-Tilt-Wing Unmanned Aerial Vehicle

The problem of designing the longitudinal flight controller for a quad-tilt-wing unmanned aerial vehicle to achieve robust performance under constraints on the controller structure is considered. An approach based on -synthesis is proposed, in which a nonparametric inverse multiplicative uncertainty description is employed to account for the uncertainties of the aircraft attitude dynamics at each operating condition, with specific reference to variations in the number of unstable modes. The resulting -synthesis problem with a frequency-dependent weighting function, the structure of which is defined a priori, is reformulated as a structured problem and solved by means of available MATLAB software. Simulation results are presented in order to show the effectiveness of the approach.

Robust output voltage control of a high gain DC–DC converter under applied load and input voltage uncertainties

In this study, using robust and adaptive approaches, two controllers are designed to compensate for the applied load and input voltage uncertainties of a high voltage gain dc–dc converter. These controllers are designed based on measuring only the output voltage of the converter. The permissible ranges of the uncertainties are calculated using dynamics equations analysis while practical limitations are considered. In the first approach, an input multiplicative uncertainty is considered for the model of the converter by linearisation of non-linear equations as well as using permissible ranges of the uncertainty. Then, a robust H∞ controller is designed. In the second approach, the converter model is considered as a first-order dynamics model with the unknown parameters and an unknown additive term while a direct robust model reference adaptive controller is designed for it. Finally, the non-linear equations of the converter are simulated and the performances of the controllers are compared with each other. The control methods are then implemented on a 200 W prototype of a converter for evaluating the simulations results.

Integrated nonlinear robust adaptive control for active front steering and direct yaw moment control systems with uncertainty observer

This paper presents an integrated nonlinear robust adaptive controller with uncertainty observer for active front wheel steering system and direct yaw moment control system. First, an integrated vehicle chassis control model is established as the nominal model with the additive and multiplicative uncertainties of the system. Secondly, an integrated nonlinear robust adaptive control law with the additive uncertainty observer is designed via Lyapunov stability theory to calculate the corrective yaw moment, and an adaptive law is designed based on projection correction method to online estimate and compensate the multiplicative uncertainty of the system. Then, the constrained optimal allocation problem of the corrective yaw moment is transformed into the nonlinear optimization problem, and the sequential quadratic programming method is used to solve the nonlinear optimization problem to coordinate active front wheel steering system and direct yaw moment control system. Finally, the performance of the proposed integrated nonlinear robust adaptive controller is verified via vehicle dynamics simulation software.

Robust decentralized proportional–integral controller design for an activated sludge process

AbstractThis paper presents the design of a decentralized proportional–integral (PI) controller for a wastewater treatment plant (WWTP). The aeration rate and the return recycle sludge rate are manipulated inputs to the WWTP process, while substrate and biomass concentration are considered as the output variables. The study is divided into two segments: A decentralized controller is designed based on the best pairing in the first segment, and in the second segment, a decoupler is developed to reduce the interactions. Decouplers are generally used to create independent loops in the multivariable control loops. Each decoupled subsystem is converted to first order plus time delay (FOPTD) model using a system identification toolbox to design independent diagonal controllers. The numerical simulations have been performed to evaluate the effectiveness of the presented methods. Furthermore, a robustness study has also been carried out by taking into account multiplicative input and output uncertainties, and it is found that the controller designed based on minimizing the integral of absolute error (IAE) criteria shows more robust.

IJFP-2018-0015 - Robust Control Design for an Inlet Metering Velocity Control System of a Linear Hydraulic Actuator

In this paper, we consider a hydraulic system in which the velocity is controlled using an inlet-metered pump. The flow of the inlet-metered pump is controlled using an inlet metering valve that is placed upstream from a fixed displacement check valve pump. Placing the valve upstream from the pump reduces the energy losses across the valve. The multiplicative uncertainty associated with uncertain parameters in an inlet metering velocity control system is studied. Six parameters are considered in the uncertainty analysis. Four of the parameters are related to the valve dynamics which are the natural frequency, the damping ratio, the static gain, and the time delay. The other two parameters are the discharge coefficient and the fluid bulk modulus. Performance requirements for the system are described in the frequency domain. Frequency domain analysis is used to determine if the closed-loop velocity control system has robust performance. The time response of the nominal system with PID and H∞ controllers were found to be similar. The H∞ controller was found to have the advantages of robust performance when considering the parametric uncertainty while not requiring integral control as in the PID control system. The PID system did not achieve robust performance.

Robust adaptive control for continuous wheel slip rate tracking of vehicle with state observer

This article proposes a novel robust adaptive wheel slip rate tracking control method with state observer. First, a modified tracking differentiator is proposed based on a combination of tangent sigmoid function with terminal attraction factor and linear function to improve convergence speed and avoid chattering phenomenon, and then, the modified tracking differentiator is used as state observer to smooth and estimate the states of the system. Second, a robust adaptive wheel slip rate tracking control law with fuzzy uncertainty observer and modified adaptive laws is derived based on Lyapunov-based method. The fuzzy uncertainty observer is used for estimating and compensating the additive uncertainty, and the modified adaptive laws are used for estimating the unknown optimal weight vector of the fuzzy uncertainty observer and the multiplicative uncertainty. Finally, the performance of the robust adaptive wheel slip rate tracking control method is verified based on the model-in-the-loop simulation system.

Robust stability analysis of real-time hybrid simulation considering system uncertainty and delay compensation

Real-time hybrid simulation (RTHS) which combines physical experiment with numerical simulation is an advanced method to investigate dynamic responses of structures subjected to earthquake excitation. The desired displacement computed from the numerical substructure is applied to the experimental substructure by a servo-hydraulic actuator in real time. However, the magnitude decay and phase delay resulted from the dynamics of the servo-hydraulic system affect the accuracy and stability of a RTHS. In this study, a robust stability analysis procedure for a general single-degree-of-freedom structure is proposed which considers the uncertainty of servo-hydraulic system dynamics. For discussion purposes, the experimental substructure is a portion of the entire structure in terms of a ratio of stiffness, mass, and damping, respectively. The dynamics of the servo-hydraulic system is represented by a multiplicative uncertainty model which is based on a nominal system and a weight function. The nominal system can be obtained by conducting system identification prior to the RTHS. A first-order weight function formulation is proposed which needs to cover the worst possible uncertainty envelope over the frequency range of interest. Then, the Nyquist plot of the perturbed system is adopted to determine the robust stability margin of the RTHS. In addition, three common delay compensation methods are applied to the RTHS loop to investigate the effect of delay compensation on the robust stability. Numerical simulation and experimental validation results indicate that the proposed procedure is able to obtain a robust stability margin in terms of mass, damping, and stiffness ratio which provides a simple and conservative approach to assess the stability of a RTHS before it is conducted.

Bullwhip effect in pricing under the revenue-sharing contract

Bullwhip effect in Pricing (BP) is the amplification or absorption of price fluctuation across the supply chain. This paper aims to derive conditions for BP under a revenue-sharing contract. For this purpose, we consider both a deterministic model and stochastic newsvendor models with additive and multiplicative uncertainties assuming both the linear and isoelastic demand forms. Contrasting the results with a no-contract case, this paper shows that the cost-pass-through and the BP ratio increase with the increase of the revenue-share percentage. Numeric simulations also showed different phases of price-variation for the linear and the isoelastic demand forms.

Falsified Model-Invariant Safety-Preserving Control With Application to Closed-Loop Anesthesia

This brief introduces a novel safety-preserving control scheme with minimal conservatism for uncertain systems. We have recently introduced the model-invariant safety verification technique that provides a formal guarantee of safety for systems with multiplicative model uncertainty. This approach requires a multi-model description of model uncertainty. The resulting safety system may be conservative for systems that do not exhibit the worst case dynamical response. In this brief, we employ model falsification to reduce conservatism of the model-invariant safety verification technique. Members of a model set that characterizes model uncertainty are falsified if discrepancy between predictions of those models and measured responses of the uncertain system is established, thereby reducing model uncertainty. To demonstrate the effectiveness of the proposed technique, we formalize a model-invariant safety system for closed-loop propofol anesthesia. The safety system maintains predicted propofol concentration in plasma as well as the patient’s blood pressure within safety bounds despite uncertainty in patient responses to propofol.

Iterative Learning Control for a Type of Modified Smith Predictor

Abstract This article proposes a novel iterative learning control (ILC) design for a type of modified Smith predictor, in particular, to control a single-input single-output unstable plant or integral process with a time delay. Frequency domain techniques are applied to synthesize the learning control law, and a sufficient condition is given to ensure robust convergence of the tracking error. Robustness of the system is studied, considering a multiplicative uncertainty. Moreover, the impact of the load disturbance over successive iterations is investigated as well. To this end, a numerical example is given to demonstrate the efficacy of the proposed approach.